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abstract: |
We study the transition to the continuum of an initially bound quantum particle in $\RR^d$, $d=1,2,3$, subjected, for $t\ge 0$, to a time periodic forcing of arbitrary magnitude. The analysis is carried out for compactly supported potentials, satisfying certain auxiliary conditions. It provides complete analytic information on the time Laplace transform of the wave function. From this, comprehensive time asymptotic properties (Borel summable transseries) follow.
We obtain in particular a criterion for whether the wave function gets fully delocalized (complete ionization). This criterion shows that complete ionization is generic and provides a convenient test for particular cases. When satisfied it implies absence of discrete spectrum and resonances of the associated Floquet operator. As an illustration we show that the parametric harmonic perturbation of a potential chosen to be any nonzero multiple of the characteristic function of a measurable compact set has this property.
Ionization, delocalization, resonances, Floquet theory, Borel summability.
author:
- |
O. Costin, R. D. Costin, and J. L. Lebowitz$^1$ [$_{\mbox{
Department of Mathematics, Rutgers University}}$]{}
title: '[Time asymptotics of the Schrödinger wave function in time-periodic potentials]{}'
---
[^1]
[[ *Dedicated to Elliott Lieb on the occasion of his 70th birthday*]{}]{}
Introduction
============
We consider the non-relativistic Schrödinger equation for the wave function $\psi(x,t), x\in\RR^d$ $$\label{eq:eqa}
i\, \frac{\partial\psi }{\partial
t}\, =\Big(-\Delta+V(x)+\Omega(x,t)\Big)\, \psi$$ where $\Omega(x,t)$ is a time-periodic external potential (not necessarily small): $$\label{eq:eqa2}
\Omega(x,t)=\Omega(x,t+2\pi/\omega), \ \omega>0$$ We take $V$ and $\Omega$ real-valued and so that $$\label{as1}
V\in L^\infty(\RR^d),\ \
\Omega\in L^\infty(\RR^d\times [0,2\pi/\omega])$$ with $\Omega\not\equiv 0$ satisfying $$\label{Fourier}
\Omega(x,t)=\sum_{j\in\ZZ}\Omega_j(x)e^{ij\omega t},\ \ \ \Omega_j(x)=\overline{\Omega_{-j}(x)};\ \ \sup_{j,x}|\Omega_j(x)| j^2<\infty$$ We set, without loss of generality, $$\label{Omega0}
\Omega_0(x)=0$$ We are interested in the behavior of solutions $\psi(x,t)$ for large $t$ when $$\label{psil2}
\psi(x,0)=:\psi_0(x)\in L^2(\RR^d), \ \ \ \int_{\RR^d}|\psi_0|^2dx=1$$ and $\psi_0$ is sufficiently regular (we assume it of class $C^4$). Of particular interest is the [*survival probability*]{} for the particle in a ball $B$ in $\RR^d$, $\int_{x\in B}|\psi|^2
dx:=\mathcal{P}_B(t)$. If $\mathcal{P}_B(t)$ approaches zero as $t\rightarrow\infty$ for all $B$, then we say that the particle escapes to infinity and [*complete ionization*]{} occurs.
While many results in the paper only require (\[as1\]) (\[Fourier\]) plus sufficient algebraic decay of $\Omega$ and $V$ for large $|x|$, some specific results later in the paper, particularly detailed analytic information, require that $V$ and $\Omega$ are compactly supported,
(V)((,t)) \[compactDa\]\
\
(\_0)(V) ((,t)) \[compactDb\]
with $${D}\subset\mathbb{R}^d \text{\ compact,} \
\mathbb{R}^d\setminus D \text{\ connected,
meas}(\partial D)=0$$ The rest of the paper will therefore be written in the context of this setting.
Nature of the results
---------------------
Under the assumptions (\[compactDb\]) $\psi(x,t)$ is obtained for large $t$ as a convergent combination of exponentials and Borel summable power series in $t^{-1/2}$.
If an additional assumption (connected to the absence of discrete spectrum of the Floquet operator) is satisfied, the long time expression of $\psi$ contains only decaying terms, cf. Theorem \[genth12\] in § \[3.8\], i.e. we get complete ionization.[^2]
We find in Proposition \[cdi\] a convenient sufficient condition for complete ionization, and show that it is satisfied by a nonperturbative example[^3] $$\label{nonp}
V(x)=V_{{D}}\,\bchi_{{D}}(x);\ \ \Omega(x,t)=2\Omega_{{D}} \,\bchi_{{D}}(x)\sin \omega t$$ where $\bchi_{{D}}$ is the characteristic function of $D$ in $d=1,2,3$ and $V_D$ and $\Omega_D$ are arbitrary nonzero constants.
We previously obtained similar results for more general potentials in $d=1$ and radially symmetric ones in $d=2,3$. See [@JPA; @CMP221; @CRM; @CMP224; @genpaper] and [@UAB] where there is a review of our previous work on this problem.
Strategy of the approach
------------------------
The key steps of our approach are outlined in §\[KS\]. The method we use is based on a study of the analytic properties of $\hat{\psi}$, the Laplace transform of $\psi$. The type and position of the singularities of $\hat{\psi}$, given in Theorem \[Pn=3\] and Lemma \[L12\], provide information about the time behavior of $\psi$; the former are obtained from an appropriate equation to which the Fredholm alternative approach applies.
Laplace transform, link with Floquet theory {#secLT}
===========================================
Exisence of a strongly differentiable unitary propagator for (\[eq:eqa\]) (see [@Reed-Simon] v.2, Theorem X.71) implies that for $\psi_0\in L^2(\RR^d)$, the Laplace transform $$\label{eq:Lap}
\hat{\psi}(\cdot,p):=\int_0^{\infty}\psi(\cdot,t)e^{-pt}dt$$ exists for $\Re(p)>0$. It satisfies the equation $$\label{eq:S-lt}
(-\Delta+V(x)-ip)\hat{\psi}(x,p)=-i\psi_0-\sum_{j\in\ZZ} \Omega_j(x)\hat{\psi}(x,p-ij\omega)$$ and the map $p\to \psi(\cdot,p)$ is $L^2$ valued analytic in the right half plane $$\label{H}
p\in\mathbb{H}=\{z:\Re(z)>0\}$$ Clearly, equation (\[eq:S-lt\]) couples $\hat{\psi}(x,p_1)$ with $\hat{\psi}(x,p_2)$ iff $(p_1-p_2)\in i\omega\ZZ$. Setting $$\label{defnp}
p=i(\sigma+n\omega)\text{ with }
\Re\,\sigma\in[0,\omega)$$ (sometimes it will be technically helpful to relax this restriction on $\sigma$) we define $y_n^{[1]}(x;\sigma)=
\hat{\psi}(x,i(\sigma+n\omega))$. Eq (\[eq:S-lt\]) now becomes a differential-difference system $$\label{eq:S-lt2}
(-\Delta+V+\sigma+n\omega)y_n^{[1]}=-i\psi_0-\sum_{j\in\ZZ}\Omega_j(x)\left(S^{-j}y^{[1]}\right)_n$$ where the shift operator $S$ is given by $$\label{Sdef}
(Sy)_n=y_{n+1}$$
Connection with Floquet theory {#cF}
------------------------------
The solution of (\[eq:eqa\]) with time periodic $\Omega$ is of course the subject of Floquet theory (see [@[30]] and [@YajPriv]-[@Reed-Simon]) and therefore our analysis connects to it in a number of ways. Let $K$ be the quasi-energy operator in Floquet theory $$\label{eq:quasien}
(Ku)(x,\theta)= \left(-i\frac{\partial}{\partial \theta} -\Delta+V(x)+\Omega(x,\theta)\right)u(x,\theta); \ x\in \RR^d,\,\theta\in S^1_{{2\pi}/{\omega}}$$ Then, letting $$\label{ftu}
u(x,\theta;\sigma)=\sum_{n\in\ZZ}y_n^{[1]}(x;\sigma)e^{in\omega\theta}$$ be the solution of the eigenvalue equation $$\label{evK}
Ku=-\sigma u$$ we get an equation for the $y_n^{[1]}$ which is identical to the homogeneous part of equation (\[eq:S-lt2\])[^4]. Solutions of (\[evK\]) with $u\in L^2(\RR^d\times
S^1_{{2\pi}/{\omega}})$ correspond to eigenfunctions of $K$.
\[R01\] If $u$ is an eigenfunction of $K$ corresponding to the eigenvalue $-\sigma$, then $ue^{-ij\omega\theta}$ is an eigenfunction with eigenvalue $-\sigma+j\omega$. For this reason it is enough to restrict $\sigma$ to the strip given in (\[defnp\]).
Complete ionization clearly requires the absence of a discrete spectrum of (\[evK\]) (Otherwise, if $u(x,\theta)$ is an eigenfunction of $K$, then $e^{i\sigma t}u(x,t)$ would be a space localized solution of the Schrödinger equation.) In a recent work [@YajPriv], Galtbayar, Jensen and Yajima proved that the opposite is also true. They obtained asymptotic series in $t^{-1/2}$ for the projection of the wave function $\psi(x,t)$ on the space orthogonal to the discrete spectrum of $K$.
Our approach via Laplace transform is different from that of [@YajPriv]. For the corresponding time evolution our results are stronger than those obtained in [@YajPriv] but apply to the more restrictive classes of $V$ and $\Omega$ satisfying (\[compactDb\]) in $d=1,2,3$. We show that the time behavior of $\psi(x,t)$ is given by a Borel summable transseries containing both power law decay and exponential terms. For potentials satisfying our condition (\[nonp\]) we show that $K$ has no discrete spectrum or resonances and all the exponentials are decaying.
More details on the connection between our work, Floquet theory and [@YajPriv] are given in §\[F22\].
Integral equation, compactness and analyticity
==============================================
Laplace space equation
----------------------
To simplify contour deformation, we first improve the decay of $\hat{\psi}$ for large $p$, by pulling the first two terms in the asymptotic behavior for large $p$ from $\hat{\psi}$. Let $\delta_{ij}=1$ if $i=j$, and $0$ otherwise, $\delta_{ij}^c=1-\delta_{ij}$ and define the operator $\mathfrak{N}$ by $$\label{deffrakN}
(\mathfrak{N}f)_n(x)=(-\Delta+V)\frac{\delta^c_{n0} f_n(x)}{\sigma+n\omega}-\delta_{n0}f_{n,x}+\sum_{k\ne n}\Omega_{n-k}(x)\frac{\delta^c_{k0} f_k(x)}{\sigma+k\omega}$$
Green function representation
-----------------------------
To pass to an integral form of the system of equations (\[eq:finy\]) we apply to them the Green function of $(-\Delta+\sigma+n\omega)$, given by $$\label{GreenF}
\Big(\mathfrak{g}_nf\Big)(x)=\int G(\kappa_n(x-x')) f(x')dx'$$ with $$\label{defkappa}
\kappa_n=\sqrt{-ip}=\sqrt{\sigma+n\omega}\ \text{(when
$p\in\mathbb{H}$, $\kappa_n$ is in the fourth quadrant)}$$ and $$\label{formulares}
G(\kappa_nx)=\left\{
\begin{array}{ccccc}\displaystyle \frac{1}{2}\kappa_n^{-1}e^{-\kappa_n|x|}\ \ & d=1\\ & \\ \displaystyle
\frac{1}{2\pi}K_0(\kappa_n|x|)\ \ & d=2\\ &\\ \displaystyle
\frac{1}{4\pi}|x|^{-1}e^{-\kappa_n|x|}\ \ & d=3 \end{array}\right.$$ (see [@Reed-Simon]) where $K_0$ is the modified Bessel function of second kind, $$\label{Bessel0}
K_{0}(x)=\int_0^{\infty}e^{-x\cosh t}dt=e^{-x}\int_0^{\infty}\frac{e^{-xs}}{\sqrt{s(s+2)}}ds$$ Note that, in the setting (\[compactDa\]), for $f$ supported in ${D}$ we have $$\label{GreenF2}
\Big(\mathfrak{g}_nf\Big)(x)=\int_{{D}} G(\kappa_n(x-x')) f(x')dx'$$ Eq. (\[deffrakN\]) in integral form becomes $$\label{eq:E}
y_n^{[1]}=-i\mathfrak{g}_n\psi_0+(\mathfrak{C}y^{[1]})_n$$ where $$\label{eq:E1}
y^{[1]}=(y_n^{[1]})_{n\in\ZZ}\ \ \text{and}\ \ (\mathfrak{C}y^{[1]})_n=-\mathfrak{g}_n\left[Vy_n^{[1]}+\sum_{j\in\ZZ}\Omega_j(x)\left(S^{-j}y^{[1]}\right)_n\right]$$ To ensure better decay with respect to $n$ we further substitute in (\[eq:E1\]) $$\label{eq:E2}
y_n^{[1]}=-i\mathfrak{g}_n\psi_0-i\psi_{1,n}+y_n$$ where $(\psi_{1,n})_{n\in\ZZ}=:\psi_1$ and $$\label{eq:E4}
\psi_1=\mathfrak{C}\left[\left(\mathfrak{g}_n\psi_0\right)_{n\in\ZZ}\right]$$ Then $y$ satisfies $$\label{eq:fint2}
y=w+\mathfrak{C}(\sigma)y$$ (We write $\mathfrak{C}$ for $\mathfrak{C}(\sigma)$ when the dependence on $\sigma$ need not be stressed.) In differential form (\[eq:fint2\]) reads $$\label{eq:finy}
(-\Delta+\sigma+n\omega)y_n=i\psi_{2,n}-Vy_n-
\sum_{j\in\ZZ}\Omega_j(x)\left(S^{-j}y\right)_n$$
The Hilbert space
-----------------
To analyze the properties of (\[eq:fint2\]) we use the Hilbert space $$\label{defH}
\mathcal{H}=l^2_{\gamma}(L^2(B))$$ where $B$ is an arbitrary ball (containing ${D}$) defined as the space of sequences $\{y_n\}_{n\in\ZZ},\,y_n\in L^2(B)$ with $$\|y\|^2_{\mathcal{H}}=\sum_{n\in\ZZ}|n|^\gamma\|y_n\|^2_{L^2(B)}<\infty$$ and adequate $\gamma$; we take for definiteness $\gamma=3/2$; larger $\gamma$ can be taken if one assumes more differentiability than (\[Fourier\]) implies. (Note that $\mathcal{H}$ is different from the Hilbert space $L^2(L^2(\RR^d))$ used in Floquet theory. See also §\[F22\].)
Strategy of the approach, continued {#KS}
-----------------------------------
As mentioned, unitarity of the evolution shows that $\hat{\psi}(\cdot,p)\in L^2(\RR^d)$ if $\Re(p)>0$. In the integral form (\[eq:fint2\]), whose solutions are in $\mathcal{H}$ when $\Re(p)>0$, the operator is, under our assumptions, compact. The solution of this equation is shown to be unique in $\mathcal{H}$ for large enough $\Re(p)$ by the contractivity of the integral operator. Uniqueness and analyticity of the solution for $p$ in the right half plane $\mathbb{H}$ follow by an application of the analytic version of the Fredholm alternative [@Reed-Simon]. We then show that the solution and thus $\hat{\psi}$ are analytic with respect to a uniformizing variable, in appropriately chosen domains containing parts of the imaginary axis. The contour of the inverse Laplace transform, $\tau\to c+i\tau;\tau\in\RR, c>0$, can then be deformed to $i\RR$ (the boundary of $\mathbb{H}$) where $\hat{\psi}$ is analytic except for a discrete set of square root branch points. The large time behavior of $\psi$ follows.
Compactness
-----------
In $d=1$, we further transform the equation, see (\[d1,2\]) in Appendix \[pd=1\], to improve the regularity of the operator at $n=0$ and $\sigma=0$.
\[L4\] Under the assumptions (\[compactDa\]), $w\in\mathcal{H}$ and $\mathfrak{C}$ is a compact operator on $\mathcal{H}$.
To show that $w\in\mathcal{H}$ we use the fact that the operators $\mathfrak{g}_n$ satisfy (see Appendix A of [@Agmon], and also §\[compactness\]). $$\label{eq:Agmon}
\sup_{n\in\ZZ}(1+|n|)^{1/2}\|\mathfrak{g}_n\|_{L^2(\mathcal{D})}<\infty$$ Then $$\label{eq:E6}
\sup_{n\in\ZZ} (1+|n|)^{1/2} \|(\mathfrak{g}_n\psi_0)_{n}\|<\infty\ \ \text{implying} \ \
\sup_{n\in\ZZ}(1+|n|)\|\psi_{1,n}\|_{L^2(\mathcal{D})}<\infty$$ In view of (\[Fourier\]) we also have $$\label{eq:E8}
\sup_{n\in\ZZ}(1+|n|)\left\|V\psi_{1,n}+\sum_{j\in\ZZ}\Omega_j(x)\left(S^{-j}y\right)_n\right\|_{L^2(\mathcal{D})}<\infty$$ implying $$\label{eq:E9}
\sup_{n\in\ZZ}(1+|n|^{3/2})\|w_n\|_{L^2(\mathcal{D})}<\infty$$
It is not difficult to check that $\mathfrak{g}_n$ is compact on $L^2(B)$ for each $n$; it is more delicate to show compactness of $\mathfrak{C}$; both properties are proved in §\[S6\].
Uniqueness
----------
\[L5\] For large enough $-\Im\sigma$, eq. (\[eq:fint2\]) has a unique solution in $\mathcal{H}$.
The proof is given in Appendix \[Unic\].
Analytic structure of $\hat\psi$
--------------------------------
\[R4\] It is convenient to introduce the uniformizing variable $\sigma=u^2$; with the natural branch of the square root, $u$ is in the fourth quadrant when $\sigma$ is in the lower half plane. In this variable, we write $\kappa_0=u$ and $\kappa_n=\sqrt{n\omega+u^2}$ for $n\ne 0$.
\[L6\] In the setting (\[compactDa\]) the operator $\mathfrak{C}(\sigma)$ is analytic in $u$ in the region $S_{\omega}=\{u:|\Re u^2|<\omega\}$ hence in $\sqrt{\sigma}$ in the strip (see Remark \[R01\]) $$\label{strip}
\Big\{\sigma:\Re(\sigma)\in(-\omega,\omega)\Big\}$$ Additionally, $\mathfrak{C}(\sigma)$ is analytic in $\sigma$ at any $\sigma\ne 0$.
In terms of $u$ we define $\kappa_0=u$, $\kappa_n=\sqrt{u^2+n\omega}$ and then $\kappa_n$ is analytic in $u$ in the simply connected region $S_\omega$ for any $n\in\ZZ$. Since in the setting (\[compactDa\]) the integral (\[GreenF2\]) is over a compact set, $D$, $\mathfrak{g}_n$ is also analytic for any $n\in\ZZ$. Analyticity of $\mathfrak{C}$ follows from the fact that $$\label{limC}
\mathfrak{C}=\lim_{N\to\infty} \mathfrak{C}_N$$ with $$\label{decC}
\Big(\mathfrak{C}_N\,\,y\Big)_n :=\left\{\begin{array}{cc}\ \ \big(\mathfrak{C}y\big)_n; \ \ &|n|\le N\\ 0 &{\text otherwise}\end{array} \right.$$ and convergence is uniform in $u$ on compact subsets of $S_{\omega}$. This is shown in Lemma \[L27\]. $\Box$
\[P4\] There exists a unique solution $y$ to (\[eq:fint2\]) and it has the same analyticity properties as $\mathfrak{C}$ if $$\label{hmg}
\text{$ ${\bf(A)} \text{ For $\Im \sigma \le 0$}\ \ \
($ v=\mathfrak{C}v$, $v\in\mathcal{H}$)}\ \Rightarrow v=0$$
. This is nothing more than the analytic Fredholm alternative (see e.g. [@Reed-Simon] Vol 1, Theorem VI.14, pp. 201). $\Box$
This formulation is convenient in determining the analytic properties of $y$ with respect to $\sigma$, instrumental for the Borel summability results stated in Proposition \[genth1\].
\[Pn=3\] If [**(A)**]{} and (\[compactDb\]) hold, then:
\(i) the solution $y$ of (\[eq:fint2\]) is meromorphic in $u$ in the disk $\{u:|u|<\sqrt{\omega}\}$, see Remark \[R4\], analytic at $u=0$ and in the fourth quadrant of $S_{\sqrt{\omega}}$. Furthermore, $y$ is analytic in $\sigma$ at any $\sigma_0\ne 0$.
\(ii) $\hat{\psi}$ is analytic in $p$ in a cut neighborhood of $i\RR$, $\{p:\Re(p)>-\epsilon\}$ with cuts toward $-\infty$ at $i
\,n\,\omega,\ n\in\ZZ$. Furthermore, in a neighborhood of $i\,n\,\omega$, see (\[defnp\]), we have $\hat{\psi}(p)=A_n(p)+B_n(p)\sqrt{\sigma}$ where $A_n$ and $B_n$ are analytic at $i\,n\,\omega$ and, for some $\epsilon<\sqrt{\omega}$ and $|u|\le \epsilon$ we have $$\label{normAB}
\sup_{n\in\ZZ,|u|<\epsilon}n^{\gamma}\Big(\|A_n(p)\|_{L^2(B)}+\| B_n(p)\|_{L^2(B)}\Big)<\infty$$
\(i) follows from Propositions \[L6\] and \[P4\], and from the link between $\hat{\psi}$ and $y_n$.
\(ii) The functions $A_n$ and $B_n$, are simply the even and odd part respectively of the analytic function $y_n(x;u)$. The estimate follows from the fact that $y(x;u)\pm
y(x;-u)\in\mathcal{H}$ is analytic in $u$ for $|u|<\epsilon$.
In Proposition \[genth1\] and Theorem \[genth12\] below we use the following result.
\[L12\] In the setting (\[compactDa\]), if $\hat{\psi}$ has a pole at $\sigma=\sigma_0\in i\RR$, then, in the variable $\sigma-\sigma_0$ if $\sigma_0\ne 0$ or $u$ if $\sigma_0=0$, the pole is simple.
This is shown in Appendix \[simpoles\]. $\Box$
\[Pex\] Condition [**(A)**]{} is satisfied for the potentials (\[nonp\]).
This is established in §\[Examples\]. $\Box$
Asymptotic expansion of $\psi$ and Borel summability {#3.8}
----------------------------------------------------
\[genth1\] In the setting (\[compactDb\]) there exist $N\in\NN$, $\{\Gamma_k\}_{k\le N}$, and $\{F_{\omega;k}(t,x)\}_{k\le N}$, $2\pi/\omega$-periodic functions of $t$, such that, for $t>0$,
$$\label{transP3}\psi(t,x)=\sum_{j\in\ZZ}e^{ij\omega
t}h_j(t,x)+\sum_{k=1}^N P_k(t)e^{-\Gamma_k t}F_{\omega;k}(t,x)$$
with $\Re\Gamma_k\ge 0$ for all $k\le N$, $P_k(t)$ are polynomials in $t$, reducing to constants if $\Re\Gamma_k=0$, and the $h_j(t,x)$ have Borel summable power series in $t^{-1/2}$ $$\label{e113}h_j(t,x)=\mathcal{LB}\sum_{k\ge k_0}h_{kj}(x)t^{-k/2}$$ with $k_0\ge 1$.
\[F2\] If $\tilde{f}$ is a formal power series, say in inverse powers of $t$, then $\tilde{F}=\mathcal{B}\tilde{f}$ is also a formal power series, defined as the term-wise inverse Laplace transform in $t$ of $\tilde{f}$. If (1) $\tilde{F}$ is convergent (2) its sum $F$ can be analytically continued along $\RR^+$ and (3) $\exists \nu$ s.t. $F(p)\in L^1(\RR^+, e^{-\nu p}dp)$, then the Laplace transform $\mathcal{L}F$ is by definition the Borel sum of $\tilde{f}$ denoted by $\mathcal{LB}\tilde{f}$. In our context we have, more precisely, $$h_j(t,x)=\int_0^\infty\, F_j(\sqrt{p},x)\, e^{-pt}\, dp\, \sim
\, \sum_{k}\, h_{kj}(x)\, {t^{-k/2}},\ t\to+\infty$$ where the functions $F_j(s,x)$ are analytic in $s$ in a neighborhood of $\RR^+$ and for any $b\in\RR$ there exist a constant $C$ such that for all $j$ and $p\in\RR^+$, $$\sup_{p\ge 0;|x|<b}|F_j(\sqrt{p},x)e^{-C|p|}| \le f_j$$ where the $f_j$ decay in $j$ faster than $j^{-2}$ under the assumption (\[Fourier\]) and factorially if $\Omega$ is a trigonometric polynomial. Thus the function series in (\[transP3\]) converges (rapidly in the latter case).
The role of condition [**(A)**]{} is described in the following result.
\[genth12\] (i) If [**(A)**]{} holds, then on the right side of (\[e113\]) and (\[transP3\]) we have $$\label{Aholds}
k_0\ge 3\ \text{ and }\ \Re\Gamma_k> 0\ \text{ for all }\ k.$$ In particular we have complete ionization of the system. (See also Proposition \[Pex\], as well as Proposition \[cdi\] and Remark \[Overdet\] below.)
\(ii) If [**(A)**]{} is not satisfied, then some $\Re\Gamma_k$ may vanish; the part of $\psi$ corresponding to these $\Gamma_k$ remains a spatially localized quasiperiodic function of $t$. Exceptionally, $k_0=1$ if [**(A)**]{} does not hold (see also [@YajPriv] and Proposition \[Fl3\]).
The proofs of Proposition \[genth1\] and Theorem \[genth12\] are sketched in Appendix \[Sketch\]. In one dimension a similar result is stated in [@UAB].
Ionization condition for compactly supported potentials
=======================================================
For the setting (\[compactDa\]) we derive a technically convenient condition implying [**(A)**]{}.
Assume $0\ne v\in\mathcal{H}$ and $ v=\mathfrak{C}v$. Then there exists a nontrivial solution in $\mathcal{H}$ to the system $$\label{hmdiff}
(-\Delta+\sigma+n\omega)y_n=-Vy_n-\sum_{j\in\ZZ}\Omega_j(x)y_{n-j}$$ We multiply (\[hmdiff\]) by $\overline{y}_n$, integrate over a ball $B$ containing ${D}$, sum over $n$ (which is legitimate since $y\in\mathcal{H}$) and take the imaginary part of the resulting expression. Noting that $$\begin{gathered}
\label{conjsym}
\overline{\sum_{j,n\in\ZZ}\Omega_j(x)y_{n-j}\overline{y_n}}=\sum_{j,n\in\ZZ}
\Omega_{-j} \overline{y}_{n-j}y_{n}=\sum_{j,n\in\ZZ}
\Omega_{j} \overline{y}_{n+j}y_{n}\\=\sum_{j,m\in\ZZ}\Omega_j(x)\overline{y_m}y_{m-j}\end{gathered}$$ so the sum (\[conjsym\]) is real, we get $$\begin{gathered}
\label{eq:nonreal}
0=\Im\left(-\sigma\sum_{n\in\ZZ}\|y_n\|^2+\int_{B}\sum_{n\in\ZZ} dx
\overline{y}_n\Delta y_n\right)\\=
-\Im\sigma\sum_{n\in\ZZ}\|y_n\|^2+\frac{1}{2i} \int_{\partial
B}\left(\sum_{n\in\ZZ}\overline{y}_n\nabla y_n-y_n\nabla
\overline{y}_n\right)\cdot \mathbf{n}\,dS\end{gathered}$$
We take $d=3$ (the analysis is simpler in one or two dimensions). It is convenient to decompose $y_n$ using spherical harmonics; we write $$\label{sph}
y_n=\sum_{l\ge 0, |m|\le l}R_{n,l,m}(r)Y_l^m(\theta,\phi).$$ The last integral in (\[eq:nonreal\]), including the prefactor, then equals $$\begin{gathered}
\label{sph2}
-8\pi\,i\, r_B^2\sum_{n\in\ZZ}\sum_{m,l}\Big[\overline{R}_{n,m,l}R'_{n,m,l}-
\overline{R'}_{n,m,l}R_{n,m,l}\Big]\\=-8\pi\,i\, r_B^2\sum_{n\in\ZZ}\sum_{m,l}W[\overline{R}_{n,m,l},R_{n,m,l}]\end{gathered}$$ where $r_B$ is the radius of $B$ and $W[f,g]$ is the Wronskian of $f$ and $g$. On the other hand, since $V$ and $\Omega$ are compactly supported, we have outside of $B$ $$\label{eq:outside}
\Delta y_n-(\sigma+n\omega)y_n=0$$ and then by (\[sph\]), $R_{n,l,m}$ satisfy for $r>r_B$ the equation $$\label{Rnlm}
R''+\frac{2}{r}R'-\frac{l(l+1)}{r^2}R=(\sigma+n\omega)R$$ where we have suppressed the subscripts. Let $g_{n,l,m}=rR_{n,l,m}$. Then for the $g_{n,l,m}$ we get $$\label{gnlm}
g''-\left[\frac{l(l+1)}{r^2}+(\sigma+n\omega)\right]g=0$$ thus $$\label{r-g}
\overline{R}R'=\frac{\overline{g}g'}{r^2}-\frac{|g|^2}{r^3}$$ and $$\label{r-g1}
r^2W[\overline{R},R]=W[\overline{g},g]=:W_n.$$ Multiplying (\[gnlm\]) by $\overline{g}$, the conjugate of (\[gnlm\]) by $g$ and subtracting, we get for $r>r_B$, $$\label{difg}
W_n'=(\sigma-\overline{\sigma})|g|^2=2i|g|^2 \Im\,\sigma$$
\[condifty\] Simple estimates using equation (\[hmg\]), the definition (\[GreenF\]) and (\[defkappa\]) imply that, for some $c_n$, $$\label{cinf}
y_n(x)=\frac{e^{-\kappa_n|x|}}{|x|}\Big(c_n(\theta,\phi)+O(|x|^{-1})\Big)\ \text{as}\ |x|\to\infty$$
Let us consider two cases of (\[hmdiff\]).
Case (i): $\Im\,\sigma\,<0$. By Remark \[condifty\] we have $$\label{ginfty}
g\sim C e^{-\kappa_n r}(1+o(1))\ \ \text{as}\ \ r\to\infty$$ There is a one-parameter family of solutions of (\[gnlm\]) satisfying (\[ginfty\]) and the asymptotic expansion can be differentiated [@Wasow]. We assume, to get a contradiction, that there exist $n$ for which $g_n\ne 0$. For these $n$ we have, using (\[ginfty\]), differentiability of this asymptotic expansion and (\[defkappa\]) that $$\label{t2}
\frac{1}{2i}\lim_{r\to\infty}|g_n|^{-2} W_n=-\Im\kappa_n\, >0.$$ It follows from (\[difg\]) and (\[t2\]) that $\frac{1}{2i}W_n$ is strictly positive for all $r>r_B$ and all $n$ for which $g_n\ne
0$. This implies that the last term in (\[eq:nonreal\]) is a sum of positive terms which shows that (\[eq:nonreal\]) cannot be satisfied.
Case (ii): $\Im\,\sigma\,=0$. For $n>0$ there exists only one solution $g$ of (\[gnlm\]) which decays at infinity (cf. Remark \[condifty\] and the discussion in Case (i)), and since (\[gnlm\]) has real coefficients this $g$ must be a (constant multiple of a) real function as well; therefore we have $W_n=0$ for $n\ge 0$.
For $n<0$, we use Remark \[condifty\] (and differentiability of the asymptotic expansion as in Case (i)) to calculate the Wronskian $W_n$ of $g,\overline{g}$ in the limit $r\to \infty$: $W_n=|c_n|^2
(1+o(1))$. Since for $\Im\,\sigma=0$, $W_n$ is constant, cf. (\[difg\]), it follows that $W_n$ is exactly equal to $|c_n|^2$. Thus, using (\[eq:nonreal\]) and (\[sph2\]) we have $$\label{IonizCond}
y_n(x)=0\ \ \text{for all}\ \ n<0 \ \text{and}\ |x|>r_B$$
\[cdi\] In the setting (\[compactDa\]), if [**(A)**]{} fails, then we have $$\label{I2}
y_n(x)=0\ \ \text{for all}\ \ n<0 \ \text{and}\ x\notin {D}$$
Outside $D$ we have $\mathfrak{O}y_n=0$, where $\mathfrak{O}$ is the elliptic operator $-\Delta+\sigma+n\omega$. The proof follows immediately from (\[IonizCond\]), by standard unique continuation results [@Hormander], [@Miranda], [@Treves] ( in fact, $\mathfrak{O}$ is analytic hypo-elliptic).
\[Overdet\] Proposition \[cdi\] points toward generic ionization under time periodic forcing. Indeed, we see from (\[I2\]) that equations (\[hmdiff\]) are formally overdetermined when $n<0$ ($y_n$ is in the domain of $\Delta$ so that, in (\[hmdiff\]), the function and “one derivative” are given on the boundary) and are expected, generically, not to have nontrivial solutions even if $y_{n}$ had to satisfy (\[eq:finy\]) for $n<0$ only.
The latter reduced problem is relatively easier to study and we used it to show that [**(A)**]{} holds in a number of settings, including the potential (\[nonp\]), see [@UAB].
There do in fact exist nongeneric potentials (though not in the class (\[as1\])) for which ionization fails [@CMP221; @[6a]; @RCL].
Connection with Floquet theory, continued {#F22}
=========================================
\[P17\] If $u$ is an eigenfunction or resonance[^5] of the operator $K$ defined in (\[eq:quasien\]) such that $u\in L^2(\RR^3\times S^1_{{2\pi}/{\omega}})$, then $u=\mathfrak{C}u$ in $\mathcal{H}$ and so [**(A)**]{} fails.
The proof is an immediate consequence of Proposition \[P161\] in Appendix \[PP21\]. Conversely, we have the following result.
\[Fl3\] We assume the setting (\[compactDa\]).
\(i) If $v=\mathfrak{C}v$ for some $\sigma_0\in(0,\omega)$ and $v\in\mathcal{H}$, then $v\in l^2(L^2(\RR^d))$ thus it is an eigenfunction of $K$.
\(ii) If $v=\mathfrak{C}v$ for $\sigma_0=0$ and $v\in\mathcal{H}$, then, for $d=3$, $v$ is of the form $v=C|x|^{-1}\delta_{n0}+\tilde{v}(x)$ where $\tilde{v}\in
l^2(L^2(\RR^3))$ (resonance of $K$).
For the same reasons as before, we focus on $d=3$.
\(i) We see from (\[GreenF2\]), (\[formulares\]) and (\[I2\]) that for all $n$ we have $y_n\in L^2(\RR^3)$. Furthermore, a straightforward calculation shows that $\|y_n\|_{L^2(\RR^3)}\le
C\|y_n\|_D $ where $C$ is independent of $n$. Proposition \[P161\] in Appendix \[PP21\] gives the necessary estimates in $n$ to complete the proof in this case.
\(ii) For $n\ne 0$ we have, for the same reasons as in (i), $y_n\in
L^2(\RR^3)$. But now, at $n=0$, since $\sigma_0=0$ the Green function (\[formulares\]) does not have enough decay to ensure $y_0\in
L^2(\RR^3)$. We have instead, for $x'\in D$ and $|x|\to\infty$, $G_0(x-x')=\frac{1}{4\pi}|x|^{-1}+O(|x|^{-2})$. The statement now follows from (\[GreenF2\]) and (\[eq:fint2\]). $\Box$
Example (\[nonp\]) {#Examples}
==================
To show that it can be effectively checked whether (\[I2\]) can be nontrivially satisfied, we consider the example (\[nonp\]). It is convenient to Fourier transform the system (\[hmdiff\]) in $x$. In view of (\[I2\]), for $n<0$, $y_n=0$ outside ${D}$. We then have, for $n<0$, $$\label{eq:FT1}
\check{y}_n:=\int_{\RR^3} y_n e^{-ik\cdot x}dx=\int_{{D}} y_n e^{-ik\cdot x}dx$$ and $$\label{eq:FT}
-k^2\check{y}_n=-k^2\int_{\RR^3} y_n e^{-ik\cdot x}dx= \int_{\RR^3}\Delta y_n e^{-ik\cdot x}dx=\int_{{D}}\Delta y_n e^{-ik\cdot x}dx$$ For the setting (\[nonp\]) and $n<-1$, (\[hmdiff\]) reads $$\label{F1}
(k^2+\sigma+n\omega)\check{y}_n=-V_{{D}}\check{y}_n+i\Omega_{{D}}\left( \check{y}_{n+1}- \check{y}_{n-1}\right)$$
\[R12\] For $n\le -1$, the functions $\check{y}_n$ are entire of exponential order one; more precisely, if $B$ is a ball containing ${D}$ we have $$\label{expord1}
|\check{y}_n(k)|\le \sqrt{{\rm Vol}({D})} \,\,e^{|k|r_B}\,\,\|y_n\|_{L^2(D)}$$
This follows immediately from the definition of $\check{y}$. (See also [@Zemanian] for a comprehensive characterization of the Fourier transform of a compactly supported distribution.)
\[entire\] The generating function $$\label{eq:defY}
Y(k,z)=\sum_{m\ge 0}\check{y}_{-m-2}(k)z^m$$ is entire in $k$ and analytic in $z$ for $|z|<1$.
. Since $y\in\mathcal{H}$ we have $$\label{bound}
\|y_n\|_{L^2(D)}\le {\rm const}\,\, |n|^{-3/2}$$ Using Remark \[R12\] the conclusion follows.
A straightforward calculation shows that $Y$ satisfies the equation $$\label{eqY}
M Y-z \frac{\partial Y}{\partial z}-i\beta \left(z
-\frac{1}{z} \right)Y=
i\beta\check{y}_{-1} +i\beta\frac{\check{y}_{-2}}{z}$$ where $$\label{defM}
M=\omega^{-1}(k^2+\sigma-2\omega+V_{{D}})$$ and $\beta=\Omega_{{D}}/\omega$. The solution of (\[eqY\]) is $$\label{sol1}
Y=z^Me^{-i\beta(z+z^{-1})}\left[C(k)-i\beta\int_0^z e^{i\beta(s+s^{-1})}
\left(\frac{\check{y}_{-1}}{s^{M+1}}
+\frac{\check{y}_{-2}}{s^{M+2}}\right)ds\right]$$ where the integral follows a path in which $0$ is approached along the negative imaginary line.
Proposition \[entire\] implies $C(k)\equiv 0$.
. It is easy to check that otherwise the limit of $Y(k,z)$ as $z\to 0$ along $i\RR^{-}$ would not exist. $\Box$
Thus $$\label{sol2}
Y(k,z)=-i\beta z^Me^{-i\beta(z+z^{-1})}\int_0^z e^{i\beta(s+s^{-1})}
\left(\frac{\check{y}_{-1}(k)}{s^{M+1}}
+\frac{\check{y}_{-2}(k)}{s^{M+2}}\right)ds$$ We now use the nontrivial monodromy of $Y$ on the Riemann surface of $\log z$, following from the integral representation (\[sol2\]). Analytic continuation around the origin gives $$\label{monodr}
i\beta^{-1}e^{i\beta(z+z^{-1})} (Y(\cdot,ze^{2\pi i})-Y(\cdot,z))=\check{y}_{-1}\oint_{\mathcal{C}} \frac{e^{i\beta(s+s^{-1})}}{s^{M+1}}ds+\check{y}_{-2}\oint_{\mathcal{C}} \frac{e^{i\beta(s+s^{-1})}}{s^{M+2}}ds$$ where $\mathcal{C}$ is the curve shown in Fig. 1.
0.5cm 0.5cm
Let $$\label{defF}
F(M)=\oint_{\mathcal{C}} \frac{e^{i\beta(s+s^{-1})}}{s^{M}}ds$$
\[013\] We have $$\label{mon2}
\check{y}_{-1}(k)F(M+1)+\check{y}_{-2}(k)F(M+2)=0$$
This follows immediately from the discussion above. $\Box$
\[014\] For every large $N\in\NN$, $F(z)$ has exactly one zero of the form $z_N=N+o(N^0)$. For large $N$ we have $F(1+z_{N})\ne 0$.
It turns out that $F(M)$ is a Bessel function of order $M$ evaluated at $2$ and a proof can be given based on this representation. However, in view of later generalizations we prefer to give a more general argument that does not rely on explicit representations.
Let $M=N+\zeta$ with $N$ a large positive integer, $\zeta$ complex with $|\zeta|=\epsilon$ and $\epsilon$ positive and small. Let $C_1$ be the counterclockwise circle $\{z:|z|=1\}$ and $L$ the segment $[0,-i]$; we write $$\begin{gathered}
\label{dec2}
F(N+\zeta)=(1-e^{-2\pi i\zeta})\int_L
\frac{e^{i\beta(s+s^{-1})}}{s^{M}}ds+\int_{C_1}
\frac{e^{i\beta(s+s^{-1})}}{s^{M}}ds\\=
2\pi i\zeta\Big(1+O(\zeta^{-1})\Big)\int_L
\frac{e^{i\beta(s+s^{-1})}}{s^{M}}ds+\int_{C_1}
\frac{e^{i\beta(s+s^{-1})}}{s^{M}}ds\end{gathered}$$ where in the integral along $L$ the principal branch of the log is used. The integral over $L$ can be estimated with Watson’s Lemma, see e.g. [@benderorszag] $$2\pi i\zeta\int_L
\frac{e^{i\beta(s+s^{-1})}}{s^{M}}ds=i\zeta (2\pi)^{3/2}\beta^{1-N} N^{N-3/2}e^{-N}e^{-\zeta\ln(\beta/N)}\Big(1+o(N^0)\Big)$$ By the Riemann-Lebesgue lemma, $\int_{C_1}\to 0$ as $N\to\infty$. We get $$\label{estF1}
F(N+\zeta)= i^M\zeta (2\pi)^{3/2}\beta^{1-N} N^{N-3/2}e^{-N}e^{-\zeta\ln(\beta/N)}\Big(1+o(N^0,\zeta^0)\Big)+o(N^0)$$ The existence of a unique simple zero at some $N+\zeta_N$ with $|\zeta_N|<\epsilon$ is a consequence of the argument principle.
The position of $\zeta_N$ can be found more accurately as follows. We have $$\label{ratio}
\zeta_N=\frac{\zeta_N}{1-e^{-2\pi i\zeta_N}}\int_{C_1}
\frac{e^{i\beta(s+s^{-1})}}{s^{N+\zeta_N}}ds\Big(\int_L
\frac{e^{i\beta(s+s^{-1})}}{s^{N+\zeta_N}}ds\Big)^{-1}$$ from which it follows that $\zeta_N=o(\text{ \rm const}^N/N!)$ which readily implies that (\[ratio\]) is contractive and that $$\label{ratio1}
\zeta_N=\frac{1}{2\pi i}\int_{C_1}
\frac{e^{i\beta(s+s^{-1})}}{s^{N}}ds\Big(\int_L
\frac{e^{i\beta(s+s^{-1})}}{s^{N}}ds\Big)^{-1}\Big(1+o(N^0)\Big)$$ Using (\[estF1\]) and the fact that the first integral in (\[ratio1\]) gives the Laurent coefficients of $e^{i\beta(s+s^{-1})}$ which can be independently estimated from the series expansion, we find that, with constants that can be calculated, $$\label{estzetaN}
|\zeta_N|=c_1 c_2^N N^{-2N+c_3}\Big(1+o(N^0)\Big)$$ Thus $\zeta_{N+1}/\zeta_N\to 0$ as $N\to \infty$ and the second part of the Proposition follows.
Relation (\[mon2\]), with $\check{y}_{-1}(k), \check{y}_{-2}(k)$ entire of exponential order one (cf. Remark \[R12\]) implies $$\label{concl1}
\check{y}_{-1}(k)= \check{y}_{-2}(k)=0 \ \forall \, k\in\CC^3$$ and then $$\label{concl12}
{y}_{n}(x)= 0\ \forall n\in\ZZ\ {\text{\rm and almost all}}\ x\in\RR^d$$
. Propositions \[013\] and \[014\] and (\[defM\]) imply that $\check{y}_{-1}(k)$ has at least $const\, R^2$ zeros in a disk of large radius $R$. Since $\check{y}_{-1}(k)$ is an entire function of exponential order one, it follows that $\check{y}_{-1}\equiv 0$ (see, e.g. [@SZ]). By (\[mon2\]) we have $\check{y}_{-2}\equiv 0$, so that ${y}_{-1}\equiv {y}_{-2}\equiv
0$ .
In the present model (\[hmdiff\]) reads $$\label{hmdiffs}
(-\Delta+\sigma+n\omega)y_n=-V_D\bchi_Dy_n-i\Omega_D\bchi_D(y_{n+1}-y_{n-1})$$ and (\[hmdiffs\]) with $n=-1$ and $n=-2$ implies $\bchi_D y_0=0$, $\bchi_D y_{-3}=0$ respectively; inductively $\bchi_D y_n=0$ for all $n$. Then $(-\Delta+\sigma+n\omega)y_n=0$ in $\RR^d\setminus\partial
D$. Since, by Proposition \[Fl3\], $y_n\in L^2(\RR^3)$ we see (for instance by taking the Fourier transform) that $\|y_n\|=0.$
\[extension\] It can be shown that condition [**(A)**]{} holds with $V_D\bchi_D$ replaced by $V_D\bchi_D+V_1(x)$ where $V_1(x)$ is bounded, not necessarily constant, with compact support disjoint from $D$. On the support of $V_1$, $\Omega$ is zero and it can be seen that for $n$ sufficiently negative $y_n$ is zero on the support of $V_1$. From this point on the arguments are very similar, but we will not pursue this here.
Compactness {#S6}
===========
Compactness of the operator $\mathfrak{g}_n$ defined in (\[GreenF2\]) {#compactness}
---------------------------------------------------------------------
The case $d=1$ is discussed in Appendix \[pd=1\]. For $d=2,3$ compactness follows from Theorem VI.23 , Vol. 1, pp. 210 of [@Reed-Simon] (for $d=3$, note that $e^{-\kappa_n|x-y|}/|x-y|\in L^2(D\times D)$).
Compactness of $\mathfrak{C}$
-----------------------------
The property (\[eq:Agmon\]) is mentioned in Appendix A of [@Agmon]. We include here an elementary proof of (\[eq:normK\]) below (which also can be refined without serious difficulty to yield the sharper estimate (\[eq:Agmon\])).
\[L22\] We have $$\label{eq:normK}
\|\mathfrak{g}_n\|{\longrightarrow} 0\ \ \ {\text{as}}\ |n|\to\infty$$ (where $\|\cdot\|$ is the ${L^2(D)\mapsto L^2(D)}$ operator norm) uniformly in $u$ in the region $S_{\omega,\epsilon,A}=\{u:|u|<A,|\Re(u^2)|<\omega -\epsilon\}$, where $A>0$ and $\epsilon$ is any small positive number.
. Relation (\[eq:normK\]) follows from a general result by Agmon [@Agmon] which provides estimates on the rate of convergence. We give below an elementary proof in our case.
We prove the result for $d=3$ (the proof is simpler in $d=1,2$, noting that for large $x$ with $\arg\,x \in (-\pi,\pi) $ we have $K_{0}(x)=\sqrt{\frac{\pi}{2}}e^{-x}x^{-1/2}(1+o(1))$). Define $$\label{defQ}
Q_{n}(x',x'')= \int_{{D}} dx
\frac{e^{-\kappa_n|x'-x|-\overline{\kappa}_{n}|x''-x|}}{|x'-x||x''-x|}$$ We have $$\begin{gathered}
\label{normK}
\|\mathfrak{g}_n\|^2= \sup_{\|f\|=1}\int_{{D}^2}Q_{n}(x',x'')f(x')\overline{f(x'')}dx'dx''\\ \le \left(\int_{{D}^2}|Q_{n}(x',x'')|^2 dx'dx'\right)^{1/2}\end{gathered}$$ The last integral goes to zero as $|n|\to\infty$. To see that, note that $$\label{rel1}
\sqrt{n\omega+u^2}=\sqrt{n\omega}+O(n^{-1/2});\ \ {\text{as}}\ n\to+\infty$$ and using the triangle inequality we get $$\begin{gathered}
\label{defQ2}
\left|Q_{n}(x',x'')\right|\le {\rm
Const\,}e^{-\sqrt{n\omega}|x'-x''|}\int_{{D}}
\frac{dx}{|x'-x||x''-x|}\\\le {\rm
Const\,}e^{-\sqrt{n\omega}|x'-x''|}\end{gathered}$$ and the conclusion, for $n\to +\infty$ follows by dominated convergence. We now focus on large negative $n$. Since $$\label{rel2}
\sqrt{n\omega+u^2}=-
i\sqrt{|n|\omega}+O(n^{-1/2});\ \ {\text{as}}\ n\to-\infty$$ it is easy to check that (\[eq:normK\]) follows once we show that $\|\mathfrak{g}_{[\nu]}\|\to 0$ as $\nu\to\infty$ where $\|\mathfrak{g}_{[\nu]}\|$ is obtained by replacing $\kappa_n$ with $i\nu$ in the definition of $\mathfrak{g}_n$. We first show, with an analogous definition of $Q_{[\nu]}$, that $$\label{eq:boundQ}
\sup_{x,x'\in{D},\nu\in\RR}\left|Q_{[\nu]}(x,x')\right|=Q_0<\infty$$ Indeed, we choose a ball $B_b$ centered at $x'$ of radius $b$ large enough so that it contains ${D}$ and write the integrals (\[defQ\]) in spherical coordinates centered at $x'$ with $x''$ on the $z$ axis; in these coordinates $|x-x'|=r$ and $|x-x''|\ge d(x,Oz)=r\sin\theta$ and thus $$|Q_{[\nu]}(x,x')|\le \int_{B_b}\frac{dx}{|x-x'||x-x''|}\le
\int_{B_b}drd\theta d\phi\le 4\pi b$$
Let $\rho(x;x',x'')=|x-x'|-|x-x''|$. We then have $|\rho(x;x',x'')|\le
|x'-x''|$ and we get $$Q_{[\nu]}(x',x'')=\int_{{D}} dx
\frac{e^{i\nu\rho(x;x',x'')}}{|x'-x||x''-x|}=
\int_{-|x''-x'|}^{|x''-x'|}e^{i\nu\rho}d\mu(\rho)$$ where the positive measure $\mu$ is defined by $$\label{eq:intgrd}
\mu(A)= \mu_{x',x''}(A)=\int_{\{x:\rho(x)\in A\}\cap {D}}\frac{dx}{{|x'-x||x''-x|}}$$ Since the integrand in (\[eq:intgrd\]) is in $L^1$, the measure $\mu
$ is absolutely continuous with respect to the Lebesgue measure $m$. We let $h(\rho;x',x'')=\frac{d\mu}{dm}$; then $h\in L^1$ and we get $$\label{Lebesgue}
Q_{[\nu]}(x',x'') =
\int_{-|x''-x'|}^{|x''-x'|}e^{i\nu\rho}h(\rho;x',x'')d\rho$$ By the Riemann-Lebesgue lemma we have[^6] $$\label{eq:Qtozero}
Q_{[\nu]}(x',x'') \to 0\ \text{as}\ \nu\to\infty$$ Now (\[eq:boundQ\]), (\[eq:Qtozero\]) and again dominated convergence implies $ \|\mathfrak{g}_{[\nu]}\|\to 0$ as $\nu\to\infty$ completing the proof.
\[L27\] Under the assumption (\[compactDa\]), the operator $\mathfrak{C}$ is compact on $\mathcal{H}$ and analytic in $u$ in $S_{\omega,\epsilon, A}$, cf. Lemma \[L22\].
Indeed, $\mathfrak{C}$ is the norm limit (\[limC\]), uniform in $u\in S_{\omega,\epsilon, A}$, where $\mathfrak{C}_N$ are compact by Lemma \[L22\] and analytic as explained in the proof of Lemma \[L6\]. We note that the operator $$\label{normOmega}
\left\|\sum_{j\in\ZZ}\Omega_j S^{-j}\right\|$$ is bounded in $\mathcal{H}$. Indeed, if we write $\langle
n\rangle:=1+|n|$ we have, for $(n,j)\in\ZZ^2$ $\langle n\rangle \le
\langle j\rangle\langle n-j\rangle$ and $$\begin{gathered}
\label{Yajima1}
\sum_{n\in\ZZ}\langle n\rangle^{\gamma}\left|\sum_{j\in\ZZ}\Omega_j y_{n-j}\right|^2\le
\sum_{n\in\ZZ}\left(\sum_{j\in\ZZ} \langle j\rangle^{\gamma/2}|\Omega_j| \langle n-j\rangle^{\gamma/2}|y_{n-j}|\right)^2 \\ =
\sum_{j_1,j_2\in\ZZ}\langle j_1\rangle^{\frac{\gamma}{2}}
|\Omega_{j_1}|
\langle j_2\rangle^{\frac{\gamma}{2}}
|\Omega_{j_2}|
\sum_{n\in\ZZ}
\langle n-j_1\rangle^{\frac{\gamma}{2}}|y_{n-j_1}|
\langle n-j_2\rangle^{\frac{\gamma}{2}}|y_{n-j_2}|\\
\le \|y\|_{l^2_\gamma}^2\|\langle j\rangle^{\gamma/2}\Omega_j\|^2_1\le C\|y\|_{l^2_\gamma}^2\end{gathered}$$ by (\[Fourier\]). $\Box$
Appendixes
==========
Compact operator formulation and proof of Lemma \[L4\] for $d=1$ {#pd=1}
----------------------------------------------------------------
We can assume without loss of generality $D\subset[-1,1]$. For $n=0$ we choose some large positive $a$ such that $\sin 2\sqrt{a}\ne 0$, denote by $f_{\pm}(x)$ the functions $e^{\mp ux}$ and let $\psi_+$ be the solution of the equation $$\label{d=1,1}
-\psi''+(a \bchi_{[-1,1]}+ u^2)\psi=0$$ (see Remark \[R4\]) with initial condition $\psi_+(1)=f_+(1)$, $\psi'_+(1)=f'_+(1)$, and similarly let $\psi_-$ be the solution of (\[d=1,1\]) with initial condition $\psi_-(-1)=f_-(-1)$, $\psi'_-(-1)=f'_-(-1)$. Since both the equation and the initial conditions are analytic in $u$ at $u=0$, so are the solutions $\psi_{\pm}$ and their Wronskian $W(u)$. It can be checked that $W(0)=\sqrt{a}\sin 2\sqrt{a}\ne 0$. In fact, taking $\tau_a=\sqrt{a-u^2}$ we have $$\label{pm}
\psi_{\pm}(x)=\tau_a^{-1}e^{-u}\Big[\tau_a\cos(\tau_a x\mp \tau_a)\mp u\sin(\tau_a x\mp \tau_a))\Big]$$ In a neighborhood of $u=0$ we write for $n\ne 0$ the same integral expression (\[eq:fint2\]), while for $n=0$ we write $$\label{d1,2}
y_0=\mathfrak{g}_{0,a}\psi_{1,0}-\mathfrak{g}_{0,a} (V +a \bchi_{[-1,1]}) y_0+i\mathfrak{g}_{0,a}\left(\sum_{j\in\ZZ}\Omega_j S^{-j}y\right)_n$$ where $$\label{eq:defg0}
W(u)(\mathfrak{g}_{0,a} f)(x)=\psi_{+}(x)\int_{-1}^x \psi_{+}(s)f(s)ds-
\psi_{-}(x)\int_{1}^x \psi_{-}(s)f(s)ds$$ With the same conventions, we now write the integral system in the form (\[eq:fint2\]). Compactness and analyticity are now shown in the same way as for $d=2,3$. $\Box$.
Proof of Lemma \[L5\] {#Unic}
---------------------
Let $\sigma=\sigma_0-2i\tau$ where $\sigma_0\in[0,\omega)$ and $\tau>0$. We show that $$\label{normc1}
\|\mathfrak{C}\|\to 0\ \ {\text as }\ \ \tau\to\infty$$ and uniqueness follows by contractivity. The calculation leading to (\[normc1\]) is quite straightforward, but we provide it for convenience. In $d=1,2$ the estimate follows from the behavior of the Green function for large argument. We then focus on $d=3$. By (\[defkappa\]) we have $$\label{eq:evalroot}
\Re(\kappa_n)=\left(\frac{1}{2}\left((\sigma_0+n\omega)^2+\sigma_0+n\omega\right)^{1/2}+\tau\right)^{1/2}$$ For $n>0$ we then have $\Re\kappa_n>\sqrt{n\omega}$ and the same calculation as for (\[defQ2\]) shows that $$\label{eq:normK1}
\|\mathfrak{g}_n\|\mathop{\longrightarrow}_{L^2(B)} 0\ \ \ {\text{as}}\ n\to +\infty$$ uniformly in $\tau$. For $n<0$ (\[eq:evalroot\]) gives $$\label{Qn23}
\left|Q_{n}(x',x'')\right|\le \left|Q_{\nu}(x',x'')\right|$$ where $$\label{eq:evalroot3}
-\nu:=\Im(\kappa_n)\to\infty \ \text{as } n\to-\infty$$ and now (\[eq:Qtozero\]) shows that $$\label{eq:normK2}
\|\mathfrak{g}_n\|\mathop{\longrightarrow}_{L^2(B)} 0\ \ \ {\text{as}}\ n\to -\infty$$ uniformly in $\tau$. We choose then $n_0$ large enough so that $$\label{eq:normK23}
\sup_{n\ge n_0,\tau>0}\|\mathfrak{g}_n\|\le \epsilon$$ For $\tau$ large enough we have, still from (\[eq:evalroot\]), $$\label{midr}
\Re(\kappa_n)>\frac{1}{2}\tau^{1/2};\ \ -n_0\le n\le n_0$$ Choosing a ball $B$ centered at $x$ containing $D$, we then have for large $\tau$ and some constants independent of $f, \tau,x$ and $n\in(-n_0,n_0)$, with the notation $\alpha=\frac{1}{2}\tau^{1/2}$, $$\begin{gathered}
\label{eq:partK2} \left|\Big(\mathfrak{g}_{n} f\Big)(x)\right|\le
\left\|\frac{e^{-\alpha|x-x'|}}{|x-x'|}\right\|_{L^2(D)}\|f\|_{L^2(D)} \\ \le
C_1\|f\|_{L^2(D)}\left\|\frac{e^{-\alpha|x-x'|}}{|x-x'|}\right\|_B\le
\frac{C_2}{\tau}\|f\|_{L^2(D)}\le\epsilon\end{gathered}$$ and the conclusion follows.
Proof of Lemma \[L12\] {#simpoles}
----------------------
Assume $\sigma_0$ is a value of $\sigma$ where invertibility of $I-\mathfrak{C}(\sigma_0)$ fails. Then $\sigma_0\in[0,\omega)$. By the Fredholm alternative we know that $I-\mathfrak{C}(\sigma)$ is invertible in some punctured neighborhood of $\sigma_0$ where the solution of (\[eq:fint2\]) is meromorphic.
(i): $\sigma_0\ne 0$. Denote $\zeta=\sigma-\sigma_0$. We rewrite (\[eq:fint2\]) in a suitable way near $\sigma_0$. We have from (\[eq:finy\]) $$\label{mod1}
(-\Delta+\sigma_0+n\omega)y_n=-i\psi_{2,n}-Vy_n-\zeta y_n+
\sum_{j\in\ZZ}\Omega_j(x)\left(S^{-j}y\right)_n$$ which we write symbolically $$\label{mod12}
\mathfrak{W}y=-\zeta y-i\psi_{2,n}$$ and from (\[eq:finy\]) and (\[eq:fint2\]) we have \*\*\*\*\*\*\*\*\*\*\*\*\* $$\label{eq:fint3}
y_n=-i\mathfrak{g}_n\psi_{2,n}-\mathfrak{g}_n \Big[V y_n-\zeta y_n-\sum_{j\in\ZZ}\Omega_j \left(S^{-j}y\right)_n\Big]$$ implying the following version of (\[eq:fint2\]), with evident notation, $$\label{it2}
y=y_0+\zeta\mathfrak{g}y +\mathfrak{C}(\sigma_0)y$$ On the other hand, $$\label{laurent}
y=\sum_{j=-M}^{\infty}c_j\zeta^j$$ with the coefficients $c_j\in\mathcal{H}$. Assume, to get a contradiction, that $M\ge 2$. Inserting in (\[it2\]) we get $$\begin{aligned}
\label{f0}
c_{-M}&=&\mathfrak{C}(\sigma_0) c_{-M}\\
c_{-M+1}&=&\mathfrak{C}(\sigma_0) c_{-M+1}+\mathfrak{g}c_{-M}\nonumber\\
\cdots\nonumber\end{aligned}$$ In differential form we have, $$\begin{aligned}
\label{f01}
\mathfrak{W}c_{-M}&=&0\\
\mathfrak{W}c_{-M+1}&=&-c_{-M}\end{aligned}$$ By (\[f0\]) and Proposition \[Fl3\] we have $c_{-M}\in
l^2(L^2(\RR^d)):=\mathcal{H}_1$ (in fact, $(c_{-M})_n$ decay at least exponentially in $|x|$). On the other hand we then have from (\[f01\]) and noting the formal self-adjointness of $\mathfrak{W}$, $$\label{mod3}\langle c_{-M}, c_{-M}\rangle=
-\Big\langle c_{-M},\mathfrak{W}c_{-M+1}\Big\rangle=
-\Big\langle\mathfrak{W} c_{-M},c_{-M+1}\Big\rangle=0$$ which is a contradiction.
\(ii) $\sigma_0=0$: there are two differences w.r.t case (i): (a) meromorphicity and Laurent expansions now use the variable $u=\sqrt{\sigma}$; and (b) $c_{-M}$ is not necessarily in $\mathcal{H}_1$ so we work with $\mathcal{H}_B=l^2(L^2(B))$ for large enough $B$. These differences can be dealt with straightforwardly, so we just outline the main steps. The Laplacian is the only ingredient of $\mathfrak{W}$ not formally self-adjoint in $\mathcal{H}_B$. Integration by parts, implicit in (\[mod3\]) brings in boundary terms of the form $$\label{eq:Bt}
\int_{\partial B}f\nabla g\cdot dS$$ where $f$ and $g$ are $c_{-M}$ or $c_{-M+1}$. Both $f$ and $g$ have decay $|x|^{-1}$ and this behavior is differentiable, as is manifest from (\[f0\]), (\[formulares\]), and (\[GreenF2\]). The contribution from the integrals (\[eq:Bt\]) is thus $O(r_B^{-1})$ which equals the norm $\|c_{-M}\|_{\mathcal{H}_B}$, clearly nondecreasing in $r_B$. This again forces $c_{-M}=0$, a contradiction. $\Box$
Sketch of the proof of Proposition \[genth1\] and Theorem \[genth12\] {#Sketch}
---------------------------------------------------------------------
We first show Theorem \[genth12\] (i). The contour of the inverse Laplace transform can be deformed as shown in Fig. 2. Pushing the contour of integration to the left brings in residues due to the meromorphic integrand, and since the kernel of the inverse Laplace transform is $e^{pt}$, residues in the left half plane give rise to decaying exponentials in $\psi(x,t)$. Uniform bounds on the Green function as $p\to -\infty$ are easy to prove. Consequently, there are only finitely many arrays of poles of $\hat{\psi}$. The contour of integration in the inverse Laplace transform can be pushed all the way to $-\infty$ in view of the exponential decay of the kernel $e^{pt}$. We are left with integrals along the sides of the cuts which, after the change of variable $p\leftrightarrow -p$ (or $p\leftrightarrow -pe^{i\alpha}$ if poles exist on the cuts), are seen to be Laplace transforms. Since $\hat{\psi}$ is analytic in $\sqrt{p+i n\omega}$, the contour deformation result shows, ipso facto, Borel summability of the asymptotic series of $\psi(x,t)$ for large $t$.
The general case is proved in a very similar way, using Lemma \[L12\]. If [**(A)**]{} does not hold, then some of the poles of the meromorphic function $y_n(\sigma)$ can be on the segment $(-\omega,\omega)$. If a pole is placed at $\sigma=0$, then analyticity in $u$ in the operator entails a singularity of the form $\sigma^{-1/2}A(\sigma)+B(\sigma)$ with $A$ and $B$ analytic, whence the conclusion.
0.5cm 0.5cm
Estimates needed for Proposition \[P17\] and Proposition \[Fl3\] {#PP21}
----------------------------------------------------------------
\[P161\] Let $y$ be a solution in $l^2(L^2(B))$ of the homogeneous equation associated to (\[eq:finy\]). Under the assumptions (\[compactDb\]), we have $$\label{1/j}
\| y_j\|_{L^2(B)}=O(j^{-2})\ {\text as } |j|\to\infty$$
Let $\epsilon$ be small enough and choose $j_0>0$ large enough (the proof is similar for $j_0<0$) so that $\|\mathfrak{g}_j\|_{L^2(B)}<\epsilon$ for all $j\ge j_0$, see (\[eq:normK\]). We consider the Banach space $\mathcal{B}_{j_0}$ of sequences of functions $\{y_j\}_{j\ge j_0}$ defined on $B$ for which the norm $$\label{NormB}
\|y\|_{j_0}:= \sup_{j\ge j_0}j^{2}\| y_j\|_{L^2(B)}$$ is finite. For $j>j_0$ we write the homogeneous part of (\[eq:finy\]) in the form $$\begin{gathered}
\label{eq:fintq}
y_j=-\mathfrak{g}_j V y_j+i\mathfrak{g}_j\Big[\sum_{m\ge 0}\Omega_{-m}
y_{m+j}(x) \\+\sum_{0\le l \le j-j_0}\Omega_l y_{j-l}(x)+\sum_{l\ge
j-j_0}\Omega_l y_{j-l}(x)\Big]
\\=-\mathfrak{g}_j V y_j+i\mathfrak{g}_j\Big[\sum_{m\ge 0}\Omega_{-m}
y_{m+j}(x)+\sum_{0\le l \le j-j_0}\Omega_l y_{j-l}(x)\Big] +E_j(x)\\=:
\Big(\mathfrak{T}_{j_0}\,\,y+E\Big)_j\end{gathered}$$ Since $\|\Omega_j\|_{L^2(B)}=O(j^{-2})$ and $y\in l^2(L^2(B))$ we see that $\|E\|_{j_0}<\infty$. It can be checked that $\mathfrak{T}_{j_0}:\mathcal{B}_{j_0}\to \mathcal{B}_{j_0}$ is bounded, that $\|\mathfrak{T}_{j_0}\|\to 0$ as $j_0\to \infty$, and thus eq. (\[eq:fintq\]) is contractive if $j_0$ is large. The Proposition follows.
. We are very grateful to K. Yajima for many useful comments and suggestions, including the argument in (\[Yajima1\]). We thank A. Soffer for helpful discussions. Work supported by NSF Grants DMS-0100495, DMS-0074924, DMR-9813268, and AFOSR Grant F49620-01-1-0154.
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[^1]: $^1$Also Department of Physics.
[^2]: While explicit information on long time behavior requires $\psi_0$ to be localized, decay for more general $\psi_0\in
L^2$ is then an immediate consequence of the unitarity of Schrödinger evolution.
[^3]: The same results hold if a bounded time-independent potential, not necessarily constant, with compact support disjoint from $D$ is added to $V$, see Remark \[extension\].
[^4]: The functional spaces are different. Proposition \[Fl3\] clarifies this question.
[^5]: See [@YajPriv] and Proposition \[Fl3\] (ii).
[^6]: Noting that the estimate of the norm of $\mathfrak{g}_n$ can only increase if extended to $L^2(B)$ where $B$ is a ball containing $D$, and that $h$ calculated in $B$ is piecewise smooth the max of $Q_{[\nu]}$ can be in fact bounded by an inverse power of $\nu$; we do not however need this refinement here.
|
---
author:
- 'F. Paolucci'
- 'G. Marchegiani'
- 'E. Strambini'
- 'F. Giazotto'
title: 'Phase-Tunable temperature amplifier'
---
Introduction
============
The discovery of thermoionic emission by Fredrick Guthrie in 1873 [@Guthrie1876] brought to the invention of the first electronic devices: the diode and triode amplifiers [@Guarnieri2012]. After more than $100$ years, the recent advances of transistor-based technology [@Pugh1991] made possible the design and production of new daily life devices. In the era of energy saving, the common goal in electronics is to increase the device efficiency in order to abate energy losses and pollutant emissions. Anyways, further developments of nowadays technology are bounded by quantum mechanical restrictions to miniaturization and by heat dissipation [@Mannhart2010]. The inescapable heat generated in solid-state nano-structures is considered detrimental in electronics. As a consequence, the ability of mastering the heat transport in such structures has been only recently investigated [@Giazotto2006], and it could lead to new concepts and capabilities. In this framework, the experimental demonstration in $2012$ of heat interference in a SQUID [@Giazotto2012] heralded the foundation of the thermal counterpart of coherent electronics: coherent caloritronics [@Martinez2014; @Mart2014]. Despite it is still distant from the ripeness of electronics, coherent caloritronics is rapidly growing through the design and the realization of thermal analogues of electronic devices, such as heat diodes [@Giazotto2012], transistors [@Fornieri2015], valves [@Strambini2014], amplifiers [@Fornieri2016] and modulators [@Giaz2012]. One of the theoretical foundations of coherent caloritronics resides in the prediction of the periodic dependence of thermal currents across a Josephson junction [@Josephson1962] on the quantum phase difference between the two superconductors [@Maki1965]. Hence, the resulting thermal modulation acquires a phase-coherent character. So far, quantum interference between Josephson-coupled superconductors has been realized through the use of a SQUID [@Clarke] or, more recently, taking advantage from a newly designed SQUIPT [@Giazotto2010; @Meschke2011; @Virtanen2016]. Thereby, the thermal transport across caloritronic devices is manipulated by a magnetic flux $\Phi$ threading a superconducting ring, and an external source of magnetic field is essential. The last requirement impeded the realization of fully thermal on-chip coherent caloritronic devices up to now. In the last two years, surprisingly large thermoelectric effects in spin-filtered superconducting tunnel junctions have been predicted [@Machon2013; @Ozaeta2014] and demonstrated [@Kolenda2016]. This discovery enables the direct transduction, for the first time at cryogenic temperatures, of temperature gradients into electrical signals.
Here we present the first on-chip fully thermal device in caloritronics: the phase-tunable temperature amplifier (PTA). Our architecture takes advantage from the closed-circuit current generated by a thermoelectric element in order to create a magnetic field which controls heat transport across a thermal nano-valve. By employing widely used materials and a geometry feasible with standard lithographic techniques, we show the basic input-to-output temperature conversion, and define several figures of merit in analogy to electronics to evaluate the performances of the temperature amplifier. The device layout may foster its use in different field of science, like quantum information [@Nielsen], thermal logics [@Li2012] and radiation detection [@Giazotto2006].
Working principle and basic behavior
====================================
The PTA is the caloritronic equivalent of the voltage amplifier in electronics [@Millman], since temperature is the thermal counterpart of electric potential. The voltage-temperature analogy is schematized in fig. \[Figure1\]-a, where the usual symbol of voltage amplifiers (blue) and the corresponding representation of temperature amplifiers (red) are depicted. A voltage amplifier is a device which produces an output signal $V_{OUT}=G~\Delta V_{IN}$ , where $G>1$ is the gain and $\Delta V_{IN}=V_{IN}-V_{REF}$ is the difference between the input signal $V_{IN}$ and the reference $V_{REF}$. Since the law of conservation of energy does not allow the creation of energy, the system requires a voltage supply $V_S$ to operate. Analogously, a temperature amplifier generates an output temperature $T_{OUT}=G~T_{IN}$, where $T_{IN}$ is the input signal. In this case, the operation power is supplied by a temperature $T_S$. Differently from electronics, where the absolute value of the signals has no physical meaning and an arbitrary reference potential is required, in caloritronics the temperature signals can take only positive values and they are always referred to zero temperature (zero energy). Thereby, the base temperature $T_{BATH}$ has a different and more complex role than a simple reference. It defines the background energy level, the operation [@Strambini2014] and the energy losses of the system due to electron-phonon interaction [@Giazotto2006]. In the following, we set $T_{BATH}=10~$mK that ensures low noise and reduced energy losses. The PTA is composed of a normal metal-ferromagnetic insulator-superconductor ($N-FI-S$) tunnel junction inductively coupled to a SQUIPT [@Giazotto2010] through a superconducting coil.
In an electronic conductor, a thermoelectric effect can be generated by breaking the electron-hole symmetry in the density of states (DOS) [@Mermin]. Recently, it has been shown that this can be efficiently realized in superconductor-based structure: i) by inducing a Zeeman spin-splitting $h_{ex}$ in the quasiparticle DOS, hence breaking the electron-hole symmetry for each spin band, ii) by selecting a specific spin band (spin-filtering) [@Giaz2008; @Ozaeta2014]. In our scheme, both the mechanisms are provided by a single ferromagnetic insulator layer of the $N-FI-S$ junction [@Giazotto2015]. A temperature gradient between the normal metal $N$ and the superconductor $S$ generates the thermoelectric signal: an open circuit thermovoltage $V_T$ in the Seebeck regime or a closed circuit thermocurrent $I_T$ in the Peltier regime [@Giazotto2015]. In our device, we take advantage of the closed circuit thermocurrent in order to create a magnetic field by means of a superconducting coil of self-inductance $L$. The superconductor is kept at $T_{BATH}$ while the normal metal is set to the input temperature $T_{IN}>T_{BATH}$, because in this configuration the provided thermocurrent exhibits a monotonic behavior with rising temperature gradient [@Giazotto2015]. Figure \[Figure1\]-c shows the dependence of $I_T$ on $T_{IN}$ for different values of $h_{ex}$. The thermocurrent is a growing function of the spin-splitting of the DOS (i.e. $h_{ex}$) and abruptly rises when the thermal gradient is greater than a critical value (in our numerical calculation $T_{IN}\geq$200 mK). The detailed description of the temperature-to-current transduction of the $N-FI-S$ junction is given in the Appendices.
We now turn our attention on the second building block of our device: the thermally biased SQUIPT. It is composed of a normal metal wire $N_1$ interrupting a superconducting ring $S_1$, as portrayed in fig. \[Figure1\]-b. Owing to the good electric contact between $N_1$ and $N_2$, the metal wire acquires a superconducting character through the superconducting proximity effect [@Holm1932]. A normal metal $N_2$ probe tunnel-coupled to the wire through a thin insulating layer acts as the output electrode of the device. A magnetic flux $\Phi$ threading the ring modulates the density of states of the proximized wire [@Petrashov1995; @leSueur2008] and, as a consequence, the electronic transport between $N_1$ and $N_2$ [@Giazotto2010; @Meschke2011]. Analogously, the temperature-biased SQUIPT has been predicted to act as a thermal nano-valve leading to a phase-dependent thermal transport between $S_1$ and $N_1$ [@Strambini2014]. The detailed theoretical description of the SQUIPT can be found in the Appendices. The thermal behavior of the nano-valve is resumed in fig. \[Figure1\]-d, where the dependence of $T_{OUT}$ on the magnetic flux $\Phi$ for different values of $T_S$ is plotted. The probe temperature is minimum at $\Phi=0$, where the energy gap is fully induced in the $N_1$ DOS. When the magnetic field is switched on, the probe temperature increases due to the closure of the minigap [@Giazotto2010], reaching a maximum at $\Phi\sim0.45~\Phi_0$ and slightly lowering for $\Phi \to \Phi_0/2$ [@Strambini2014]. Furthermore, the maximum value of $T_{OUT}$ increases with $T_S$ while its modulation with $\Phi$ softens for large values of the supply temperature. Notably, thermal transport across the SQUIPT is phase-dependent, because it is modulated by the superconducting macroscopic phase difference across the proximized wire [@Strambini2014].
The architecture of the PTA requires to couple these two building blocks. This goal is achieved by means of a superconducting coil of inductance $L$ connected to the thermoelectric element (see fig. \[Figure1\]-b). By placing the thermal nano-valve in the center of this coil is possible to drive the SQUIPT by means of the static magnetic flux generated by the coil. The magnetic flux through the SQUIPT is $\Phi=M~I_T$, where $M$ is the mutual inductance between the coil and the SQUIPT. This assembly permits to relate the input $T_{IN}$ with the output $T_{OUT}$ temperature. As typically done in electronics, it is useful to introduce a parameter which sets the input corresponding to the maximum operating output required (here $T_{OUT_{MAX}}$). This quantity is tipically called sensitivity (here we use the symbol $Sens$). As already seen (fig 1-d), the temperature of the output probe $N_2$ increases monotonically with the flux for values smaller than $\sim0.45\Phi_0$, where it reaches a maximum. Furthermore, the thermocurrent, hence the flux, increases monotonically with the input temperature. If we define $I_{T_{MAX}}$ as the current generated by the thermoelectric temperature for $T_{IN}=Sens$, the coupling required is $M=0.45\Phi_0/I_{T_{MAX}}$ and the output is a growing function of the input signal, as normally required to an amplifier. Note that the coupling inductance scales inversely with the sensitivity (i.e. $Sens\sim 1/M$). This is not surprising: if we consider a high operating temperature (high $Sens$), a low thermocurrent is sufficient to perform the job.
The basic behavior of the temperature amplifier is illustrated in fig. \[Figure2\]-a, where the dependence of the output temperature $T_{OUT}$ on the input temperature $T_{IN}$ is depicted for a supply temperature $T_S=250~$mK and for different sensitivities $Sens$. Note that both the minimum and the maximum output temperature are independent on $Sens$. The minimum temperature is obtained at null input signal, i.e. when the normal layer $N$ of the $N-FI-S$ element is at the bath temperature $T_{IN}=T_{BATH}$. For this reason, we refer to it as noise temperature $T_{Noise}$. The maximum, by definition, is obtained at $T_{IN}=Sens$, corresponding to a flux $0.45\Phi_0$. The horizontal dotted black line sets the minimum value of the output active range $OAR=T_{OUT_{MAX}}-T_{{OUT}_{MIN}}$ (i.e. the interval where the output varies with the input signal), defined as $T_{{OUT}_{MIN}}=T_{Noise}+10\%T_{Noise}$. The size of the OAR is independent on $Sens$ (for our simulation parameters is approximately 130 mK). The independence of the OAR on the $Sens$ may appear surprising at first. However it is easy to understand once it is realized that the OAR is only related on the valve (SQUIPT) operation, whereas $Sens$ only affects the coupling required between the thermoelectric and the valve. On the other hand, $T_{OUT}$ calculated at a specific $T_{IN}$ drops by increasing $Sens$, because the $I_T$ is independent of the sensitivity, and the inducting coupling $M$ lowers by increasing $Sens$.
The supply temperature $T_S$ has a great influence on the behavior of the PTA, because it defines the minimum and the maximum values of $T_{OUT}$, as illustrated in fig. \[Figure2\]-b. For values of $T_S$ comparable to the critical temperature $T_{C-S_1}$ of the ring of the SQUIPT, $T_{OUT}$ depends only weakly on $T_{IN}$, because the energy gap of the ring $\Delta_{S_1}$ closes and the proximized wire assumes an almost metallic character for every value of the magnetic flux $\Phi$ (i.e input temperature $T_{IN}$). By lowering $T_S$ the superconducting pairing potential rises and the flux $\Phi$ successfully modulates thermal transport across the SQUIPT in the complete range $0-0.45~\Phi_0$, hence the output temperature varies with all the values of the input signal (see the traces for $T_S=450-150~$mK in fig. \[Figure2\]-b). When $T_S\leq0.1~ T_{C-S_1}$ the thermal broadening of the Fermi distribution $k_B T_S$ is small compared to the energy gap of the ring, and the phase dependence of the thermal transport becomes dominant only when the energy gap is almost fully suppressed, i.e. $\Phi \to 0.45~\Phi_0$. Thereby, the output temperature is exclusively modulated for $T_{IN} \approx Sens$ and the output signal can be lower than the input, as shown for $T_S=60~$mK in fig. \[Figure2\]-b. The ensemble of these behaviors leads to the conclusion that the temperature amplifier efficiently works when $0.1~ T_{C-S_1}\leq T_S \leq 0.4~ T_{C-S_1}$.
The most relevant parameter for an amplifier is the gain $G$, which is plotted in fig. \[Figure2\]-c as a function of the $T_{IN}$ for different values of $Sens$ and $T_S=250~$mK. The gain is independent on $Sens$ for $T_{IN}=T_{BATH}$, because $T_{Noise}$ is only determined by $T_S$. On the contrary, $G$ strongly depends on $Sens$ when the output temperature resides in the $OAR$ (i.e. $T_{OUT}\geq T_{OUT_{MIN}}$). In particular, $G$ lowers by increasing sensitivity at fixed $T_{IN}$, and $G(T_{IN}=Sens)$ drops for rising $Sens$, because $M$ scales inversely with the sensitivity and the maximum output signal is exclusively controlled by $T_S$ (see fig. \[Figure2\]-a). For a given $Sens$, the gain grows with $T_{IN}$ when the amplifier is in the active output mode, i.e. the values of $G$ above the black dotted line in fig. \[Figure2\]-c. This behavior is the result of the joint action of the temperature-to-current conversion due to the thermoelectric element and the dependence of the thermal transport across the SQUIPT on the magnetic flux.
Depending on the requirements, one can opt for low values of $T_S$ in order to increase the $OAR$ or choose high values of $T_S$ to maximize $G$. Since the behavior of the device is satisfactory both in terms of gain and output active range only in a limited range of supply temperatures, the use of materials with higher critical temperature for the ring of the SQUIPT could be beneficial in terms of device performances. Higher values of $T_{C-S_1}$ would guarantee wider $OAR$ and larger $G$ at $T_{IN}=Sens$. The maximum value of the gain in the active region at the optimal constant ratio $T_{C-S_1}/T_S\approx5.2$ rises linearly with the critical temperature of the SQUIPT for every value of $Sens$, as depicted in fig. \[Figure2\]-d. Therefore, the PTA could potentially be used both at higher values of $T_S$ and $T_{IN}$ ensuring large $G$ and wide $OAR$ too.
Figures of merit
================
In full analogy with electronics, we define particular figures of merit for the temperature amplifier. First of all, in our system the input-to-output thermal impedance $Z_{IN-OUT}^{th}$ is infinite. This arises from the double thermal-to-electrical-to-thermal transduction which ensures perfect heat decoupling between the input load and the output signal. Thereby, no heat current flows directly from the input lead to the output electrode.
Another important parameter is the input amplification range that represents the interval of the input signal for which the output resides in the $OAR$. The length of this interval $IAR$ is defined as: [$$IAR=Sens-T_{IN}(T_{OUT_{MIN}}),
\label{eq:DeltaTin}$$ ]{} where $T_{IN}(T_{OUT_{MIN}})$ is the value of the input temperature corresponding to the minimum value of the $OAR$. The $IAR$ is a function both of the $T_S$ and of $Sens$, as illustrated in fig. \[Figure3\]-a. For small values of $T_S$ the $OAR$ is small and, hence, the $IAR$ is not extended too. By rising the supply temperature the $IAR$ enlarges till $T_S$ reaches about $250~$mK. A further increase of the supply temperature yields a softening of $\Delta_{S_1}(T)$, and a consequent compression of $OAR$, as already elucidated above. The reduction of the $OAR$ is mirrored in a narrowing of the $IAR$. The non-monotonic behavior of the $IAR$ with the $Sens$ comes from the competition between the two terms on the right side of Eq. (1) and can be ascribed to the thermoelectric element. One the one hand, the increase of $Sens$ naturally enlarges the $IAR$ by widening the total input temperature range. On the other hand, $I_T$ rapidly rises with $T_{IN}$, as illustrated in fig. \[Figure1\]-c. The resulting magnetic flux $\Phi$ is modulated only for values of the input temperature approaching $Sens$, because $M$ is small and for the thermocurrents typical of narrow temperature gradients the flux always tends to zero. The latter effect manifests itself in lowering $IAR$ for increasing $Sens$ (see fig. \[Figure3\]-a).
In our amplifier, the temperature is the potential used in the amplification. Hence we can define the efficiency $\eta$ as: [$$\eta=\frac{T_{OUT_{MAX}}}{T_S}\times 100,
\label{eq:Eta}$$ ]{} where $T_{OUT_{MAX}}=T_{OUT}(T_{IN}=Sens)$. The efficiency reaches $\sim95\%$ for very small supply temperatures and monotonically decreases with rising $T_S$, as plotted in fig. \[Figure3\]-b. The drop of $\eta$ can be explained with the closure of $\Delta_{S_1}$ and the growth of the losses through the phonons resulting from the temperature increase [@Giazotto2006]. In the region of best performances in terms of $OAR$, $G$ and $IAR$ (represented with the yellow rectangle in fig. \[Figure3\]-b) the efficiency ranges from $\sim90\%$ to $\sim60\%$. These large $\eta$ values are comparable to analogous commercial electronic amplifiers.
The $OAR$ provides a first and reliable estimate of the useful interval of the output signal. A more complete analysis employs the output dynamic range $DR$ defined as: [$$DR=20\times \log\left( \frac{T_{OUT_{MAX}}+T_{Noise}}{T_{Noise}}\right).
\label{eq:DR}$$ ]{} The $DR$ widens by increasing supply temperature up to $T_S=150~$mK, because $T_{OUT_{MAX}}$ rises while $T_{Noise}$ is almost unaffected (as shown in the inset of fig. \[Figure3\]-c). Further increase of $T_S$ enlarges the noise with a steeper rate, while $T_{OUT_{MAX}}$ tends to level to a constant value. As a consequence, $DR$ decreases for values of $T_S$ approaching the SQUIPT critical temperature. Despite that, the PTA reaches the maximum performances in terms of $DR$ in the optimal region in terms of the other figures of merit, as depicted by the turquoise rectangle in fig. \[Figure3\]-c.
Finally, we consider the differential gain, defined as: [$$DG=\frac{\mathrm d T_{OUT}}{\mathrm d T_{IN}}.
\label{eq:dG}$$ ]{} At a fixed sensitivity, $DG$ displays a bell-like shape, as shown in fig. \[Figure3\]-d. The height, width and position of the peak are sensitivity-dependent. In particular, for small and large values of $Sens$ the peak is high and narrow, while for intermediate sensitivities the peak is low and broad in $T_{IN}$. Since $DG$ is always greater than zero, the output signal is always a monotonically growing function of the input, as required for an amplifier.
Conclusions
===========
We have proposed the phase-tunable temperature amplifier, which is the caloritronic counterpart of the voltage amplifier in electronics. The pivotal architecture proposed in this work constitutes the first fully-thermal on-chip device in coherent caloritronics, because the magnetic field necessary to control the thermal nano-valve (SQUIPT) is self-generated by the use of a thermoelectric element ($N-FI-S$ junction). The operating principle and the performances have been studied in detail paying specific attention to the experimental feasibility of geometry and material composition. The predicted input-to-output temperature conversion provides a maximum gain $G\approx11$ at small input signals which is mainly limited by the superconducting critical temperature $T_{C-S_1}$ of the Al-based nano-valve. In addition, we defined several figures of merit in full analogy with voltage amplifiers obtaining remarkable results especially in terms of output dynamic range $DR$ and efficiency $\eta$. The authors acknowledge the European Research Council under the European Unions Seventh Framework Programme (FP7/2007-2013)/ERC Grant No. 615187 - COMANCHE and the MIUR under the FIRB2013 Grant No. RBFR1379UX - Coca for partial financial support. The work of E.S. is funded by a Marie Curie Individual Fellowship (MSCA-IFEF-ST No. 660532-SuperMag).
Appendices
==========
N-FI-S junction
---------------
The thermoelectric is a tunnel junction made of a normal metal $N$ at temperature $T_{IN}$, a ferromagnetic insulator $FI$ and a superconductor $S$ at $T_{BATH}$. The $FI$ layer operates a double action: it behaves as a spin filter with polarization $P=(G_{\uparrow}-G_{\downarrow})/(G_{\uparrow}+G_{\downarrow})$ where $G_{\uparrow}$ and $G_{\downarrow}$ are the spin up and spin down conductances [@Moodera2007], and it causes the spin-splitting of the DOS of the superconductor by the interaction of its localized magnetic moments with the conducting quasiparticles in $S$ through an exchange field $h_{ex}$. Since the exchange interaction in a superconductor decays over the coherence length $\xi_0$ [@Tokuyasu1988], we assume $S$ thinner than $\xi_0$ and a spatially homogeneous spin-splitted DOS [@Giaz2008]: [$$\tag{A1}
N_{\uparrow,\downarrow}(E)=\frac{1}{2} \left| \Re\left\lbrack \frac{E+i\Gamma \pm h_{ex}}{\sqrt{(E+i\Gamma \pm h_{ex})^2-\Delta^2}} \right\rbrack \right|,
\label{eq:DOSsplit}$$ ]{} where $E$ is the energy, $\Gamma$ is the Dynes parameter accounting for broadening [@Dynes1984], and $\Delta(T_{BATH}, h_{ex})$ is the temperature and exchange field-dependent superconducting energy gap. The pairing potential is calculated self-consistently from the BCS equation [@Tinkham; @Giaz2008]: [$$\tag{A2}
\begin{split}
\ln \left(\frac{\Delta_0}{\Delta} \right)=\int_0^{\hbar\omega_D}\frac{f_+(E)+f_-(E)}{\sqrt{E^2+\Delta^2}}dE
\end{split}
\label{eq:GapEq}$$ ]{} where $f_{\pm}(E)= \left \{ 1+\exp\frac{\sqrt{E^2+\Delta^2}\mp h_{ex}}{k_BT_{BATH}} \right \}^{-1}$ is the Fermi distribution of the electrons, $\omega_D$ is the Debye frequency of the superconductor, $\Delta_0$ is the zero-field and zero-temperature superconducting gap, and $k_B$ is the Boltzmann constant. The tunnel thermocurrent in the closed circuit configuration is only due to the temperature gradient and takes the form: [$$\tag{A3}
\begin{split}
&I_T=\frac{1}{eR_T}\int_{-\infty }^{\infty }\left\lbrack N_+(E)+PN_-(E)\right\rbrack \\
&\left\lbrack f_N(E,T_{IN})-f_S(E,T_{BATH}) \right\rbrack dE,
\end{split}
\label{eq:It}$$ ]{} where $e$ is the electron charge, $R_T$ is the tunnel resistance in the normal state, $N_{\pm}(E)=N_{\uparrow}(E)\pm N_{\downarrow}(E) $, $f_{N}(E,T_{IN})= \left \lbrack 1+\exp\left( E/k_BT_{IN} \right)\right \rbrack^{-1}$ and $f_{S}(E,T_{BATH})= \left \lbrack 1+\exp\left( E/k_BT_{BATH} \right)\right \rbrack^{-1}$ are the metal and superconductor Fermi functions, respectively.
Temperature-biased SQUIPT
-------------------------
We model the SQUIPT as a superconducting ring $S_1$ interrupted by a one-dimensional normal metal wire $N_1$ ($l\gg w,t$ where $l$, $w$ and $t$ are the wire length, width and thickness, respectively). The superconducting properties acquired by the wire through the proximity effect [@Holm1932] has been shown to be modulated by the magnetic flux $\Phi$ threading the ring [@Giazotto2010; @Meschke2011]. Similarly, it has been recently shown that by interrupting the $S_1$-loop of the SQUIPT with a superconducting wire $S_2$, the superconducting properties of the latter are tuned by the magnetic flux threading the loop [@Virtanen2016]. Here we consider a hybrid superconductor-normal metal SQUIPT. Finally, a normal metal $N_2$ probe is tunnel-coupled to the wire through a thin insulating layer, and acts as output electrode. The DOS of the wire $N_{wire}$ is the real part of the quasi-classical retarded Green’s function $g^R$ [@Rammer1986] obtained by solving the one-dimensional Usadel equation [@Usadel1970]. In the short junction limit (i.e. when $E_{Th}=\hbar D/l^2\gg \Delta_{0_{S_1}}$, where $E_{Th}$ is the Thouless energy, $\hbar$ is the reduced Planck constant and $D$ is the wire diffusion coefficient) the proximity effect is maximized, and the DOS is expressed by [@Strambini2014; @Giazotto2010]: [$$\tag{A4}
\begin{split}
N_{wire}(E,\Phi)=\left| \Re\left\lbrack \frac{E-iE_{Th}\gamma g_s}{\sqrt{(E-iE_{Th}\gamma g_s)^2+\left\lbrack E_{Th}\gamma f_s \cos \left( \frac{\pi \Phi}{\Phi_0} \right) \right\rbrack^2}} \right\rbrack \right|.
\end{split}
\label{eq:DOSwire}$$ ]{} Above, $\gamma=R_{N_1}/R_{S_1N_1}$ is the transmissivity of the $S_1N_1$ contact (with $R_{N_1}$ resistance of the normal wire and $R_{S_1N_1}$ resistance of the $S_1N_1$ interface), $g_S(E)=\frac{E+i\Gamma_{S_1}}{\sqrt{(E+i\Gamma_{S_1})^2-\Delta_{S_1}^2}}$ is the coefficient of the phase-independent part of the DOS (with $\Gamma_{S_1}$ Dynes broadening parameter [@Dynes1984] and $\Delta_{S_1}$ BCS temperature dependent energy gap [@Tinkham]), $f_S(E)=\frac{\Delta_{S_1}}{\sqrt{(E+i\Gamma_{S_1})^2-\Delta_{S_1}^2}}$ is the coefficient of the phase-dependent part of the DOS, and $\Phi_0=2.068\times10^{-15}~$Wb is the magnetic flux quantum. The heat current $J$ tunneling from the $S_1N_1$ ring to the $N_2$ probe has been theoretically [@Giazotto2006; @Strambini2014] and experimentally [@Giazotto2012; @Mart2014; @Fornieri2015] shown to depend on the temperatures of the ring $T_S$ and of the normal electrode $T_{OUT}$ through: [$$\tag{A5}
\begin{split}
&J(T_S,T_{OUT},\Phi)= \\
&\frac{2}{e^2R_{T_1}}\int_{0}^{\infty}N_{wire}(E) \left\lbrack f_0(E,T_S)-f_0(E, T_{OUT}) \right\rbrack E dE
\end{split}
\label{eq:Jsquipt}$$ ]{} where $f_0(E, T)=\left\lbrack1+\exp (E/k_BT)\right\rbrack^{-1}$ is the Fermi distribution of the quasiparticles in the ring for $T=T_S$ and in the probe for $T=T_{OUT}$. The steady-state temperature of the probe $T_{OUT}$ depends on the thermal current flowing from $S_1N_1$ to $N_2$ and on the exchange mechanism occurring in $N_2$. Below $\sim1~$K the relaxation is mainly due to electron-phonon coupling [@Giazotto2006] and can be quantified as $J_{e-ph, N_2}(T_{OUT}, T_{BATH})=\Sigma~ V \left(T_{OUT}^n-T_{BATH}^n\right)$, where $\Sigma$ is the electron-phonon coupling constant, $V$ is the volume of the probe and the exponent $n$ depends on the disorder of the system. For metals, in the clean limit $n=5$, while in the dirty limit $n=4,6$ [@Giazotto2006; @Strambini2014]. At the steady state by setting a constant temperature of the superconducting ring $T_S$ the output temperature of the nano-valve $T_{OUT}$ can be obtained by solving the following balance equation: [$$\tag{A6}
\begin{split}
-J(T_S, T_{OUT},\Phi)+J_{e-ph, N_2}(T_{OUT}, T_{BATH})=0.
\end{split}
\label{eq:BalEq}$$ ]{}
Materials and geometry
----------------------
The thermoelectric element is composed of $15~$nm of Cu as $N$, $1~$nm of EuS as $FI$ and $3~$nm of Al as $S$. Within this geometry the Al layer has typically: $T_C\approx3~$K, $\Delta_0\approx 456~\mu$eV and $\Gamma=1\times10^{-4}\Delta_0$. We consider an EuS layer characterized by: $P=0.95$, $h_{ex}=0.45~\Delta_0$ and $R_T=0.1~\Omega$. The superconducting coil originating the magnetic flux is made of $10~$nm thick aluminum and it is embedded in $10~$nm of Al$_2$O$_3$. The SQUIPT is made of a copper $N_1$ wire ($l=100~$nm, $w=30~$nm, $t=30~$nm) of diffusivity $D=1\times10^{-2}~$m$^2/$s, and of a $150~$ nm thick Al $S_1$ ring of radius $r_{SQUIPT}=5~\mu$m with $\Delta_{0-S_1}=200~\mu$eV, $T_{C-S_1}\approx1.32~$K and $\Gamma_{S_1}=1\times10^{-4}\Delta_{0-S_1}$. The transmissivity of the $S_1N_1$ contact is $\gamma=33$. The AlMn probe is tunnel-coupled to the proximized wire through a $1~$nm thick aluminum oxide layer ($R_{T_1}=100~$k$\Omega$). The parameters of the AlMn electrode are: $\Sigma=4\times10^{9}~$WK$^{-6}$m$^{-3}$ [@Fornieri2015], $V=1\times10^{-20}~$m$^3$ and $n=6$ [@Giazotto2006; @Fornieri2015].
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abstract: 'The idea that the quantum space-time of microphysics may be fractal everywhere was intensively investigated recently, and several authors have presented the geodesic equations of different fractal space - times. In the present work we obtain the geodesic and the geodesic deviation equations in fractal space-times by using the Bazanski method. We also extend this approach to obtain the equations of motion for spinning and spinning charged particles in the above-mentioned spaces, in a similar way to their counterparts in Riemannian geometry.'
author:
- 'M. E. Kahil'
- 'T. Harko'
title: 'Path and Path Deviation equations of Fractal Space-Times: A Brief Introduction'
---
Introduction
============
The proposal that the space-time in which the evolution of the microscopic objects takes place may be a fractal has attracted a lot of attention recently [@NoSc84; @No94; @No98; @No05; @Ag05; @AgGo06; @Go06; @No06]. A fractal structure is a manifestation of the universality of self-organization processes, a result of a sequence of spontaneous symmetry breaking. In the fractal space-time model the typical trajectories of quantum particles are continuous, but non-differentiable, and can be characterized by a fractal dimension that jumps from $D=1$ at large length scale to $D=2$ at small length scale. The fractal dimension $D=2$ is the fractal dimension of the Brownian motion, or, equivalently, of a Markov-Wiener process. As suggested by Nelson [@Ne66], quantum mechanics can also be interpreted as assuming that any particle is subjected to an underlying Brownian motion of unknown origin, which is described by two (forward and backward) Wiener processes: when combined together, they wield the complex nature of the wave function and they transform Newton’s equation of dynamics into the Schrodinger equation.
An alternative way for the description of the motion was introduced by Bazanski [@Ba89], an approach that has the advantage of providing both the equations of the geodesics as well as the geodesic deviation equation. It is the purpose of the present paper to generalize the Bazanski approach to the case of the fractal space-times.
Fractal space-times and the Schrodinger equation
================================================
Let us consider a fractal curve $f(x)$ between two points $A$ and $B$ that is continuous, but nowhere differentiable. Such a curve has an infinite length. This is a direct consequence of the Lebesgue theorem, which states that a finite length curve is almost everywhere (i.e., except a set of points with null dimension) differentiable [@NoSc84; @No06]. Another important property of a fractal curve is that between any two points of the curve we can get a curve with the same properties as the initial curve, that is, a continuous, nowhere differentiable and infinite length curve. Therefore fractal curves are almost self-similar everywhere.
In the differentiable case the usual definition of the derivative of a given function are given by $$\frac{df}{dt}=\lim_{\Delta t\rightarrow +0}\frac{f\left( t+\Delta
t\right) -f(t)}{\Delta t}=\lim_{\Delta t\rightarrow
-0}\frac{f\left( t\right) -f(t-\Delta t)}{\Delta t},$$ and one can pass from one definition to the other by the transformation $%
\Delta t\rightarrow -\Delta t$. The differentiable nature of the space-time implies the local differential (proper) time reflection invariance. In the non-differentiable case two functions $df_{+}/dt$ and $df_{-}/dt$ are defined as explicit functions of $t$ and $dt$, $$\frac{df_{+}}{dt}=\lim_{\Delta t\rightarrow +0}\frac{f\left(
t+\Delta t,\Delta t\right) -f(t,\Delta t)}{\Delta
t},\frac{df_{-}}{dt}=\lim_{\Delta
t\rightarrow -0}\frac{f\left( t,\Delta t\right) -f(t-\Delta t,\Delta t)}{%
\Delta t},$$ with the plus sign corresponding to the forward process, while the minus sign corresponds to the backward process. In other words, the non-differentiable nature of the space-times implies the breaking of the local differential (proper) time reflection invariance. If we apply the definition of the derivatives to the coordinate functions, we obtain $dX_{\pm }^{i}=dx_{\pm }^{i}+d\xi
_{\pm }^{i} $, where $dx_{\pm }^{i}$ are the usual classical variables, and $d\xi _{\pm }^{i}$ are the non-differentiable variables. By taking the average of these equations we obtain $\left\langle dX_{\pm }^{i}\right\rangle =\left\langle dx_{\pm
}^{i}\right\rangle $, since $\left\langle d\xi _{\pm
}^{i}\right\rangle =0$. If we denote by $d\vec{x}_{+}/dt=\vec{v}_{+}$ the forward speed and by $d\vec{x}_{-}/dt=\vec{v}_{-}$ the backward speed, then $%
\left( \vec{v}_{+}+\vec{v}_{-}\right) /2$ may be considered as the differentiable (classical speed), while $\left( \vec{v}_{+}-\vec{v}%
_{-}\right) /2$ is the non-differentiable speed. These two quantities can be combined in a single quantity if we introduce the complex speed $\vec{V}%
=\delta \vec{x}$, where $\delta
=\left(d_{+}+d_{-}\right)/2dt-i\left(d_{+}-d_{-}\right)/2dt$ [@No94; @No98; @No05; @Ag05; @AgGo06; @Go06; @No06].
By considering that the continuous but non-differentiable curve of motion is immersed in a three-dimensional space, any function $f\left( X^{i},t\right) $ can be expanded into a Taylor series as $df=f\left( X^{i}+dX^{i},t+dt\right) -f\left( X^{i},t\right)
=\left[ (\partial /\partial X^{i})dX^{i}+(\partial /\partial
t)dt\right] f\left( X^{i},t\right) $. By assuming that the mean values of the function $f$ and of its derivatives coincide with themselves we obtain $d_{\pm }f/dt=\partial f/\partial
t+\vec{v}_{\pm }\cdot \nabla f_{\pm }$, while the operator $\delta
$ is given by $\delta f/dt=\partial f/\partial t+\vec{V}\cdot
\nabla f$ [@No94; @No98; @No05; @Ag05; @AgGo06; @Go06; @Wa99].
We can now apply the principle of the scale covariance, which postulates that the passage from classical (differentiable) mechanics to the non-differentiable mechanics can be realized by replacing the standard time derivative $df/dt$ by the complex operator $\delta /dt$. Therefore in a covariant form the equation of geodesics of the fractal space-time can be written as $$\frac{\delta \vec{V}}{dt}=\frac{\partial \vec{V}}{\partial
t}+\vec{V}\cdot \nabla \vec{V}=0.$$
By considering that the fluid is irrotational, $\nabla \times
\vec{V}=0$, by introducing the complex speed potential $\phi $ so that $\vec{V}=\nabla \phi $, and by assuming that $\phi =-2iD\ln
\psi $, where $D$ is a constant, the equation of motion of the fluid takes the form of the Schrodinger equation, $$D^{2}\Delta \psi +iD\frac{\partial \psi }{\partial t}-U\psi =0,$$ where $U=D^{2}\Delta \ln \psi $. $D$ defines the differential-non-differential transition, that is, the transition from the explicit scale dependence to scale independence.
In order to study gravitational phenomena in fractal space-times it is necessary to extend the concept of metric by taking into account the fluctuating character of the paths. This corresponds to the passage from Special Scale Relativity to General Scale Relativity. The line element between two neighboring points in a fractal geometry can be described as $$d\tilde{s}^{2}=\tilde{g}_{\mu \nu }dX^{\mu }dX^{\nu
}=\tilde{g}_{\mu \nu }\left( dx^{\mu }+d\xi^{\mu}\right)\left( dx^{\nu }+d\xi^{\nu }\right) ,$$ leading to a generalized metric in a curved fractal space-time of the form $$\tilde{g}_{\mu \nu }\left( x,t\right) =g_{\mu \nu }\left(
x,t\right) +\gamma _{\mu \nu }\sqrt{\left( \frac{\lambda
_{c}}{dx^{\mu }}\right) \left( \frac{\lambda _{c}}{
dx^{\nu}}\right) },$$ where $\gamma_{\mu \nu}$ is described as the first approximation in terms of stochastic variables [@No06]. Based on this fluctuating metric one can obtain the affine connection of the fractal space-time as $\tilde{\Gamma}%
_{jk}^{i}=\Gamma _{jk}^{i}+\chi _{jk}^{i}$, where $\Gamma
_{jk}^{i}$ is the usual Christoffel connection, and $\chi
_{jk}^{i}$ is the fluctuating part. The mean values of the affine connection satisfy the conditions $%
\left\langle \tilde{\Gamma}_{jk}^{i}\right\rangle =\Gamma _{jk}^{i}$ and $%
\left\langle \chi _{jk}^{i}\right\rangle =0$. Similarly one can define the curvature tensor $\tilde{R}_{jkl}^{i}=R_{jkl}^{i}+\Xi _{jkl}^{i}$, so that $%
\left\langle \tilde{R}_{jkl}^{i}\right\rangle =R_{jkl}^{i}$ and $%
\left\langle \Xi _{jkl}^{i}\right\rangle =0$.
The Bazanski approach in fractal space-times
============================================
Geodesic and geodesic deviation equations can be obtained simultaneously by applying the action principle on the Bazanski Lagrangian [@Ba89; @Wa95; @Wa99; @Ka06]: $$L= g_{\alpha \beta} U^{\alpha} \frac{D \Psi^{\beta}}{Ds},$$ where $D/Ds$ is the covariant derivative. By taking the variation with respect to the deviation vector ${\Psi^{\rho}}$ and with the unit tangent vector $U^{\rho}$ one obtains the geodesic equation and the geodesic deviation equation respectively, $$\frac{dU^{\alpha}}{ds} +{{\alpha} \brace {\mu \nu}}U^{\mu}U^{\nu}=0 ,\frac{%
D^{2}\Psi^{\alpha}}{Ds^{2}} = R^{\alpha}_{. \beta \gamma \delta}
U^{\beta}U^{\gamma} \Psi^{\delta},$$ where ${{\alpha} \brace {\mu \nu}}$ is the Christoffel symbol of the second kind $R^{\alpha}_{\beta \gamma \delta}$ is the Riemann- Christoffel curvature tensor. In the case of the fractal space-times we propose the following form of the Bazanski Lagrangian $L$, $$L=g_{\mu \nu }\tilde{V}^{\mu }\frac{\hat{D}_{\pm }\tilde{\Psi}^{\nu }}{Ds}%
+f_{\mu }\tilde{\Psi}^{\nu },$$ where $f_{\mu}= (e/m) \tilde{F}_{\mu \nu} V^{\nu} + (1/2m)\tilde{%
R}_{\mu \nu \gamma \delta }S^{\gamma \delta}V^{\nu}$, and the covariant scale derivatives in the fractal space are defined as $$\frac{\tilde{D}\tilde{V}}{\tilde{D}s}=\frac{D\tilde{V}^{\alpha }}{Ds}+\tilde{%
V}^{\mu }\cdot D_{\mu }\tilde{V}^{\alpha }-i\mu \left( D^{\mu
}D_{\mu }+\xi R\right) \tilde{V}^{\alpha },$$ and $$\frac{\tilde{D}_{\pm }\left( V^{a}+iU^{a}\right)
}{Ds}=\frac{D\left( V^{a}+iU^{a}\right) }{Ds}+\left( V^{\mu
}+iU^{\mu }\right) \cdot D_{\mu \pm }\left( V^{a}+iU^{a}\right)
-i\mu \left( \partial _{\pm }^{\mu }\partial _{\mu \pm }\right)
\left( V^{a}+iU^{a}\right),$$ respectively.
By using the Bazanski approach we can immediately obtain the Lorentz force equation, the Papapetrou equation (corresponding to the motion of spinning particles) and the Dixon equation (describing the motion of spinning particles in electromagnetic fields as $$\frac{\hat{D}\tilde{V}^{\mu }}{\hat{D}s}=\frac{e}{m}\tilde{F}_{\nu }^{\mu }%
\tilde{V}^{\nu },\frac{\hat{D}\tilde{V}^{\mu }}{\hat{D}s}=\frac{1}{2m}\tilde{%
R}_{\nu \sigma \rho }^{\mu }\tilde{S}^{\sigma \rho }\tilde{V}^{\nu },\frac{%
\hat{D}\tilde{V}^{\mu }}{\hat{D}s}=\frac{1}{2m}\tilde{R}_{\nu
\sigma \rho }^{\mu }\tilde{S}^{\sigma \rho }\tilde{V}^{\nu
}+\frac{e}{m}\tilde{F}_{\nu }^{\mu }\tilde{V}^{\nu }.$$
The quantum covariant derivative in fractal space-times can be generalized as $$\frac{\tilde{D}}{\tilde{D}s}=\left[ \tilde{V}^{0}D_{0}-\frac{i\lambda _{c}}{2%
}\left( D^{0}D_{0}+\xi R\right) \right] =0.$$ The geodesic equation of motion is $\tilde{D}\tilde{V}^{0}/\tilde{D}s=0$. By putting $\tilde{V}_{\mu
}=i\lambda D_{\mu }\ln \Psi $, we obtain the Klein-Gordon equation for a free particle in a curved space as $$\lambda ^{2}D^{\mu }\ln \Psi D_{\mu }D_{\rho }\ln \Psi
+\frac{\lambda _{c}^{2}}{2}\left( D^{\mu }D_{\mu }D_{\rho }\ln
\Psi +\xi RD_{\rho }\ln \Psi \right) =0.$$
The geodesic deviation equation is given by $$\begin{aligned}
\frac{\hat{D}^{2}\tilde{\Psi}^{\alpha }}{\hat{D}s^{2}}&=&\frac{\hat{D}}{\hat{%
D}s}\left[ \frac{\hat{D}\tilde{\Psi}^{\alpha
}}{\hat{D}s}+\tilde{V}^{\mu }\cdot D_{\mu }\tilde{\Psi}^{\alpha
}-i\mu \left( D^{\mu }D_{\mu }+\xi
R\right) \tilde{\Psi}^{\alpha }\right] = \nonumber \\
&&\tilde{R}_{\beta \gamma \delta }^{\alpha }\tilde{V}^{\beta }\tilde{V}%
^{\gamma }\tilde{\Psi}^{\delta }=\left( R_{\beta \gamma \delta
}^{\alpha
}+\Xi _{\beta \gamma \delta }^{\alpha }\right) \tilde{V}^{\beta }\tilde{V}%
^{\gamma }\tilde{\Psi}^{\delta }.\end{aligned}$$
Let us introduce now the covariant derivation operator with respect to the coordinates as $D\Psi _{\alpha }/Ds=D_{\alpha }\ln
\Phi \left( x,\Psi \right) $ and the covariant derivation operator with respect to the deviation vector $\tilde{V}^{\alpha }=D_{\Psi
^{\alpha }}\ln \Phi \left( x,\Psi \right) $. Then the geodesic deviation equation in afractal space-time is given by $$\begin{aligned}
\frac{\hat{D}^{2}D_{\alpha }\ln \Phi }{\hat{D}s^{2}} &=&\frac{\hat{D}}{\hat{D%
}s}\left[ \frac{\hat{D}\left( D_{\alpha }\ln \Phi \right) }{\hat{D}s}%
+D_{\Psi ^{\alpha }}\ln \Phi \cdot D_{\mu }D_{\alpha }\ln \Phi
-i\mu \left(
D^{\mu }D_{\mu }+\xi R\right) D_{\alpha }\ln \Phi \right] = \nonumber \\
&&\left( R_{\beta \gamma \delta }^{\alpha }+\Xi _{\beta \gamma
\delta }^{\alpha }\right) D^{\Psi ^{\beta }}\ln \Phi D^{\Psi
^{\gamma }}\ln \Phi D^{\Psi ^{\delta }}\ln \Phi .\end{aligned}$$
Discussions and final remarks
=============================
The theory of scale relativity extends Einstein’s principle of relativity to scale transformations of resolutions, and it gives up the concept of differentiability of space-time. Its main result is the reformulation of quantum mechanics from its first principle. In a fractal space time a small increment of displacement of the non differentiable four-coordinates along one of the geodesics can be generally decomposed into its mean and a fluctuating term. In the present paper we have obtained the path equations in general scale relativity, as described by a fractal Riemannian geometry, and we have combined the geodesic equations and the Schrodinger/ Klein-Gordon equations in a single equation, which can be reduced to each of them separately if and only if one uses the averaging procedure and solve the problem of the integrability of the affine connection due to the curvature of the space time. Also, we have obtained a quantum analog of the geodesic deviation equation as defined in a Fractal Space-Time.
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abstract: 'The strong subadditivity inequality of von Neumann entropy relates the entropy of subsystems of a tripartite state $\rho_{ABC}$ to that of the composite system. Here, we define $\boldsymbol{T}^{(a)}(\rho_{ABC})$ as the extent to which $\rho_{ABC}$ fails to satisfy the strong subadditivity inequality $S(\rho_{B})+S(\rho_{C}) \le S(\rho_{AB})+S(\rho_{AC})$ with equality and investigate its properties. In particular, by introducing auxiliary subsystem $E$, we consider any purification $|\psi_{ABCE}\rangle$ of $\rho_{ABC}$ and formulate $\boldsymbol{T}^{(a)}(\rho_{ABC})$ as the extent to which the bipartite quantum correlations of $\rho_{AB}$ and $\rho_{AC}$, measured by entanglement of formation and quantum discord, change under the transformation $B\rightarrow BE$ and $C\rightarrow CE$. Invariance of quantum correlations of $\rho_{AB}$ and $\rho_{AC}$ under such transformation is shown to be a necessary and sufficient condition for vanishing $\boldsymbol{T}^{(a)}(\rho_{ABC})$. Our approach allows one to characterize, intuitively, the structure of states for which the strong subadditivity is saturated. Moreover, along with providing a conservation law for quantum correlations of states for which the strong subadditivity inequality is satisfied with equality, we find that such states coincides with those that the Koashi-Winter monogamy relation is saturated.'
author:
- Razieh Taghiabadi
- Seyed Javad Akhtarshenas
- Mohsen Sarbishaei
title: 'Reexamination of strong subadditivity: A quantum-correlation approach '
---
*Introduction.—*Correlations between different subsystems of a classical or quantum composite system result to inequalities relating the entropy of various subsystems to that of the composite system. For a given state $\rho_{AB}$ of the quantum system $\mathcal{H}_{AB}$, consisting of two subsystems $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$, the *subadditivity* (SA) states that [@WehrlRMP1978] $$\label{SA}
S(\rho_{AB})\le S(\rho_{A})+S(\rho_{B}),$$ where $\rho_{A}={{\mathrm {Tr}}}_{B}{(\rho_{AB})}$ and $\rho_{B}={{\mathrm {Tr}}}_{A}{(\rho_{AB})}$ are states of the subsystems and $S(\rho)=-{{\mathrm {Tr}}}{\rho\log{\rho}}$ is the von Neumann entropy of $\rho$. The inequality implies that any correlation between subsystems decreases the amount of information needed to specify one of the subsystems once we know the other one [@ZyczkowskiBook2006]. The equality holds if and only if the subsystems are uncorrelated, i.e., $\rho_{AB}=\rho_{A}\otimes \rho_{B}$. It turns out therefore that the extent to which the state $\rho_{AB}$ fails to satisfy the SA with equality is a measure of total correlations (classical+quantum) and is defined as the *mutual information* [@NielsenBook2000] $$\label{MI}
I(\rho_{AB})=S(\rho_{A})+S(\rho_{B})-S(\rho_{AB}).$$ A stronger inequality holds when the composite system is composed of three subsystems. Suppose $\rho_{ABC}$ is a quantum state of the composite system $\mathcal{H}_{ABC}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C}$. The *strong subadditivity* (SSA), which was conjectured for quantum systems by Lanford and Robinson [@LanfordJMP1968] and proved by Lieb and Ruskai [@LiebRuskaiJMP1975], states that $$\label{SSA1}
S(\rho_{ABC})+S(\rho_{C}) \le S(\rho_{AC})+S(\rho_{BC}).$$ Clearly, by choosing $\mathcal{H}_C=\mathbb{C}$, Eq. recovers the subadditivity relation . The SSA inequality is equivalent to $$\label{SSA2}
S(\rho_{B})+S(\rho_{C}) \le S(\rho_{AB})+S(\rho_{AC}),$$ which can be seen by adding an auxiliary subsystem $\mathcal{H}_{E}$ such that $\rho_{ABCE}$ is a pure state. The SSA inequality plays a crucial role in quantum information theory. It appears almost everywhere in quantum information theory from Holevo bound on the accessible information in the quantum ensemble [@HolevoPIT1973; @SchumacherPRL1996; @RogaPRL2010], properties of coherent information [@SchumacherPRA1996; @BarnumPRA1998; @LloydPRA1997], definition of squashed entanglement [@ChristandlJMP2004; @CarlenLMP2012], monogamy of quantum correlations [@KoashiPRA2004; @FanchiniPRA2011; @YangPRA2013], to some features of quantum discord such as condition for nullity [@DattaARXIV1003.5256], and condition for saturating the upper bound [@XiPRA2012]. Note that the SSA inequality can be written also as [@NielsenBook2000] $$\label{SSA2p}
0 \le S_{\rho_{AB}}(A \mid B)+S_{\rho_{AC}}(A \mid C),$$ where $$\label{CE}
S_{\rho_{AB}}(A \mid B)=S(\rho_{AB})-S(\rho_{B}),$$ is the *quantum conditional entropy* which, unlike its classical counterpart, can take negative values. This latter form of SSA emphasises how highly SSA is nontrivial in the quantum case in a sense that although each term on the right-hand side of can be negative, both of them cannot be negative simultaneously.
Contrary to the SA inequality, the characterization of states for which the SSA is saturated is not trivial. Obviously, when the global state is factorized, i.e., $\rho_{ABC}=\rho_{A}\otimes \rho_{B}\otimes \rho_{C}$, the equality holds but the converse is not true in general. Petz [@PetzCMP1986] and Ruskai [@RuskaiJMP2002] have provided algebraic criteria to check that if any given state satisfies the inequality with equality, however, their description do not characterize the structure of such states. An important progress in providing the structure of states for which the SSA inequality is satisfied with equality is given in Ref. [@HaydenCMP2004]. Hayden [*et al.* ]{}have shown that the state $\rho_{ABC}$ satisfies the SSA inequality with equality if and only if there is a decomposition of the subsystem $\mathcal{H}_C$ into a direct orthogonal sum of tensor products as $\mathcal{H}_C=\bigoplus_{j} \mathcal{H}_{C_j^L}\otimes \mathcal{H}_{C_j^R}$ such that $\rho_{ABC}=\bigoplus_{j} q_j\rho_{AC_j^L}\otimes \rho_{C_j^RB}$, with states $\rho_{AC_j^L}$ on $\mathcal{H}_A\otimes \mathcal{H}_{C_j^L}$ and $\rho_{C_j^RB}$ on $\mathcal{H}_{C_j^R}\otimes \mathcal{H}_B$, and a probability distribution $\{q_j\}$. On the basis of the results of [@HaydenCMP2004], the structure of states for which the SSA inequality is saturated is given in [@ZhangCTP2015].
As it is mentioned above any deviation of SA from equality refers to some correlations existing in the bipartite state. It is therefore natural to ask the question: Is it possible to express deviation from equality of the SSA inequality to the existence of some kind of correlations. In this paper, we address this issue and define the quantity $\boldsymbol{T}^{(a)}(\rho_{ABC})$ as the extent to which the SSA inequality deviates from equality in terms of two different aspects of quantum correlations, i.e., entanglement of formation (EOF) [@BennettPRA1996] and quantum discord (QD) [@ZurekPRL2001; @HendersonJPA2001]. Our main results are Theorems \[TheoremDeltaCorrelations\] and \[TheoremSSAcondition\] and also Corollary \[Corollary-Results\]. To be specific, Theorem \[TheoremDeltaCorrelations\] shows that $\boldsymbol{T}^{(a)}(\rho_{ABC})$ can be expressed by means of the extent to which the bipartite quantum correlations of $\rho_{AB}$ and $\rho_{AC}$, measured by EOF and QD, change under the transformation $B\rightarrow BE$ and $C\rightarrow CE$, where $E$ is an auxiliary subsystem that purifies $\rho_{ABC}$. In Theorem \[TheoremSSAcondition\], we use this measure and characterize the structure of states for which the SSA inequality is satisfied with equality. Our approach provides an information-theoretic aspect for such states, that is, $\boldsymbol{T}^{(a)}(\rho_{ABC})=0$ if and only if quantum correlations of $\rho_{AB}$ and $\rho_{AC}$ do not change under the above transformation. This, however, can be regarded as a kind of conservation law for quantum correlations, i.e., if $\boldsymbol{T}^{(a)}(\rho_{ABC})=0$ then the sum of entanglement of formation of $\rho_{AB}$ and $\rho_{AC}$ is equal to the sum of their quantum discord. Moreover, we find that the class of states saturating the Koashi-Winter inequality coincides with those states that the SSA inequality is satisfied with equality.
*Monogamy of quantum correlations.—*Quantum entanglement and quantum discord are two aspects of quantum correlations defined respectively within entanglement-separability paradigm and an information-theoretic perspective. Quantum entanglement is defined as those correlations that cannot be generated by local operations and classical communication [@WernerPRA1989]. Several measures have been proposed to quantify quantum entanglement, the most important one is the *entanglement of formation* (EOF) [@WoottersPRL1998], defined for a bipartite state $\rho_{AB}$ as $$E(\rho_{AB})=\min\sum_{i}p_i E(\psi_i),$$ where minimum is taken over all pure state decompositions $\rho_{AB}=\sum_{i}p_i{|\psi_i\rangle}{\langle \psi_i|}$ and $E(\psi_i)=S({{\mathrm {Tr}}}_{B}[{|\psi_i\rangle}{\langle \psi_i|}])$. However, quantum entanglement cannot capture all nonclassical correlations of a composite state in a sense that a composite mixed state may exhibit some quantum correlations even if it is disentangled. From various measures proposed for this different aspect of quantum correlation, *quantum discord* (QD) has received a great deal of attention. For a bipartite state $\rho_{AB}$, quantum discord is defined as the difference between two classically equivalent but quantum mechanically different definitions of quantum mutual information $$\label{QD}
D^{(B)}(\rho_{AB})=I(\rho_{AB})-J^{(B)}(\rho_{AB}),$$ where $J^{(B)}(\rho_{AB})=\max_{\{\Pi_i^B\}}J^{\{\Pi_i^{B}\}}(\rho_{AB})$ is the classical correlation of the state $\rho_{AB}$. Moreover [@ZurekPRL2001] $$\label{CC}
J^{\{\Pi_i^{B}\}}(\rho_{AB})=S(\rho_{A})-\sum_ip_iS(\rho_{AB}|\Pi_i^{B}),$$ where $\{\Pi_i^{B}\}$ is the set of projection operators on the subsystem $B$, and $S(\rho_{AB}|\Pi_i^{B})$ is the conditional entropy of $A$ when measurement is performed on $B$ and the $i$-th outcome is obtained with probability $p_i=\mathrm{Tr}[\Pi_i^{B}\rho_{AB}\Pi_i^{B}]$. We notice here that QD is not, in general, symmetric under the swap of the two parties, $A\leftrightarrow B$.
For a general tripartite state $\rho_{ABC}$ there exists a monogamic relation between EOF and QD of its corresponding bipartite mixed states, the so-called Koashi-Winter (K-W) relation [@KoashiPRA2004] $$\label{KWinequality}
E(\rho_{AB})\leq D^{(C)}(\rho_{AC})+S_{\rho_{AC}}({A\mid C}).$$ When $\rho_{ABC}$ is pure, the inequality is saturated and $\rho_{ABC}$ is called a purification of the mixed states $\rho_{AB}$ and $\rho_{AC}$. In this case the state $\rho_{AC}$ is called $B$-complement to $\rho_{AB}$ and, similarly, $\rho_{AB}$ is called $C$-complement to $\rho_{AC}$. For this particular case of pure global state $\rho_{ABC}$, the following *quantum conservation law*, as it is called by Fanchini [*et al.* ]{}[@FanchiniPRA2011], is obtained $$\label{ConservationLaw}
E(\rho_{AB})+E(\rho_{AC})=D^{(B)}(\rho_{AB})+D^{(C)}(\rho_{AC}).$$ Moreover, Cen [*et al.* ]{}[@CenPRA2011] have used the K-W relation and proposed a schema to quantify the QD and EOF and their ordering relation. In particular, they have characterized the QD of an arbitrary two-qubit state reduced from pure three-qubit states and a class of rank 2 mixed states of $4\times 2$ systems. In [@BaiPRA2013], the authors used the K-W relation and explored the monogamy property of the square of QD in mutipartite systems. They have shown that the square QD is monogamous for three-qubit pure states.
*Deviation from equality of strong subadditivity.—*Let $\boldsymbol{T}^{(a)}(\rho_{ABC})$ denotes the degree to which $\rho_{ABC}$ fails to saturate SSA, i.e., the difference between the right-hand side and the left-hand side of Eq. $$\begin{aligned}
\label{DeltaARight-Left}
\boldsymbol{T}^{(a)}(\rho_{ABC})=S(\rho_{AB})+S(\rho_{AC})-S(\rho_{B})-S(\rho_{C}).\end{aligned}$$ Then the following Lemma states that $\boldsymbol{T}^{(a)}(\rho_{ABC})$ is a concave function of its input state $\rho_{ABC}$.
\[LemmaDeltaConcavity\] Suppose $\{p_k,\rho_{ABC}^k\}_{k}$ is an ensemble of states generating $\rho_{ABC}$, that is, $\rho_{ABC}=\sum_{k}^Kp_k\rho_{ABC}^k$. Then $$\label{DeltaConcavity}
\boldsymbol{T}^{(a)}(\rho_{ABC})\ge \sum_{k}^Kp_k \boldsymbol{T}^{(a)}(\rho_{ABC}^k).$$ Moreover, the equality holds if the marginal states $\{\rho_{B}^k\}_k$ and $\{\rho_{C}^k\}_k$ are mutually orthogonal, i.e., $\rho_{B}^k\perp \rho_{B}^{k^\prime}$ and $\rho_{C}^k\perp \rho_{C}^{k^\prime}$ for $k\ne k^\prime$.
For state $\rho=\sum_{k}^Kp_k\rho^k$, acting on $\mathcal{H}$, define the *Holevo quantity* $\chi_{\rho}=S(\rho)-\sum_kp_kS(\rho^k)$. The inequality then easily obtained by noting that for any bipartite state $\rho_{AB}$ we have $\chi_{\rho_{AB}}\ge \chi_{\rho_B}$ [@SchumacherPRL1996; @NielsenBook2000] (For a different proof of see Ref. [@NielsenBook2000]). For the second claim of the Lemma note that for any state $\rho=\sum_{k}p_k\rho^k$ we have $S\left(\sum_{k}p_k\rho^k\right)\le H(p_k)+\sum_{k}p_kS(\rho^k)$, with equality if and only if the states $\{\rho^k\}_k$ have support in orthogonal subspaces $\mathcal{H}_{k}$ of the Hilbert space $\mathcal{H}=\bigoplus_{k=1}^{K}\mathcal{H}_{k}$ [@NielsenBook2000], i.e., they are mutually orthogonal in a sense that $\rho^k\perp \rho^{k^\prime}$ for $k\ne k^\prime$. Using this we find that the orthogonality condition for the marginal states $\{\rho_{AB}^k\}_k$, $\{\rho_{AC}^k\}_k$, $\{\rho_{B}^k\}_k$, and $\{\rho_{C}^k\}_k$ implies the equality of the inequality . The proof becomes complete if recalling that for any two bipartite states $\rho_{AB}$ and $\rho_{AB}^\prime$, the orthogonality of parts implies the orthogonality of whole, i.e., $\rho_{B}\perp\rho_{B}^\prime$ implies $\rho_{AB}\perp\rho_{AB}^\prime$.
The following corollary is immediately obtained from Lemma \[LemmaDeltaConcavity\].
\[CorollaryZeroDelta\] (i) Let $\rho_{ABC}$ be a state with vanishing $\boldsymbol{T}^{(a)}(\rho_{ABC})$, i.e., $\boldsymbol{T}^{(a)}(\rho_{ABC})=0$. Then for any ensemble $\{p_k,\rho_{ABC}^k\}_{k}$ giving rise to $\rho_{ABC}$ we have $\boldsymbol{T}^{(a)}(\rho_{ABC}^k)=0$ for $k=1,\cdots,K$. (ii) Moreover, any ensemble $\{p_k,\rho_{ABC}^k\}_{k}$ with $\boldsymbol{T}^{(a)}(\rho_{ABC}^k)=0$ generates a state $\rho_{ABC}=\sum_{k}^{K}p_k\rho_{ABC}^k$ with $\boldsymbol{T}^{(a)}(\rho_{ABC})=0$ provided that the marginal states $\{\rho_{B}^k\}_k$ and $\{\rho_{C}^k\}_k$ are mutually orthogonal.
*Strong subadditivity versus quantum correlations.—*Consider two bipartite states $\rho_{AB}$ and $\rho_{AC}$ with the same tripartite extension $\rho_{ABC}$, i.e., $\rho_{AB}={{\mathrm {Tr}}}_{C}(\rho_{ABC})$ and $\rho_{AC}={{\mathrm {Tr}}}_{B}(\rho_{ABC})$. Let ${|\psi_{ABCE}\rangle}$ be any purification of $\rho_{ABC}$, where $E$ is an auxiliary subsystem. Define $\widetilde{B}=BE$ and $\widetilde{C}=CE$, i.e., $\mathcal{H}_{\widetilde{B}}=\mathcal{H}_B\otimes \mathcal{H}_E$ and $\mathcal{H}_{\widetilde{C}}=\mathcal{H}_C\otimes \mathcal{H}_E$. Armed with these definitions, let us apply the K-W relation to the pure states $\rho_{AB\widetilde{C}}$ and $\rho_{A\widetilde{B}C}$ and the mixed state $\rho_{ABC}$ to get $$\begin{aligned}
\label{KWrhoC}
E(\rho_{AB})&=&D^{(\widetilde{C})}(\rho_{A\widetilde{C}})+S_{\rho_{A\widetilde{C}}}(A \mid \widetilde{C}),
\\ \label{KWrhoB}
E(\rho_{A\widetilde{B}})&=&D^{(C)}(\rho_{AC})+S_{\rho_{AC}}(A \mid C),
\\ \label{KWrho}
E(\rho_{AB})&\le &D^{(C)}(\rho_{AC})+S_{\rho_{AC}}(A \mid C).\end{aligned}$$ The same relations hold if we exchange $B \leftrightarrow C$; denoting them with (\[KWrhoC\]$'$), (\[KWrhoB\]$'$), and (\[KWrho\]$'$), respectively. Theorem \[TheoremDeltaCorrelations\] provides a relation for $\boldsymbol{T}^{(a)}(\rho_{ABC})$ in terms of quantum correlations.
\[TheoremDeltaCorrelations\] $\boldsymbol{T}^{(a)}(\rho_{ABC})$ can be expressed in terms of the quantum correlations of the aforementioned bipartite states $$\begin{aligned}
\label{DeltaACorrelations}\nonumber
&&\hspace{-8mm}\boldsymbol{T}^{(a)}(\rho_{ABC}) \\ \nonumber
&=&\left[E(\rho_{A\widetilde{B}})-E(\rho_{AB})\right]+\left[D^{(\widetilde{C})}(\rho_{A\widetilde{C}})- D^{(C)}(\rho_{AC})\right] \\ \nonumber
&=&\left[E(\rho_{A\widetilde{C}})-E(\rho_{AC})\right]+\left[D^{(\widetilde{B})}(\rho_{A\widetilde{B}})- D^{(B)}(\rho_{AB})\right] \\ \nonumber
&=&\left[E(\rho_{A\widetilde{B}})+E(\rho_{A\widetilde{C}})\right]-\left[D(\rho_{AB})+D(\rho_{AC})\right] \\ \label{DeltaACorrelations}
&=& \left[D(\rho_{A\widetilde{B}})+D(\rho_{A\widetilde{C}})\right]-\left[E(\rho_{AB})+E(\rho_{AC})\right].\end{aligned}$$ Moreover $$\begin{aligned}
\label{deltaE}
\delta^{(a)}_x(E)=E(\rho_{A\widetilde{X}})-E(\rho_{AX}) &\ge & 0,\end{aligned}$$ for $X=B,C$.
By subtracting Eq. from , and using that for any pure state $\rho_{XYZ}$ one can write $S_{\rho_{XZ}}(X \mid Z)=-S_{\rho_{XY}}(X \mid Y)$, we find the first equality of Eq. . The third equality is obtained by adding Eq. to (\[KWrhoB\]$'$). The second and fourth equalities are obtained just by swapping $B\leftrightarrow C$. Moreover, the nonnegativity of Eq. follows easily by subtracting Eq. from .
Theorem claims that for a given $\rho_{ABC}$ with *a prior* bipartite quantum correlations of $A$ with other subsystems $B$ and $C$, $\boldsymbol{T}^{(a)}(\rho_{ABC})$ quantifies the increased quantum correlations caused by the transformations $B\rightarrow\widetilde{B}=BE$ and $C\rightarrow\widetilde{C}=CE$. This feature of $\boldsymbol{T}^{(a)}(\rho_{ABC})$ allows one to find, along with other results, the structure of states for which $\boldsymbol{T}^{(a)}(\rho_{ABC})$ achieves its upper or lower bounds. For the former, note that $$\begin{aligned}
\label{DeltaUpperBound}
\boldsymbol{T}^{(a)}(\rho_{ABC}) & \le &
\max\left[E(\rho_{A\widetilde{B}})+E(\rho_{A\widetilde{C}})\right] \\ \nonumber
&-&\min\left[D(\rho_{AB})+D(\rho_{AC})\right] \\ \nonumber
&\le & 2S(\rho_{A})\le 2\log{d_A},\end{aligned}$$ which obtained from the third equality of Eq. and the fact that for any bipartite state $\rho_{XY}$, we have $E(\rho_{XY})\le \min\{S(\rho_{X}),S(\rho_{Y})\}$ and $D^{(Y)}(\rho_{XY})\ge 0$. A simple investigation shows that the last inequality is saturated if and only if $$\label{DeltaUpperBoundRho}
\rho_{ABC}=\mathbb{I}_{d_A}/d_{A}\otimes \rho_{BC},$$ where $\mathbb{I}_{d_A}$ denotes the unity matrix of $\mathcal{H}_A$ and $d_{A}=\dim{\mathcal{H}_A}$. This follows from the fact that the minimum in Eq. is obtained when the prior quantum discord of $A$ with $B$ and $C$ are zero, and the maximum is achieved when the transformed states $\rho_{A\widetilde{B}}$ and $\rho_{A\widetilde{C}}$ are maximally entangled states. Both extermums will be attained simultaneously if and only if $\rho_{A}$ is maximally mixed state and factorized from the rest of the system, so that $A$ does not possess any prior correlation with $B$ and $C$ at all. Although Eq. can be obtained by means of Eqs. and , the above investigation shows the role of quantum correlations more clearly.
*Conditions for $\boldsymbol{T}^{(a)}(\rho_{ABC})=0$.—*Theorem \[TheoremDeltaCorrelations\] states that a state satisfies SSA with equality if and only if the quantum correlations of $A$ with $B$ and $C$ do not change under the transformation $B\rightarrow\widetilde{B}$ and $C\rightarrow\widetilde{C}$. This happens, in particular, whenever $\rho_{ABC}$ is pure, but this is not the only case that $\boldsymbol{T}^{(a)}(\rho_{ABC})$ vanishes. Indeed, it is not difficult to see that any state of the form $\rho_{ABC}={|\psi_{AY}\rangle}{\langle \psi_{AY}|}\otimes \rho_{Z}$ possesses quantum correlations which are invariant under transformations $B\rightarrow \widetilde{B}$ and $C\rightarrow \widetilde{C}$.
\[LemmaSSAcondition\] Let $Y$ be a partition of $BC$ and $Z$ its complement, i.e., $\mathcal{H}_{Y}\otimes\mathcal{H}_{Z}=\mathcal{H}_{B}\otimes \mathcal{H}_{C}$. More precisely, let $\mathcal{H}_{B}$ and $\mathcal{H}_{C}$ can be tensor producted as $\mathcal{H}_{B}=\mathcal{H}_{B^L}\otimes\mathcal{H}_{B^R}$ and $\mathcal{H}_{C}=\mathcal{H}_{C^L}\otimes\mathcal{H}_{C^R}$, respectively, and define $\mathcal{H}_{Y}=\mathcal{H}_{B^L}\otimes\mathcal{H}_{C^L}$ and $\mathcal{H}_{Z}=\mathcal{H}_{B^R}\otimes\mathcal{H}_{C^R}$. Then for any state of the form $$\label{PsiRhoProduct1}
\rho_{ABC}={|\psi_{AY}\rangle}{\langle \psi_{AY}|}\otimes \rho_{Z},$$ we have $\boldsymbol{T}^{(a)}(\rho_{ABC})=0$.
Recall that purification produces entanglement between the system under consideration and the auxiliary system if and only if the original system is impure [@NielsenBook2000]. Any purification of $\rho_{ABC}$ leads to a purification of $\rho_{Z}$ as $\rho_{\widetilde{Z}}$ where $\widetilde{Z}=ZE$ for the auxiliary subsystem $E$. In turn, it change $\rho_{ABC}$ to $\rho_{ABCE}={|\psi_{AY}\rangle}{\langle \psi_{AY}|}\otimes \rho_{\widetilde{Z}}$. Clearly, this does not create any correlation between the first and the second part of $\rho_{ABC}$.
Although the above Lemma provides a sufficient condition for equality of SSA, it is not necessary in general. However, it provides building blocks for the structure of states for which the SSA inequality is satisfied with equality. Indeed, for a fixed pure state ${|\psi_{AY}\rangle}$, it is not difficult to see that the set of states with structure given by Eq. forms a convex subset of the set of all states. It follows therefore that *only* ensembles of the form $\{p_k,{|\psi^k_{AY_k}\rangle}{\langle \psi^k_{AY_k}|}\otimes \rho^k_{Z_k}\}_{k}$ can realize a state with $\boldsymbol{T}^{(a)}(\rho_{ABC})=0$. Theorem \[TheoremSSAcondition\] provides the necessary and sufficient conditions on $Y_k$ and $Z_k$, in order to achieve states with $\boldsymbol{T}^{(a)}(\rho_{ABC})=0$.
\[TheoremSSAcondition\] For a given $\rho_{ABC}$ we have $\boldsymbol{T}^{(a)}(\rho_{ABC})=0$ if and only if $\rho_{ABC}$ can be expressed as $$\label{RhoABC-RhoABCk}
\rho_{ABC}=\sum_{k=1}^Kp_k\rho_{ABC}^k,$$ such that the marginal states $\{\rho_{B}^k\}_k$ and $\{\rho_{C}^k\}_k$ are mutually orthogonal and $\rho_{ABC}^k$ has the following form $$\label{PsiRhoProductK}
\rho_{ABC}^k={|\psi^k_{AY_k}\rangle}{\langle \psi^k_{AY_k}|}\otimes \rho^k_{Z_k}.$$ Here $\rho_{ABC}^k$ is defined on $\mathcal{H}_{A}\otimes \mathcal{H}_{B_k}\otimes \mathcal{H}_{C_k}$, and $Y_k$ is a partition of $B_kC_k$ and $Z_k$ denotes its complement in such a way that $\mathcal{H}_{Y_k}=\mathcal{H}_{B_k^L}\otimes \mathcal{H}_{C_k^L}$ and $\mathcal{H}_{Z_k}=\mathcal{H}_{B_k^R}\otimes \mathcal{H}_{C_k^R}$.
The sufficient condition is a simple consequence of Corollary \[CorollaryZeroDelta\]-(ii) and Lemma \[LemmaSSAcondition\]. To prove the necessary condition, let $\rho_{ABC}$ be a state such that $\boldsymbol{T}^{(a)}(\rho_{ABC})=0$. It follows from Corollary \[CorollaryZeroDelta\]-(i) that $\rho_{ABC}=\sum_k p_k\rho_{ABC}^k$ implies $\boldsymbol{T}^{(a)}(\rho_{ABC}^k)=0$ for $k=1,\cdots,K$, as such, $\rho_{ABC}^k$ takes the form given by . It remains only to prove that $\rho_{B}^k \perp \rho_{B}^{k^\prime}$ and $\rho_{C}^k \perp \rho_{C}^{k^\prime}$ for $k\ne k^\prime$. Using the spectral decomposition $\rho_{Z_k}^k=\sum_{i_k}\mu_{k}^{i_k}{|\mu_{Z_k}^{i_k}\rangle}{\langle \mu_{Z_k}^{i_k}|}$ and introducing the auxiliary subsystem $E$ with an orthogonal decomposition $\mathcal{H}_E=\bigoplus_{k}^{K}\mathcal{H}_{E_k}$ and corresponding orthonormal basis $\{{|\lambda_{E_k}^{i_k}\rangle}\}_{i_k}$, for $k=1,\cdots,K$, we arrive at the following purification for $\rho_{ABC}$ $$\label{PsiABCE}
{|\Psi_{ABCE}\rangle}=\sum_{k}^{K}\sqrt{p_k}{|\psi^k_{AY_k}\rangle}\otimes {|\phi^k_{Z_kE_k}\rangle},$$ where ${|\phi^k_{Z_kE_k}\rangle}=\sum_{i_k}\sqrt{\mu_{k}^{i_k}}{|\mu_{Z_k}^{i_k}\rangle}{|\lambda_{E_k}^{i_k}\rangle}$. Tracing out $C$ and defining $\widetilde{B}=BE$, we find $$\begin{aligned}
\label{RhoABE} \nonumber
\rho_{A\widetilde{B}}&=&\sum_{k,k^\prime}^{K}\sqrt{p_kp_{k^\prime}}
{{\mathrm {Tr}}}_{C}\left[{|\psi^k_{AY_k}\rangle}{\langle \psi^{k^\prime}_{AY_{k^\prime}}|}\otimes {|\phi^k_{Z_kE_k}\rangle}{\langle \phi^{k^\prime}_{Z_{k^\prime}E_{k^\prime}}|}\right] \\
\nonumber
&=&\sum_{k\ne k^\prime}^{K}\sqrt{p_kp_{k^\prime}}
{{\mathrm {Tr}}}_{C}\left[{|\psi^k_{AY_k}\rangle}{\langle \psi^{k^\prime}_{AY_{k^\prime}}|}\otimes {|\phi^k_{Z_kE_k}\rangle}{\langle \phi^{k^\prime}_{Z_{k^\prime}E_{k^\prime}}|}\right]
\\ \label{RhoABE}
&+&\sum_{k}^{K}p_k \sigma_{AB_k^L}^k\otimes \varrho_{B_k^RE_k}^k.\end{aligned}$$ Here we have defined $\sigma_{AB_k^L}^k={{\mathrm {Tr}}}_{C_k^L}{|\psi^k_{AY_k}\rangle}{\langle \psi^k_{AY_k}|}$ and $\varrho_{B_k^RE_k}^k={{\mathrm {Tr}}}_{C_k^R}\left[{|\phi^k_{Z_kE_k}\rangle}{\langle \phi^{k}_{Z_{k}E_{k}}|}\right]$ as states on $\mathcal{H}_{A}\otimes \mathcal{H}_{B_k^L}$ and $\mathcal{H}_{B_k^R}\otimes \mathcal{H}_{E}$, respectively. Now, tracing out $C$ from $\rho_{ABC}$ of Eq. , we get $$\begin{aligned}
\label{RhoAB}
\rho_{AB}=\sum_{k}^{K}p_k
\sigma_{AB_k^L}^k \otimes \varrho_{B_k^R}^k,\end{aligned}$$ where $\varrho_{B_k^R}^k={{\mathrm {Tr}}}_{C_{k}^R}\rho_{Z_k}^k$. Comparing this with $\rho_{A\widetilde{B}}$ given by Eq. , one can easily see that $E(\rho_{A\widetilde{B}})=E(\rho_{AB})$ if and only if $\rho_{A\widetilde{B}}$ can be written as $$\begin{aligned}
\label{RhoABE2}
\rho_{A\widetilde{B}}=\sum_{k}^{K}p_k
\sigma_{AB_k^L}^k \otimes \varrho_{B_k^RE_k}^k.\end{aligned}$$ This happens if and only if the first term of the second equality of vanishes, i.e., ${{\mathrm {Tr}}}_{C}\left[{|\psi^k_{AY_k}\rangle}{\langle \psi^{k^\prime}_{AY_{k^\prime}}|}\otimes {|\phi^k_{Z_kE_k}\rangle}{\langle \phi^{k^\prime}_{Z_{k^\prime}E_{k^\prime}}|}\right]=0$. This, in turn implies $\rho_{C}^k \perp \rho_{C}^{k^\prime}$ for $k\ne k^\prime$, where $\rho_{C}^k={{\mathrm {Tr}}}_{AB}[{|\psi^k_{AY_k}\rangle}{\langle \psi^k_{AY_k}|}\otimes{|\mu_{Z_k}^{i_k}\rangle}{\langle \mu_{Z_k}^{i_k}|}]$ and $\rho_{C}^{k^\prime}$ is defined similarly. In a same manner we find that the condition $E(\rho_{A\widetilde{C}})=E(\rho_{AC})$ leads to $\rho_{B}^k \perp \rho_{B}^{k^\prime}$ for $k\ne k^\prime$. Using this, it follows that $D^{(\widetilde{B})}(\rho_{A\widetilde{B}})=D^{(B)}(\rho_{AB})$, which completes the proof.
The following Corollary is immediately obtained from Theorem \[TheoremSSAcondition\].
\[Corollary-Results\] (i) $\rho_{ABC}$ satisfies the SSA inequality with equality if and only if it satisfies the Koashi-Winter relation with equality, i.e., $S(\rho_{B})+S(\rho_{C}) = S(\rho_{AB})+S(\rho_{AC})$ implies $E(\rho_{AB})= D^{(C)}(\rho_{AC})+S_{\rho}({A\mid C})$ and $E(\rho_{AC})= D^{(B)}(\rho_{AB})+S_{\rho}({A\mid B})$, and vice versa.\
(ii) If $\rho_{ABC}$ satisfies the SSA inequality with equality then it satisfies the quantum conservation law, i.e., $S(\rho_{B})+S(\rho_{C}) = S(\rho_{AB})+S(\rho_{AC})$ implies $E(\rho_{AB})+E(\rho_{AC})=D^{(B)}(\rho_{AB})+D^{(C)}(\rho_{AC})$.
Note that we have considered only situation that the SSA inequality is saturated identically. Indeed, as it is clear from the first line of Eq. , vanishing $\boldsymbol{T}^{(a)}(\rho_{ABC})$ may happen even for nonzero $E(\rho_{A\widetilde{B}})- E(\rho_{AB})$, due to the possibility that $D^{(\widetilde{C})}(\rho_{A\widetilde{C}})- D^{(C)}(\rho_{AC})$ takes negative value. This *approximate* case that a state *almost* saturates SSA inequality is also addressed in Ref. [@HaydenCMP2004] for the SSA inequality , and it is proved in [@BrandaoCMP2011] that in this case $\rho_{ABC}$ is well approximated by structure given in [@HaydenCMP2004]. However, a look at Eq. shows that such approximate case does not happen if $\delta^{(a)}_x(D)=D^{(\widetilde{X})}(\rho_{A\widetilde{X}})- D^{(X)}(\rho_{AX})\ge 0$ for at least one choice of $X=B,C$. For example, when the conservation law holds then $\delta^{(a)}_B(E)+\delta^{(a)}_C(E)=\delta^{(a)}_B(D)+\delta^{(a)}_C(D)$, implies that both $\delta^{(a)}_B(D)$ and $\delta^{(a)}_C(D)$ cannot be negative.
*Examples.—*In order to investigate how the theorems work, we provide some illustrative examples. (i) First, let us consider the tripartite mixed state $\rho^1_{ABC}={|\psi^1_A\rangle}{\langle \psi^1_A|}\otimes\varrho^1_{BC}$. Since $A$ is factorized from the rest of the system, so that $E(\rho^1_{AB})=D^{(C)}(\rho^1_{AC})=0$. On the other hand, any purification of this state leads to $\rho^1_{ABC}\rightarrow{|\psi^1_{ABCE}\rangle}{\langle \psi^1_{ABCE}|}={|\psi^1_A\rangle}{\langle \psi^1_A|}\otimes {|\psi^1_{BCE}\rangle}{\langle \psi^1_{BCE}|}$, where ${|\psi^1_{BCE}\rangle}$ is a purification of $\varrho^1_{BC}$. It follows that, $A$ is factorized also from $\widetilde{B}=BE$ and $\widetilde{C}=CE$, so that $E(\rho^1_{A\widetilde{B}})=D^{(\widetilde{C})}(\rho^1_{A\widetilde{C}})=0$. Therefore, according to the firs line of Eq. , $\boldsymbol{T}^{(a)}(\rho^1_{ABC})=0$. Moreover, for this mixed state the K-W relation is satisfied with equality, reduces in this case to $0=0+0$.
\(ii) Now, as the second example, consider the tripartite mixed state $\rho^2_{ABC}={|\psi^2_{AB}\rangle}{\langle \psi^2_{AB}|}\otimes\varrho^2_{C}$. Purification of this state leads to $\rho^2_{ABC}\rightarrow {|\psi^2_{AB}\rangle}{\langle \psi^2_{AB}|}\otimes {|\psi^2_{CE}\rangle}{\langle \psi^2_{CE}|}$, with ${|\psi^2_{CE}\rangle}$ as a purification of $\varrho^2_{C}$. Clearly, we find $E(\rho^2_{A\widetilde{B}})=E(\rho^2_{AB})=S(\rho^2_{A})$ and $D^{(\widetilde{C})}(\rho^2_{A\widetilde{C}})=D^{(C)}(\rho^2_{AC})=0$, where $\rho^2_{A}={{\mathrm {Tr}}}[{|\psi^2_{AB}\rangle}{\langle \psi^2_{AB}|}]$. In this case, we arrive again at $\boldsymbol{T}^{(a)}(\rho^1_{ABC})=0$, and an equality for the K-W relation .
\(iii) We present the final example as a convex combination of the states given above, i.e., $$\begin{aligned}
\label{Exampleiii}
\rho_{ABC}&=&p_1\rho^1_{ABC}+p_2\rho^2_{ABC} \\ \nonumber
&=&p_1{|\psi^1_A\rangle}{\langle \psi^1_A|}\otimes\varrho^1_{BC}+p_2{|\psi^2_{AB}\rangle}{\langle \psi^2_{AB}|}\otimes\varrho^2_{C},\end{aligned}$$ where $p_1+p_2=1$. Moreover, we set $\dim{\mathcal{H}_A}=2$ and $\dim{\mathcal{H}_B}=\dim{\mathcal{H}_C}=4$, and define $$\begin{aligned}
{|\psi^1_{A}\rangle}&=&\alpha_1{|0_A\rangle}+\beta_1{|1_A\rangle}, \\
{|\psi^2_{AB}\rangle}&=&\alpha_2{|0_A0_B\rangle}+\beta_2{|1_A\phi_B\rangle},\end{aligned}$$ where ${|\phi_B\rangle}=a{|1_B\rangle}+b{|2_B\rangle}$ and $$\begin{aligned}
\varrho^1_{BC}&=&\lambda_1{|2_B2_C\rangle}{\langle 2_B2_C|}+(1-\lambda_1){|3_B3_C\rangle}{\langle 3_B3_C|}, \\
\varrho^2_{C}&=&\lambda_2{|0_C\rangle}{\langle 0_C|}+(1-\lambda_2){|1_C\rangle}{\langle 1_C|}.\end{aligned}$$ Without loss of generality we assume that all parameters are real. In this case, we find $$\begin{aligned}
\label{ExampleiiiAB}
\rho_{AB}&=&p_1{|\psi^1_A\rangle}{\langle \psi^1_A|}\otimes\varrho^1_{B}+p_2{|\psi^2_{AB}\rangle}{\langle \psi^2_{AB}|}, \\ \label{ExampleiiiAC}
\rho_{AC}&=&p_1{|\psi^1_A\rangle}{\langle \psi^1_A|}\otimes\varrho^1_{C}+p_2\sigma^2_A\otimes\varrho^2_{C},\end{aligned}$$ where $\sigma^2_A={{\mathrm {Tr}}}_B[{|\psi^2_{AB}\rangle}{\langle \psi^2_{AB}|}]=\alpha_2^2{|0_A\rangle}{\langle 0_A|}+\beta_2^2{|1_A\rangle}{\langle 1_A|}$. Using Eq. , one can easily calculate $\boldsymbol{T}^{(a)}(\rho_{ABC})$, where after some simplification takes the form $$\begin{aligned}
\label{ExampleiiiT}\nonumber
\boldsymbol{T}^{(a)}(\rho_{ABC})&=&\sum_{j=1}^4(-)^j\mu_j\log{\mu_j} \\
&-&p_2\beta_2^2\log{p_2\beta_2^2}+p_2\log{p_2}.\end{aligned}$$
![(Color online) (a) $\boldsymbol{T}^{(a)}(\rho_{ABC})$ in terms of $\beta_2$ and $\lambda_1$, for $b=1/\sqrt{2}$. (b) $\boldsymbol{T}^{(a)}(\rho_{ABC})$ in terms of $\beta_2$ and $b$, for $\lambda_1=1/2$. In both plots, we assumed $\alpha_1=1/\sqrt{2}$ and $p_1=\lambda_2=1/2$.[]{data-label="Figure1"}](fig1){width="10cm"}
Here $$\begin{aligned}
\mu_{1,3}&=&\frac{1}{2}\left[p_1\lambda_1+p_2\pm\sqrt{(p_1\lambda_1-p_2)^2+4p_1p_2\beta_1^2\gamma^2}\right], \\
\mu_{2,4}&=&\frac{1}{2}\left[p_1\lambda_1+p_2\beta_2^2\pm\sqrt{(p_1\lambda_1-p_2\beta_2^2)^2+4p_1p_2\gamma^2}\right],\end{aligned}$$ where $\gamma=\sqrt{\lambda_1} b\beta_2$. Simple investigation of Eq. shows that $\boldsymbol{T}^{(a)}(\rho_{ABC})$ vanishes if and only if $\gamma=0$, i.e., one of the parameters $\lambda_1$, $b$, or $\beta_2$ vanishes. In Fig. \[Figure1\] we have plotted $\boldsymbol{T}^{(a)}(\rho_{ABC})$ in terms of the pairs $\{\beta_2,\; \lambda_1\}$ and $\{\beta_2, \; b\}$. The figure shows clearly that $\boldsymbol{T}^{(a)}(\rho_{ABC})$ approaches zero whenever one of the parameters $\lambda_1$, $b$, or $\beta_2$ approaches zero. Now, let us turn our attention to Theorem \[TheoremSSAcondition\] and gain a better understanding of this Theorem. To this aim, first note that each term of Eq. has vanishing $\boldsymbol{T}^{(a)}(\rho_{ABC})$, i.e., $\boldsymbol{T}^{(a)}(\rho^1_{ABC})=\boldsymbol{T}^{(a)}(\rho^2_{ABC})=0$. Moreover, using $$\begin{aligned}
\rho^1_{B}&=& \lambda_1{|2_B\rangle}{\langle 2_B|}+(1-\lambda_1){|3_B\rangle}{\langle 3_B|}, \\
\rho^2_{B}&=&\alpha_2^2{|0_B\rangle}{\langle 0_B|}+\beta_2^2{|\phi_B\rangle}{\langle \phi_B|}, \\
\rho^1_{C}&=&\lambda_1{|2_C\rangle}{\langle 2_C|}+(1-\lambda_1){|3_C\rangle}{\langle 3_C|}, \\
\rho^2_C&=&\varrho^2_C,\end{aligned}$$ one can easily see that $\rho^1_B\rho^2_B=\lambda_1b\beta_2^2 {|2_B\rangle}{\langle \phi_B|}$ and $\rho^1_C\rho^2_C=0$. This implies that the required conditions of Theorem \[TheoremSSAcondition\] are satisfied if and only if $\lambda_1 b \beta_2=0$, which is in complete agreement with the result obtained from Eq. .
We continue with this example and apply Theorem \[TheoremDeltaCorrelations\] and Corollary \[Corollary-Results\] to evaluate QD and EOF of $2\times 4$ mixed bipartite states $\rho_{AB}$ and $\rho_{AC}$, reduced from *mixed* tripartite state $\rho_{ABC}$ of Eq. . Equation shows that $\rho_{AC}$ is separable and, since $\rho^1_{C}\perp\rho^2_C$, we get [@ZurekPRL2001] $$\label{ExampleiiiCorrelationAC}
E(\rho_{AC})=D^{(C)}(\rho_{AC})=0,$$ which holds for arbitrary values of $\boldsymbol{T}^{(a)}(\rho_{ABC})$. However, for $\boldsymbol{T}^{(a)}(\rho_{ABC})=0$, i.e., $\lambda_1 b \beta_2=0$, we can use the Corollary \[Corollary-Results\] and write $$\begin{aligned}
\label{ExampleiiiCorrelationAB}
E(\rho_{AB})=D^{(B)}(\rho_{AB})=S_{\rho_{AC}}(A\mid C).\end{aligned}$$ Here, the first equality is obtained from Eq. and the conservation law \[Corollary-Results\]-(ii), and the second equality comes from the K-W relation \[Corollary-Results\]-(i). Furthermore, denoting an orthonormal basis for the auxiliary subsystem $E$ with $\{{|\lambda_E\rangle}\}_{\lambda=0}^{3}$, one can provide the following purification for $\rho_{ABC}$ $$\begin{aligned}
\nonumber
{|\Psi_{ABCE}\rangle}&=&\sqrt{p_1}{|\psi^1_A\rangle}\left[\sqrt{\lambda_1}{|2_B2_C2_E\rangle}+\sqrt{1-\lambda_1}{|3_B3_C3_E\rangle}\right] \\
&+&\sqrt{p_2}{|\psi^2_{AB}\rangle}\left[\sqrt{\lambda_2}{|0_C0_E\rangle}+\sqrt{1-\lambda_2}{|1_C1_E\rangle}\right].\end{aligned}$$ Using this we find $$\begin{aligned}
\label{ExampleiiiRhoABE}
\rho_{A\widetilde{B}}&=&p_1{|\psi^1_A\rangle}{\langle \psi^1_A|}\otimes\varrho^1_{BE}+p_2{|\psi^2_{AB}\rangle}{\langle \psi^2_{AB}|}\otimes\varrho^2_{E}, \\ \label{ExampleiiiRhoACE}
\rho_{A\widetilde{C}}&=&p_1{|\psi^1_A\rangle}{\langle \psi^1_A|}\otimes\varrho^1_{CE}+p_2\sigma^2_{A}\otimes\varrho^2_{CE} \\ \nonumber
&+&\sqrt{p_1p_2\lambda_1}b\left({|1_A\rangle}{\langle \psi^1_A|}\otimes {|\psi_{CE}\rangle}{\langle 2_C2_E|}\right. \\ \nonumber
&& \qquad\qquad+\left.{|\psi^1_A\rangle}{\langle 1_A|}\otimes {|2_C2_E\rangle}{\langle \psi_{CE}|}\right),\end{aligned}$$ where we have defined $$\begin{aligned}
\varrho^1_{BE}&=&\lambda_1{|2_B2_E\rangle}{\langle 2_B2_E|}+(1-\lambda_1){|3_B3_E\rangle}{\langle 3_B3_E|}, \\
\varrho^1_{CE}&=&\lambda_1{|2_C2_E\rangle}{\langle 2_C2_E|}+(1-\lambda_1){|3_C3_E\rangle}{\langle 3_C3_E|}, \\
\varrho^2_{CE}&=&\lambda_2{|0_C0_E\rangle}{\langle 0_C0_E|}+(1-\lambda_2){|1_C1_E\rangle}{\langle 1_C1_E|}, \\
\varrho^2_{E}&=&\lambda_2{|0_E\rangle}{\langle 0_E|}+(1-\lambda_2){|1_E\rangle}{\langle 1_E|}, \\
{|\psi_{CE}\rangle}&=&\sqrt{\lambda_2}{|0_C0_E\rangle}+\sqrt{1-\lambda_2}{|1_C1_E\rangle}.\end{aligned}$$ The bipartite states and are obtained for arbitrary values of $\boldsymbol{T}^{(a)}(\rho_{ABC})$. However, if we set one of the parameters $\lambda_1$, $b$, or $\beta_2$ equal to zero, we get $\boldsymbol{T}^{(a)}(\rho_{ABC})=0$ and one can use the benefits of invariance of quantum correlations of $\rho_{AB}$ and $\rho_{AC}$ under the transformation $B\longrightarrow BE$ and $C\longrightarrow CE$. In this case, invoking Eqs. and , one can write $$\begin{aligned}
E(\rho_{A\widetilde{B}})&=&D^{(\widetilde{B})}(\rho_{A\widetilde{B}})=S_{\rho_{AC}}(A \mid C), \\ E(\rho_{A\widetilde{C}})&=&D^{(\widetilde{C})}(\rho_{A\widetilde{C}})=0,\end{aligned}$$ which are valid as far as $b\lambda_1\beta_2=0$.
*Conclusion.—*We have defined $\boldsymbol{T}^{(a)}(\rho_{ABC})$ as the extent to which the tripartite state $\rho_{ABC}$ fails to saturate SSA inequality. An important feature of our approach is the possibility of writing $\boldsymbol{T}^{(a)}(\rho_{ABC})$ as the amount by which the bipartite quantum correlations of $\rho_{AB}$ and $\rho_{AC}$ change under the transformation $B\rightarrow \widetilde{B}=BE$ and $C\rightarrow \widetilde{C}=CE$, with $E$ as an auxiliary subsystem purifying $\rho_{ABC}$. This feature of $\boldsymbol{T}^{(a)}(\rho_{ABC})$ seems remarkable since it provides a simple method to find the structure of states for which the SSA inequality is saturated by its lower and upper bounds. The concavity property of $\boldsymbol{T}^{(a)}(\rho_{ABC})$ with respect to its input reveals that a state with vanishing $\boldsymbol{T}^{(a)}(\rho_{ABC})$ can be realized *only* by those ensembles $\{p_k,\rho_{ABC}^k\}_k$ for which $\boldsymbol{T}^{(a)}(\rho_{ABC}^k)=0$. We have characterized such ensembles by means of invariance of quantum correlations of $\rho_{AB}$ and $\rho_{AC}$ under the transformation $B\rightarrow \widetilde{B}$ and $C\rightarrow \widetilde{C}$. It turns out that $\boldsymbol{T}^{(a)}(\rho_{ABC}^k)=0$ if and only if the subsystem $A$ lives in a *pure* state, so that it is not affected under purification of $\rho^k_{ABC}$. On the other hand, the upper bound $\boldsymbol{T}^{(a)}(\rho_{ABC})=2\log{d_A}$ is saturated if and only if the subsystem $A$ participates in the purification of $\rho_{ABC}$ as possible as it can, i.e., $\rho_{A}$ is maximally mixed and factorized from the rest of the system. Intuitively, the contribution of the subsystem $A$ in the purification of $\rho_{ABC}$ plays a central role in a sense that the amount by which the quantum correlations of $\rho_{AB}$ and $\rho_{AC}$ changes under the above transformation depends on the extent to which the subsystem $A$ shares its degrees of freedom in the purification. It happens that, the more contribution the subsystem $A$ has in the purification of $\rho_{ABC}$, the more quantum correlations will be shared between $A$ and the subsystems $\widetilde{B}$ and $\widetilde{C}$, leading to a greater value for $\boldsymbol{T}^{(a)}(\rho_{ABC})$.
Moreover, the approach presented in this paper explores that the class of states for which the SSA inequality is saturated coincides with those that the K-W inequality is saturated. This generalizes, to the best of our knowledge, the previous results for pure states to those with vanishing $\boldsymbol{T}^{(a)}(\rho_{ABC})$. Interestingly, the condition $\boldsymbol{T}^{(a)}(\rho_{ABC})=0$ exhausts such states. In addition, we found that if $S(\rho_{B})+S(\rho_{C}) = S(\rho_{AB})+S(\rho_{AC})$ then $E(\rho_{AB})+E(\rho_{AC})=D^{(B)}(\rho_{AB})+D^{(C)}(\rho_{AC})$, which is a possible extension of the so-called quantum conservation law previously obtained in [@FanchiniPRA2011] for pure states. Due to the widespread use of the SSA inequality in quantum information theory, it is hoped that a quantum correlation description of SSA inequality should shed some light on the several inequalities obtained from it. In particular, our results may have applications in monogamy inequalities, squashed entanglement, Holevo bounds, coherent information, and study of open quantum systems.
The authors would like to thank Fereshte Shahbeigi and Karol Życzkowski for helpful discussion and comments. This work was supported by Ferdowsi University of Mashhad under grant 3/28328 (1392/07/15).
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abstract: 'We calculate the Schiff moment of the nucleus [$^{199}{\rm Hg}$]{}, created by $\pi$NN vertices that are odd under parity (P) and time–reversal (T). Our approach, formulated in diagrammatic perturbation theory with important core–polarization diagrams summed to all orders, gives a close approximation to the expectation value of the Schiff operator in the odd–A Hartree–Fock–Bogoliubov ground state generated by a Skyrme interaction and a weak P– and T–odd pion–exchange potential. To assess the uncertainty in the results, we carry out the calculation with several Skyrme interactions, the quality of which we test by checking predictions for the isoscalar–E1 strength distribution in [$^{208}{\rm Pb}$]{}, and estimate most of the important diagrams we omit.'
author:
- 'J. H. de Jesus'
- 'J. Engel'
title: 'Time–Reversal–Violating Schiff Moment of [$^{199}{\rm Hg}$]{}'
---
Introduction {#sec:intro}
============
The existence of a permanent[^1] electric dipole moment (EDM) in leptons, neutrons or neutral atoms is direct evidence for time–reversal (T) violation. Because of the CPT theorem, the search for EDMs can provide us valuable information about sources of CP violation. Though a phase in the Cabibbo–Kobayashi–Maskawa matrix is enough to account the level of CP violation in kaon and B–meson decays, it cannot explain the observed matter/anti–matter asymmetry in the Universe. Physics that can (and new physics at the weak scale more generically) should produce EDMs not far from current upper limits.
So far no EDMs have been observed, but experiments are continually improving. Here we are interested in the conclusions that can be drawn from the measured upper limit [@romalis01] in [$^{199}{\rm Hg}$]{}, a diamagnetic atom. The largest part of its EDM most likely comes from T violation in the nucleus, caused by a T–violating (and parity–violating) component of the nucleon–nucleon interaction. The atomic EDM is generated by the subsequent interaction of the nucleus with the electrons.
That interaction is more subtle than one might think. If the nucleus and electrons were non–relativistic point–particles interacting solely via electrostatic forces, the electrons would rearrange in response to a nuclear EDM to cancel it essentially exactly. Fortunately, as was shown by Schiff [@schiff63], the finite size of the nucleus leads to a residual atomic EDM. It turns out, however, that the relevant nuclear quantity is not the nuclear EDM but rather the nuclear “Schiff moment” $$\label{eqn:sm}
S~\equiv~\langle\Psi_0|S^z|\Psi_0\rangle~,$$ which is the nuclear ground–state expectation value, in the substate $|\Psi_0\rangle$ with angular momentum projection $M_J$ equal to the angular momentum $J$, of the $z$–component of the “Schiff operator” $$\label{eqn:op}
\bm{S}~=~\frac{e}{10}\,\sum_{p=1}^Z\left(r^2_p-\frac{5}{3}\,\langle{r}^2
\rangle_{\rm
ch}\right)\bm{r}_p~.$$ Here $e$ is the charge of the proton, $\langle r^2\rangle_{\rm ch}$ is the mean squared radius of the nuclear charge distribution, and the sum is restricted to protons.
For the Schiff moment to exist, P and T must be violated by the nuclear Hamiltonian. We assume that whatever its ultimate source, the T violation works its way into a meson–mediated P– and T–violating NN interaction generated from a Feynman graph containing a meson propagator, the usual strong meson–NN strong vertex and a (much weaker) P– and T–violating meson–NN vertex. The second vertex can take three different forms in isospin space. References [@herczeg87; @griffiths91; @towner94] showed that short–range nuclear correlations and a fortuitous sign make the contribution of $\rho$– and $\omega$–exchange to the interaction small compared to that of pion–exchange if the T–violating coupling constants of the different mesons are all about the same, and so we neglect everything but pion exchange. The most general P– and T–odd NN potential then has the form ($\hbar=c=1$) $$\begin{aligned}
\label{eqn:wpt}
W(\bm{r}_a-\bm{r}_b)&=&-\frac{{\rm g}m_\pi^2}{8\pi m_{\rm N}}
\left\{\left[{\rm \bar{g}}_0\left(\bm{\tau}_a\cdot\bm{\tau}_b\right)
-\frac{{\rm \bar{g}}_1}{2}\left(\tau_a^z+\tau_b^z\right)+
{\rm \bar{g}}_2\left(3\tau_a^z\tau_b^z-\bm{\tau}_a\cdot\bm{\tau}_b\right)
\right]\left(\bm{\sigma}_a-\bm{\sigma}_b\right)-\right.\nonumber\\
&&\mbox{}\left.-\frac{{\rm \bar{g}}_1}{2}\left(\tau_a^z-\tau_b^z\right)
\left(\bm{\sigma}_a+\bm{\sigma}_b\right)\right\}\cdot
\left(\bm{r}_a-\bm{r}_b
\right)\frac{{\rm exp}\left(-m_\pi|\bm{r}_a-\bm{r}_b|\right)}
{m_\pi|\bm{r}_a-\bm{r}_b|^2}\left[1+\frac{1}{m_\pi|\bm{r}_a-\bm{r}_b|}
\right]~,\end{aligned}$$ where $m_\pi$ is the mass of the pion, $m_{\rm N}$ that of the nucleon, $\tau^z|p\rangle=-|p\rangle$, ${\rm g} \equiv 13.5$ is the strong $\pi$NN coupling constant, and the $\bar{\rm g}_i$ are the isoscalar ($i=0$), isovector ($i=1$), and isotensor ($i=2$) PT–odd $\pi$NN coupling constants. A word of caution here: more than one sign convention for the $\bar{\rm g}$’s is in use. Our $\bar{\rm
g}_0$ and $\bar{\rm g}_1$ are defined with a sign opposite to those used by Flambaum et. al [@flambaum86; @flambaum02] and by Dmitriev et. al [@dmitriev03; @dmitriev05].
The goal of this paper is to calculate the dependence of the Schiff moment of [$^{199}{\rm Hg}$]{} on the T–violating $\pi$NN couplings (we leave the dependence of these couplings on fundamental sources of CP violation to others) so that models of new physics can be quantitatively constrained. An accurate calculation is not easy because the Schiff moment depends on the interplay of the Schiff operator with complicated spin– and space–dependent correlations induced by the the two–body interaction $W$. In the early calculation by Flambaum, Khriplovich and Sushkov almost two decades ago [@flambaum86], the correlations were taken to be admixtures of simple 1–particle 1–hole excitations into a Slater determinant produced by a one–body Wood–Saxon potential.
More recent work [@dmitriev03; @dmitriev05] made significant improvements by treating the correlations in the RPA after generating an approximately self–consistent one–body potential. However, that work used only the relatively schematic Landau–Migdal interaction (in addition to $W$) in the RPA and mean–field equations, and did not treat pairing self consistently. The reliance on a single strong interaction makes it difficult to analyze uncertainty. Such an analysis seems to be particularly important in [$^{199}{\rm Hg}$]{}, the system with the best experimental limit on its EDM. The calculated Schiff moment of Refs. [@dmitriev03; @dmitriev05] in that nucleus depends extremely weakly on the isoscalar coefficient ${\rm \bar{g}}_0$, a result of coincidentally precise cancellations among single–particle and collective excitations. They might be less precise when other interactions are used.
Here we make several further improvements. Our mean field, which we calculate in $^{198}$Hg before treating core polarization by the valence nucleon, includes pairing and is exactly self consistent. Pairing changes the RPA to the quasiparticle–RPA (QRPA), a continuum version of which we use to obtain ground–state correlations. Most importantly, we carry out the calculation with several sophisticated (though still phenomenological) Skyrme NN interactions, the appropriateness of which we explore by examining their ability to reproduce measured isoscalar–E1 strength (generated by the isoscalar component of the Schiff operator) in $^{208}$Pb. The use and calibration of more than one such force allows us to get a handle on the uncertainty in our final results.
The rest of this paper is organized as follows: Section II describes our approach and the Skyrme interactions we use, and includes their predictions for strength distributions that bear on the Schiff moment. Section III presents our results and an analysis of their uncertainty, including a calculation in the simpler nucleus $^{209}$Pb that allows us to check the size of effects we omit in [$^{199}{\rm Hg}$]{}. Section IV is a conclusion.
Procedure for evaluating Schiff moments {#sec:method}
=======================================
Method
------
Our Schiff moment is a close approximation to the expectation value of the Schiff operator in the completely self–consistent one–quasiparticle ground state of [$^{199}{\rm Hg}$]{}, constructed from a two–body interaction that includes both a Skyrme potential and the P– and T–violating potential $W$. It is an approximation because we do not treat $W$ in a completely self consistent way, causing an error that we estimate to be small in the Section \[sec:results\]. In addition, we do not actually carry out the mean–field calculation in [$^{199}{\rm Hg}$]{} itself. Instead, we start from the HF+BCS ground–state of the even–even nucleus [$^{198}{\rm Hg}$]{} and add a neutron in the $2p_{1/2}$ level. We then treat the core–polarizing effects of this neutron in the QRPA. A self–consistent core with QRPA core polarization is completely equivalent to a fully self–consistent odd–A calculation [@brown70]. We omit one part of the QRPA core polarization, again with an estimate showing its contribution to be insignificant.
A good way to keep track of the two interactions and their effects is to formulate the calculation (and corrections to it) as a sum of Goldstone–like diagrams, following the shell–model effective–operator formalism presented, e.g., in Ref.[@ellis97]. The one difference between our diagrams and he usual “Brandow” kind is that our fermion lines will represent BCS quasiparticles rather than pure particles or holes. Our diagrams reduce to the familiar kind in the absence of pairing.
We begin, following a spherical HF+BCS calculation in [$^{198}{\rm Hg}$]{} (in a 20–fm box with mixed volume and surface pairing fixed as in Ref. [@bender02]), by dividing the Hamiltonian into unperturbed and residual parts. The unperturbed part, expressed in the quasiparticle basis, is $$\label{eqn:H_0}
H_0 = T + V_{00} + V_{11} ~,$$ where $T$ is the kinetic energy and $V$ the Skyrme interaction, with subscripts that refer to the numbers of quasiparticles the operator creates and destroys. The residual piece[^2] is $$\label{eqn:H_res}
H_{\rm res} = W + V_{22} + V_{13}+V_{31}+V_{04}+V_{40}~.$$ The interaction $W$ can also be expanded in terms of quasiparticle creation and annihilation operators; all the terms are included in $H_{\rm res}$, though $W_{00}$ vanishes because $W$ is a pseudoscalar operator. The valence “model space” of effective–operator theory is one–dimensional: a quasiparticle with $u=0$ and $v=1$ (i.e. a particle, since it is not part of a pair) in the the $a\equiv (2p_{1/2},m=1/2)$ level. The unperturbed ground state $|\Phi_a\rangle$ is simply this one–quasiparticle state. Then the expectation value of $S^z$, Eq. (\[eqn:sm\]), in the full correlated ground state $|\Psi_{a}\rangle \equiv |\Psi_0\rangle$ is given by $$\langle\Psi_{{a}}|S^z|\Psi_{{a}}\rangle~=~{\cal N}^{-1}
\langle\Phi_{{a}}|\left[1+H_{\rm
res}\left(\frac{Q}{\epsilon_{{a}}-H_0}\right)+\cdots\right]S^z
\left[1+\left(
\frac{Q}{\epsilon_{{a}}-H_0}\right)H_{\rm
res}+\cdots\right]|\Phi_{{a}}\rangle~.
\label{eqn:per4}$$ Here $\epsilon_{{a}}$ is the single–quasiparticle energy of the valence nucleon, the operator $Q$ projects onto all other single–quasiparticle states, ${\cal N}$ is a normalization factor that, we will argue later, is very close to one, and the dots represent higher–order terms in $H_{\rm res}$. To evaluate the expression, we write $S^z$ in the quasiparticle basis as $S^z=S_{11}+S_{02} + S_{20}$ ($S_{00}$ vanishes for the same reason as $W_{00}$).
![First–order quasiparticle diagrams contributing to the Schiff moment. Diagrams (i), (ii), (iv) and (v) do not contribute if the valence nucleon is a neutron, which is the case in [$^{199}{\rm Hg}$]{}. Diagram (vi) is the complex conjugate of diagram (iii).[]{data-label="fig:order1"}](figure1)
The zeroth–order contribution to the Schiff moment in Eq.(\[eqn:per4\]) vanishes because the Schiff operator cannot connect two states with the same parity, and also because the Schiff operator acts only on protons while the valence particle in [$^{199}{\rm Hg}$]{} is a neutron. (There are no center–of–mass corrections to the effective charges [@sushkov84].) The terms that are first order in $H_{\rm res}$ do not include the strong interaction $V$ because it has a different parity from the Schiff operator. Thus the lowest order contribution to the Schiff moment is $$\langle\Psi_{{a}}|S^z|\Psi_{{a}}\rangle^{\rm
first-order}~=~\langle-|q_{{a}}\left[W\left(\frac{Q}{\epsilon_{{a}}-H_0}\right)
S^z\right]
q^{\dag}_{{a}}|-\rangle+{\rm c.c.}~, \label{eqn:order1}$$ where $q^{\dag}_{{a}}$ is the creation operator for a quasiparticle in the valence level ${{a}}$ and $|-\rangle$ is the no–quasiparticle BCS vacuum describing the even–even core, so that $|\Phi_a\rangle$ is just $q^{\dag}_a|-\rangle$. The contribution of Eq. (\[eqn:order1\]) in an arbitrary nucleus can be represented as the sum of the diagrams in Fig. \[fig:order1\], the rules for which we give in the Appendix[^3]. In [$^{199}{\rm Hg}$]{}, because the valence particle is a neutron, only diagrams (iii) and (vi) are nonzero. We can interpret diagram (iii) as the Schiff operator exciting the core to create a virtual three–quasiparticle state, which is then de–excited back to the ground state when the valence neutron interacts with the core through $W$. This diagram and its partner (vi) are what was evaluated by Flambaum et al. [@flambaum86], though their mean field was a simple Wood–Saxon potential, their $W$ was a zero–range approximation that didn’t include exchange terms, and they neglected pairing.
. The broken line represents the action of the Schiff operator (as in Fig.\[fig:order1\]), the zig-zag line represents the P– and T–violating interaction (also as in Fig. \[fig:order1\]), and the looped line represents a generic Skyrme interaction.[]{data-label="fig:QRPA"}](figure2)
Core polarization, implemented through a version of the canonical–basis QRPA code reported in Ref. [@terasaki05] (with residual spurious center–of–mass motion removed following Ref.[@agrawal03]), can be represented by a subset of the higher–order diagrams. Because $W$ is so weak, we need only include it in first order. The higher order terms in $V$ that we include have the effect of replacing the two–quasiparticle bubble in (iii) and (vi) of Fig. \[fig:order1\] with chains of such bubbles (see Fig. \[fig:QRPA\]), as well adding diagrams in which the QRPA bubble chains are excited through a strong interaction of the core with the valence neutron. We therefore end up evaluating diagrams labeled A, B1, and B2, in Fig. \[fig:orderRPA\] (plus two more of type B in which the interaction $W$ is below the Schiff operator). The explicit expression for diagram A, the first on the left in the figure, is $$\label{eqn:diagA}
\langle\Psi_{{a}}|S^z|\Psi_{{a}}\rangle_{\rm diag-A}~=~-\sum_\lambda
\sum_{{{k}}>{{l}}}\sum_{k^{\prime}>l^{\prime}}Z^{\lambda\ast}_{{{k}}{{l}}}
\langle -|S^z_{02}|{{k}}{{l}}\rangle
Z^\lambda_{k^{\prime}l^{\prime}}\langle {{a}}k^{\prime}
l^{\prime} |W_{31}|{{a}}\rangle
{\cal E}_\lambda^{-1}~.$$
![QRPA diagrams contributing to the Schiff moment. The filled bubble represents an infinite sum of quasiparticle bubbles, including all the forward and backward amplitudes. The two B diagrams have partners (not shown) in which $W$ acts below $S^z$.[]{data-label="fig:orderRPA"}](figure3)
Here, $Z^\lambda_{{{k}}{{l}}}\equiv
X^\lambda_{{{k}}{{l}}}+Y^\lambda_{{{k}}{{l}}}$ represents the QRPA amplitudes (the sum appears because the matrix elements of all our operators are real) and ${\cal E}_\lambda$ is the energy of the collective state $\lambda$ in [$^{198}{\rm Hg}$]{}. The quasiparticle matrix elements $\langle
-|S^z_{02}|{{k}}{{l}}\rangle$ and $\langle
{{a}}{{k}}{{l}}|W_{31}|{{a}}\rangle$ are related to the usual particle matrix elements $\langle {{k}}|S^z|{{l}}\rangle$ and $\langle
{{a}}{{k}}|W|{{a}}{{l}}\rangle$ through the transformations discussed in the Appendix. In the absence of QRPA correlations, the $X$ and $Y$ amplitudes are 1 or 0, and Eq. (\[eqn:diagA\]) reduces to that associated with diagrams (iii) and (vi) of Fig. \[fig:order1\].
Diagrams B1 and B2 of Fig. \[fig:orderRPA\] have the explicit expressions $$\begin{aligned}
\langle\Psi_{{a}}|S^z|\Psi_{{a}}\rangle_{\rm diag-B1}&=&
-2\sum_\lambda\sum_{{i}}\sum_{{{k}}>{{l}}}\sum_{k^{\prime}>l^{\prime}}
\langle {{a}}|W_{11}|{{i}}\rangle Z^{\lambda\ast}_{{{k}}{{l}}}
\langle {{i}}|V_{13}|{{a}}{{k}}{{l}}\rangle Z^\lambda_{k{\prime}l^{\prime}}
\langle k^{\prime}l^{\prime}|S^z_{20}|-
\rangle(\epsilon_{{a}}-\epsilon_{{i}})^{-1}
{\cal E}_\lambda^{-1}~,
\label{eqn:diagB1}\\
\langle\Psi_{{a}}|S^z|\Psi_{{a}}\rangle_{\rm diag-B2}
&=&-2\sum_\lambda\sum_{{i}}\sum_{{{k}}>{{l}}}\sum_{k^{\prime}>l^{\prime}}\langle
{{a}}|W_{11}|{{i}}\rangle Z^{\lambda\ast}_{{{k}}{{l}}}\langle
-|S^z_{02}|{{k}}{{l}}\rangle Z^\lambda_{k^{\prime}l^{\prime}}\langle
{{i}}k^{\prime}l^{\prime}|V_{31}|{{a}}\rangle
(\epsilon_{{a}}-\epsilon_{{i}})^{-1}({\cal E}_\lambda
-\epsilon_{{a}}+\epsilon_{{i}})^{-1}~.
\label{eqn:diagB2}\end{aligned}$$ The factor 2 accounts for diagrams not shown in Fig.\[fig:orderRPA\] in which $W$ acts [*below*]{} the QRPA bubble, and the $\epsilon$’s are quasiparticle energies. The difference between Eq. (\[eqn:diagB1\]) and Eq. (\[eqn:diagB2\]) is mainly in the three-quasiparticle intermediate states.
A complete QRPA calculation that is first order in $W$ would also include versions of diagram A in which $W$ trades places with one of the $V$’s in the bubble sum. We don’t evaluate such diagrams but estimate their size (which we find to be small) rom calculations in the simpler nucleus [$^{209}{\rm Pb}$]{} in the next section. We also use that nucleus to examine other low–order diagrams not included in the bubble sum of diagram A.
Why do we expect the QRPA subset of diagrams to be sufficient? The reason is that they generally account for the collectivity of virtual excitations in a reliable way when calculated with Skyrme interactions. We illustrate this statement below with some calculations of isoscalar–E1 strength in [$^{208}{\rm Pb}$]{}.
Interactions
------------
We carry out the calculation with 5 different Skyrme interactions: our preferred interaction SkO$^{\prime}$ [@bender02; @reinhard99] (preferred for reasons discussed in Ref. [@engel03]), and the older, commonly used interactions SIII [@beiner75], SkM$^*$ [@bartel82], SLy4 [@chabanat98], and SkP [@dobaczewski84]. To get some idea of how well they will work, we calculate the strength distribution of the isoscalar–E1 operator $$\label{eqn:isgdr}
\bm{D}_0~=\sum_{p=1}^Zr_p^2\bm{r}_p+
\sum_{n=1}^Nr_n^2\bm{r}_n~.$$ This operator is interesting because it is the isoscalar version of the Schiff operator (the isoscalar version of the second term in the Schiff operator acts only on the center of mass and so doesn’t appear in $\bm{D}_0$). The isoscalar–E1 strength, measured, e.g., in [$^{208}{\rm Pb}$]{} [@clark01], seems to fall mainly into two peaks. The high–energy peak, related to the compressibility coefficient ${\rm
K}_\infty$, [@davis97; @hamamoto98; @colo00; @vretenar00; @clark01; @abrosimov02; @shlomo02], is observed to lie between 19 and 23 MeV, depending on the experimental method used [@davis97; @clark01]. Recent interest has focused on a smaller but still substantial low–energy peak around $12~{\rm MeV}$, which has been studied theoretically in the RPA [@colo00; @vretenar00] as well as experimentally [@clark01].
Figure \[fig:isgdrpb\] shows the predictions of several Skyrme interactions in the RPA, with widths of 1 MeV introduced by hand following Ref. [@hamamoto98], for the isoscalar–E1 strength distribution in [$^{208}{\rm Pb}$]{}. The figure also shows the locations of the measured low– and high–energy peaks of Davis et al. [@davis97] and of Clark et al. [@clark01]. Nearly all self–consistent RPA calculations, including ours (except with SkP) over–predict the energy of the larger peak by a few MeV [@colo00; @clark01]. SIII does a particularly poor job. The predicted low–energy strength is closer to experiment, though usually a little too low. Table \[tb:isgdrpb\] summarizes the situation. Unfortunately, the data are not precise enough to extract much more than the centroids of the two peaks. Since the Schiff–strength distribution in [$^{199}{\rm Hg}$]{}helps determine the Schiff moment, it would clearly be useful to have better data, either in that nucleus or a nearby one such as [$^{208}{\rm Pb}$]{}.
The isovector–E1 strength distribution also bears on the Schiff moment through the second term of the Schiff operator, but it is well understood experimentally and generally reproduced fairly well by Skyrme interactions.
low (MeV) high (MeV)
----------------------- -- -------------- --------------
SkM$^\star$ $11.0$ $25.3$
SkP $10.0$ $23.4$
SIII $11.6$ $28.3$
SLy4 $11.4$ $26.4$
SkO$^\prime$ $10.3$ $24.8$
Experiment [@davis97] $-$ $22.4\pm0.5$
Experiment [@clark01] $12.2\pm0.6$ $19.9\pm0.8$
: Comparison between experimental and theoretical results for the centroids of the low- and high-energy peaks in the distribution of isoscalar E1 strength in [$^{209}{\rm Pb}$]{}. The experimental results are from Refs. [@clark01] and [@davis97]. (Ref. [@davis97] identifies only the high-energy peak.) The theoretical distributions are from self-consistent HF+RPA calculations with five different Skyrme interactions.[]{data-label="tb:isgdrpb"}
Results and estimate of uncertainty {#sec:results}
===================================
Results with several forces
---------------------------
The Schiff moment can be written as $$S=a_0 ~ {\rm g \bar{g}}_0 +a_1 ~ {\rm g \bar{g}}_1 + a_2 ~ {\rm g
\bar{g}}_2
~,$$ where ${\rm \bar{g}}_i$ are the P– and T–odd $\pi$NN coupling constants and all the nuclear physics is summarized by the three coefficients $a_i$. We present our results for these coefficients by showing the effects, in turn, of several improvements on early calculations.
The first calculations of Schiff moments [@flambaum86], as noted above, correspond to our first–order diagrams (iii) and (vi) of Fig. \[fig:order1\] but with no pairing, with a simple Wood–Saxon potential in place of a self–consistent mean field, and with the zero–range limit of the direct part of $W$. The results of Ref. [@flambaum86] are given here in the first line of Tab.\[tb:result\].
$a_0$ $a_1$ $a_2$
-------------------- --------- --------- ---------
Ref. [@flambaum86] $0.087$ $0.087$ $0.174$
Naive limit $0.095$ $0.095$ $0.190$
Diagram A only $0.018$ $0.034$ $0.031$
Full result $0.010$ $0.074$ $0.018$
: Calculated coefficients $a_i$ from Ref. [@flambaum86] and with the Skyrme interaction SkO$^{\prime}$ in several limits (see text). The full result is in the last line.[]{data-label="tb:result"}
When we repeat the calculation, evaluating diagrams (iii) and (vi) with $W$ approximated by its direct part in the zero–range limit and with the mean field from the Skyrme interaction SkO$^\prime$ (so that the only differences in the calculations are the one–body potential and BCS paring), we get the coefficients in second line of the table.
The finite range of the potential reduces the $a_i$ from these zero–range values by 30–40%, depending on the Skyrme interaction used. Exchange terms, when the range is finite, decrease $a_0$ by a few percent, have no effect on $a_1$ and increase $a_2$ by half the amount they decrease $a_0$.
The three coefficients in lines 1 and 2 of the table are not independent; the isotensor coefficient is exactly two times larger than the isovector and isoscalar coefficients. Because the valence neutron must excite core protons to couple to the Schiff operator through (iii) and (vi) of Fig. \[fig:order1\], only the neutron–proton part of $W$ contributes, and under the assumptions of no core spin, no exchange terms, and zero–range, $W$ reduces to $$W^{\rm contact}_{\rm direct}(\bm{r}_{n}-\bm{r}_p)~=~-\frac{1}
{2m_\pi^2m_{\rm
N}}({\rm g}{\rm \bar{g}}_0+{\rm g}{\rm \bar{g}}_1+2{\rm g}{\rm
\bar{g}}_2)\bm{\sigma}_n\cdot\bm{\nabla}_n\delta(\bm{r}_{n}-
\bm{r}_p)~,
\qquad {\rm no~core~spin}
\label{eqn:4wpt}$$ Thus, in the approach of Flambaum et al. [@flambaum86], the Schiff moment is a function of a single parameter, usually called $\eta_{np}$. Exchange terms add another independent parameter and QRPA bubbles bring in a third. This last term arises because the valence neutron, besides exciting a proton quasiparticle pair in the core, can now excite neutron quasiparticle pairs that annihilate and create proton pairs inside the bubble that then couple to the Schiff operator (see Fig. \[fig:QRPA\]). Thus, in our complete calculation, particularly when the B diagrams are included, the three coefficients $a_i$ are independent.
The collectivity of the core turns out to be very important. As can be seen in the third line of Tab. \[tb:result\], when the single–particle bubble in the calculation above is replaced by the full QRPA bubble sum to give diagram A, all three $a_i$ shrink [*substantially*]{}. The reason is that the Schiff strength is pushed on average to higher energies, both in the low–lying and high–lying analogs of the isoscalar–E1 distribution. (The high–lying peak actually is replaced by two peaks, the higher of which is at about 38 MeV. There is no peak corresponding to the giant isovector–E1 resonance, as was shown in Ref. [@engel99a].) The reduction is greater in the isoscalar and tensor channels — a factor of 4 to 6 depending on the Skyrme interaction — than in the isovector channel, where it is a factor of 2 to 3. Figure \[fig:civ\] shows the integral of the contribution to diagram A as a function of core–excitation energy (so that at large energy the lines approach the value for the diagram) for $a_1$, with and without the bubble sum, for all 5 forces.
The reason for the difference in the size of the reduction is that the ${\rm \bar{g}}_0$ and ${\rm \bar{g}}_2$ parts of the interaction affect protons and neutrons in opposite ways (see, e.g., Eq. (9) of Ref. [@engel03]), causing a destructive interference, while the ${\rm \bar{g}}_1$ part affects them in the same way. This difference is absent from the single–quasiparticle picture because neutron excitations of the core don’t play a role there. Another way of saying the same thing is that when the neutrons and protons are affected in the same way, the second (dipole–like) term in the Schiff operator, Eq. (\[eqn:op\]), contributes very little because the center of mass and center of charge move together. In the other two channels the contribution of the second term is similar in magnitude and opposite in sign to that of the first term so that, as Fig. \[fig:parts\] shows, the net value is smaller.
The type–B diagrams (see Fig. \[fig:QRPA\]) are important corrections to diagram A. The effective weak isoscalar and isotensor one–body potentials (i.e. the tadpole) contribute with opposite sign from that of the isovector potential; again, see Eq.(9) of [@engel03], which displays the direct part of the one–body potential explicitly. The sign turns out to be opposite that of diagram A in the isoscalar and isotensor channels, further suppressing $a_0$ and $a_2$, and the same as diagram A in the isovector channel, largely counteracting the suppression by collectivity in that diagram. The net result for SkO$^{\prime}$ is in the last line of Tab. \[tb:result\]; for the other forces the net results appear in Tab. \[tb:skyrme\]. The isovector coefficient $a_1$ ends up not much different from the early estimate of Ref. [@flambaum86] but the isoscalar coefficient $a_0$ is smaller by a factor of about 9 to 40 and the isotensor coefficient $a_2$ by a factor of about 7 to 16.
$a_0$ $a_1$ $a_2$
-------------- --------- --------- ---------
SkM$^\star$ $0.009$ $0.070$ $0.022$
SkP $0.002$ $0.065$ $0.011$
SIII $0.010$ $0.057$ $0.025$
SLy4 $0.003$ $0.090$ $0.013$
SkO$^\prime$ $0.010$ $0.074$ $0.018$
: Full coefficients $a_i$ in [$^{199}{\rm Hg}$]{} for the five different Skyrme interactions used here. The units are $e~{\rm fm}^3$.[]{data-label="tb:skyrme"}
Uncertainty and final result
----------------------------
The several Skyrme interactions we use all give different results, but the spread in numbers is about a factor of four in the isoscalar channel, two in the isotensor channel, and much less for the large isovector coefficient $a_1$. It is possible that all the interactions are systematically deficient, but we have no evidence for that. In any event, the effective interaction is not the only source of uncertainty. We have evaluated only a subset of all diagrams, and although it is not obvious whether all the rest should be evaluated with effective interactions that are determined through Hartree–Fock– or RPA–based fits, there are some that should certainly be included and we would like to estimate their size.
The diagrams labeled C and D in Fig. \[fig:others\] are the leading terms in bubble chains that would result from including $W$ in the Hartree–Fock calculation (C) and in the QRPA calculation (D) in [$^{198}{\rm Hg}$]{}. We evaluated both sets in the simple nucleus [$^{209}{\rm Pb}$]{} (which has no pairing at the mean–field level), and found that diagram C can be nearly as large as the type–B diagrams in the isovector and isotensor channels. The same is true of diagram D in the isotensor channel. In that nucleus, however, diagram A is much larger than all the others and essentially determines the $a_i$. In [$^{199}{\rm Hg}$]{}, we only evaluated diagram C, but found that even though diagrams A and B can cancel there, they do so the most in the isoscalar channel, where the diagrams C and D are smallest. In the end diagram C never amounts to more than 10% of the sum of the A and B diagrams, and usually amounts to much less. Including the higher order (QRPA) terms in the bubble chain will only reduce the diagram-C contribution, so we conclude that it can be neglected. We are not positive that the same statement is true of diagram D, but unless it is much larger in [$^{199}{\rm Hg}$]{} than in [$^{209}{\rm Pb}$]{} (none of the other diagrams are), it can be neglected too.
The diagram labeled E represents a correction from outside our framework that is of the same order as the terms we include. We evaluated it in [$^{209}{\rm Pb}$]{}; it is uniformly smaller than those of type C and D. Unless the situation is very different in [$^{199}{\rm Hg}$]{}, it can be neglected as well. The fact that these extra diagrams are all small is not terribly surprising; they all bring in extra energy denominators and/or interrupt the collective bubble.
![Diagrams we did not include in our calculation in [$^{199}{\rm Hg}$]{}but the value of which we estimated through calculations in [$^{209}{\rm Pb}$]{}(and [$^{199}{\rm Hg}$]{} in the case of diagram C). We have omitted the labels on the the interactions; they are the same as in the earlier figures.[]{data-label="fig:others"}](figure7)
We have also not included the normalization factor ${\cal N}$ in Equation (\[eqn:per4\]). When calculated to second order in [$^{209}{\rm Pb}$]{}, it is about 1.05, independent of the Skyrme interaction used. Though this factor could be larger in RPA order because of low–lying phonons, most strength is pushed up by the RPA and we do not anticipate a large increase. It is reasonable to assume these statements are true in [$^{199}{\rm Hg}$]{} as well.
At short distances the NN potential is strongly repulsive and the associated short–range correlations should be taken into account. Reference [@griffiths91], however, reports that the correlations reduce matrix elements of the effective one–body P– and T–violating pion–exchange potential only by about 5%, and in Ref. [@dobaczewski05], which calculates the Schiff moment of [$^{225}{\rm Ra}$]{}, their effects are smaller than 10%. We are not missing much by neglecting them, though we would be if we included a $\rho$–meson exchange potential.
When all is said and done, the uncertainty is dominated by our uncertainty in the effective interaction. Our preferred interaction is SkO$^{\prime}$, which (to repeat) gives the result $$S_{^{199}{\rm Hg}}^{\rm SkO^{\prime}}~=~0.010{\rm g\bar{g}}_0+
0.074{\rm g\bar{g}}_1+
0.018{\rm g\bar{g}}_2~~~[e~{\rm fm}^3]~.
\label{eqn:myresults2}$$ If instead we average the results from the five interactions, we get $$S_{^{199}{\rm Hg}}^{\rm ave}~=~0.007{\rm g\bar{g}}_0+0.071{\rm g\bar{g}}_1+
0.018{\rm g\bar{g}}_2~~~[e~{\rm fm}^3]~,
\label{eqn:averesults}$$ The range of results in Tab. \[tb:skyrme\] is a measure of the uncertainty.
As noted in the introduction, Refs. [@dmitriev03; @dmitriev05] contain a similar calculation. They report $$S_{^{199}{\rm Hg}}^{\rm Ref.\ [8]}~=~0.0004{\rm
g\bar{g}}_0+0.055{\rm g\bar{g}}_1+ 0.009{\rm g\bar{g}}_2~~~[e~{\rm
fm}^3]~,\label{eqn:hgdmitri}$$ the most striking aspect of which is the isoscalar coefficient $a_0$; it is more than an order of magnitude smaller than our preferred value and five times smaller than the smallest coefficient produced by any of our interactions. We see no fundamental reason for such serious suppression, and suspect that the same cancellation we observe is coincidentally more precise in the single Hg calculation of Refs. [@dmitriev03; @dmitriev05]. The authors applied their method to other nuclei, but did not find the same level of suppression in any of them. Even the cancellation produced in our calculations by SkP and SLy4 seems coincidentally severe. Though it is possible that other realistic Skyrme interactions would produce still smaller coefficients, we have a hard time imagining it.
Conclusions {#sec:conclusions}
===========
Our goal has been a good calculation of the dependence of the Schiff moment of [$^{199}{\rm Hg}$]{}, the quantity that determines the electric dipole moment of the corresponding atom, on three P– and T–violating $\pi$NN coupling constants. The current experimental limit on the dipole moment of the [$^{199}{\rm Hg}$]{} atom, $|d| < 2.1\times 10^{-28}
e~{\rm cm}$, together with the theoretical results of Ref.[@dzuba02], $d=-2.8 \times 10^{-17}(S/e~{\rm fm}^3)~e~{\rm cm}$, yields the constraint $|S|<7.5\times 10^{-12} e~{\rm fm}^3$. The $a_i$ calculated in this paper then give a constraint on the three ${\rm \bar{g}}_i$.
In obtaining the $a_i$ we have included what we believe to be most of the important physics, including a pion–exchange P– and T–violating interaction, collective effects that are known to renormalize strength distributions of Schiff–like operators, pairing at the mean–field level, self–consistency, and finally, several different Skyrme interactions. The last of these, together with an examination of effects we omitted, allows us to give the first real discussion of uncertainty for a calculation in this experimentally important nucleus.
We conclude that while the isovector coefficient $a_1$ is not very different from the initial estimate of Ref. [@flambaum86], the isoscalar coefficient, which determines the limit one can set on the QCD parameter $\bar{\theta}$, is smaller by between about 9 and 40 (with the former our preferred value) and the isotensor parameter $a_2$ by a factor between about 7 and 16 (with our preferred value about 10). The uncertainty in these numbers comes primarily from our lack of knowledge about the effective interaction. There is good reason to make better measurements of low–lying dipole strength, particularly in the isoscalar channel. They would help to unravel the details of nuclear structure that determine the Schiff moment.
We thank J. Dobaczewski and J. Terasaki for helpful discussions. This work was supported in part by the U.S. Department of Energy under grant DE-FG02-97ER41019 and by the Fundação para a Ciência e a Tecnologia (Portugal). J. H. de Jesus thanks the Institute for Nuclear Theory at the University of Washington for its hospitality and the Department of Energy for partial support during the completion of this work.
Appendix {#sec:appendix}
========
Rules for quasiparticle diagrams in the uncoupled basis
-------------------------------------------------------
There are some differences between our rules for quasiparticle diagrams and the usual rules for particle–hole diagrams. The main one is that one– and two–body operators are written in a quasiparticle basis and do not conserve quasiparticle number, leading to different expressions for matrix elements. An example is the generic quasiparticle operator $O_{20}$, which contains two quasiparticle creation and no destruction operators. Its matrix elements will be written $\langle kl|O_{20}|-\rangle$, which means that it creates two quasiparticle states $|k\rangle|l\rangle$ out of the quasiparticle vacuum $|-\rangle$.
In what follows, “in" refers to lines with arrows pointing toward the vertex and “out" to lines pointing away from it. A diagram should be read from top to bottom, and from left to right. The rules are then:
1. Each operator $O_{11}$ contributes $\langle {\rm out}|O_{11}|{\rm in}\rangle$;
2. Each operator $O_{20}$ contributes $\langle {\rm out,out^{\prime}}|O_{20}|-\rangle$; because the diagram is read from the left, the label “out" is on the line that is further to the left;
3. Each operator $O_{02}$ contributes $\langle-|O_{02}|{\rm in,in^{\prime}}\rangle$;
4. Each operator $O_{22}$ contributes $\langle {\rm out,out^{\prime}}|O_{22}|
{\rm in,in^{\prime}}\rangle$;
5. Each operator $O_{31}$ contributes $\langle {\rm out,out^{\prime},
out^{\prime\prime}} |O_{31}|{\rm in}\rangle$;
6. Each operator $O_{13}$ contributes $\langle {\rm out} |O_{13}|{\rm in,in^{\prime},
in^{\prime\prime}}\rangle$;
7. Each operator $O_{40}$ contributes $\langle {\rm out,out^{\prime},out^{\prime\prime}
,out^{\prime\prime\prime}}|O_{40}|-\rangle$;
8. Each operator $O_{04}$ contributes $\langle-|O_{04}|{\rm in,in^{\prime},
in^{\prime\prime},in^{\prime\prime\prime}}\rangle$;
9. The diagram should be summed over all intermediate states;
10. Energy denominators are evaluated by operating with $(\epsilon_a-H_0)^{-1}$ between the action of every two operators in the diagram, giving $[\epsilon_a+\sum_k\epsilon_k]^{-1}$, where $\epsilon_k$ are quasiparticle energies;
11. The phase for each diagram is $(-)^{n_l}$, where $n_l$ is the number of closed loops.
12. A factor of $1/2$ is included for each pair of lines that start at the same vertex and end at the same vertex.
Folded diagrams with additional rules occur in general, but we do not discuss them here.
Matrix elements of quasiparticles operators
-------------------------------------------
We first summarize some important quantities involving the quasiparticle creation and annihilation operators $q^{\dag}$ and $q$, which are defined in terms of the usual particle operators $a^{\dag}$ and $a$ by $$\begin{aligned}
\left\{\begin{array}{c}q_k=u_ka_k-v_k\tilde{a}_k^{\dag}\\
q_k^{\dag}=u_ka_k^{\dag}-v_k
\tilde{a}_k\end{array}\right .,\nonumber\\
\label{eqn:quasiparticles2}\\
\left\{\begin{array}{c}\tilde{q}_k=u_k\tilde{a}_k+v_ka_k^{\dag}\\
\tilde{q}_k^{\dag}
=u_k\tilde{a}_k^{\dag}+v_ka_k\end{array}\right ..\nonumber\\\nonumber\end{aligned}$$ Here $$\tilde{q}_k~\equiv~\tilde{q}_{l_kj_km_k}~=~(-)^{l_k+j_k+m_k}q_{l_kj_k-m_k}~=~
(-)^{l_k+j_k+m_k}q_{-k}~. \label{eqn:qpproperties1}$$ From the anti–commutation rules for $a^{\dag}$ and $a$, we derive the following anti–commutation rules for the quasiparticle operators in Eq. (\[eqn:quasiparticles2\]) $$\begin{aligned}
\begin{array}{c}
\label{eqn:qpproperties2}
\{q_k,q_l\}=\{q^{\dag}_k,q^{\dag}_l\}=\{\tilde{q}_k,
\tilde{q}_l\}=\{\tilde{q}^{\dag}_k,
\tilde{q}^{\dag}_l\}=\{\tilde{q}_k,q_l\}=\{\tilde{q}^{\dag}_k,q^{\dag}_l\}=0\\
\{q_k,q^{\dag}_l\}=\{\tilde{q}_k,\tilde{q}^{\dag}_l\}=\delta_{kl}\\
\{\tilde{q}_k,q^{\dag}_l\}=(-)^{l_k+j_k+m_k}\delta_{-kl}
\end{array}\\\nonumber\end{aligned}$$ Using definition (\[eqn:quasiparticles2\]) and properties (\[eqn:qpproperties1\]) and (\[eqn:qpproperties2\]), one can write a one–body operator in second quantization as $$T=\sum_{kl}T_{kl}a^{\dag}_ka_l=T_0+T_{11}+T_{20}+T_{02}~,
\label{eqn:qpoperator1}$$ where $$\begin{aligned}
T_0&=&\sum_kv^2_kT_{kk}~, \\
T_{11}&=&\sum_{kl}T_{kl}(u_ku_lq^{\dag}_kq_l-v_kv_l
\tilde{q}^{\dag}_l\tilde{q}_k)~,\\
T_{20}&=&\sum_{kl}T_{kl}u_kv_lq^{\dag}_k\tilde{q}^{\dag}_l~,
\label{eqn:t20}\\
T_{02}&=&\sum_{kl}T_{kl}u_lv_k\tilde{q}_kq_l~.\\\nonumber
\label{eqn:qpoperator3}\end{aligned}$$ Here, the subscripts indicate the number of quasiparticle creation and annihilation operators involved. In the same way, a two–body operator takes the form $$\begin{aligned}
V&=&-\frac{1}{4}\sum_{klij}(V_{klij}-V_{klji})a^{\dag}_ka^{\dag}_la_ia_j=-
\frac{1}{4}\sum_{klij}
\overline{V}_{klij}a^{\dag}_ka^{\dag}_la_ia_j=\nonumber\\
&=&V_0+V_{11}+V_{20}+V_{02}+V_{22}+V_{31}+V_{13}+V_{40}+V_{04}~,
\label{eqn:qpoperator4}\end{aligned}$$ where $$\begin{aligned}
V_0&=&\frac{1}{4}\sum_{kl}\left(u_kv_ku_lv_lP_kP_l
\overline{V}_{k-kl-l}+2v_k^2v_l^2
\overline{V}_{klkl}\right)~, \\
V_{11}&=&\frac{1}{2}\sum_{kli}u_kv_lu_iv_iP_i
\overline{V}_{kli-i}\left(q^{\dag}_k
\tilde{q}_l+\tilde{q}^{\dag}_lq_k\right)\nonumber\\
&&+\mbox{}\sum_{kli}v_k^2\overline{V}_{klki}
\left(u_lu_iq^{\dag}_lq_i-v_lv_i\tilde{q}^{\dag}_i
\tilde{q}_l\right)~,\\
V_{20}&=&\frac{1}{4}\sum_{kli}u_iv_iP_i
\overline{V}_{kli-i}\left(u_ku_lq^{\dag}_kq^{\dag}_l+v_kv_l
\tilde{q}^{\dag}_l\tilde{q}^{\dag}_k\right)\nonumber\\
&&+\mbox{}\sum_{kli}u_kv_l^2v_i
\overline{V}_{klil}q^{\dag}_k\tilde{q}^{\dag}_i,\\
V_{22}&=&\frac{1}{4}\sum_{klij}\overline{V}_{klij}
\left(u_ku_lu_iu_jq^{\dag}_kq^{\dag}_lq_jq_i+v_kv_lv_iv_j
\tilde{q}^{\dag}_j\tilde{q}^{\dag}_i
\tilde{q}_k\tilde{q}_l\right)\nonumber\\
&&+\mbox{}\sum_{klij}u_kv_lu_jv_i\overline{V}_{klij}q^{\dag}_k
\tilde{q}^{\dag}_i
\tilde{q}_lq_j~,\\
V_{31}&=&\frac{1}{2}\sum_{klij}u_kv_l\overline{V}_{klij}
\left(u_iu_jq^{\dag}_iq^{\dag}_j
\tilde{q}^{\dag}_lq_k+v_iv_jq^{\dag}_k\tilde{q}^{\dag}_j\tilde{q}^{\dag}_i
\tilde{q}_l\right)~,\\
V_{40}&=&\frac{1}{4}\sum_{klij}u_ku_lv_iv_j
\overline{V}_{klij}q^{\dag}_kq^{\dag}_l
\tilde{q}^{\dag}_j\tilde{q}^{\dag}_i~,
\label{eqn:qpoperator2}\end{aligned}$$ with $P_k=(-)^{l_k+j_k+m_k}$, $V_{ba}=(V_{ab})^\dagger$ and $\overline{V}_{abcd}=V_{abcd}-V_{abdc}$.
The matrix elements of quasiparticle operators are related to those of the usual one– and two–body operators. We show how this works for the operator $T_{20}$; the generalization to other operators follows automatically. $T_{20}$ adds two quasiparticles to any state. From $|ij\rangle=q^{\dag}_iq^{\dag}_j|-\rangle$ and Eq. (\[eqn:t20\]), we write $$\langle
ij|T_{20}|-\rangle~=~\sum_{kl}T_{kl}u_kv_l\langle-|q_jq_iq^{\dag}_k
\tilde{q}^{\dag}_l|-\rangle~.
\label{eqn:t20a}$$ Since $$\langle-|q_jq_iq^{\dag}_k\tilde{q}^{\dag}_l|-\rangle~=~P_l
[-\delta_{jk}\delta_{-li}+ \delta_{-lj}\delta_{ik}]~,
\label{eqn:t20b}$$ Eq. (\[eqn:t20a\]) becomes $$\langle ij|T_{20}|-\rangle~=~-P_i u_jv_i\langle j|T|-i\rangle+P_j
u_iv_j \langle i|T|-j\rangle~. \label{eqn:t20ac}$$
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[^1]: Permanent or static, rather than induced.
[^2]: The BCS transformation makes $V_{02}$ and $V_{20}$ zero.
[^3]: These rules are similar to the ordinary Brandow/Goldstone–diagram rules but since fermion lines represent quasiparticles, their number need not be conserved at each interaction. In addition, we have only upward going lines because there are no quasiholes.
|
---
author:
- 'C. Guidorzi, C. Clemens, S. Kobayashi, J. Granot, A. Melandri, P. D’Avanzo, N. P. M. Kuin, A. Klotz, J. P. U. Fynbo, S. Covino, J. Greiner, D. Malesani, J. Mao, C. G. Mundell, I. A. Steele, P. Jakobsson, R. Margutti, D. Bersier, S. Campana, G. Chincarini, V. D’Elia, D. Fugazza, F. Genet, A. Gomboc, T. Krühler, A. Küpcü Yoldaş, A. Moretti, C. J. Mottram, P. T. O’Brien, R. J. Smith, G. Szokoly, G. Tagliaferri, N. R. Tanvir, N. Gehrels'
nocite:
- '[@DellaValle06; @Fynbo06; @Galyam06]'
- '[@Zhang07_rev]'
- '[@Donaghy06; @Granot05_2]'
- '[@Barthelmy05]'
- '[@Ghisellini09]'
- '[@Burrows05]'
- '[@Roming05]'
- '[@Mao08a]'
- '[@Mao08b]'
- '[@Bloom08]'
- '[@Klotz08c]'
- '[@Zerbi01]'
- '[@Greiner08]'
- '[@Norris05]'
- '[@Meszaros06]'
- '[@Starling08]'
- '[@Lazzati02; @Guidorzi05]'
- '[@KumarPiran00; @Granot03; @Johannesson06]'
- '[@Zhang03; @Kumar03; @Gomboc08b]'
title: 'Rise and fall of the X-ray flash 080330: an off-axis jet?'
---
[X-ray flashes (XRFs) are a class of gamma-ray bursts (GRBs) with the peak energy of the time-integrated $\nu\,F_\nu$ spectrum, $E_{\rm p}$, typically below 30 keV, whereas classical GRBs have $E_{\rm p}$ of a few hundreds keV. Apart from $E_{\rm p}$ and the systematically lower luminosity, the properties of XRFs, such as the duration or the spectral indices, are typical of the classical GRBs. Yet, the nature of XRFs and the differences from that of GRBs are not understood. In addition, there is no consensus on the interpretation of the shallow decay phase observed in most X-ray afterglows of both XRFs and GRBs.]{} [We examine in detail the case of XRF 080330 discovered by Swift at the redshift of $1.51$. This burst is representative of the XRF class and exhibits an X-ray shallow decay. The rich and broadband (from NIR to UV) photometric data set we collected across this phase makes it an ideal candidate to test the off-axis jet interpretation proposed to explain both the softness of XRFs and the shallow decay phase.]{} [We present prompt $\gamma$-ray, early and late NIR/visible/UV and X-ray observations of the XRF 080330. We derive a spectral energy distribution from NIR to X-ray bands across the shallow/plateau phase and we describe the temporal evolution of the multi-wavelength afterglow within the context of the standard afterglow model.]{} [The multi-wavelength evolution of the afterglow is achromatic from $\sim$$10^2$ s out to $\sim$$8\times10^{4}$ s. The energy spectrum from NIR to X-ray is nicely fitted with a simple power-law, $F_\nu\propto\nu^{-\beta_{\rm ox}}$, with $\beta_{\rm ox}=0.79\pm0.01$ and negligible rest-frame dust extinction. The light curve can be modelled either by a piecewise power-law or by the combination of a smoothly broken power law with an initial rise up to $\sim$600 s, a plateau lasting up to $\sim$2 ks, followed by a gradual steepening to a power-law decay index of $\sim$$2$ out to 82 ks. At this point, there appears a bump modelled with a second component, while the corresponding optical energy spectrum, $F_\nu\propto\nu^{-\beta_{\rm o}}$, reddens by $\Delta\beta_{\rm o}=0.26\pm0.06$.]{} [A single-component jet viewed off-axis explains the light curve of XRF 080330, the late time reddening as due to the reverse shock of an energy injection episode, and its being an XRF. Other possibilities, such as the optical rise marking the pre-deceleration of the fireball within a wind environment, cannot be definitely excluded, but seem somewhat contrived. We rule out the dust decreasing column density swept up by the fireball as the explanation of the rise of the afterglow.]{}
Introduction {#sec:intro}
============
Time-integrated photon spectra of long gamma-ray bursts (GRBs) can be adequately fitted with a smoothly broken power law [@Band93], whose low-energy and high-energy indices, $\alpha_{\rm B}$ and $\beta_{\rm B}$, have median values of $-1$ and $-2.3$, respectively [@Preece00; @Kaneko06]. The corresponding $\nu$$F_\nu$ spectrum peaks at $E_{\rm p}$, the so-called peak energy, whose rest frame value is found to correlate with other relevant observed intrinsic properties, such as the isotropic-equivalent radiated $\gamma$-ray energy, $E_{\rm iso}$ [@Amati02], or its collimation-corrected value, $E_{\gamma}$ [@Ghirlanda04]. In the BATSE catalogue, the $E_{\rm p}$ distribution clusters around $300$ keV with a $\sim$$100$ keV width [@Kaneko06].
When observations of GRBs in softer energy bands than BATSE became available thanks to BeppoSAX and HETE-2, a new class of soft GRBs with $E_{\rm p}$$\lesssim30$ keV, so named X-ray flashes (XRFs), was soon discovered [@Heise01; @Barraud03]. GRBs with intermediate softness, called X-ray rich (XRR) bursts, were also observed, with $30$ keV $\la$ $E_{\rm p}$ $\la$$100$ keV [@Sakamoto05]. These soft GRBs share the same temporal and spectral properties, aside from the systematically lower $E_{\rm p}$, with the classical GRBs both for the prompt [@Frontera00; @Barraud03; @Amati04] and, partially, the afterglow emission [@Sakamoto05; @Dalessio06; @Mangano07]. Moreover, they were found to obey the $E_{\rm p}$–$E_{\rm iso}$ correlation [@Amati07] discovered for classical GRBs, extending it all the way down to $E_{\rm p}$ values of a few keV and forming a continuum [@Sakamoto05]. Like classical GRBs, also XRFs have been found to be associated with SNe [@Campana06; @Pian06] and therefore connected with the collapse of massive stars. The comparison also holds for the cases in which apparently no associated SN was found both for classical GRBs (e.g. Della Valle et al. 2006, Fynbo et al. 2006, Gal-Yam et al. 2006) and for XRFs [@Levan05].
A number of different models have been proposed in the literature to explain the nature of XRRs and XRFs (e.g., see the review of Zhang 2007): i) standard GRBs viewed well off the axis of the jet, thus explaining the softness as due to a larger viewing angle and a lower Doppler factor [@Yamazaki02; @Granot02; @Granot05_2]; ii) two coaxial jets with different opening angles (wide and narrow), $\theta_{\rm w}>\theta_{\rm n}$, and viewed at an angle $\theta_{\rm v}$, $\theta_{\rm n}<\theta_{\rm v}<\theta_{\rm w}$ [@Peng05]; iii) the “dirty fireball” model characterised by a small value of the bulk Lorentz factor due to a relatively high baryon loading of the fireball [@Dermer00]; iv) distribution of high Lorentz factors with low contrast of the colliding shells [@Mochkovitch04]. In the off-axis interpretation, a number of different models of the structure and opening angle of the jet have been proposed (e.g. Granot et al. 2005; Donaghy 2006).
The advent of Swift [@Gehrels04] has made it possible to collect a large sample of early X-ray afterglow light curves of GRBs. Concerning the XRRs and XRFs, the 15–150 keV energy band of the Swift Burst Alert Telescope (BAT; Barthelmy et al. 2005) and its relatively large effective area still allow to detect them, although those with $E_{\rm p}$ of a few keV are disfavoured with respect to BeppoSAX and HETE-2 instruments [@Sakamoto08]. Thanks to Swift it is possible to study the early X-ray afterglow properties of these soft events. Like hard GRBs, also XRFs occasionally exhibit X-ray flares [@Romano06]. @Sakamoto08 analysed a sample of XRFs, XRRs and classical GRBs detected with Swift and found some evidence for an average X-ray afterglow luminosity of XRFs being roughly half that of classical GRBs and some differences between the average X-ray afterglow light curves.
An unexpected discovery of Swift is the shallow decay phase experienced by most of X-ray afterglows between a few $10^2$ up to $10^3$–$10^4$ s after the trigger time [@Tagliaferri05; @Nousek06; @Zhang06]. Several interpretations have been put forward (e.g. see Ghisellini et al. 2009 for a brief review). Among them, some invoke continuous energy injection to the fireball shock front through refreshed shocks [@Nousek06; @Zhang06], depending on the progenitor period of activity: a long- or short-lived powering mechanism, either in the form of a prolonged, continuous energy release ($L(t)\propto$$t^{-q}$), or via discrete shells whose $\Gamma$ distribution is a steep power-law. For instance, in the cases of GRB 050801 and GRB 070110 a newly born millisecond magnetar was suggested to power the flat decay observed in the optical and X-ray bands [@Depasquale07; @Troja07]. Alternatively, geometrical models interpret the shallow decay as the delayed onset of the afterglow observed from viewing angles outside the edge of a jet [@Granot02; @Salmonson03; @Granot05; @Eichler06]. Other models invoke two-component jets viewed off the axis of the narrow component, also invoked to explain the late time observations of GRB 030329 [@Berger03]. In particular, this model would explain the initial flat decay observed in XRF 030723, dominated by the wide component, followed by a late rebrightening peaking at $\sim$16 days and interpreted as due to the deceleration and lateral expansion of the narrow component [@Huang04; @Butler05], although alternative explanations for this in terms of a SN have also been proposed [@Fynbo04; @Tominaga04].
Other models explain the shallow decay as due to a temporal evolution of the fireball micro-physical parameters [@Ioka06; @Granot06]; scattering by dust located in the circumburst medium [@Shao07]; “late prompt” activity of the inner engine, keeping up a prolonged emission of progressively lower power and Lorentz factor shells, which radiate at the same distance as for the prompt emission [@Ghisellini07; @Ghisellini09]; a dominating reverse shock in the X-ray band propagating through late shells with small Lorentz factors [@Genet07; @Uhm07]. @Yamazaki09 suggests that the plateau and the following standard decay phases are an artifact of the choice of $t_0$, provided that the engine activity begins before the trigger time by $\sim$$10^3$–$10^4$ s.
XRF 080330 was promptly discovered by the Swift-BAT and automatically pointed with the X-Ray Telescope (XRT; Burrows et al. 2005) and the Ultraviolet/Optical Telescope (UVOT; Roming et al. 2005) as shown in Figure \[f:multi\_lc\]. In this work we present a detailed analysis of the Swift data, from the prompt $\gamma$-ray emission to the X-ray and optical afterglow and combine it with the large multi-filter data set collected from the ground, encompassing a broad band, from NIR to UV wavelengths, and spanning from one minute out to $\sim$3 days post burst. The main properties exhibited by XRF 080330 are the rise of the optical afterglow up to $\sim$300 s, followed by a shallow decay also present in the X-ray, after which it gradually steepens, and either a possible late time ($\sim10^5$ s) brightening (Fig. \[f:fluxall\_beulore\]) or a sharp break (Fig. \[f:fluxall\_beu3\]). The richness of the multi-wavelength data collected throughout the rise-flat top-steep decay allows us to constrain the broadband energy spectrum of the shallow decay phase as well as its spectral evolution. Moreover, it is possible to constrain the optical flux extinction due to dust along the line of sight and, in particular, near the progenitor. This GRB is a good benchmark for the proposed models of XRFs sources and of their link with the classical GRBs through the common properties, such as the flat decay phase.
The paper is organised as follows: Sects. \[sec:obs\] and \[sec:an\] report the observations, data reduction and analysis, respectively. We report our multi-wavelength combined analysis in Sect. \[sec:multi\]. In Sect. \[sec:disc\] we discuss our results in the light of the models proposed in the literature and Sect. \[sec:conc\] reports our conclusions.
Throughout the paper, times are given relative to the BAT trigger time. The convention $F(\nu,t)\propto\nu^{-\beta}\,t^{-\alpha}$ is followed, where the spectral index $\beta$ is related to the photon index $\Gamma=\beta+1$. We adopted the standard cosmology: $H_0=70$kms$^{-1}$Mpc$^{-1}$, $\Omega_\Lambda=0.7$, $\Omega_{\rm M}=0.3$.
All the quoted errors are given at 90% confidence level for one interesting parameter ($\Delta\chi^2=2.706$), unless stated otherwise.
Observations {#sec:obs}
============
XRF 080330 triggered the [Swift]{}-BAT on 2008 March 30 at 03:41:16 UT. The $\gamma$-ray prompt emission in the 15–150 keV energy band consisted of a multiple–peak structure with a duration of about 60 s [@Mao08a]. An uncatalogued, bright and fading X-ray source was promptly identified by XRT. From the initial 100-s finding chart taken with the UVOT telescope in the White filter from 82 s the optical counterpart was initially localised at RA $= 11^{\rm h}$ $17^{\rm m}$ $04\fs51$, Dec. $=+30^{\circ}$ $37^{\prime}$ $22\farcs1$ (J2000), with an error radius of $1\farcs0$ (1$\sigma$; Mao et al. 2008a). During the observations, the Swift star trackers failed to maintain a proper lock resulting in a drift which affected the observations and accuracy of early reports. We finally refined the position from the UVOT field match to the USNO–B1 catalogue: RA $= 11^{\rm h}$ $17^{\rm m}$ $04\fs52$, Dec. $=+30^{\circ}$ $37^{\prime}$ $23\farcs5$ (J2000), with an error radius of $0\farcs3$ (1$\sigma$; Mao et al. 2008b), consistent with the position derived from ground telescopes (e.g., PAIRITEL, Bloom & Starr 2008).
The Télescopes à Action Rapide pour les Objets Transitoires (TAROT; Klotz et al. 2008c) began observing at $20.4$ s ($4.5$ s after the notice) and discovered independently the optical counterpart during the rise with $R$$\sim$$16.8$ at 300 s [@Klotz08a]. TAROT went on observing until the dawn at $1.4$ ks [@Klotz08b].
The Rapid Eye Mount[^1] (REM; Zerbi et al. 2001) telescope reacted promptly and began observing at 55 s and detected the optical afterglow in $R$ band [@Davanzo08]. The optical counterpart was promptly detected also by other robotic telescopes, such as ROTSE–IIIb [@Schaefer08; @Yuan08], PROMPT [@Schubel08] and RAPTOR; the latter in particular observed a $\sim$$10$-s long optical flash of $R=17.46\pm0.22$ at 60 s contemporaneous with the last $\gamma$-ray pulse [@Wren08].
The Liverpool Telescope (LT) began observing at 181 s. The optical afterglow was automatically identified by the LT-TRAP GRB pipeline [@Guidorzi06] with $r'$$\sim$$17.3$ [@Gomboc08a], thus triggering the multi-colour imaging observing mode in the $g'r'i'$ filters which lasted up to the dawn at $4.9$ ks.
The Faulkes Telescope North (FTN) observations of XRF 080330 were carried out from $8.4$ to $9.1$ hr and again from $31.8$ to $33.9$ hr with deep $r'$ and $i'$ filter exposures.
The Gamma-Ray Burst Optical and Near-Infrared Detector (GROND; Greiner et al. 2008) started simultaneous observations in $g'r'i'z'JHK$ filters of the field of GRB 080330 at $3.1$ minutes and detected the afterglow with $J=15.92\pm0.04$ and $H=15.46\pm0.11$ from the first 240-s of effective exposure [@Clemens08]. A spectrum of XRF 080330 was acquired at $46$ minutes with the Nordic Optical Telescope (NOT). The identification of absorption features allowed to measure the redshift, which turned out to be $z=1.51$ [@Malesani08]. This was soon confirmed by the spectra taken with the Hobby-Eberly Telescope [@Cucchiara08]. The Galactic reddening along the line of sight to the GRB is $E_{B-V}=0.017$ [@Schlegel98]. The corresponding extinction in each filter was estimated through the NASA/IPAC Extragalactic Database extinction calculator[^2]: $A_{UVW1}=0.120$, $A_U=0.090$, $A_B=0.071$, $A_g=0.064$, $A_V=0.055$, $A_r=0.047$, $A_R=0.044$, $A_I=0.032$, $A_i=0.035$, $A_z=0.022$, $A_J=0.015$, $A_H=0.010$, $A_K=0.006$.
Data reduction and analysis {#sec:an}
===========================
Gamma–ray data {#sec:gamma}
--------------
The BAT data were processed with the [heasoft]{} package (v.6.4) adopting the ground-refined coordinates provided by the BAT team [@Markwardt08]. The BAT detector quality map was obtained by processing the nearest-in-time enable/disable map of the detectors.
The top panel of Figure \[f:bat\_lc\] shows the 15–150 keV mask-weighted light curve of XRF 080330 as recorded by BAT, expressed as counts per second per fully illuminated detector for an equivalent on-axis source. The solid line displayed in Fig. \[f:bat\_lc\] corresponds to the result of fitting the profile from $-1.2$ to $100$ s with a combination of four pulses [@Markwardt08] as modelled by Norris et al. (2005; hereafter N05 model). Table \[t:BAT\_N05\] reports the corresponding derived parameters: $t_{\rm p}$ (peak time), $A$ (15–150 keV peak flux), $\tau_{\rm r}$ (rise time), $\tau_{\rm d}$ (decay time), $w$ (pulse width), $k$ (pulse asymmetry) and the model fluence in the 15–150 keV band. The goodness of the fit is $\chi^2/{\rm dof}=375/379$. Parameter uncertainties were derived by propagation starting from the best-fit parameters and taking into account their covariance. We tried to apply the same analysis to the light curves of the resolved energy channels to investigate temporal lags and, more generally, the dependence of the parameters on energy; however, because of the faintness and softness of the signal, we could not constrain the parameters in a useful way.
[lccccccc]{} Pulse & $t_{\rm p}$ & $A$ & $\tau_{\rm r}$ & $\tau_{\rm d}$ & $w$ & $k$ & Fluence\
& (s) & ($10^{-8}$ erg cm$^{-2}$ s$^{-1}$) & (s) & (s) & (s) & & ($10^{-8}$ erg cm$^{-2}$)\
1 & $0.5\pm0.2$ & $4.6\pm0.6$ & $0.79\pm0.18$ & $1.30\pm0.26$ & $2.09\pm0.29$ & $0.24\pm0.16$ & $8.6\pm1.0$\
2 & $4.2\pm0.6$ & $2.0\pm0.3$ & $1.04\pm0.64$ & $4.71\pm1.30$ & $5.75\pm1.44$ & $0.64\pm0.20$ & $10.8\pm1.9$\
3 & $7.5\pm0.2$ & $2.5\pm0.7$ & $0.32\pm0.19$ & $0.84\pm0.43$ & $1.16\pm0.46$ & $0.45\pm0.32$ & $2.7\pm0.8$\
4 & $56.2\pm1.2$ & $1.3\pm0.2$ & $2.52\pm1.26$ & $10.5\pm2.3$ & $13.0\pm2.4$ & $0.61\pm0.18$ & $15.5\pm2.3$\
In addition, the energy spectra in the 15–150 keV band were extracted using the tool [batbinevt]{}. We applied all the required corrections: we updated them through [batupdatephakw]{} and generated the detector response matrices using [batdrmgen]{}. Then we used [batphasyserr]{} in order to account for the BAT systematics as a function of energy. Finally we grouped the energy channels of the spectra by imposing a 3$\sigma$ (or 2-$\sigma$ when the S/N was too low) threshold on each grouped channel. We fitted the resulting photon spectra, $\Phi(E)$ (ph cm$^{-2}$s$^{-1}$keV$^{-1}$), with a power law with pegged normalisation ([pegpwrlw]{} model under [xspec]{} v.11.3.2). We extracted several spectra in different time intervals: over $T_{90}$, total, spanning the bunch of the first three pulses, the fourth pulse alone and that around the peak, determined on a minimum significance criterion. The results are reported in Table \[t:BAT\_XRT\_spec\]. The time-averaged spectral index is $\beta_\gamma=1.65\pm0.51$ with a total fluence of $S(15-150~{\rm keV})=(3.6\pm0.8)\times10^{-7}$ erg cm$^{-2}$ and a $0.448$-s peak photon flux of $(1.0\pm0.2)$ ph cm$^{-2}$ s$^{-1}$, in agreement with previous results [@Markwardt08]. The bottom panel of Fig. \[f:bat\_lc\] shows marginal evidence for a soft–to–hard evolution: $\beta_\gamma$ passes from $2.0\pm0.5$ (first three pulses) to $1.4\pm0.5$ (fourth pulse).
Following @Sakamoto08, a GRB is classified as an XRR (XRF) depending on whether the fluence ratio $S(25-50~{\rm keV})/S(50-100~{\rm keV})$ is lower (greater) than $1.32$. The fluence ratio of XRF 080330, $1.5_{-0.3}^{+0.7}$, places it among the XRFs, although still compatible with being an XRR burst. Although from BAT data alone we could not measure the peak energy of the time-integrated $\nu\,F_\nu$ spectrum, we tried to fit with a smoothed broken power law model [@Band93] by fixing the low-energy index $\alpha_{\rm B}$ to -1 [@Kaneko06], given that $\Gamma=\beta_{\gamma}+1>2$ and is very likely dominated by the high-energy index $\beta_{\rm B}$. This way we derived the following constraint: $E_{\rm p}<35$ keV, in agreement with the upper limit to its rest-frame (intrinsic) value, $E_{\rm p,i}=E_{\rm p}\,(1+z)<88$ keV, obtained by @Rossi08.
Recently, @Sakamoto09 calibrated a method aimed at estimating $E_{\rm p}$ from the $\Gamma_\gamma$ as measured with BAT, provided that $1.3$$<$$\Gamma$$<$$2.3$. In the case of XRF 080330, the confidence interval on $\Gamma$, $2.65\pm0.51$, marginally overlaps with the allowed range; however, since $E_{\rm p}$ is anti-correlated with $\Gamma$ and the lower limit on $\Gamma$ lies within the usable range, we can derive an upper limit to $E_{\rm p}$ from the relation of @Sakamoto09, which turns out to be $30$ keV, in agreement with our previous value. These results are fully consistent with a previous preliminary analysis [@Markwardt08].
We constrained $E_{\rm iso}$ in the rest-frame 1–$10^4$ keV band using the upper limit on $E_{\rm p}$ of $35$ keV. Following the prescriptions by @Amati02 and @Ghirlanda04, we found $E_{\rm iso}<2.2\times10^{52}$ ergs. Combined with $E_{\rm p,i}<88$ keV, this places XRF 080330 in the $E_{\rm p,i}$–$E_{\rm iso}$ space consistently with the Amati relation [@Amati02; @Amati06].
![[*Top panel*]{}: 15–150 keV BAT mask-weighted light curve (binning time of 0.512 s). The thick solid line shows the result of fitting the profile with four pulses modelled with Norris profiles [@Norris05]. [*Bottom panel*]{}: spectral index $\beta_\gamma$ as a function of time. []{data-label="f:bat_lc"}](080330_f1.eps){width="8.5cm"}
X–ray data {#sec:X}
----------
The XRT data were processed using the [heasoft]{} package (v.6.4). We ran the task [xrtpipeline]{} (v.0.11.6) applying calibration and standard filtering and screening criteria. Data from 77 to 134 s were acquired in Windowed Timing (WT) mode and the following in Photon Counting (PC) mode due to the faintness of the source. Events with grades 0–2 and 0–12 were selected for the two modes, respectively. XRT observations went on up to $5.9\times10^5$ s, with a total net exposure time of $30.6$ ks. The XRT analysis was performed in the 0.3–10 keV energy band.
Source photons were extracted from WT mode data in a rectangular region 40 pixels along the image strip (20 pixel wide) centred on the source, whereas the background photons were extracted from an equally-sized region with no sources.
Firstly, we extracted the first orbit PC data from 136 to 331 s, where the point spread function (PSF) of the source looked unaffected by the spacecraft drifting, and extracted the following refined position: RA$= 11^{\rm h}$ $17^{\rm m}$ $04\fs68$, Dec. $=+30^{\circ}$ $37^{\prime}$ $24\farcs8$ (J2000), with an error radius of $4.0$ arcsec [@MaoGuidorzi08]. We corrected these data for pile-up by extracting source photons from an annular region centred on the above position and with inner and outer radii of 4 and 30 pixels (1 pixel$\mbox{}=2\farcs36$), respectively. The background was estimated from a three-circle region with a total area of $30.3\times10^3$ pixel$^{2}$ away from any source present in the field. Finally, we re-extracted the source photons over the entire first orbit within a larger circular region centred on the same position and with a radius of 40 pixels, to compensate for the drifting. The light curve of the full first orbit data (PC mode) was then corrected so as to match the previous one correctly produced in the 136–331 s sub-interval.
The resulting 0.3–10 keV light curve is shown in Fig. \[f:multi\_lc\] (black empty triangles). It was binned so as to achieve a minimum signal to noise ratio (SNR) of 3. The data taken in following orbits were not enough to provide a significant detection and only a 3$\sigma$ upper limit was obtained.
The X-ray curve can be fitted with a broken power law, with the following parameters: $\alpha_{\rm x,1}=4.8\pm0.4$, $t_{\rm b}=163_{-10}^{+9}$ s, $\alpha_{\rm x,2}=0.26\pm0.10$ ($\chi^2/{\rm dof}=66/70$). The last upper limit clearly requires a further break. We set a lower limit on $\alpha_{\rm x,3}$ by connecting the end of first orbit data with the late upper limit under the assumption that the second break occurred at the beginning of the data gap. This turned into $\alpha_{\rm x,3}>1.3$ ($\ge$3$\sigma$ confidence). The later the second break time, the steeper the final decay.
We extracted the 0.3–10 keV spectrum in two different time intervals: i) “XRT-WT” interval, from $77$ to $134$ s (WT mode), corresponding to the initial steep decay; ii) “plateau” interval, from $423$ to $1507$ s (PC mode) corresponding to the following flat decay (or “plateau”) phase. Source and background spectra were extracted from the same regions as the ones used for the light curve for the corresponding time intervals and modes. The ancillary response files were generated using the task [xrtmkarf]{}. Spectral channels were grouped so as to have at least 20 counts per bin. Spectral fitting was performed with [xspec]{} (v. 11.3.2).
We modelled both spectra with a photoelectrically absorbed power law (model [wabs$\cdot$zwabs$\cdot$pow]{}), adopting the photoelectric cross section by @Morrison83. The first column density was frozen to the weighted average Galactic value along the line of sight to the GRB, $N_{\rm H}^{\rm (Gal)}=1.23\times10^{20}$ cm$^{-2}$ [@Kalberla05], while the second rest-frame column density, $N_{{\rm H},z}$, was left free to vary. While during the steep decay we found no evidence for significant rest-frame absorption, with a 90% confidence limit of $N_{{\rm H},z}<1.4\times10^{21}$ cm$^{-2}$, in the plateau spectrum we found only marginal evidence for it, $N_{{\rm H},z}=1.6_{-1.5}^{+1.8}\times10^{21}$ cm$^{-2}$. The spectral index, $\beta_{\rm x}$, varies from $1.06_{-0.09}^{+0.10}$ to $0.80_{-0.15}^{+0.16}$: the significance of this change is $\sim2.3$ $\sigma$. The best-fit parameters are reported in Table \[t:BAT\_XRT\_spec\].
Near–UV/Visible UVOT data {#sec:UVOT}
-------------------------
The Swift UVOT instrument started observing on 2008 March 30 at 03:42:19 UT, 63 s after the BAT trigger, with a $9.37$-s settling exposure. Since the detectors are powered up during this exposure, the effective exposure time may be less than reported. We checked the brightness in this exposure with later exposures, to confirm that no correction was needed. The first $99.7$-s finding chart exposure started at 03:42:39 UT in the white filter in event mode followed by a $399.8$-s exposure in the $V$ filter, also in event mode. Due to the loss of lock by the spacecraft star trackers, the attitude information was incorrect. In order to process the data, [xselect]{} was used to extract images for short time intervals. The length of the interval was chosen short enough that the drift of the spacecraft was mostly within $7\arcsec$, and at most $14\arcsec$, but long enough to get a reasonably accurate measurement. A source region was placed over the position of the source, making checks for consistency with the position of nearby stars, and the magnitudes were determined using the ftool [uvotevtlc]{}. In most cases, an aperture with a radius of $5\arcsec$ was used, and three measurements used a slightly larger aperture. No aperture correction was made, since the source shape in those cases was very elongated. The measured magnitudes were converted back to the original count rates, using the UVOT calibration [@Poole08]. These were subsequently converted to fluxes using the method of @Poole08, but for an incident power-law spectrum with $\beta$$=$$0.8$ and for a redshift $z$$=$$1.51$.
NIR/Visible ground-based data {#sec:opt}
-----------------------------
Robotically triggered observations with the LT began at 181 s leading to the automatic identification by the GRB pipeline LT-TRAP [@Guidorzi06] of the optical afterglow at the position RA $= 11^{\rm h}$ $17^{\rm m}$ $04\fs48$, Dec. $=+30^{\circ}$ $37^{\prime}$ $23\farcs8$ (J2000; 1$\sigma$ error radius of $0.2$ arcsec). This is consistent within $1.6$ $\sigma$ with the refined UVOT position. The afterglow was seen during the end of the rise with $r'=17.3\pm0.1$ and subsequently decay [@Gomboc08a]. Following two initial sequences of $3\times10$ s each in the $r'$ filter during the detection mode (DM), the multi-colour imaging observing mode (MCIM) in the $g'r'i'$ filters was automatically selected. Observations carried on up to $4.9$ ks.
The FTN observations of XRF 080330 were carried out from $8.4$ to $9.1$ hr and again from $31.8$ to $33.9$ hr with deep $r'$ and $i'$ filter exposures as part of the [*RoboNet 1.0*]{} project[^3] [@Gomboc06].
Calibration was performed against five non-saturated field stars with preburst SDSS photometry [@Cool08], by adopting their PSF to adjust the zero point of the single images. Photometry was carried out using the Starlink GAIA software. Magnitudes were converted into flux densities (mJy) following @Fukugita96. Results are reported in Table \[tab:photom\].
Optical $R$-band observations of the afterglow of GRB 080330 were carried out with the REM telescope equipped with the ROSS optical spectrograph/imager on 2008 March 30, starting about 55 seconds after the burst [@Davanzo08]. We collected 38 images with typical exposures times of 30, 60 and 120 s, covering a time interval of about $0.5$ hours. Image reduction was carried out by following the standard procedures: subtraction of an averaged bias frame, division by a normalised flat frame. The astrometry was fitted using the USNOB1.0[^4] catalogue. We grouped our images into 18 bins in order to increase the signal-to-noise ratio (SNR) and performed aperture photometry with the SExtractor package [@Bertin96] for all the objects in the field. In order to minimise the systematics, we performed differential photometry with respect to a selection of local isolated and non-saturated standard stars. The calibration of NOT images taken with the $R$ filter was performed with respect to the converted magnitudes in the $R$-band of the selected set of stars used for the calibration of LT and GROND images in the SDSS passbands. We transformed the $r'$ and $i'$ magnitudes of the calibration stars [@Cool08] into $R$ and $I$ magnitude following the filter transformations of @Krisciunas98. Hereafter, the magnitudes shown are not corrected for Galactic extinction, whilst fluxes as well as all the best-fit models are. When the models are plotted together with magnitudes, the correction for Galactic extinction is removed from the models.
[lcrrccccc]{} Interval & Energy band & Start time & Stop time & $\beta$ & $N_{\rm H,z}$ & Mean flux & $A_{V,z}^{a}$ & $\chi^2$/dof\
& (keV) & (s) & (s) & & ($10^{21}$ cm$^{-2}$) & (erg cm$^{-2}$ s$^{-1}$) & &\
$T_{90}$ & 15–150 & $0.0$ & $67.0$ & $1.44\pm0.46$ & – & $(3.3\pm0.8)\times10^{-7}$ & – & $1.44/6$\
Total & 15–150 & $-2.0$ & $90.0$ & $1.65\pm0.51$ & – & $(3.6\pm0.8)\times10^{-7}$ & – & $5.6/7$\
Pulses 1–3 & 15–150 & $-2.0$ & $20.0$ & $2.0\pm0.5$ & – & $(2.2\pm0.5)\times10^{-7}$ & – & $1.82/7$\
Pulse 4 & 15–150 & $52.9$ & $90.0$ & $1.4\pm0.5$ & – & $(1.4\pm0.5)\times10^{-7}$ & – & $5.8/5$\
Peak & 15–150 & $0.384$ & $0.832$& $1.1\pm0.6$ & – & $(1.0\pm0.2)^{b}$ & – & $6.1/6$\
XRT-WT & 0.3–10 & $77$ & $134$ & $1.06_{-0.09}^{+0.10}$&$<1.4$& $(4.1\pm0.3)\times10^{-10}$ & – & $28.3/31$\
Plateau & 0.3–10 & $423$ & $1507$ & $0.80_{-0.15}^{+0.16}$ & $1.6_{-1.5}^{+1.8}$ & $(2.3\pm0.3)\times10^{-11}$ & – & $17.2/24$\
SED 2 & opt–X & $186.8$ & $269.4$ & $0.74\pm0.03$& $[2.7]$ & – & $< 0.04$ & $8.2/6$\
SED 3 & opt–X & $423$ & $1507$ & $0.79\pm0.01$ & $2.7\pm0.8$ & – & $< 0.02$ & $32/34$\
SED 4 & opt & $78117$ & $93620$ & $0.85\pm0.30$ & – & – & $0.10_{-0.06}^{+0.14}$ & $3.4/4$\
SED 2 & opt & $186.8$ & $269.4$ & $0.61\pm0.13$ & – & – & $[0]$ & $3.1/3$\
SED 3 & opt & $423$ & $1507$ & $0.74\pm0.05$ & – & – & $[0]$ & $7.9/8$\
SED 4 & opt & $78117$ & $93620$ & $1.05\pm0.06$ & – & – & $[0]$ & $4.8/5$\
$^{a}$ Rest-frame extinction obtained by modelling the SED with an SMC profile as parametrised by @Pei92.\
$^{b}$ Peak photon flux in units of ph cm$^{-2}$ s$^{-1}$.\
Spectroscopy {#sec:NOT_spec}
------------
Starting at $\approx$$46$ min we obtained a 1800 s spectrum with a low resolution grism and a 1.3 arcsec wide slit covering the spectral range from about 3500 to 9000 [Å]{} at a resolution of 14 [Å]{} with the NOT (Fig. \[f:NOT\_spec\]). The airmass was about 1.8 at the start of the observations. The spectrum was reduced using standard methods for bias subtraction, flat-fielding and wavelength calibration using an Helium-Neon arc spectrum. The rms of the residuals in the wavelength calibration were about 0.3 [Å]{}. The spectrum was flux-calibrated using an observation with the same setup of the spectrophotometric standard star HD93521. Table \[t:spectrum\] reports the identified lines.
{width="17cm"}
Multi-wavelength combined analysis {#sec:multi}
==================================
Panchromatic light curve {#sec:multi_lc}
------------------------
Figure \[f:multi\_lc\] displays the light curves of the prompt emission (15–150 keV) and of the 0.3–10 keV and NIR/visible/UV afterglow derived from our data sets plus some points taken from RAPTOR [@Wren08]. High-energy fluxes (magnitudes) are referred to the right-hand (left-hand) y-axis. First of all, we note that the peak time of the last $\gamma$-ray pulse (Table \[t:BAT\_N05\]) is contemporaneous with the optical flash detected by RAPTOR, reported at $58.9\pm2.5$ s [@Wren08].
The initial steep decay observed by XRT is a smooth continuation of the last $\gamma$-ray pulse and is thus the tail of the prompt GRB emission, and likely to correspond to its high-latitude emission. Most notably, during the X-ray steep decay the optical flux is seen to rise up to $\sim$$300$ s and finally a simultaneous plateau is reached at both energy bands, lasting up to $\sim$$1500$ s, when the X-ray observations stopped. This strongly suggests that the plateau is emission from a region which is physically distinct from that responsible for the prompt emission and its tail (the rapid decay phase).
As discussed in Sect. \[sec:SED\], the afterglow does not show evidence for spectral evolution throughout the observations, except for late epochs ($t\sim10^5$ s), when there is evidence for reddening. The achromatic nature of the afterglow light curve allows for a multi-wavelength simultaneous fit of twelve light curves, where only the normalisations are left free to vary independently from each other. We consider all of the available passbands: $K$, $H$, $J$, $z'$, $i'$, $r'$, $V$, $g'$, $B$, $U$, $UWV1$ and X-ray, respectively. The latter curve is fitted from 300 s onward, so as to exclude the initial steep decay. Hereafter we present two alternative combinations of models, both providing a reasonable description of the flux temporal evolution. In both cases we had to add a 2% systematics to all of the measured uncertainties to account for some residual variability with respect to the models, in order to have acceptable $\chi^2$ values and correspondingly acceptable parameters’ uncertainties.
### Multiple smoothly broken power law {#sec:beu3}
A possible description of the light curves is offered by a multiple broken power law (Fig. \[f:fluxall\_beu3\]). This has the advantage of a more straightforward interpretation in terms of the standard fireball evolution model due to synchrotron emission. We started from the parametrisation by @Beuermann99 and added two more breaks to finally provide a sufficiently detailed description. The fitting function is given by eq. (\[eq:beu3\]). $$\displaystyle F(t)\,= \,\frac{F_0}{\left[\left(t/t_{b1}\right)^{n\,\alpha_1} +
\left(t/t_{b1}\right)^{n\,\alpha_2} + \left(t/t_{b2}\right)^{n\,\alpha_3}
+ \left(t/t_{b3}\right)^{n\,\alpha_4} \right]^{1/n}}
\label{eq:beu3}$$ The free parameters are the normalisation constant (different for each curve), $F_0$, three break time constants, $t_{bi}$ ($i=1,2,3$), four power-law indices, $\alpha_1<\alpha_2<\alpha_3<\alpha_4$, the smoothness $n$. Apart from the normalisations, all the curves share the same parameters. Overall, the free parameters and the degrees of freedom (dof) total 20 and 184, respectively.
Equation (\[eq:beu3\]) looks like a piecewise power law only in the following regime: $t_{b1}\ll\,t_{b2}\ll\,t_{b3}$, where each individual term takes over at well separate epochs. The light curve of XRF 080330 fits in this case, as proven by the best-fit results (first line of Table \[t:panchro\_fit\]) and shown in Fig. \[f:fluxall\_beu3\]. The effective break times, $t_{b1,{\rm eff}}$, $t_{b2,{\rm eff}}$, $t_{b3,{\rm eff}}$, i.e. the times at which the model (\[eq:beu3\]) changes the power-law regime, are simply given by $t_{b1,{\rm eff}}=t_{b1}$, $t_{b2,{\rm eff}}=(t_{b1}^{\alpha_2}/t_{b2}^{\alpha_3})^{1/(\alpha_2-\alpha_3)}$, $t_{b3,{\rm eff}}=(t_{b2}^{\alpha_3}/t_{b3}^{\alpha_4})^{1/(\alpha_3-\alpha_4)}$.
The goodness of the fit in terms of $\chi^2/{\rm dof}$ is 212/184, corresponding to a non-rejectable P-value of $7.7$%. The normalisation constants for the different bands are the following ($\mu$Jy): $F_K=1077_{-59}^{+54}$, $F_H=942_{-50}^{+45}$, $F_J=769_{-44}^{+40}$, $F_z=643_{-30}^{+26}$, $F_i=554_{-25}^{+19}$, $F_r=464_{-21}^{+16}$, $F_V=418_{-33}^{+34}$, $F_g=362_{-16}^{+12}$, $F_B=362\pm68$, $F_U=277\pm45$, $F_{UVW1}=129\pm52$, while the X-ray normalisation, expressed in flux units in the 0.3–10 keV band instead of flux density, is $F_{\rm x}=(3.7\pm0.3)\times10^{-11}$ erg cm$^{-2}$ s$^{-1}$. The effective break times are found be $t_{b1,{\rm eff}}=317$ s, $t_{b2,{\rm eff}}=1850$ s and $t_{b3,{\rm eff}}=82.4$ ks, respectively.
The bottom panel of Fig. \[f:fluxall\_beu3\] shows the residuals of the $r'$ curve with respect to the model; the displayed uncertainties do not include the 2% systematics added by the fitting procedure. We note that between $6$ and $7\times10^3$ s the model overpredicts the flux by 2–4$\sigma$ with respect to the measured values, corresponding to a $\sim\,0.1$ magnitude difference. However, the later points seem to rule out a steeper decay than the modelled one. Alternatively, one might interpret this as suggestive of a steeper decay followed by a second component thus causing a late flux enhancement. This possibility motivated us to provide an alternative description, described in the next Sect. \[sec:beulore\].
### A two-component model: late-time brightening {#sec:beulore}
In Figure \[f:fluxall\_beulore\] we modelled the first part ($t<10^4$ s) with a simple smoothly broken power law with a single break time, $t_{b1}$, and two power-law indices, $\alpha_1$ ($\alpha_2$), taking over for at $t\ll t_{b1}$ ($t\gg t_{b1}$). In order to model the later data points, we had to add a further component. A Lorentzian proved successful in this respect, so that the complete model used is given by the following equation. $$\displaystyle F(t)\,= \,\frac{F_{0,r}}{\left[\left(t/t_{b1}\right)^{n\,\alpha_1} +
\left(t/t_{b1}\right)^{n\,\alpha_2}\right]^{1/n}}\, + \,
\frac{F_{\rm {\sc L},r}}{1 + \left[2\left(t-t_{\rm c}\right)/t_{\rm w}\right]^2}
\label{eq:beulore}$$ This was used to fit the $r'$ curve. The free parameters are the normalisation constant, $F_{0,r}$, the break time, $t_{b1}$, two power-law indices, $\alpha_1$ and $\alpha_2$, the smoothness $n$, the Lorentzian normalisation, $F_{\rm {\sc L},r}$, the peak time, $t_{\rm c}$ and its width, $t_{\rm w}$. The time-integrated flux density of the latter component is $\pi\,F_{\rm {\sc L},r}\,t_{\rm w}/2$. The two terms of eq. (\[eq:beulore\]) peak at $t_{p1}=t_{b1}(-\alpha_1/\alpha_2)^{1/[n(\alpha_2-\alpha_1)]}$ and $t_{p2}=t_{\rm c}$, respectively. Each of the light curves of the remaining filters were fitted with a free scaling factor with respect to the $r'$ curve as modelled by eq. (\[eq:beulore\]). The free parameters and the dof total 19 and 185, respectively.
The best-fit result is shown in Fig. \[f:fluxall\_beulore\], while the second line of Table \[t:panchro\_fit\] reports the corresponding best fit values. The fit is good: $\chi^2/{\rm dof}=187/185$. The scaling factors for the remaining bands are the following: $f_K=2.31\pm0.12$, $f_H=2.03\pm0.10$, $f_J=1.65\pm0.09$, $f_z=1.40\pm0.04$, $f_i=1.20\pm0.03$, $f_V=0.90\pm0.08$, $f_g=0.78\pm0.02$, $f_B=0.77_{-0.15}^{+0.19}$, $f_U=0.59_{-0.10}^{+0.12}$, $f_{UVW1}=0.28_{-0.11}^{+0.17}$, while the X-ray normalisation is still $F_{\rm x}=(3.7\pm0.3)\times10^{-11}$ erg cm$^{-2}$ s$^{-1}$. The two components peak at $t_{p1}=600$ s and $t_{p2}=34.4$ ks, respectively.
We also tried to model the second component with a rising and falling smoothly broken power law instead of a Lorentzian. However, this brings in too many free parameters, such as the slope of the rise, so unless one finds reasons to fix some of them to precise values, the fit with such a component turns into highly undetermined parameters.
[rrrrrrrrrrrr]{} $\alpha_1$ & $t_{b1}$ & $\alpha_2$ & $t_{b2}$ & $\alpha_3$ & $t_{b3}$ & $\alpha_4$ & $n$ & $t_{\rm c}$ & $t_{\rm w}$ & $F_{\rm {\sc L},r}$ & $\chi^2$/dof\
& (s) & & (s) & & (ks) & & & (ks) & (ks) & ($\mu$Jy) &\
$-0.56_{-0.33}^{+0.24}$ & $317_{-76}^{+151}$ & $0.15_{-0.07}^{+0.09}$ & $1456_{-46}^{+67}$ & $1.08\pm0.02$ & $23.8_{-3.1}^{+3.2}$ & $3.51_{-0.34}^{+0.37}$ & $5.4_{-1.3}^{+1.9}$ & – & – & – & $212/184$\
$-0.38_{-0.23}^{+0.22}$ & $2480_{-900}^{+1420}$ & $2.02_{-0.75}^{+0.85}$ & – & – & – & – & $0.49_{-0.28}^{+0.61}$ & $34.4_{-8.1}^{+10.6}$ & $72.7_{-12.2}^{+14.6}$ & $11.9_{-2.7}^{+3.5}$ & $187/185$\
Spectral Energy Distribution {#sec:SED}
----------------------------
Figure \[f:SEDall\] displays four SEDs we derived in as many different time intervals (see shaded bands in Fig. \[f:multi\_lc\]):
1. SED 1 includes the last $\gamma$-ray pulse and the optical flash detected by RAPTOR [@Wren08], around 60 s;
2. SED 2 corresponds to the final part of the optical rise, coinciding with the final part of the X-ray steep decay, spanning from $186.8$ to $269.4$ s;
3. SED 3 has the broadest wavelength coverage and corresponds to the plateau phase, from $\sim$$400$ to $\sim$$1500$ s.
4. SED 4 includes NIR/visible measurements around the possible late time break in the light curve (Fig. \[f:fluxall\_beu3\]), at $\sim$$10^5$ s.
![GRB rest-frame SEDs 1 to 4 (shown with asterisks, triangles, squares and diamonds, respectively). The dashed line shows the best-fitting power-law model of SED 2: $\beta_{\rm ox}=0.79\pm0.01$ and $A_{V,z}<0.02$. X-ray data are not absorption-corrected.[]{data-label="f:SEDall"}](multi_sed.eps){width="8.5cm"}
To construct SED 1 we made use of the RAPTOR measurement [@Wren08], a UVOT upper limit in the $V$ band and of the BAT spectrum of the fourth pulse. Figure \[f:SED1\] displays this SED: the solid line shows the best fit with a smoothed broken power law used to fit the high-energy photon spectra of the prompt emission of GRBs [@Band93].
![GRB rest-frame SED 1 from observed optical to $\gamma$-ray during the optical flash concomitant with the last $\gamma$-ray pulse at $\sim60$ s. The solid line shows the best-fitting smoothed broken power law [@Band93] with the following parameters: $\alpha_{\rm B}=-1.1$, $\beta_{\rm B}=-2.35$ and $E_{\rm p,i}=71$ keV.[]{data-label="f:SED1"}](sed_pulse4_rest15Hz.ps){height="8.5cm"}
The best-fitting parameters are the following: $\alpha_{\rm B}=-1.12$, $\beta_{\rm B}=-2.35$ and $E_{\rm p,i}=71$ keV ($\chi^2/{\rm dof}=5.8/5$) consistent with the limit on $E_{\rm p,i}$ derived in Sect. \[sec:gamma\]. The Band indices are photon indices, so the corresponding energy indices are $0.12$ and $1.35$, respectively. While $\beta_{\rm B}$ was constrained by the BAT data themselves, we solved the coupled indetermination $\alpha_{\rm B}$–$E_{\rm p,i}$ by initially freezing the low-energy index $\alpha_{\rm B}$ to the typical value of $-1$ [@Kaneko06] and then leaving it to vary. The above minimum $\chi^2$ was so found. Although this does not break the degeneracy of both parameters (for every $E_{\rm p,i}<88$ keV there is a value of $\alpha_{\rm B}$ for which an acceptable fit is given), here it is shown that the extrapolation of a typical Band model fitting the spectrum of the last pulse matches the optical flux observed during the flash. However, because of the lack of measurement of $\alpha_{\rm B}$ from $\gamma$-ray data, the optical flux matched by the extrapolation of the Band model may still be accidental.
The time interval of SED 2 corresponds to the first $JHK$ GROND frames and spans from $186.8$ to $269.4$ s (see Fig. \[f:multi\_lc\]). It consists of contemporaneous $Vr'JHK$ frames as well as X-rays.
![GRB rest-frame SED 2 corresponding to the final part of the rise of the optical afterglow.[]{data-label="f:SED2"}](sed1_rest15Hz.ps){height="8.5cm"}
The NIR-to-X SED 2 can be fitted with a single power-law with $\beta_{\rm ox}=0.74\pm0.03$ and negligible dust extinction. The optical data alone can be fitted with an unextinguished power law with $\beta_{\rm o}=0.61\pm0.13$ (Table \[t:BAT\_XRT\_spec\]).
SED 3, taken during the plateau, is the richest one including all of the passbands considered in this work, but the $\gamma$-rays (see Fig. \[f:multi\_lc\]). Our NIR values are consistent with the $JHK$ points of @Bloom08. Given the steadiness of the light curve and the evidence for no significant colour change along the plateau, the SED so obtained is fairly robust.
![GRB rest-frame SED from observed NIR to X-ray during the plateau from $400$ to $1500$ s. The solid line shows the best-fit model (SMC profile) corresponding to a single unextinguished power law with spectral index $\beta=0.79\pm0.01$ and negligible local-frame extinction, $A_{V,z}<0.02$ (see text).[]{data-label="f:SED3"}](sed_1e3s_rest15Hz_panchrofit.ps){height="8.5cm"}
The multi-wavelength fitting of the light curves of Figs. \[f:fluxall\_beu3\] and \[f:fluxall\_beulore\] (Sect. \[sec:multi\_lc\]) is dominated by the data points along the plateau phase. Thus, we built a SED using the best-fit normalisations (Sects. \[sec:beu3\] and \[sec:beulore\]) and calculated at the time of $10^3$ s. We found the same results with improved uncertainties, due to the stronger constraints imposed by a multi-band fitting. The result of this SED is displayed in Figure \[f:SED3\]. The solid line shows the best-fit model obtained adopting a rest-frame SMC-extinguished [@Pei92], X-ray photoelectrically absorbed power law with $\beta_{\rm ox}=0.79\pm0.01$. We note that the point corresponding to the UVW1 filter nicely agrees with the Lyman absorption at the GRB redshift. The rest-frame optical extinction was found to be negligible and a very tight limit could be derived, $A_{V,z}<0.02$. Thanks to the more precise estimate obtained on $\beta_{\rm ox}$, the estimate of $N_{{\rm H},z}$ improved correspondingly: $(2.7\pm0.8)\times10^{21}$ cm$^{-2}$. These results are consistent with what was obtained from the 0.3–10 keV spectrum alone and the accuracy of the estimates benefited significantly from the inclusion of NIR/visible data. Fitting the optical data alone, the result is similar: no need for a significant amount of extinction and, more importantly, the same index: $\beta_{\rm o}=\beta_{\rm ox}=\beta_{\rm x}$ (Table \[t:BAT\_XRT\_spec\]), thus ruling out the possibility of a significant reddening due to dust along the line of sight within the host galaxy.
As a consequence, two properties are inferred:
- a negligible dust column density in the circumburst environment and along the line of sight to the GRB through the host galaxy;
- a single power-law component accounting for the (observed) NIR to X-ray radiation, pointing to a single emission mechanism with no breaks in between. Furthermore, a single power-law spectrum implies an achromatic evolution, consistently with the observations, while the other way around is not true.
The epoch of SED 4 is $\sim$$80$ ks, i.e. around the final break (aftermath of the late brightening) following the light curve description given in Sect. \[sec:beu3\] (Sect. \[sec:beulore\]) and shown in Fig. \[f:fluxall\_beu3\] (Fig. \[f:fluxall\_beulore\]). This includes optical data and is shown in Fig. \[f:SEDall\] (diamonds). Data can be fitted either with a single unextinguished power-law with $\beta_{\rm o}=1.05\pm0.06$ ($A_{V,z}$ fixed to 0) or, alternatively, with $\beta_{\rm o}=0.85\pm0.30$ and some extinction, $A_{V,z}=0.10_{-0.06}^{+0.14}$. An increase of the extinction along with time seems hard to explain physically, so we are led to favour a true reddening at this time. Compared with the previous SEDs (Table \[t:BAT\_XRT\_spec\]), SED 4 is redder by $\Delta\beta_{\rm o}=0.26\pm0.06$ (significance of $\sim$$6\times10^{-5}$). We point out that the reddening is independent of the fit choice, as demonstrated from the comparison of the bare power-law indices with no dust correction between the earlier and later spectra (Table \[t:BAT\_XRT\_spec\]).
Discussion {#sec:disc}
==========
The 15–150 keV fluence and peak flux of XRF 080330 are typical of other XRFs detected by Swift. The X-ray afterglow flux places XRF 080330 in the low end of the distribution of the GRBs observed by Swift, similarly to the majority of XRFs [@Sakamoto08]. Moreover, the observed X-ray flux of XRF 080330 lies in the low end of both the XRFs sample of Swift considered by @Sakamoto08 and of the XRFs sample of BeppoSAX of @Dalessio06. The X-ray afterglow of XRF 080330 does require a remarkable steepening after the shallow phase, with $\alpha_{\rm x,3}>1.3$ regardless of the light curve modelling (Fig. \[f:multi\_lc\]). This decay is typical of classical GRBs (or steeper), but is in contrast to the fairly shallow decays found by @Sakamoto08 for their sample of Swift XRFs. The optical flux of the XRF 080330 afterglow is within 1$\sigma$ of the distribution of the BeppoSAX XRFs sample of @Dalessio06 at 40 ks post burst.
The coincidence of the steep decay observed in the X-ray light curve, that is a smooth continuation of the last $\gamma$-ray pulse, suggests that this corresponds to its high-latitude emission or the so-called “curvature effect” [@Fenimore96; @Kumar00; @Dermer04]. In the case of a thin shell emitting for a short time, the closure relation expected between temporal and spectral indices is $\alpha=\beta+2$, with the time origin $t_0$ reset to the ejection time of the related pulse, earlier than the onset by about 3–4 times the width of the pulse. This still holds even if the emission occurs over a finite range of radii [@Genet08], though in that case the ratio of the ejection to pulse onset time difference and the pulse width becomes smaller ($\sim\,1$ for $\Delta\,R\sim\,R$).
We fitted the X-ray decay up to $210$ s with the form $F(t)+{\bf k\,}(t-t_0)^{-\alpha_{\rm x}}$, where the parameters of $F(t)$ (eq. \[eq:beu3\]) were frozen to their corresponding best-fit values obtained in Sect. \[sec:beu3\], and the power-law parameters $t_0$, $\alpha_{\rm x}$ and its normalisation ${\bf k}$ were left free to vary. We obtained: $t_0=53_{-18}^{+9}$ s and $\alpha_{\rm x}=2.4_{-0.5}^{+0.9}$ s ($\chi^2/{\rm dof}=22/27$), as shown by the solid line in Fig. \[f:xrt\_curvature\]. During the steep decay, it is $\beta_{\rm x}=1.06_{-0.09}^{+0.10}$ (Table \[t:BAT\_XRT\_spec\]). The curvature relation is fully satisfied and we note that $t_0$ does correspond to the time of the last pulse. Replacing eq. (\[eq:beu3\]) with eq. (\[eq:beulore\]) for the underlying component, $F(t)$, the best-fit parameters do not change to a noticeable degree.
![0.3–10 keV steep decay curve. The solid line shows the best-fitting model of the form $F(t)+k\,(t-t_0)^{-\alpha_{\rm x}}$, where the parameters of $F(t)$ (eq. \[eq:beu3\]) were frozen to their best-fit values obtained by the multi-band fit of the afterglow component (Sect. \[sec:beu3\]) and with $k$, $t_0$ and $\alpha_{\rm x}$ free parameters. Both terms, $F(t)$ and the power-law, are shown separately by the dotted and dashed curves, respectively.[]{data-label="f:xrt_curvature"}](xrt_curvature.eps){width="8.5cm"}
The optical afterglow of XRF 080330 exhibited a slow rise up to $\sim\,300$ s, followed by plateau out to $\sim\,2\times\,10^3$ s, after which it decayed within a typical power-law index of about $1.1$ approximately out to a few $10^4$ s. Then, a sharp break to a decay index of $3.5_{-0.3}^{+0.4}$ occurred concurrently with an optical reddening (Sect. \[sec:beu3\]; Fig. \[f:fluxall\_beu3\]). Alternatively, after the plateau a more gradual transition to a power-law decay index of $2.0\pm0.8$ set in, followed by a smooth, red bump (Sect. \[sec:beulore\]; Fig. \[f:fluxall\_beulore\]). We discuss each phase separately in the following subsections.
Optical afterglow rise {#sec:rise}
----------------------
In the context of the fireball model (e.g. Mészáros 2006 and references therein), the possibility that the peak of the optical afterglow emission corresponds to the passage of the peak synchrotron frequency is ruled out by the lack of spectral evolution: $\beta_{\rm o}$ should evolve from negative ($-1/3$) to positive values, while we find no evidence for $\beta_{\rm o}$ changing before $\sim$$8\times10^4$ s.
Another possible interpretation of the optical peak is the onset of the afterglow, as for GRB 060418 and GRB 060607A [@Molinari07; @Jin07] and, possibly, for XRF 071010A as well [@Covino08]. In the case of XRF 080330, the rise during the pre-deceleration of the fireball within an ISM is much shallower than $\alpha\sim-3$, expected at frequencies between $\nu_{\rm m}$ and $\nu_{\rm c}$ [@Sari99; @Granot05; @Jin07]. A wind environment would fit in a better way the slow rise of XRF 080330. Under these assumptions and in the thin shell case as the duration of the GRB is much shorter than the deceleration time, we can estimate the initial Lorentz factor, $\Gamma_0$ (approximately, twice as large as the Lorentz factor at the peak), in a wind-shaped density profile, $n(r)=A\,r^{-s}$ ($A$ is constant), with $s=2$, from the peak time and the $\gamma$-ray radiated energy, $E_{\rm iso}$ [@Chevalier00; @Molinari07]. For consistency, only the two-component model (Sect. \[sec:beulore\]; Fig. \[f:fluxall\_beulore\]) must be considered. In this case, we take the peak time of the first component, $t_{p1}=600$ s: assuming $\eta=0.2$ (radiative efficiency), $A=3\times10^{35}$ cm$^{-1}$, it turns out $\Gamma_0<80$, and a corresponding deceleration radius smaller than $7\times10^{16}$ cm. As in the case of XRF 071010A, the initial Lorentz factor is smaller than those found for classical GRBs. There is no evidence for the presence of a reverse shock; should the injection frequency of the reverse shock lie within the optical passbands, it would dominate the optical flux and exhibit a fast ($\sim$$t^{-2.1}$) decay [@Kobayashi07], not observed here. Nonetheless, this can still be the case if the injection frequency lies below the optical bands [@Jin07; @Mundell07]. A weak point of this interpretation is that the case $s=2$ is ruled out: $\alpha=s(p+5)/4-3=0.79\pm0.01$ [@Granot05]. Inverting this relation, a value of $s=1.4\pm0.1$ is required to explain the observed $\alpha=-0.4\pm0.2$. This argument, together with the absence of reverse shock, whose $F_{\nu,{\rm max}}$ should be much larger compared with the forward shock by a factor of $\sim\,\Gamma$ (although see above), makes the interpretation of a deceleration through a wind environment somewhat contrived.
In the context of a single jet viewed off-axis [@Granot05_2], the rising part of the XRF 080330 curve is explained by the emission coming from the edge of the jet: as the bulk Lorentz factor $\Gamma$ decreases, the beaming cone gets progressively wider, thus resulting in a rising flux. The peak in the light curve is reached when it is $\Gamma\,\sim\,1/(\theta_{\rm obs}-\theta_0)$, where $\theta_{\rm obs}$ and $\theta_0$ are the viewing and jet opening angles, respectively. According to the optical afterglow classification given by @Panaitescu08, XRF 080330 belongs to the class of slow-rising and peaking after 100 s events. Those authors found a possible anti-correlation between the peak flux and the peak time for a number of fast-rising afterglows, followed also by the slow-rising class and, in this respect, XRF 080330 is no exception. The suggested interpretation of the rise is either the pre-deceleration synchrotron emission or the emergence of a highly collimated outflow seen off-axis. In the latter case, assuming a power-law angular distribution of the kinetic energy, $\mathcal{E}(\theta)\propto(\theta/\theta_{\rm c})^{-q}$ ($q>0$), high values for $q$ correspond to slower rises and dimmer peak fluxes, for a fixed off-axis viewing angle ($\theta_{\rm obs}=2\,\theta_{\rm c}$). In the former case, the anti-correlation is ascribed to different circumburst environment densities for different events: XRF 080330, because of the negligible dust extinction, would lie in the high-peak flux region, which is not the case. This favours the interpretation of an off-axis jet whose angular distribution of energy quickly drops away from the jet axis.
An example of another XRF whose optical counterpart showed a very similar behaviour is XRR 030418. The rise of this XRR, for which only an upper limit to its redshift ($z<5$) was obtained [@Dullighan03], has been interpreted as due to the decreasing extinction along with time, caused by the dust column density crossed by the fireball during its expansion [@Rykoff04]. Figure \[f:cfr\_030418\] shows the light curve compared with that of XRF 080330.
![$r$-band afterglow of XRF 080330 compared with the XRR 030418 (empty circles), for which there is only an upper limit to its redshift, $z<5$ [@Dullighan03]. Data of XRR 030418 have been taken from @Rykoff04 [@Ferrero03; @Dullighan03]. The solid line is the best fit of the XRF 080330 $r$ curve of Sect. \[sec:beu3\], while the dashed line is the best fit obtained with the same model applied to XRR 030418.[]{data-label="f:cfr_030418"}](obsfr_flux_080330_vs_030418.eps){width="8.5cm"}
The solid line shows the best fit to the $r'$ curve of XRF 080330 of Sect. \[sec:beu3\], while the dashed line shows the same model applied to the XRR 030418 data. XRR 030418 displays a steeper rise ($\alpha_1=-1.5$), which strongly depends on the zero time and could be the same as that of XRF 080330 if the time origin was moved by $(190\pm50)$ s forward in time (lab frame). However, there is nothing around this time in the $\gamma$-ray light curve of XRR 030418. Apart from the different slopes of the rise and the lack of a late-time steepening in the case of XRR 030418, the plateau and post-plateau decay look very similar. If both XRFs are caused by the same process, from the spectral (lack of) evolution XRF 080330 during the rise-plateau-initial decay phases we can rule out the decreasing dust column density hypothesis.
Plateau {#sec:plateau}
-------
From the SED extracted around the plateau no break is found between optical and X-ray frequencies, with $\beta_{\rm ox}=0.79\pm0.01$. In the regime of slow cooling it is reasonable to assume that both optical ($\nu_{\rm o}$) and X-ray (${\nu_{\rm x}}$) frequencies lie between the injection ($\nu_{\rm m}$) and the cooling ($\nu_{\rm c}$) frequencies: $\nu_{\rm m}<\nu_{\rm o}<\nu_{\rm x}<\nu_{\rm c}$ [@Sari98]. The power-law index of the electron energy distribution, $p$, is given by $\beta_{\rm ox}=(p-1)/2$, yielding $p=2.58\pm0.02$, fully within the range of values found for other bursts (e.g. Starling et al. 2008). The temporal decay index depends on the density profile: the ISM (wind) case predicts a value of $\alpha=3(p-1)/4=1.18\pm0.02$ ($\alpha=3p/4-1/4=1.68\pm0.02$). After the plateau, depending on the light curve modelling, the measured decay index is either $\alpha=\alpha_{3}=1.08\pm0.02$ (Sect. \[sec:beu3\]; Fig. \[f:fluxall\_beu3\]) or $\alpha=\alpha_{2}=2.0\pm0.8$ (Sect. \[sec:beulore\]; Fig. \[f:fluxall\_beulore\]). While the multiple smoothly broken power-law description (Sect. \[sec:beu3\]) definitely rules out the wind environment, both environments are still possible in the two-component model (Sect. \[sec:beulore\]), mainly because of the poorly measured decay index, $\alpha_2$. If one interprets the flat decay as due to energy injection [@Nousek06; @Zhang06], the corresponding index would be $q\sim0.3$.
In the off-axis jet interpretation, even if we consider an initially uniform sharp-edged jet, the shocked external medium at the sides of the jet has a significantly smaller Lorentz factor than near the head of the jet, and therefore its emission is not strongly beamed away from off-beam lines of sight. As a result, either an early very shallow rise or decay is expected for a realistic jet structure and dynamics [@Eichler06]. In the case of XRF 030723, @Granot05_2 showed that, for $\theta_{\rm obs}\sim\,2\,\theta_0$, an initial plateau is expected in the light curve.
Our observations of the afterglow rise of XRF 080330 rule out the interpretation proposed by @Yamazaki09 of the plateau as due to an artifact of the choice of the reference time, as all the other rising curves do.
Jet break {#sec:jetbreak}
---------
According to the light curve description of Sect. \[sec:beu3\] shown in Fig. \[f:fluxall\_beu3\], for which only an ISM environment is possible (Sect. \[sec:plateau\]), after the plateau phase the light curve is expected to approach the on-axis light curve with $\alpha=3(p-1)/4$. The late-time steepening observed around $8\times10^4$ s, estimated to be $\Delta\alpha=\alpha_4-\alpha_3=2.4\pm0.4$, cannot be produced by the passage of the cooling frequency $\nu_{\rm c}$ through the optical, as that is expected to be as small as $\Delta\alpha=1/4$ (ISM/wind).
In the off-axis jet interpretation, assuming a value for $\theta_0$ of a few degrees, another advantage of this interpretation is the steep late time decay (at $\gtrsim$ 1 day) as a consequence of joining the post jet break on-axis light curve. According to the light curve modelling given in Sect. \[sec:beulore\] shown in Fig. \[f:fluxall\_beulore\], the gradual steepening following the plateau corresponds to the post-jet break emission: the observed power-law decay index, $\alpha_2=2.0\pm0.8$ is consistent with the expected $\alpha$ $=$ $p$ $=$ $2.6$. We note that the relatively sharp jet break favours the ISM environment. Overall, in the context of an off-axis viewing angle interpretation, the light curve suggests that $\theta_{\rm obs} \sim (1.5-2)\theta_0$ as well as an early jet break (at $\lesssim\,1\;$day), which in turn implies a narrow jet with a half-opening angle of the order of a few degrees, $\theta_0\,\sim\,0.05$. As a simple feasibility check, we note that for $\theta_{\rm obs} < 2\theta_0$ the ratio of the on-axis to off-axis $E_{\gamma,{\rm iso}}$ is equal to $\delta^2$ (assuming that the observed energy range includes $E_{\rm p}$ where most of the energy is radiated), where $\delta$ is the ratio of their corresponding Doppler factors and therefore of their $E_{\rm p}$ [@Eichler04]. In our case, for an observed off-axis $(1+z)E_{\rm p} \sim 60\;$keV, an on-axis value of $\sim 1\;$MeV would require $\delta = 1+[\Gamma_0(\theta_{\rm obs}-\theta_0)]^2 \sim 17$ and $\Gamma_0(\theta_{\rm obs}-\theta_0) \sim 4$, which for $\theta_{\rm obs}-\theta_0 \sim (0.5-1)\theta_0$ and $\theta_0 \sim 0.05$ gives $\Gamma_0 \sim 80-160$. Here $\Gamma_0$ is the initial Lorentz factor at the edge of the jet. More realistically, the jet would not be perfectly uniform with extremely sharp edges, and instead $\Gamma_0$ is expected to be lower at the outer edge of the jet and larger near its center (where it could easily reach several hundreds in our illustrative example here). In this case $\delta^2\sim300$ so that the observed $E_{\gamma,{\rm iso}}$ in the $15-150\;$keV range, which is $2\times 10^{51}\;$erg, would imply an on-axis value for $E_{\rm\gamma,iso}$ of $\sim10^{54}\;$erg, which for a narrow jet with $\theta_{\rm obs} \sim 0.05$ would correspond to a true energy of the order of $10^{51}\;$erg. This demonstrates that this scenario can work for reasonable values of the physical parameters. We point out that the estimate of the on-axis $E_{\gamma,{\rm iso}}$ of $\sim10^{54}$ erg is for a wide energy range containing $E_{\rm p}$, since in our illustrative example most of the energy is released within the observed range. A more accurate estimate of the break time and of the corresponding opening angle is difficult, due to the degeneracy involved in the light curve modelling (Figs. \[f:fluxall\_beu3\] and \[f:fluxall\_beulore\]). Table \[t:main\_prop\] summarises the main properties of XRF 080330.
[lc]{} Name & Value\
&\
$z$ & $1.51$\
$S(15-150~{\rm keV})$ & $(3.6\pm0.8)\times10^{-7}$ erg cm$^{-2}$\
$P(15-150~{\rm keV})$ & $(1.0\pm0.2)$ ph cm$^{-2}$ s$^{-1}$\
$E_{\rm p}$ & $<35$ keV\
$E_{\rm p,i}=E_{\rm p}\, (1+z)$ & $<88$ keV\
$E_{\rm iso}$ ($15$–$150$ keV, obs frame) & $(2.1\pm0.5)\times10^{51}$ ergs\
$E_{\rm iso}$ ($1$–$10^4$ keV, GRB frame) & $<2.2\times10^{52}$ ergs\
$t_{\rm jet}$ (obs frame) & $\lesssim1$ day\
$\theta_0$ (jet opening angle)$^{\rm (a)}$ & few degrees\
$\theta_{\rm obs}$ (viewing angle)$^{\rm (a)}$ & $\sim (1.5-2)\theta_0$\
$^{\rm (a)}$ Under the assumption of a single off-axis jet.
Late time red bump {#sec:bump}
------------------
Overall, the two-component description of the light curves of Sect. \[sec:beulore\] shown in Fig. \[f:fluxall\_beulore\] appears to be slightly favoured over the multiple smoothly broken power-law of Fig. \[f:fluxall\_beu3\]. So, irrespective of the nature of the rise and plateau, we speculate on the possible nature of the second component, modelled in Fig. \[f:fluxall\_beulore\] as a late time bump. Clearly, a SN bump, such as that possibly observed in the light curve of XRF 030723 [@Fynbo04], is ruled out mainly because that is expected to peak days later, which is incompatible with one single day after XRF 080330; not to mention the too high redshift of XRF 080330 for a 1998bw-like SN to be detected.
Alternatively, a density bump seems a viable solution, given that $\nu_{\rm o}<\nu_{\rm c}$ (e.g. Lazzati et al. 2002; Guidorzi et al. 2005), although the explanation of the observed contemporaneous reddening requires ad hoc assumptions, such as the case of GRB 050721, which showed similar properties to XRF 080330 (same $\beta_{\rm ox}$ with no breaks between optical and X-ray, late time redder optical bump). In that case, the observed reddening was explained as due to the presence of very dense clumps surviving the GRB radiation and with a small covering factor [@Antonelli06].
If the late reddening is due to the passage of $\nu_{\rm c}$ through the optical bands, in addition to what argued in Sect. \[sec:jetbreak\], another weak point of the multiple smoothly broken power-law description of Fig. \[f:fluxall\_beu3\] (Sect. \[sec:beu3\]) is the chromatic change of $\Delta\beta_{\rm o}=0.26\pm0.06$ we observe in the optical bands around $8\times10^4$ s. The passage of the cooling frequency does not explain it: the observed reddening would be 0.5, i.e. twice as much. This could still be the case, if our measurement might have taken place in the course of the spectral change, as the broken power-law spectrum is a simple approximation. However, since it is $\nu_{\rm c}>\nu_{\rm x}$ during the plateau because of the unbroken power-law spectrum between optical and X-ray, $\nu_{\rm c}$ would have decreased very rapidly, thus making this option not reasonable: if at $10^3$ s it is $\nu_{\rm c}>\nu_{\rm x}=10^{18}$ Hz, at $10^5$ s it should be $\nu_{\rm c}>10^{17}$ Hz $\gg\,\nu_{\rm o}$, because $\nu_{\rm c}\propto\,t^{-1/2}$ (ISM case), thus this possibility is to be ruled out. Likewise, in the wind case it is $\nu_{\rm c}\propto\,t^{1/2}$. The times $t_{K}$ and $t_{g}$, at which $\nu_{\rm c}$ would cross the most redward and blueward filters, $K$ and $g'$, would differ by a factor of $(\nu_g/\nu_K)^2\sim20$, which looks incompatible with the light curves. Furthermore, the observed reddening rules out the wind case, as $\beta_{\rm o}$ should decrease from $p/2$ to $(p-1)/2$.
Although an energy injection to the blast wave (forward shock) can explain the bump feature in the light curve, it is difficult to explain the reddening if we consider only the forward shock emission, as we have discussed. A possible explanation for the reddening is that the rebrightening is due to the short-lived ($\Delta\,t\sim\,t$) reverse shock of a slow shell which caught up with the shock front and increased its energy through a refreshed shock (e.g. Kumar & Piran 2000; Granot et al. 2003; Jóhannesson et al. 2006): since that shock is going into a shell of ejecta, rather than the external medium, it can have a much larger $\epsilon_B$ (magnetised fireball: Zhang et al. 2003; Kumar & Panaitescu 2003; Gomboc et al. 2008b) and, therefore, a lower $\nu_{\rm c}$, quite naturally; for an ISM where $\nu_{\rm c}$ decreases with time, then $\nu_{\rm c}$ of the reverse shock could be around the optical for reasonable model parameter values. The need for a separate component to explain a chromatic break in the light curve was also suggested in the case of GRB 061126 [@Gomboc08b]. Notably, the final steepening after $10^5$ s, which in the modelling of Sect. \[sec:beu3\] (Fig. \[f:fluxall\_beu3\]) is described with $\alpha_4=3.5_{-0.3}^{+0.4}$, is compatible with the high-latitude emission of the reverse shock: $\alpha=\beta_{\rm o,late}+2=3.05\pm0.06$. The jet break might happen slightly earlier than the break time in the optical light curve.
A somewhat more contrived way to explain the bump is the appearance of a second narrower jet in the two-component jet model, as proposed for XRF 030723 [@Huang04]. In this model, the viewing angle, is within or slightly off the cone of the wide jet and outside the narrow jet. The plateau phase would reflect the deceleration of the wide jet [@Granot06]. Depending on the isotropic-equivalent kinetic energy of the wide and narrow jet, $E_{\rm w,iso}$ and $E_{\rm n,iso}$, on the jet opening angles, $\theta_{0,{\rm w}}$ and $\theta_{0,{\rm n}}$, on the initial bulk Lorentz factors, $\gamma_{0,{\rm w}}$ and $\gamma_{0,{\rm n}}$, respectively, as well as on the viewing angle, the afterglow emission of either component is dominant at different times. According to the results of @Huang04, the light curve of XRF 080330 could be qualitatively explained as follows: the first component obtained in Sect. \[sec:beulore\] (Fig. \[f:fluxall\_beulore\]) represents the contribution of the wide jet dominating at early times: the rise could be due either to the afterglow onset (in a wind environment) or to a viewing angle slightly beyond the wide jet opening angle: $\theta_{\rm obs}\gtrsim\theta_{0,{\rm w}}$. The appearance of the second component would mark the deceleration and lateral expansion of the narrow jet, the peak time corresponding to the case $\gamma_{\rm n}\sim1/\theta_{\rm obs}$. Unlike XRF 030723, which showed a relatively sharp late-time peak, the bump exhibited by XRF 080330 looks less sharp and pronounced. Although this might suggest a relatively lower energy of the narrow jet compared to XRF 030723, yet we cannot exclude the case $E_{\rm w,iso}\ll\,E_{\rm n,iso}$. The ratio of the observed energies of XRF 080330, of about $0.6$ according to the modelling of Sect. \[sec:beulore\], corresponds to the ratio between the early and the late time emissions of the wide and narrow jets, respectively. Depending on the values of $\theta_{\rm obs}$, $\theta_{0,{\rm w}}$, $\theta_{0,{\rm n}}$, a comparable ratio of observed energies, such as that observed for XRF 080330, can still be obtained in the case of a much more energetic narrow jet, $E_{\rm w,iso}\ll\,E_{\rm n,iso}$.
Such a model turned out to be successful in accounting for the naked-eye GRB 080319B [@Racusin08]: in that case, the two collimation-corrected energies were comparable, while the isotropic-equivalent energy of the narrow jet ($\theta_{0,{\rm n}}=0.2^{\circ}$) was about 400 times larger than that of the wide jet ($\theta_{0,{\rm w}}=4.0^{\circ}$).
However, in the context of the two-component model the explanation of the late time reddening simultaneously with the appearance of the narrow component emission requires two different values of $p$ for the two jets, which does not look reasonable on a physical ground. Another option is that the cooling frequency of the second jet might be around the optical at the time of the bump: however, if the shock micro-physics parameters ($p$, $\epsilon_B$, $\epsilon_e$), are the same for the two shocks, as expected on physical grounds, and obviously the external medium is the same, then the only thing that is different as far as $\nu_{\rm c}$ is concerned, is $E_{\rm iso}$. Since the presence of the bump requires $E_{\rm n,iso}>E_{\rm w,iso}$, then this would not work for the wind case, where $\nu_{\rm c}\propto\,E_{\rm iso}^{1/2}$. Even in the ISM case, where $\nu_{\rm c}\propto\,E_{\rm iso}^{-1/2}$, the fact that $\nu_{\rm c}>\nu_{\rm x}$ at $10^3$ s, and therefore $\nu_{\rm c}>10^{17}$ Hz at $10^5$ s for the wide jet, requires $E_{\rm n,iso}/E_{\rm w,iso}\gtrsim10^4$ (which is very extreme) in order for $\nu_{\rm c,n}(10^5~{\rm s})$ to be around $10^{15}$ Hz (required by the observed reddening). Therefore, while this could work in principle, in practise it requires extreme parameters. In particular, the amplitude of the bump suggests that $E_{\rm n}\sim\,E_{\rm w}$, and therefore from the required $E_{\rm n,iso}/E_{\rm w,iso}\gtrsim10^4$ it would follow $\theta_{\rm n}/\theta_{\rm w}\lesssim\,10^{-2}$, which seems pushed to the extreme.
Overall, these considerations make the single off-axis jet interpretation much more compelling.
Conclusions {#sec:conc}
===========
XRF 080330 is representative of the XRR and XRF classes of soft GRBs. Its $\gamma$- and X-ray properties of both prompt and high-energy afterglow emission place it in the low-flux end of the distribution. The multi-band (NIR through UV) optical curve showed an initial rise up to $\sim$300 s, followed by a $\sim$2-ks long plateau, temporally coinciding with the canonical flat decay of X-ray afterglows of all kinds of GRBs, followed by a gradual steepening and a possible jet break. We provided two alternative descriptions of the light curve: a piecewise power-law with three break times, the last of which occurring around $8\times10^4$ s and followed by a sharp steepening, with the power-law decay index changing from $1.1$ to $3.5_{-0.3}^{+0.4}$. The SED from NIR to X-ray wavelength is fitted with a simple power-law with $\beta_{\rm ox}=0.8$ and negligible GRB-frame extinction, $A_V<0.02$ adopting a SMC-like profile, with no evidence for chromatic evolution during the rise, plateau and early ($<8\times10^4$ s) decay phases. However, after the possible late time break we observe a reddening in the optical bands of $\Delta\beta_{\rm o}=0.26\pm0.06$, which cannot be accounted for in terms of the synchrotron spectrum evolution of a standard afterglow model, unless a different description of the light curve is considered. In the alternative model of the light curves, we identified two distinct components: the first is modelled with a smoothly broken power law and fits the rise plateau and early decay of the afterglow, while the second, taking over around $8\times10^4$ s, is modelled with an energy injection episode peaking at $34_{-8}^{+11}$ ks and with a time-integrated energy of $\sim$60% that of the first component.
The X-ray light curve consists of the initial steep decay, which is likely the high-latitude emission of the last $\gamma$-ray pulse. At the same time, the optical afterglow rises up to a plateau, temporally coincident with the X-ray flat decay. In this case, we collected strong evidence that the emission mechanism during this phase is the same from optical to X-rays and is consistent with synchrotron emission of a decelerating fireball with an electron energy distribution power-law index of $p=2.6$.
The lack of spectral evolution throughout the rise, plateau and early decay argue against a temporally decreasing dust column density claimed to explain similar optical light curves of past soft bursts.
The optical rise ($\alpha\sim-0.4$) is too slow for the afterglow onset within a uniform circumburst medium, but could still be the case if a wind environment is considered. In this case, under standard assumptions we constrained the Lorentz factor of the fireball to be smaller than 80, thus confirming the scenario of XRFs as less relativistic GRBs. However, we found that the interpretation of a single-component off-axis jet with an opening angle of a few degrees and a viewing angle about twice as large, can explain the observations: this not only accounts for the light curve morphology, but also explains the soft nature of XRF of the $\gamma$-ray prompt event. The reddening observed at $8\times10^4$ s can be interpreted as the short-lived reverse shock of an energy injection caused by a slow shell which caught up with the fireball shock front, also responsible for the contemporaneous bump in the light curve. A two-component jet could also work, but would introduce more free parameters and would require extreme conditions.
The interpretation of the late bump as produced by a density enhancement in the medium swept up by the fireball cannot be ruled out, although the reddening seems to require ad hoc explanations. In this case, as shown by @Nakar07, it is hard to produce a flux enhancement with density inhomogeneities, although it is not excluded given the lack of a sharp rise in this bump.
Overall, the XRF 080330 optical and X-ray afterglows properties have also been observed in many other GRBs [@Panaitescu08]. This both supports the view of a common origin of XRFs and classical GRBs, which form a continuum and do not call for distinct mechanisms. The importance of a prompt multi-wavelength coverage of the early phases of a GRB is clearly demonstrated in the case of XRF 080330.
This work is supported by ASI grant I/R/039/04 and by the Ministry of University and Research of Italy (PRIN 2005025417). JG gratefully acknowledges a Royal Society Wolfson Research Merit Award. DARK is funded by the DNRF. PJ acknowledges support by a Marie Curie European Re-integration Grant within the 7th European Community Framework Program under contract number PERG03-GA-2008-226653, and a Grant of Excellence from the Icelandic Research Fund. We gratefully acknowledge the contribution of the Swift team members at OAB, PSU, UL, GSFC, ASDC, MSSL and our sub-contractors, who helped make this mission possible. We acknowledge Sami-Matias Niemi for executing the NOT observations. CG is grateful to A. Kann for his reading and comments.
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-------------------------------- --------------------------------- -------- --------- ---------------------------
$\lambda_{\textrm{obs}}$ \[Å\] $\lambda_{\textrm{rest}}$ \[Å\] $z$ Feature EW$_{\textrm{obs}}$ \[Å\]
3889.4 1548.2/1550.8 1.5101 4.3$\pm$0.7
4028.8 1608.5 1.5047 FeII 2.0$\pm$0.7
4198.2 1670.8 1.5127 AlII 2.0$\pm$0.7
5093.5 2026.1 1.5139 4.3$\pm$0.4
5190.3 2062.2 1.5166 CrII
5190.3 2062.7 1.5166 ZnII
5883.4 2344.2 1.5100 FeII 1.4$\pm$0.5
5983.0 2382.8 1.5109 FeII 2.2$\pm$0.5
6497.5 2586.7 1.51 FeII
6527.0 2600.2 1.51 FeII
7031.1 2796.3/2803.5 1.5112 MgII 5.2$\pm$0.3
-------------------------------- --------------------------------- -------- --------- ---------------------------
[^1]: [http://www.rem.inaf.it/]{}
[^2]: [http://nedwww.ipac.caltech.edu/forms/calculator.html]{}
[^3]: [http://www.astro.livjm.ac.uk/RoboNet/]{}
[^4]: [http://www.nofs.navy.mil/data/fchpix/]{}
|
---
abstract: 'Code summarization and code search have been widely adopted in software development and maintenance. However, few studies have explored the efficacy of unifying them. In this paper, we propose [$TranS^{3}$]{}, a transformer-based framework to integrate code summarization with code search. Specifically, for code summarization, [$TranS^{3}$]{} enables an actor-critic network, where in the actor network, we encode the collected code snippets via transformer- and tree-transformer-based encoder and decode the given code snippet to generate its comment. Meanwhile, we iteratively tune the actor network via the feedback from the critic network for enhancing the quality of the generated comments. Furthermore, we import the generated comments to code search for enhancing its accuracy. To evaluate the effectiveness of [$TranS^{3}$]{}, we conduct a set of experimental studies and case studies where the experimental results suggest that [$TranS^{3}$]{} can significantly outperform multiple state-of-the-art approaches in both code summarization and code search and the study results further strengthen the efficacy of [$TranS^{3}$]{} from the developers’ points of view.'
author:
- Wenhua Wang
- Yuqun Zhang
- Zhengran Zeng
- Guandong Xu
bibliography:
- 'ref.bib'
title:
- '[$TranS^{3}$]{}: A Transformer-based Framework for Unifying Code Summarization and Code Search'
- '[$TranS^{3}$]{}: A Transformer-based Framework for Unifying Code Summarization and Code Search'
---
Introduction {#sec:introduction}
============
Code summarization and code search have become increasingly popular in software development and maintenance [@leclair2019neural; @movshovitz2013natural; @sridhara2010towards; @lv2015codehow; @gu2018deep; @yao2019coacor], because they can help developers understand and reuse billions of lines of code from online open-source repositories and thus significantly enhance software development and maintenance process [@yao2019coacor]. In particular, since much of the software maintenance effort is spent on understanding the maintenance task and related software source code [@lientz1980software], effective and efficient documentation is quite essential to provide high-level descriptions of program tasks for software maintenance. To this end, code summarization aims to automatically generate natural language comments for documenting code snippets [@moreno2017automatic]. On the other hand, over years various open-source and industrial software systems have been rapidly developed where the source code of these systems is typically stored in source code repositories. Such source code can be treated as important reusable assets for developers because they can help developers understand how others addressed similar problems for completing their program tasks, e.g., testing [@deeproad; @mutationtesting; @regressiontesting; @simulee1; @deepbillboard1; @iottesting; @simulee2], fault localization [@deepfl; @prfl; @prfl1], program repair and synthesis [@bytecode; @history; @sttt1; @aucs], in multiple software development domains [@smartvm; @mssurvey; @bigvm]. Correspondingly, there also raises a strong demand for an efficient search process through a large codebase to find relevant code for helping programming tasks. To this end, code search refers to automatically retrieving relevant code snippets from a large code corpus given natural language queries.
The recent research progress towards code summarization and code search can mainly be categorized to two classes: *information-retrieval*-based approaches and *deep-learning*-based approaches. To be specific, the *information-retrieval*-based approaches derive the natural language clues from source code, compute and rank the similarity scores between them and source code/natural language queries for recommending comments/search results [@wong2013autocomment; @movshovitz2013natural; @lu2015query; @lv2015codehow]. The *deep-learning*-based approaches use deep neural networks to encode source code/natural language into a hidden space, and utilize neural machine translation models for generating code comments and computing similarity distance to derive search results [@hu2018summarizing; @chen2018neural; @gu2018deep; @akbar2019scor; @yao2019coacor].
Based on the respective development of code summarization and code search techniques, we infer that developing a unified technique for optimizing both domains simultaneously is not only mutually beneficial but also feasible. In particular, on one hand, since the natural-language-based code comments can reflect program semantics to strengthen the understanding of the programs [@leclair2019neural], adopting them in code search can improve the matching process with natural language queries [@yao2019coacor]. Accordingly, injecting code comments for code search is expected to enhance the search results [@Scholer2014Query]. On the other hand, the returned search results can be utilized as an indicator of the accuracy of the generated code comments to guide their optimization process. Moreover, since code summarization and code search can share the same technical basis as mentioned above, it can be inferred that it is feasible to build a framework to unify and advance both the domains. Therefore, it is essential to integrate code summarization with code search.
Although integrating code summarization with code search can be promising, there remains the following challenges that may compromise its performance: (1) state-of-the-art code summarization techniques render inferior accuracy. According to the recent advances in code summarization [@wong2013autocomment; @iyer2016summarizing; @hu2018deep], the accuracy of the generated code comments appears to be inferior for real-world applicability (around 20% in BLEU-1 with many well-recognized benchmarks). Integrating such code comments might lead to inaccuracies of matching natural language queries and further compromise the performance of code search. (2) how to effectively and efficiently integrate code summarization with code search remains challenging. Ideally, the goal of integrating code summarization with code search is to optimize the performance of both domains rather than causing trade-offs. Moreover, such integration is expected to introduce minimum overhead. To this end, it is essential to propose an effective and efficient integration approach.
To tackle the aforementioned problems, in this paper, we propose a framework, namely [$TranS^{3}$]{} for optimizing both code summarization and code search based on a recent NLP technique—transformer [@vaswani2017attention]. Unlike the traditional CNN-based approaches that suffer from long-distance dependency problem [@wang2016dimensional] and RNN-based approaches that suffer from excessive load imposed by sequential computation [@huang2013accelerating], transformer advances in applying the self-attention mechanism which can parallelize the computation and preserve the integral textual weights for encoding to achieve the optimal accuracy of text representation [@vaswani2017attention].
[$TranS^{3}$]{} consists of two components: the code summarization component and code search component. Specifically, the code summarization component is initialized by preparing a large-scale corpus of annotated $ <code; comment> $ pairs to record all the code snippets with their corresponding comments as the training data. Next, we extract the semantic granularity of the training programs for constructing a tree-transformer to encode the source code into hidden vectors. Furthermore, such annotated pair vectors are injected into our deep reinforcement learning model, i.e., the actor-critic framework, for the training process, where the actor network is a formal encoder-decoder model to generate comments given the input code snippets; and the critic network evaluates the accuracy of the generated comments according to the ground truth (the input comments) and give feedback to the actor network. At last, given the resulting trained actor network and a code snippet, its corresponding comment can be generated. Given a natural language query, the code search component is launched by encoding the natural language query, the generated code comments, and the code snippets into the vectors respectively via transformer and tree-transformer. Next, we compute similarity scores between query/code vectors and query/comment vectors for deriving and optimizing their weighted $scores$. Eventually, we rank all the code snippets according to their $score$s for recommending the search results. The underlying transformer in [$TranS^{3}$]{} can enhance the quality of the generated code and thus strengthen the code search results by importing the impact from the generated comments. Moreover, since the code search component applies the encoder trained by the code summarization component without incurring extra training process, its computing overhead can be maintained minimum.
To evaluate the effectiveness and efficiency of [$TranS^{3}$]{}, we conduct a set of experiments based on the GitHub dataset in [@Barone2017A] which includes over 120,000 code snippets of Python functions and their corresponding comments. The experimental results suggest that [$TranS^{3}$]{} can outperform multiple state-of-the-art approaches in both code summarization and code search, e.g., [$TranS^{3}$]{} can significantly improve the code summarization accuracy from 47.2% to 141.6% in terms of BLEU-1 and the code search accuracy from 5.1% to 28.8% in terms of MRR compared with the selected state-of-the-art approaches. In addition, we also conduct case studies for both code summarization and code search where the study results further verify the effectiveness of [$TranS^{3}$]{}.
In summary, the main contributions of this paper are listed as follows:
- **Idea.** To the best of our knowledge, we build the first transformer-based framework for integrating code summarization and code search, namely [$TranS^{3}$]{}, that can optimize the accuracy of both domains.
- **Technique.** To precisely represent the source code, we design a transformer-based encoder and a tree-transformer-based encoder for encoding code and comments by injecting the impact from the semantic granularity of well-formed programs.
- **Evaluation.** To evaluate [$TranS^{3}$]{}, we conduct a substantial number of experiments based on real-world benchmarks. The experimental results suggest that [$TranS^{3}$]{} can outperform several existing approaches in terms of accuracy of both code summarization and code search. In addition, we also conduct empirical studies with developers. The results suggest that the quality of the generated comments and search results are widely acknowledged by developers.
The reminder of this paper is organized as follows. Section \[sec:background\] illustrates some preliminary background techniques. Section \[sec:example\] gives an example to illustrate our motivation for unifying code summarization and code search. Section \[sec:approach\] elaborates the details of our proposed approach. Section \[sec:experiment\] demonstrates the experimental and study results and analysis. Section \[sec:threats\] introduces the threats to validity. Section \[sec:relatedwork\] reviews the related work. Section \[sec:conclusion\] concludes this paper.
Background {#sec:background}
==========
In this section, we present the preliminary background techniques relevant to [$TranS^{3}$]{}, including language model, transformer, and reinforcement learning, which are initialized by introducing basic notations and terminologies. Let $\mathbf{x}=(x_1, x_2,\ldots, x_{|\mathbf{x}|})$ denote the code sequence of one function, where $ x_t$ represents a token of the code, e.g., ... “*def*”, “*fact*”, or “*i*” in a Python statement “*def fact(i)*:”. Let $\mathbf{y}=(y_1, y_2,\ldots, y_{|\mathbf{y}|})$ denote the sequence of the generated comments, where [$|\mathbf{y}|$]{} denotes the sequence length. Let $T$ denote the maximum step of decoding in the encoder-decoder framework. We use notation $y_{l \ldots m}$ to represent the comment subsequence $y_l, \ldots , y_m$ and $\mathcal{D}=\{(\mathbf{x}_{N},\mathbf{y}_{N})\}$ as the training dataset, where $N$ is the size of training set.
Language Model
--------------
A language model refers to the decoder of neural machine translation which is usually constructed as the probability distribution over a particular sequence of words. Assuming such sequence with its length $ T $, the language model defines $p(y_{1:T})$ as its occurrence probability which is usually computed based on the conditional probability from a window of $n$ predecessor words, known as $n$-gram [@wang2016bugram], as shown in Equation \[conditionprobability\].
$$p(y_{1:T})=\prod_{t=1}^{i=T}p(y_t|y_{1:t-1})\approx \prod_{t=1}^{t=T}p(y_t|y_{t-(n-1):t-1})
\label{conditionprobability}$$
While the $n$-gram model can only predict a word based on a fixed number of predecessor words, a neural language model can use predecessor words with longer distance to predict a word based on deep neural networks which include three layers: an input layer which maps each word $ x_t $ to a vector, a recurrent hidden layer which recurrently computes and updates a hidden state $ h_t $ after reading $ x_t $, and an output layer which estimates the probabilities of the subsequent words given the current hidden state. In particular, the neural network reads individual words from the input sentence, and predicts the subsequent word in turn. For the word $y_{t}$, the probability of its subsequent word $y_{t+1}$, $p(y_{t+1}|y_{1:t})$ can be computed as in Equation \[predictstate\]:
$$p(y_{t+1}|y_{1:t}) = g(\mathbf{h}_t)
\label{predictstate}$$
where $g$ is a stochastic output layer (e.g., a softmax for discrete outputs) that generates output tokens with the hidden state $\mathbf{h}_t$ computed as Equation \[hiddenstate\]:
$$\mathbf{h}_t = f(\mathbf{h}_{t-1}, w(x_t))
\label{hiddenstate}$$
where $w(x_t)$ denotes the weight of the token $x_t$.
Transformer
-----------
Many neural machine translation approaches integrate the attention mechanism with sequence transduction models for enhancing the accuracy. However, the encoding networks are still exposed with challenges. To be specific, the CNN-based encoding networks are subjected to long-distance dependency issues and the RNN-based encoding networks are subjected to the long-time computation. To address such issues, transformer [@vaswani2017attention] is proposed to effectively and efficiently improve the sequence representation by adopting the self-attention mechanism only. Many transformer-based models e.g., BERT [@devlin2018bert], ERNIE [@sun2019ernie], XLNET [@yang2019xlnet], have been proposed and verified to dramatically enhance the performance of various NLP tasks such as natural language inference , text classification, and retrieval question answering [@Bowman2015A; @Voorhees2001The].
Transformer consists of $ N $ identical layers where each layer consists of two sub-layers. The first sub-layer realizes a multi-head self-attention mechanism, and the second sub-layer is a simple, position-wise fully connected feed-forward neural network, as shown in Figure \[fig:transformer\]. Note that the output of the first sub-layer is input to the second sub-layer and the outputs of both the sub-layers need to be normalized prior to the subsequent process.
![The Transformer Model Architecture.[]{data-label="fig:transformer"}](./transformer.pdf){width="47.00000%"}
### Self-attention Mechanism
The attention function can be described as mapping a query and a set of key-value pairs to an output, where the query, keys, values, and output are all vectors. The output is computed as a weighted sum of the values, where the weight assigned to each value is computed by a compatibility function of the query with the corresponding key.
The input consists of queries, keys and values of the dimension $ d_k $. Accordingly, transformer computes the dot products of the query with all keys, divides each resulting element by $ \sqrt{d_k}$, and applies a softmax function to obtain the weights on the values. In practice, we simultaneously compute the attention function on a set of queries which are packed together into a matrix $ Q $. In addition, the keys and values are also packed together into matrices $ K $ and $ V $. Therefore, the matrix of outputs can be computed as:
$$Attention(Q,K,V) = softmax(\dfrac{QK^{T}}{\sqrt{d_k}})V
\label{attention}$$
Instead of implementing a single attention function, transformer adopts a multi-head attention which allows the model to jointly attend to information from different representation subspaces at different positions.
The self-attention mechanism derives the relationships between the current input token and all the other tokens to determine the current token vector for the final input representation. By taking advantage of the overall token weights, such mechanism can dramatically alleviate the long-distance dependency problem caused by the CNN-based transduction models, i.e., compromising the contributions of the long-distance tokens. Moreover, the multi-head self-attention mechanism can parallelize the computation and thus resolve the excessive computing overhead caused by the RNN-based transduction models which sequentially encode the input tokens.
### Position-wise Feed-Forward Neural Network
In addition to multi-head self-attention sub-layers, each of the layers contains a fully connected feed-forward neural network, which is applied to each position separately. Since transformer contains no recurrence or convolution, in order to utilize the order of the sequence, transformer injects “positional encodings” to the input embedding.
Since transformer has been verified to be dramatically effective and efficient in encoding word sequences, we infer that by representing code as a sequence, transformer can also be expected to excel in the encoding efficacy. Therefore, in [$TranS^{3}$]{}, we adopt transformer as the encoder.
Reinforcement Learning for Code Summarization
---------------------------------------------
In code summarization, reinforcement learning (RL)[@Thrun2005Reinforcement] refers to interacting with the ground truth, learning the optimal policy from the reward signals, and generating texts in the testing phase. It can potentially solve the exposure bias problem introduced by the maximum likelihood approaches which is used to train the RNN model. Specifically in the inference stage, a typical RNN model generates a sequence iteratively and predicts next token conditioned on its previously predicted ones that may never be observed in the training data [@yu2017seqgan]. Such a discrepancy between training and inference can become cumulative along with the sequence and thus prominent as the length of sequence increases. While in the reinforcement-learning-based framework, the reward, other than the probability of the generated sequence, is calculated to give feedback to train the model to alleviate such exposure bias problem. Such text generation process can be viewed as a Markov Decision Process (MDP) $\{state, action, policy, reward\}$. Specifically in the MDP settings, $state$ $\mathbf{s}_t$ at time $t$ consists of the code snippets $\mathbf{x}$ and the predicted words ${y_0,y_1,\ldots,y_t}$. The $action$ space is defined as the dictionary $\mathcal{Y}$ where the words are drawn, i.e., $y_t \subset \mathcal{Y}$. Correspondingly, the $state$ transition function $P$ is defined as $\mathbf{s}_{t+1} = \{\mathbf{s}_t, y_{t}\}$, where the $action$ (word) $y_{t}$ becomes a part of the subsequent $state$ $\mathbf{s}_{t+1}$ and the $reward$ $r_{t+1}$ can be derived. The objective of the generation process is to find a $policy$ that iteratively maximizes the expected $reward$ of the generated sentence sampled from the model’s $policy$, as shown in Equation \[rnncode\],
$$\underset{\theta}{\max}\mathcal{L}(\theta) = \underset{\theta}{\max}\mathbb{E}_{\underset{\hat{\mathbf{y}}\sim P_{\theta}(\cdot|\mathbf{x})}{\mathbf{x}\sim \mathcal{D}}}[R(\hat{\mathbf{y}},\mathbf{x})]
\label{rnncode}$$
where $\theta$ is the policy parameter to be learned, $\mathcal{D}$ is the training set, $\hat{\mathbf{y}}$ denotes the predicted $action$s/words, and $R$ is the reward function.
To learn the policy, many approaches have been proposed, which are mainly categorized into two classes [@sutton1998introduction]: (1) the policy-based approaches (e.g., Policy gradients [@williams1992simple]) which optimize the policy directly via policy gradient and (2) the value-based approaches (e.g., Q-learning [@watkins1992q]) which learn the Q-function, and at each time the agent selects the action with the highest Q-value. It has been verified that the policy-based approaches may suffer from a variance issue and the value-based approaches suffer from a bias issue [@keneshloo2018deep]. To address such issues, the Actor-Critic learning approach is proposed [@Konda2003Actor] to combine the strengths of both policy- and value-based approaches where the actor chooses an action according to the probability of each action and the critic assigns the value to the chosen action for speeding up the learning process for the original policy-based approaches.
In this paper, we adopt the actor-critic learning model for code summarization of [$TranS^{3}$]{}.
Illustrative Example {#sec:example}
====================
In this section, we use a sample Python code snippet to illustrate our motivation for unifying code summarization and code search. Figure \[fig:example\] shows the Python code snippet, the comment generated by our approach and its associated natural language query in our dataset. Traditional code search approaches usually compute the similarity scores of the query vector and the code snippet vectors for recommending and returning the relevant code snippets. On the other hand, provided the comment information, it is plausible to enhance the code search results by enabling an additional mapping process between the query and the comments corresponding to the code snippets. For example, in Figure \[fig:example\], given the query “*get the recursive list of target dependencies*”, although the code snippet can provide some information such as “*dependencies*”, “*target*”, which might be helpful for being recommended, its efficacy can be compromised due to the disturbing information such as “*dicts*”, “*set*”, “*pending*” in the code snippet. It is expected to enhance the search result by integrating the comment information during the searching process when it has the identical “*target*”, “*dependencies*” with the query. To this end, we infer that a better code search result can be expected if high-quality comment information can be integrated in the code search process.
![An example Python code snippet and the corresponding query and generated comment.[]{data-label="fig:example"}](./example.pdf){width="48.00000%"}
{width="70.00000%"}
The Approach of [$TranS^{3}$]{} {#sec:approach}
===============================
We formulate the research problem of integrating code summarization with code searchas as follows:
- First, we attempt to find a policy that generates a sequence of words $\mathbf{y}=(y_1, y_2,\ldots, y_{|\mathbf{y}|})$ from dictionary $\mathcal{Y}$ to annotate the code snippets in the corpus as their comments. Next, given a natural language query $\mathbf{x}=(x_1, x_2,\ldots, x_{|\mathbf{x}|})$, we aim to find the code snippets that can satisfy the query under the assistance of the generated comments.
![An example of Self-Attention Mechanism.[]{data-label="fig:attention"}](./attention.pdf){width="32.00000%"}
To address such research problem, we propose [$TranS^{3}$]{} with its framework shown in Figure \[fig:overview\], where Figure \[fig:overview\](a) presents the code summarization part and Figure \[fig:overview\](b) presents the code search part.
Transformer- and Tree-Transformer-based Encoder {#sec:encoder}
-----------------------------------------------
In [$TranS^{3}$]{}, we utilize transformer to build the encoder. Specifically, we develop the transformer-based encoder to encode the comments, the query and each program statement. Moreover, we develop a tree-transformer-based encoder that exploits the semantic granularity information of programs for enhancing the program encoding accuracy.
**Transformer-based Encoder.** The transformer-based encoder is initialized by embedding the input tokens into vectors via word embedding [@Mikolov2013Efficient]. Specifically, we tokenize the natural language comments/queries based on their intervals and the code based on a set of symbols, i.e., [{ . , " ’ : \* () ! - \_ (space)}]{}. Next, we apply word embedding to derive each token vector in one input sequence. Furthermore, for each token vector $x_i$, we derive its representation according to the self-attention mechanism as follows: (1) deriving the query vector $ q_i$, the key vector $k_i$, and the value vector $v_i$ by multiplying $x_i$ with a randomly-generated matrix, (2) computing the scores of $x_i$ from all the input token vectors by the dot product of $ q_i \cdot k_j $, where $j\epsilon [1, n]$ and $n$ denotes the number of input tokens, (3) dividing the scores of $x_i$ by $\sqrt{d_k}$ where $d_k$ denotes the dimension number of $k_i$ and normalizing the results by softmax to obtain the weights (contributions) of all the input token vectors, (4) multiplying such weights and their corresponding value vectors to obtain an interim vector space $v^{'}$, and (5) summing all the vectors in $v^{'}$ for deriving the final vector of $x_i$, $z_i$. As a result, all the token vectors are input to the feed-forward neural network to obtain the final representation vector of the input sequence of natural language comment.
We use Figure \[fig:attention\] as an example to illustrate how the transformer-based encoder works, where the token vectors of “*Software*” and “*Engineering*” are embedded as $x_1$ and $x_2$ respectively. For $x_1$, its corresponding $q_1$, $k_1$, and $v_1$ are derived in the beginning. Next, its scores from all the token vectors, i.e., $x_1$ and $x_2$, can be computed by $q_1 \cdot k_1$ (112) and $q_1 \cdot k_2$ (96). Assuming $d_k$ is 64, by dividing the resulting dot products by $\sqrt{d_k}$ and normalizing, the weights of $x_1$ and $x_2$ can be computed as 0.88 and 0.12. At last, $z_1$ can be derived by 0.88\*$v_1$ + 0.12\*$v_2$.
**Tree-Transformer-based Code Encoder.** It can be observed that well-formed source code can reflect the program semantics through its representations, e.g., the indents of Python. In general, in a well-formed program, the statement with fewer indents tends to indicate more abstracted semantics than the one with longer indents. Therefore, we infer that incorporating the indent-based semantic granularity information for encoding can inject program semantics for program comprehension and thus be promising to enhance the encoding accuracy. Such injection can potentially be advanced when leveraging transformer. In particular, in addition to the original self-attention mechanism which determines the token vector score by only importing the token-level weights, statement-level impacts can be injected by analyzing statement indents, obtaining the semantic hierarchy of the code, and realizing the hierarchical encoding process.
**Input** : ordered tree ()
**Output**: vector representation of the tree
node\_list $\leftarrow$ root node **return** Transformer(node\_list); node\_list.append(PostOrderTraverse($i$’s children)) return Transformer(node\_list)
In this paper, we design a tree-transformer-based encoder that incorporates indent-based semantic granularity for encoding programs. Firstly, we construct an ordered tree according to the indent information of a well-formed program. In particular, by reading the program statements in turn, we initialize the tree by building the root node out of the function definition statement. Next, we iteratively label each of the subsequent statements with an indent index assigned by counting the indents such that the statements with the same indent index $ i $ are constructed as the ordered sibling nodes and the preceding statement above such statement block with the indent index $i-1$ is constructed as their parent node. Secondly, we encode each node (i.e., each statement) of the tree into a vector by transformer. At last, we build the tree-transformer accordingly to further encode all the vector nodes of the tree for obtaining the code snippet representations. Specifically, we traverse the tree in a post-order manner. Assuming a node $n_i$ and its parent node $n_j$, if $n_i$ is a leaf node, we replace the vector of $n_j$, namely $V_{n_j}$ by the vector list {$V_{n_i}$, $V_{n_j}$} and subsequently traverse $n_j$’s other child nodes; otherwise, we traverse $n_i$’s child nodes. Next, we encode node $n_j$ with the updated vector list {$V_{n_i}$, $V_{n_j}$} by transformer when it has no child nodes. The tree-transformer encoding process is shown as Algorithm \[alg:tree\].
Figure \[fig:tree\] illustrates indent-based tree representation of the code snippet given in Figure \[fig:example\]. We use this example to describe how the tree-transformer-based encoder works. Specifically in Figure \[fig:example\], we construct nodes “Dependencies = set()”, “pending=set(roots)”, “while pending:” and “return list(...)” as siblings because they are assigned with the same indents and one-shorter-indent preceding statement “def DeepDependencyTargets(target\_dicts, roots):”, which is constructed as their parent node. Then, we encode all the statement nodes into vectors by transformer respectively. Next, as the root’s child nodes “Dependencies = set()” and “pending=set(roots)” are leaf nodes, we replace the root vector by the vector list of them three. Then, since the root’s child node “while pending:” is not the leaf node, we first encode its child node “if (r in dependencies):” with “continue” by transformer, and then encode the resulting vector with the siblings of “while pending:” and “if (r in dependencies):” together by transformer. At last, we encode the root node with all its child nodes to obtain the final representation of this code snippet.
Code Summarization
------------------
Initialized by collecting code snippets with their associated comments and forming $ <code; comment>$ pairs for training the code summarization model, the code summarization component is implemented via reinforcement learning (i.e., the actor-critic framework), where the actor network establishes an encoder-decoder mechanism to derive code comments and the critic network iteratively provides feedback for tuning the actor network. In particular, the actor network leverages a transformer-based or a tree-transformer-based encoder to encode the collected code into hidden space vectors and applies a transformer-based decoder to decode them to natural language comments. Next, by computing the similarity between the generated and the ground-truth comments, the critic network iteratively provides feedback for tuning the actor network. As a result, given a code snippet, its corresponding natural language comment can be generated based on the trained code summarization model.
![The Source Code and the Tree Structure.[]{data-label="fig:tree"}](./tree.pdf){width="37.00000%"}
### Actor Network.
The actor network is composed of an encoder and decoder.
**Encoder.** We construct the tree representation of the source code and establish a tree-transformer, described as Section \[sec:encoder\], to encode the source code into hidden space vectors for the code representation.
**Decoder.** After obtaining the code snippet representations, [$TranS^{3}$]{} implements the decoding process for them, i.e., generating comments from the hidden space, to derive their associated natural language comments.
The decoding process is launched by generating an initial decoding state $s_0 = \{x\}$ by encoding the given code snippet. At step $t$, state $s_t$ is generated to maintain the source code snippet and the previously generated words $y_{1...t-1}$, i.e., $s_t = \{x, y_{1...t-1}\}$. Specifically, the previously generated words $y_{1...t-1}$ are encoded into a vector by transformer and subsequently concatenated with state $s_{t-1}$. Our approach predicts the $t$th word by using a softmax function. Let $p(y_t|\mathbf{s}_t)$ denote the probability distribution of the $t$th word $y_t$ in the state $\mathbf{s}_t$, we can obtain the following equation: $$p(y_t|\mathbf{s}_t)=softmax(\mathbf{W}_s\mathbf{s}_t+\mathbf{b}_s)$$ Next, we update $\mathbf{s}_{t}$ to $\mathbf{s}_{t+1}$ to generate the next word. This process is iterated till it exceeds the max-step or generates the end-of-sequence (EOS) token for generating the whole comment corresponding to the code snippet.
### Critic Network
To enhance the accuracy of the generated code comments, [$TranS^{3}$]{} applies a critic network to approximate the value of the generated comments at time $t$ to issue a feedback to tune the network iteratively. Unlike the actor network which outputs a probability distribution, the critic network outputs a single value on each decoding step. To illustrate, given the generated comments and the reward function $r$, the value function $V$ is defined to predict the total reward from the state $\mathbf{s}_t$ at time $t$, which is formulated as follows,
$$\label{eq:value}
V(\mathbf{s}_t)=\mathbb{E}_{\overset{\mathbf{s}_{t+1:T},}{y_{t:T}}}\left [\sum_{l=0}^{T-t}r_{t+l}|y_{t+1},\cdots,y_{T}, \mathbf{h} \right]$$
where $T$ is the max step of decoding and $\mathbf{h}$ is the representation of code snippet. By applying the reward function, we can obtain an evaluation score (e.g., BLEU) when the generation process of the comment sequences is completed. Such process is terminated when the associated step exceeds $T$ or generates the end-of-sequence (EOS) token. For instance, a BLEU-based reward function can be calculated as:
$$\label{eq:bleu}
r=exp(\frac{1}{N} * \sum_{i=1}^{N}logp_n )$$
where $p_n=\frac{\sum_{n-gram \in c} count(n-gram)}{\sum_{n-gram\in c^{'}}count(n-gram)}$, and $c$ is the generated comment an $c^{'}$ is the ground truth.
### Model Training
For the actor network, the training objective is to minimize the negative expected reward, which is defined as $\mathcal{L}(\theta) =- \mathbb{E}_{y_{1,\ldots,T}\sim \pi}(\sum_{l=t}^{T}r_t)$, where $\theta$ is the parameter set of the actor network. Defining policy as the probability of a generated comment, we adopt the policy gradient approach to optimize the policy directly, which is widely used in reinforcement learning.
The critic network attempts to minimize the following loss function, $$\mathcal{L}(\phi) = \frac{1}{2}\left \| V(\mathbf{s}_t) - V_\phi(\mathbf{s}_t) \right \|^2$$ where $V(\mathbf{s}_t) $ is the target value, $V_\phi(\mathbf{s}_t)$ is the value predicted by the critic network with its parameter set $\phi$. Eventually, the training for comment generation is completed after $\mathcal{L}(\phi)$ converges.
Denoting all the parameters as $\Theta=\{\theta, \phi \}$, the total loss of our model can be represented as $\mathcal{L}(\Theta)=\mathcal{L}(\theta)+ \mathcal{L}(\phi)$. We employ stochastic gradient descend with the diagonal variant of AdaGrad [@duchi2011adaptive] to tune the parameters of [$TranS^{3}$]{} for optimizing the code summarization model.
Code Search
-----------
Given a natural language query, [$TranS^{3}$]{} encodes all the code snippets and the generated comments into vector sets by tree-transformer-based encoder and transformer-based encoder respectively, and encodes the query into a vector by transformer-based encoder. Next, we compute the similarity scores between the query vector and the vectors in both the code snippets vector set and the generated comments vector set. At last, we rank all the code snippets to recommend the search results derived from the linear combination of the two similarity score sets which are trained for optimality.
As shown in Figure \[fig:overview\] (b), we encode the code snippets and the generated comments into vector spaces $\{V_c\} $ and $\{V_s\}$ by the tree-transformer-based encoder and transformer-based encoder respectively. We also encode the given natural language query into a vector $V_q$ by transformer-based encoder. Next, we compute the similarity scores between $V_q$ and the vectors of the code snippet $i$ from both $\{V_c\} $ and $\{V_s\}$ as $sim(V_q, V_{ci})$ and $ sim(V_q, V_{si})$. Furthermore, we derive the weighted score of the code snippet $i$, $score_i$, by linearly combining $sim(V_q, V_{ci})$ and $ sim(V_q, V_{si})$, as shown in Equation \[eq:score\]. Eventually, we rank all the code snippets according to their $score$s for recommending the search results,
$$\label{eq:score}
score(Q, C) = \beta * sim(V_q, V_{ci}) + (1-\beta) * sim(V_q, V_{si})$$
where $ \beta $ is a parameter that ranges from 0 to 1 and determined after training, $sim()$ is computed by consine. Specially, given the query $q_i$, the training objective is to ensure $score(q_i, c_i) > score(q_i, c_j)$, where $c_j$ demonstrates the code snippets in the dataset expect $ c_i$.
Evaluation {#sec:experiment}
==========
We conduct a set of extensive experiments on the effectiveness and efficiency of *[$TranS^{3}$]{}* in terms of both the code summarization and code search components compared with state-of-the-art approaches.
Experimental Setups {#sec_dataset}
-------------------
To evaluate the performance of our proposed approach, we use the dataset presented in [@Barone2017A] where over 120,000 $<$code;comment$>$ pairs are collected from various Python projects in GitHub [@github] with 50,400 code tokens and 31,350 comment tokens in its vocabulary respectively. For cross validation, we shuffle the original dataset and use the first 60% for training, 20% for validation, and the rest for testing.
In our experiments, the word embedding size is set to 1280, the batch size is set to 2048, the layer size is set to 6, and the head number is set to 8. We pretrain both actor network and critic network with 20000 steps each, and train the actor-critic network with 100000 steps simultaneously. For the code search part, the comments in the dataset are utilized for the query. All the experiments in this paper are implemented with Python 3.5, and run on a computer with a 2.8 GHz Intel Core i7 CPU, 64 GB 1600 MHz DDR3 RAM, and a Saturn XT GPU with 24 GB memory running RHEL 7.5.
Result Analysis {#sec_result}
---------------
### Code summarization
To evaluate the code summarization component of [$TranS^{3}$]{}, we select several state-of-the-art approaches, i.e., Hybrid-DeepCom [@hu2019deep], CoaCor [@yao2019coacor], and AutoSum [@wan2018improving] for performance comparison with [$TranS^{3}$]{}. In particular, Hybrid-DeepCom [@hu2019deep] utilizes the AST sequence converted by traversing the AST as the code representation and input the AST sequence to the GRU-based NMT for code summarization via combining lexical and structure information. CoaCor [@yao2019coacor] utilizes the plain text of source code and an LSTM-based encoder-decoder framework for code summarization. AutoSum [@wan2018improving] utilizes a tree-LSTM-based NMT model and inputs the code snippet as plain text to the code-generation model with reinforcement learning for performance enhancement. Similarly, the evaluation for [$TranS^{3}$]{} is also designed to explore the performance of its different components, where [$TranS^{3}$]{}$_{base}$ adopts the transformer-based encoder; [$TranS^{3}$]{}$_{tree}$ adopts the tree-transformer-based encoder for source code; and [$TranS^{3}$]{}$_{tree+RL}$, i.e., the complete [$TranS^{3}$]{}, utilizes the tree-transformer-based encoder for source code and reinforcement learning for further enhancing the code summarization model.
We evaluate the performance of all the approaches based on four widely-used evaluation metrics adopted in neural machine translation and image captioning: BLEU [@papineni2002bleu], METEOR [@banerjee2005meteor], ROUGE [@lin2004rouge] and CIDER [@vedantam2015cider]. In particular, BLEU measures the average n-gram precision on a set of reference sentences with a penalty for short sentences. METEOR evaluates how well the results capture content from the references via recall which is computed via stemming and synonymy matching. ROUGE-L imports account sentence level structure similarity and identifies the longest co-occurrence in sequential n-grams. CIDER is a consensus-based evaluation protocol for image captioning that evaluates the similarity of the generated comments and the ground truth.
Approaches **BLEU-1** **METEOR** **ROUGE-L** **CIDER**
----------------------------- ------------ ------------ ------------- -----------
Hybrid-DeepCom 15.60 6.09 14.33 51.88
CoaCor 25.60 9.52 29.38 78.11
AutoSum 25.27 9.29 39.13 75.01
[$TranS^{3}$]{}$_{base}$ 27.69 10.26 41.87 81.01
[$TranS^{3}$]{}$_{tree}$ 32.05 11.74 45.92 84.56
[$TranS^{3}$]{}$_{tree+RL}$ **37.69** **13.52** **51.27** **87.24**
: Code summarization results with different metrics. (Best scores are in boldface.)[]{data-label="tab:summarization"}
[|p[0.4cm]{}||p[3.3cm]{}|p[3cm]{}|p[9cm]{}|]{} & **Issue link** &
--------------- --
**Generated**
**comment**
--------------- --
& **Feedback**\
1 & <https://github.com/mikunit567/GAE/issues/1> & Validate a given xsrf token by retrieving it. & *“Yes, this is correct. Validate a retrieved XSRF from the memory cache and then with the token perform an associated action.”*\
2 & <https://github.com/hamzafaisaljarral/scoop/issues/1> & Iterates through the glob nodes. & *“Yup you have got that right but for better understanding you have to look into django-shop documentation and look into django-cms documentation as well.”*\
3 & <https://github.com/rumd3x/PSP-POC/issues/1> & Combine two lists in a list. & *“The `pstats` package is used for creating reports from data generated by the Profiles class. The `add_callers` function is supposed to take a `source` list, and a `target` list, and return `new_callers` by combining the call results of both target and source by adding the call time.”*\
Table \[tab:summarization\] demonstrates the code summarization results of all the approaches in terms of the selected metrics. While the compared approaches achieve close performances, e.g., around 20% in terms of BLEU-1, [$TranS^{3}$]{} can approximate 38%. In particular, we can obtain the following detailed findings. First, we can observe that [$TranS^{3}$]{} can significantly outperform all the compared approaches in terms of all the evaluated metrics. For instance, the complete [$TranS^{3}$]{}, i.e., [$TranS^{3}$]{}$_{tree+RL}$ can outperform all the compared approaches from 47.2% to 141.6% in terms of BLEU-1. Such performance advantages can indicate the superiority of the transformer-enabled self-attention mechanism over the mechanisms, including the attention mechanism, that are adopted in other RNN-based approaches, because the self-attention mechanism can effectively capture the impacts of the overall text on all the tokens of the input sequences for better reflecting their semantics and thus optimizing the language model weights. Next, we can verify that each component of [$TranS^{3}$]{} is effective for enhancing the performance. For instance, by applying the tree-transformer-based encoder, [$TranS^{3}$]{}$_{tree}$ can dramatically outperform [$TranS^{3}$]{}$_{base}$ that only applies the transformer-based encoder by 15.7% in terms of BLEU-1. We can verify that our tree transformer based on identifying and leveraging the indent-based program semantic granularity can effectively strengthen the language model by augmenting the semantic level information for tokens. Moreover, by applying reinforcement learning, [$TranS^{3}$]{}$_{tree+RL}$ outperforms [$TranS^{3}$]{}, i.e., [$TranS^{3}$]{}$_{tree}$ by 17.5% in terms of BLEU-1, which can further verify the strength of reinforcement learning as verified in [@yao2019coacor; @wan2018improving]. Note that the performance of certain approaches, e.g., Hybrid-DeepCom, dramatically differs from its original performance in [@hu2019deep] mainly because of the training data and programming language differences.
We also conduct a set of case studies to further evaluate the effectiveness of [$TranS^{3}$]{}. In particular, we first collect Python projects from GitHub and input them to our [$TranS^{3}$]{}-trained model for generating their corresponding comments. Next, we issue such generated comments to the corresponding developers for their evaluations on the quality of the generated comments. In total, we received [24]{} responses, among which [11]{} developers confirmed the correctness of the generated comments to summarize their code snippets. In addition, [5]{} developers extended detailed explanations of the associated code which also expose their support to our generated comments. The rest responses are unrelated to the correctness of our generated comments. Table \[tab:study\] presents selected examples of the developer feedback where the first and second case indicate that the developers confirm the correctness of our generated comments while the third case reveals that the developer is supportive to the generated comment though he did not directly present it.
### Code search
To evaluate the effectiveness of the code search component of [$TranS^{3}$]{}, we select several state-of-the-art approaches for comparison. Firstly, for the aforementioned approaches Hybrid-DeepCom, AutoSum, and CoaCor, we utilize the generated comments of those approaches as the input for the code search part of [$TranS^{3}$]{}. In addition to further utilizing them for code search, we also compare [$TranS^{3}$]{} with DeepCS [@gu2018deep] which utilizes RNN to encode code and query and compute the distance between the code vector and the query vector for returning the code snippets with the closest vectors. The performance of code search is evaluated in terms of four widely-used metrics: MRR (Mean Reciprocal Rank) [@Craswell2009Mean], nDCG (normalized Discounted Cumulative Gain) [@Wang2013A] and Success Rate@k [@Xuan2016Relationship], where MRR measures the average reciprocal ranks of results given a set of queries and the reciprocal rank of a query is computed as the inverse of the rank of the first hit result; nDCG considers the ranking of the search results which evaluates the usefulness of result based on its position in the result list; and Success Rate@k measures the percentage of queries for which more than one correct result exist in the top $k$ ranked results.
Approaches **MRR** **nDCG** **SR@5** **SR@10**
----------------------------- ----------- ----------- ----------- ----------- --
DeepCS 48.41 58.85 57.44 66.78
CoaCor 59.33 67.51 **67.05** 73.58
Hybrid-DeepCom 50.92 59.92 60.52 68.35
AutoSum 57.68 63.52 63.43 70.16
[$TranS^{3}$]{}$_{base}$ 58.43 65.13 63.28 70.85
[$TranS^{3}$]{}$_{tree}$ 60.57 68.43 65.16 74.13
[$TranS^{3}$]{}$_{tree+RL}$ **62.37** **70.62** 66.95 **75.21**
: Code search accuracy compared with baselines. (Best scores are in boldface.)[]{data-label="tab:search"}
Table \[tab:search\] shows the code search result comparisons between our proposed approach and the aforementioned baselines where we can observe that [$TranS^{3}$]{} can outperform all the other approaches in terms of all the evaluate metrics. Specifically, in terms of MRR, [$TranS^{3}$]{}$_{tree+RL}$ can outperform all the other approaches from 5.12% to 28.8%. Compared with the code summarization results, the advantages of [$TranS^{3}$]{} over the same adopted approaches on code search dramatically shrinks which can be discussed as follows: (1) the code search metrics are naturally subject to less distinguishable results than the code summarization metrics. For Hybrid-DeepCom, AutoSum, and [$TranS^{3}$]{} which all utilize the generated comments to strengthen their code search performance, their adopted code summarization metrics are essentially based on word frequency which generally are fine-grained, e.g., BLEU-based metrics, while their code search metrics are generally based on coarse-grained query-wise comparisons. Therefore, the code summarization metrics tend to result in distinguishable results for different techniques because they are likely to reflect the trivial difference between two generated comments. However, their corresponding code search results might not be that distinguishable because the two generated comments might be trained to result in the result in the identical code rankings. For instance, suppose two code summarization approaches generate the comments “*returns the path of the target dependencies*” and “*derive a target-dependency list*” respectively. While they can be used to represent the same code snippets, they may result in different BLEU scores because they consist of different words. However, if they are used for code search, they can both rank the code snippet of Figure 2 on the top and thus result in the identical score in terms of the code search metrics. (2) CoaCor [@yao2019coacor] can approach a close performance to [$TranS^{3}$]{} because its rewarding mechanism utilizes the search accuracy to guide the code annotation generation and search modeling directly. However, We can observe that [$TranS^{3}$]{} significantly outperforms CoaCor in terms of code summarization (by 47.2%). Therefore, to bridge such performance gap, CoaCor has to pay extra effort for enhancing its modeling process while [$TranS^{3}$]{} can limit its effort in training the model once and for all for optimizing both code summarization and code search.
We also conduct a case study to evaluate the effectiveness of [$TranS^{3}$]{}. We organized 5 postgraduate students and 5 developers with certain Python background. We designed 15 programming tasks where each participant is asked to choose 3 tasks for code search using [$TranS^{3}$]{} as well as our benchmark. Two example tasks are listed as follows:
- Task 1: Remove all the files in a directory.
- Task 2: Sends a message to the admins.
Then, they are asked to evaluate if the searched code snippets can solve the tasks or are helpful for solving them, by giving a score on a five-point Likert scale (strongly agree is 5 and strongly disagree is 1). For the 10 participants, the average Likert score is 3.167 (with standard deviation of 1.472), which indicates that in general, the efficacy of [$TranS^{3}$]{} can be acceptable.
Threats to Validity {#sec:threats}
===================
There are several threats to validity of our proposed approach and its results, which are presented as follows.
The main threat to internal validity is the potential defects in the implementation of our techniques. To reduce such threat, we adopted a commonly-used benchmark with over 120,000 Python functions for evaluating the effectiveness and efficiency of our proposed approach and several existing approaches for comparison. Moreover, to ensure the fair comparison, we directly downloaded the optimized models of the existing approaches for comparison.
The threats to external validity mainly lie in the dataset quality and the evaluation metrics of our experiments. On one hand, the quality of the training data, i.e., the $ <code; comment> $ pairs adopted in our experiment was not evaluated. Among the over 120,000 python functions, it is likely that part of the poor-quality data can taint the training results. However, since (1) all the approaches were evaluated in the identical benchmark, and (2) the adopted evaluation metrics measure the performance of the approaches by word frequency where the corresponding performance difference among the approaches can indicate their word mapping levels, we can also infer that given high-quality training data, the performance distribution of all the approaches are likely to maintain consistency, where [$TranS^{3}$]{} can still outperform the other approaches in terms of the word-frequency metrics. Moreover, the performance of the tree-transformer-based encoder heavily relies on the quality of program forms. However, the experimental results indicate that the tree-transformer-based encoder can achieve better performance than the transformer-based encoder regardless the quality of the program forms. On the other hand, the word-frequency-based metrics cannot fully reflect the the semantic correctness of the approaches. To reduce such threat, we adopted a set of empirical studies such that developers can feedback for the quality of our code summarization and code search results. The positive study results can strengthen the validity of the effectiveness and efficiency of [$TranS^{3}$]{}.
Related Work {#sec:relatedwork}
============
Code Summarization
------------------
The code summarization techniques can be mainly categorized as information-retrieval-based approaches and deep-learning-based approaches.
**Information-retrieval-based approaches.** Wong et al. [@wong2013autocomment] proposed AutoComment which leverages code-description mappings to generate description comments for similar code segments matched in open-source projects. Similarly they also apply code clone detection techniques to find similar code snippets and extract comments from the similar code snippets [@wong2015clocom]. Movshovitz-Attias et al. [@movshovitz2013natural] predicted comments from Java source files using topic models and n-grams. Haiduc et al. [@Haiduc2010On] combined IR techniques, i.e., Vector Space Model and Latent Semantic Indexing, to generate terms-based summaries for Jave classes and methods.
**Deep-learning-based approaches.** The deep-learning-based approaches usually leverage Recurrent Neural Networks (RNNs) or Convolution neural networks (CNNs) with the attention mechanism. For instance, Iyer et al. [@iyer2016summarizing] proposed to use RNN with an attention mechanism—CODE-NN to produce comments for C\# code snippets and SQL queries. Allamanis et al. [@allamanis2016convolutional] proposed an attentional CNN on the input tokens to detect local time-invariant and long-range topical attention features to summarize code snippets into function name-like summaries. Considering the API information, Hu et al. [@hu2018summarizing] proposed TL-CodeSum to generate summaries by capturing semantics from the source code with the assistance of API knowledge. Chen et al. [@chen2018neural] proposed BVAE which utilizes C-VAE to encode code and L-VAE to encode natural language. In addition to such encoder-decoder-based approaches, Wan et al. [@wan2018improving; @drlcomment] drew on the insights of deep reinforcement learning to alleviate the exposure bias issue by integrating exploration and exploitation into the whole framework. Hu et al. [@hu2018deep] proposed DeepCom which takes AST sequence converted by traversing the AST as the input of NMT and they also extended this work by considering hybrid lexical and syntactical information in [@hu2019deep]. Leclair et al. [@leclair2019neural] combined words from code with code structure from AST, which allows the model to learn code structure independent of the text in code.
Code Search
-----------
Code search techniques also mainly consists of information-retrieval-based approaches and deep-learning-based approaches.
**Information-retrieval-based approaches.** Hill et al. [@Hill2014NL] proposed CONQUER which integrates multiple feedback mechanisms into the search results view. Some approaches proposed to extend the queries, for example, Lu et al. [@lu2015query] proposed to extend queries with synonyms generated from WordNet and then match them with phrases extracting from code identifiers to obtain the search results. Lv et al. [@lv2015codehow] designed a API understanding component to figure out the potential APIs and then expand the query with the potential APIs and retrieve relevant code snippets from the codebase. Similarly, Raghothaman et al. [@raghothaman2016swim] proposed *swim*, which first suggests an API set given a query by the natural language to API mapper that is extracted from clickthrough data in search engine, and then generates code using the suggested APIs by the synthesizer.
**Deep-learning-based approaches** The deep learning-based approaches usually encode the code snippets and natural language query into a hidden vector space, and then train a model to make the corresponding code and query vector more similar in the hidden space. Gu et al. [@gu2018deep] proposed DeepCS, which reads code snippets and embeds them into vectors. Then, given a query, it returns the code snippets with the nearest vectors to the query. Luan et al. [@luan2018aroma] proposed Aroma, which takes a code snippet as input, assembles a list of method bodies that contain the snippet, clusters and intersects those method bodies to offer code recommendations. Different from the above approaches, Akbar et al. [@akbar2019scor] presented a framework that incorporates both ordering and semantic relationships between the terms and builds one-hot encoding model to rank the retrieval results. Chen et al. [@chen2018neural] proposed BVAE, which includes C-VAE and L-VAE to encode code and query respectively, based on which semantic vector for both code and description and generate completely. Yao et al. [@yao2019coacor] proposed CoaCor, which designs a rewarding mechanism to guide the code annotation model directly based on how effectively the generated annotation distinguishes the code snippet for code retrieval.
**Other approaches.** Sivaraman et al. [@sivaraman2019active] proposed ALICE, which integrates active learning and inductive logic programming to incorporate partial user feedback and refine code search patterns. Takuya et al. [@takuya2011spontaneous] proposed Selene, which uses the entire editing code as query and recommends code based on a associative search engine. Lemons et al. [@Lemos2007CodeGenie] proposed a test-driven code search and reuse approach, which searches code according to the behavior of the desired feature to be searched.
Conclusion {#sec:conclusion}
==========
In this paper, we propose [$TranS^{3}$]{}, which is a transformer-based framework to integrate code summarization with code search. Specifically, [$TranS^{3}$]{} enables an actor-critic network for code summarization. In the actor network, we build transformer- and tree-transformer-based encoder to encode code snippets and decode the given code snippet to generate their comments. Meanwhile, we utilize the feedback from the critic network to iteratively tune the actor network for enhancing the quality of the generated comments. Furthermore, we import the generated comments to code search for enhancing its accuracy. We conduct a set of experimental studies and case studies to evaluate the effectiveness of [$TranS^{3}$]{}, where the experimental results suggest that [$TranS^{3}$]{} can significantly outperform multiple state-of-the-art approaches in both code summarization and code search and the study results further strengthen the efficacy of [$TranS^{3}$]{} from the developers’ points of view.
|
---
abstract: 'We give a rigorous characterization of what it means for a programming language to be [*memory safe*]{}, capturing the intuition that memory safety supports [*local reasoning about state*]{}. We formalize this principle in two ways. First, we show how a small memory-safe language validates a [*noninterference*]{} property: a program can neither affect nor be affected by unreachable parts of the state. Second, we extend separation logic, a proof system for heap-manipulating programs, with a “memory-safe variant” of its [*frame rule*]{}. The new rule is stronger because it applies even when parts of the program are buggy or malicious, but also weaker because it demands a stricter form of separation between parts of the program state. We also consider a number of pragmatically motivated variations on memory safety and the reasoning principles they support. As an application of our characterization, we evaluate the security of a previously proposed dynamic monitor for memory safety of heap-allocated data.'
author:
- Arthur Azevedo de Amorim
- Cătălin Hricu
- 'Benjamin C. Pierce'
- |
Arthur Azevedo de Amorim^1^ Cătălin Hricu^2^ Benjamin C. Pierce^1^\
^1^University of Pennsylvania ^2^Inria Paris
bibliography:
- 'refs.bib'
- 'mp.bib'
- 'safe.bib'
title: The Meaning of Memory Safety
---
=1 input[texdirectives]{}
Introduction {#sec:introduction}
============
Memory safety, and the vulnerabilities that follow from its absence [@Szekeres2013], are common concerns. So what [is]{} it, exactly? Intuitions abound, but translating them into satisfying formal definitions is surprisingly difficult [@Hicks:memory-safety].
In large part, this difficulty stems from the prominent role that informal, everyday intuition assigns, in discussions of memory safety, to a range of errors related to memory [*mis*]{}use—buffer overruns, double frees, etc. Characterizing memory safety in terms of the absence of these errors is tempting, but this falls short for two reasons. First, there is often disagreement on which behaviors qualify as errors. For example, many real-world C programs intentionally rely on unrestricted pointer arithmetic [@MemarianMLNCWS16], though it may yield undefined behavior according to the language standard [@ISO:C99 §6.5.6]. Second, from the perspective of security, the critical issue is not the errors themselves, but rather the fact that, when they occur in unsafe languages like C, the program’s ensuing behavior is determined by obscure, low-level factors such as the compiler’s choice of run-time memory layout, often leading to exploitable vulnerabilities. By contrast, in memory-safe languages like Java, programs can attempt to access arrays out of bounds, but such mistakes lead to sensible, predictable outcomes.
Rather than attempting a definition in terms of bad things that cannot happen, we aim to formalize memory safety in terms of [*reasoning principles*]{} that programmers can soundly apply in its presence (or conversely, principles that programmers should [*not*]{} naively apply in unsafe settings, because doing so can lead to serious bugs and vulnerabilities). Specifically, to give an account of [*memory*]{} safety, as opposed to more inclusive terms such as “type safety,” we focus on reasoning principles that are common to a wide range of stateful abstractions, such as records, tagged or untagged unions, local variables, closures, arrays, call stacks, objects, compartments, and address spaces.
What sort of reasoning principles? Our inspiration comes from [ *separation logic*]{} [@Reynolds:2002], a variant of Hoare logic designed to verify complex heap-manipulating programs. The power of separation logic stems from *local reasoning* about state: to prove the correctness of a program component, we must argue that its memory accesses are confined to a *footprint*, a precise region demarcated by the specification. This discipline allows proofs to ignore regions outside of the footprint, while ensuring that [arbitrary]{} invariants for these regions are preserved during execution.
The locality of separation logic is deeply linked to memory safety. Consider a hypothetical jpeg decoding procedure that manipulates image buffers. We might expect its execution not to interfere with the integrity of an unrelated window object in the program. We can formalize this requirement in separation logic by proving a specification that includes only the image buffers, but not the window, in the decoder’s footprint. Showing that the footprint is respected would amount to checking the bounds of individual buffer accesses, thus enforcing memory safety; conversely, if the decoder is not memory safe, a simple buffer overflow might suffice to tamper with the window object, thus violating locality and potentially paving the way to an attack.
Our aim is to extend this line of reasoning beyond conventional separation logic, seeking to encompass settings such as ML, Java, or Lisp that enforce memory safety automatically without requiring complete correctness proofs—which can be prohibitively expensive for large code bases, especially in the presence of third-party libraries or plugins over which we have little control. The key observation is that memory safety forces code to respect a natural footprint: the set of its reachable memory locations (reachable with respect to the variables it mentions). Suppose that the jpeg decoder above is written in Java. Though we may not know much about its input-output behavior, we can still assert that it cannot have any effect on the window object simply by replacing the detailed reasoning demanded by separation logic by a simple inaccessibility check.
Our *first contribution* is to formalize local reasoning principles supported by an ideal notion of memory safety, using a simple language () to ground our discussion. We show three results () that explain how the execution of a piece of code is affected by extending its initial heap. These results lead to a *noninterference* property (), ensuring that code cannot affect or be affected by unreachable memory. In , we show how these results yield a variant of the frame rule of separation logic (), which embodies its local reasoning capabilities. The two variants have complementary strengths and weaknesses: while the original rule applies to unsafe settings like C, but requires comprehensively verifying individual memory accesses, our variant does not require proving that every access is correct, but demands a stronger notion of separation between memory regions. These results have been verified with the Coq proof assistant [@coq-manual].[^1] Our *second contribution* () is to evaluate pragmatically motivated relaxations of the ideal notion above, exploring various trade-offs between safety, performance, flexibility, and backwards compatibility. These variants can be broadly classified into two groups according to reasoning principles they support. The stronger group gives up on some secrecy guarantees, but still ensures that pieces of code cannot modify the contents of unreachable parts of the heap. The weaker group, on the other hand, leaves gaps that completely invalidate reachability-based reasoning.
Our *third contribution* () is to demonstrate how our characterization applies to more realistic settings, by analyzing a heap-safety monitor for machine code [@pump_asplos2015; @micropolicies2015]. We prove that the abstract machine that it implements also satisfies a noninterference property, which can be transferred to the monitor via refinement, modulo memory exhaustion issues discussed in . These proofs are also done in Coq.[^2]
We discuss related work on memory safety and stronger reasoning principles in , and conclude in . While memory safety has seen prior formal investigation (e.g. [@NagarakatteZMZ09; @SwamyHMGJ06]), our characterization is the first phrased in terms of reasoning principles that are valid when memory safety is enforced automatically. We hope that these principles can serve as good criteria for formally evaluating such enforcement mechanisms in practice. Moreover, our definition is self-contained and does not rely on additional features such as full-blown capabilities, objects, module systems, etc. Since these features tend to depend on some form of memory safety anyway, we could see our characterization as a common core of reasoning principles that underpin all of them.
An Idealized Memory-Safe Language {#sec:imp}
=================================
Our discussion begins with a concrete case study: a simple imperative language with manual memory management. It features several mechanisms for controlling the effects of memory misuse, ranging from the most conventional, such as bounds checking for spatial safety, to more uncommon ones, such as assigning unique identifiers to every allocated block for ensuring temporal safety.
Choosing a language with manual memory management may seem odd, since safety is often associated with garbage collection. We made this choice for two reasons. First, most discussions on memory safety are motivated by its absence from languages like C that also rely on manual memory management. There is a vast body of research that tries to make such languages safer, and we would like our account to apply to it. Second, we wanted to stress that our characterization does not depend fundamentally on the mechanisms used to enforce memory safety, especially because they might have complementary advantages and shortcomings. For example, manual memory management can lead to more memory leaks; garbage collectors can degrade performance; and specialized type systems for managing memory [@SwamyHMGJ06; @Rust] are more complex. After a brief overview of the language, we explore its reasoning principles in .
Language Overview
-----------------
Command Description
----------------------------- ------------------
$x \gets e$ Local assignment
$x \gets [e]$ Read from heap
$[e_1] \gets e_2$ Heap assignment
$x \gets \calloc(e_{size})$ Allocation
$\cfree(e)$ Deallocation
$\cskip$ Do nothing
$\cifte{e}{c_1}{c_2}$ Conditional
$\cwhiledo{e}{c}$ Loop
$c_1; c_2$ Sequencing
[ $$\begin{aligned}
{2}
s \in \St & \teq \Ls \times \M && \text{ (states)} \\
l \in \Ls & \teq \var \partfunfin \V && \text{ (local stores)} \\
m \in \M & \teq \I \times \Z \partfunfin \V && \text{ (heaps)} \\
v \in \V & \teq \Z \uplus \B \uplus \{\nil\} \uplus \I \times \Z && \text{ (values)} \\
\Ot & \teq \St \uplus \{ \oerror \} && \text{ (outcomes)}\end{aligned}$$ $$\begin{aligned}
\I & \teq \text{a countably infinite set} \\
X \partfunfin Y & \teq \text{finite partial functions $X \partfun Y$}\end{aligned}$$ ]{}
summarizes the language syntax and other basic definitions. Expressions $e$ include variables $x \in \var$, numbers $n \in \Z$, booleans $b \in \B$, an invalid pointer $\nil$, and various operations, both binary (arithmetic, logic, etc.) and unary (extracting the offset of a pointer). We write $[e]$ for dereferencing the pointer denoted by $e$.
Programs operate on states consisting of two components: a *local store*, which maps variables to values, and a *heap*, which maps pointers to values. Pointers are not bare integers, but rather pairs $(i, n)$ of a *block identifier* $i \in \I$ and an offset $n \in \Z$. The offset is relative to the corresponding block, and the identifier $i$ need not bear any direct relation to the physical address that might be used in a concrete implementation on a conventional machine. (That is, we can equivalently think of the heap as mapping each identifier to a separate array of heap cells.) Similar structured memory models are widely used in the literature, as in the CompCert verified C compiler [@LeroyB08] and other models of the C language [@KangHMGZV15], for instance.
We write $\lsb c \rsb(s)$ to denote the outcome of running a program $c$ in an initial state $s$, which can be either a successful final state $s'$ or a fatal run-time error. Note that $\lsb c \rsb$ is partial, to account for non-termination. Similarly, $\lsb e \rsb (s)$ denotes the result of evaluating the expression $e$ on the state $s$ (expression evaluation is total and has no side effects). The formal definition of these functions is left to the Appendix; we just single out a few aspects that have a crucial effect on the security properties discussed later.
#### Illegal Memory Accesses Lead to Errors {#illegal-memory-accesses-lead-to-errors .unnumbered}
The language controls the effect of memory misuse by raising errors that stop execution immediately. This contrasts with typical C implementations, where such errors lead to unpredictable *undefined behavior*. The main errors are caused by reads, writes, and frees to the current memory $m$ using *invalid pointers*—that is, pointers $p$ such that $m(p)$ is undefined. Such pointers typically arise by offsetting an existing pointer out of bounds or by freeing a structure on the heap (which turns all other pointers to that block in the program state into dangling ones). In common parlance, this discipline ensures both *spatial* and *temporal* memory safety.
#### Block Identifiers are Capabilities {#block-identifiers-are-capabilities .unnumbered}
Pointers can only be used to access memory corresponding to their identifiers, which effectively act as capabilities. Identifiers are set at allocation time, where they are chosen to be fresh with respect to the entire current state ([i.e.,]{}the new identifier is not associated with any pointers defined in the current memory, stored in local variables, or stored on the heap). Once assigned, identifiers are immutable, making it impossible to fabricate a pointer to an allocated block out of thin air. This can be seen, for instance, in the semantics of addition, which allows pointer arithmetic but does not affect identifiers: $$\begin{aligned}
\lsb e_1 + e_2 \rsb(s) & \teq
\begin{cases}
n_1 + n_2 & \text{if $\lsb e_i\rsb(s) = n_i$} \\
(i,n_1+n_2) & \text{if $\lsb e_1\rsb(s) = (i,n_1)$ and $\lsb e_2\rsb(s) = n_2$} \\
\nil & \text{otherwise}
\end{cases}\end{aligned}$$ $$\begin{aligned}
\lsb e_1 + e_2 \rsb(s) & \teq
\begin{cases}
n_1 + n_2 & \text{if $\lsb e_i\rsb(s) = n_i$} \\
(i,n_1+n_2) & \text{if $\lsb e_1\rsb(s) = (i,n_1)$} \\
& \text{and $\lsb e_2\rsb(s) = n_2$} \\
\nil & \text{otherwise}
\end{cases}\end{aligned}$$ For simplicity, nonsensical combinations such as adding two pointers simply result in the $\nil$ value. A real implementation might represent identifiers with hardware tags and use an increasing counter to generate identifiers for new blocks (as done by Dhawan [[*et al.*]{}]{}[@pump_asplos2015]; see \[sec:monitor\]); if enough tags are available, every identifier will be fresh.
#### Block Identifiers Cannot be Observed {#block-identifiers-cannot-be-observed .unnumbered}
Because of the freshness condition above, identifiers can reveal information about the entire program state. For example, if they are chosen according to an increasing counter, knowing what identifier was assigned to a new block tells us how many allocations have been performed. A concrete implementation would face similar issues related to the choice of physical addresses for new allocations. (Such issues are commonplace in systems that combine dynamic allocation and information-flow control [@AmorimCDDHPPPT16].) For this reason, our language keeps identifiers opaque and inaccessible to programs; they can only be used to reference values in memory, and nothing else. We discuss a more permissive approach and its consequences in .
Note that hiding identifiers doesn’t mean we have to hide *everything* associated with a pointer: besides using pointers to access memory, programs can also safely extract their offsets and test if two pointers are equal (which means equality for both offsets and identifiers). Our Coq development also shows that it is sound to compute the size of a memory block via a valid pointer.
#### New Memory is Always Initialized {#new-memory-is-always-initialized .unnumbered}
Whenever a memory block is allocated, all of its contents are initialized to $0$. (The exact value does not matter, as long it is some constant that is not a previously allocated pointer.) This is important for ensuring that allocation does not leak secrets present in previously freed blocks; we return to this point in .
Reasoning with Memory Safety {#sec:reasoning}
============================
Having presented our language, we now turn to the reasoning principles that it supports. Intuitively, these principles allow us to analyze the effect of a piece of code by restricting our attention to a smaller portion of the program state. A first set of *frame theorems* (\[thm:frame-ok\], \[thm:frame-loop\], and \[thm:frame-error\]) describes how the execution of a piece of code is affected by extending the initial state on which it runs. These in turn imply a noninterference property, , guaranteeing that program execution is independent of inaccessible memory regions—that is, those that correspond to block identifiers that a piece of code does not possess. Finally, in , we discuss how the frame theorems can be recast in the language of separation logic, leading to a new variant of its frame rule ().
Basic Properties of Memory Safety {#sec:frame-theorems}
---------------------------------
$$\begin{aligned}
{2}
(l_1,m_1) \cup (l_2, m_2) & \teq (l_1 \cup l_2, m_1 \cup m_2) && \text{ (state union)}\\
(f \cup g)(x) & \teq
\begin{cases}
f(x) & \text{if $x \in \dom(f)$} \\
g(x) & \text{otherwise}
\end{cases} && \text{ (partial function union)} \\
\blocks(l, m) & \teq \{ i \in \I \mid \exists n, (i, n) \in \dom(m) \}
&& \text{ (identifiers of live blocks)} \\
\ids(l, m) & \teq \blocks(l, m) && \text{ (all identifiers in state)} \\
& \cup\{ i \mid \exists x, n, l(x) = (i, n) \} && \\
& \cup \{ i \mid \exists p, n, m(p) = (i, n) \} && \\
\vars(l, m) & \teq \dom(l) && \text{ (defined local variables)} \\
\vars(c) & \teq \text{local variables of program $c$} && \\
X \fresh Y & \teq (X \cap Y = \emptyset) && \text{ (disjoint sets)} \\
\pi \cdot s & \teq \text{rename identifiers with permutation $\pi$}\end{aligned}$$
summarizes basic notation used in our results. By *permutation*, we mean a function $\pi : \I \to \I$ that has a two-sided inverse $\pi^{-1}$; that is, $\pi \circ \pi^{-1} = \pi^{-1} \circ \pi = \mathsf{id}_{\I}$. Some of these operations are standard and omitted for brevity.[^3]
The first frame theorem states that, if a program terminates successfully, then we can extend its initial state almost without affecting execution.
\[thm:frame-ok\] Let $c$ be a command, and $s_1$, $s_1'$, and $s_2$ be states. Suppose that $\lsb c\rsb(s_1) = s_1'$, $\vars(c) \subseteq \vars(s_1)$, and $\blocks(s_1) \fresh \blocks(s_2)$. Then there exists a permutation $\pi$ such that $\lsb c\rsb(s_1 \cup s_2) = \pi \cdot s_1' \cup s_2$ and $\blocks(\pi \cdot s_1') \fresh \blocks(s_2)$.
The second premise, $\vars(c) \subseteq \vars(s_1)$, guarantees that all the variables needed to run $c$ are already defined in $s_1$, implying that their values do not change once we extend that initial state with $s_2$. The third premise, $\blocks(s_1) \fresh \blocks(s_2)$, means that the memories of $s_1$ and $s_2$ store disjoint regions. Finally, the conclusion of the theorem states that (1) the execution of $c$ does not affect the extra state $s_2$ and (2) the rest of the result is almost the same as $s_1'$, except for a permutation of block identifiers.
Permutations are needed to avoid clashes between identifiers in $s_2$ and those assigned to regions allocated by $c$ when running on $s_1$. For instance, suppose that the execution of $c$ on $s_1$ allocated a new block, and that this block was assigned some identifier $i \in \I$. If the memory of $s_2$ already had a block corresponding to $i$, $c$ would have to choose a different identifier $i'$ for allocating that block when running on $s_1 \cup s_2$. This change requires replacing all occurrences of $i$ by $i'$ in the result of the first execution, which can be achieved with a permutation that swaps these two identifiers. [^4]
The proof of relies crucially on the facts that programs cannot inspect identifiers, that memory can grow indefinitely (a common assumption in formal models of memory), and that memory operations fail on invalid pointers. Because of the permutations, we also need to show that permuting the initial state $s$ of a command $c$ with any permutation $\pi$ yields the same outcome, up to some additional permutation $\pi'$ that again accounts for different choices of fresh identifiers.
\[thm:renaming\] Let $s$ be a state, $c$ a command, and $\pi$ a permutation. There exists $\pi'$ such that:
c(s) =
&\
&\
’ s’ &
A similar line of reasoning yields a second frame theorem, which says that we cannot make a program terminate just by extending its initial state.
\[thm:frame-loop\] \
Let $c$ be a command, and $s_1$ and $s_2$ be states. If $\lsb c\rsb(s_1) = \bot$, $\vars(c) \subseteq \vars(s_1)$, and $\blocks(s_1) \fresh \blocks(s_2)$, then $\lsb c\rsb(s_1 \cup s_2) = \bot$.
The third frame theorem shows that extending the initial state also preserves erroneous executions. Its statement is similar to the previous ones, but with a subtle twist. In general, by extending the state of a program with a block, we might turn an erroneous execution into a successful one—if the error was caused by accessing a pointer whose identifier matches that new block. To avoid this, we need a different premise ($\ids(s_1) \fresh \blocks(s_2)$) preventing any pointers in the original state $s_1$ from referencing the new blocks in $s_2$—which is only useful because our language prevents programs from forging pointers to existing regions. Since $\blocks(s) \subseteq \ids(s)$, this premise is stronger than the analogous ones in the preceding results.
\[thm:frame-error\] \
Let $c$ be a command, and $s_1$ and $s_2$ be states. If $\lsb c\rsb(s_1) = \oerror$, $\vars(c)\subseteq \vars(s_1)$, and $\ids(s_1) \fresh \blocks(s_2)$, then $\lsb c\rsb(s_1 \cup s_2) = \oerror$.
Memory Safety and Noninterference
---------------------------------
The consequences of memory safety analyzed so far are intimately tied to the notion of *noninterference* [@GoguenM82]. In its most widely understood sense, noninterference is a *secrecy* guarantee: varying secret inputs has no effect on public outputs. Sometimes, however, it is also used to describe *integrity* guarantees: low-integrity inputs have no effect on high-integrity outputs. In fact, both guarantees apply to unreachable memory in our language, since they do not affect code execution; that is, execution (1) cannot modify these inaccessible regions (preserving their integrity), and (2) cannot learn anything meaningful about them, not even their presence (preserving their secrecy).
\[cor:noninterference\] Let $s_1$, $s_{21}$, and $s_{22}$ be states and $c$ be a command. Suppose that $\vars(c) \subseteq \vars(s_1)$, that $\ids(s_1) \fresh \blocks(s_{21})$ and that $\ids(s_1) \fresh \blocks(s_{22})$. When running $c$ on the extended states $s_1 \cup s_{21}$ and $s_1 \cup s_{22}$, only one of the following three possibilities holds: (1) both executions loop ($\lsb c\rsb(s_1 \cup s_{21}) = \lsb c\rsb(s_1 \cup s_{22}) = \bot$); (2) both executions terminate with an error ($\lsb c\rsb(s_1 \cup s_{21}) = \lsb c\rsb(s_1 \cup s_{22}) = \oerror$); or (3) both executions successfully terminate without interfering with the inaccessible portions $s_{21}$ and $s_{22}$ (formally, there exists a state $s_1'$ and permutations $\pi_1$ and $\pi_2$ such that $\lsb c\rsb(s_1 \cup s_{2i}) = \pi_i \cdot s_1' \cup s_{2i}$ and $\ids(\pi_i \cdot s_1') \fresh \blocks(s_{2i})$, for $i = 1, 2$).
- Both executions loop: $\lsb c\rsb(s_1 \cup s_{21}) {=} \lsb c\rsb(s_1 \cup
s_{22}) {=} \bot$;
- both executions terminate with an error:\
$\lsb c\rsb(s_1 \cup s_{21}) = \lsb
c\rsb(s_1 \cup s_{22}) = \oerror$; or
- both executions successfully terminate without interfering with the inaccessible portions $s_{21}$ and $s_{22}$. Formally, there exists a state $s_1'$ and permutations $\pi_1$ and $\pi_2$ such that $\lsb c\rsb(s_1 \cup
s_{2i}) = \pi_i \cdot s_1' \cup s_{2i}$ and $\ids(\pi_i \cdot s_1') \fresh
\blocks(s_{2i})$, for $i = 1, 2$.
Consider the result of executing $c$ on $s_1$. If $\lsb c\rsb(s_1) = \bot$, we apply twice using $s_{21}$ and $s_{22}$ as the unreachable states (recall that $\ids(s_1) \fresh \blocks(s_{2i})$ implies $\blocks(s_1) \fresh \blocks(s_{2i})$). If $\lsb c\rsb(s_1) = \oerror$, it suffices to apply twice. And finally, if $\lsb
c\rsb(s_1) = s_1'$, we just apply twice.
Noninterference is often formulated using an *indistinguishability relation* on states, which expresses that one state can be obtained from the other by varying its secrets. We could have equivalently phrased the above result in a similar way. Recall that the hypothesis $\ids(s_1) \fresh \blocks(s_2)$ means that memory regions stored in $s_2$ are unreachable via $s_1$. Then, we could call two states “indistinguishable” if the reachable portions are the same (except for a possible permutation). In , the connection with noninterference will provide a good benchmark for comparing different flavors of memory safety.
Memory Safety and Separation Logic {#sec:separation}
----------------------------------
We now explore the relation between the principles identified above, especially regarding integrity, and the local reasoning facilities of separation logic. Separation logic targets specifications of the form $\triple{p}{c}{q}$, where $p$ and $q$ are predicates over program states (subsets of $\St$). For our language, this could roughly mean $$\begin{aligned}
\forall s \in p, &\, \vars(c) \subseteq \vars(s) \Rightarrow \lsb c\rsb (s)
\in q \cup \{\bot\}.\end{aligned}$$ That is, if we run $c$ in a state satisfying $p$, it will either diverge or terminate in a state that satisfies $q$, but it will not trigger an error. Part of the motivation for precluding errors is that in unsafe settings like C they yield undefined behavior, destroying all hope of verification.
Local reasoning in separation logic is embodied by the *frame rule*, a consequence of . Roughly, it says that a verified program can only affect a well-defined portion of the state, with all other memory regions left untouched.[^5]
\[thm:frame-rule\] Let $p$, $q$, and $r$ be predicates over states and $c$ be a command. The rule $$\inferrule*[Right=Frame]
{ \independent(r, \modvars(c)) \\ \triple{p}{c}{q} }
{ \triple{p * r}{c}{q * r} }$$ is sound, where $\modvars(c)$ is the set of local variables modified by $c$, $\independent(r, V)$ means that the assertion $r$ does not depend on the set of local variables $V$ $$\forall l_1\,l_2\,m, (\forall x \notin V, \; l_1(x) = l_2(x))
\Rightarrow (l_1, m) \in r \Rightarrow (l_2, m) \in r,$$ and $p * r$ denotes the *separating conjunction* of $p$ and $r$: $$\{ (l, m_1 \cup m_2) \mid (l, m_1) \in p, (l, m_2) \in r,
\blocks(l, m_1) \fresh \blocks(l, m_2) \}.$$
As useful as it is, precluding errors during execution makes it difficult to use separation logic for *partial verification*: proving [*any*]{} property, no matter how simple, of a nontrivial program requires detailed reasoning about its internals. Even the following seemingly vacuous rule is unsound in separation logic: $$\inferrule*[Right=Taut]{ }{\triple{p}{c}{\ctrue}}$$ For a counterexample, take $p$ to be $\ctrue$ and $c$ to be some arbitrary memory read $x \gets [y]$. If we run $c$ on an empty heap, which trivially satisfies the precondition, we obtain an error, contradicting the specification.
Fortunately, our memory-safe language—in which errors have a sensible, predictable semantics, as opposed to wild undefined behavior—supports a variant of separation logic that allows looser specifications of the form $\triple{p}{c}{q}_e$, defined as $$\begin{aligned}
\forall s \in p, &\, \vars(c) \subseteq \vars(s) \Rightarrow \lsb c\rsb (s)
\in q \cup \{\bot, \oerror\}.\end{aligned}$$
These specifications are weaker than their conventional counterparts, leading to a subsumption rule: $$\inferrule{ \triple{p}{c}{q} }{ \triple{p}{c}{q}_e}$$
Because errors are no longer prevented, the <span style="font-variant:small-caps;">Taut</span> rule $\triple{p}{c}{\ctrue}_e$ Because errors are no longer prevented, the <span style="font-variant:small-caps;">Taut</span> rule $$\inferrule*[Right=Taut]{ }{\triple{p}{c}{\ctrue}_e}$$ becomes sound, since the $\ctrue{}$ postcondition now means that any outcome whatsoever is acceptable. Unfortunately, there is a price to pay for allowing errors: they compromise the soundness of the frame rule. The reason, as hinted in the introduction, is that preventing run-time errors has an additional purpose in separation logic: it forces programs to act locally—that is, to access only the memory delimited their pre- and postconditions. To see why, consider the same program $c$ as above, $x \gets [y]$. This program clearly yields an error when run on an empty heap, implying that the triple $\triple{\mathsf{emp}}{c}{x = 0}_e$ $$\triple{\mathsf{emp}}{c}{x = 0}_e$$ is valid, where the predicate $\mathsf{emp}$ holds of any state with an empty heap and $x = 0$ holds of states whose local store maps $x$ to $0$. Now consider what happens if we try to apply an analog of the frame rule to this triple using the frame predicate $y \mapsto 1$, which holds in states where $y$ contains a pointer to the unique defined location on the heap, which stores the value $1$. After some simplification, we arrive at the specification $\triple{y \mapsto 1}{c}{x = 0 \wedge y \mapsto 1}_e$, $$\triple{y \mapsto 1}{c}{x = 0 \wedge y \mapsto 1}_e,$$ which clearly does not hold, since executing $c$ on a state satisfying the precondition leads to a successful final state mapping $x$ to $1$.
For the frame rule to be recovered, it needs to take errors into account. The solution lies on the reachability properties of memory safety: instead of enforcing locality by preventing errors, we can use the fact that memory operations in a safe language are automatically local—in particular, local to the identifiers that the program possesses.
\[thm:weak-frame-rule\] Under the same assumptions as , the following rule is sound $$\inferrule*[Right=SafeFrame]
{ \independent(r, \modvars(c)) \and \triple{p}{c}{q}_e }
{ \triple{p \triangleright r}{c}{q \triangleright r}_e }$$ where $p \mathrel{\triangleright} r$ denotes the *isolating conjunction* of $p$ and $r$, defined as $$\{ (l, m_1 \cup m_2) \mid (l, m_1) \in p, (l, m_2) \in r,
\ids(l, m_1) \fresh \blocks(l, m_2) \}.$$ $$\begin{aligned}
\{ (l, m_1 \cup m_2) \mid & \;(l, m_1) \in p, (l, m_2) \in r, \\
& \;\ids(l, m_1) \fresh \blocks(l, m_2) \}.
\end{aligned}$$
The proof is similar to the one for the original rule, but it relies additionally on . This explains why the isolating conjunction is needed, since it ensures that the fragment satisfying $r$ is unreachable from the rest of the state.
Discussion
----------
As hinted by their connection with the frame rule, the theorems of are a form of local reasoning: to reason about a command, it suffices to consider its reachable state; *how* this state is used bears no effect on the unreachable portions. In a more realistic language, reachability might be inferred from additional information such as typing. But even here it can probably be accomplished by a simple check of the program text.
For example, consider the hypothetical jpeg decoder from . We would like to guarantee that the decoder cannot tamper with an unreachable object—a window object, a whitelist of trusted websites, etc. The frame theorems give us a means to do so, provided that we are able to show that the object is indeed unreachable; additionally, they imply that the jpeg decoder cannot directly extract any information from this unreachable object, such as passwords or private keys.
Many real-world attacks involve direct violations of these reasoning principles. For example, consider the infamous Heartbleed attack on OpenSSL, which used out-of-bounds reads from a buffer to leak data from completely unrelated parts of the program state and to steal sensitive information [@DurumericKAHBLWABPP14]. Given that the code fragment that enabled that attack was just manipulating an innocuous array, a programmer could easily be fooled into believing (as probably many have) that that snippet could not possibly access sensitive information, allowing that vulnerability to remain unnoticed for years.
Finally, our new frame rule only captures the fact that a command cannot influence the heap locations that it cannot reach, while our noninterference result () captures not just this integrity aspect of memory safety, but also a secrecy aspect. We hope that future research will explore the connection between the secrecy aspect of memory safety and (relational) program logics.
Relaxing Memory Safety {#sec:relaxations}
======================
So much for formalism. What about reality?
Strictly speaking, the security properties we have identified do not hold of any real system. This is partly due to fundamental physical limitations—real systems run with finite memory, and interact with users in various ways that transcend inputs and outputs, notably through time and other side channels.[^6] A more interesting reason is that real systems typically do not impose all the restrictions required for the proofs of these properties. Languages that aim for safety generally offer relatively benign glimpses of their implementation details (such accessing the contents of uninitialized memory, extract physical addresses from pointers or compare them for ordering) in return for significant flexibility or performance gains. In other systems, the concessions are more fundamental, to the extent that it is harder to clearly delimit what part of a program is unsafe: the SoftBound transformation [@NagarakatteZMZ09], for example, adds bounds checks for C programs, but does not protect against memory-management bugs; a related transformation, CETS [@NagarakatteZMZ10], is required for temporal safety.
In this section, we enumerate common relaxed models of memory safety and evaluate how they affect the reasoning principles and security guarantees of . Some relaxations, such as allowing pointers to be forged out of thin air, completely give up on reachability-based reasoning. Others, however, retain strong guarantees for integrity while giving up on some secrecy, allowing aspects of the global state of a program to be observed. For example, a system with finite memory () may leak some information about its memory consumption, and a system that allows pointer-to-integer casts () may leak information about its memory layout. Naturally, the distinction between integrity and secrecy should be taken with a grain of salt, since the former often depends on the latter; for example, if a system grants privileges to access some component when given with the right password, a secrecy violation can escalate to an integrity violation!
Forging Pointers {#sec:forging-pointers}
----------------
Many real-world C programs rely on using use integers as pointers. If this idiom is allowed without restrictions, then robust local reasoning is compromised, as every memory region may be reached from anywhere in the program. It is not surprising that languages that strive for memory safety either forbid this kind of pointer forging or confine it to clearwell-delimited unsafe fragments.
More insidiously, and perhaps surprisingly, similar dangers also lurk in the stateful abstractions of some systems that are widely regarded as “memory safe.” JavaScript, for example, allows code to access *arbitrary* global variables by indexing an associative array with a string, a feature that enables many serious attacks [@caja; @FournetSCDSL13; @TalyEMMN11; @MeyerovichL10]. One might argue that global variables in JavaScript are “memory unsafe” because they fail to validate local reasoning: even if part of a JavaScript program does not explicitly mention a given global variable, it might still change this variable or the objects it points to. Re-enabling local reasoning requires strong restrictions on the programming style [@BhargavanDM13; @caja; @FournetSCDSL13].
Observing Pointers {#sec:observing-pointers}
------------------
The language of maintains a complete separation between pointers and other values. In reality, this separation is often only enforced in one direction. For example, some tools for enforcing memory safety in C [@NagarakatteZMZ09; @DeviettiBMZ08] allow pointer-to-integer casts [@KangHMGZV15] (a feature required by many low-level idioms [@cheri_asplos2015; @MemarianMLNCWS16]); and the default implementation of `hashCode()` in Java leaks address information. To model such features, we can extend the syntax of expressions with a form $\cast(e)$, the semantics of which are defined with some function $\lsb \cast\rsb : \I\times \Z \to \Z$ for converting a pointer to an integer: $$\begin{aligned}
\lsb\cast(e)\rsb(s) & = \lsb\cast\rsb(\lsb e\rsb(s)) \qquad \text{ if $\lsb
e\rsb(s) \in \I \times \Z$}\end{aligned}$$
Note that the original language included an operator for extracting the offset of a pointer. Their definitions are similar, but have crucially different consequences: while offsets do not depend on the identifier, allocation order, or other low-level details of the language implementation (such as the choice of physical addresses when allocating a block), all of these could be relevant when defining the semantics of $\cast$. The three frame theorems (\[thm:frame-ok\], \[thm:frame-loop\], and \[thm:frame-error\]) are thus lost, because the state of unreachable parts of the heap may influence integers observed by the program. An important consequence is that secrecy is weakened in this language: an attacker could exploit pointers as a side-channel to learn secrets about data it shouldn’t access.
Nevertheless, *integrity* is not affected: if a block is unreachable, its contents will not change at the end of the execution. (This result was also proved in Coq.)
\[thm:integrity-noninterference\] Let $s_1$, $s_2$, and $s'$ be states and $c$ a command such that $\vars(c)
\subseteq \vars(s_1)$, $\ids(s_1) \fresh \blocks(s_2)$, and $\lsb c\rsb(s_1
\cup s_2) = s'$. Then we can find $s_1' \in \St$ such that $s' = s_1' \cup
s_2$ and $\ids(s_1') \fresh \blocks(s_2)$.
The stronger noninterference result of showed that, if pointer-to-integer casts are prohibited, changing the contents of the unreachable portion $s_2$ has no effect on the reachable portion, $s_1'$. In contrast, Theorem \[thm:integrity-noninterference\] allows changes in $s_2$ to influence $s_1'$ in arbitrary ways in the presence of these casts: not only can the contents of this final state change, but the execution can also loop forever or terminate in an error.
To see why, suppose that the jpeg decoder of is part of a web browser, but that it does not have the required pointers to learn the address that the user is currently visiting. Suppose that there is some relation between the memory consumption of the program and that website, and that there is some correlation between the memory consumption and the identifier assigned to a new block. Then, by allocating a block and converting its pointer to a integer, the decoder might be able to infer useful information about the visited website [@JanaS12a]. Thus, if $s_2$ denoted the part of the state where that location is stored, changing its contents would have a nontrivial effect on $s_1'$, the part of the state that the decoder does have access to. We could speculate that, in a reasonable system, this channel can only reveal information about the layout of unreachable regions, and not their contents. Indeed, we conjecture this for the language of variant of our language considered in this subsection.
Finally, it is worth noting that simply excluding casts might not suffice to prevent this sort of vulnerability. Recall that our language takes both offsets and identifiers into account for equality tests. For performance reasons, we could have chosen a different design that only compares physical addresses, completely discarding identifiers. If attackers know the address of a pointer in the program—which could happen, for instance, if they have access to the code of the program and of the allocator—they can use pointer arithmetic (which is generally harmless and allowed in our language) to find the address of other pointers. If $x$ holds the pointer they control, they can run, for instance, $$y \gets \calloc(1); \cifte{x + 1729 = y}{\ldots}{\ldots},$$ to learn the location assigned to $y$ and draw conclusions about the global state.
Uninitialized Memory {#sec:uninitialized}
--------------------
Safe languages typically initialize new variables and objects. But this can degrade performance, leading to cases where this feature is dropped—including standard C implementations, safer alternatives [@NagarakatteZMZ09; @DeviettiBMZ08], OCaml’s `Bytes.create` primitive, or Node.js’s `Buffer.allocUnsafe`, for example.
The problem with this concession is that the entire memory becomes relevant to execution, and local reasoning becomes much harder. By inspecting old values living in uninitialized memory, an attacker can learn about parts of the state they shouldn’t access and violate secrecy. This issue would become even more severe in a system that allowed old pointers or other capabilities to occur in re-allocated memory in a way that the program can use, since they could yield access to restricted resources directly, leading to potential integrity violations as well. (The two examples given above—OCaml and Node.js—do not suffer from this issue, because any preexisting pointers in re-allocated memory are treated as bare bytes that cannot be used to access memory.)
Dangling Pointers and Freshness {#sec:freshness}
-------------------------------
Another crucial issue is the treatment of dangling pointers—references to previously freed objects. Dangling pointers are problematic because there is an inherent tension between giving them a sensible semantics (for instance, one that validates the properties of ) and obtaining good performance and predictability. Languages with garbage collection avoid the issue by forbidding dangling pointers altogether—heap storage is freed only when it is unreachable. In the language of \[sec:imp\], besides giving a well-defined behavior to the use of dangling pointers (signaling an error), we imposed strong freshness requirements on allocation, mandating not only that the new identifier not correspond to any existing block, but also that it not be present [*anywhere else*]{} in the state.
To see how the results of are affected by weakening freshness, suppose we run the program $x \gets \calloc(1); z \gets (y = x)$ $$x \gets \calloc(1); z \gets (y = x)$$ on a state where $y$ holds a dangling pointer. Depending on the allocator and the state of the memory, the pointer assigned to $x$ could be equal to $y$. Since this outcome depends on the entire state of the system, not just the reachable memory, now fail. Furthermore, an attacker with detailed knowledge of the allocator could launder secret information by testing pointers for equality. Weakening freshness can also have integrity implications, since it becomes harder to ensure that blocks are properly isolated. For instance, a newly allocated block might be reachable through a dangling pointer controlled by an attacker, allowing them to access that block even if they were not supposed to.
Some practical solutions for memory safety use mechanisms similar to our language’s, where each memory location is tagged with an identifier describing the region it belongs to [@ClauseDOP07; @pump_asplos2015]. Pointers are tagged similarly, and when a pointer is used to access memory, a violation is detected if its identifier does not match the location’s. However, for performance reasons, the number of possible identifiers might be limited to a relatively small number, such as 2 or 4 [@ClauseDOP07] or 16 [@m7negative]. In addition to the problems above, since multiple live regions can share the same identifier in such schemes, it might be possible for buffer overflows to lead to violations of secrecy and integrity as well.
Although we framed our discussion in terms of identifiers, the issue of freshness can manifest itself in other ways. For example, many systems for spatial safety work by adding base and bounds information to pointers. In some of these [@DeviettiBMZ08; @NagarakatteZMZ09], dangling pointers are treated as an orthogonal issue, and it is possible for the allocator to return a new memory region that overlaps with the range of a dangling pointer, in which case the new region will not be properly isolated from the rest of the state.
Finally, dangling pointers can have disastrous consequences for overall system security, independently of the freshness issues just described: freeing a pointer more than once can break allocator invariants, enabling attacks [@Szekeres2013].
Infinite Memory {#sec:infinite-memory}
---------------
Our idealized language allows memory to grow indefinitely. But real languages run on finite memory, and allocation fails when programs run out of space. Besides enabling denial-of-service attacks, finite memory has consequences for secrecy. does not hold in a real programming language as is, because an increase in memory consumption can cause a previously successful allocation to fail. By noticing this difference, a piece of code might learn something about the *entire* state of the program. How problematic this is in practice will depend on the particular system under consideration.
A potential solution is to force programs that run out of memory to terminate immediately. Though this choice might be bad from an availability standpoint, it is probably the most benign in terms of secrecy. We should be able to prove an *error-insensitive* variant of , where the only significant effect that unreachable memory can have is to turn a successful execution or infinite loop into an error. Similar issues arise for information-flow control IFC mechanisms that often cannot prevent secrets from influencing program termination, leading to *termination-insensitive* notions of noninterference. Unfortunately, even an error-insensitive result might be too strong for real systems, which often make it possible for attackers to extract multiple bits of information about the global state of the program—as previously noted in the IFC literature [@askarov08:tini_leaks_more_than_1_bit]. Java, for example, does not force termination when memory runs out, but triggers an exception that can be caught and handled by user code, which is then free to record the event and probe the allocator with a different test. And most languages do not operate in batch mode like ours does, merely producing a single answer at the end of execution; rather, their programs continuously interact with their environment through inputs and outputs, allowing them to communicate the exact amount of memory that caused an error.
This discussion suggests that, if size vulnerabilities are a real concern, they need to be treated with special care. One approach would be to limit the amount of memory an untrusted component can allocate(as done for instance by Yang and Mazières [@Yang:2014]) [@Yang:2014], so that exhausting the memory allotted to that component doesn’t reveal information about the state of the rest of the system (and so that also global denial-of-service attacks are prevented). A more speculative idea is to develop *quantitative* versions [@Smith09; @BackesKR09] of the noninterference results discussed here that apply only if the total memory used by the program is below a certain limit.
Side-channel Attacks {#sec:physics}
--------------------
As often done in the information-flow control literature, our main results assume the code does not leak information through side-channels. In practice, attackers may learn secrets about unreachable memory regions by observing differences in execution time caused by caches, which are normally shared by all the code. While the attacker model considered in this paper does not try to address such side-channel attacks, one should be able to use the previous research on the subject to protect against them or limit the damage they can cause [@Smith09; @BackesKR09; @ZhangAM12; @StefanBYLTRM13].
Case Study: A Memory-safety Monitor {#sec:micro-policy}
===================================
To demonstrate the applicability of our characterization, we use it to analyze a tag-based monitor proposed by Dhawan [[*et al.*]{}]{}to enforce heap safety for low-level code [@pump_asplos2015]. In prior work [@micropolicies2015], we and others showed that an idealized model of the monitor correctly implements a higher-level abstract machine with built-in memory safety—a bit more formally, every behavior of the monitor is also a behavior of the abstract machine. Building upon this work, we prove that this abstract machine satisfies a noninterference property similar to . We were also able to prove that a similar result holds for a lower-level machine that runs a so-called “symbolic” representation of the monitor—although we had to slightly weaken the result to account for memory exhaustion (cf. ), since the machine that runs the monitor has finite memory, while the abstract machine has infinite memory. If we had a verified machine-code implementation of this monitor, it would be possible to prove a similar result for it as well.
Tag-based Monitor {#sec:monitor}
-----------------
We content ourselves with a brief overview of Dhawan [[*et al.*]{}]{}’s monitor [@pump_asplos2015; @micropolicies2015], since the formal statement of the reasoning principles it supports are more complex than the one for the abstract machine from \[sec:abstract\], on which we will focus. Following a proposal by Clause *et al.* [@ClauseDOP07], Dhawan [[*et al.*]{}]{}’s monitor enforces memory safety for heap-allocated data by checking and propagating *metadata tags*. Every memory location receives a tag that uniquely identifies the allocated region to which that location belongs (akin to the identifiers in ), and pointers receive the tag of the region they are allowed to reference. The monitor assigns these tags to new regions by storing a monotonic counter in protected memory that is bumped on every call to `malloc`; with a large number of possible tags, it is possible to avoid the freshness pitfalls discussed in . When a memory access occurs, the monitor checks whether the tag on the pointer matches the tag on the location. If they do, the operation is allowed; otherwise, execution halts. The monitor instruments the allocator to make set up tags correctly. Its implementation achieves good performance using the *PUMP*, a hardware extension accelerating such micro-policies for metadata tagging [@pump_asplos2015].
Abstract Machine {#sec:abstract}
----------------
The memory-safe abstract machine [@micropolicies2015] operates on two kinds of values: machine words $w$, or pointers $(i, w)$, which are pairs of an identifier $i \in \I$ and an offset $w$. We use $\W$ to denote the set of machine words, and $\V$ to denote the set of values. Machine states are triples $(m, {\mathit{rs}}, {\mathit{pc}})$, where (1) $m \in \I \partfunfin \V^*$ is a *memory* mapping identifiers to lists of values; (2) ${\mathit{rs}}\in \R \partfunfin \V$ is a *register bank*, mapping register names to values; and (3) ${\mathit{pc}}\in \V$ is the *program counter*.
- $m \in \I \partfunfin \V^*$ is a *memory*, which maps identifiers to lists of values;
- ${\mathit{rs}}\in \R \partfunfin \V$ is a *register bank*, mapping registers (elements of a finite set $\R$) to values; and
- ${\mathit{pc}}\in \V$ is the *program counter*.
$\Nop$, $\Const~w~r_d$, $\Mov~r_s~r_d$, $\Binop_\oplus~r_1~r_2~r_d$, $\Load~r_p~r_d$, $\Store~r_p~r_s$, $\Jump~r$, $\Jal~r$, $\Bnz~r~w$, $\Halt$
The execution of an instruction is specified by a step relation $s \to s'$. If there is no $s'$ such that $s \to s'$, we say that $s$ is stuck, which means that a fatal error occurred during execution. On each instruction, the machine checks if the current program counter is a pointer and, if so, tries to fetch the corresponding value in memory. The machine then ensures that this value is a word that correctly encodes an instruction and, if so, acts accordingly. The instructions of the machine, representative of typical RISC architectures, allow programs to perform binary and logical operations, move values to and from memory, and branch. The instructions of the machine, representative of typical RISC architectures, are summarized in . Programs can perform binary operations ($\Binop$), move values to and from memory ($\Load$, $\Store$), and branch ($\Jump$, $\Jal$, $\Bnz$). The machine is in fact fairly similar to the language of . Some operations are overloaded to manipulate pointers; for example, adding a pointer to a word is allowed, and the result is obtained by adjusting the pointer’s offset accordingly. Accessing memory causes the machine to halt when the corresponding position is undefined.
In addition to these basic instructions, the machine possesses a set of special *monitor services* that can be invoked as regular functions, using registers to pass in arguments and return values. There are two services $\calloc$ and $\cfree$ for managing memory, and one service $\mathsf{eq}$ for testing whether two values are equal. The reason for using separate monitor services instead of special instructions is to keep its semantics closer to the more concrete machine that implements it. While instructions include an equality test, it cannot replace the $\mathsf{eq}$ service, since it only takes physical addresses into account. As argued in , such comparisons can be turned into a side channel: comparing out-of-bounds pointers to different blocks reveals information about the global state of the allocator. To prevent this, testing two pointers for equality directly using the corresponding machine instruction results in an error if the pointers have different block identifiers.
Verifying Memory Safety
-----------------------
The proof of memory safety for this abstract machine mimics the one carried for the language in . We use similar notations as before: $\pi
\cdot s$ means renaming every identifier that appears in $s$ according to the permutation $\pi$, and $\ids(s)$ is the finite set of all identifiers that appear in the state $s$. A simple case analysis on the possible instructions yields analogs of (we don’t include an analog of because we consider individual execution steps, where loops cannot occur). We show single-step versions for simplicity, but the results generalize easily to multiple steps.
\[thm:mp-renaming\] Let $\pi$ be a permutation, and $s$ and $s'$ be two machine states such that $s \to s'$. There exists another permutation $\pi'$ such that $\pi \cdot s
\to \pi' \cdot s'$.
\[thm:mp-frame-ok\] Let $(m_1, {\mathit{rs}}, {\mathit{pc}})$ be a state of the abstract machine, and $m_2$ a memory. Suppose that $\ids(m_1, {\mathit{rs}}, {\mathit{pc}}) \fresh \dom(m_2)$, and that $(m_1, {\mathit{rs}},
{\mathit{pc}}) \to (m', {\mathit{rs}}', {\mathit{pc}}')$. Then, there exists a permutation $\pi$ such that $\ids(\pi \cdot m', \pi \cdot {\mathit{rs}}, \pi \cdot {\mathit{pc}}) \fresh \dom(m_2)$ and $(m_2 \cup m_1, {\mathit{rs}}, {\mathit{pc}}) \to (m_2 \cup \pi \cdot m', \pi \cdot {\mathit{rs}}', \pi \cdot
{\mathit{pc}}')$. $$\ids(\pi \cdot m', \pi \cdot {\mathit{rs}}, \pi \cdot {\mathit{pc}}) \fresh \dom(m_2) \text{ and}$$ $$(m_2 \cup m_1, {\mathit{rs}}, {\mathit{pc}}) \to (m_2 \cup \pi \cdot m', \pi \cdot {\mathit{rs}}', \pi \cdot
{\mathit{pc}}').$$
\[thm:mp-frame-error\] Let $(m_1, {\mathit{rs}}, {\mathit{pc}})$ be a machine state, and $m_2$ a memory. If $\ids(m_1, {\mathit{rs}}, {\mathit{pc}}) \fresh \dom(m_2)$, and $(m_1, {\mathit{rs}}, {\mathit{pc}})$ is stuck, then $(m_2 \cup m_1, {\mathit{rs}}, {\mathit{pc}})$ is also stuck.
Once again, we can combine these properties to obtain a proof of noninterference. Our Coq development includes a complete statement. Once again, by combining these properties, we obtain a proof of noninterference.
\[cor:mp-noninterference\] Let $s = (m_1, {\mathit{rs}}, {\mathit{pc}})$ be a state, and $m_{21}$ and $m_{22}$ two memories. Suppose that $\ids(m_1, {\mathit{rs}}, {\mathit{pc}}) \fresh \dom(m_{2i})$ for $i = 1, 2$. When trying to run $s$ by adding the extra memories $m_{2i}$, only the following two possibilities can arise.
- Both $(m_{21} \cup m_1, {\mathit{rs}}, {\mathit{pc}})$ and $(m_{22} \cup m_1, {\mathit{rs}}, {\mathit{pc}})$ are stuck; or
- both states successfully step without interfering with the inaccessible portions $m_{21}$ and $m_{22}$. Formally, there exists a state $s' = (m',
{\mathit{rs}}', {\mathit{pc}}')$, and permutations $\pi_1$ and $\pi_2$ such that $$(m_{2i} \cup
m_1, {\mathit{rs}}, {\mathit{pc}}) \to (m_{21} \cup \pi_i \cdot m', \pi_i \cdot {\mathit{rs}}', \pi_i \cdot
{\mathit{pc}}) \text{ and}$$ $$\ids(\pi_i \cdot s') \fresh \dom(m_{2i})\text{, for }i = 1, 2.$$
Discussion
----------
The reasoning principles supported by the memory-safety monitor have an important difference compared to the ones of . In the memory-safe language, reachability is relative to a program’s local variables. If we want to argue that part of the state is isolated from some code fragment, we just have to consider that fragment’s local variables—other parts of the program are still allowed to access the region. The memory-safety monitor, on the other hand, does not have an analogous notion: an unreachable memory region is useless, since it remains unreachable by all components forever.
It seems that, from the standpoint of noninterference, heap memory safety *taken in isolation* is much weaker than the guarantees it provides in the presence of other language features, such as local variables. Nevertheless, the properties studied above suggest several avenues for strengthening the mechanism and making its guarantees more useful. The most obvious one would be to use the mechanism as the target of a compiler for a programming language that provides other (safe) stateful abstractions, such as variables and a stack for procedure calls. A more modest approach from the point of view of formal verificationwould be to add other state abstractions to the mechanism itself. Besides variables and call stacks, if the mechanism made code immutable and separate from data, a simple check would suffice to tell whether a code segment stored in memory references a given privileged register. If the register is the only means of reaching a memory region, we should be able to soundly infer that that code segment is independent of that region. On a last note, although the abstract machine we verified is fairly close to our original language, the dynamic monitor that implements it using tags is quite different (\[sec:monitor\]). In particular, the monitor works on a machine that has a flat memory model, and keeps track of free and allocated memory using a protected data structure that stores block metadata. It was claimed that reasoning about this base and bounds information was the most challenging part of the proof that the monitor implements the abstract machine [@micropolicies2015]. For this reason, we believe that this proof can be adapted to other enforcement mechanisms that rely solely on base and bounds information—for example, fat pointers [@LowFat2013; @DeviettiBMZ08] or SoftBound [@NagarakatteZMZ09]—while keeping a similar abstract machine as their specification, and thus satisfying a similar noninterference property. This gives us confidence that our memory safety characterization generalizes to other settings.
Related Work {#sec:related-work}
============
The present work lies at the intersection of two areas of previous research: one on formal characterizations of memory safety, the other on reasoning principles for programs. We review the most closely related work in these areas.
#### Characterizing Memory Safety {#characterizing-memory-safety .unnumbered}
Many formal characterizations of memory safety originated in attempts to reconcile its benefits with low-level code. Generally, these works claim that a mechanism is safe by showing that it prevents or catches typical temporal and spatial violations. Examples in the literature include: Cyclone [@SwamyHMGJ06], a language with a region-basedtype system for safe manual memory management; CCured [@ccured_toplas2005], a program transformation that adds temporal safety to C by refining its pointer type to distinguish between with various degrees of safety; Ivory [@ElliottPWHBSSL15] an embedding of a similar “safe-C variant” into Haskell; SoftBound [@NagarakatteZMZ09], an instrumentation technique for C programs for spatial safety, including the detection of bounds violations within an object; CETS [@NagarakatteZMZ10], a compiler pass for preventing temporal safety violations in C programs, including accessing dangling pointers into freed heap regions and stale stack frames; the memory-safety monitor for the PUMP [@pump_asplos2015; @micropolicies2015], which formed the basis of our case study in ; and languages like Mezzo [@pottier-protzenko-13] and Rust [@Turon17], whose guarantees extend to preventing data races [@BalabonskiPP14]. Similar models appear in formalizations of C [@LeroyB08; @Krebbers15], which need to rigorously characterize its sources of undefined behavior—in particular, instances of memory misuse.
Either explicitly or implicitly, these works define memory errors as attempts to use a pointer to access a location that it was not meant to access—for example, an out-of-bounds or free one. This was noted by Hicks [@Hicks:memory-safety], who, inspired by SoftBound, proposed to define memory safety as an execution model that tracks what part of memory each pointer can access. Our characterization is complementary to these accounts, in that it is *extensional*: its data isolation properties allow us to reason directly about the observable behavior of the program. Furthermore, as demonstrated by our application to the monitor of and the discussions on , it can be adapted to various enforcement mechanisms and variations of memory safety.
#### Reasoning Principles {#reasoning-principles .unnumbered}
Separation logic [@Reynolds:2002; @Yang:2002] has been an important source of inspiration for our work. The logic’s frame rule enables its local reasoning capabilities and imposes restrictions that are similar to those mandated by memory-safe programming guidelines. As discussed in , our reasoning principles are reminiscent of the frame rule, but use reachability to guarantee locality in settings where memory safety is enforced automatically. In separation logic, by contrast, locality needs to be guaranteed for each program individually by comprehensive proofs.
Several works have investigated similar reasoning principles for a variety of program analyses, including static, dynamic, manual, or a mixture of those. Some of these are formulated as expressive logical relations, guaranteeing that programs are compatible with the framing of state invariants; representative works include: L${}^{\mbox{\small 3}}$ [@AhmedFM07], a linear calculus featuring strong updates and aliasing control; the work of Benton and Tabereau [@Benton:2009] on a compiler for a higher-order language; and the work of Devriese *et al.* [@DevriesePB16] on object capabilities for a JavaScript-like language. Other developments are based on proof systems reminiscent of separation logic with rules that guarantee isolation; these include Yarra [@SchlesingerPSWZ14], an extension of C that allows programmers to protect the integrity of data structures marked as *critical*; the work of Agten *et al.* [@Agten0P15], which allows mixing unverified and verified components by instrumenting the program to check that required assertions hold at interfaces; and the logic of Swasey *et al.* [@SwaseyGD17] for reasoning about object capabilities.
Unlike our work, these developments do not propose reachability-based isolation as a general *definition* of memory safety, nor do they attempt to analyze how their reasoning principles are affected by common variants of memory safety. Furthermore, many of these other works—especially the logical relations—rely on encapsulation mechanisms such as closures, objects, or modules that go beyond plain memory safety. Memory safety alone can only provide complete isolation, while encapsulation provides finer control, allowing some interaction between components, while guaranteeing the preservation of certain state invariants. In this sense, one can see memory-safety reasoning as a special case of encapsulation reasoning. Nevertheless, it is a practically relevant special case that is interesting on its own, since when reasoning about an encapsulated component, one must argue explicitly that the invariants of interest are preserved by the private operations of that component; memory safety, on the other hand, guarantees that *any* invariant on unreachable parts of the memory is automatically preserved.
Perhaps closer to our work, Maffeis *et al.* [@MaffeisMT10] show that their notion of “authority safety” guarantees isolation, in the sense that a component’s actions cannot influence the actions of another component with disjoint authority. Their notion of authority behaves similarly to the set of block identifiers accessible by a program in our language; however, they do not attempt to connect their notion of isolation to the frame rule, noninterference, or traditional notions of memory safety.
Morrisett *et al.* [@Morrisett:1995] state a correctness criterion for garbage collection based on program equivalence. Some of the properties they study are similar to the frame rule, describing the behavior of code running in an extended heap. However, they use this analysis to justify the validity of deallocating objects, rather than studying the possible interactions between the extra state and the program in terms of integrity and secrecy.
Other works attempt to characterize protection schemes that are weaker than full memory safety. Juglaret *et al.* [@JuglaretHAEP16] propose a correctness criterion for compiling compartmentalized programs, which allows memory-safety violations to occur within each compartment, but bounds the effect of such violations on other compartments of the program. Their criterion is reminiscent of the traditional notion of *full abstraction*, which guarantees that contextually equivalent programs remain equivalent after compilation. Abadi and Plotkin [@AbadiP12] develop an address-space randomization scheme for a simple compiler, and prove a *probabilistic* full-abstraction result for it. While full abstraction and related properties guarantee that certain security properties of programs are preserved by compilation, these works do not consider whether these properties encompass the type of isolation guarantee analyzed here.
Mezzo [@pottier-protzenko-13] is a concurrent dialect of ML which rules out errors such as data races while enabling certain idioms that are usually not possible in a purely functional setting, such as strong updates (changing the *type* of a reference when assigning to it) and gradual, stateful initialization of immutable data structures. Thus, it provides some guarantees that are beyond what is usually seen as defining memory safety. The soundness proofs for a fragment of the type system [@BalabonskiPP14] include a progress and preservation result, as well as the absence of data races, in the sense that every piece of data can only be written by at most one thread concurrently.
Conclusions and Future Work {#sec:conclusion}
===========================
We have explored the consequences of memory safety for reasoning about programs, formalizing intuitive principles that, we argue, capture the essential distinction between memory-safe systems and memory-unsafe ones. We showed how the reasoning principles we identified apply to a recent dynamic monitor for heap memory safetyof low-level code.
The systems studied in this paper have a simple storage model: the language of has just global variables and flat, heap-allocated arrays, while the monitor of doesn’t even have variables or immutable code. Realistic programming platforms, of course, offer much richer stateful abstractions, including, for example, procedures with stack-allocated local variables as well as structured objects with contiguously allocated sub-objects. In terms of memory safety, these systems have a richer vocabulary for describing resources that programs can access, and programmers could benefit from isolation-based local reasoning involving these resources.
For example, in typical safe languages with procedures, the behavior of a procedure should depend only on its arguments, the global variables it uses, and the portions of the state that are reachable from these values; if the caller of that procedure has a private object that is not passed as an argument, it should not affect or be affected by the call. Additionally, languages such as C allow for objects consisting of contiguously allocated sub-objects for improved performance. Some systems for spatial safety [@NagarakatteZMZ09; @DeviettiBMZ08] allow *capability downgrading*—that is, narrowing the range of a pointer so that it can’t access outside of a sub-object’s bounds. It would be interesting to refine our model to take these features into account. In the case of the monitor of , such considerations could lead to improved designs or to the integration of the monitor inside a secure compiler. Conversely, it would be interesting to derive finer security properties for relaxations of memory safety like the ones discussed in . Some inspiration could come from the information-flowIFC literature, where quantitative noninterference results provide bounds on the probability that some secret is leaked, the rate at which it is leaked, how many bits are leaked, etc. [@BackesKR09; @Smith09].
The main goal of this work was to understand, formally, the benefits of memory safety for informal and partial reasoning, and to evaluate a variety of weakened forms of memory safety in terms of which reasoning principles they preserve. However, our approach may also suggest ways to improve program verification. One promising idea is to leverage the guarantees of memory safety to obtain proofs of program correctness modulo unverified code that could have errors, in contexts where complete verification is too expensive or not possible ([e.g.,]{}for programs with a plugin mechanism).
#### Acknowledgments {#acknowledgments .unnumbered}
We are grateful to Antal Spector-Zabusky, Greg Morrisett, Justin Hsu, Michael Hicks, Nick Benton, Yannis Juglaret, William Mansky, and Andrew Tolmach for useful suggestions on earlier drafts. This work is supported by NSF grants Micro-Policies (1513854) and DeepSpec (1521523), DARPA SSITH/HOPE, and ERC Starting Grant SECOMP (715753).
Appendix {#appendix .unnumbered}
========
$$\begin{array}{rclr}
\oplus & ::= & {+} \mid {\times} \mid {-} \mid {=} \mid {\leq} \mid {\cand} \mid {\cor} & \text{(operators)}\\
e & ::= & x \in \var \mid b \in \B \mid n \in \Z & \text{(expressions)} \\
& \mid & e_1 \oplus e_2 \mid \cnot\,e \mid \coffset\,e \mid \nil & \\
c & ::= & \cskip \mid c_1; c_2 & \text{(commands)}\\
& \mid & \cifte{e}{c_1}{c_2} & \\
& \mid & \cwhiledo{e}{c} & \\
& \mid & x \gets e \mid x \gets [e] \mid [e_1] \gets e_2 & \\
& \mid & x \gets \calloc(e) \mid \cfree(e)
\end{array}$$ [ $$\begin{aligned}
s \in \St & \teq \Ls \times \M & \text{(states)} \\
l \in \Ls & \teq \var \partfunfin \V & \text{(local stores)} \\
m \in \M & \teq \I \times \Z \partfunfin \V & \text{(heaps)} \\
v \in \V & \teq \Z \uplus \B \uplus \{\nil\} \uplus \I \times \Z & \text{(values)} \\
\Ot & \teq \St \uplus \{ \oerror \} & \text{(outcomes)}\end{aligned}$$ $$\begin{aligned}
\I & \teq \text{some countably infinite set} \\
X \partfunfin Y & \teq \text{partial functions $X \partfun Y$ with finite domain}\end{aligned}$$]{}
[$$\begin{aligned}
{2}
\lsb x\rsb(l, m)
&&&\teq
\begin{cases}
l(x) & \text{if $x \in \dom(l)$} \\
\nil & \text{otherwise} \\
\end{cases} \\
\lsb b\rsb(s) &&&\teq b \\
\lsb n\rsb(s) &&&\teq n \\
\lsb \nil\rsb(s) &&&\teq \nil \\
\lsb e_1 + e_2\rsb(s)
&&&\teq
\begin{cases}
n_1 + n_2
& \text{if $\lsb e_1\rsb(s) = n_1$ and $\lsb e_2\rsb(s) = n_2$} \\
(i, n_1 + n_2)
& \text{if $\lsb e_1\rsb(s) = (i, n_1)$ and $\lsb e_2\rsb(s) = n_2$} \\
& \text{or $\lsb e_1\rsb(s) = n_1$ and $\lsb e_2\rsb(s) = (i, n_2)$} \\
\nil
& \text{otherwise}
\end{cases} \\
\lsb e_1 - e_2\rsb(s)
&&&\teq
\begin{cases}
n_1 - n_2
& \text{if $\lsb e_1\rsb(s) = n_1$ and $\lsb e_2\rsb(s) = n_2$} \\
(i, n_1 - n_2)
& \text{if $\lsb e_1\rsb(s) = (i, n_1)$ and $\lsb e_2\rsb(s) = n_2$} \\
\nil
& \text{otherwise}
\end{cases} \\
\lsb e_1 \times e_2\rsb(s)
&&&\teq
\begin{cases}
n_1 \times n_2
& \text{if $\lsb e_1\rsb(s) = n_1$ and $\lsb e_2\rsb(s) = n_2$} \\
\nil
& \text{otherwise}
\end{cases} \\
\lsb e_1 = e_2\rsb(s)
&&&\teq (\lsb e_1\rsb(s) = \lsb e_2\rsb(s)) \\
\lsb e_1 \leq e_2\rsb(s)
&&&\teq
\begin{cases}
n_1 \leq n_2
& \text{if $\lsb e_1\rsb(s) = n_1$ and $\lsb e_2\rsb(s) = n_2$} \\
\nil
& \text{otherwise}
\end{cases} \\
\lsb e_1 \cand e_2\rsb(s)
&&&\teq
\begin{cases}
b_1 \wedge b_2
& \text{if $\lsb e_1\rsb(s) = b_1$ and $\lsb e_2\rsb(s) = b_2$} \\
\nil
& \text{otherwise}
\end{cases} \\
\lsb e_1 \cor e_2\rsb(s)
&&&\teq
\begin{cases}
b_1 \vee b_2
& \text{if $\lsb e_1\rsb(s) = b_1$ and $\lsb e_2\rsb(s) = b_2$} \\
\nil
& \text{otherwise}
\end{cases} \\
\lsb \cnot\;e\rsb(s)
&&&\teq
\begin{cases}
\neg b
& \text{if $\lsb e\rsb(s) = b$} \\
\nil
& \text{otherwise}
\end{cases} \\
\lsb \coffset\;e\rsb(s)
&&&\teq
\begin{cases}
n
& \text{if $\lsb e\rsb(s) = (i, n)$} \\
\nil
& \text{otherwise}
\end{cases}\end{aligned}$$ ]{}
[$$\begin{aligned}
{2}
\bind(f, \bot) &&&\teq \bot
\\
\bind(f, \oerror) &&&\teq \oerror
\\
\bind(f, (I, l, m))
&&&\teq
\begin{cases}
(I \cup I', l', m')
& \text{if $f(l, m) = (I', l', m')$} \\
\oerror
& \text{if $f(l, m) = \oerror$} \\
\bot
& \text{otherwise}
\end{cases}
\\
\cif(b, x, y)
&&&\teq
\begin{cases}
x
& \text{if $b = \ctrue$} \\
y
& \text{if $b = \cfalse$} \\
\oerror & \text{otherwise}
\end{cases}\end{aligned}$$ ]{}
\_[+]{}(l, m) (, l, m)
c\_1; c\_2\_[+]{}(l, m) (c\_2\_[+]{}, c\_1\_[+]{}(l, m))
\_[+]{}(l, m) (e(l, m), c\_1 \_[+]{}(l, m), c\_2 \_[+]{}(l, m))
\_[+]{} (f(l, m). (e(l, m), (c\_[+]{}, f(l, m)), (, l, m)))
x e\_[+]{}(l, m) (, l\[x e(l, m)\], m)
x \_[+]{}(s)
(, l\[x v\], m) &\
&
e\_2 \_[+]{}(s)
(, l, m\[(i, n) e\_2(l, m)\]) &\
&
x (e)\_[+]{}(l, m)
({i}, l\[x (i, 0)\], m\[(i, k) 0 0 k < n\]) &\
&
(e)\_[+]{}(l, m)
(, l, m\[(i, k) k \]) &\
&
\[sec:semantics\]
This appendix defines the language of more formally. summarizes the syntax of programs and repeats the definition of program states. The syntax is standard for a simple imperative language with pointers.
defines expression evaluation, $\lsb e \rsb : \St \to \V$. Variables are looked up in the local-variable part of the state (for simplicity, heap cells cannot be dereferenced in expressions; the command $x \gets [e]$ puts the value of a heap cell in a local variable). Constants (booleans, numbers, and the special value $\nil$ used to simplify error propagation) evaluate to themselves. Addition and subtraction can be applied both to numbers and to combinations of numbers and pointers (for pointer arithmetic); multiplication only works on numbers. Equality is allowed both on pointers and on numbers. Pointer equality compares both the block identifier and its offset, and while this is harder to implement in practice than just comparing physical addresses, this is needed for not leaking information about pointers (see ). The special expression $\coffset$ extracts the offset component of a pointer; we introduce it to illustrate that for satisfying our memory characterization pointer offsets do not need to be hidden (as opposed to block identifiers). The less-than-or-equal operator only applies to numbers—in particular, pointers cannot be compared. However, since we can extract pointer offsets, we can compare those instead.
The definition of command evaluation employs an auxiliary partial function that computes the result of evaluating a program along with the set of block identifiers that were allocated during evaluation. Formally, $\lsb c\rsb_{+} : \St \partfun \Ot_{+}$, where $\Ot_{+}$ is an extended set of outcomes defined as $\power_\fin(\I) \times \St \uplus \{\oerror\}$. We then set $$\begin{aligned}
\lsb c \rsb(l, m) & =
\begin{cases}
(l', m') & \text{if $\lsb c\rsb_{+}(l, m) = (I, l', m')$} \\
\oerror & \text{if $\lsb c\rsb_{+}(l, m) = \oerror$} \\
\bot & \text{if $\lsb c\rsb_{+}(l, m) = \bot$}
\end{cases} \\
\finalids(l, m) & =
\begin{cases}
\ids(l, m) \setminus I & \text{if $\lsb c\rsb_{+}(l, m) = (I, l', m')$} \\
\emptyset & \text{otherwise}
\end{cases}\end{aligned}$$
To define $\lsb c \rsb_{+}$, we first endow the set $\St \partfun \Ot_{+}$ with the partial order of program approximation: $$f \sqsubseteq g \ \ \teq\ \ \forall s, f(s) \neq \bot \Rightarrow f(x) = g(x)$$ This allows us to define the semantics of iteration (the rule for $\cwhiledo{e}{c}$) in a standard way using the Kleene fixed point operator $\codeface{fix}$.
The definition of $\lsb c \rsb_{+}$ appears in , where several of the rules use a $\bind$ operator () to manage the “plumbing” of the sets of allocated block ids between the evaluation of one subcommand and the next. The rules for $\cif$ and $\cwhile$ also use an auxiliary operator $\cif$ (also defined in ) that turns non-boolean guards into errors.
The evaluation rules for $\cskip$, sequencing, conditionals, $\codeface{while}$, and assignment are standard. The rule for heap lookup, $x \gets [e]$, evaluates $e$ to a pointer and then looks it up in the heap, yielding an error if $e$ does not evaluate to a pointer or if it evaluates to a pointer that is invalid, either because its block id is not allocated or because its offset is out of bounds. Similarly, the heap mutation command, $[e_1] \gets e_2$, requires that $e_1$ evaluate to a pointer that is valid in the current memory $m$ (i.e., such that looking it up in $m$ yields something other than $\bot$). The allocation command $x \gets \calloc(e)$ first evaluates $e$ to an integer $n$, then calculates the next free block id for the current machine state ($\freshf(\ids(l, m))$); it yields a new machine state where $x$ points to the first cell in the new block and where a new block of $n$ cells is added the heap, all initialized to $0$. Finally, $\cfree(e)$ evaluates $e$ to a pointer and yields a new heap where every cell sharing the same block id as this pointer is undefined.
[^1]: The proofs are available at <https://github.com/CCS492/MemorySafety>. at: <https://github.com/arthuraa/memory-safe-language>.
[^2]: Available at <https://github.com/micro-policies/micro-policies-coq/tree/master/memory_safety>.
[^3]: The renaming operation $\pi \cdot s$, in particular, can be derived formally by viewing $\St$ as a nominal set over $\I$ [@Pitts:2013] obtained by combining products, disjoint unions, and partial functions.
[^4]: It would have been possible to use arbitrary functions from identifiers to identifiers, instead of permutations; however, this would complicate some of the statements, since we would have to prevent different identifiers from aliasing after a renaming. Similar issues motivated the use of permutations in the theory of nominal sets [@Pitts02].
[^5]: Technically, the frame rule requires a slightly stronger notion of specification, accounting for permutations of allocated identifiers; our Coq development has a more precise statement.
[^6]: Though the attacker model considered in this paper does not try to address such side-channel attacks, one should be able to use the previous research on the subject to protect against them or limit the damage they can cause [@Smith09; @BackesKR09; @ZhangAM12; @StefanBYLTRM13].
|
---
abstract: |
The focus of this paper is the random sequences in the form $\{X_{0},X_{1},$ $X_{n}=X_{n-2}+X_{n-1},n=2,3,..\dot{\}},$ referred to as Fibonacci Sequence of Random Variables (FSRV). The initial random variables $X_{0}$ and $X_{1} $ are assumed to be absolutely continuous with joint probability density function (pdf) $f_{X_{0},X_{1}}.$ The FSRV is completely determined by $X_{0} $ and $X_{1}$ and the members of Fibonacci sequence $\digamma
\equiv \{0,1,1,2,3,5,8,13,21,34,55,89,144,...\}.$ We examine the distributional and limit properties of the random sequence $X_{n},n=0,1,2,... $ . Key words. Random variable, distribution function, probability density function, sequence of random variables.
author:
- |
Ismihan Bayramoglu\
Department of Mathematics, Izmir University of Economics, Izmir, Turkey\
E-mail: ismihan.bayramoglu@ieu.edu.tr
title: A note on Fibonacci Sequences of Random Variables
---
Introduction
============
Let $\{\Omega ,\digamma ,P\}$ be a probability space and $X_{i}\equiv
X_{i}(\omega ),\omega \in \Omega ,i=0,1$ be absolutely continuous random variables defined on this probability space with joint probability density function (pdf) $f_{X_{0},X_{1}}(x,y).$ Consider a sequence of random variables $X_{n}\equiv X_{n}(\omega ),n\geq 1$ given in $\{\Omega ,\digamma
,P\}$ defined as $\{X_{0},X_{1},$ $X_{n}=X_{n-2}+X_{n-1},$ $n=2,3,..\}.$ We call this sequence “the Fibonacci Sequence of Random Variables”. It is clear that $X_{2}=X_{0}+X_{1},$ $X_{3}=X_{0}+2X_{1},...$ and for any $n=0,1,2,...$ we have $X_{n}=a_{n-1}X_{0}+a_{n}X_{1},$ where $\
\{a_{n}=a_{n-2}+a_{n-1},n=2,3,...;a_{0}=0,a_{1}=1,a_{2}=1$ $\}$ is the Fibonacci sequence $\digamma \equiv \{0,1,1,2,3,5,8,13,21,34,55,89,144,...\}.
$ It is also clear that the Fibonacci Sequence of Random Variables (FSRV) $X_{n},n=0,1,2,...$ is the sequence of dependent random variables based on initial random variables $X_{0}$ and $X_{1},$ which fully defined by the members of the Fibonacci sequence $\digamma $. We are interested in the behavior of FSRV, i.e. the distributional properties of $X_{n}$ and joint distributions of $X_{n}$ and $X_{n+k\text{ }}$ for any $n$ and $k.$ In the Appendix Figure A1 and Figure A2, we present some examples of realizations of FSRV in the case of independent random variables $X_{0}$ and $X_{1}$ having * Uniform(0,1)* distribution and Standard normal distribution with the R codes provided.
This paper is organized as follows. In Section 2, the probability density function of $X_{n}$ is considered, followed by a discussion of two cases where $X_{0}$ and $X_{1}$ have exponential and uniform distributions, respectively. Then, there is an investigation of limit behavior of ratios of some characteristics of pdf of $X_{n}$ for large $n.$ In the considered examples, the ratio of maximums of the pdfs, modes and expected values of consecutive elements of FSRV converge to golden ratio $\varphi \equiv \frac{1-\sqrt{5}}{2}=1,6180339887...$ . The ratio $X_{n+1}/X_{n}$ and normalized sums of $X_{n}$’s for large $n$ are discussed in Section 3. In Section 4, the focus is on the joint distributions of $X_{n}$ and $X_{n+k},$ for $2\leq k\leq n$ and on the prediction of $X_{n+k}$ given $X_{n}.$
Distributions
=============
Consider $\ X_{n}=a_{n-1}X_{0}+a_{n}X_{1},$ $n=0,1,2,...$ , where $X_{0}$ and $X_{1}$ are absolutely continuous random variables with joint pdf $f_{X_{0},X_{1}}(x,y),$ $(x,y)\in
\mathbb{R}
^{2}$ and $a_{n},n=0,1,2,...$ is the Fibonacci sequence. Denote by $f_{0}$ and $f_{1}$ the marginal pdf’s of $X_{0}$ and $X_{1},$ respectively.
\[Theorem 1\]The pdf of $X_{n}$ is $$f_{X_{n}}(x)=\frac{1}{a_{n}a_{n-1}}\int\limits_{-\infty }^{\infty
}f_{X_{0},X_{1}}(\frac{x-t}{a_{n-1}},\frac{t}{a_{n}})dt. \label{a000}$$
If $X_{0}$ and $X_{1}$ are independent, then $$f_{X_{n}}(x)=\frac{1}{a_{n}a_{n-1}}\int\limits_{-\infty }^{\infty
}f_{X_{0}}(\frac{x-t}{a_{n-1}})f_{X_{1}}(\frac{t}{a_{n}})dt. \label{a00}$$
Equations (\[a000\]) and (\[a00\]) are straightforward results of distributions of linear functions of random variables (see eg., Feller (1971), Ross (2002), Gnedenko (1978), Skorokhod (2005))
\[Case 1\]Exponential distribution. Let $X_{0}$ and $X_{1}$ be independent and identically distributed (iid) random variables having exponential distribution with parameter $\lambda =1.$ Then the pdf of $X_{n}$ is $$\begin{aligned}
f_{X_{n}}(x) &=&\frac{1}{a_{n-2}}\left\{ \exp \left( \frac{xa_{n-2}}{a_{n-1}a_{n}}\right) -1\right\} \exp (-\frac{x}{a_{n-1}}),x\geq 0,\text{ }n=3,4,...\text{ } \label{a001} \\
f_{X_{2}}(x) &=&x\exp (-x),x\geq 0. \notag\end{aligned}$$Below in Figure 1, the graphs of $f_{X_{n}}(x)$ for different values of $n$are presented. 
----------------------------------------------------------------------------
Figure 1. Graphs of $f_{X_{n}}(x),$ $n=2,3,4,5,6,7,8,$ given in (\[a001\])
----------------------------------------------------------------------------
The expected value of $X_{n}$ is$$\begin{aligned}
EX_{n} &=&\frac{1}{a_{n-2}}\left( \int\limits_{0}^{\infty }x\exp \left(
-x\left( \frac{a_{n}-a_{n-2}}{a_{n-1}a_{n}}\right) \right)
dx-\int\limits_{0}^{\infty }x\exp (-\frac{x}{a_{n-1}})\right) dx \\
&=&\frac{1}{a_{n-1}}\left( \frac{a_{n}^{2}a_{n-1}^{2}}{(a_{n}-a_{n-2})^{2}}-a_{n-1}^{2}\right) =a_{n+1}.\end{aligned}$$and variance is $$\begin{aligned}
Var(X_{n}) &=&\frac{1}{a_{n-2}}\left( \int\limits_{0}^{\infty }x^{2}\exp
\left( -x\left( \frac{a_{n}-a_{n-2}}{a_{n-1}a_{n}}\right) \right)
dx-\int\limits_{0}^{\infty }x^{2}\exp (-\frac{x}{a_{n-1}})\right) dx \\
&=&a_{2n-1}.\end{aligned}$$
\[Theorem 2\] Let $M_{n}=\underset{0<x<\infty }{\max }f_{X_{n}}(x)$ and $x_{n}^{\ast }=\underset{0<x<\infty }{\arg \max }f_{X_{n}}(x)$ be the maximum of $f_{X_{n}}(x)$ and mode of $\ X_{n},$ $n=2,3,...,$ respectively. Then $$\begin{aligned}
\underset{n\rightarrow \infty }{\lim }\frac{M_{n}}{M_{n+1}} &=&\underset{n\rightarrow \infty }{\lim }\frac{x_{n+1}^{\ast }}{x_{n}^{\ast }}=\underset{n\rightarrow \infty }{\lim }\frac{E(X_{n+1})}{E(X_{n})}=\varphi \text{ and }
\\
\underset{n\rightarrow \infty }{\lim }\frac{Var(X_{n+1})}{Var(X_{n})}
&=&\varphi ^{2},\end{aligned}$$where$$\varphi \equiv \frac{1-\sqrt{5}}{2}=1,6180339887...$$is the golden ratio.
The following can easily be verified $$\frac{d}{dx}f_{X_{n}}(x)=\frac{(-\frac{x}{a_{n-1}})(e^{\frac{xa_{n-2}}{a_{n-1}a_{n}}}-1)}{a_{n-2}a_{n-1}}+\frac{e^{-\frac{x}{a_{n-1}}}e^{\frac{xa_{n-2}}{a_{n-1}a_{n}}}}{a_{n-1}a_{n}}=0. \label{a010}$$The equation (\[a010\]) has unique solution $$x_{n}^{\ast }=\frac{a_{n-1}a_{n}\ln \left( \frac{a_{n}}{a_{n}-a_{n-2}}\right) }{a_{n-2}}.$$Therefore $X_{n}$ is unimodal and we have $$\begin{aligned}
M_{n} &=&f_{X_{n}}(x_{n}^{\ast })=\frac{1}{a_{n}-a_{n-2}}\left( \frac{a_{n}}{a_{n}-a_{n-2}}\right) ^{-\frac{a_{n}}{a_{n-2}}} \\
M_{n+1} &=&f_{X_{n+1}}(x^{\ast })=\frac{1}{a_{n+1}-a_{n-1}}\left( \frac{a_{n+1}}{a_{n+1}-a_{n-1}}\right) ^{-\frac{a_{n+1}}{a_{n-1}}}\end{aligned}$$and using$$\underset{n\rightarrow \infty }{\lim }\frac{a_{n+\alpha }}{a_{n}}=\varphi
^{\alpha }$$we obtain $$\underset{n\rightarrow \infty }{\lim }\frac{M_{n}}{M_{n+1}}=\frac{x_{n+1}^{\ast }}{x_{n}^{\ast }}=\varphi .$$
\[Case 2\]Uniform distribution. Let $X_{0}$ and $X_{1}$ be iid with $Uniform(0,1)$ distribution. Then from (\[a00\]) we obtain $$\begin{aligned}
f_{X_{n}}(x) &=&\frac{1}{a_{n}a_{n-1}}\int\limits_{0}^{a_{n}}f_{X_{0}}(\frac{x-t}{a_{n-1}})dt=\frac{a_{n-1}}{a_{n}a_{n-1}}\int\limits_{0}^{a_{n}}dF_{X_{0}}(\frac{x-t}{a_{n-1}}) \notag \\
&=&\frac{1}{a_{n}}\left\{ F_{X_{0}}(\frac{x}{a_{n-1}})-F_{X_{0}}(\frac{x-a_{n}}{a_{n-1}})\right\} \notag \\
&=&\left\{
\begin{array}{cc}
0, & x<0\text{ and }x>a_{n}+a_{n-1} \\
\frac{x}{a_{n}a_{n-1}} & 0\leq x\leq a_{n-1} \\
\frac{1}{a_{n}} & a_{n-1}\leq x\leq a_{n} \\
\frac{1}{a_{n}}(1-\frac{x-a_{n}}{a_{n-1}}) & a_{n}\leq x\leq a_{n}+a_{n-1}\end{array}\right. . \label{a002}\end{aligned}$$
---------------------------------------------------------------------
Figure 2. Graphs of $f_{X_{n}}(x),n=5,6,7,8,9,$ given in (\[a002\])
---------------------------------------------------------------------
\
\
\
It can be easily verified that $E(X_{n})=\frac{a_{n-1}+a_{n}}{2}=\frac{a_{n+1}}{2}$ and $var(X_{n})=\frac{a_{n-1}^{2}+a_{n}^{2}}{12}.$
One can observe that $\ $ $f_{X_{n}}(x)$ is not unimodal, $f_{X_{n}}(x)$ is constant in the interval $(a_{n-1},a_{n})$ and $\ M_{n}=\underset{0<x<1}{\max }f_{X_{n}}(x)=\frac{1}{a_{n-1}}$ , $\ \underset{0<x<1}{\inf \arg \min }f_{X_{n}}(x)=a_{n-1},$ $\underset{0<x<1}{\sup \arg \min }f_{X_{n}}(x)=a_{n}$.
It is not difficult to observe that the similar to Theorem 1 results hold also in this case.
Large $n$ and normalized Fibonacci sequence of random variables
===============================================================
Let $\ X_{n}=a_{n-1}X_{0}+a_{n}X_{1},$ $n=0,1,2,...$ be FSRV, where $X_{0}$ and $X_{1}$ are absolutely continuous random variables with joint pdf $f_{X_{0},X_{1}}(x,y),$ $(x,y)\in
\mathbb{R}
^{2}.$ Consider the sequence of random variables $Z_{n}\equiv \frac{X_{n+1}}{X_{n}},n=1,2,...$ . One has
$$Z_{n}(\omega )=\frac{X_{n+1}(\omega )}{X_{n}(\omega )}=\frac{a_{n+1}X_{1}(\omega )+a_{n}X_{0}(\omega )}{a_{n}X_{1}(\omega
)+a_{n-1}X_{0}(\omega )}=\frac{\frac{a_{n+1}}{a_{n}}X_{1}(\omega
)+X_{0}(\omega )}{X_{1}(\omega )+\frac{a_{n-1}}{a_{n}}X_{0}(\omega )}=\frac{\frac{a_{n+1}}{a_{n}}X_{1}(\omega )+X_{0}(\omega )}{X_{1}(\omega )+\frac{1}{\frac{a_{n}}{a_{n-1}}}X_{0}(\omega )}.$$
Since $\underset{n\rightarrow \infty }{\lim }\frac{a_{n+1}}{a_{n}}=\varphi
,$ ($\varphi =\frac{1-\sqrt{5}}{2}=1,6180339887...$ is the golden ratio), it follows that $$\begin{aligned}
Z_{n}(\omega ) &\rightarrow &\varphi ,\text{ pointwise in } \\
\Omega _{1} &=&\{\omega :\varphi X_{1}+X_{0}\neq 0\text{ and }\varphi
X_{1}+X_{0}\neq \infty \}\subset \Omega \text{ }\end{aligned}$$For the normalized FSRV, the following limit relationship is valid.
\[Theorem 3\]Let $E(X_{i})=\mu _{i},Var(X_{i})=\sigma _{i}^{2},i=0,1$ and $$Y_{n}(\omega )\equiv Y_{n}=\frac{X_{n}-E(X_{n})}{\sqrt{Var(X_{n})}}=\frac{X_{0}+\frac{a_{n}}{a_{n-1}}X_{1}-(\mu _{0}+\frac{a_{n}}{a_{n-1}}\mu _{1})}{\sqrt{\sigma _{0}^{2}+\frac{a_{n}^{2}}{a_{n-1}^{2}}\sigma _{1}^{2}}},\text{ }\omega \in \Omega .$$Then, $$Y_{n}\overset{}{\rightarrow }Y\equiv \frac{X_{0}+\varphi X_{1}-(\mu
_{0}+\varphi \mu _{1})}{\sqrt{\sigma _{0}^{2}+\varphi ^{2}\sigma _{1}^{2}}},\text{ as }n\rightarrow \infty \text{ for all }\omega \in \Omega .$$The limiting random variable $Y\equiv Y(\omega )$ has distribution function (cdf) $$P\{Y\leq x\}=P\{X_{0}+\varphi X_{1}\leq x\sqrt{\sigma _{0}^{2}+\varphi
^{2}\sigma _{1}^{2}}+(\mu _{0}+\varphi \mu _{1})\}. \label{cc1}$$
It is clear that the pdf of $X_{0}+\varphi X_{1}$ is $$f_{X_{0}+\varphi X_{1}}(x)=\frac{1}{\varphi }\int\limits_{0}^{\infty
}f_{X_{0}}(x-t)f_{X_{1}}(\frac{t}{\varphi })dt. \label{aa1}$$and the pdf of $Y$ is then $$f_{Y}(x)=\sqrt{\sigma _{0}^{2}+\varphi ^{2}\sigma _{1}^{2}}f_{X_{0}+\varphi
X_{1}}(x\sqrt{\sigma _{0}^{2}+\varphi ^{2}\sigma _{1}^{2}}+(\mu _{0}+\varphi
\mu _{1})) \label{aaa1}$$
\[Example 1\]Let $\ X_{0}$ and $X_{1}$ be iid random variables having exponential distribution with parameter $\lambda =1,$ then from (\[aa1\]) we have $$\begin{aligned}
f_{X_{0}+\varphi X_{1}}(x) &=&\frac{1}{\varphi }\int\limits_{0}^{x}\exp
(-x-t)\exp (t-\frac{t}{\varphi })dt \\
&=&\frac{\exp (-x)}{\varphi -1}[\exp (x(1-\frac{1}{\varphi }))-1].\end{aligned}$$Therefore, $$\begin{aligned}
P\{Y &\leq &x\}=P\{X_{0}+\varphi X_{1}\leq x\sqrt{\sigma _{0}^{2}+\varphi
^{2}\sigma _{1}^{2}}+(\mu _{0}+\varphi \mu _{1})\} \\
&=&\int\limits_{0}^{c(x)}\left\{ \frac{\exp (-t)}{\varphi -1}(\exp (t(1-\frac{1}{\varphi }))-1)\right\} dt,\end{aligned}$$where $c(x)=x\sqrt{\sigma _{0}^{2}+\varphi ^{2}\sigma _{1}^{2}}+(\mu
_{0}+\varphi \mu _{1}).$ And the pdf is $$\begin{aligned}
f_{Y}(x) &=&\left\{
\begin{tabular}{ll}
$\sqrt{\sigma _{0}^{2}+\varphi ^{2}\sigma _{1}^{2}}\left\{ \frac{\exp (-c(x))}{\varphi -1}\left[ \exp \left( c(x)(1-\frac{1}{\varphi })\right) -1\right]
\right\} ,$ & $x\geq -\frac{(\mu _{0}+\varphi \mu _{1})}{\sqrt{\sigma
_{0}^{2}+\varphi ^{2}\sigma _{1}^{2}}}$ \\
$0$ & $Otherwise$\end{tabular}\right. \\
&=&\left\{
\begin{tabular}{ll}
$\sqrt{1+\varphi ^{2}}\left\{ \frac{\exp (-c(x))}{\varphi -1}\left[ \exp
\left( c(x)(1-\frac{1}{\varphi })\right) -1\right] \right\} ,$ & $x\geq -\frac{1+\varphi }{\sqrt{1+\varphi ^{2}}}$ \\
$0$ & $Otherwise$\end{tabular}\right. .\end{aligned}$$
---------------------------------------
Figure 3. The graph of pdf $f_{Y}(x)$
---------------------------------------
\
\
\[Example 2\]Let $X_{0}$ and $X_{1}$ be independent random variables with $Uniform(0,1)$ distribution. Then from (\[aa1\]) we have $$f_{X_{0}+\varphi X_{1}}(x)=\left\{
\begin{array}{ccc}
\frac{x}{\varphi }, & 0\leq x\leq 1 & \\
\frac{1}{\varphi }, & 1\leq x\leq \varphi & \\
\frac{1-x}{\varphi }+1, & \varphi \leq x\leq 1+\varphi & \\
0, & elsehwere &
\end{array}\right. .$$This is a trapezoidal pdf with graph given below in Figure 4.\
\
------------------------------------------------------
Figure 4. The graph of $f_{X_{0}+\varphi X_{1}}(x)$
------------------------------------------------------
\
\
\
To find the distribution of limiting random variable $Y,$ we consider $$P\{Y\leq x\}=P\{X_{0}+\varphi X_{1}\leq x\sqrt{\sigma _{0}^{2}+\varphi
^{2}\sigma _{1}^{2}}+(\mu _{0}+\varphi \mu _{1})\}$$It is clear that $$\begin{aligned}
\mu _{0} &=&\mu _{1}=1/2,\text{ }\sigma _{0}^{2}=\sigma _{1}^{2}=1/12, \\
a &=&\sqrt{\sigma _{0}^{2}+\varphi ^{2}\sigma _{1}^{2}}=\sqrt{\frac{1+\varphi ^{2}}{12}},b=\mu _{0}+\varphi \mu _{1}=\frac{1+\varphi }{2}\end{aligned}$$ and the cdf of $Y$ is $$\begin{aligned}
F_{Y}(x) &=&P\{Y\leq x\}=P\{X_{0}+\varphi X_{1}\leq ax+b\}= \\
&&\left\{
\begin{tabular}{lll}
$0$ & $x\leq -\frac{b}{a}$ & \\
$\frac{1}{\varphi }\int\limits_{0}^{ax+b}udu=\frac{(ax+b)^{2}}{2\varphi },$
& $-\frac{b}{a}\leq x\leq \frac{1-b}{a}$ & \\
$\frac{1}{2\varphi }+\frac{1}{\varphi }\int\limits_{1}^{ax+b}du=\frac{1}{2\varphi }+\frac{ax+b-1}{\varphi },$ & $\frac{1-b}{a}\leq x\leq \frac{\varphi -b}{a}$ & \\
$\begin{array}{c}
\frac{1}{2\varphi }+\frac{1}{\varphi }+\frac{1}{\varphi }\int\limits_{\varphi }^{ax+b}(\frac{1-u}{\varphi }+1)du \\
=\frac{2ax+2b+2ax\varphi +2b\varphi -a^{2}x^{2}-2axb-b^{2}-\varphi ^{2}-1}{2\varphi }\end{array}$ & $\frac{\varphi -b}{a}\leq x\leq \frac{1+\varphi -b}{a}$ & \\
$1$ & $x\geq \frac{1+\varphi -b}{a}.$ &
\end{tabular}\right.\end{aligned}$$ The pdf of $Y$ is $$f_{Y}(x)=\left\{
\begin{array}{cccc}
0, & x<-\frac{b}{a}\text{ \ or }x>\frac{1+\varphi -b}{a} & & \\
\frac{(ax+b)a}{\varphi }, & -\frac{b}{a}<x\leq \frac{1-b}{a} & & \\
\frac{a}{\varphi } & \frac{1-b}{a}<x\leq \frac{\varphi -b}{a} & & \\
\frac{a(1+\varphi -b-ax)}{\varphi } & \frac{\varphi -b}{a}<x\leq \frac{1+\varphi -b}{a} & &
\end{array}\right. .$$\
\
Limits of normalized sums of Fibonacci sequence of random variables
-------------------------------------------------------------------
Here we are interested in the limiting behavior of sums of members of FSRV. Consider $S_{n}=\sum\limits_{i=0}^{n}X_{i}$. We have $$\begin{aligned}
S_{n} &=&X_{0}+X_{1}+\cdots +X_{n}=X_{0}+X_{1}+\sum\limits_{i=2}^{n}X_{i} \\
&=&X_{0}+X_{1}+\sum\limits_{i=2}^{n}(a_{i-1}X_{0}+a_{i}X_{1}) \\
&=&X_{0}+X_{1}+X_{0}\sum\limits_{i=2}^{n}a_{i-1}+X_{1}\sum\limits_{i=2}^{n}a_{i} \\
&=&X_{0}+X_{1}+X_{0}\sum\limits_{i=1}^{n-1}a_{i}+X_{1}(\sum\limits_{i=1}^{n}a_{i}-a_{1}) \\
&=&X_{0}+X_{1}+X_{0}(a_{n+1}-1)+X_{1}(a_{n+2}-1-a_{1}) \\
&=&a_{n+1}X_{0}+(a_{n+2}-1)X.\end{aligned}$$Since$$\sum\limits_{i=1}^{n}a_{i}=a_{n+2}-1.$$Therefore$$\begin{aligned}
S_{n} &=&X_{0}+X_{1}+\cdots +X_{n} \\
&=&a_{n+1}X_{0}+(a_{n+2}-1)X_{1}.\end{aligned}$$The pdf of $S_{n}$ is $$f_{S_{n}}(x)=\frac{1}{a_{n+1}(a_{n+1}-1)}\int\limits_{-\infty }^{\infty
}f_{X_{0}}(\frac{x-t}{a_{n+1}})f_{X_{1}}(\frac{t}{a_{n+2}-1)})dt. \label{q3}$$
\[Theorem 4\]Under conditions of Theorem 3 for a sequence $X_{0},X_{1},$ $X_{n}=a_{n-1}X_{0}+a_{n}X_{1},$ $n=2,3,...$ $\ \ $we have$$\begin{aligned}
ES_{n} &=&a_{n+1}\mu _{0}+(a_{n+2}-1)\mu _{1} \\
Var(S_{n}) &=&a_{n+1}^{2}\sigma _{0}^{2}+(a_{n+2}-1)^{2}\sigma _{1}^{2}\end{aligned}$$$$\frac{S_{n}-E(S_{n})}{\sqrt{var(S_{n})}}\rightarrow Y\text{ as }n\rightarrow
\infty ,\text{ for all }\omega \in \Omega ,$$where $Y$ has cdf (\[cc1\]).
Indeed, $$\begin{aligned}
&&\frac{S_{n}-E(S_{n})}{\sqrt{var(S_{n})}} \\
&=&\frac{X_{0}+(\frac{a_{n+2}}{a_{n+1}}-\frac{1}{a_{n+1}})X_{1}-(\mu _{0}+(\frac{a_{n+2}}{a_{n+1}}-\frac{1}{a_{n+1}})\mu _{1}}{\sqrt{\sigma _{0}^{2}+(\frac{a_{n+2}}{a_{n+1}}-\frac{1}{a_{n+1}})^{2}\sigma _{1}^{2}},} \\
&&\overset{}{\rightarrow }\frac{X_{0}+\varphi X_{1}-(\mu _{0}+\varphi \mu
_{1})}{\sqrt{\sigma _{0}^{2}+\varphi ^{2}\sigma _{1}^{2}}}=Y,\text{ as }n\rightarrow \infty .\end{aligned}$$
\[Example 3\]Let $X_{0}$ and $X_{1}$ be iid exponential(1) random variables. Then the pdf of $S_{n}$ is $$\begin{aligned}
f_{S_{n}}(x) &=&\frac{1}{a_{n+1}(a_{n+2}-1)}\int\limits_{0}^{x}\exp (\frac{x-t}{a_{n+1}})\exp (\frac{t}{a_{n+2}-1)})dt \notag \\
&=&\frac{\exp (-\frac{x}{a_{n+1}})}{a_{n+1}-a_{n+2}+1}\left( 1-\exp
(-x\left( \frac{1}{a_{n+2}-1}-\frac{1}{a_{n+1}}\right) \right) . \label{q3a}\end{aligned}$$
Joint distributions of $X_{n}$ and $X_{n+k}$
============================================
Next, we focus on the joint distributions of $X_{n}=a_{n-1}X_{0}+a_{n}X_{1}$ and $X_{n+k}=a_{n+k-1}X_{0}+a_{n+k}X_{1},$ for $k\geq 1.$
\[Theorem 5\]The joint pdf of $X_{n}$ and $X_{n+k}$ is$$\begin{aligned}
&&f_{X_{n},X_{n+k}}(y_{0},y_{1}) \notag \\
&=&\frac{1}{a_{k}}f_{X_{0},X_{1}}(\frac{a_{n+k}y_{0}-y_{1}a_{n}}{(-1)^{n}a_{k}},\frac{a_{n-1}y_{1}-a_{n+k-1}y_{0}}{(-1)^{n}a_{k}}).
\label{bb2}\end{aligned}$$
Let$$\left\{
\begin{array}{c}
y_{0}=a_{n-1}x_{0}+a_{n}x_{1} \\
y_{1}=a_{n+k-1}x_{0}+a_{n+k}x_{1}\end{array}\right. . \label{b1}$$The Jacobian of this linear transformation is $J=a_{n-1}a_{n+k}-a_{n}a_{n+k-1}$ and the solution of the system of equations (\[b1\]) is $$\left\{
\begin{array}{c}
x_{0}=(a_{n+k}y_{0}-y_{1}a_{n})/(a_{n-1}a_{n+k}-a_{n}a_{n+k-1}) \\
x_{1}=(a_{n-1}y_{1}-a_{n+k-1}y_{0})/(a_{n-1}a_{n+k}-a_{n}a_{n+k-1})\end{array}\right. .$$Therefore, the joint pdf of $X_{n}$ and $X_{n+k}$ is $$\begin{aligned}
&&f_{X_{n},X_{n+k}}(y_{0},y_{1}) \notag \\
&=&\frac{1}{\left\vert a_{n-1}a_{n+k}-a_{n}a_{n+k-1}\right\vert }f_{X_{0},X_{1}}(\frac{a_{n+k}y_{0}-y_{1}a_{n}}{a_{n-1}a_{n+k}-a_{n}a_{n+k-1}}, \notag \\
&&\frac{a_{n-1}y_{1}-a_{n+k-1}y_{0}}{a_{n-1}a_{n+k}-a_{n}a_{n+k-1}}).
\label{b2}\end{aligned}$$Using the d’Ocagne’s identity (see e.g. Dickson (1966)) $\
a_{m}a_{n+1}-a_{m+1}a_{n}=(-1)^{n}a_{m-n}$ we have $J=a_{n-1}a_{n+k}-a_{n}a_{n+k-1}=-(a_{n+k-1}a_{n}-a_{n+k}a_{n-1})=(-1)^{n}a_{k}.
$ Therefore,$$\begin{aligned}
&&f_{X_{n},X_{n+k}}(y_{0},y_{1}) \\
&=&\frac{1}{a_{k}}f_{X_{0},X_{1}}(\frac{a_{n+k}y_{0}-y_{1}a_{n}}{(-1)^{n}a_{k}},\frac{a_{n-1}y_{1}-a_{n+k-1}y_{0}}{(-1)^{n}a_{k}}).\end{aligned}$$
If $X_{0}$ and $X_{1}$ are independent then $$\begin{aligned}
&&f_{X_{n},X_{n+k}}(x,y) \notag \\
&=&\frac{1}{a_{k}}f_{X_{0}}\left( \frac{a_{n+k}x-ya_{n}}{(-1)^{n}a_{k}}\right) f_{X_{1}}\left( \frac{a_{n-1}y-a_{n+k-1}x}{(-1)^{n}a_{k}}\right) .
\label{bb3}\end{aligned}$$
\[Example 4\]Let $X_{0}$ and $X_{1}$ be iid exponential(1) random variables, $n=4,k=3.$ Then $a_{n+k}=a_{7}=13,$ $a_{n+k-1}=a_{6}=8,$ $a_{n-1}=a_{3}=2,$ $a_{n}=a_{4}=3$ $\ $and $a_{k}=a_{3}=2.$ Then from ([bb3]{}) $$\begin{aligned}
&&f_{X_{4},X_{7}}(x,y) \notag \\
&=&\left\{
\begin{array}{cc}
\begin{array}{c}
\frac{1}{2}\exp (-(13/2)x+(3/2)y) \\
\times \exp (-y+4x),\end{array}
& x\geq 0\text{ and }4x\leq y\leq 13/3x \\
0 & otherwise\end{array}\right. . \notag \\
&=&\left\{
\begin{array}{cc}
\frac{1}{2}\exp (-(5/2)x)\exp (y/2) & x\geq 0\text{ and }4x\leq y\leq 13/3x
\\
0 & otherwise\end{array}\right. \label{c4}\end{aligned}$$The marginal pdf’s are $$f_{X_{4}}(x)=\left\{
\begin{array}{cc}
e^{-\frac{x}{3}}-e^{-\frac{x}{2}}, & x\geq 0 \\
0, & otherwise\end{array}\right. \label{m1}$$and$$f_{X_{7}}(x)=\left\{
\begin{array}{cc}
\frac{1}{5}\left( e^{-\frac{x}{13}}-e^{-\frac{x}{8}}\right) , & x\geq 0 \\
0, & otherwise\end{array}\right. .$$
\[Example 5\]Let $X_{0}$ and $X_{1}$ be independent uniform(0,1) random variables. Again, let $n=4,k=3.$ Then $a_{n+k}=a_{7}=13,$ $a_{n+k-1}=a_{6}=8,$ $a_{n-1}=a_{3}=2,$ $a_{n}=a_{4}=3$ $\ $and $a_{k}=a_{3}=2.$ Then $$\begin{aligned}
&&f_{X_{n},X_{n+k}}(x,y) \notag \\
&=&\frac{1}{a_{k}}f_{X_{0}}\left( \frac{a_{n+k}x-ya_{n}}{(-1)^{n}a_{k}}\right) f_{X_{1}}\left( \frac{a_{n-1}y-a_{n+k-1}x}{(-1)^{n}a_{k}}\right)
\notag \\
&=&\left\{
\begin{array}{cc}
\frac{1}{a_{k}} & 0\leq \frac{a_{n+k}x-ya_{n}}{(-1)^{n}a_{k}}\leq 1,0\leq
\frac{a_{n-1}y-a_{n+k-1}x}{(-1)^{n}a_{k}}\leq 1 \\
0, & otherwise\end{array}\right. \label{c1}\end{aligned}$$(To check whether (\[c1\]) is a pdf, we need to show $\int\limits_{0}^{1}\int\limits_{0}^{1}f_{X_{n},X_{n+k}}(x,y)dxdy=1.$ Indeed, $$\begin{aligned}
&&\int\limits_{0}^{1}\int\limits_{0}^{1}f_{X_{n},X_{n+k}}(x,y)dxdy \\
&=&\frac{1}{a_{k}}\underset{0\leq \frac{a_{n+k}x-ya_{n}}{(-1)^{n}a_{k}}\leq
1,0\leq \frac{a_{n-1}y-a_{n+k-1}x}{(-1)^{n}a_{k}}\leq 1}{\int \int }dxdy \\
&=&\left\{
\begin{array}{c}
\begin{array}{c}
a_{n+k}x-ya_{n}=t,a_{n-1}y-a_{n+k-1}x=s \\
x=\frac{ta_{n-1}+sa_{n}}{(-1)^{n}a_{k}},y=\frac{sa_{n+k}+ta_{n+k-1}}{(-1)^{n}a_{k}}\end{array}
\\
t\leq (-1)^{n}a_{k},s\leq (-1)^{n}a_{k} \\
J=\left\vert
\begin{array}{cc}
\frac{a_{n-1}}{(-1)^{n}a_{k}} & \frac{a_{n}}{(-1)^{n}a_{k}} \\
\frac{a_{n+k-1}}{(-1)^{n}a_{k}} & \frac{a_{n+k}}{(-1)^{n}a_{k}}\end{array}\right\vert =\frac{a_{n-1}a_{n+k}-a_{n}a_{n+k-1}}{(-1)^{2n}a_{k}^{2}}=\frac{(-1)^{n}a_{k}}{(-1)^{2n}a_{k}^{2}}\end{array}\right\} \\
&=&\frac{1}{a_{k}}\int\limits_{0}^{(-1)^{n}a_{k}}\int\limits_{0}^{(-1)^{n}a_{k}}\frac{1}{\left\vert (-1)^{n}a_{k}\right\vert }dxdy=1.)\end{aligned}$$ For $n=4$ and $k=3,$ the $$\begin{aligned}
&&f_{X_{4},X_{7}}(x,y) \notag \\
&=&\frac{1}{2}f_{X_{0}}\left( \frac{13x-3y}{2}\right) f_{X_{1}}\left( \frac{2y-8x}{2}\right) \notag \\
&=&\left\{
\begin{array}{cc}
\frac{1}{2}, & 0\leq \frac{13x-3y}{2}\leq 1,0\leq \frac{2y-8x}{2}\leq 1 \\
0, & otherwise\end{array}\right. . \label{c2}\end{aligned}$$
Prediction of future values
===========================
It is well known that with respect to squared error loss, the best unbiased predictor of $X_{n+k},$ given $X_{n}$ is$$E\{X_{n+k}\mid X_{n}\}.$$Let $$\begin{aligned}
g(x) &=&E\{X_{n+k}\mid X_{n}=x\} \notag \\
&=&\frac{1}{f_{X_{n}}(x)}\int\limits_{-\infty }^{\infty
}yf_{X_{n},X_{n+k}}(x,y)dy, \label{cc2}\end{aligned}$$then $E\{X_{n+k}\mid X_{n}\}=g(X_{n}).$ Using (\[a000\]) and (\[bb2\]) $\ $ from (\[cc2\]) one can easily calculate the best predictor of $X_{n+k},$ given $X_{n}.$
\[Example 6\]Let $X_{0}$ and $X_{1}$ be independent exponential(1) random variables. Let $n=4,k=3.$ Then $a_{n+k}=a_{7}=13,$ $a_{n+k-1}=a_{6}=8,$ $a_{n-1}=a_{3}=2,$ $a_{n}=a_{4}=3$ $\ $and $a_{k}=a_{3}=2 $ as in Example 4. Then from (\[c4\]) we can write $$\begin{aligned}
g(x) &=&\frac{1}{e^{-\frac{x}{3}}-e^{-\frac{x}{2}}}\int\limits_{4x}^{13/3x}y\frac{1}{2}\exp (-(5/2)x)\exp (y/2)dy \\
&=&\frac{1}{3}\frac{12e^{-x/2}-6e^{-x/2}+6e^{-x/3}-13e^{-x/3}}{e^{-x/2}-e^{-x/3}} \\
&=&4x-2-\frac{x}{3(e^{-x/6}-1)},\end{aligned}$$
Therefore, $$X_{7}\simeq 4X_{4}-2-\frac{X_{4}}{3(e^{-X_{4}/6}-1).}.$$
In this note, we considered the sequence of random variables $\{X_{0},X_{1},$ $X_{n}=X_{n-2}+X_{n-1},$ $n=2,3,..\}$ which is equivalent to $\left\{
X_{0},X_{1},X_{n}=\
a_{n-1}X_{0}+a_{n}X_{1},n=2,3,...\right\} ,$ where $X_{0}$ and $X_{1}$ are absolutely continuous random variables with joint pdf $f_{X_{0},X_{1}},$ and $a_{n}=a_{n-1}+a_{n-2},$ $n=2,3,...$ $(a_{0}=0,$ $a_{1}=1)$ is the Fibonacci sequence. In the paper, the sequence $X_{n},n=0,1,2,...$ is referred to as the Fibonacci Sequence of Random Variables. We investigated the limiting properties of some ratios and normalizing sums of this sequence. For exponential and uniform distribution cases, we derived some interesting limiting properties that reduce to the golden ratio and also investigated the joint distributions of $X_{n}$ and $X_{n+k}.$ The considered random sequence has benefical properties and may be worthy of attention associated with random sequences and autoregressive models.
[9]{} Dickson. L. E. (1966)* History of the Theory of Numbers* Volume 1, New York: Chelsea.
Gnedenko, B.V. (1978) *The Theory of Probability*, Mir Publishers, Moscow.
Feller, W. (1971) *An Introduction to Probability Theory and Its Applications*, Volume 2, John Wiley & Sons Inc. , New York, London, Sydney.
Melham, R.S. and Shannon, A.G. (1995) *A generalization of the Catalan identity and some consequences*, The Fibonacci Quarterly 33, 82–84, 1995.
Ross, S. (2016) *A First Course in Probability*. Prentice-Hall Inc. , NJ.
Skorokhod, A.V. (2005) *Basic Principles and Applications of Probability Theory*, Springer.
Appendix
========
For illustration of the behaviour of FSRV, the simulated values of random variables $X_{0}$ and $X_{1}$ from uniform (0,1) and standard normal distribution are obtained. The corresponding codes in R are also given. The corresponding code in R for uniform(0,1) distribution is:\
\
$>$a$<-$seq(1:10); for (i in 3:10) a\[i\]=a\[i-1\]+a\[i-2\]; x$<-$runif(10); y$<-$runif(10); z$<-$numeric(10); for (i in 2:10) z\[i\]=a\[i\]\*x\[i\]+a\[i-1\]\*x\[i-1\]; c$<-$seq(1:10); plot(c,z,col=“red”,bg=“yellow”,pch=22,bty=“l”);\
\
The corresponding code in R for standard normal distribution is:\
\
$>$a$<-$seq(1:10); for (i in 3:10) a\[i\]=a\[i-1\]+a\[i-2\]; x$<-$rnorm(10); y$<-$rnorm(10); z$<-$numeric(10); for (i in 2:10) z\[i\]=a\[i\]\*x\[i\]+a\[i-1\]\*x\[i-1\]; c$<-$seq(1:10); plot(c,z,col=“red”,bg=“yellow”,pch=22,bty=“l”);\
\
|
---
abstract: 'By the Wolff’s cluster Monte Carlo simulations and numerical minimization within a mean field approach, we study the low temperature phase diagram of water, adopting a cell model that reproduces the known properties of water in its fluid phases. Both methods allows us to study the water thermodynamic behavior at temperatures where other numerical approaches –both Monte Carlo and molecular dynamics– are seriously hampered by the large increase of the correlation times. The cluster algorithm also allows us to emphasize that the liquid–liquid phase transition corresponds to the percolation transition of tetrahedrally ordered water molecules.'
author:
- 'Marco G. Mazza'
- Kevin Stokely
- Elena Strekalova
- 'H. Eugene Stanley'
- Giancarlo Franzese
title: 'Cluster Monte Carlo and numerical mean field analysis for the water liquid–liquid phase transition'
---
Introduction
============
Water is possibly the most important liquid for life [@sitges] and, at the same time, is a very peculiar liquid [@Debenedetti-JPCM03]. In the stable liquid regime its thermodynamic response functions behave qualitatively differently than a typical liquid. The isothermal compressibility $K_T$, for example, has a minimum as a function of temperature at $T=46~^\circ$C, while for a typical liquid $K_T$ monotonically decreases upon cooling. Water’s anomalies become even more pronounced as the system is cooled below the melting point and enters the metastable supercooled regime [@DebenedettiStanley].
Different hypothesis have been proposed to rationalize the anomalies of water [@Franzese-JPCM08]. All these interpretations, but one, predict the existence of a liquid–liquid phase transition in the supercooled state, consistent with the experiments to date [@Franzese-JPCM08] and supported by different models [@Debenedetti-JPCM03].
To discriminate among the different interpretations, many experiments have been performed [@angell2008]. However, the freezing in the temperature-range of interest can be avoided only for water in confined geometries or on the surface of macromolecules [@Franzese-JPCM08; @Stanley_etal]. Since experiments in the supercooled region are difficult to perform, numerical simulations have played an important role in recent years to help interpret the data. However, also the simulations at very low temperature $T$ are hampered by the glassy dynamics of the empirical models of water [@slow; @kfsPRL2008]. For these reasons is important to implement more efficient numerical simulations for simple models, able to capture the fundamental physics of water but also less computationally expensive. Here we introduce the implementation of a Wolff’s cluster algorithm [@wolff] for the Monte Carlo (MC) simulations of a cell model for water [@fs]. The model is able to reproduce all the different scenarios proposed to interpret the behavior of water [@kevin] and has been analyzed (i) with mean field (MF) [@fs; @Franzese-JPCM07; @kumar-JPCM08], (ii) with Metropolis MC simulations [@fmsPRE03; @kfsPRL2008] and (iii) with Wang-Landau MC density of state algorithm [@marques-PRE07]. Recent Metropolis MC simulations [@kfsPRL2008] have shown that very large times are needed to equilibrate the system as $T\rightarrow 0$, as a consequence of the onset of the glassy dynamics. The implementation of the Wolff’s clusters MC dynamics, presented here, allows us to (i) drastically reduce the equilibration times of the model at very low $T$ and (ii) give a geometrical characterization of the regions of correlated water molecules (clusters) at low $T$ and show that the liquid–liquid phase transition can be interpreted as a percolation transition of the tetrahedrally ordered clusters.
![A pictorial representation of five water molecules in 3$d$. Two hydrogen bonds (grey links) connect the hydrogens (in blue) of the central molecule with the lone electrons (small gray lines) of two nearest neighbor (n.n.) molecules. A bond index (arm) with $q=6$ possible values is associated to each hydrogen and lone electron, giving rise to $q^4$ possible orientational states for each molecule. A hydrogen bond can be formed only if the two facing arms of the n.n. molecules are in the same state. Arms on the same molecule interact among themselves to mimic the O-O-O interaction that drives the molecules toward a tetrahedral local structure.[]{data-label="picture"}](mssf-CPC-fig1.eps)
The model
=========
The system consists of $N$ particles distributed within a volume $V$ in $d$ dimensions. The volume is divided into $N$ cells of volume $v_i$ with $i\in[1,N]$. For sake of simplicity, these cells are chosen of the same size, $v_i=V/N$, but the generalization to the case in which the volume can change without changes in the topology of the nearest–neighbor (n.n.) is straightforward. By definition, $v_i\geq
v_0$, where $v_0$ is the molecule hard-core volume. Each cell has a variable $n_i=0$ for a gas-like or $n_i=1$ for a liquid-like cell. We partition the total volume in a way such that each cell has at least four n.n. cells, e.g. as in a cubic lattice in 3$d$ or a square lattice in 2$d$. Periodic boundary conditions are used to limit finite–size effects.
The system is described by the Hamiltonian [@fs] $$\begin{gathered}
\mathscr{H} = - \epsilon \sum_{\langle i,j \rangle} n_i n_j
-J \sum_{\langle ij \rangle} n_i n_j
\delta_{\sigma_{ij},\sigma_{ji}} +\\
-J_\sigma \sum_i n_i \sum_{(k,l)_i} \delta_{\sigma_{ik},\sigma_{il}},
\label{ham}\end{gathered}$$ where $\epsilon>0$ is the strength of the van der Waals attraction, $J>0$ accounts for the hydrogen bond energy, with four (Potts) variables $\sigma_{ij}=1,\ldots,q$ representing bond indices of molecule $i$ with respect to the four n.n. molecules $j$, $\delta_{a,b}=1$ if $a=b$ and $\delta_{a,b}=0$ otherwise, and $\langle i,j\rangle$ denotes that $i$ and $j$ are n.n. The model does not assume a privileged state for bond formation. Any time two facing bond indices (arms) are in the same (Potts) state, a bond is formed. The third term represents an intramolecular (IM) interaction accounting for the O–O–O correlation [@Ricci-Chaplin], locally driving the molecules toward a tetrahedral configuration. When the bond indices of a molecule are in the same state, the energy is decreased by an amount $J_\sigma\geqslant0$ and we associate this local ordered configuration to a local tetrahedral arrangement [@note]. The notation $(k,l)_i$ indicates one of the six different pairs of the four bond indices of molecule $i$ (Fig.\[picture\]).
Experiments show that the formation of a hydrogen bond leads to a local volume expansion [@Debenedetti-JPCM03]. Thus in our system the total volume is $$\label{vol}
V = N v_0 + N_{HB} v_{HB},$$ where N\_[HB]{}\_[<i,j>]{} n\_i n\_j \_[\_[ij]{},\_[ji]{}]{} is the total number of hydrogen bonds, and $v_{HB}$ is the constant specific volume increase due to the hydrogen bond formation.
Mean–field analysis
===================
In the mean–field (MF) analysis the macrostate of the system in equilibrium at constant pressure $P$ and temperature $T$ ($NPT$ ensemble) may be determined by a minimization of the Gibbs free energy per molecule, $g \equiv (\langle \mathscr{H} \rangle + PV -
TS)/N_w$, where $$\label{num}
N_w = \sum_i n_i$$ is the total number of liquid-like cells, and $S=S_n+S_\sigma$ is the sum of the entropy $S_n$ over the variables $n_i$ and the entropy $S_\sigma$ over the variables $\sigma_{ij}$.
A MF approach consists of writing $g$ explicitly using the approximations $$\begin{aligned}
\sum_{<ij>} n_i n_j &\longrightarrow 2 N n^2 \\
\sum_{<ij>} n_i n_j \delta_{\sigma_{ij}, \sigma_{ji}} &\longrightarrow 2 N n^2 p_\sigma \\
\sum_{i} n_i \sum_{(k,l)_i} \delta_{\sigma_{ik}, \sigma_{il}}
&\longrightarrow 6 N n p_\sigma\end{aligned}$$ where $n=N_w/N$ is the average of $n_i$, and $p_\sigma$ is the probability that two adjacent bond indices $\sigma_{ij}$ are in the appropriate state to form a hydrogen bond.
Therefore, in this approximation we can write $$\begin{aligned}
V & = N v_0 + 2 N n^2 p_\sigma v_{HB},\\
\langle \mathscr{H} \rangle & = -2\left[\epsilon n+\left(J n+ 3
J_\sigma\right)p_\sigma \right] n N .\end{aligned}$$
The probability $p_\sigma$, properly defined as the thermodynamic average over the whole system, is approximated as the average over two neighboring molecules, under the effect of the mean-field $h$ of the surrounding molecules $$\label{psig1}
p_\sigma = \left< \delta_{\sigma_{ij},\sigma_{ji}} \right>_h .$$
The ground state of the system consists of all $N$ variables $n_i=1$, and all $\sigma_{ij}$ in the same state. At low temperatures, the symmetry will remain broken, with the majority of the $\sigma_{ij}$ in the preferred state. We associate this preferred state to the tetrahedral order of the molecules and define $m_\sigma$ as the density of the bond indices in the tetrahedral state, with $0 \leq m_\sigma \leq 1$. Therefore, the number density $n_\sigma$ of bond indices $\sigma_{ij}$ is in the tetrahedral state is $$n_\sigma={{1 + (q-1)m_\sigma} \over q}.$$
Since an appropriate form for $h$ is [@fs] $$h = 3 J_\sigma n_\sigma,$$ we obtain that ${{3J_\sigma} \over q} \leq h \leq 3J_\sigma$.
The MF expressions for the entropies $S_n$ of the $N$ variables $n_i$, and $S_\sigma$ of the $4Nn$ variables $\sigma_{ij}$, are then [@Franzese-JPCM07] $$\begin{gathered}
S_\sigma = -k_B4Nn[n_\sigma\log(n_\sigma) + \\
(1-n_\sigma)\log(1-n_\sigma) + \log(q-1)] ,\end{gathered}$$ where $k_B$ is the Boltzmann constant.
Equating $$\label{psig2}
p_\sigma \equiv n_\sigma^2 + {{(1-n_\sigma)^2} \over {q-1}} ,$$ with the approximate expression in Eq. (\[psig1\]), allows for solution of $n_\sigma$, and hence $g$, in terms of the order parameter $m_\sigma$ and $n$.
By minimizing numerically the MF expression of $g$ with respect to $n$ and $m_\sigma$, we find the equilibrium values $n^{(eq)}$ and $m_\sigma^{(eq)}$ and, with Eqs. (\[num\]) and (\[vol\]), we calculate the density $\rho$ at any $(T,P)$ and the full equation of state. An example of minimization of $g$ is presented in Fig. \[mf\] where, for the model’s parameters $J/\epsilon=0.5$, $J_\sigma/\epsilon=0.05$, $v_{HB}/v_0=0.5$, $q=6$, a discontinuity in $m_\sigma^{(eq)}$ is observed for $Pv_0/\epsilon>0.8$. As discussed in Ref.s [@fs; @fmsPRE03] this discontinuity corresponds to a first order phase transition between two liquid phases with different degree of tetrahedral order and, as a consequence, different density. The higher $P$ at which the change in $m_\sigma^{(eq)}$ is continuous, corresponds to the pressure of a liquid–liquid critical point (LLCP). The occurrence of the LLCP is consistent with one of the possible interpretations of the anomalies of water, as discussed in Ref. [@Franzese-JPCM07]. However, for different choices of parameters, the model reproduces also the other proposed scenarios [@kevin].
![Numerical minimization of the molar Gibbs free energy $g$ in the mean field approach. The model’s parameters are $J/\epsilon=0.5$, $J_\sigma/\epsilon=0.05$, $v_{HB}/v_0=0.5$ and $q=6$. In each panel we present $g$ (dashed lines) calculated at constant $P$ and different values of $T$. The thick line crossing the dashed lines connects the minima $m_\sigma^{(eq)}$ of $g$ at different $T$. Upper panel: $Pv_0/\epsilon=0.7$, for $T$ going from $k_BT/\epsilon=0.06$ (top) to $k_BT/\epsilon=0.08$ (bottom). Middle panel: $Pv_0/\epsilon=0.8$, for $T$ going from $k_BT/\epsilon=0.05$ (top) to $k_BT/\epsilon=0.07$ (bottom). Lower panel: $Pv_0/\epsilon=0.9$, for $T$ going from $k_BT/\epsilon=0.04$ (top) to $k_BT/\epsilon=0.06$ (bottom). In each panel dashed lines are separated by $k_B \delta
T/\epsilon=0.001$. In all the panels $m_\sigma^{(eq)}$ increases when $T$ decreases, being 0 (marking the absence of tetrahedral order) at the higher temperatures and $\simeq 0.9$ (high tetrahedral order) at the lowest temperature. By changing $T$, $m_\sigma^{(eq)}$ changes in a continuous way for $Pv_0/\epsilon=0.7$ and $0.8$, but discontinuous for $Pv_0/\epsilon=0.9$ and higher $P$. []{data-label="mf"}](mssf-CPC-fig2a.eps "fig:") ![Numerical minimization of the molar Gibbs free energy $g$ in the mean field approach. The model’s parameters are $J/\epsilon=0.5$, $J_\sigma/\epsilon=0.05$, $v_{HB}/v_0=0.5$ and $q=6$. In each panel we present $g$ (dashed lines) calculated at constant $P$ and different values of $T$. The thick line crossing the dashed lines connects the minima $m_\sigma^{(eq)}$ of $g$ at different $T$. Upper panel: $Pv_0/\epsilon=0.7$, for $T$ going from $k_BT/\epsilon=0.06$ (top) to $k_BT/\epsilon=0.08$ (bottom). Middle panel: $Pv_0/\epsilon=0.8$, for $T$ going from $k_BT/\epsilon=0.05$ (top) to $k_BT/\epsilon=0.07$ (bottom). Lower panel: $Pv_0/\epsilon=0.9$, for $T$ going from $k_BT/\epsilon=0.04$ (top) to $k_BT/\epsilon=0.06$ (bottom). In each panel dashed lines are separated by $k_B \delta
T/\epsilon=0.001$. In all the panels $m_\sigma^{(eq)}$ increases when $T$ decreases, being 0 (marking the absence of tetrahedral order) at the higher temperatures and $\simeq 0.9$ (high tetrahedral order) at the lowest temperature. By changing $T$, $m_\sigma^{(eq)}$ changes in a continuous way for $Pv_0/\epsilon=0.7$ and $0.8$, but discontinuous for $Pv_0/\epsilon=0.9$ and higher $P$. []{data-label="mf"}](mssf-CPC-fig2b.eps "fig:") ![Numerical minimization of the molar Gibbs free energy $g$ in the mean field approach. The model’s parameters are $J/\epsilon=0.5$, $J_\sigma/\epsilon=0.05$, $v_{HB}/v_0=0.5$ and $q=6$. In each panel we present $g$ (dashed lines) calculated at constant $P$ and different values of $T$. The thick line crossing the dashed lines connects the minima $m_\sigma^{(eq)}$ of $g$ at different $T$. Upper panel: $Pv_0/\epsilon=0.7$, for $T$ going from $k_BT/\epsilon=0.06$ (top) to $k_BT/\epsilon=0.08$ (bottom). Middle panel: $Pv_0/\epsilon=0.8$, for $T$ going from $k_BT/\epsilon=0.05$ (top) to $k_BT/\epsilon=0.07$ (bottom). Lower panel: $Pv_0/\epsilon=0.9$, for $T$ going from $k_BT/\epsilon=0.04$ (top) to $k_BT/\epsilon=0.06$ (bottom). In each panel dashed lines are separated by $k_B \delta
T/\epsilon=0.001$. In all the panels $m_\sigma^{(eq)}$ increases when $T$ decreases, being 0 (marking the absence of tetrahedral order) at the higher temperatures and $\simeq 0.9$ (high tetrahedral order) at the lowest temperature. By changing $T$, $m_\sigma^{(eq)}$ changes in a continuous way for $Pv_0/\epsilon=0.7$ and $0.8$, but discontinuous for $Pv_0/\epsilon=0.9$ and higher $P$. []{data-label="mf"}](mssf-CPC-fig2c.eps "fig:")
The simulation with the Wolff’s clusters Monte Carlo algorithm
==============================================================
To perform MC simulations in the $NPT$ ensemble, we consider a modified version of the model in which we allow for continuous volume fluctuations. To this goal, (i) we assume that the system is homogeneous with all the variables $n_i$ set to 1 and all the cells with volume $v=V/N$; (ii) we consider that $V\equiv V_{MC}+N_{HB}v_{HB}$, where $V_{MC}\geqslant N v_0$ is a dynamical variable allowed to fluctuate in the simulations; (iii) we replace the first (van der Waals) term of the Hamiltonian in Eq. (\[ham\]) with a Lennard-Jones potential with attractive energy $\epsilon>J$ and truncated at the hard-core distance U\_W(r)
&\
&\
\[LJ\] where $r_0\equiv(v_0)^{1/d}$; the distance between two n.n. molecules is $(V/N)^{1/d}$, and the distance $r$ between two generic molecules is the Cartesian distance between the center of the cells in which they are included.
The simplification (i) could be removed, by allowing the cells to assume different volumes $v_i$ and keeping fixed the number of possible n.n. cells. However, the results of the model under the simplification (i) compares well with experiments [@Franzese-JPCM07]. Furthermore, the simplification (i) allows to drastically reduce the computational cost of the evaluation of the $U_W(r)$ term from $N(N-1)$ to $N-1$ operations. The changes (i)–(iii) modify the model used for the mean field analysis and allow off-lattice MC simulations for a cell model in which the topology of the molecules (i.e. the number of n.n.) is preserved. The comparison of the mean field results with the MC simulations show that these changes do not modify the physics of the system.
We perform MC simulations with $N=2500$ and $N=10000$ molecules, each with four n.n. molecules, at constant $P$ and $T$, in 2d, and with the same parameters used for the mean field analysis. To each molecules we associate a cell on a square lattice. The Wolff’s algorithm is based on the definition of a cluster of variables chosen in such a way to be thermodynamically correlated [@clusters; @coniglio]. To define the Wolff’s cluster, a bond index (arm) of a molecule is randomly selected; this is the initial element of a stack. The cluster is grown by first checking the remaining arms of the same initial molecule: if they are in the same Potts state, then they are added to the stack with probability $p_{\rm same}\equiv
\min\left[1,1-\exp(-\beta J_\sigma)\right]$ [@wolff], where $\beta\equiv(k_BT)^{-1}$. This choice for the probability $p_{\rm same}$ depends on the interaction $J_\sigma$ between two arms on the same molecule and guarantees that the connected arms are thermodynamically correlated [@coniglio]. Next, the arm of a new molecule, facing the initially chosen arm, is considered. To guarantee that connected facing arms correspond to thermodynamically correlated variables, is necessary [@clusters] to link them with the probability $p_{\rm facing}\equiv \min\left[1,1-\exp(-\beta J')\right]$ where $J'\equiv J-P v_{HB}$ is the $P$–dependent effective coupling between two facing arms as results from the enthalpy $\mathscr{H} + PV$ of the system. It is important to note that $J'$ can be positive or negative depending on $P$. If $J'>0$ and the two facing arms are in the same state, then the new arm is added to the stack with probability $p_{\rm facing}$; if $J'<0$ and the two facing arms are in different states, then the new arm is added with probability $p_{\rm facing}$ [@note2]. Only after every possible direction of growth for the cluster has been considered the values of the arms are changed in a stochastic way; again we need to consider two cases: (i) if $J'>0$, all arms are set to the same new value \^[new]{}=( \^[old]{}+) q where $\phi$ is a random integer between 1 and $q$; (ii) if $J'<0$, the state of every single arm is changed (rotated) by the same random constant $\phi \in [1,\dots q]$ \_i\^[new]{}=( \_i\^[old]{}+) q . In order to implement a constant $P$ ensemble we let the volume fluctuate. A small increment $\Delta r/r_0=0.01$ is chosen with uniform random probability and added to the current radius of a cell. The change in volume $\Delta V\equiv
V^{\rm new}-V^{\rm old}$ and van der Waals energy $\Delta E_W$ is computed and the move is accepted with probability $\min\left(1,\exp\left[-\beta\left(\Delta E_W+P\Delta V-T \Delta
S\right)\right]\right)$, where $\Delta S\equiv
-Nk_B\ln(V^{new}/V^{old})$ is the entropic contribution.
Monte Carlo correlation times
=============================
The cluster MC algorithm described in the previous section turns out to be very efficient at low $T$, allowing to study the thermodynamics of deeply supercooled water with quite intriguing results [@mazza]. To estimate the efficiency of the cluster MC dynamics with respect to the standard Metropolis MC dynamics, we evaluate in both dynamics, and compare, the autocorrelation function of the average magnetization per site $M_i\equiv\frac{1}{4}\sum_j\sigma_{ij}$, where the sum is over the four bonding arms of molecule $i$.
C\_M(t)\_i .
For sake of simplicity, we define the MC dynamics autocorrelation time $\tau$ as the time, measured in MC steps, when $C_M(\tau)=1/e$. Here we define a MC step as $4N$ updates of the bond indices followed by a volume update, i.e. as $4N+1$ steps of the algorithm.
![Comparison of the autocorrelation function $C_M(t)$ for the Metropolis (circles) and Wolff (squares) implementation of the present model. We show the temperatures $k_BT/\epsilon=0.11$ (top panel), $k_BT/\epsilon=0.09$ (middle panel), $k_BT/\epsilon=0.06$ (bottom panel), along the isobar $Pv_0/\epsilon=0.6$ close to the LLCP for $N=50\times 50$.\[corrfu\]](T0.11-P0.6-L50.eps "fig:") ![Comparison of the autocorrelation function $C_M(t)$ for the Metropolis (circles) and Wolff (squares) implementation of the present model. We show the temperatures $k_BT/\epsilon=0.11$ (top panel), $k_BT/\epsilon=0.09$ (middle panel), $k_BT/\epsilon=0.06$ (bottom panel), along the isobar $Pv_0/\epsilon=0.6$ close to the LLCP for $N=50\times 50$.\[corrfu\]](T0.09-P0.6-L50.eps "fig:") ![Comparison of the autocorrelation function $C_M(t)$ for the Metropolis (circles) and Wolff (squares) implementation of the present model. We show the temperatures $k_BT/\epsilon=0.11$ (top panel), $k_BT/\epsilon=0.09$ (middle panel), $k_BT/\epsilon=0.06$ (bottom panel), along the isobar $Pv_0/\epsilon=0.6$ close to the LLCP for $N=50\times 50$.\[corrfu\]](T0.06-P0.6-L50.eps "fig:")
In Fig. \[corrfu\] we show a comparison of $C_M(t)$ for the Metropolis and Wolff algorithm implementations of this model for a system with $N=50\times
50$, at three temperatures along an isobar below the LLCP, and approaching the line of the maximum, but finite, correlation length, also known as Widom line $T_W(P)$ [@Franzese-JPCM07]. In the top panel, at $T\gg T_W(P)$ ($k_BT/\epsilon=0.11$, $Pv_0/\epsilon=0.6$), we find a correlation time for the Wolff’s cluster MC dynamics $\tau_{\rm W}\approx3\times10^3$, and for the Metropolis dynamics $\tau_{\rm M}\approx10^6$. In the middle panel, at $T>T_W(P)$ ($k_BT/\epsilon=0.09$, $Pv_0/\epsilon=0.6$) the difference between the two correlation times is larger: $\tau_{\rm W}\approx2.5\times10^3$, $\tau_{\rm M}\approx3\times10^6$. The bottom panel, at $T\simeq T_W(P)$ ($k_BT/\epsilon=0.06$, $Pv_0/\epsilon=0.6$) shows $\tau_{\rm
W}\approx3.7\times10^2$, while $\tau_{\rm M}$ is beyond the accessible time window ($\tau_{\rm M}>10^7$).
Since as $T\rightarrow 0$ the system enters a glassy state [@kfsPRL2008], the efficiency $\tau_{\rm M}/
\tau_{\rm W}$ grows at lower $T$ allowing the evaluation of thermodynamics averages even at $T\ll T_C$ [@mazza]. In particular, the cluster MC algorithm turns out to be very efficient when approaching the Widom line in the vicinity of the LLCP, with an efficiency of the order of $10^4$. We plan to analyze in a systematic way how the efficiency $\tau_{\rm M}/
\tau_{\rm W}$ grows on approaching the LLCP. This result is well known for the standard liquid-gas critical point [@wolff] and, on the basis of our results, could be extended also to the LLCP. However, this analysis is very expensive in terms of CPU time and goes beyond the goal of the present work. Nevertheless, the percolation analysis, presented in the next section, helps in understanding the physical reason for this large efficiency.
The efficiency is a consequence of the fact that the average size of Wolff’s clusters changes with $T$ and $P$ in the same way as the average size of the regions of correlated molecules [@coniglio], i.e. a Wolff’s cluster statistically represents a region of correlated molecules. Moreover, the mean cluster size diverges at the critical point with the same exponent of the Potts magnetic susceptibility [@coniglio], and the clusters percolate at the critical point, as we will discuss in the next section.
Percolating clusters of correlated molecules
============================================
The efficiency of the Wolff’s cluster algorithm is a consequence of the exact relation between the average size of the finite clusters and the average size of the regions of thermodynamically correlated molecules. The proof of this relation at any $T$ derives straightforward from the proof for the case of Potts variables [@coniglio]. This relation allows to identify the clusters built during the MC dynamics with the correlated regions and emphasizes (i) the appearance of heterogeneities in the structural correlations [@hetero], and (ii) the onset of percolation of the clusters of tetrahedrally ordered molecules at the LLCP [@percolation], as shown in Fig. \[clus\].
![Three snapshots of the system with $N=100\times 100$, showing the Wolff’s clusters of correlated water molecules. For each molecule we show the states of the four arms and associate different colors to different arm’s states. The state points are at pressure close to the critical value $P_C$ ($Pv_0/\epsilon=0.72\simeq P_Cv_0/\epsilon$) and $T>T_C$ (top panel, $k_BT/\epsilon=0.0530$), $T\simeq T_C$ (middle panel, $k_BT/\epsilon=0.0528$), $T<T_C$ (bottom panel, $k_BT/\epsilon=0.0520$), showing the onset of the percolation at $T\simeq T_C$. \[clus\]](mssf-CPC-fig3a.eps "fig:"){width="42.00000%" height="38.00000%"} ![Three snapshots of the system with $N=100\times 100$, showing the Wolff’s clusters of correlated water molecules. For each molecule we show the states of the four arms and associate different colors to different arm’s states. The state points are at pressure close to the critical value $P_C$ ($Pv_0/\epsilon=0.72\simeq P_Cv_0/\epsilon$) and $T>T_C$ (top panel, $k_BT/\epsilon=0.0530$), $T\simeq T_C$ (middle panel, $k_BT/\epsilon=0.0528$), $T<T_C$ (bottom panel, $k_BT/\epsilon=0.0520$), showing the onset of the percolation at $T\simeq T_C$. \[clus\]](mssf-CPC-fig3b.eps "fig:"){width="42.70000%" height="38.00000%"} ![Three snapshots of the system with $N=100\times 100$, showing the Wolff’s clusters of correlated water molecules. For each molecule we show the states of the four arms and associate different colors to different arm’s states. The state points are at pressure close to the critical value $P_C$ ($Pv_0/\epsilon=0.72\simeq P_Cv_0/\epsilon$) and $T>T_C$ (top panel, $k_BT/\epsilon=0.0530$), $T\simeq T_C$ (middle panel, $k_BT/\epsilon=0.0528$), $T<T_C$ (bottom panel, $k_BT/\epsilon=0.0520$), showing the onset of the percolation at $T\simeq T_C$. \[clus\]](mssf-CPC-fig3c.eps "fig:"){width="42.00000%" height="38.00000%"}
A systematic percolation analysis [@clusters] is beyond the goal of this report, however configurations such as those in Fig. \[clus\] allow the following qualitative considerations. At $T>T_C$ the average cluster size is much smaller than the system size. Hence, the structural correlations among the molecules extends only to short distances. This suggests that the correlation time of a local dynamics, such as Metropolis MC or molecular dynamics, would be short on average at this temperature and pressure. Nevertheless, the system appears strongly heterogeneous with the coexistence of large and small clusters, suggesting that the distribution of correlation times evaluated among molecules at a given distance could be strongly heterogeneous. The clusters appear mostly compact but with a fractal surface, suggesting that borders between clusters can rapidly change.
At $T\simeq T_C$ there is one large cluster, in red on the right of the middle panel of Fig. \[clus\], with a linear size comparable to the system linear extension and spanning in the vertical direction. The appearance of spanning clusters shows the onset of the percolation geometrical transition. At this state point the correlation time of local, such as Metropolis MC dynamics or molecular dynamics would be very slow as a consequence of the large extension of the structurally correlated region. On the other hand, the correlation time of the Wolff’s cluster dynamics is short because it changes in one single MC step the state of all the molecules in clusters, some of them with very large size. Once the spanning cluster is formed, it breaks the symmetry of the system and a strong effective field acts on the molecules near its border to induce their reorientation toward a tetrahedral configuration with respect the molecules in the spanning cluster.
As shown in Fig.3, the spanning cluster appears as a fractal object, with holes of any size. The same large distribution of sizes characterizes also the finite clusters in the system. The absence of a characteristic size for the clusters (or the holes of the spanning cluster) is the consequence of the fluctuations at any length-scale, typical of a critical point.
At $T<T_C$ the majority of the molecules belongs to a single percolating cluster that represents the network of tetrahedrally ordered molecules. All the other clusters are small, with a finite size that corresponds to the regions of correlated molecules. The presence of many small clusters gives a qualitative idea of the heterogeneity of the dynamics at these temperatures.
Summary and conclusions
=======================
We describe the numerical solution of mean field equations and the implementation of the Wolff’s cluster MC algorithm for a cell model for liquid water. The mean field approach allows us to estimate in an approximate way the phase diagram of the model at any state point predicting intriguing new results at very low $T$ [@mazza].
To explore the state points of interest for these predictions the use of standard simulations, such as molecular dynamics or Metropolis MC, is not effective due to the onset of the glassy dynamics [@kfsPRL2008]. To overcome this problem and access the deeply supercooled region of liquid water, we adopt the Wolff’s cluster MC algorithm. This method, indeed, allows to greatly accelerate the autocorrelation time of the system. Direct comparison of Wolff’s dynamics with Metropolis dynamics in the vicinity of the liquid-liquid critical point shows a reduction of the autocorrelation time of a factor at least $10^4$.
Furthermore, the analysis of the clusters generated during the Wolff’s MC dynamics allows to emphasize how the regions of tetrahedrally ordered molecules build up on approaching the liquid–liquid critical point, giving rise to the backbone of the tetrahedral hydrogen bond network at the phase transition [@percolation]. The coexistence of clusters of correlated molecules with sizes that change with the state point gives a rationale for the heterogeneous dynamics observed in supercooled water [@hetero].
Acknowledgments
===============
We thank Andrew Inglis for introducing one of the authors (MGM) to VPython, Francesco Mallamace for discussions, NSF grant CHE0616489 and Spanish MEC grant FIS2007-61433 for support.
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---
abstract: 'We study the timely issue of charge order checkerboard patterns observed in a variety of cuprate superconductors. We suggest a minimal model in which strong quantum fluctuations in the vicinity of a single antiferromagnetic quantum critical point generate the complexity seen in the phase diagram of cuprates superconductors and, in particular, the evidenced charge order. The Fermi surface is found to fractionalize into hotspots and antinodal regions, where physically different gaps are formed. In the phase diagram, this is reflected by three transition temperatures for the formation of pseudogap, charge density wave, and superconductivity (or quadrupole density wave if a sufficiently strong magnetic field is applied). The charge density wave is characterized by modulations along the bonds of the CuO lattice with wave vectors connecting points of the Fermi surface in the antinodal regions. These features, previously observed experimentally, are so far unique to the quantum critical point in two spatial dimensions and shed a new light on the interplay between strongly fluctuating critical modes and conduction electrons in high-temperature superconductors.'
author:
- 'H. Meier$^{1,2}$, C. Pépin$^{3}$, M. Einenkel$^{2}$, and K. B. Efetov$^{2,3,\Diamond}$'
title: Cascade of phase transitions in the vicinity of a quantum critical point
---
Introduction
============
High-temperature (high-$T_{c}$) cuprate superconductors [@mueller; @nagaosa; @np] rank among the most complex materials ever discovered. Despite the rich diversity within the cuprate family, all compounds share common features such as the antiferromagnetic Mott insulator phase at zero or small doping. Magnetic fluctuations are ubiquitously present in all compounds of the cuprate family. Upon hole-doping of the copper-oxide planes, they become superconductors at unusually high transition temperatures $T_c$. Ultimately, at intermediate doping, they exhibit the enigmatic pseudo-gap phase characterized by a gap observed in transport and thermodynamics up to a temperature $T^*>T_c$.
In the last years, incommensurate charge modulations have been reported in many of the families’ compounds. These modulations form a checkerboard pattern and possibly also break nematicity. [@wise; @davis; @yazdani; @julien; @ghiringhelli; @chang; @achkar; @leboeuf; @blackburn] Complimentary to each other, these experiments demonstrate that this order is different from stripe spin-charge modulations predicted earlier [@zaanen; @machida], observed in La-compounds [@tranquada], and discussed in numerous publications (see, e.g., Refs. ), as well as from the $d$-wave order proposed in Ref. .
Among the simplest properties common to all the cuprate compounds is the presence of strong antiferromagnetic fluctuations due to the proximity of a doping-driven quantum phase transition between an antiferromagnetic and normal metal phase. Approaching the complexity of the cuprates from the perspective of this *universal* singularity, we provide an extensive study of a single antiferromagnetic two-dimensional quantum critical point (QCP). Proximity to quantum phase transitions [@sachdev; @sachkeim] is generally believed important to explain the intriguing behavior of high-$T_c$ cuprates [@nagaosa; @np; @mm], heavy fermions [@lrvw], or doped ferromagnets [@mac].
Our study unveils that this QCP triggers a cascade of phase transitions with symmetries different from those of the parent transition. These phases include $d$-wave superconductivity, a checkerboard structure of quadrupole density wave (QDW), a charge density wave (CDW) with another checkerboard structure turned by $45^\circ$ with respect to the former, and the “pseudogap state” which lacks any long range order. The additional charge order (CDW) arises due to interaction of electrons with superconducting fluctuations in situations when superconductivity itself is destroyed. To the best of our knowledge, formation of CDW due to superconducting fluctuations has not been considered previously. The complexity of the phase diagram is recovered out of a single original QCP using a low energy effective theory describing interaction between low energy fermions and paramagnons, which represent the quantum fluctuations of the antiferromagnetic order parameter. This unexpected result enriches the conventional picture [@sachdev; @sachkeim] of a single QCP and may provide new insights into the pseudogap phase of hole-doped cuprates.
Physical picture {#sec:picture}
================
Before delving into details of the microscopic derivation, let us first develop the physical picture and phenomenology. In Sec. \[sec:micro\] we provide a microscopic study to back up the physical picture, and finally, in Sec. \[sec:cuprates\] we address the question how our results may help to understand physical phenomena observed in the high-$T_c$ cuprates.
Spin-fermion model and pseudogap state
--------------------------------------
We adopt the two-dimensional spin-fermion model [@ac] for the antiferromagnetic QCP as the “minimal model” in which we seek to understand the diversity of the non-magnetic phases. As has been known for a while [@ac; @acs], this model features a superconducting instability of the normal metal state. More recently, linearizing the quasiparticle spectrum near so-called “hotspots”, Metlitski and Sachdev pointed out [@ms2] an $\mathrm{SU}(2)$ particle-hole symmetry of the effective Lagrangian that might lead to another instability toward a “bond order” state. About two years later, it has been noticed [@emp] that, in fact, a state with a complex order parameter comprising both superconductivity and an unusual charge order forms below a certain $T^*$. These phenomena significantly expand the earlier effective picture [@hertz] of free but Landau-damped paramagnons.
![(a) Brillouin zone and Fermi surface. Quantum critical paramagnons single out eight hotspots that we organize in two quartets ($\mathrm{L} =1$ and $\mathrm{L} =2$). (b) Extended model of hotspot (red) and antinodal states (blue). Non-singular paramagnons with wave vectors $\mathbf{K}_{1},\ldots,\mathbf{K}_{4}$ mediate the interaction between hotspot and antinodal states. (c) Cooper pair generation at antinodes $\mathbf{A}$ and $\mathbf{B}$.[]{data-label="fig01"}](fig01.eps){width="\linewidth"}
In the model considered, spin-$\tfrac{1}{2}$ fermion quasiparticles $\psi=(\psi_\uparrow,\psi_\downarrow)$, which occupy states close to the Fermi surface shown in Fig. \[fig01\](a), couple to paramagnons $\boldsymbol{\phi}=(\phi^x,\phi^y,\phi^z)$ with propagator $$\begin{aligned}
\big\langle\phi^\alpha_{\omega,\mathbf{q}}\phi^\beta_{-\omega,-\mathbf{q}} \big\rangle
&=
\frac{\delta_{\alpha\beta}}{c^{-2}\omega^2+(\mathbf{q}-\mathbf{Q})^2+\xi_{\mathrm{AF}}^{-2}}
\ , \label{a01}\end{aligned}$$ where $c$ is the velocity of paramagnon excitations. At the QCP, the length $\xi_{\mathrm{AF}}$ diverges so that the paramagnon propagator becomes singular at the antiferromagnetic ordering wave vector $\mathbf{Q}=(\pm \pi
/a,\pm \pi /a)$, where $a$ is the lattice constant of the Cu layer. Quasiparticles emitting or absorbing such singular paramagnons exist only in the vicinity of eight hotspots, see Fig. \[fig01\]. In a first approximation, we thus focus on these hotspots.
The energy scale $\Gamma\sim\lambda^2$, where $\lambda$ is fermion–paramagnon coupling constant, determines a temperature $T^*\sim 0.1\Gamma$, below which a complex order with competing charge and $d$-wave superconducting suborders shows up [@emp] and completely changes major properties of the system. The order parameter in this regime can be represented in the form $b_0 u$, where $b_0\sim\Gamma$ is an amplitude and $u$ an unitary matrix in particle-hole space, $$\begin{aligned}
u= \left(
\begin{array}{cc}
\Delta _{\mathrm{QDW}} & \Delta _{\mathrm{SC}} \\
-\Delta _{\mathrm{SC}}^* & \Delta _{\mathrm{QDW}}^*
\end{array}
\right) \ . \label{k0}\end{aligned}$$ In this matrix, $\Delta_{\mathrm{QDW}}$ and $\Delta_{\mathrm{SC}}$ are complex amplitudes for charge order and superconductivity, respectively. Unitarity imposes $|\Delta_{\mathrm{QDW}}|^{2}+|\Delta_{\mathrm{SC}}|^{2}=1$. In fact, there are two independent order parameters of the form of Eq. (\[k0\]), one for each of the two quartets of hotspots, Fig. \[fig01\](a), which in the “hotspot-only” approximation are effectively decoupled.
The charge order competing with superconductivity is characterized by a quadrupole moment spatially modulated with wave vectors $\mathbf{Q}_1$ and $\mathbf{Q}_2$, see Fig. \[fig02\]. These wave vectors connect hotspots opposite to each other with respect to the center of the Brillouin zone but are equivalently represented as in the inset of Fig. \[fig02\]. The resulting checkerboard structure of this *quadrupole-density wave* (QDW) is shown in Fig. \[fig02\](a). The QDW (or, equivalently, “bond order”) instability for wave vectors $\mathbf{Q}_{1,2}$ has been recently confirmed in an unrestricted Hartree-Fock study. [@sachdev13a]
The matrix order parameter $u$, Eq. (\[k0\]), obtained from mean-field equations at temperatures $T<T^*$ is highly degenerate. [@emp] At low enough temperatures, this degeneracy is lifted by curvature and magnetic field effects, the former favoring superconductivity, the latter QDW. [@mepe] At high enough temperatures (but still below $T^*$) thermal fluctuations restore the degeneracy and thus establish a *pseudogap phase* without a specific long-range order. The effective $\mathrm{O}(4)$ non-linear $\sigma$-model for fluctuations of $u$ (derived in Ref. ) as well as a more recent $\mathrm{O}(6)$-model [@sachdev13b] show in many aspects a good agreement with experiments.
Antinodal states
----------------
The nontrivial order parameter (\[k0\]) has been derived taking into account only interactions mediated by the critical modes with momenta $\sim\mathbf{Q}$ corresponding to the strongest antiferromagnetic fluctuations. Fermi surface regions beyond the hotspots have not yet been touched by the theoretical treatment and remained gapless in the “hotspot-only” approximation.
For the superconducting suborder, however, it is clear that the gap should cover the entire Fermi surface [@norman], with the exception of the nodes of the $d$-wave gap function that are situated at the intercept points of the Fermi surface and the diagonals of the Brillouin zone. In particular, we expect a significant superconducting gap also at the so-called *antinodes* situated at the zone edges, see Fig. \[fig01\](b). The main result of the present study is that superconductivity is not the only possible order close to the antinodes, and we are going to show that another charge order (CDW) can appear in this region and challenge superconductivity there. Favorably for CDW, opposite antinodes are effectively nested for a singular interaction. Flatness of the antinodal Fermi surface is not requisite but may enhance this effect.
To be specific, we extend the study of the spin-fermion model by considering both hotspots and antinodes, see Fig. \[fig01\](b). In the leading approximation, fermion quasiparticles located close to the antinodes interact with hotspot fermions by exchanging non-singular paramagnons with propagator $\langle\phi^{j}\phi^{j}\rangle \simeq [(\Delta K)^{2} + \xi_{\mathrm{AF}}^{-2}]^{-1}$ where $\Delta K = |\mathbf{K}_1-\mathbf{Q}|$ is the distance between hotspots and nearest antinodes, cf. Fig. \[fig01\]. This interaction is clearly weaker than the interaction between hotspots connected by $\mathbf{Q}$. On the other hand, it allows quantum criticality to spread into the so far untouched antinodal regions. The smallness of the non-singular propagators justifies a perturbative treatment, whereas singular paramagnons have to be fully accounted for.
We now discuss the effects due to the paramagnon-mediated interaction between hotspot and antinodal quasiparticles. We begin with the case of established superconductivity at the hotspots and then, more interestingly, for the case of hotspots gapped by QDW or pseudogapped hotspots.
Below $T_c$, hotspot fermions form Cooper pairs. In this case, we may neglect fluctuations and replace pairs of hotspot fermion fields by their mean-field average $\Delta_\mathrm{SC} \sim b_0\langle\psi^\dagger_{\uparrow,1}\psi^\dagger_{\downarrow,3}\rangle$. Let us consider two antinodal quasiparticles situated, e.g., at antinodes $\mathbf{A}$ and $\mathbf{B}$, see Fig. \[fig01\](c). Virtually exchanging a paramagnon with wave vector $\mathbf{K}_1$, they are scattered to hotspots $\mathbf{1}$ and $\mathbf{3}$. There, they are affected by the established superconducting order $\Delta_\mathrm{SC}$ and thus form Cooper pairs themselves. Interestingly, a similar virtual process is impossible for the particle-hole suborder (QDW) since in this case both particle and hole would have to emit a paramagnon of the same wave vector.
As a result, the explicit mean-field analysis, cf. Eq. (\[026\]), yields a superconducting gap at the antinodes, which by a factor of $$\begin{aligned}
\alpha \sim \frac{\Gamma^2}{v^2\big[(\Delta K)^2 + \xi_\mathrm{AF}^{-2}]} \label{k01}\end{aligned}$$ is smaller than the hotspot gap. Notably, cf. again Eq. (\[026\]), the antinodal gap has $d$-wave symmetry. Moreover, by continuity the antinodal superconductivity fixes the relative phase of the so far decoupled superconducting suborders of the two hotspot quartets in Fig. \[fig01\](a), ensuring overall $d$-wave symmetry of the superconducting order parameter. We note that this mean-field result actually does not require separating the Fermi surface into hotspots and antinodal regions and has been obtained with full momentum resolution. [@norman]
Charge density wave
-------------------
When hotspot superconductivity is destroyed by either thermal fluctuations or a strong magnetic field, the superconducting gap at the hotspots has zero mean, $\langle\Delta_{\mathrm{SC}}\rangle=0$, implying absence of antinodal superconductivity as well. However, antinodal quasiparticles still couple to non-zero superconducting fluctuations $\Delta_{\mathrm{SC}}(\mathbf{r},\tau)$ induced at the antinodes by the same mechanism that produced the antinodal superconducting gap in the preceding section.
In this situation, the superconducting fluctuations mediate an effective interaction between antinodal fermions. Close to the transition, the mass $\xi^{-2}_{\mathrm{SC}}$ of superconducting fluctuations is small and the effective interaction becomes critical. This also leads to effective nesting of opposite antinodes. As a result, this situation is remarkably similar to the initial situation of hotspot fermions interacting via critical paramagnons. While quantum-critical paramagnons reorganize the ground state of hotspot quasiparticles into the pseudogap state, the critical superconducting fluctuations play a very similar role at the antinodes and trigger in analogy a transition to another phase. This repeated triggering of orders thus constitutes a *cascade* of phase transitions.
The order parameter formed at the antinodes is pure particle-hole pairing. It cannot be a form of superconductivity because it has to be “orthogonal” to the superconducting fluctuations that mediate the effective interaction. Furthermore, particle-hole pairing at antinodes $\mathbf{A}$ and $\mathbf{B}$, see Fig. \[fig01\](b), is independent from particle-hole pairing at $\mathbf{C}$ and $\mathbf{D}$. This can be seen as, e.g., wave vectors $\mathbf{K}_1$ and $\mathbf{K}_2$ mediate interactions at antinodes $\mathbf{A}$ and $\mathbf{B}$ but have no meaning for $\mathbf{C}$ and $\mathbf{D}$, where involved paramagnons carry wave vectors $\mathbf{K}_3$ and $\mathbf{K}_4$. Invariance under rotations of $90^\circ$ then inevitably leads to a bidirectional *charge density wave* (CDW) order at the antinodes.
The explicit analysis (see Sec. \[sec:micro\]) follows the same steps as the mean-field scheme of Ref. for the pseudogap state. This leads us to a similar universal mean-field equation, see Eq. (\[045\]), with all relevant energies measured in units of the energy $$\begin{aligned}
\Gamma_{\mathrm{CDW}}\sim\alpha^2\Gamma
\label{h01}\end{aligned}$$ with $\alpha \ll 1$ defined in Eq. (\[k01\]). A non-zero CDW gap exists at temperatures $T<T_{\mathrm{CDW}} \sim 0.1\Gamma _{\mathrm{CDW}}$. In realistic cuprate systems, we may expect $T_{c}<T_{\mathrm{CDW}}<T^*$ as well as comparable energy scales, $\Gamma_{\mathrm{CDW}}\sim\Gamma$. The calculation of charge density $\rho(\mathbf{r})$ in the CDW phase leads to a spatial modulation of the form $$\begin{aligned}
\rho_{\mathrm{CDW}}(\mathbf{r})
\sim \frac{e\Gamma^2_{\mathrm{CDW}}}{v^2}
\big\{
\cos(\mathbf{Q}_{x}\mathbf{r}+\varphi_{x})
+\cos(\mathbf{Q}_{y}\mathbf{r}+\varphi_{y})
\big\}\ .
\label{c02}\end{aligned}$$ The wave vectors $\mathbf{Q}_{x}$ and $\mathbf{Q}_{y}$ (see Fig. \[fig02\]) connect opposite antinodes and correspond to a *modulation along the bonds* of the Cu lattice. The resulting pattern is a checkerboard as shown in Fig. \[fig02\](b), similarly to the pattern of QDW shown in Fig. \[fig02\](a). Notably, the CDW and QDW patterns are turned by $45^{\circ}$ with respect to each other. Variables $\varphi_{x,y}$ denote offset phases. Figure \[fig02\](c) summerizes the results of our study by providing a sketch of the emergent orders as a function of the position on the Fermi surface.
![Checkerboard charge order for the pseudogap suborder of (a) QDW and (b) antinodal CDW. Modulation vectors $\mathbf{Q}_i$ giving the periods $\mathbf{R}_{i}=2\protect\pi\mathbf{Q}_{i}/|\mathbf{Q}_{i}|^2$ are shown in the inset. (c) Qualitative dependence of the superconducting and charge order gaps on the position on the Fermi surface (HS = hotspots, AN = antinodes).[]{data-label="fig02"}](fig02.eps){width="\linewidth"}
Microscopic analysis {#sec:micro}
====================
Effective Lagrangian
--------------------
We begin our microscopic analysis by developing a convenient and compact notation for the subsequent calculations. We are mainly interested in the low-lying excitations close to the hotspots and antinodes, which we numerate according to Fig. \[fig01\](b) with numbers $j=1,\ldots,8$ and capital letters $J=\mathrm{A},\ldots,\mathrm{D}$, respectively. In this spirit, we represent a general quasiparticle field $\psi(\mathbf{r})$ as $$\begin{aligned}
\psi(\mathbf{r}) &= \sum_{j=1}^8 \mathrm{e}^{\mathrm{i}\mathbf{p}_j\mathbf{r}}\psi_j(\mathbf{r})
+ \sum_{J=\mathrm{A}}^{\mathrm{D}} \mathrm{e}^{\mathrm{i}\mathbf{p}_J\mathbf{r}}\chi_J(\mathbf{r})
\label{h11}\ ,\end{aligned}$$ where $\mathbf{p}_j$ and $\mathbf{p}_J$ denote the positions of hotspots $j$ and antinodes $J$, respectively, in the Brillouin zone. The fields for hotspot quasiparticles $\psi_j$ and for antinodal ones $\chi_J$ fluctuate only slowly in space on scales much larger than the lattice constant $a$.
Following Ref. , we introduce three pseudospin sectors $\mathrm{L}\otimes\Lambda\otimes\Sigma$ to organize the hotspot states, $$\begin{aligned}
\boldsymbol{\psi} =
\left(
\begin{array}{c}
\left(
\begin{array}{c}
\left(
\begin{array}{c}
\psi_{1}\\
\psi_{2}
\end{array}
\right)_\Sigma
\\
\left(
\begin{array}{c}
\psi_{3}\\
\psi_{4}
\end{array}
\right)_\Sigma
\end{array}
\right)_\Lambda\\
\left(
\begin{array}{c}
\left(
\begin{array}{c}
\psi_{5}\\
\psi_{6}
\end{array}
\right)_\Sigma
\\
\left(
\begin{array}{c}
\psi_{7}\\
\psi_{8}
\end{array}
\right)_\Sigma
\end{array}
\right)_\Lambda
\end{array}
\right)_{\mathrm{L}}\label{002}\ .\end{aligned}$$ Inspecting the structure defined in Eq. (\[002\]), we see that the sector ${\mathrm{L}}$ organizes the hotspots in the two quartets along the diagonals of the Brillouin zone, cf. Fig. \[fig01\](a). Sector $\Lambda$ distinguishes inside each of the quartets the two pairs of hotspots connected by the antiferromagnetic ordering wave vector $\mathbf{Q}$. Finally, the pseudospin $\Sigma$ corresponds to the two hotspots within each of such pairs. The antinodal fields are similarly combined into $$\begin{aligned}
\boldsymbol{\chi} =
\left(
\begin{array}{c}
\left(
\begin{array}{c}
\chi_{\mathrm{A}}\\
\chi_{\mathrm{B}}
\end{array}
\right)_\Upsilon\\
\left(
\begin{array}{c}
\chi_{\mathrm{C}}\\
\chi_{\mathrm{D}}
\end{array}
\right)_\Upsilon
\end{array}
\right)_\Xi
\label{001}\ ,\end{aligned}$$ where $\Xi$ and $\Upsilon$ are two more pseudospins for the four antinodes in Fig. \[fig01\](b). Operators acting on these various pseudospin spaces are conveniently expanded in Pauli matrices denoted by, e.g., $\Upsilon_1$ for the first Pauli matrix in $\Upsilon$ space. Each of the field components $\psi_j$ and $\chi_J$ in Eqs. (\[002\]) and (\[001\]) is itself a spinor for the physical spin, for which we use as usual the Pauli matrix notation $\boldsymbol{\sigma}=(\sigma_1,\sigma_2,\sigma_3)$.
In the approximation of linearized Fermi surfaces close to hotspots and antinodes, the non-interacting part $\mathcal{L}_0$ of the Lagrangian reads $$\begin{aligned}
\mathcal{L}_0 =
\boldsymbol{\chi}^\dagger
\big(
\partial_\tau
- \mathrm{i}\hat{\mathbf{v}}\nabla
\big)
\boldsymbol{\chi}
+
\boldsymbol{\psi}^\dagger
\big(
\partial_\tau
- \mathrm{i}\hat{\mathbf{V}}\nabla
\big)
\boldsymbol{\psi}
\ ,\label{i11}\end{aligned}$$ where the velocity operator for the antinodal states reads $$\begin{aligned}
\hat{\mathbf{v}} = - \frac{v}{2}\big[
\Upsilon_3(1+\Xi_3)\mathbf{e}_x
+
\Upsilon_3(1-\Xi_3)\mathbf{e}_y
\big]
\ .\label{002d}\end{aligned}$$ Herein, $\mathbf{e}_x$ and $\mathbf{e}_y$ are unit vectors in the directions of Cu bonds and $v$ is the (antinodal) Fermi velocity. The hotspot velocity operator $\hat{\mathbf{V}}$ is a little more complicated. Since we do not use this operator in the present study directly, we refer the reader to Ref. .
During the analysis, it will be convenient to study charge and superconducting correlations on equal footing. Therefore, we introduce another pseudospin $\tau$ distinguishing particle and hole states, $$\begin{aligned}
\Psi = \frac{1}{\sqrt{2}}
\left(
\begin{array}{c}
\boldsymbol{\psi}\\
\mathrm{i}\sigma_2\boldsymbol{\psi}^*
\end{array}
\right)_\tau
\ , \quad
{\mathrm{X}}= \frac{1}{\sqrt{2}}
\left(
\begin{array}{c}
\boldsymbol{\chi}\\
\mathrm{i}\sigma_2\boldsymbol{\chi}^*
\end{array}
\right)_\tau
\label{002g}\ .\end{aligned}$$ The matrix $C = -\tau_2\sigma_2$ allows for a definition of charge-conjugation $$\begin{aligned}
\bar{\Psi} = \Psi^\mathrm{t} C
\ , \quad
\bar{{\mathrm{X}}} = {\mathrm{X}}^\mathrm{t} C
\label{002i}\ .\end{aligned}$$ In particle-hole space notation, the Lagrangian (\[i11\]) becomes $$\begin{aligned}
\mathcal{L}_0 &=
-\bar{{\mathrm{X}}}
\big(
\partial_\tau
- \mathrm{i}\hat{\mathbf{v}}\nabla
\big)
{\mathrm{X}}-\bar{\Psi}
\big(
\partial_\tau
- \mathrm{i}\hat{\mathbf{V}}\nabla
\big)
\Psi
\label{i21}\ ,\end{aligned}$$ which concludes the non-interacting part of the effective theory.
In order to incorporate the interaction mediated by paramagnons $\boldsymbol{\phi}$ into the model, we again single out the relevant modes. These are those harmonics of the field $\boldsymbol{\phi}$ with wave vector close to $\mathbf{Q}$ for hotspot–hotspot interaction and wave vectors at $\mathbf{K}_1,\ldots,\mathbf{K}_4$ for hotspot–antinode interactions, see Fig. \[fig01\](b). We assume that $\mathbf{K}_1-\mathbf{K}_2$ is not an inverse lattice vector, which for a general curved Fermi surface is the correct assumption.
In the compact notation, the Lagrangian for interaction at wave vectors $\sim\mathbf{Q}$ is written as[@emp] $$\begin{aligned}
\mathcal{L}_{\mathrm{int},\mathbf{Q}} &= \lambda\ \bar{\Psi}\Sigma_1(\boldsymbol{\phi}_0\boldsymbol{\sigma})\Psi\ .
\label{hm012}\end{aligned}$$ Here $\lambda$ is the coupling constant for the paramagnon–fermion interaction. The general correlation function for $\boldsymbol{\phi}$, Eq. (\[a01\]), translates to the correlation $$\begin{aligned}
\big\langle\phi^\alpha_{0,\omega,\mathbf{q}}\phi^\beta_{0,-\omega,-\mathbf{q}} \big\rangle
&=
\frac{\delta_{\alpha\beta}}{c^{-2}\omega^2+\mathbf{q}^2+\xi_{\mathrm{AF}}^{-2}}
\label{hm013}\end{aligned}$$ for the field $\boldsymbol{\phi}_0$ entering Eq. (\[hm012\]).
For the interaction at wave vectors $\mathbf{K}_1,\ldots,\mathbf{K}_4$, we introduce fields $\boldsymbol{\phi}_{\pm k}$ that are related to field $\boldsymbol{\phi}$ of Eq. (\[a01\]) as $$\begin{aligned}
\boldsymbol{\phi}_{\pm k,\mathbf{q},\omega} &= \boldsymbol{\phi}_{\pm\mathbf{K}_k+\mathbf{q},\omega}
\label{hm014}\end{aligned}$$ with correlations $$\begin{aligned}
\big\langle\phi^\alpha_{k,\omega,\mathbf{q}}\phi^\beta_{-k,-\omega,-\mathbf{q}} \big\rangle
&\simeq
\frac{\delta_{\alpha\beta}}{(\Delta K)^2+\xi_{\mathrm{AF}}^{-2}}
\ . \label{h013}\end{aligned}$$ While hotspot–hotspot paramagnons $\boldsymbol{\phi}_{0}$ become critical at the antiferromagnetic QCP ($\xi_{\mathrm{AF}}\rightarrow\infty$), paramagnons $\boldsymbol{\phi}_{\pm k}$ are effectively static as $\Delta K=|\mathbf{K}_1-\mathbf{Q}|\gg |\mathbf{q}|,c^{-1}\omega$ at low energies. The corresponding Lagrangian reads $$\begin{aligned}
\mathcal{L}_{\mathrm{int},\mathbf{K}} &=
2\lambda \sum_{k=1}^4\Big(
\bar{\Psi} T_k(\boldsymbol{\phi}_k\boldsymbol{\sigma}) {\mathrm{X}}+
\bar{{\mathrm{X}}}\ T_k^\mathrm{t}(\boldsymbol{\phi}_k\boldsymbol{\sigma})\Psi
\Big)
\ ,
\label{006}\end{aligned}$$ where matrices $T_k$ describe the various scattering processes between hotspots and antinodes. They are given by $$\begin{aligned}
T_1 &=
\left(
\begin{array}{cc}
t_{3}^{\mathrm{B}} & 0 \\
0 & t_{1}^{\mathrm{A}}
\end{array}
\right)_\tau\ ,\quad
T_2 =
\left(
\begin{array}{cc}
t_{6}^{\mathrm{A}} & 0 \\
0 & t_{8}^{\mathrm{B}}
\end{array}
\right)_\tau\ ,\nonumber\\
T_3 &=
\left(
\begin{array}{cc}
t_{7}^{\mathrm{D}} & 0 \\
0 & t_{5}^{\mathrm{C}}
\end{array}
\right)_\tau\ ,\quad
T_4 =
\left(
\begin{array}{cc}
t_{4}^{\mathrm{C}} & 0 \\
0 & t_{2}^{\mathrm{D}}
\end{array}
\right)_\tau \ ,
\label{007}\end{aligned}$$ where the $8\times 4$ matrices $t_j^J$ are defined by $\boldsymbol{\psi}^\dagger t_j^J \boldsymbol{\chi}=\psi_j^\dagger\chi_J$. While non-trivial effects due to the hotspot Lagrangian $\mathcal{L}_{\mathrm{int},\mathbf{Q}}$, Eq. (\[hm012\]), have been extensively studied in Refs. , we are now in a position to extend the physical picture by effects emerging in the antinodal region, which couple nontrivially to the hotspots via Lagrangian $\mathcal{L}_{\mathrm{int},\mathbf{K}}$, Eq. (\[006\]).
Emerging orders
---------------
### Pseudogap state
Coupling between hotspot fermions and quantum-critical paramagnons $\boldsymbol{\phi}_0$, Eq. (\[hm013\]), has been studied for a long time. In Ref. , it was shown that close to the QCP ($\xi_{\mathrm{AF}}\rightarrow\infty$) below a temperature $T^*\sim\Gamma\sim\lambda^2$, an unusual order parameter composed of two competing suborders appears. These are superconductivity with complex amplitudes $\Delta_\mathrm{SC}^1$ and $\Delta_\mathrm{SC}^2$ and a charge order of a spatially modulated quadrupole moment (quadrupole density wave, QDW) with amplitudes $\Delta_\mathrm{QDW}^1$ and $\Delta_\mathrm{QDW}^2$, cf. Eq. (\[k0\]). Upper indices refer to the two decoupled quartets of hotspots given by ${\mathrm{L}}=1$ and ${\mathrm{L}}=2$ states, respectively, cf. Fig. 1(a). This order, hereafter referred to as “pseudogap”, constitutes a stable saddle-point manifold in the theory $\mathcal{L}_0+\mathcal{L}_{\mathrm{int},\mathbf{Q}}$. We incorporate it in terms of a mean-field term that replaces $\mathcal{L}_{\mathrm{int},\mathbf{Q}}$, Eq. (\[hm012\]), in the model. This term is given by $$\begin{aligned}
\mathcal{L}_\mathrm{PG} &=
\bar{\Psi}\
b(\mathrm{i}\partial_\tau) \mathcal{O}_\mathrm{PG}
\Psi
\ ,\label{i22}\end{aligned}$$ where $b(\varepsilon)$ is a function of fermionic Matsubara frequencies $\varepsilon$ and $\mathcal{O}_\mathrm{PG}$ is a matrix in the pseudospin spaces that reflects the symmetry of the order parameter. It reads[@emp] $$\begin{aligned}
\mathcal{O}_\mathrm{PG} &=\mathrm{i}\Sigma_3
\left(
\begin{array}{cc}
\left(
\begin{array}{cc}
0 & u_1\\
- u_1^\dagger & 0
\end{array}
\right)_\Lambda & 0
\\
0 &
\left(
\begin{array}{cc}
0 & u_2\\
- u_2^\dagger & 0
\end{array}
\right)_\Lambda
\end{array}
\right)_{{\mathrm{L}}}\ .
\label{024}\end{aligned}$$ Here, $u_1$ and $u_2$ are $\mathrm{SU}(2)$ matrices in particle-hole space for each of the two quartets of hotspots.
Let us expand the $u_j$ in particle-hole space Pauli matrices $\tau_i$, $$\begin{aligned}
u_j = \Delta^j_0 + \mathrm{i}\big(
\Delta^j_1 \tau_1+\Delta^j_2 \tau_2+\Delta^j_3 \tau_3
\big)
\ ,\label{025}\end{aligned}$$ so that $\Delta^j_{\mathrm{QDW}}=\Delta^j_0+\mathrm{i}\Delta^j_3$ and $\Delta_{\mathrm{SC}}^j=\Delta^j_1+\mathrm{i}\Delta^j_2$. Numbers $\Delta^j_n$ are real and satisfy the constraint $\sum_{n=0}^3 [\Delta^j_n]^2=1$ imposed by unitarity. At low energies, we may approximate [@emp] the function $b(\varepsilon)$ as a (positive) constant, $b(\varepsilon)\simeq b_0$.
Study of fluctuations [@emp] of the pseudogap $b(\varepsilon)\mathcal{O}$ shows that below a temperature $T_c<T^*$, one of the suborders —QDW or superconductivity— is suppressed, provided symmetry-breaking effects such as curvature of the Fermi surface are included in the consideration. In the absence of the magnetic field, finite curvature makes the composite order parameter prefer superconductivity as the ground state, whereas a sufficiently strong magnetic field can make a charge modulated state (QDW) energetically more favourable.[@mepe] Between $T_c$ and $T^*$, neither are capable of forming a long-range order and the system is in a regime of strong thermal fluctuations between the two suborders.
### Antinodal superconductivity
Averaging the Lagrangian (\[006\]) over the paramagnon fluctuations $\boldsymbol{\phi}_k$, Eq. (\[h013\]), yields an effective $4$-point interaction vertex $$\begin{aligned}
\mathcal{L}_\mathrm{int}
&=
-\frac{4\lambda^2}{(\Delta K)^2}\sum_{k=1}^4
\bar{{\mathrm{X}}}\tau_1 T_k^\mathrm{t}\tau_1\boldsymbol{\sigma}\Psi\bar{\Psi}\boldsymbol{\sigma}\tau_1T_k\tau_1{\mathrm{X}}\ .
\label{022}\end{aligned}$$ The model $\mathcal{L}_0+\mathcal{L}_\mathrm{PG}+\mathcal{L}_\mathrm{int}$, Eqs. (\[i21\]), (\[i22\]), and (\[022\]), is the effective model our subsequent study on the physics at the antinodes is based on.
In a mean-field scheme to decouple the interaction $\mathcal{L}_\mathrm{int}$, Eq. (\[022\]), we replace the $\Psi\bar{\Psi}$ operator by its mean-field correlation function, which by Eqs (\[i21\]) and (\[i22\]) is given by $$\begin{aligned}
\langle\Psi\bar{\Psi}\rangle_{\mathrm{m.f.}} = \frac{J(T)}{4\pi}\ \mathcal{O}\ .
\label{023}\end{aligned}$$ The function $J(T)$ is defined as $$\begin{aligned}
J(T)= \frac{\Omega T}{v}\sum_{\varepsilon}
\frac{b(\varepsilon)}
{\sqrt{\varepsilon^2+b^2(\varepsilon)}}
\label{023a}\end{aligned}$$ and $\Omega \sim \lambda^2/v$ is the volume of the hotspot, cf. Ref. . Inside the pseudogap regime, the function $J(T)\sim \lambda^4/v^2$ is in a good approximation independent of the temperature $T$, while it turns to zero when $T$ approaches $T^*$.
Inserting Eq. (\[023\]) into Eq. (\[022\]) yields $\mathcal{L}_\mathrm{int}\simeq \mathcal{L}_{\mathrm{m.f.}}$ with the mean-field Lagrangian given by $$\begin{aligned}
\mathcal{L}_{\mathrm{m.f.}}&=
\frac{3\lambda^2J(T)}{\pi (\Delta K)^2}
\big[
\bar{{\mathrm{X}}}\
\Xi_3\Upsilon_1
\big\{
\Delta_1
\tau_1 +
\Delta_2
\tau_2
\big\}\ {\mathrm{X}}\big]\ .
\label{026}\end{aligned}$$ Herein, $\Delta_1=(\Delta_1^1-\Delta_1^2)/2$ and $\Delta_2=(\Delta_2^1-\Delta_2^2)/2$ form the effective amplitude $\Delta_{\mathrm{SC}}=\Delta_1 + \mathrm{i}\Delta_2$ of the hotspot superconductivity. Importantly, in this mean-field treatment, only the superconducting suborder of the hotspot pseudogap gives a contribution, while the QDW does not effectively couple to the fields $\bar{{\mathrm{X}}}$ and ${\mathrm{X}}$ so that it does not play a direct role at the antinodes. Equation (\[026\]) thus demonstrates that the hotspot superconductivity induces a superconducting order parameter at the antinodes by the same mechanism sketched in Fig. \[fig01\](c) and discussed in Sec. \[sec:picture\]. The presence of $\Xi_3$ reflects the $d$-wave symmetry of the superconducting order. The order parameter of antinodal superconductivity is maximal if $$\begin{aligned}
\Delta_{1,2}^1 = -\Delta_{1,2}^2\ ,
\label{027}\end{aligned}$$ which should be energetically the favoured configuration. Note that the matching condition (\[027\]) reduces the $\mathrm{O}(4)\times\mathrm{O}(4)$ symmetry of the hotspot order to a constrained $\mathrm{O}(6)$ model, cf. Ref. .
Let us estimate the strength of the superconducting gap induced at the antinodes. According to Eq. (\[023a\]), we estimate $J(T)$ inside the pseudogap as $J(T)\sim \lambda^4/v^2$, which is smaller than the high energy scale given by the momentum distance $\Delta K$ between hotspots and antinodes. Thus, while the hotspot pseudogap is of order $\Gamma\sim\lambda^2$, the induced antinodal superconducting gap is of order $\lambda^2 [\lambda^4/(v\Delta K)^2]\sim \alpha \Gamma \ll \Gamma$, cf. Eq. (\[k01\]). We emphasize once more that the antinodal superconductivity is induced only if the hotspot system is in the superconducting state.
### Antinodal charge-density wave order
Let us now address the case when hotspot superconductivity is destroyed by either thermal fluctuations above $T_c$ (pseudogap state) or by a strong enough magnetic field at arbitrary temperature. In the latter case, we obtain QDW at $T<T_c$ or the pseudogap state at $T>T_c$ instead of the superconductor. Then, the mean-field decoupling in Eq. (\[026\]) does not induce a finite gap at the antinodes as $\Delta_1=\Delta_2=0$. However, superconducting fluctuations are still present even if $\langle\Delta_1(\mathbf{r},\tau)\rangle=\langle\Delta_2(\mathbf{r},\tau)\rangle=0$. These fluctuations have been studied with the help of a non-linear $\sigma$-model in Ref. .
At not too high temperatures above the superconducting critical temperature $T_c$ at zero field or below $T_c$ in a sufficiently strong magnetic field destroying the superconductivity, the superconducting fluctuations $\Delta_\mathrm{SC}(\mathbf{r},\tau)$ are small and the $\sigma$-model yields the effective Lagrangian $$\begin{aligned}
\mathcal{L}_{\mathrm{fluct}} \simeq \dfrac{g\lambda^2}{2}
\big(
|\partial_\mu\Delta_\mathrm{SC}|^2 + \xi^{-2}_{\mathrm{SC}}\ |\Delta_\mathrm{SC}|^2
\big)
\label{041}\end{aligned}$$ with $\partial_\mu = (u^{-1}\partial_\tau,\nabla)$, $g\sim 1$ a coupling constant, and $u\sim v$ the velocity of the fluctuation modes. For $T>T_c$ it is not easy to carry out explicit calculations in the pseudogap state. However, it is well-known[@zinn] that there is no phase transition in the two-dimensional fully isotropic O(4)-symmetric $\sigma$-model as all excitations have a gap. In our situation this means that correlation functions of superconducting fluctuations can still formally be obtained from Eq. (\[041\]) but the constants entering this equations have now to be considered as effective parameters whose values can hardly be calculated analytically. In the subsequent analysis, we assume that the length $\xi_{\mathrm{SC}}$ diverges on the critical line separating the superconducting region from QDW or pseudogap phase.
In the Gaussian approximation of Eq. (\[041\]), we immediately integrate the fluctuation modes out of the Lagrangian (\[026\]) (, where $\Delta_\mathrm{SC}=\Delta_1+\mathrm{i}\Delta_2$ is now assumed to fluctuate both in space and time). Then, we obtain the effective interaction between the antinodal fermions, $$\begin{aligned}
&\mathcal{L}_{\mathrm{int},\mathrm{fluct}}
=
-\frac{9\lambda^2 J^2(T)}{\pi^2 g (\Delta K)^4}
\sum_{j=1}^{2}
\big(
\bar{{\mathrm{X}}}(\mathbf{r},\tau)\
\Xi_3\Upsilon_1
\tau_j
\ {\mathrm{X}}(\mathbf{r},\tau)
\big)\nonumber\\
&\times \Phi(\mathbf{r}-\mathbf{r}',\tau-\tau')
\big(
\bar{{\mathrm{X}}}(\mathbf{r}',\tau')\
\Xi_3\Upsilon_1
\tau_j
\ {\mathrm{X}}(\mathbf{r}',\tau')
\big)
\ ,
\label{043}\end{aligned}$$ where $$\begin{aligned}
\Phi_{\mathbf{q},\omega} =
\frac{1}{u^{-2}\omega^2 + \mathbf{q}^2+\xi^{-2}_{\mathrm{SC}}}
\label{044}\end{aligned}$$ is the propagator of superconducting fluctuations. At the transition, $\xi_\mathrm{SC}\rightarrow\infty$ and this propagator is singular in the infrared limit, which makes the antinodal points effectively hot. Moreover, opposite antinodes are effectively nested. We emphasize, though, that, in analogy with the hotspot fermions interacting via critical paramagnons, this effective nesting is due to the singular form of the propagator of superconducting fluctuations in the vicinity of the superconductor transition where the length $\xi_{\mathrm{SC}}$ diverges. This does not necessarily require a geometrically flat Fermi surface at the antinodes.
*The interaction (\[043\]) generates an instability toward charge-density wave (CDW) order.* Indeed, the Lagrangian (\[043\]) for the interaction of antinodal fermions has effectively the same form as the effective interaction induced by paramagnons that is responsible for the formation of the pseudogap. We thus introduce a CDW order parameter in the Lagrangian, $$\begin{aligned}
\mathcal{L}_\mathrm{CDW} &= \bar{{\mathrm{X}}}b_{\mathrm{CDW}}(\mathrm{i}\partial_\tau)\mathcal{O}_\mathrm{CDW}{\mathrm{X}}\label{hm031}\ ,\end{aligned}$$ and obtain in analogy with Ref. the mean-field equation
$$\begin{aligned}
b_{\mathrm{CDW}}(\varepsilon)\mathcal{O}_\mathrm{CDW}
=
-\frac{9\lambda^2 J^2(T)}{\pi^2 g (\Delta K)^4}
\sum_{j=1}^{2}T\sum_{\varepsilon',\mathbf{k}'}
\Phi_{\mathbf{k}',\varepsilon-\varepsilon'}
\Xi_3\Upsilon_1
\tau_j
\
\frac{b_{\mathrm{CDW}}(\varepsilon')\mathcal{O}_\mathrm{CDW}}{\varepsilon'^2+(v\mathbf{k}')^2+b^2_{\mathrm{CDW}}(\varepsilon')}
\
\Xi_3\Upsilon_1
\tau_j
\ .
\label{hm032}\end{aligned}$$
Deriving Eq. (\[hm032\]) has required that $\mathcal{O}_\mathrm{CDW}$ anticommutes with the velocity operator $\hat{\mathbf{v}}$, Eq. (\[002d\]), which implies $\{\mathcal{O}_\mathrm{CDW},\Upsilon_3\}=0$ and $[\mathcal{O}_\mathrm{CDW},\Xi_3]=0$. In addition, we assume the normalization $\mathcal{O}_\mathrm{CDW}^2=1$. Furthermore, in order to compensate for the minus sign in Eq. (\[hm032\]), we need to impose that $\{\mathcal{O}_\mathrm{CDW},\Upsilon_1\tau_1\}=0$ and $\{\mathcal{O}_\mathrm{CDW},\Upsilon_1\tau_2\}=0$. Summarizing all these constraints, the antinodal order parameter becomes $$\begin{aligned}
\mathcal{O}_\mathrm{CDW} =
\left(
\begin{array}{cc}
\Delta_x' \Upsilon_1\tau_3 + \Delta_x'' \Upsilon_2 & 0 \\
0 & \Delta_y' \Upsilon_1\tau_3 + \Delta_y'' \Upsilon_2
\end{array}
\right)_\Xi
\label{044}\ .\end{aligned}$$ Parameters $\Delta_x'$ and $\Delta_x''$ play the roles of real and imaginary parts for the order parameter of CDW in $x$-direction while $\Delta_y'$ and $\Delta_y''$ do so for the $y$-direction. They satisfy the nonlinear constraints $[\Delta_x']^2+[\Delta_x'']^2=1$ and $[\Delta_y']^2+[\Delta_y'']^2=1$.
Measuring all quantities of dimension of energy in units of $$\begin{aligned}
\Gamma_{\mathrm{CDW}}&=\frac{18u\lambda^2 J^2}{\pi^2 g v(\Delta K)^4}
\label{hm041}\ ,\end{aligned}$$ we derive from Eq. (\[hm032\]) a *universal* self-consistency equation for the CDW amplitude $b_{\mathrm{CDW}}(\varepsilon)$, $$\begin{aligned}
\bar{b}_{\mathrm{CDW}}(\varepsilon)
&= \bar{T}\sum_{\bar{\varepsilon}'}
\frac{1}{|\bar{\varepsilon}-\bar{\varepsilon}'|}
\frac{\bar{b}_{\mathrm{CDW}}(\bar{\varepsilon}')}
{\sqrt{\bar{\varepsilon}'^2 + \bar{b}_{\mathrm{CDW}}^2(\bar{\varepsilon}')}}
\ .
\label{045}\end{aligned}$$ In this equation, all quantities $z$ of dimension energy enter in the form $\bar{z}=z/\Gamma_{\mathrm{CDW}}$, The energy scale $\Gamma_{\mathrm{CDW}}\sim \alpha^2 \Gamma$, cf. Eq. (\[k01\]), is smaller than both the pseudogap energy scale $\sim\Gamma$ and the antinodal superconducting gap $\sim \alpha\Gamma$, which appears when the pseudogap has ordered into the superconducting suborder. Numerical investigation of Eq. (\[045\]) indicates non-zero solutions for $b_{\mathrm{CDW}}(T,\varepsilon)$ below a temperature $T_\mathrm{CDW} \approx 0.09 \Gamma_\mathrm{CDW}$. Figure \[fig06\] shows the (interpolated) amplitude $b_{\mathrm{CDW}}(T,0)$ as a function of temperature $T$.
![Dimensionless charge-density gap $\bar{b}_\mathrm{CDW}=b_\mathrm{CDW}/\Gamma_\mathrm{CDW}$ as a function of dimensionless temperature $\bar{T}$ interpolated to the frequency $\varepsilon=0$. A CDW order appears below the temperature $T_\mathrm{CDW}\approx 0.09\Gamma_\mathrm{CDW}$. []{data-label="fig06"}](fig03.eps){width="0.7\linewidth"}
Calculating the charge density in the presence of the order parameter $\mathcal{O}_\mathrm{CDW}$, Eq. (\[044\]), we obtain formula (\[c02\]) for the bidirectional CDW modulation, $$\begin{aligned}
\rho_{\mathrm{CDW}}(\mathbf{r})
\sim \frac{e\Gamma_{\mathrm{CDW}}^2}{v^2}
\big\{
\cos(\mathbf{Q}_x\mathbf{r}+\varphi_x)+
\cos(\mathbf{Q}_y\mathbf{r}+\varphi_y)
\big\}
\label{046}\ ,\end{aligned}$$ where $\varphi_x$ and $\varphi_y$ denote the phases of the CDW order in $x$ and $y$ directions, respectively. Thus, the charge density is modulated with the wave vectors $\mathbf{Q}_x$ and $\mathbf{Q}_y$ connecting two opposite antinodal points. This contrasts the modulations of the quadrupole-density $D_{xx}$ generated[@emp] at the hotspots in the presence of QDW, $$\begin{aligned}
D_{xx}(\mathbf{r}) &\sim e\big\{|\Delta_{\mathrm{QDW}}^1|\cos(\mathbf{Q}_1\mathbf{r}+\varphi_1)
\nonumber\\
&\qquad +
|\Delta_{\mathrm{QDW}}^2|\cos(\mathbf{Q}_2\mathbf{r}+\varphi_2)\big\}
\label{047}\ .\end{aligned}$$ QDW wave vectors $\mathbf{Q}_1$ and $\mathbf{Q}_2$ are turned by $45^{\circ}$ and longer than the CDW wave vectors by a factor roughly given by $\sqrt{2}$. Both orders form checkerboards as illustrated in Fig. \[fig02\]. Figure \[fig02\](c) shows the type of particle-hole order, i.e. whether QDW or CDW, as a qualitative function of the position on the Fermi surface. Whereas within our model hotspot and antinodal regions are separated, we expect in realistic systems regions of small overlap of the two orders in between.
Cuprate physics {#sec:cuprates}
===============
![Qualitative phase diagram summarizing the results of Ref. and the present work for zero magnetic field. Close to the antiferromagnetic (AF) QCP, $\xi_{\mathrm{AF}}^{-2}=0$, and upon lowering the temperature, the systems develops first at $T^*$ the instability toward the fluctuating pseudogap state (PG) characterized by the order parameter of Eq. (\[k0\]). At lower temperatures $T<T_\mathrm{CDW}<T^*$, strong superconducting fluctuations induce a transition toward charge density wave (CDW) formed at the antinodes. Finally, below $T_c$, the particle-particle suborder of the pseudogap prevails due to curvature effects and establishes $d$-wave superconductivity.[]{data-label="fig07"}](fig04.eps){width="0.8\linewidth"}
We now address the phase diagram of cuprates in the proximity of the antiferromagnetic QCP. We emphasize that our theory applies only to the “metallic” side of the antiferromagnet–normal metal phase transition. The regions of too low doping are thus excluded in the following discussion. In the region of intermediate doping, suppression of carrier density below a crossover temperature $T^*$ observed in NMR measurements [@warren; @alloul] was the first evidence for the existence of a “pseudogap” in the electron spectrum. In contrast, $d$-wave superconductivity appears only below a considerably lower temperature $T_{c}$. In our theory, $T^*$ is associated with the crossover to the strongly fluctuating $\mathrm{O}(4)$-symmetric composite order (superconductivity and QDW) close to the hotspots. [@emp] The phase diagram, see Fig. \[fig07\], is further enriched by the formation of CDW order with wave vectors $\mathbf{Q}_{x,y}$ (Fig. \[fig02\]) at the edge of the Brillouin zone. Also the emergence of the CDW order is ultimately due to the proximity to the QCP. The additional phase transition is expected to occur at a temperature $T_{\mathrm{CDW}}$ inside the pseudogap phase, $T_{c}<T_{\mathrm{CDW}}<T^*$.
The charge modulation observed in various recent experiments [@wise; @davis; @yazdani; @julien; @ghiringhelli; @chang; @achkar; @leboeuf; @blackburn] has been attributed [@emp; @sachdev13a] to the existence of QDW (or “bond order”) correlations. This picture is, in principle, in agreement with NMR results [@julien] and sound propagation measurements [@leboeuf; @shekhter]. However, STM studies [@wise; @davis; @yazdani] of BSCCO and experiments with hard [@chang; @blackburn] and resonant soft [@ghiringhelli; @achkar] X-ray scattering on YBCO have revealed a charge modulation along the bonds of the Cu lattice with modulation vectors close to $\mathbf{Q}_{x,y}$, which are the CDW wave vectors. Moreover, QDW has a vanishing Fourier transform near even Bragg peaks. Therefore, STM and hard X-ray experiments can hardly be expected to detect the QDW modulation.
The seeming contradiction is resolved when we include the CDW, Eq. (\[c02\]), in the Cu lattice. Then, this explains the experimental results [@wise; @davis; @yazdani; @julien; @ghiringhelli; @chang; @achkar; @leboeuf; @blackburn]. CDW appears below a critical temperature $T_{\mathrm{CDW}}$ that can be considerably lower than $T^*$, in line with the results of the hard X-ray experiment of Ref. . In addition, Hall effect measurements [@leboeuf_hall] indicate a reconstruction of the Fermi surface that is attributed to the formation of CDW. The transition temperatures $T_{\mathrm{CDW}}$ of these two experiments agree with each other. Evidence for a transition below $T^*$ and related to CDW has also been found recently in a Raman scattering study. [@letacon_dec13] The dual effect of the two modulations (QDW and CDW) on the two species of atoms in the CuO plane is a characteristic of our theory and might be tested via resonant soft X-ray scattering.
Very recent STM and resonant elastic X-ray experiments [@comin; @yazdani_dec13] on BSCCO confirm the CDW wave vectors’ orientation along the bonds but indicate that they connect hotspots rather than antinodes. In our model, we expect CDW to set in at wave vectors as soon as the QDW gap is small. In realistic systems, this may indeed happen already not very far from the hotspots, possibly enhanced by reconstruction of the Fermi surface. Details behind this physics are clearly beyond the range of our “minimal model” and left for a separate study.
The emergence of various gaps in $\mathbf{k}$-space around the Fermi surface has been reported in Raman scattering on Bi-2212 and Hg-1201 compounds. [@sacuto] It was demonstrated that in overdoped samples the superconducting gap spreads all over the Fermi surface. In contrast, in underdoped samples the coherent Cooper pairs are observed mostly near the nodes, whereas the gap at the antinodes is mainly of a non-superconducting origin. This effect can naturally be explained within our picture because the hotspots move to nodes with decreasing the doping and the superconducting gap at the antinodes should decrease. At the same time, the CDW gap grows at the antinodes thus pushing away" the Cooper pairs.
We note that after our work has been completed and distributed as a preprint on arXiv, a work discussing the issue of the rotation of the charge order wave vector by $45^{\circ}$ has appeared. [@chubukov2014] A solution of mean-field equations for a new CDW suggested in the latter work, although very interesting, is not stable against formation of SC/QDW order of Ref. below its transition temperature $T^*$. As a result, new preemptive states predicted in Ref. may be possible only in the vicinity of $T^*$.
Conclusion
==========
Extending the analysis of the spin-fermion model for the two-dimensional antiferromagnetic QCP to the antinodal regions, we find below the pseudogap temperature $T^*$ another transition to a bidirectional CDW induced at the zone edge by superconducting fluctuations. The physics behind this transition is determined by pseudogap physics emerging at the hotspots. Our theory thus shows how a complexity of offspring phases arises out of the single QCP. The results enable us to address recently observed charge order features in the phase diagram of the high-$T_c$ cuprates.
K.B.E. acknowledges support by the Chaire Blaise Pascal award of the Région Île-de-France. H.M. acknowledges the Yale Prize Postdoctoral Fellowship. Financial support (K.B.E., H.M., and M.E.) by SFB/TR12 of DFG is gratefully appreciated.
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---
abstract: 'The complex scaled hyperspherical adiabatic expansion method is used to compute momentum and energy distributions of the three $\alpha$-particles emerging from the decay of low-lying $^{12}$C-resonances. The large distance continuum properties of the wave functions are crucial and must be accurately calculated. We discuss separately decays of natural parity states: two $0^+$, one $1^{-}$, three $2^+$, one $3^-$, two $4^+$, one $6^+$, and one of each of unnatural parity, $1^{+}$, $2^-$, $3^+$, $4^-$. The lowest natural parity state of each $J^{\pi}$ decays predominantly sequentially via the $^{8}$Be ground state whereas other states including unnatural parity states predominantly decay directly to the continuum. We present Dalitz plots and systematic detailed momentum correlations of the emerging $\alpha$-particles.'
author:
- 'R. Álvarez-Rodríguez$\:^1$, A.S. Jensen$\:^1$, E. Garrido$\:^2$, D.V. Fedorov$\:^1$, H.O.U. Fynbo$\:^1$'
title: 'Momentum distributions of $\alpha$-particles from decaying low-lying $^{12}$C-resonances'
---
Introduction
============
The low-lying resonance states of $^{12}$C have been studied over many years both theoretically and experimentally, motivated partly by their astrophysical importance [@mor56; @tak70; @fri71; @ueg77; @pic97; @des02; @nef04; @fedt04; @kur07; @fre07a]. Surprisingly, many issues are still not really understood, e.g. the energies, angular momenta, structure and decay properties of the resonances. Completely open questions still remain on the $2^+$ resonances. Morinaga conjectured in the fifties that a $2^+$ state should exist around 9 MeV as a member of the rotational band with the $0^+$ resonance at 7.65 MeV as band-head [@mor56]. Several experiments recently provided new results [@joh03; @ito04; @dig05] but unfortunately no agreement has yet been reached for the position and width of the first $2^+$ resonance.
Attempts to obtain information about the spectrum from decay measurements immediately face the problem that only the final state is observed. Properties of the initial state must then be reconstructed from the momentum distributions of the three fragments after the decay. Both initial state and the intermediate paths connecting initial and final states are not observables. These configurations can therefore only be described through model interpretations. This is somewhat different in reaction experiments, where information can in addition be extracted from properties of outgoing particles in transfer or scattering reaction.
If we assume that the initial state is a resonance populated one way or another, and that its decay is independent of the previous history. This is a simplification decoupling the formation from the decay in analogy to compound nuclear reactions. The decay process can then be viewed as a stationary wave function connecting initial and final states through a continuous series of intermediate configurations. This is equivalent to a time dependent process where the initial state, formed at small distances, evolves through the intermediate configurations and results in the final state at large distances. This implies a steady state outgoing flux described precisely by the stationary resonance wave function.
Thus the resonance wave function can be interpreted as reflecting the decay mechanisms. Two principally different modes are traditionally considered, i.e. sequential decay via an intermediate two-body configuration, and decay directly into the three-body continuum. In both cases the final state is embedded in the three-body continuum and the modes can only be distinguished if the momentum distributions carry unique information characterizing one of the modes. Otherwise the distinction becomes fluent or a matter of an artificial, although perhaps more precise, model definition. Previous approaches to describe this type of observables have been performed mainly for the $1^+$ states, e.g. Faddeev calculations [@tak], R-matrix computations, which describe their decay as sequential [@bal], and Kurchatov fitting, which describes them as direct or democratic [@kor].
The purpose of this paper is to present $\alpha$-particle momentum distributions and Dalitz plots [@dal53] after decays of all the computed $^{12}$C-resonances [@alv07a] below the proton separation threshold at an excitation energy of 15.96 MeV, where only $3\alpha$-decay is possible. These distributions should help to establish spins and parities of the yet unknown levels.They provide then information about structures of initial and intermediate states. Combined with the measurements a more complete picture of the $^{12}$C-spectrum and the decay mechanisms should then emerge. In section 2 we first sketch the theoretical framework and the choice of interactions. The results are presented and discussed in section 3 for both unnatural and natural parity states. Section 4 contains a summary and the conclusions.
Theoretical framework
=====================
The resonances decay into three particles, therefore we need a theoretical tool to describe this three-body continuum structure. We employ the hyperspherical complex rotated [@ho02; @fed03] adiabatic expansion [@nie01] in coordinate space to compute bound states and resonances. This method is able to deal with several simultaneously bound and nearly bound two-body states in different subsystems. Relatively large distances can often be calculated accurately with a specific choice of basis and partial waves. The Fourier transform of the wave function provides the observable momentum distributions. The three-body model consisting of $\alpha$-particles requires interactions which reproduce energies and scattering properties of the $\alpha$-$\alpha$ system.
Practical procedure
-------------------
We describe $^{12}$C as a 3$\alpha$-cluster system at all distances. We use Faddeev equations and solve them in coordinate space using the adiabatic hyperspherical expansion method [@nie01; @gar05b; @fed03]. The hyperspherical coordinates consist of the hyperradius $\rho$ and five generalized angles. The angular Faddeev decomposed wave functions, $\Phi_{nJM}=\sum_{i=1}^3 \Phi_{nJM}^{(i)}$, are chosen for each $\rho$ as the eigenfunctions of the angular part of the complex scaled ($\vec r \rightarrow \vec r \exp(i\theta)$) Faddeev equations $$(T_\Omega-\lambda_n) \Phi_{nJM}^{(i)}+\frac{2m}{\hbar^2} \rho^2V_i
\Phi_{nJM} = 0 \qquad i=1,2,3\;.
\label{fadeq}$$ $T_\Omega$ is the angular part of the kinetic energy operator and $V_i$ is the potential between particles $j$ and $k$, being {$i,j,k$} a cyclic permutation of {$1,2,3$}. The total wave function, $\Psi^{JM}$, is expanded on the hyper-angular eigenfunctions, i.e. $$\Psi^{JM} = \frac{1}{\rho^{5/2}}\sum_n f_n (\rho) \Phi_{nJM}
(\rho,\Omega)\;,$$ where the $\rho$-dependent expansion coefficients, $f_n (\rho)$, are the hyperradial wave functions obtained from the coupled set of hyperradial equations $$\begin{aligned}
\left(-\frac{\partial^2}{\partial \rho^2}+ \frac{15/4}{\rho^2}+
\frac{2m}{\hbar^2} [W_n(\rho)+V_{3b}(\rho)-E]\right) \:f_n(\rho)
&&\nonumber \\ = \sum_ {n^\prime =1}^ \infty \hat
P_{nn^\prime}f_{n^\prime} (\rho) &&\;.
\label{radial}\end{aligned}$$ $W_n(\rho)$ are the angular eigenvalues of the three-body system Hamiltonian with fixed $\rho$, $V_{3b}$ is the three-body potential, $E$ is the three-body energy and $P_{nn^\prime}$ are the non adiabatic terms. The eigenvalues $W_n(\rho)$ of the angular equations eq. (\[fadeq\]) serve as effective potentials.
In order to obtain the resonances we use the complex scaling method. According to this method, the energy and width of a resonance state are associated with the complex eigenvalues of a certain analytically continued Hamiltonian operator. The appropriate operator results from the rotation of the position vectors of the ordinary Hamiltonian into the complex coordinate plane $$\vec r \to \vec r \: e^{i\:\theta} \qquad \theta > 0, \mathrm{ real} \;.$$ This gives rise to the complex-rotated Hamiltonian $$H_\theta (\vec r) = H (\vec r \: e^{i\:\theta} )\;.$$
The complex energy of a resonance corresponds to a pole in the momentum-space wave function, while in coordinate-space this form corresponds to a large-distance asymptotic wave function consisting of outgoing waves. In other words, the three-body resonance corresponds to a complex energy solution $E_0=E_R-i\:E_I$ of the system (\[radial\]) with the asymptotic boundary condition of an outgoing wave in every channel $n$ $$f_n(\rho \to \infty) = C_n\: e^{+i\:\kappa\rho}\;,
\label{boundary}$$ where $C_n$ is an asymptotic normalization coefficient and $\kappa =
\sqrt {2mE/\hbar^2}$ is the three-body momentum or the conjugate of $\rho$. It has been seen that this boundary condition determines that the scattering matrix has a pole at the complex energy $E_0$, being $E_R$ the position of the resonance and $\Gamma = 2 E_I$ its width.
The $^{12}$C-resonances are not necessarily of three-body character even though this by definition must be the case at large distances for $3\alpha$-decay. We use the three-body model also at small distances because, like in Gamow’s theory of $\alpha$-decay, the detailed structure at small distances is not important for the decay properties which only require the proper description of the emerging three particles. We use the three-body short-range potential to adjust the corresponding small-distance part of the effective potential to reproduce the correct resonance energies which are all-decisive for decay properties as evident in the probability for tunneling through a barrier.
At intermediate distances the three $\alpha$-particles are formed and the potential has a barrier that determines the partial width of the resonance. At large distances the resonance wave functions contain information about distributions of relative energies between the three particles after the decay. These properties are connected to the many-body properties at small distances via preformation factors, as in $\alpha$-decay. An adjustment of the resonance energy is then needed. After complex rotation the resonance wave function is characterized by an exponential fall-off at large distance. Thus the crucial information is found in relative sizes of the very small values, $f_n$, of the resonance at large distances which are very difficult to compute accurately especially when the Coulomb interaction is present.
Momentum distributions
----------------------
The complex scaled coordinate space resonance wave function should be rotated back to real coordinates and Fourier transformed to provide the observable momentum distributions. Unfortunately the corresponding integral is not convergent and a regularization procedure has to be applied. The origin is simply that the resulting wave function should be a non-normalizable outgoing plane wave at large distances. We overcome this problem with the Zeldovic regularization procedure which is well defined for short-range interactions [@fed03]. In total this amounts to using the angular part of the coordinate space wave function at a large hyperradius, but interpreted as the momentum space wave function. Inclusion of the Coulomb interaction is achieved by treating it as an ordinary potential up to a large value of the hyperradius and then extrapolate the diagonal parts of the adiabatic wave functions with the numerically obtained Coulomb and centrifugal potentials.
Two different cases must be treated, i.e. sequential and direct decays distinguished theoretically by the structure of the adiabatic wave functions [@alv07b]. Direct decay is characterized by structures where all particles are far apart and as the hyperradius increases all distances increase proportionally. The Zeldovic regularized Fourier transform of the resonance wave function gives in this case the momentum distributions [@fed04].
Sequential decay is characterized by a wave function describing a bound-state like structure of two close-lying particles supplemented by the third particle far away. For a complex scaled wave function such a structure would be that of a two-body resonance, provided the rotation angle $\theta$ is larger than the angle corresponding to the energy and width of this resonance. These structures approach two-body bound state configurations as the hyperradius increases.
However, Fourier transformed and rotated back to the real axis, the wave function should at large distances approach the description of the third particle (plane or Coulomb wave) leaving the decaying resonance which has the given two-body energy and width. This is two sequential two-body decays, hence the characterizing notation. The resulting momentum distributions cannot be obtained from the rotated wave function but should instead be calculated from the correct physical description of two two-body decays. This results in a Breit-Wigner distribution for the third particle with a width equal to the sum of two-body and initial three-body resonance widths peaking around the energy found by subtracting the two-body from the three-body energies.
Interactions
------------
The basic ingredients are the two-body interactions,$V_i$, between particles $j$ and $k$, where {$i,j,k$} is a cyclic permutation of {$1,2,3$}. First $V_i$ must reproduce the low-energy two-body scattering properties which can be obtained independently for each partial wave resulting in angular momentum dependent or non-local interactions. We rely on the experience gained previously especially through [@alv07a], and we choose an Ali-Bodmer potential [@ali66] slightly modified in order to reproduce the s-wave resonance of $^8$Be. The phase shifts are essentially unchanged and reproduce $\alpha-\alpha$ scattering data but in order to describe sequential decays properly the two-body subsystems must also have the correct energies. In total we use a potential given as $$\begin{aligned}
V_{\alpha \alpha} &=& \left( 125 \hat P_{l=0} + 20 \hat P_{l=2} \right)
e^{-r^2/1.53^2} \nonumber \\
&&- 30.18 e^{-r^2/2.85^2}\;,\end{aligned}$$ where lengths are in fm and strengths are in MeV. The operators $\hat P_l$ project on angular momentum.
The three-body resonance energy and wave function can now be computed but the energy usually does not coincide with the measured value. It may be close, indicating that the three-body structure is nearly correct. Then only fine-tuning is needed due to the neglected smaller three-body effects of polarization or excitations of intrinsic particle degrees of freedom or off-shell effects. We emphasize that only three-body effects are missing since the two-body data already is reproduced by the phenomenological two-body interactions. We then correct the energy by including a diagonal three-body short-range interaction chosen to be Gaussian in hyperradius, i.e. $V_{3b}=
S\exp(-\rho^2/b^2)$. The structures of the resonances are then maintained [@fed96]. A larger range corresponding to a third order power law is not selected as e.g. in [@tho00] where it is used to compensate for the limitation in Hilbert space due to the hyperharmonic expansion in only one Jacobi coordinate. Our better basis confines the three-body interaction to be genuinely of short-range character.
In the actual parameter choice we prefer to maintain the same values of $b$ and $S$ for different states with the same angular momentum and parity $J^{\pi}$ but allow variation with $J^{\pi}$. To see the systematic behavior we then decided to fix $b=6$ fm corresponding to the hyperradius obtained when the three alphas are touching in an equilateral triangle. The strength $S$ is then adjusted to reproduce one of the observed resonance energies. The main dependence is indirect through the variation of the three-body energy and much less through the shape of the total potential [@alv07a]. In this way we attempt to separate the effects of the initial many-body structure from the symmetries related to the angular momentum conservation. The strongest influence is expected from Coulomb potentials and centrifugal barriers.
Computed distributions
======================
We find $^{12}$C-resonances below 15.96 MeV for most angular momenta $J \leq 6$ and all parities, i.e. two $0^+$, three $2^+$, two $4^+$, and one of each of $1^\pm$, $2^-$, $3^\pm$, $4^-$ and $6^+$ [@alv07a]. Their structures were described in detail in [@alv07a] including the variation with possible interaction parameters. However, only small and intermediate distance properties are important for energies, widths and partial wave decomposition. The final state momentum distributions after decay arise from the large distance properties which are much more difficult to determine numerically.
The procedure is to compute ratios of radial wave functions at large distances. This supplies the relative weights on the contributions from each of the adiabatic wave functions. First we have to remove the contributions from the wave functions corresponding to population of two-body resonances. These fractions must be computed as consecutive two-body decays and their contributions added to the remaining results from direct decays which are found by absolute square of the wave function at a large hyperradius followed by integration over the unobserved angular variables.
The asymptotic large-distance behavior should be reached by increasing the partial waves and the basis size used. This convergence can be tested by showing independence of the results with variation of the largest value of the hyperradius. Failing the test implies that the basis size is too small, or contrarily a larger hyperradius can be compensated by a larger basis producing the same result. It is then economical to get stability for a hyperradius and basis as small as possible. In most cases we find that the asymptotic behavior is reached for hyperradii larger than about 60 fm. There is a small variation of the distributions from 70 to 100 fm, and we have chosen 80 fm as the value of $\rho$ where the energy distributions are computed.
The results fall in two groups of natural and unnatural parity states, e.g. implying that sequential decay through $^8$Be($0^+$) is either allowed or forbidden by conservation of angular momentum and parity. Decay through $^8$Be($2^+$) is possible in both cases but this state is rather broad and the result would be hard to distinguish from direct decay. We see no indication of population of this channel in the numerical results. To optimize the accuracy we then maintain as small a scaling angle as possible consistent with distinct separation of the three-body resonance from the background continuum.
Unnatural parity states
-----------------------
{width="18cm"}
These states are $1^+$, $2^-$, $4^-$ and $3^+$ and our basis describes them as decaying directly although analyses of measured distributions employ interpretations as sequential through $^8$Be($2^+$) [@dig05; @dig06; @fyn03].
The lowest $1^+$ state was briefly discussed previously in [@alv07b; @alv07d]. Experimentally two $1^+$ states, isospin 0 and 1, are known but we find only one reflecting that we are confined to isospin 0 by using $\alpha$-particles as building blocks. Both states are very far from resembling $\alpha$-cluster states. Still the decays of both states must proceed through the same $\alpha$-cluster configurations, although the weights on the adiabatic potentials might differ from state to state. Underlying many-body effects are beyond the present model but we can pinpoint the neglected effects, i.e. the preformation factors established at small distances where the many-body problem is constraint into a three-body problem, and better three-body potential to account for the transition between the $N$- and three-body degrees of freedom at short and large distances, respectively. For these reasons the contributions from the individual adiabatic potentials could differ for decays of these two $1^+$ states of different isospin.
We first focus on the isospin 1 state at an excitation energy of 14.98 MeV ($7.70$ MeV above threshold). We adjust the three-body potential and compute the energy distributions shown in Fig.\[fig1\]. The upper part exhibits the Dalitz plot and the lower part projects the distribution on the axis with one $\alpha$-particle energy. The latter is computed by using Monte Carlo integration over all phase space directly from the wave function.
The measured distributions [@bal74] are very uncertain first of all because the lower-lying isospin zero $1^+$ state at 12.70 MeV (5.42 MeV above threshold) also is populated via feeding from a gamma transition between the two $1^+$ states. This contribution is not easily removed from the existing data to allow a clean comparison. A better analysis or a new experiment measuring the $\alpha$-decay of the T=1 $1^+$ state in complete kinematics is required. Our computed result is almost identical to the distributions, measured and calculated, for the 12.70 MeV state [@fyn03; @alv07b] if the difference in available energy is corrected for. The distributions in Fig.\[fig1\] are then direct prediction based on the assumptions that the isospin zero components in both states are equally populated and decay through the same mechanism.
A test of this prediction would provide interesting information about the dynamics of isospin mixing. Two extremes can be imagined, i.e. the same isospin $0$ components can be present from small to large distance resulting in the same distribution, or different complicated many-body structures at small distances clusterize into $\alpha$-particles around the nuclear surface and proceed to detection at large distances. We know that the partial decay widths for both states are much smaller than predicted from the cluster model [@alv07a] but this information does not prohibit the momentum distributions from being almost identical.
In the computation we find only one of each state of even $J$ and negative parity, i.e. one $2^-$ one $4^-$ state. The experimental spectrum has two states of $2^-$ where the highest at 13.26 MeV (5.98 MeV above threshold) only is tentatively assigned to have $2^-$ [@azj] while no $4^-$ state is found experimentally. It is then tempting to believe that this state at 13.26 MeV really is a $4^-$ state as indicated by our computations [@alv07c]. This new spin-parity assignment has also been suggested recently in [@fre07].
One way to decide which spin and parity is correct is to measure the momentum distributions of the fragments emerging after decay. Usually this carries distinct signatures of the angular momentum of the decaying state. In [@jac72; @ant75] it is shown that, even within non-sophisticated theoretical models, the basic signatures of the angular momentum are present in the experimental data. We first turn to the energy distributions in Fig.\[fig1\]. Both $2^-$ and $4^-$ show very similar distributions but the peaks appear at slightly higher values for the $4^-$ state. However, the two-dimensional Dalitz plots differ more from each other. Both have the triangular symmetry but the $2^-$ resonance have virtually nothing in between these peaks in contrast to the somewhat more smeared out distributions of the $4^-$ resonance.
In the computations we find also a $3^+$ resonance for which there is no experimental evidence, but it has been suggested in [@fre07] to assign these quantum numbers to the state at about an excitation energy of 13.35 MeV. Theoretically a $3^+$ state has also been found in [@ueg77]. With a reasonable three-body strength, -20 MeV, placing the state at 14.40 MeV (7.13 MeV above threshold), we find the energy distributions in fig. \[fig1\]. The distribution is very broad but peaked at intermediate energies. This is seen to arise from a Dalitz plot distribution with a small hole in the middle surrounded by a close-lying dense circle and a much larger diffuse distribution.
The angular momentum may leave an even more distinct signature in the angular distributions, shown in Fig.\[fig2\], of the directions between two particles and their center of mass and the third particle. The information is then directly about the corresponding angular momentum denoted as $\ell_y$, i.e. the angular momentum of the third particle relative to the center of mass of the other two with the relative angular momentum $\ell_x$. We see that the angular distribution patterns are quite different for different states. The $1^+$ distribution shows two broad peaks separated by a minimum with vanishing probability at an angle of $\pi/2$.
This reflects that the partial wave components in the angular wave function are a linear combination of only $(\ell_x, \ell_y) = (2,2),
(4,4)$ each coupled to the resulting value of $1$ [@alv07b]. Choosing the specific directional angles of $\phi_x = \phi_y =
\theta_x=0$ and $\theta_y = \pi/2$ we find that only projection quantum numbers of $m_x =0$ and $m_y =0, \pm 2, \pm 4$ give non-vanishing contributions. This is only consistent with a projection of the total angular momentum $M= m_x + m_y$ since $M=m_y=\pm 1$ gives zero. However when all projections are zero the Clebsch-Gordan coupling coefficient is also zero. The observables in Fig.\[fig2\] reveal information about the intrinsic angular momenta used to construct the wave function.
In contrast both $2^-$, $3^+$ and $4^-$ have peaks in the distributions at $\pi/2$. The different shapes can be traced back to the different partial wave decomposition computed and discussed in [@alv07b], i.e. $2^-$ has about 40% to 60% of $\ell_y =1,3$, while $4^-$ is dominated by $\ell_y =3$, and $3^+$ has about twice as much $\ell_y =2$ as $\ell_y =4$. These features are clearly distinguishable demonstrating that these observables can be used to determine the large-distance structure of these resonances. The initial state can still only be determined through the theoretical information about the dynamical evolution of the resonances.
The one-dimensional distributions in Fig.\[fig1\] can be used to extract the distributions of how far the three particles are from each other [@alv08a]. This is visualized by a triangle with a particle in each corner moving apart from their common center of mass. In particular the distributions of the ratio, $x/y$, of the distances between two particles, $x$, and their center of mass and the third particle, $y$, are shown in Fig.\[fig3\]. Since all Jacobi systems are identical we do not have to distinguish between Jacobi sets. Unfortunately the symmetric wave function then do not allow distinction between these identical particles.
With several peaks as for the $1^+$ resonance the interpretation is obvious, namely that each peak contains one $\alpha$-particle. The triangular geometric structure for the decay of this isospin one $1^+$ state then corresponds to side ratios of 2.2:1.8:1 of the triangle. For the other unnatural parity states only one broad peak is seen close to the value 1. For an equilateral triangle the $x/y$-ratio is $2/\sqrt{3} \approx 1.15$ which then is the only value where a narrow peak is possible. Otherwise a broader peak must cover overlapping distributions deviating somewhat from the equilateral triangle and corresponding to similar but less symmetric configurations.
Natural parity states
---------------------
{width="18cm"}
{width="18cm"}
These states are $0^+$, $1^-$, $2^+$, $3^-$, $4^+$ and $6^+$. They can decay via the energetically favorable $^8$Be($0^+$) which asymptotically must be described by one of the adiabatic potentials with the $^8$Be+$\alpha$ structure. The signature is simply that this potential approaches the complex energy of the $^8$Be($0^+$) resonance. The radial resonance wave functions at large distances determine the population fractions for each of the adiabatic potentials. In particular we can find the fraction of decay proceeding sequentially through this $0^+$-state, and furthermore we can compute the related distributions as two consecutive two-body decays.
The result is one peak close to an energy of $E_{max}=2E_{\alpha}/3$ with a width roughly equal to the width of the decaying state, and a broader square-like peak at an energy of about $E_{max}/4$ determined by kinematics. Here we assumed vanishing energy and width of $^8$Be($0^+$), otherwise the peak positions and widths should be modified. The Dalitz plots should also reflect these features by showing one high-energy, almost vertical, single-$\alpha$ distribution, and two separated (for each of the other $\alpha$-particles) more horizontal distributions corresponding to a broader peak after projection on the single $\alpha$-energy $x$-axis.
The angular distribution from sequential decay through $^8$Be($0^+$) must reflect the behavior of the angular momentum $\ell_y$ precisely as for ordinary decays of a quantum state of given angular momentum. The direct decay is expected to give a relatively broad distribution shifted from the central value at half the maximum energy by an appropriate average over the combinations of angular momentum phase space factors. This can also be interpreted geometrically as an expanding triangular configuration with given side ratios.
$J^\pi$ E$_{\alpha \alpha \alpha}$ (MeV) E$_{exc}$ (MeV) sequential direct
--------- ---------------------------------- ----------------- ------------ --------
$0^+_1$ 0.38 7.66 95% 5%
$2^+_1$ 1.38 8.66 97% 3%
$3^-$ 2.33 9.60 96% 4%
$4^+_1$ 3.25 10.52 92% 8%
$1^-$ 3.61 10.88 70% 30%
$0^+_2$ 3.95 11.22 59% 41%
$2^+_2$ 4.48 11.76 15% 85%
$2^+_3$ 6.49 13.76 4% 96%
$4^+_2$ 6.83 14.10 20% 80%
$6^+$ 7.13 14.40 5% 95%
: \[tab1\] Energy above the triple-$\alpha$ threshold, excitation energy and estimated amount of sequential via $^8$Be($0^+$) and direct decays for the natural parity states of $^{12}$C. If necessary we label the resonances with increasing energy above threshold.
First we extract the percentage of sequential decay via $^8$Be($0^+$) and direct decay for the natural parity states, see table \[tab1\]. The lowest-lying natural parity states of each $J^{\pi}$ ($0^+$, $2^+$, $3^-$ and $4^+$ states with excitation energies 7.66 MeV, 8.66 MeV, 9.60 MeV and 10.52 MeV) seem to be completely dominated by decays via $^8$Be($0^+$). In contrast, the highest-lying $2^+$ state at 13.76 MeV excitation energy and the $6^+$ state at 14.40 MeV excitation energy only have small fractions decaying through the $^8$Be ground state. In the remaining cases ($1^-$, $0^+$, $2^+$ and $4^+$ states with excitation energies 10.88 MeV, 11.22 MeV, 11.76 MeV and 14.10 MeV) both mechanisms are comparable.
The lowest of the two $0^+$ resonances is the so-called Hoyle state, which plays an important role in nuclear astrophysics. According to our computation, it decays almost entirely sequentially. Very little is left for the direct decay which therefore is not shown. The experimental distribution is also consistent with complete domination of sequential decay as in our computation [@alv07b]. We also omit the other three natural parity resonances dominated by sequential decays. We concentrate instead on the 6 resonances where a substantial amount is direct decay. These distributions are shown in Figs. \[fig4\] and \[fig5\] after removal of the contributions from the sequential decay through the $^8$Be ground state. The experimental analyses can extract the trivial contribution from the decay through the $^8$Be ground state. It is therefore straightforward to make a comparison with the experiment. Both Dalitz plots and projected single-$\alpha$ energy distributions are shown.
The higher-lying $0^+$ resonance has a large width of about 3.5 MeV. On top of this difficulty the population through beta-decay of the corresponding energy region leads to violation of the independent approximation of formation and decay of the resonance. The main effect is a shift in energy of the resonance position. In any case this state has a significant probability of decaying directly into the three body continuum. This part, shown in fig. \[fig4\], exhibits a triangular structure in the Dalitz plot, but now we find one low-energy $\alpha$-particle and two of moderate energies. This is in almost complete contrast to the sequential decay where one energy is high and two are small.
Next we focus on our results in connection with the existence and position of low-lying $2^+$ resonances which still is an open question for the $^{12}$C nucleus. The old suggestion is that the Hoyle state should be the band-head followed by a $2^+$ state at around 10 MeV [@mor56]. There are experimental indications for the existence of such a state [@fyn03] but no consensus has so far been reached. On the other hand other theoretical models, also cluster models, find three $2^+$ resonances in this energy region [@kan06]; in [@ueg77] two $2^+$ excited states are found in this region, while in [@des02] one $2^+$ state appears below 12.3 MeV excitation energy. We find rather different structures for these three states, still all of $\alpha$-cluster structure. Each of them is dominated by its own adiabatic wave function corresponding to three different low-lying adiabatic potentials with differing partial wave decomposition [@alv07a]. Most likely these states are hidden behind broad states of roughly the same energy. They are therefore extremely difficult to distinguish from the background in any of the experiments.
Their decay properties also vary substantially, e.g. the percentage of sequential decay through $^8$Be($0^+$), see table \[tab1\]. The lowest state almost exclusively decays sequentially while the other two mostly decay directly. In fig. \[fig4\] we see that the direct parts give very broad distributions. For the second $2^+$ resonance all three $\alpha$-particles emerge with large probability with similar kinetic energies. For the third $2^+$ resonance the distribution is more diffuse and the energies are more unevenly divided resulting in a structured but relatively broad distribution.
We continue with the $1^-$ state, both Dalitz plot and one-dimensional projection are shown in Fig. \[fig5\] for the 30% decaying directly into the three-body continuum. A similar triangular structure as for the second $0^+$ resonance is seen although substantially more smeared out resulting in two overlapping broad peaks after projection on the $x$-axis.
Both the $3^-$ resonance and the lowest of the two $4^+$ resonances are almost completely dominated by sequential decay. The second of the $4^+$ resonances gives a rather diffuse distribution of kinetic energy of the $\alpha$-particles, see fig. \[fig5\]. It resembles somewhat the distribution from the third $2^+$ resonance except that the small probability holes in the Dalitz plot now also are smeared out. This distribution is again almost the opposite of the sequential decay distribution with one high and two low energy particles. The $6^+$ resonance has a symmetric distribution extending about 1 MeV around a central region where all energies are roughly equal.
![The angular distributions of the directions between two particles and their center of mass and the third particle for the $(0_2^+, 1^-, 2_2^+, 2_3^+, 4_2^+, 6^+)$-resonances in Figs.\[fig4\] and \[fig5\]. We have performed a Monte Carlo integration over the phase space. The sequential part is removed as in figs. \[fig4\] and \[fig5\]. We label as in table \[tab1\].[]{data-label="fig6"}](figure6.ps){width="8.5cm"}
![The distributions of the ratio of the distances between two particles and their center of mass and the third particle for the $(0_2^+, 1^-, 2_2^+, 2_3^+, 4_2^+, 6^+)$-resonances in Fig.\[fig4\]. The sequential part is removed as in figs. \[fig4\] and \[fig5\]. We label as in table \[tab1\].[]{data-label="fig7"}](figure7.ps){width="8.5cm"}
We now turn to the other type of information found in the angular distributions which exhibit the correlated directions of emergence. Obviously the sequential decay through the $^8$Be($0^+$) state must be with the third $\alpha$-particle in the opposite directions of $^8$Be. The only information here is then about the partial wave component ($\ell_y$) of that third particle relative to $^8$Be. Angular momentum conservation then requires the total angular momentum $J =
\ell_y$. Thus the most interesting new information is contained in the directly decaying parts shown in Fig.\[fig6\]. These distributions also vary from state to state reflecting the structure in terms of partial waves as discussed in [@alv07a].
The distribution corresponding to $0_2^+$-state is essentially from the isotropic distribution of $\ell_y=0$ modified by a smaller contribution from $\ell_y=2$ with maxima at $\pi/4$ and $3\pi/4$ separated by zero probability at $\pi/2$. The distribution corresponding to $1^-$ shows two peaks separated by a small minimum at $\pi/2$. The largest partial waves are here $\ell_y=1,3$. The angular distributions of both the second and third $2^+$ resonance seem to contain a narrow peak on top of a broader one. These structures are due to large contributions from $\ell_y=0$ supplemented by contributions from $\ell_y=2$ and $\ell_y=4$, respectively. Finally, the distributions form $4^+_2$ and $6^+$ both exhibit one smooth, and for $6^+$ also relatively narrow, peak around $\pi$/2. The partial wave structures of these states are mainly $\ell_y=2$, and $\ell_y=2,4$, respectively
We again attempt to extract the geometric structure of the dominating triangular decay configurations. The results for the ratio between one pair of particles and their center of mass and the third particle are shown in Fig.\[fig7\]. They are all rather similar with a relatively broad peak around 1, but for $1^-$ and $2_3^+$ with more structure at a larger ratio suggesting another peak. As in Fig.\[fig3\] the peaks must cover overlapping distributions to correspond to an almost equilateral triangle. In the case of $0^+$ a very broad peak appears around 3, and the other two peaks are around 0.8. This gives rise to an obtuse triangle with side ratios 1.7:1:1.
Summary and conclusions
=======================
We have computed the $\alpha$-particle momentum distributions of 14 three-body decaying low-lying $^{12}$C many-body resonances with 10 different angular momenta and parities. The results are exhibited as single $\alpha$ energy distributions and energy correlations of Dalitz plots. We assume that the decays of the resonances are independent of their formation as for compound nuclear reactions. We use a three-$\alpha$ cluster model to describe all states even at small distances where the cluster model sometimes fails badly and the many-body structure is indispensable for a structure computation. The idea is, the same as for the classical $\alpha$-emission, that three $\alpha$-particles must be formed at small or intermediate distances as they emerge at large distances after the decay. Thus the small distance properties should only supply boundary conditions and impose energy and angular momentum conservation. This we mock up in the $3\alpha$ cluster model by a three-body interaction adjusted to reproduce the resonance energy. Again a simple analogy is found in the preformation factors in $\alpha$-emission.
An extreme example is the isospin 1 state which cannot be formed by $\alpha$-clusters. Its $\alpha$-decay width is consequently very small but still the resulting distributions are with the present assumptions predicted to be essentially the same as the $1^+$ isospin 0 state.
For three-body decays the interest, and the complication, is how the energy is shared between the three particles. This is determined by the “dynamic evolution” of the resonances, i.e. by the change in structure from small to large distances. To a large extent the decisive properties are symmetries from angular momentum and parity conservation. The resulting momentum distributions carry information about both initial resonance state and the intermediate configurations (decay mechanisms). The only energetically allowed two-body structure is the ground state of $^8$Be. Sequential decay through this state is dominating for natural parity states for the lowest resonance of a given angular momentum. The momentum distributions for the fractions decaying directly are predicted for all resonances below the proton separation threshold.
Whenever possible we give a geometric description of the parts decaying directly to the three-body continuum. This is expressed as side ratios of the $\alpha$-particles emerging in a triangle.
The Dalitz plots and $\alpha$-energy distributions differ from state to state. A complementary observable is the correlation between the direction of one particle and the center of mass of the other two. These distributions could be used to assign spin and parity to these decaying states as soon as sufficiently accurate experimental data become available. The directly measured angular distribution must contain information about the angular momentum of one particle with respect to the center of mass of the other two particles at large distances. Since several partial waves may contribute this information is not unique, and may have to be supplemented with other information. Furthermore, the uncertainty remains of how the measured large-distance properties reflect the small and intermediate-distance structures of the resonance wave function. Only a theoretical model can provide this connection.
In conclusion, we provide systematic and detailed decay information (fraction of sequential decay, Dalitz plots, single-$\alpha$ energy distributions, momentum direction correlations), which can be compared to upcoming experimental data, for each of the 14 lowest $^{12}$C resonances decaying by $3\alpha$-emission.
[**Acknowledgments**]{}
R.A.R. acknowledges support by a post-doctoral fellowship from Ministerio de Educación y Ciencia (Spain).
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---
abstract: |
We give lower bounds for the size of linearization discs for power series over $\mathbb{C}_p$. For quadratic maps, and certain power series containing a ‘sufficiently large’ quadratic term, we find the exact linearization disc. For finite extensions of $\mathbb{Q}_p$, we give a sufficient condition on the multiplier under which the corresponding linearization disc is maximal (i.e. its radius coincides with that of the maximal disc in $\mathbb{C}_p$ on which $f$ is one-to-one). In particular, in unramified extensions of $\mathbb{Q}_p$, the linearization disc is maximal if the multiplier map has a maximal cycle on the unit sphere. Estimates of linearization discs in the remaining types of non-Archimedean fields of dimension one were obtained in [@Lindahl:2004; @Lindahl:2009; @Lindahl:2009eq].
Moreover, it is shown that, for any complete non-Archimedean field, transitivity is preserved under analytic conjugation. Using results by Oxtoby [@Oxtoby:1952], we prove that transitivity, and hence minimality, is equivalent the unique ergodicity on compact subsets of a linearization disc. In particular, a power series $f$ over $\mathbb{Q}_p$ is minimal, hence uniquely ergodic, on all spheres inside a linearization disc about a fixed point if and only if the multiplier is maximal. We also note that in finite extensions of $\mathbb{Q}_p$, as well as in any other non-Archimedean field $K$ that is not isomorphic to $\mathbb{Q}_p$ for some prime $p$, a power series cannot be ergodic on an entire sphere, that is contained in a linearization disc, and centered about the corresponding fixed point.
author:
- |
Karl-Olof Lindahl\
School of Mathematics and Systems Engineering\
Växjö University, 351 95, Växjö, Sweden\
`Karl-Olof.Lindahl@vxu.se`
title: 'Estimates of linearization discs in $p$-adic dynamics with application to ergodicity[^1]'
---
[**Mathematics Subject Classification (2000):**]{} 32P05, 32H50, 37F50, 37B05, 37A50 [**Key words:**]{} dynamical system, conjugation, linearization, $p$-adic numbers, non-Archimedean field
Introduction
============
In this paper we study iteration of power series $f$ defined over $\mathbb{C}_p$, the completion of the algebraic closure of the $p$-adic numbers $\mathbb{Q}_p$. As in complex dynamics (i.e. iteration of complex-valued analytic functions, see e.g. [@Carleson/Gamelin:1991; @Milnor:2000; @Beardon:1991]), the main features of the dynamics under $f\in \mathbb{C}_p[[x]]$ is determined by the character of the periodic points of $f$, i.e. the modulus of the multiplier at the periodic points. A periodic fixed point point $x_0$ may be either *attracting*, *indifferent* or *repelling* depending on whether the multiplier $\lambda=f'(x_0)$ is inside, on or outside the unit sphere. In this paper we consider non-resonant (i.e. $\lambda $ not a root of unity) indifferent fixed points.
A power series over a complete valued field of the form $$f(x)=\lambda (x-x_0) + \text{(higher order terms)}$$ is said to be linearizable at the fixed point $x_0$ if there exists a convergent power series solution $g$ to the following form of the Schröder functional equation (SFE) $$\label{schroder functional equation}
g\circ f(x)=\lambda g(x), \quad \lambda =f'(x_0),$$ which conjugates $f$ to its linear part in some neighborhood of $x_0$. By the non-Archimedean Siegel theorem of Herman and Yoccoz [@Herman/Yoccoz:1981], as in the complex field case [@Siegel:1942], the condition $$\label{Siegel condition}
|1- \lambda^n|\geq Cn^{-\beta} \quad\text{for some real numbers
$C,\beta >0$},$$ on $\lambda$ is sufficient for convergence also in the non-Archimedean field case. Their theorem applies to the multi-dimensional case. In dimension one, the condition (\[Siegel condition\]) is always satisfied for non-resonant multipliers in fields of characteristic zero, i.e. the $p$-adic case studied in this paper, and the equal characteristic case of studied in [@Lindahl:2009eq]. This is not always true in fields of prime characteristic as shown in [@Lindahl:2004; @Lindahl:2009].
As shown by Herman and Yoccoz, in the two-dimensional $p$-adic case there also exist examples where the Siegel condition is not satisfied and the corresponding conjugacy diverges. The multi-dimensional $p$-adic case has been taken further by Viegue in his thesis [@Viegue:2007]. In this paper we only consider the one-dimensional non-resonant $p$-adic case so the conjugacy always converges.
The conjugacy function $g$ is unique if we specify the image and derivative at $x_0$. It is custom to assume that $g(x_0)=0$ and $g'(x_0)=1$. By the local invertibility theorem, $g$ has a local inverse $g^{-1}$ at $x_0$. We will refer to the *(indifferent) linearization disc* of $f$ about $x_0$, denoted by $\Delta _f(x_0)$, as the largest disc $U\subset
\mathbb{C}_p$, with $x_0\in U$, such that (\[schroder functional equation\]) holds for all $x\in U$, and $g$ converges and is one-to-one on $U$. The possibly larger disc, on which the the semi-conjugacy (\[schroder functional equation\]) holds, will be referred to as the *semi-disc*.
Note that, by definition, $f$ must be one-to-one on the linearization disc $\Delta _f(x_0)$. Moreover, the full conjugacy $$\label{full conjugacy}
g\circ f\circ g^{-1}(x)=\lambda x,$$ is valid for all $x\in g(\Delta _f(x_0))$. Let $f^{\circ n}$ denote the $n$-fold composition of $f$ with itself. On $g(\Delta _f(x_0))$ we have $g\circ f^{\circ n}\circ
g^{-1}(x)=\lambda ^n x$. Hence, there is a one-to-one correspondence between orbits under $f$ and the multiplier map $T_{\lambda}: x\mapsto \lambda x$, on $\Delta_f(x_0)$ and $g(\Delta_f(x_0))$, respectively. In particular, since $\lambda $ is not a root of unity, $f$ can have no periodic points on the linearization disc, except the fixed point $x_0$. However, the semi-disc may contain other periodic points as well, as manifest in the papers [@Arrowsmith/Vivaldi:1994; @Pettigrew/Roberts/Vivaldi:2001]. In fact, the semi-disc is contained in the *quasi-periodicity domain* of $f$, defined as the interior of the set of points on the projective line $\mathbb{P}(\mathbb{C}_p)=\mathbb{C}_p\cup \{\infty\}$ that are recurrent by $f$. In the case that $f$ is a rational function, Rivera-Letelier [@Rivera-Letelier:2000] gave several characterizations of the quasi-periodicity domain of $f$ and described its local and global dynamics. In particular, he proved that analytic components of the domain of quasi-periodicity, which are $p$-adic analogues of Siegel discs and Herman rings in complex dynamics, are open affinoids (that is, they have simple geometry), and contains infinitely many indifferent periodic points.
Our aim in this paper is three-fold. First, we obtain lower (sometimes optimal) bounds for the size of linearization discs for $f\in\mathbb{C}_p[[x]]$. These estimates extend results on quadratic polynomials over $\mathbb{Q}_p$ by Ben-Menahem [@Ben-Menahem:1988], and Thiran, Verstegen, and Weyers [@Thiran/EtAL:1989], and for certain polynomials with maximal multipliers over the $p$-adic integers $\mathbb{Z}_p$ by Pettigrew, Roberts and Vivaldi [@Pettigrew/Roberts/Vivaldi:2001], as well as results on small divisors in $\mathbb{C}_p$ by Khrennikov [@Khrennikov:2001a].
Second, we prove that transitivity (the existence of a dense orbit) is preserved under analytic conjugation into linearization discs over an arbitrary complete non-Archimedean field. Using results by Oxtoby [@Oxtoby:1952], the transitivity of $f$ on compact subsets of a linearization disc is proven to be equivalent to the ergodicity and unique ergodicity of $f$.
Third, when the dynamics is defined over the $p$-adic numbers $\mathbb{Q}_p$, we give necessary and sufficient conditions on the multiplier, that $f$ is transitive, hence uniquely ergodic, on spheres inside the linearization disc. These results generalize results obtained by Bryk and Silva [@Bryk/Silva:2003], and by Gundlach, Khrennikov, and Lindahl [@Gundlach/Khrennikov/Lindahl:2001:a], for monomials $f:x\mapsto cx^n$, and for 1-Lipschitz power series by Anashin [@Anashin:2006]. On the other hand, we also prove that transitivity is not possible on a whole sphere in any proper extension of $\mathbb{Q}_p$. Results on the transitivity and ergodic breakdown of the $p$-adic multiplier map $x\mapsto \lambda x$ were obtained by Oselies and Zieschang [@Oselies/Zieschang:1975], and by Coelho and Parry [@Coelho/Parry:2001].
A classification of measure-preserving transformations of compact-open subsets of non-Archimedean local fields were obtained recently by Kingsbery, Levin, Preygel and Silva [@KingsberyLevinPreygelSilva:2009]. They show that if a $C\sp 1$ transformation $T$ is measure-preserving when restricted to a compact-open set $X$ then $X$ can be written as a disjoint union of invariant compact-open sets such that $T$ restricted to each such set is either a local isometry or topologically and measurably conjugate to an ergodic Markov transformation. Concerning polynomials, the question of ergodicity is also answered, except in the case where the polynimial is $1$-Lipschitz, as in the present paper (the power series $f$ is certainly $1$-Lipschitz on the entire linearization disc). Non-1-Lipschitz functions were also studied in [@AnashinKhrennikov:2009].
Let us also mention some related works on measure preserving transformations on the Berkovich space, which is a much larger space than the $p$-adics. The Berkovich space provides a bridge between non-archimedean and complex dynamics. The works [@RumelyBaker:2004; @FavreRivera-Letelier:2004; @FavreRivera-Letelier:2006], construct a natural invariant measure for a wide class of rational functions, similar to existing constructions in complex dynamics.
Further results on the properties of the dynamics on $p$-adic linearization discs are provided in [@Arrowsmith/Vivaldi:1994; @Pettigrew/Roberts/Vivaldi:2001]. Estimates for linearization discs in prime characteristic were obtained in [@Lindahl:2004; @Lindahl:2009], and for fields of charactersitic zero in the equal charactersitic case [@Lindahl:2009eq]. See [@Lindahl:2004], for further comments on the non-Archimedean problem of linearization and its relation to the complex field case.
The construction of cojugacies in $p$-adic dynamics is related to standard and well-established techniques of local arithmetic geometry, see e.g. Lubin [@Lubin:1994] and the construction of local canonical heights by Call and Silverman [@CallSilverman:1993], and Hsia [@Hsia:1996]. For indifferent, non-resonant, fixed points the conjugacy function is related to the ‘logarithm’ of the theory of one-parameter formal Lie-groups defined over the $p$-adics [@Arrowsmith/Vivaldi:1994; @Lubin:1994]. As in [@Lubin:1994], the Lie-logarithm is constructed as the limit $$\label{Lie-logarithm}
\lim_{n\to\infty}\frac{f^{\circ p^n}-id_x}{p^n},$$ and is, up to a constant, the quotient between the conjugacy function $g$ and its derivative $g'$. The Lie-logarithm contains useful information about the dynamics of $f$. In particular, its roots are periodic points of $f$ [@Lubin:1994]. See Li [@Li:1996a; @Li:1996c] for various results on this matter, including the counting of periodic points of $p$-adic power series. Rivera-Letelier [@Rivera-Letelier:2000] proved if $f$ is a rational function, then the Lie-logarithm converges uniformly on the entire domain of quasi-periodicity.
For some additional references on non-Archimedean dynamics and its relationship, similarities, and differences with respect to the Archimedean theory of complex dynamics, see e.g. [@Arrowsmith/Vivaldi:1993; @Benedetto:2000c; @Benedetto:2001a; @Benedetto:2002; @Benedetto:2003a; @Bezivin:2004a; @Bezivin:2004b; @Hsia:1996; @Hsia:2000; @Khrennikov:2001a; @Rivera-Letelier:2003; @DeSmedtKhrennikov:1997; @AnashinKhrennikov:2009; @Khrennikov/Nilsson:2004; @Khrennikov:2003ryssbok; @DragovichKhrennikovMihajlovic:2007; @KhrennikovMukhamedovMendes:2007; @KhrennikovSvensson:2007; @Khrennikov:2003nauk; @Svensson:2005; @NilssonNyqvist:2004]. Applications of $p$-adic numbers have been proposed in coding theory [@CalderbankSloane:1995], round off errors [@Bosio/Vivaldi:2000], random number generation [@WoodcockSmart], and in biochemistry and physics [@AvetisovBikulovKozyrevOsipov:2002; @BaakeMoodySchlottmann:1998; @Khrennikov:1997; @Khrennikov:2004a; @RammalToulouseVirasoro:1986].
Summary of results
==================
Our most general result on the size of a linearization disc in $\mathbb{C}_p$ can be stated in the following way (see also Theorem \[theorem general estimate\] and Lemma \[lemma upper bound Siegel and isometry\]).
\[theorem linearization disc cp general\] Let $f\in\mathbb{C}_p[[x]]$ have an indifferent fixed point $x_0$, with multiplier $\lambda =f'(x_0)$, not a root of unity. Suppose that $f$ has the following expansion about $x_0$ $$\label{definition power series about x0}
f(x)=x_0+\lambda (x-x_0)+\sum_{i\geq 2}a_i(x-x_0)^i, \quad
\textrm{with } a=\sup_{i\geq 2}|a_i|^{1/(i-1)}.$$ Then, the linearization disc $\Delta_f(x_0)$, satisfies $D_{\sigma(\lambda,a)}(x_0)\subseteq \Delta_f(x_0)\subseteq
\overline{D}_{1/a}(x_0)$, where $\sigma(\lambda,a)$ is defined by (\[definition sigma\]). Moreover, if the conjugacy function $g$ converges on the closed disc $\overline{D}_{\sigma(\lambda,a)}(x_0)$, then $\Delta_f(x_0)\supseteq \overline{D}_{\sigma(\lambda,a)}(0)$. In particular, $f$ can have no periodic points in the punctured open disc $D_{\sigma(\lambda,a)}(x_0)\setminus \{x_0\}$.
The proof is based on estimates of the coefficients of the conjugacy function $g$. Applying a result of Benedetto [@Benedetto:2003a] (Proposition \[proposition-discdegree\] below), on these estimates we find a lower bound for the region of convergence of the inverse $g^{-1}$, and hence of the linearization disc.
Note that the estimate $\sigma=\sigma(\lambda,a)$ depends only on $\lambda$ and the real number $a$. To find the exact size of the linearization disc we do in general need more information about the coefficients of $f$. However, for a large class of quadratic polynomials, and certain power series containing a ‘sufficiently large’ quadratic term, we prove that $$\tau=|1-\lambda|^{-1/p}\sigma(\lambda,a)$$ is the exact radius of the linearization disc. More precisely, our main result can be stated in the following way (see also Theorem \[theorem quadratic polynomials\]).
\[thmA quadratic maps\] Let $p$ be an odd prime. Let $$f(x)=x_0+\lambda (x-x_0)+
a(x-x_0)^2\in\mathbb{C}_p[x-x_0],$$ with $\lambda$ not a root of unity. Suppose that $p^{-1}<|1-\lambda |<1$. Then, the linearization disc $\Delta_f(x_0)$ is equal to the disc $D_{\tau(\lambda,a)}(x_0)$, where the radius $\tau(\lambda,a)=|1-\lambda |^{-1/p}\sigma(\lambda,a)$.
This result is extended in Theorem \[theorem quadratic power series\] to power series containing a ‘sufficiently large’ quadratic term. We also give sufficient conditions, there being a fixed point on the ‘boundary’ of the linearization disc, i.e. the sphere $S_{\tau}(x_0)$ about $x_0$ of radius $\tau$.
Note that $\tau(\lambda,a)<1/a$. Hence, at least in this case, the linearization disc cannot contain the maximal disc $D_{1/a}(x_0)$ on which $f$ is one-to-one.
The relatively complicated expression for $\sigma$ stems from the presence of $p^s$th roots of unity in the punctured disc $D_1(1)\setminus \{1\}$, as described in Section \[section geometry and roots of uni ty C p\]. Some properties of $\sigma$ are discussed in Section \[section asympt behav siegel rad\]. In particular, we prove the following result.
\[theorem asymptotic behavior sigma\] Let $|\alpha -\lambda ^m|$ be fixed. Then, the estimate $\sigma$ of the radius of the linearization disc goes to $1/a$ as $m$ or $s$ goes to infinity. If $s$ and $m$ are fixed, then $\sigma \to 0$ as $|\alpha -\lambda
^m|\to 0$.
We now turn to the special case when the dynamics is restricted to $\mathbb{Q}_p$ and its finite extensions. In $\mathbb{Q}_p$, there are no $p^s$th roots of unity in $D_1(1)\setminus \{1\}$, and $\sigma $ takes a simpler form.
\[theoremA linearization disc in Qp\] Let $p$ be an odd prime, and let $f\in\mathbb{Q}_p[[x-x_0]]$ be of the form (\[definition power series about x0\]). Let $\Delta _f(x_0,\mathbb{Q}_p)=\Delta _f(x_0)\cap \mathbb{Q}_p$ be the corresponding linearization disc in $\mathbb{Q}_p$. Then, $\Delta_f(x_0,\mathbb{Q}_p)\supseteq
D_{\sigma_1}(x_0,\mathbb{Q}_p)$, where $$\sigma_1=a^{-1}p^{-\frac{1}{m(p-1)}}|1-\lambda ^m|^{\frac{1}{m}},$$ and $m\geq 1$ is the smallest integer such that $|1-\lambda
^m|<1$. Furthermore, if $|1-\lambda ^m|=p^{-1}$ and $m=p-1$, then $\Delta_f(x_0,\mathbb{Q}_p)$ is either the open or closed disc of radius $1/a$ about $x_0$. In particular, if either $\max_{i\geq 2 }|a_i|^{1/(i-1)}$ is attained (as for polynomials) or $f$ diverges on $S_{1/a}(x_0,\mathbb{Q}_p)$, then $\Delta_f(x_0,\mathbb{Q}_p)=D_{1/a}(x_0,\mathbb{Q}_p)$.
Note that the condition $|1-\lambda ^m|=p^{-1}$ and $m=p-1$, imply that $\lambda $ has a maximal cycle modulo $p^2$ in the sense that it is a generator of the group of units $(\mathbb{Z}/p^2\mathbb{Z})^{*}$. In this case $\lambda $ is said to be *maximal*.
\[theoremA linearization disc in Q2\]
Let $f\in\mathbb{Q}_2[[x-x_0]]$ be of the form (\[definition power series about x0\]). Then, the following two statements hold:
1. If $|1-\lambda |<1/2$, then the linearization disc $\Delta_f(x_0,\mathbb{Q}_2)$ contains the open disc of radius $\sigma_1= |1-\lambda |/2a $ about $x_0$.
2. If $|1-\lambda |=1/2$, then the linearization disc $\Delta_f(x_0,\mathbb{Q}_2)$ contains the open disc of radius $\sigma_3=\sqrt{ |1+\lambda|}/a$ about $x_0$.
\[theoremA maximal linearization disc in K \] Let $K$ be a finite extension of $\mathbb{Q}_p$ of degree $n$, with ramification index $e$, residue field $k$ of degree $[k:\mathbb{F}_p]=n/e$, and uniformizer $\pi$. Let $f\in K[[x]]$ be a power series of the form (\[definition power series about x0\]) and $\alpha$ a root of unity such that there is no closer root of unity to $\lambda^{p^{n/e}-1}$ than $\alpha$.
Suppose that $\lambda$ has a maximal cycle modulo $\pi^2 $ and $$\log_p e \leq (p^{n/e} -3)p/(p-1)- \nu\left ( \frac{\alpha - \lambda ^{p^{n/e}-1}}{1-\lambda^{p^{n/e}-1}}
\right )
+\log_p(p-1),$$ where $\nu$ is the valuation. Then, the linearization disc $\Delta_f(x_0,K)=\Delta_f(x_0)\cap K$ is maximal in the sense that $\Delta_f(x_0,K)$ is either the open or closed disc of radius $1/a$. In particular, if either $\max_{i\geq 2 }|a_i|^{1/(i-1)}$ is attained (as for polynomials) or $f$ diverges on $S_{1/a}(x_0)$, then $\Delta_f(x_0,K)=D_{1/a}(x_0,K)$.
Note that if the ramification index $e$ is not divisible by $p-1$, then $\alpha =1$ so that the $\nu$-term vanishes in this case. Also note that the linearization disc may be maximal even if $\lambda$ does not have a maximal cycle modulo $\pi^2 $, see Theorem \[theorem maximal linearization disc in K \].
In the final section of this paper we note some facts concerning transitivity, minimality and ergodicity on linearization discs. In particular, we show that transitivity is preserved under analytic conjugation into a linearization disc. More precisely.
\[thmA transitivity preserved\] Let $K$ be a complete non-Archimedean field. Suppose that the power series $f(x)=x_0 +
\lambda (x-x_0) +O((x-x_0)^2)\in K[[x-x_0]]$ is analytically conjugate to $T_{\lambda}$, on the linearization disc $\Delta_f(x_0)$ in $K$, via a conjugacy function $g$, with $g(x_0)=0$ and $|g'(x_0)|=1$. Suppose also that the subset $X\subseteq
\Delta_f(x_0)$ is invariant under $f$. Then, the following statements hold:
1) $f$ is transitive on $X$ if and only if $T_{\lambda}$ is transitive on $g(X)$.
2) If $X$ is compact and $f$ is transitive on $X$, then $f$ is minimal on $X$. Moreover, $f(X)=X$ and $g(X)=T_{\lambda}(g(X))$.
Moreover, if $X$ is compact, the following are equivalent
In fact, the minimality of $f$ is equivalent to its unique ergodicity.
\[thmA minimlity ergodicity subset\] Let $K$, $f$, $\Delta
_f(x_0)$, and $g$ be as in Theorem \[thmA transitivity preserved\]. Suppose that the subset $X\subset \Delta_f(x_0)$ is non-empty, compact and invariant under $f$. The following statements are equivalent:
1. $T_{\lambda} : g(X)\to g(X) $ is minimal.
2. $f: X\to X$ is minimal.
3. $f: X\to X$ is uniquely ergodic.
4. $f$ is ergodic for any $f$-invariant measure $\mu$ on the Borel sigma-algebra $\mathcal{B}(X)$ that is positive on non-empty open sets.
The unique invariant measure $\mu$ is the normalized Haar measure $\mu$ for which the measure of a disc is equal to the radius of the disc.
Note that the conjugacy function $g$ maps spheres in the linearization disc into spheres about the origin. In $\mathbb{Q}_p$, the multiplier map $T_{\lambda}:x\mapsto \lambda x $ is minimal on each sphere $S$ about the origin if and only if $\lambda$ is a generator of $(\mathbb{Z}/p^2\mathbb{Z})^{*}$. Moreover, if $\lambda$ is a generator of $(\mathbb{Z}/p^2\mathbb{Z})^{*}$, then as a consequence of Theorem \[theoremA linearization disc in Qp\], the linearization disc $\Delta _f(x_0,\mathbb{Q}_p)$ includes the the open disc $D_{1/a}(0)$.
\[thmA ergodic discs in Qp\] Let $p$ be an odd prime, and let the series $f\in\mathbb{Q}_p[[x-x_0]]$ be of the form (\[definition power series about x0\]). Let $S\subset
\mathbb{Q}_p$ be a non-empty sphere of radius $r<1/a$ about $x_0$, i.e. $r$ is an integer power of $p$. Then, the following statements are equivalent:
1. $\lambda $ is a generator of $(\mathbb{Z}/p^2\mathbb{Z})^{*}$.
2. $f:S\to S$ is minimal.
3. $f: S\to S$ is uniquely ergodic.
4. $f$ is ergodic for any $f$-invariant measure $\mu$ on the Borel sigma-algebra $\mathcal{B}(S)$ that is positive on non-empty open sets.
By Theorem \[theoremA linearization disc in Qp\] the estimate of the radius $1/a$ is maximal in the sense that there exist examples of such $f$, which either diverges on the sphere $S_{1/a}(x_0)$ or satisfy $f(x)=x_0$ for at least one $x\in S_{1/a}(x_0)$. We have, however, not been able to rule out the possibility that in some cases we may allow $r=1/a$, see Lemma \[lemma f one-to-one\].
Also note that if $\lambda\in S_1(0)$ is not a generator of $(\mathbb{Z}/p^2\mathbb{Z})^{*}$, then $T_{\lambda }$ and hence $f(x)=\lambda x +O(x^2)$ may still be minimal on some subset of a sphere. A complete classification of the ergodic breakdown of $\mathbb{Q}_p$ with respect to $T_{\lambda}$ is given in [@Oselies/Zieschang:1975].
We also note (lemma \[lemma transitivity multiplier\]) that in a finite proper extension of $\mathbb{Q}_p$, a power series cannot be ergodic on an entire sphere, that is contained in a linearization disc, and centered about the corresponding fixed point. In fact, if $K$ is a non-Archimedean field, then ergodicity on a linearization sphere is only possible if $K$ is isomorphic to a field of $p$-adic numbers. For transitivity to occur, $K$ must be locally compact. Therefore, $K$ is either a $p$-adic field or a field of prime characteristics. Let $K$ be a locally compact field of prime characteristc, with uniformizer $\pi$. If $x\in K$ and $x \equiv 1 \mod \pi$, then $x ^{p^n} \equiv 1\mod \pi ^{p^n}$. As a consequence, $T_{\lambda}$ cannot be transitive on a sphere in $K$, see Lemma \[lemma non transitivity multiplier char p\].
\[theorem non-archimedean ergodic disc\] Let $K$ be a complete non-Archimedean field and let $f$ be holomorphic on a disc $U$ in $K$. Suppose that $f$ has a linearization disc $\Delta\subset U$ and $S\subset \Delta $ is a sphere about the corresponding fixed point $x_0\in K$. Then $f$: $S\to S$ is ergodic if and only if $K$ is isomorphic to $\mathbb{Q}_p$ and the multiplier is a generator of the group of units $(\mathbb{Z}/p^2\mathbb{Z})^*$. Furthermore, if $K=\mathbb{Q}_p$ and $\lambda $ is a generator of the group of units $(\mathbb{Z}/p^2\mathbb{Z})^*$, then the radius of $\Delta$ is $1/a$ (considered as a disc in $\mathbb{Q}_p$).
Preliminaries {#section preliminaries}
=============
Throughout this paper $K$ is a non-Archimedean field, complete with respect to a nontrivial absolute value $|\cdot |$. That is, $|\cdot |$ is a multiplicative function from $K$ to the nonnegative real numbers with $|x|=0$ precisely when $x=0$, satisfying the following strong or ultrametric triangle inequality: $$\label{sti}
|x+y| \leq \max[|x|,|y|],\quad\text{for all $x,y\in K$},$$ and nontrivial in the sense that it is not identically $1$ on $K^*$, the set of all nonzero elements in $K$. One useful consequence of ultrametricity is that for any $x,y\in
K$ with $|x|\neq |y|$, the inequality (\[sti\]) becomes an equality. In other words, if $x,y\in K$ with $|x|<|y|$, then $|x+y|=|y|$.
In this context it is standard to denote by $\mathcal{O}$, the ring of integers of $K$, given by $\mathcal{O}=\{x\in K : |x|\leq 1\}$, by $\mathcal{M}$ the unique maximal ideal of $\mathcal{O}$, given by $\mathcal{M}=\{x\in K: |x|<1\}$, and by $k$ the corresponding *residue field* $$k=\mathcal{O}/\mathcal{M}.$$ Note that if $K$ has positive characteristic $p$, then also char $k =p$; but if char $K=0$, then $k$ could have characteristic $0$ or $p$. Note also that if $x,y\in\mathcal{O}$ reduce to *residue classes* $\overline{x},\overline{y}\in k$, then $|x-y|$ is $1$ if $\overline{x}\neq\overline{y}$, and it is strictly less than $1$ otherwise.
In this paper we mainly consider the case when $K$ is either a $p$-adic field, i.e. a finite extension of a field of $p$-adic numbers $\mathbb{Q}_p$, or a field of complex $p$-adic numbers $\mathbb{C}_p$. Recall that the $p$-adic numbers are constructed in the following way. For any prime $p$, there is a unique absolute value on $\mathbb{Q}$ such that $|p|=1/p$. The field $\mathbb{Q}_p$ of $p$-adic rationals is defined to be the corresponding completion of $\mathbb{Q}$; $\mathbb{C}_p$ is then the completion of an algebraic closure of $\mathbb{Q}_p$. Let us also remark that the residue field of $\mathbb{Q}_p$ is the field $\mathbb{F}_p$, of $p$ elements, whereas the the residue field of $\mathbb{C}_p$ is the algebraic closure of $\mathbb{F}_p$.
Given $K$ with absolute value $|\cdot|$ we define the *value group* as the image $$\label{def-value group}
|K^{*}|=\{|x|:x\in K^* \}.$$ Note that, since $|\cdot |$ is multiplicative, $|K^*|$ is a multiplicative subgroup of the positive real numbers. We will also consider the full image $|K|=|K^*|\cup\{0\}$. The absolute value $|\cdot |$ is said to be *discrete* if the value group is cyclic, that is if there is a $\emph{uniformizer}$ $\pi\in K$ such that $|K^{*}|=\{|\pi |^n: n\in
\mathbb{Z}\}$. Note that if $K=\mathbb{Q}_p$, then $p$ is a uniformizer of $K$, and the value group consists of all integer powers of $p$. If $K=\mathbb{C}_p$, then $|K^{*}|$ consists of all rational powers of $p$. In particular, the absolute value on $\mathbb{C}_p$ is not discrete.
Recall that $K$ is locally compact (w.r.t. $|\cdot |$) if and only if (i) $|\cdot |$ is discrete, and (ii) the residue field $k$ is finite. If $K$ is a $p$-adic field, then $K$ is locally compact and each integer $x\in \mathcal{O}$ has a unique representation as a Taylor series in $\pi$ of the form $$\label{equation expansion}
x=\sum_{i=0}^{\infty}x_i\pi^i, \quad x_i\in \mathcal{R},$$ where $ \mathcal{R}$ is a complete system of representatives of the residue field $k$.
Given a prime $p$, a $p$-adic number $x$ can be expressed in base $p$ as $$x=\sum_{k=\nu}^{\infty}x_kp^k, \quad x_k\in\{ 0, ..., p-1 \},$$ for some integer $\nu$ such that $x_{\nu}\neq 0$ and $x_k=0$ for all $k<\nu$. The absolute value of $x$ is given by $|x|=p^{-\nu}$. If $x$ is an integer, its $p$-adic expansion contains no negative powers of $p$ and hence $|x|\leq 1$.
For future reference, let us note the following lemma.
\[lemma order of p in factorial\] Given a rational number $x$, denote by $\lfloor x\rfloor$ the integer part of $x$. Let $n\geq 1$ be an integer and let $S_n$ be the sum of the coefficients in the $p$-adic expansion of $n$. Then, $$\label{equation order of p in factorial}
\nu(n!)=\frac{n-S_n}{p-1}\leq \frac{n-1}{p-1},$$ with equality if $n$ is a power of $p$. Consequently, for all integers $a\geq 1$, $$\label{limit order of p in factorial}
\frac{\nu(\left\lfloor \frac{n}{a}\right \rfloor!)} {n} \to
\frac{1}{a(p-1)},$$ as $n$ goes to infinity.
For a proof of (\[equation order of p in factorial\]), the reader can consult [@Schikhof:1984 Lemma 25.5].
Furthermore, to each finite extension $K$ of $\mathbb{Q}_p$ of degree $n$, there is an associated *residue class degree* $f=[k:\mathbb{F}_p]$, and a *ramification index* $e$ such that $$\label{def-ramification index}
|K^*|=\{p^{l/e}: l\in\mathbb{Z}\}.$$ For example, by adjoining $\sqrt p$ to $\mathbb{Q}_p$ we get a ramified extension with ramification index $e=2$. The degree of the extension $n=[K:\mathbb{Q}_p]$, the residue class degree $f$, and the ramification index $e$ satisfy the relation $$n=e\cdot f.$$ A finite extension of degree $n$ is called *unramified*, if $e=1$ (or equivalently, $f=n$), and *ramified*, if $e>1$ (or equivalently, $f<n$).
For more information on $p$-adic numbers and their field extensions the reader can consult [@Gouvea:1997].
Non-Archimedean discs {#section non-Archimedean
discs}
---------------------
Let $K$ be a complete non-Archimedean field. Given an element $x\in K$ and real number $r>0$ we denote by $D_{r}(x)$ the open disc of radius $r$ about $x$, by $\overline{D}_r(x)$ the closed disc, and by $S_{r}(x)$ the sphere of radius $r$ about $x$. To omit confusion, we sometimes write $D_r(x,K)$ rather than $D_{r}(x)$ to emphasize that the disc is considered as a disc in $K$.
If $r\in|K^*|$ (that is if $r$ is actually the absolute value of some nonzero element of $K$), we say that $D_{r}(x)$ and $\overline{D}_r(x)$ are *rational*. Note that $S_r(x)$ is non-empty if and only if $\overline{D}_r(x)$ is rational. If $r\notin |K^*|$, then we will call $D_{r}(x)=\overline{D}_r(x)$ an *irrational* disc. In particular, if $a\in K\subset
\mathbb{C}_p$ and $r=|a|^s$ for some rational number $s\in\mathbb{Q}$, then $D_{r}(x)$ and $\overline{D}_r(x)$ are rational considered as discs in the algebraic closure $\mathbb{C}_p$. However, they may be irrational considered as discs in $K$. Note that all discs are both open and closed as topological sets, because of ultrametricity. However, as we will see in Section \[section non-Archimedean power series\] below, power series distinguish between rational open, rational closed, and irrational discs.
Non-Archimedean power series {#section non-Archimedean power series}
----------------------------
Let $K$ be a complete non-Archimedean field with absolute value $|\cdot |$. Let $f$ be a power series over $K$ of the form $$f(x)=\sum_{i=0}^{\infty}a_i(x-\alpha )^i, \quad a_i\in K.$$ Then, $f$ converges on the open disc $D_{R_f}(\alpha )$ of radius $$\label{radius of convergence}
R_f = \frac{1}{\limsup |a_i| ^{1/i}},$$ and diverges outside the closed disc $\overline{D}_{R_f}(\alpha )$ in $K$. The power series $f$ converges on the sphere $S_{R_f}(\alpha )$ if and only if $$\lim_{i\to\infty}|a_i| R_f ^i=0.$$
Let $U\subset K$ be a disc, let $\alpha\in U$ and let $f:U\to K$. We say that $f$ is [**holomorphic**]{} on $U$ if we can write $f$ as a power series $$f(x)=\sum_{i=0}^{\infty}a_i(x-\alpha)^i\in K[[x-\alpha]]$$ which converges for all $x\in U$.
Holomorphicity is well-defined since, contrary to the complex field case, it does not matter which $\alpha\in U$ we choose in the definition of holomorphicity, see e.g. [@Schikhof:1984].
The basic mapping properties of non-Archimedean power series on discs are given by the following generalization by Benedetto [@Benedetto:2003a], of the Weierstrass Preparation Theorem [@BoschGuntzerRemmert:1984; @FresnelvanderPut:1981; @Koblitz:1984].
\[proposition-discdegree\] Let $K$ be algebraically closed. Let $f(x)=\sum_{i=0}^{\infty}a_i(x-\alpha)^i$ be a nonzero power series over $K$ which converges on a rational closed disc $U=\overline{D}_R(\alpha)$, and let $0<r\leq R$. Let $V=\overline{D}_r(\alpha)$ and $V'=D_r(\alpha)$. Then $$\begin{aligned}
s &=& \max\{|a_i|r^i:i\geq 0\},\\
d &=& \max\{i\geq 0:|a_i|r^i=s\},\quad and\\
d'&=& \min\{i\geq 0:|a_i|r^i=s\}
\end{aligned}$$ are all attained and finite. Furthermore,
a. $s\geq |f'(x_0)|\cdot r$.
b. if $0\in f(V)$, then $f$ maps $V$ onto $\overline{D}_s(0)$ exactly $d$-to-1 (counting multiplicity).
c. if $0\in f(V')$, then $f$ maps $V'$ onto $D_s(0)$ exactly $d'$-to-1 (counting multiplicity).
We will consider the case $a_0=0$ in more detail. For our purpose, it is then often more convenient to state Proposition \[proposition-discdegree\] in the following way.
\[proposition one-to-one\] Let $K$ be algebraically closed and let $h(x)=\sum_{i=1}^{\infty}c_i(x-\alpha )^i$ be a power series over $K$.
1. Suppose that $h$ converges on the rational closed disc $\overline{D}_R(\alpha)$. Let $0<r\leq R$ and suppose that $$\label{ck inequality one-to-one}
|c_i|r^i\leq |c_1|r\quad \text{ for all } i\geq 2 .$$ Then, $h$ maps the open disc $D_{r}(\alpha )$ one-to-one onto $D_{|c_1|r}(0)$. Furthermore, if $$d = \max\{i\geq 1:|c_i|{r}^i=|c_1| r\},$$ then $h$ maps the closed disc $\overline{D}_{r}(\alpha )$ onto $\overline{D}_{|c_1|r}(0)$ exactly $d$-to-1 (counting multiplicity).
2. Suppose that $h$ converges on the rational open disc $D_R(\alpha )$ (but not necessarily on the sphere $S_R(0)$). Let $0<r\leq R$ and suppose that $$|c_i|r^i \leq |c_1|r\quad \text{ for all } i\geq 2 .$$ Then, $h$ maps $D_{r}(\alpha )$ one-to-one onto $D_{|c_1|r}(0)$.
As a consequence of Proposition \[proposition-discdegree\], $f$ satisfies the following Lipschitz condition.
\[proposition lipschitz\] Let $f$ be a non-constant power series defined on a disc $U\subset K$ of radius $r>0$, and suppose that $f(U)$ is a disc of radius $s>0$. Then for any $x,y\in U$, $$|f(x)-f(y)|\leq \frac{s}{r}|x-y|.$$
Also note the following non-Archimedean analogue of the Complex Koebe $1/4$-Theorem.
\[proposition koebe\] Let $K$ be algebraically closed. Let $f$ be a power series over $K$ which is convergent and one-to-one on a disc $U\subset K$, with $0\in U$. Suppose that $f(0)=0$ and $f'(0)=1$. Then $f(U)=U$.
If $f$ and $U$ satisfy the condition of the Keoebe theorem, then by the Lipschitz condition in Proposition \[proposition lipschitz\], $f:U \to U $ is not only bijective but also isometric. We have the following lemma.
\[lemma indiff fixed point disc\] Let $K$ be algebraically closed. Let $f$ be a power series over $K$, which converges and is one-to-one on a disc $U\subset K$. Suppose that there is an element $x_0\in U$ such that $f(x_0)=x_0$ and $|f'(x_0)|=1$. Then $f:U\to U$ is bijective and isometric.
First, assume that $U$ is rational closed. Consider the function $h(x)=f(x)-x_0$. By definition, $h$ is also one-to-one on $U$. Moreover, $h(x_0)=0$ and $|h'(x_0)|=1$. Thus, in view of Proposition \[proposition-discdegree\], $h(U)$ is a rational closed disc and the radius of $h(U)$ is the same as that of $U$. It follows that $f(U)$ is rational closed and that the radius of $f(U)$ is the same as that of $U$. Because both $U$ and $f(U)$ contain $x_0$, we have $f(U)=U$. The remaining case is when $U$ is open. Write $U$ as the union $\cup U_i$ of rational closed discs containing $x_0$. Then $f(U)=\cup f(U_i)=\cup U_i=U$.
Next, we show that $f:U\to U$ is isometric. As the radius of $h(U)$ is the same as that of $U$, we have by Proposition \[proposition lipschitz\] that $|h(x)-h(y)|\leq |x-y|$. On the other hand, since $h:U\to h(U)$ is bijective, we have $$|x-y|=|h^{-1}\circ h(x)- h^{-1}\circ h(y)|\leq |h(x)-h(y)|.$$ Consequently, $|h(x)-h(y)|=|x-y|$ so that $h:U\to h(U)$, and hence $f:U\to U$, is isometric.
In fact, a power series $f$ over a complete non-Archimedean field $K$, is always one-to-one (and hence isometric) on some non-empty disc about an indifferent fixed point $x_0\in K$. This is a consequence of the local invertibility theorem [@Schikhof:1984]. The maximal such disc is given by the following lemma.
\[lemma f one-to-one\] Let $K$ be algebraically closed. Let $f\in K[[x]]$ be convergent on some non-empty disc about $x_0\in K$. Suppose that $f(x_0)=x_0$ and $|f'(x_0)|=1$, and write $$f(x)=x_0+\lambda (x-x_0)+\sum_{i\geq 2}a_i(x-x_0)^i, \quad
a=\sup_{i\geq 2}|a_i|^{1/(i-1)}.$$ Let $M$ be the largest disc, with $x_0\in M$, such that $f:M\to M$ is bijective (and hence isometric). Then $M=D_{1/a}(x_0)$ if either $\max_{i\geq 2 }|a_i|^{1/(i-1)}$ is attained (as for polynomials) or $f$ diverges on $S_{1/a}(x_0)$. Otherwise, $M=\overline{D}_{1/a}(x_0)$.
Because $f$ is convergent, we must have $$a=\sup_{i\geq 2}|a_i|^{1/(i-1)}<\infty.$$ Moreover, $f$ is certainly convergent on the open disc $D_{1/a}(x_0)$. As in the proof of Lemma \[lemma indiff fixed point disc\], it is sufficient to consider the mapping properties of the map $h(x)=f(x)-x_0$.
First, in view of Proposition \[proposition one-to-one\], $h:D_{1/a}(x_0)\to D_{1/a}(0)$ is one-to-one, since by definition $$|a_i|(1/a)^i\leq 1/a= |a_1|(1/a).$$
Second, if $h$ converges on the closed disc $\overline{D}_{1/a}(x_0)$ and $\max_{i\geq 2 }|a_i|^{1/(i-1)}$ is attained for some $i\geq 2$, then $$d = \max\{i\geq
1:|a_i|{(1/a)}^i=|a_1| (1/a)\}\geq 2.$$ By Proposition \[proposition one-to-one\], $h$ is not one-to-one on $\overline{D}_{1/a}(x_0)$.
Third, if $h$ converges on the closed disc $\overline{D}_{1/a}(x_0)$ and $\max_{i\geq 2 }|a_i|^{1/(i-1)}$ is never attained. Then, $$|a_i|(1/a)^{i}<1/a=|a_1|(1/a),$$ for all $i\geq 2 $, so that $d=1$. In other words, $h:\overline{D}_{1/a}(x_0)\to \overline{D}_{1/a}(0)$ is one-to-one in this case. However, $h$ cannot be one-to-one on any (rational) disc strictly containing $\overline{D}_{1/a}(x_0)$; if $r<a=\sup_{i\geq 2}|a_i|^{1/(i-1)}$, then $|a_N|^{1/(N-1)}\geq r$ and hence $$|a_N|(1/r)^{N}\geq 1/r=|a_1|(1/r),$$ for some $N\geq 2$. This completes the proof.
It follows from the proof above that if $f$ converges on the sphere $S_{1/a}(x_0)$ but fails to be one-to-one there, then there is a point $x\in S_{1/a}(x_0)$ such that $f(x)=x_0=f(x_0)$. This is always the case when $f$ is a polynomial.
That $f$ may diverge on $S_{1/a}(x_0)$ follows since, for example, the power series $f(x)=\lambda x + \sum_{i=2}^{\infty}(a_2)^{i-1}x^i$ converges if and only if $|x|<1/|a_2|=1/a$.
Furthermore, for every $x\in M$, $|f(x)-x_0|=|x-x_0|$ and hence all spheres in $M$ are invariant under $f$.
\[remark Lemma bijective\] Recall that the discs $D_{1/a}(0)$ and $\overline{D}_{1/a}(0)$ are rational if and only if $a=\sup_{i\geq 2}|a_i|^{1/(i-1)}\in |K|$. If the maximum $a=\max_{i\geq 2}|a_i|^{1/(i-1)}$ exists, and $K$ is algebraically closed, then $a\in|K|$. This is always the case if $f$ is a polynomial. If $f$ is not a polynomial and the maximum fails to exist we may have $\sup_{i\geq 2} |a_i|^{1/(i-1)}\notin
|K|$. Let $K=\mathbb{C}_p$. Let $\beta $ be an irrational number and let $p_n/q_n$ be the $n$-th convergent of the continued fraction expansion of $\beta$. Let the sequence $\{a_i\in
\mathbb{Q}_p\}_{i\geq 2}$ satisfy $$|a_i|= \left \{
\begin{array}{ll}
p^{p_n}, & \textrm{if \quad $i-1=q_n$ and $p_n/q_n<\beta $},\\
0, & \textrm{otherwise}.
\end{array}\right.$$ Then, $$\sup_{i\geq 2} |a_i|^{1/(i-1)}=p^{\beta}\notin
|K|=\{p^r:r\in\mathbb{Q} \}\cup \{0\}.$$
For more information on non-Archimedean power series the reader can consult [@Schikhof:1984]. From a dynamical point of view, the paper [@Benedetto:2003a] contains many useful results on non-Archimedean analogues of complex analytic mapping theorems relevant for dynamics.
Linearization discs
-------------------
The results above have some important implications for linearization discs. We use the following definition of a linearization disc. Let $K$ be a complete non-Archimedean field. Suppose that $f\in K[[x]]$ has an indifferent fixed point $x_0\in K$, with multiplier $\lambda =f'(x_0)$, not a root of unity. By [@Herman/Yoccoz:1981], there is a unique formal power series solution $g$, with $g(x_0)=0$ and $g'(x_0)=1$, to the following form of the Schröder functional equation $$g\circ f(x)=\lambda g(x).$$ If the formal solution $g$ converges on some non-empty disc about $x_0$, then the corresponding *linearization disc* of $f$ about $x_0$, denoted by $\Delta _f(x_0)$, is defined as the largest disc $U\subset K$, with $x_0\in U$, such that the Schröder functional equation holds for all $x\in U$, and $g$ converges and is one-to-one on $U$. We will often refer to $g$ as the *conjugacy function*.
This notion of a linearization disc is well-defined since by the proof of Lemma \[lemma f one-to-one\], there always exist a largest disc on which $g$ is one-to-one (provided that $g$ is convergent). Also note that by the non-Archimedean Siegel theorem [@Herman/Yoccoz:1981] and the fact that $\mathbb{C}_p$ is of characteristic zero, the formal solution $g$ always converges if the state space $K=\mathbb{C}_p$.
As one might expect from previous results, both $f$ and the conjugacy $g$ turn out to be one-to-one and isometric on a non-Archimedean linearization disc.
\[lemma linearization disc isometry\] Let $K$ be algebraically closed. Suppose that $f\in K[[x]]$ has a linearization disc $\Delta_f(x_0)$ about $x_0\in K$. Let $g$, with $g(x_0)=0$ and $g'(x_0)=1$, be the corresponding conjugacy function. Then, both $g:\Delta_f(x_0)\to g(\Delta_f(x_0))$ and $f:\Delta_f(x_0)\to \Delta_f(x_0)$ are bijective and isometric. In particular, if $x_0=0$, then $g(\Delta_f(x_0))=\Delta_f(x_0)$. Furthermore, $\Delta_f(x_0)\subseteq M\subseteq \overline{D}_{1/a}(x_0)$, where $M$ and $a$ are defined as in Lemma \[lemma f one-to-one\].
By the conjugacy relation $g\circ f(x)=\lambda g(x)$ and the fact that the map $g:\Delta _f(x_0)\to g(\Delta _f(x_0))$ is one-to-one, $f:\Delta _f(x_0)\to \Delta _f(x_0)$ is also one-to-one and hence bijective and isometric by Lemma \[lemma indiff fixed point disc\].
Recall that $g(x_0)=0$ and $g'(x_0)=1$. That $g:\Delta_f(x_0)\to
g(\Delta_f(x_0))$ is bijective and isometric then follows by same arguments as those applied to $h$, in the proof of Lemma \[lemma indiff fixed point disc\].
As a consequence, the radius of a linearization disc $\Delta_f(x_0)$ is equal to to that of $g(\Delta_f(x_0))$. In particular, the radius of a linearization disc is independent of the location of the fixed point $x_0$. Therefore, we shall, without loss of generality, henceforth assume that $x_0=0$.
The forthcoming sections are very much devoted to estimates of the maximal disc on which $g$ is one-to-one. Before dealing with this more delicate problem, note the following remark.
All the results in this and the previous section, except for Proposition \[proposition-discdegree\], hold also in the case that $K$ is not algebraically closed, with the modification that the mappings are are one-to-one but not necessarily surjective. However, with certain restrictions on the multiplier $\lambda$, e.g. $f:\Delta_f(x_0)\cap \mathbb{Q}_p \to \Delta_f(x_0)\cap
\mathbb{Q}_p$ may also be surjective, see Corollary \[corollary surjective in Qp\]. In fact, as stated in Theorem \[theorem-minimalitypreserved\], if $f$ is transitive on a compact subset $X$ of a linearization disc, then $f(X)=X$.
The formal solution
-------------------
As noted in the previous section, we may, without loss of generality, assume that $f$ has its fixed point at the origin, and that $f\in \mathcal{F}_{\lambda,a}$, as defined below. Let $\lambda\in \mathbb{C}_p$ be such that $$|\lambda|=1, \quad\textrm{but } \lambda ^n\neq 1,\quad \forall
n\geq 1,$$ and let $a$ be a real number. We shall associate with the pair $(\lambda,a)$ a family $\mathcal{F}_{\lambda,a}$ of power series defined by $$\mathcal{F}_{\lambda,a}:=\left\{\lambda x
+\sum a_ix^i\in\mathbb{C}_p[[x]]:a=\sup_{i\geq 2}|a_i|^{1/(i-1)}\right\}.$$ It follows that each $f\in \mathcal{F}_{\lambda,a}$ is convergent on $D_{1/a}(0)$, and by Lemma \[lemma f one-to-one\] $f:D_{1/a}(0)\to D_{1/a}(0)$ is bijective and isometric.
As $\mathbb{C}_p$ is of characteristic zero, we may, by the non-Archimedean Siegel theorem [@Herman/Yoccoz:1981], associate with $f$ a unique convergent power series solution $g$ to the Scröder functional equation, of the form $$g(x)=x + \sum_{k\geq 2}b_kx^k,$$ and a corresponding linearization disc about the origin $$\Delta_f:=\Delta _f(0).$$ Recall that by Lemma \[lemma linearization disc isometry\], since $x_0=0$, the linearization disc $\Delta_f$ is the largest disc $U\subset
\mathbb{C}_p$ about the origin such that the full conjugacy $g\circ f \circ g^{-1}(x)=\lambda x$ holds for all $x\in U$.
Given $f\in \mathcal{F}_{\lambda,a}$, Lemma \[lemma linearization disc isometry\] yields the following concerning $\Delta_f$.
\[lemma upper bound Siegel and isometry\] Let $f\in \mathcal{F}_{\lambda,a}$. Then $f$ has a linearization disc $\Delta_f$ about the origin in $\mathbb{C}_p$. Let $g$, with $g(0)=0$ and $g'(0)=1$, be the corresponding conjugacy function. Then, the following two statements hold:
1) Both $g:\Delta_f\to \Delta_f$ and $f:\Delta_f\to \Delta_f$ are bijective and isometric.
2) $\Delta_f\subseteq \overline{D}_{1/a}(0)$. If $a=\max_{i\geq 2}
|a_i|^{1/(i-1)}$ or $f$ diverges on the sphere $S_{1/a}(0)$, then $\Delta_f\subseteq D_{1/a}(0)$.
Our results on lower bounds for linearization discs are based on the following lemma.
\[lemma bk estimate indiff\] Let $f\in \mathcal{F}_{\lambda,a}$. Then, the coefficients of the conjugacy function $g$ satisfy $$\label{b_k estimate by prod 1-lambda n}
|b_k|\leq \left( \prod_{n=1}^{k-1}|1-\lambda ^n| \right
)^{-1}a^{k-1},$$ for all $k\geq 2$.
The coefficients of the conjugacy $g$ must satisfy the recurrence relation $$\label{bk-equation}
b_k=\frac{1}{\lambda (1-\lambda^{k-1})}\sum_{l=1}^{k-1}b_l(\sum\frac{l!}{\alpha_1!\cdot ...\cdot
\alpha_k!}a_1^{\alpha_1}\cdot ...\cdot a_k^{\alpha_k})$$ where $\alpha _1,\alpha _2,\dots,\alpha _k$ are nonnegative integer solutions of $$\label{index-equations}
\left\{\begin{array}{ll}
\alpha_1+...+\alpha_k=l,\\
\alpha_1+2\alpha_2...+k\alpha_k=k,\\
1\leq l\leq k-1.
\end{array}
\right.$$ Note that the factorial factors $l!/\alpha_1!\cdot\dots\cdot\alpha_k!$ are always integers and thus of modulus less than or equal to $1$. Also recall that $|a_i|\leq a^{i-1}$. It follows that $$|b_k|\leq \left( \prod_{n=1}^{k-1}|1-\lambda ^n| \right )^{-1}
a^{\alpha},$$ for some integer $\alpha$. In view of equation (\[index-equations\]) we have $$\sum_{i=2}^{k}(i-1)\alpha_i=k-l.$$ Consequently, since $|a_i|\leq a^{i-1}$, we obtain $$\label{estimate prod a_i}
\prod_{i=2}^k|a_i|^{\alpha_i}\leq
\prod_{i=2}^{k}a^{(i-1)\alpha_i}=a^{k-l}.$$ Now we use induction over $k$. By definition $b_1=1$ and, according to the recursion formula (\[bk-equation\]), $|b_2|\leq
|1-\lambda |^{-1}|a_2|\leq |1-\lambda |^{-1}|a|$. Suppose that $$|b_l|\leq \left( \prod_{n=1}^{l-1}|1-\lambda ^n| \right
)^{-1}a^{l-1}$$ for all $l<k$. Then $$|b_k|\leq \left( \prod_{n=1}^{k-1}|1-\lambda ^n| \right
)^{-1}a^{l-1}\max\left\{ \prod_{i=2}^k|a_i|^{\alpha_i}\right \},$$ and the lemma follows by the estimate (\[estimate prod a\_i\]).
In the following sections we show how to calculate the distance $|1-\lambda ^n |$ for an arbitrary integer $n\geq 1$. Applying Proposition \[proposition one-to-one\] to the estimate in the above lemma we can then estimate the disc on which the conjugacy function $g$ is one-to-one.
Geometry of the unit sphere and the roots of unity {#section geometry and
roots of uni ty C p}
--------------------------------------------------
Let $\Gamma$ be the group of all roots of unity in $\mathbb{C}_p$. It has the important subgroup $\Gamma_u$ ($u$ for unramified), given by $$\label{Gammau}
\Gamma_u=\{\xi\in\mathbb{C}_p:\textrm{ } \xi ^m=1\textrm{
for some $m$ not divisible by $p$}\}.$$
\[proposition gamma u\] The unit sphere $S_1(0)$ in $\mathbb{C}_p$ decomposes into the disjoint union $$S_1(0)=\cup _{\xi \in \Gamma_u}D_1(\xi).$$ In particular, $\Gamma_u\cap D_1(1)=\{ 1\}$ and consequently $|1-\xi|=1$ for all $\xi \neq 1$. To each $\lambda \in S_1(0)$ there is a unique $\xi\in \Gamma_u$ and $h\in D_1(1)$ such that $\lambda =\xi h$.
See [@Schikhof:1984 p. 103].
Let us note that $\Gamma_u$ is isomorphic to the multiplicative subgroup $\overline{\mathbb{F}}_p\setminus\{0\}$ of the residue field in $\mathbb{C}_p$.
Another important subgroup of $\Gamma$ is $\Gamma_r$ ($r$ for ramified), given by $$\label{Gammar}
\Gamma_r=\{\zeta\in\mathbb{C}_p:\textrm{ }\zeta^{p^s}=1 \textrm{
for some integer $s\geq 0$}\}.$$ By elementary group theory $\Gamma _u\cap\Gamma _r =\{1\}$ and $\Gamma =\Gamma _u\times\Gamma _r$, see e.g. the paper [@Schikhof:1984 p. 103]). Most importantly, the $p^s$th roots of unity in $\Gamma_r $ are located on spheres about the point $x=1$ of radius $R(t)$, where $$\label{definition R(s) extended}
R(t):=\left\{\begin{array}{ll}
0, & \textrm{if \quad $t=0$,}\\
p^{-\frac{1}{p^{t-1}(p-1)}}, & \textrm{if \quad $t\geq 1$.}
\end{array}\right.$$ This fact is fundamental for our estimates of linearization discs.
\[proposition gamma r\] $\Gamma_r\subset D_1(1)$. If $\zeta\in\Gamma_r$ is a primitive $p^s$th root of unity for some $s\geq 0$, then $$|1-\zeta|=R(s).$$ Moreover, if $\zeta_1,\zeta_2\in S_{R(s)}(1)$ and $\zeta_1\neq\zeta_2$, then $$|\zeta_1-\zeta_2|=R(s).$$ Furthermore, for each $s\geq 1$, there are $p^s-p^{s-1}$ different roots of unity on the sphere $S_{R(s)}(1)$.
See for example [@Escassut:1995].
As a consequence we have the following lemma.
\[lemma closest root of unity\] Let $\lambda\in D_1(1)$ be not a root of unity. Then, there exist $\alpha\in\Gamma_r$ such that $|\alpha-\lambda|\leq |\gamma
-\lambda|$, for all $\gamma\in\Gamma$. Furthermore, if $|1-\lambda|\neq R(s)$ for every $s\geq 0$. Then, $\alpha =1$ is the only root of unity with this property.
Transitivity of the multiplier map
----------------------------------
In this section, $K$ is a finite extension of $\mathbb{Q}_p$ of degree $[K:\mathbb{Q}_p]=e\cdot f$, with ramification index $e$, residue field degree $f=[k:\mathbb{F}_p]$, and uniformizer $\pi$.
Let $\lambda\in \mathbb{C}_p$ be an element on the unit sphere $S_1(0)$. We are concerned with calculating the distance $$|1-\lambda ^n|$$ for each integer $n\geq 1$. In view of Proposition \[proposition gamma u\] there is an integer $m$, not divisible by $p$, such that $\lambda
= \xi h$ for some $m$th root of unity $\xi\in\Gamma_u$ and $h\in
D_1(1)$. In other words, the following integer exists $$\label{definition of m}
m=m(\lambda):=\min\{n\in \mathbb{Z}:n\geq 1, |1-\lambda ^n|<1 \}.$$ Also note that, since the residue field is of characteristic $p>0$, $m$ is not divisible by $p$. In fact, if $\lambda $ belongs to a finite algebraic extension $K$ of $\mathbb{Q}_p$, then $$1\leq m\leq p^f-1,$$ where $p^f$ is the number of elements in the residue field $k$ of $K$. In particular, if $\lambda\in\mathbb{Q}_p$ we have $1\leq m\leq p-1$.
\[lemma mod pi\] Let $K$ be a finite extension of $\mathbb{Q}_p$, with uniformizer $\pi$ and ramification index $e$. Let $x \in K$, and $\delta =\min \{1+e,p \}$. If $x \equiv 1 \mod \pi$, then we have $x ^p \equiv 1\mod \pi ^{\delta}$. Moreover, if $x \not\equiv 1\mod \pi ^{2}$, then $x^l\not\equiv 1\mod \pi ^2$, $1\leq l\leq p-1$.
First, suppose $x\in 1+O(\pi)$. Then $$x^p\in(1+O(\pi))^p=1+pO(\pi)+\sum_{k=2}^{p-1}\binom{p}{k}O(\pi^k)
+O(\pi^p)=1+O(\pi^{\delta}),$$ where the last equality follows from the fact that $|p|=|\pi^{e}|$.
Second, suppose $x\in1+ a_1\pi +O(\pi^2)$, $a_1\neq 0$, and $1\leq l\leq p-1$. Then $$x^l\in(1+a_1\pi +O(\pi^2))^l=
\sum_{k=0}^{l}\binom{l}{k}(a_1\pi +O(\pi^2))^k
=1+la_1\pi +O(\pi^2).$$
Using the terminology from [@Pettigrew/Roberts/Vivaldi:2001], we say that an element $\lambda $ on the unit sphere in $K $ is *primitive* if $m=p^f-1$, and that $\lambda $ is *maximal* if in addition, $\lambda^{p^f-1}\not\equiv 1\mod \pi^2$ (so that $|1-\lambda ^m|=|\pi |=p^{-1/e}$).
Note that [@Pettigrew/Roberts/Vivaldi:2001] only consider the case $\lambda\in \mathbb{Q}_p$ for odd primes, whereas we also consider extensions of $\mathbb{Q}_p$, including the case $p=2$. Therefore, a maximal $\lambda$ does not always give a dense orbit as explained below.
It follows from Lemma \[lemma mod pi\] that if $\lambda$ is maximal, then the multiplication map $T_{\lambda}:\lambda \mapsto \lambda x$ has cycle length $p^f-1$ modulo $\pi$, and length $p(p^f-1)$ modulo $\pi ^2, \dots$, $\pi^{\delta}$. As a consequence, $T_{\lambda}$ cannot act as a permutation modulo $\pi^{3}$ if $e\geq 2$. In particular, $T_{\lambda}$ cannot have a dense orbit on the unit sphere in $K$ in this case. We have the following Lemma.
\[lemma transitivity multiplier\] Let $K$ be a proper extension of $\mathbb{Q}_p$ and let $\lambda\in K$, with $|\lambda |=1$. Then the map $T_{\lambda}:$ $x\mapsto\lambda x$ , cannot be transitive on any sphere about the origin in $K$. Furthermore, if $\lambda \in\mathbb{Q}_p$, then $T_{\lambda}$ is transitive on a sphere about the origin (in fact all spheres inside the unit disc) in $\mathbb{Q}_p$ if and only if $\lambda$ is maximal and $p$ is odd.
Let $[K:\mathbb{Q}_p]=e\cdot f\geq 2$. First, suppose that $f>1$. By Lemma \[lemma mod pi\] $\lambda ^{p(p^f-1)}\equiv 1 \mod \pi^2$. Consequently, $T_{\lambda}$ cannot act as a permutation on all $p^{f}(p^f-1)$ elements in the group of units modulo $\pi^2$.
Second, suppose $f=1$ but $e>1$. By Lemma \[lemma mod pi\] $\lambda ^{p(p-1)}\equiv 1 \mod \pi^3$. Consequently, $T_{\lambda}$ cannot act as a permutation on all $p^{2}(p-1)$ units modulo $\pi^3$.
The last statement of the lemma follows from the fact that a integer $\lambda$ is a generator of the group of units modulo $p^k$ for every integer $k\geq 2$ if and only if $\lambda$ is a generator modulo $p^2$, which happen if and only if $\lambda $ is maximal and $p$ is odd.
### Non-transitivity in characteristic $p$
To motivate Theorem \[theorem non-archimedean ergodic disc\], we also show that the multiplier map can never be transitive on a whole sphere for fields of prime characteristics.
\[proposition mod pi char p\] Let $K$ be a locally compact field of prime characteristc, with uniformizer $\pi$. If $x\in K$ and $x \equiv 1 \mod \pi$, then $x ^{p^n} \equiv 1\mod \pi ^{p^n}$ for all integers $n\geq 0$.
Suppose $x\in 1+O(\pi)$. Then $$x^p\in(1+O(\pi))^{p^n}=1+p^{n}O(\pi)+\sum_{i=2}^{p-1}\binom{p^{n}}{i}O(\pi^i)
+O(\pi^{p^n})=1+O(\pi^{p^n}).$$
\[lemma non transitivity multiplier char p\] Let $K$ be a locally compact field of prime characteristic and let $\lambda\in K$, with $|\lambda |=1$. Then, the map $T_{\lambda}$: $x\mapsto\lambda x$ , cannot be transitive on any sphere about the origin in $K$.
By local compactness of $K$, there is a uniformizer $\pi\in K$, and hence $T_{\lambda}$ is transitive on the unit sphere if and only if it is transitive on the group of units modulo $\pi^n$ for every integer $n\geq 2$. In particular, $T_{\lambda}$ has to be transitive modulo $\pi^{p^2}$ which is impossible by the following arguments. As $K$ is locally compact, the residue field $k$ is finite. Let $c$ be the cardinality of $k$. By definition $c\geq p$, and hence there are $(c-1)p^{p^2-1}>p^2$ units modulo $\pi^{p^2}$. On the other hand, by Proposition \[proposition mod pi char p\], $\lambda ^{p^2} \equiv 1\mod \pi ^{p^2}$. Consequently, $T_{\lambda}$ cannot be transitive modulo $\pi^{p^2}$ and hence not on the unit sphere or any other sphere about the origin in $K$.
Arithmetic of the multiplier
----------------------------
\[lemma distance 1-lambda m char 0,p\] Let $\lambda \in \mathbb{C}_p$ be an element on the unit sphere, but not a root of unity. Let $m$ be defined by (\[definition of m\]), and let $s$ be the integer for which $ R(s)\leq |1-\lambda ^m|< R(s+1) $. Then, the following three statements hold:
1. If $m$ does not divide $n$, then $|1-\lambda^n |=1$.
2. If $m$ is a divisor of $n$ and $0\leq s\leq \nu(n)$ we have $$\left |1 -\lambda ^n \right |= \left\{\begin{array}{ll}
|n|p^s|1-\lambda ^m|^{p^s}, & \textrm{if $ R(s)<|1-\lambda ^m|< R(s+1) $,}\\
|n|p^s|1-\lambda ^m|^{p^s-1}|\alpha -\lambda ^m|, & \textrm{if $|1-\lambda ^m|= R(s) $.}
\end{array}\right.$$ Here $\alpha\in \Gamma _r$ is chosen so that $|\alpha-\lambda^m|\leq |\gamma -\lambda^m|$, for all $\gamma\in\Gamma$.
3. If $m$ is a divisor of $n$ and $s>\nu(n)$ so that $ |1-\lambda ^m|> R(\nu (n))$, then $$\left |1 -\lambda ^n \right |=|1-\lambda ^m|^{p^{\nu(n)}}.$$
In the second statement of the lemma, $|n|p^s\leq 1$ since we assume that $s\leq \nu(n)$ in this case.
\[remark lemma distance 1-lambda m char 0,p\] By Lemma \[lemma closest root of unity\], we may have $|\alpha
-\lambda ^m|< |1-\lambda ^m|$ only if $\lambda ^m$ belong to the same sphere about $1$ as $\alpha\in \Gamma _r$. In all other cases we may choose $\alpha = 1$. In particular, we always have $|\alpha -\lambda
^m|\leq |1-\lambda ^m|$.
If $\lambda \in \mathbb{C}_2$ and $|1-\lambda
^m|=2^{-1/(2-1)}=2^{-1}$, then the ‘closest’ root of unity $\alpha
=-1=\sum_{k=0}^{\infty}2^k$.
It is enough to consider the case $m=1$. This proof is based on the factorization of the polynomial $\lambda ^n-1$ from which we find $$\label{basicfactorequation}
\left |\lambda ^n - 1 \right |=\prod _{\theta ^n=1}\left |\lambda
- \theta \right |.$$ As noted in Section \[section geometry and roots of uni ty C p\] the roots of unity in $\mathbb{C}_p$ is the direct product $\Gamma
=\Gamma _u\times\Gamma _r$. This representation enables us to write $\left |\lambda ^n - 1 \right |$ in the form $$\label{splitedfactorequation}
\left |\lambda ^n - 1 \right |= \left |\lambda - 1 \right | \prod
_{\zeta ^n=1}\left |\lambda - \zeta \right | \prod _{\xi
^n=1}\left |\lambda - \xi \right | \prod _{(\zeta\xi )^n=1}\left
|\lambda - \zeta\xi \right |,$$ where $\zeta\in \Gamma_r\setminus \{1\}$ and $\xi\in\Gamma_u\setminus\{1\}$. Recall that we assume that $\lambda\in D_1(1)$. In view of Proposition \[proposition gamma u\], $\xi,(\zeta\xi)\notin D_1(1)$ and consequently $|\lambda
-\xi|=|\lambda - \zeta\xi|=1$. Moreover, for $n=ap^{\nu(n)}$, we have that $\zeta^n=1$ if and only if $\zeta^{p^{\nu(n)}}=1$. It follows that (\[splitedfactorequation\]) can be reduced to $$\label{simplifiedfactorequation}
\left |\lambda ^n - 1 \right |= \left |\lambda - 1 \right | \prod
_{\zeta ^{p^{\nu(n)}}=1}\left |\lambda - \zeta \right |.$$ If $p$ does not divide $n$ so that $\nu(n)=0$, then $|1-\lambda
^n|=|1-\lambda|$ as required.
In the remaining cases $\nu(n) \geq 1$ and we have to take the factors $|\lambda-\zeta|$ into account. Note that $|\lambda
-\zeta|=|(\lambda -1) +(1-\zeta)|$ and by ultrametricity $$\label{distance to roots}
\left |\lambda - \zeta \right |=\max\{\left |\lambda - 1 \right |,\left |1-\zeta \right |\},$$ if $\left |\lambda - 1 \right |\neq\left |1 -\zeta \right |$. Thus we can compute (\[simplifiedfactorequation\]) by counting the number of roots $\zeta$ that are closer and farther to $1$ compared to $\lambda $, respectively.
Recall the following facts from Proposition \[proposition gamma r\]. If $\zeta\in\Gamma_r\setminus\{1\}$ is a *primitive* $p^s$th root of unity for some $s\geq 1$, then $\left |1 - \zeta \right
|=p^{-r_s}$ where $r_s=1/(p^s-p^{s-1})$. The sphere $S_{p^{-r_s}}(1)$ contains $p^s-p^{s-1}=1/r_s$ roots of unity. Note that these spheres have radii ordered as $p^{-r_1}<p^{-r_2}<...<1$.
Now we consider the case $|1-\lambda|< p^{-r_1}$. In this case $\lambda$ is closer to $1$ than any of the roots $\zeta$ and therefore, in view of (\[distance to roots\]), $|\lambda - \zeta
|=|1 -\zeta |$ for every $\zeta\in\Gamma_r\setminus\{1\}$. From (\[simplifiedfactorequation\]) we thus have that $$\left |\lambda ^n - 1 \right |= \left |\lambda - 1 \right
|(p^{-r_1})^{1/{r_1}}\cdot ...\cdot
(p^{-r_{\nu(n)}})^{1/{r_{\nu(n)}}}=|\lambda -1|p^{-{\nu(n)}}$$ as required.
Now we consider the case $p^{-r_{s}}<|\lambda -1|<p^{-r_{s+1}}$, $1\leq s\leq \nu(n)$. We have in view of (\[distance to roots\]) that $|\lambda-\zeta|=|\lambda-1|$ for all $\zeta$ such that $|1-\zeta|<|\lambda -1|$. This is the case for $p^s$ roots. All other roots are further from $1$ than $\lambda $ is. For these roots $|\lambda -\zeta|=|1-\zeta|$. Hence, the right-hand side of (\[simplifiedfactorequation\]) becomes $$\label{distance-rs+1<r<rs}
|\lambda -1|^{p^s} (p^{-r_{s+1}})^{1/{r_{s+1}}}\cdot ...\cdot
(p^{-r_{\nu(n)}})^{1/{r_{\nu(n)}}}=|1-\lambda|^{p^s}p^{-({\nu(n)}-s)}$$ as required.
Now we consider the case $|\lambda -1|=p^{-r_s}$ for some $1\leq
s\leq \nu(n)$. Let $\alpha\in S_{p^{-r_s}}(1)$ be such that $|\alpha -\lambda| \leq |\zeta -\lambda|$ for all $\zeta\in\Gamma_r$. Note that $|\zeta -\lambda|\leq p^{-r_s}$ for all $\zeta\in S_{p^{-r_s}}(1)$ and that $\alpha$ is unique if and only if $|\lambda - \alpha |<p^{-r_s}$. By Proposition \[proposition gamma r\], $|\lambda -\zeta|=p^{-r_s}$ for all $\zeta\neq \alpha$ on the sphere $S_{p^{-r_{s}}}(1)$. For the right-hand side of (\[simplifiedfactorequation\]) we obtain $$|\lambda -1|^{p^s-1}|\lambda -\alpha |
(p^{-r_{s+1}})^{1/{r_{s+1}}}\cdot ...\cdot
(p^{-r_{\nu(n)}})^{1/{r_{\nu(n)}}},$$ Consequently, $$\label{distance-r=rs}
|\lambda ^n -1|=|\lambda -1|^{p^s-1}|\lambda-\alpha |p^{-(\nu(n)-s
)}$$ as required.
Finally, we consider $|\lambda -1|>p^{-r_{\nu(n)}}$. In this case $|\lambda -1|>|1-\zeta|$ for all $\zeta$ that are $p^{\nu(n)}$th roots of unity. Consequently, $|\lambda - \zeta|=|\lambda- 1|$ and we obtain $$|\lambda ^n-1|=|\lambda -1|^{p^{\nu(n)}},$$ as proposed in the lemma. This completes the proof.
Estimates of linearization discs
================================
We will estimate the size of the linearization disc for a power series $f\in \mathcal{F}_{\lambda,a}$. The estimates are divided into three different cases according to the three sections below.
Case I {#section case 1}
------
In this section we assume that $$\label{equation lambda case 1}
R(0)<|1-\lambda ^m|<R(1).$$ In what follows $\sigma_1$ will be the real number defined by $$\label{definition of rho1}
\sigma_1:=a^{-1}p^{-\frac{1}{m(p-1)}}|1-\lambda ^m|^{\frac{1}{m}}.$$
\[lemma case 1\] Suppose $\lambda$ is such that $R(0)<|1-\lambda^m|<R(1)$. Then, $$\label{prod ineq case 1}
\left( \prod_{n=1}^{k-1}|1-\lambda ^n| \right )^{-1}a^{k-1}\leq
p^{-\frac{1}{p-1}}\sigma_1^{-(k-1)},$$ with equality if $(k-1)/m$ is an integer power of $p$.
By Lemma \[lemma distance 1-lambda m char 0,p\] $$\label{distance 1- lambda n case 1}
\left |1 -\lambda ^n \right |= \left\{\begin{array}{ll}
1, & \textrm{if \quad $m\nmid n$,}\\
|n||1-\lambda ^m|, & \textrm{if \quad $m\mid n$,}
\end{array}\right.$$ in this case. Let $N=\lfloor k-1/m \rfloor$ denote the integer part of $k-1/m$. Then, by (\[distance 1- lambda n case 1\]) $$\label{product 1- lambda case 1}
\prod_{n=1}^{k-1}|1-\lambda ^n|=\left |N !\right ||1-\lambda
^m|^{N }.$$ By Lemma \[lemma order of p in factorial\] $$|N!|\geq p^{-\frac{N-1}{p-1}}\geq p^{-\frac{k-1}{m(p-1)}
+\frac{1}{p-1}},$$ where each inequality become an equality if $(k-1)/m$ is an integer power of $p$. It follows by (\[product 1- lambda case 1\]) that $$\left( \prod_{n=1}^{k-1}|1-\lambda ^n| \right )^{-1}a^{k-1}\leq
p^{-\frac{1}{p-1}}\sigma_1^{-(k-1)},$$ with equality if $(k-1)/m$ is an integer power of $p$.
We will prove the following theorem.
\[theorem case 1\] Let $f\in \mathcal{F}_{\lambda,a}$ and suppose $\lambda $ is such that $R(0)<|1-\lambda^m|<R(1)$. Then, the linearization disc $\Delta_f\supseteq D_{\sigma_1}(0)$. Moreover, if the conjugacy function $g$ converges on the closed disc $\overline{D}_{\sigma_1}(0)$, then $\Delta_f\supseteq
\overline{D}_{\sigma_1}(0)$.
In view of Lemma \[lemma order of p in factorial\] and Lemma \[lemma case 1\] we have $$\left ( \limsup |b_k|^{1/k} \right)^{-1}\geq \sigma_1.$$ This implies that $g$ converges on the open disc of radius $\sigma_1$. Moreover, by Lemma \[lemma case 1\] $$|b_k|\sigma_1^k\leq p^{-\frac{1}{p-1}}\sigma_1
<\sigma_1=|b_1|\sigma_1.$$ It follows by Proposition \[proposition one-to-one\] that $g:D_{\sigma_1}(0)\to D_{\sigma_1}(0)$ is a bijection. The strict inequality $|b_k|\sigma_1^k<|b_1|\sigma_1$ implies that, if $g$ converges on the closed disc $\overline{D}_{\sigma_1}(0)$, then $g:\overline{D}_{\sigma_1}(0)\to\overline{D}_{\sigma_1}(0)$ is bijective. Recall that by Lemma \[lemma f one-to-one\] $f:D_{1/a}(0)\to D_{1/a}(0)$ is a bijection. Moreover, $1/a>\sigma_1$. Consequently, the linearization disc $\Delta_f$ includes the disc $D_{\sigma_1}(0)$ or $\overline{D}_{\sigma_1}(0)$, depending on whether the conjugacy function $g$ converges on the closed disc $\overline{D}_{\sigma_1}(0)$ or not.
The theorem has some important consequences for linearization discs in $\mathbb{Q}_p$. In fact, as we will see below, a linearization disc in $\mathbb{Q}_p$ may coincide with the maximal disc on which $f$ is one-to-one and even with the region of convergence of $f$. Recall that the value group $|\mathbb{Q}_p^*|$ contains only integer powers of $p$. This implies that if $\lambda\in \mathbb{Q}_p$, then $|1-\lambda ^m|\leq p^{-1}<p^{-1/(p-1)}$ if $p>2$. Consequently, Theorem \[theorem case 1\] applies if $\lambda \in
\mathbb{Q}_p$ for some prime $p>2$. In particular, if $f$ is a power series over $\mathbb{Q}_p$, we have the following corollary.
\[corollary p-adic linearization disc\] Let $f\in \mathcal{F}_{\lambda,a}\cap\mathbb{Q}_p[[x]]$ for some odd prime $p$. Let $\Delta _f(0,\mathbb{Q}_p)=\Delta_f\cap \mathbb{Q}_p$ be the corresponding linearization disc, about the origin, in $\mathbb{Q}_p$. Then, $\Delta_f(0,\mathbb{Q}_p) \supseteq D_{\sigma_1}(0,\mathbb{Q}_p)$. Moreover, if the conjugacy function $g$ converges on the closed disc $\overline{D}_{\sigma_1}(0)$, then the linearization disc $\Delta_f(0,\mathbb{Q}_p)\supseteq \overline{D}_{\sigma_1}(0,\mathbb{Q}_p)$. Furthermore, if $\lambda$ is maximal, then, the linearization disc $\Delta_f(0,\mathbb{Q}_p)$ is maximal in the sense that $\Delta_f(0,\mathbb{Q}_p)$ is either the open or closed disc of radius $1/a$. In particular, if either $\max_{i\geq 2 }|a_i|^{1/(i-1)}$ is attained (as for polynomials) or $f$ diverges on $S_{1/a}(0,\mathbb{Q}_p)$, then $\Delta_f(0,\mathbb{Q}_p)=D_{1/a}(0,\mathbb{Q}_p)$.
If $\lambda$ is maximal, then $|1-\lambda ^m|=p^{-1}$ and $m=p-1$. Consequently, $D_{\sigma_1}(0)=\overline{D}_{\sigma_1}(0) =D_{1/a}(0)$, considered as discs in $\mathbb{Q}_p$.
Moreover, a power series $f\in \mathcal{F}_{\lambda,a}\cap
\mathbb{Q}_p$ may diverge on $S_{1/a}(0)$. For example, the power series $f(x)=\lambda x + \sum_{i=2}^{\infty}(a_2)^{i-1}x^i$ converges if and only if $|x|<1/|a_2|=1/a$.
Too see that $f$ may have a zero on the sphere $S_{1/a}(0)$, consider the following example. Let $f(x)=\lambda x +a_2x^2$. Then $a=|a_2|$. But $x=-\lambda /a_2\in\mathbb{Q}_p$ is a zero of $f$ located on the sphere $S_{1/a}(0)$ in $\mathbb{Q}_p$.
If $p$ is an odd prime and $f(x)=\lambda x +O(x^2)$ is a power series over $\mathbb{Z}_p$, with multiplier $\lambda $ such that $|1-\lambda
^m|=p^{-1}$, and $m=p-1$. Then, the linearization disc in $\mathbb{Q}_p$ includes the open unit disc $D_{1}(0)$. This result was also obtained in [@Pettigrew/Roberts/Vivaldi:2001 Proposition 2.2 ].
Let $K$ be an unramified field extension of $\mathbb{Q}_p$. Then the value group $|K^*|$ contains only integer powers of $p$. Hence, if $\lambda \in K$, then Theorem \[theorem case 1\] applies and we have the following corollary.
\[corollary case 1\] Corollary \[corollary p-adic linearization disc\] holds for any unramified extension $K$ of $\mathbb{Q}_p$.
The case $p=2$, will be treated in case III below.
Case II
-------
In this section we assume that $s\geq 1$ and $$\label{equation lambda case 2}
R(s)<|1-\lambda ^m|< R(s+1).$$ In what follows $\sigma_2$ will be the real number defined by $$\label{definition of rho2}
\sigma_2:=a^{-1}p^{-\frac{1}{m(p-1)p^{s} }} |1-\lambda ^m|^{
\frac{1}{m}(1 + \frac{p-1}{p} s ) }.$$
\[lemma case 2\] Suppose $\lambda $ satisfies (\[equation lambda case 2\]) for some $s\geq 1$. Then, $$\label{prod ineq case 2}
\left( \prod_{n=1}^{k-1}|1-\lambda ^n| \right )^{-1}a^{k-1}\leq
p^{-\frac{1}{p-1}}\sigma_2^{-(k-1)},$$ with equality if $(k-1)/mp^{s+1}$ is an integer power of $p$.
By lemma \[lemma distance 1-lambda m char 0,p\] $$\left |1 -\lambda ^n \right |= \left\{\begin{array}{ll}
1, & \textrm{if \quad $m\nmid n$,}\\
|n|p^s|1-\lambda ^m|^{p^s}, & \textrm{if \quad $m p^{s+1} \mid n$,}\\
|1-\lambda ^m|^{p^{\nu(n)}}, & \textrm{if \quad $m\mid n$ but
$mp^{s+1}\nmid n$}
\end{array}\right.$$ in this case. Throughout this proof $N$ will be the integer $$\label{definition of N case 1}
N=\left \lfloor \frac{k-1}{mp^{s+1}} \right \rfloor.$$ Note that $$\prod_{mp^{s+1}\mid n}^{k-1}|n|= | mp^{s+1}\cdot 2mp^{s+1}\cdot
\dots |= | N!||mp^{s+1}|^{N},$$ and since $p\nmid m$, $$\label{product |n| mp^s+1|n}
\prod_{mp^{s+1}\mid n}^{k-1}|n|= | N !|p^{-(s+1)N}.$$ Moreover, $$\label{product p^s mp^s+1|n}
\prod_{mp^{s+1}\mid n}^{k-1}p^s|1-\lambda ^m|^{p^s}= p^{sN}
|1-\lambda ^m|^{p^sN},$$ and $$\label{product p^nu p^s+1}
\prod_{\substack{ mp^{s+1}\nmid n \\ m\mid n} }^{k-1}|1-\lambda
^m|^{p^{\nu(n)}}= |1-\lambda ^m|^{ \sum_{j=0}^{s} \left (\left
\lfloor \frac{k-1}{mp^{j}}\right \rfloor - \left \lfloor
\frac{k-1}{mp^{j+1}}\right \rfloor \right )p^j }.$$ Combining the three products (\[product |n| mp\^s+1|n\]), (\[product p\^s mp\^s+1|n\]), and (\[product p\^nu p\^s+1\]) we obtain $$\label{product 1 -lambda case 2}
\prod_{n=1}^{k-1}|1-\lambda ^n|= |N! |p^{-N} |1-\lambda
^m|^{\Sigma_1},$$ where $$\Sigma_1 =p^sN+ \sum_{j=0}^s\left (\left \lfloor
\frac{k-1}{mp^{j}}\right \rfloor - \left \lfloor
\frac{k-1}{mp^{j+1}}\right \rfloor \right )p^j.$$ Simplifying, we obtain $$\Sigma_1=\left \lfloor \frac{k-1}{m} \right \rfloor +
\sum_{j=1}^s\left\lfloor \frac{k-1}{mp^j}\right \rfloor
(p^j-p^{j-1}).$$ Consequently, $$\Sigma_1 \leq \frac{k-1}{m}\left (1+s\frac{p-1}{p}\right ),$$ with equality if $(k-1)/mp^{s+1}$ is an integer power of $p$. By Lemma \[lemma order of p in factorial\], $$\left |N!\right |p^{-N}\geq
p^{-\frac{N-1}{p-1}-N}=p^{-N\frac{p}{p-1} +\frac{1}{p-1}}\geq
p^{-\frac{k-1}{mp^s(p-1)} +\frac{1}{p-1}},$$ where each inequality become an equality if $(k-1)/mp^{s+1}$ is an integer power of $p$. Applying these estimates to the identity (\[product 1 -lambda case 2\]) we obtain the inequality (\[prod ineq case 2\]) as required.
By similar arguments as those applied in the proof of Theorem \[theorem case 1\], we obtain the following result.
\[theorem case 2\] Let $f\in \mathcal{F}_{\lambda,a}$ and suppose $\lambda $ satisfies $$R(s)<|1-\lambda |<R(s+1),$$ for some integer $s\geq 1$. Then, $\Delta_f\supseteq
D_{\sigma_2}(0)$. Moreover, if the conjugacy function $g$ converges on the closed disc $\overline{D}_{\sigma_2}(0)$, then $\Delta_f\supseteq \overline{D}_{\sigma_2}(0)$.
Case III
--------
In this section it will be assumed that $s\geq 1$ and $$\label{equation lambda case 3}
|1-\lambda ^m|= R(s).$$ In what follows $\sigma_3$ will be the real number defined by $$\label{definition of rho3}
\sigma_3:=\sigma_2\cdot\left ( \frac{|\alpha-\lambda
^m|}{|1-\lambda ^m |} \right )^{1/mp^s},$$ where $\alpha\in \Gamma_r$ is chosen such that $|\alpha-\lambda^m|\leq |\gamma -\lambda^m|$, for all $\gamma\in\Gamma$.
\[lemma case 3\] Suppose $\lambda $ satisfies (\[equation lambda case 3\]) for some $s\geq 1$. Then $$\label{prod ineq case 3}
\left( \prod_{n=1}^{k-1}|1-\lambda ^n| \right )^{-1}a^{k-1}\leq
p^{-\frac{1}{p-1}}\sigma_3^{-(k-1)},$$ with equality if $(k-1)/mp^{s}$ is an integer power of $p$.
By Lemma \[lemma distance 1-lambda m char 0,p\] $$\left |1 -\lambda ^n \right |= \left\{\begin{array}{ll}
1, & \textrm{if \quad $m\nmid n$,}\\
|n|p^s|1-\lambda ^m|^{p^s-1}|\alpha -\lambda ^m|, & \textrm{if \quad $m p^{s} \mid n$,}\\
|1-\lambda ^m|^{p^{\nu(n)}}, & \textrm{if \quad $m\mid n$ but
$mp^{s}\nmid n$}
\end{array}\right.$$ in this case. Throughout this proof we let $M$ be the integer $$M=\left \lfloor \frac{k-1}{mp^s} \right \rfloor.$$ Note that $$\label{product |n| mp^s|n}
\prod_{mp^{s}\mid n}^{k-1}|n|= |M!|p^{-sM},$$ $$\label{product p^s mp^s|n}
\prod_{mp^{s}\mid n}^{k-1}p^s|1-\lambda ^m|^{p^s}=
p^{sM}|1-\lambda ^m|^{(p^s-1)M},$$ $$\label{product alpha mp^s|n}
\prod_{mp^{s}\mid n}^{k-1}|\alpha -\lambda ^m|= |\alpha - \lambda
^m|^{M},$$ and $$\label{product p^nu p^s}
\prod_{\substack{ mp^{s}\nmid n \\ m\mid n} }^{k-1}|1-\lambda
^m|^{p^{\nu(n)}}= |1-\lambda ^m|^{ \sum_{j=0}^{s-1} \left (\left
\lfloor \frac{k-1}{mp^{j}}\right \rfloor - \left \lfloor
\frac{k-1}{mp^{j+1}}\right \rfloor \right )p^j }.$$ Combining the three products (\[product |n| mp\^s|n\]), (\[product p\^s mp\^s|n\]), (\[product alpha mp\^s|n\]), and (\[product p\^s mp\^s|n\]) we obtain $$\label{product 1 -lambda case 3}
\prod_{n=1}^{k-1}|1-\lambda ^n|=|M!||1-\lambda ^m|^{\Sigma_2}\left
( \frac{|\alpha -\lambda ^m|}{|1-\lambda ^m|} \right )^{M},$$ where $$\Sigma_2=p^sM+\sum_{j=0}^{s-1} \left (\left \lfloor
\frac{k-1}{mp^{j}}\right \rfloor - \left \lfloor
\frac{k-1}{mp^{j+1}}\right \rfloor \right )p^j.$$ Simplifying, we obtain $$\Sigma_2=\Sigma_1=\left \lfloor \frac{k-1}{m} \right \rfloor +
\sum_{j=1}^s\left\lfloor \frac{k-1}{mp^j}\right \rfloor
(p^j-p^{j-1}).$$ As in the proof of Lemma \[lemma case 2\] we have $$\Sigma_1 \leq \frac{k-1}{m}\left (1+s\frac{p-1}{p}\right ),$$ with equality if $(k-1)/mp^{s}$ is an integer power of $p$. By Lemma \[lemma order of p in factorial\], $$\left |M!\right |\geq p^{-\frac{M-1}{p-1}}\geq
p^{-\frac{k-1}{mp^s(p-1)} +\frac{1}{p-1}},$$ where each inequality become an equality if $(k-1)/mp^{s}$ is an integer power of $p$. Furthermore, by definition $$|\alpha -\lambda^ m|\leq |1-\lambda ^m|=R(s).$$ Applying these estimates to the identity (\[product 1 -lambda case 3\]) we obtain the inequality (\[prod ineq case 3\]) as required.
By similar arguments as those applied in the proof of Theorem \[theorem case 1\], we obtain the following result.
\[theorem case 3\] Let $f\in \mathcal{F}_{\lambda,a}$ and suppose $$|1-\lambda ^m|=R(s),$$ for some integer $s\geq 1$. Then, $\Delta_f\supseteq
D_{\sigma_3}(0)$. Moreover, if the conjugacy function $g$ converges on the closed disc $\overline{D}_{\sigma_3}(0)$, then $\Delta_f\supseteq \overline{D}_{\sigma_3}(0)$.
\[corollary case 3\] Let $K$ be an unramified extension of $\mathbb{Q}_2$. Let $f$ be of the form $$f(x)=\lambda (x-x_0)+\sum a_i(x-x_0)^i\in K[[x-x_0]] , a=\sup |a_i|^{1/(i-1)}.$$ Then, the following two statements hold:
1. If $|1-\lambda |<1/2$, then the linearization disc $\Delta_f(x_0,K)$ contains the open disc of radius $\sigma_1= |1-\lambda |/2a $ about $x_0$.
2. If $|1-\lambda |=1/2$, then the linearization disc the linearization disc $\Delta_f(x_0,K)$ contains the open disc of radius $\sigma_3=\sqrt{ |1+\lambda|}/a$ about $x_0$.
As $p=2$ we must have $m=1$ and since $K$ is unramifed we must have $s\leq1$. The first statement is then a direct consequence of Theorem \[theorem case 1\]. As to the second statement, $|1-\lambda| =1/2$, Theorem \[theorem case 3\] applies with $m=1$, $s=1$ and $\alpha =-1$. Hence, $\sigma_3=\sqrt{ |1+\lambda|}/2^{3/2}a$, and since $K$ is unramified we may as well exclude the factor $2^{3/2}$.
Note that $\sqrt{ |1+\lambda|}\leq 1/2$, so even if $\lambda $ is maximal (as in the second statement of Corollary \[corollary case 3\] above), it seems that the radius of the linearization disc in $\mathbb{Q}_2$ may not be the maximal, $1/a$ as obtained for $\mathbb{Q}_p$ with $p$ odd in Corollary \[corollary p-adic linearization disc\].
Statement of the general estimate
---------------------------------
Suppose $\lambda ^m$ belongs to the annulus $$\{z\in\mathbb{C}_p:R(s)<|1-z|<R(s+1)\}.$$ Then, in view of Lemma \[lemma closest root of unity\], $\alpha=1$ is the closest root of unity to $\lambda ^m$. It follows that $\sigma_3=\sigma_2$ in this case. Consequently, the estimate $\sigma_3$ of the radius of the linearization disc, holds for all $\lambda $ such that $R(s)\leq |1-\lambda ^m|<R(s+1)$ for some $s\geq 1$. Furthermore, if we put $s=0$ and $\alpha =1$, then $\sigma_3=\sigma_1$.
Hence, $\sigma_3$ may serve as a general bound if we include the case $s=0$. Recall that, by definition, $R(0)=0$. Our estimates can thus be summarized according to the following theorem.
\[theorem general estimate\] Let $f\in \mathcal{F}_{\lambda,a}$. Suppose $\lambda $ is not a root of unity, and $$R(s)\leq |1-\lambda ^m|<R(s+1),$$ for some $s\geq 0$. Then, the linearization disc $\Delta_f\supseteq
D_{\sigma}(0)$ where
$$\label{definition sigma}
\sigma=\sigma(\lambda,a):=a^{-1}R(s+1)^{\frac{1}{m}} |1-\lambda
^m|^{ \frac{1}{m}(1 + \frac{p-1}{p} s ) }\left (
\frac{|\alpha-\lambda ^m|}{|1-\lambda ^m |} \right )^{1/mp^s}.$$
Moreover, if the conjugacy function converges on the closed disc $\overline{D}_{\sigma}(0)$, then $\Delta_f\supseteq
\overline{D}_{\sigma}(0)$.
Asymptotic behavior of the estimate of the radius of the linearization disc {#section asympt behav siegel rad}
---------------------------------------------------------------------------
In this section we consider the following question. What happens to the estimate $$\sigma=a^{-1}p^{-\frac{1}{m(p-1)p^{s} }} |1-\lambda ^m|^{
\frac{1}{m}(1 + \frac{p-1}{p} s ) }\left ( \frac{|\alpha-\lambda
^m|}{|1-\lambda ^m |} \right )^{1/mp^s},$$ of the radius of the linearization disc, as $m$ or $s$ goes to infinity? We will show that in each of these two cases $\sigma$ approach $1/a$. As stated in Lemma \[lemma upper bound Siegel and isometry\], for $f\in
\mathcal{F}_{\lambda,a}$, the value $1/a$ is the maximal radius of a linearization disc. On the other hand, if $s$ and $m$ are fixed, then $\sigma \to 0$ as $|\alpha -\lambda ^m|\to 0$.
The result is based on the following Lemma.
\[lemma bound for sigma\] For all $s\geq 0$ $$\sigma\geq a^{-1}p^{-\frac{1+s(p-1)}{m(p-1)p^s}}|1-\lambda
^m|^{\frac{1}{m}}\left ( \frac{|\alpha-\lambda ^m|}{|1-\lambda ^m
|} \right )^{1/mp^s}.$$ In particular, $$\sigma\geq a^{-1}p^{-\frac{1}{m(p-1)}}|\alpha -\lambda
^m|^{\frac{1}{m}}.$$
Recall that for $s=0$, we have that $|\alpha -\lambda
^m|=|1-\lambda|$. This completes the case $s=0$.
For $s\geq 1$, we have $|1-\lambda ^m|\geq p^{-1/(p^{s-1}(p-1))}$, and by the definition of $\sigma$ $$\sigma\geq a^{-1}p^{-\frac{1+s(p-1)}{m(p-1)p^s}}|1-\lambda
^m|^{\frac{1}{m}}\left ( \frac{|\alpha-\lambda ^m|}{|1-\lambda ^m
|} \right )^{1/mp^s}.$$ Recall from Remark \[remark lemma distance 1-lambda m char 0,p\] that we always have $|\alpha -\lambda ^m |\leq |1-\lambda ^m|$. Hence, we also obtain the following bound for $\sigma$ $$\sigma\geq a^{-1}p^{-\frac{1}{m(p-1)}}|\alpha -\lambda
^m|^{\frac{1}{m}},$$ as required.
Note that if $|1-\lambda ^m|\neq R(s)$ for all $s\geq 0$, then $\alpha =1$. Hence, by increasing $s$, we push $\lambda ^m$ further away from the closest root of unity, and make $\sigma $ closer to its maximum value $1/a$. Loosely speaking, according to the the Lemma, the farther $\lambda ^m$ is from the ‘closest’ root of unity, the closer $\sigma$ is to its maximum value $1/a$. On the other hand, if $s$ and $m$ are fixed, then $\sigma \to 0$ as $|\alpha -\lambda ^m|\to 0$. In particular, we have the following theorem.
\[theorem asymptotic behavior sigma\] Let $|\alpha -\lambda ^m|$ be fixed. Then, the estimate $\sigma$ of the radius of the linearization disc goes to $1/a$ as $m$ or $s$ goes to infinity. If $s$ and $m$ are fixed, then $\sigma \to 0$ as $|\alpha -\lambda
^m|\to 0$.
Maximal linearization discs in finite extensions of $\mathbb{Q}_p$
------------------------------------------------------------------
Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $f\in \mathcal{F}_{\lambda,a}\cap K[[x]]$. The disc $$\Delta_f(0,K)=\Delta_f\cap K$$ will be refered to as the corresponding linearization disc in $K$. We say that $\Delta_f(0,K)$ is *maximal* if it contains the open disc $D_{1/a}(0,K)=D_{1/a}(0)\cap K$.
\[theorem maximal linearization disc in K \] Let $K$ be a finite extension of $\mathbb{Q}_p$, with ramification index $e$. Let $f\in \mathcal{F}_{\lambda,a}\cap K[[x]]$ and let $s$ be the integer for which $R(s)<|1-\lambda^m|\leq R(s+1)$. Let $\epsilon$ be the integer satisfying $\nu(1-\lambda^m )=\epsilon/e$. Suppose that $$\label{general condition maximal linearization disc}
s < \left (\frac{m}{\epsilon }-2\right)\frac{p}{p-1} - \nu\left ( \frac{\alpha - \lambda ^{m}}{1-\lambda^{m}}
\right ).$$ Then, the linearization disc $\Delta_f(x_0,K)$ is maximal. In particular, if either $\max_{i\geq 2 }|a_i|^{1/(i-1)}$ is attained (as for polynomials) or $f$ diverges on $S_{1/a}(x_0,K)$, then $\Delta_f(x_0,K)=D_{1/a}(x_0,K)$.
We first consider the case $R(s)<|1-\lambda^m|< R(s+1)$. Consequently, $\epsilon p^{s-1}(p-1)<e<\epsilon p^{s}(p-1)$. In view of (\[definition sigma\]) with $\alpha = 1$, $$\nu(a\sigma)=\frac{1}{m(p-1)p^s}+ \frac{\epsilon(1+ s(p-1)/p)}{em}=
\frac{e+\epsilon p^s(p-1)(1+s(p-1)/p)}{em(p-1)p^s},$$ and since $e<\epsilon p^s(p-1)$ we have $$\nu (a\sigma)<\frac{1}{e}\cdot \frac{\epsilon (2+s(p-1)/p)}{m}.$$ It follows that $$\nu(a\sigma)<1/e$$ if $s<p(m/\epsilon -2)/(p-1)$ as required.
We now consider the case $|1-\lambda^m|= R(s+1)$. Hence, $e=\epsilon p^{s}(p-1)$. In view of (\[definition sigma\]), $$\nu(a\sigma)=\frac{1}{m(p-1)p^{s+1}}+ \frac{\epsilon(1+ (s+1)(p-1)/p)}{em}+\frac{\nu}{mp^{s+1}}.$$ Hence, $$\nu(a\sigma)=
\frac{e+\epsilon p^{s+1}(p-1)(1+(s+1)(p-1)/p)+e\nu (p-1)}{em(p-1)p^{s+1}},$$ and since $e=\epsilon p^s(p-1)$ we have $$\nu (a\sigma)=\frac{1}{e}\cdot \frac{\epsilon p(1/p+1+(s+1)(p-1)/p)+(p-1)\nu }{mp},$$ or equivalently, $$\nu (a\sigma)=\frac{1}{e}\cdot \frac{\epsilon (2+s(p-1)/p)+\nu (p-1)/p }{m}.$$ It follows that $$\nu(a\sigma)<1/e$$ if $s<p(m/\epsilon -2)/(p-1)-\nu$ as required.
One might ask whether the condition (\[general condition maximal linearization disc\]) is really necessary or just a consequence of lack of precision in our estimates of the radius of the linearization disc. However, in section \[section quadratic case\] (Corollary \[corollary quadratic power series\]) we show that there are examples where the condition (\[general condition maximal linearization disc\]) is not satisfied and the corresponding linearization disc is strictly contained in the disc $D_{1/a}(0)$.
Let $K$ be a finite extension of $\mathbb{Q}_p$ of degree $n$, with ramification index $e$, and residue field $k$ of degree $[k:\mathbb{F}_p]=n/e$. Let $\lambda $ be maximal, $f\in \mathcal{F}_{\lambda,a}\cap K[[x]]$, and let $s$ be the integer for which $R(s)<|1-\lambda^{p^{n/e}-1}|\leq R(s+1)$. Suppose that $$\label{condition maximal linearization disc lambda maximal}
s < (p^{n/e} - 3)\frac{p}{p-1} - \nu\left ( \frac{\alpha - \lambda ^{p^{n/e}-1}}{1-\lambda^{p^{n/e}-1}}
\right ).$$Then, the linearization disc $\Delta_f(0,K)$ is maximal. In particular, if either the maximum $\max_{i\geq 2 }|a_i|^{1/(i-1)}$ is attained (as for polynomials) or $f$ diverges on $S_{1/a}(0,K)$, then $\Delta_f(0,K)=D_{1/a}(0,K)$.
Recall that since $\lambda $ is maximal we have $m=p^{n/e}-1$ and moreover $|1-\lambda^{p^{n/e}-1}|=p^{-1/e}$. The corollary then follows from theorem \[theorem maximal linearization disc in K \] with $\epsilon =1$ and $m=p^{n/e}-1$.
For example, if $K$ is an unramified extension, then $e=1$ and $s=0$. Furthermore, if $\lambda$ is maximal and $p$ is odd, then $\alpha=1$. Hence, if $p>3$, the condition (\[condition maximal linearization disc lambda maximal\]) holds and the linearization disc is maximal. On the other hand, if $p=3$, then condition (\[condition maximal linearization disc lambda maximal\]) is not satisfied. However, by Corollary \[corollary case 1\] we know that the linearization disc is maximal also for $p=3$. This shows that the condition (\[condition maximal linearization disc lambda maximal\]) is not necessary in this case.
The quadratic case {#section quadratic case}
==================
To see the results at work we provide examples, where we can find the exact size of the linearization disc. We begin with quadratic polynomials, and then show how we can extend this result to power series containing a ‘sufficiently large’ quadratic term. We also give sufficient conditions, on the multiplier $\lambda$, that there is a fixed point on the ‘boundary’ of a linearization disc for quadratic polynomials.
Given a quadratic polynomial $f$ of the form $f(x)=\lambda
x+a_2x^2\in\mathbb{C}_p[x]$, the radius of the corresponding linearization disc can be estimated by $\sigma$, defined by $(\ref{definition sigma})$. If $\lambda $ is located inside the annulus $\{z: p^{-1}<|1-z|<1\}$, we can actually find the exact size of the linearization disc.
\[theorem quadratic polynomials\] Let $p$ be an odd prime and let $\lambda\in\mathbb{C}_p$, not a root of unity, belong to the annulus $\{z:p^{-1}<|1-z|<1\}$. Let $f$ be a quadratic polynomial of the form $f(x)=\lambda
x+a_2x^2\in\mathbb{C}_p[x]$. Then, the coefficients of the conjugacy $g$ satisfy $$\label{bk quadratic simplified}
|b_{k}|=\frac{|a_2|^{k-1} |1-\lambda|^{ \lfloor \frac{k-1}{p}
\rfloor }} {\prod_{n=1}^{k-1}|1-\lambda^n|}.$$ Moreover, the linearization disc about the origin, $\Delta_f=D_{\tau}(0)$, where the radius $\tau=|1-\lambda|^{-1/p}\sigma$.
We first prove that the coefficients of the conjugacy, defined by the recursive formula (\[bk-equation\]), satisfy the identity (\[bk quadratic simplified\]).
Recall that one consequence of ultrametricity is that for any $x,y\in \mathbb{K}$ with $|x|\neq |y|$, the inequality (\[sti\]) becomes an equality. In other words, if $x,y\in \mathbb{K}$ with $|x|<|y|$, then $|x+y|=|y|$. The idea of the proof is to find a dominating term in the right hand side of (\[bk-equation\]) which is strictly greater than all the others. Then, the absolute value of the coefficient $b_k$, of the conjugacy $g$, is equal to the absolute value of the dominating term. The proof is similar to that performed in [@Lindahl:2004 p 760–761], for fields of prime characteristic.
Since the term $l!/(\alpha_1!\alpha_2!)$ is always an integer $$|\frac{l!}{\alpha_1!\alpha_2!}|\leq1,$$ with equality if and only if $l!/(\alpha_1!\alpha_2!)$ is not divisible by $p$. We will show that most of the time the $b_{k-1}$-term is the greatest. In fact, $$\label{equation factorial k-1}
\frac{(k-1)!}{\alpha_1!\alpha_2!}=k-1.$$ As $k$ runs from $1, \dots, p$, the number $k-1$ will never be divisible by $p$. Recall that we assume in this proof that $|1-\lambda |<1$ (so that $m=1$). Hence, $|1-\lambda ^n|<1$, for all integers $n\geq 1$. Therefore, the $b_{k-1}$-term will be strictly greater than all the other terms in the right hand side of (\[bk-equation\]), and thus by the ultrametric triangle inequality (\[sti\]), we have $$\label{bk1}
|b_k|=\frac{|b_{k-1}||k-1||a_2|}{|1-\lambda^{k-1}|}=\frac{|a_2|^{k-1}}{|1-\lambda^{k-1}||1-\lambda^{k-2}|\dots
|1- \lambda|}.$$ But if $k=p+1$, then $|k-1|= p^{-1}$ so that for $l=k-1$ in (\[bk-equation\]), we obtain $$\label{bk-1 term k-1 divisible by p}
|b_{k-1}|\left|\frac{(k-1)!}{\alpha_1!\alpha_2!}\right||a_2|=\frac{p^{-1}|a_2|^{p}}{|1-\lambda^{p-1}||1-\lambda^{p-2}|\dots
|1-\lambda|}$$ Then, the $b_{k-2}$-term will dominate. In fact, $$\label{faculty k-2}
\left |\frac{(k-2)!}{\alpha_1!\alpha_2!}\right
|=\left|\frac{(k-2)(k-3)}{2}\right|=1, \text{ if } p|k-1, \text{ }
p>2.$$ As a consequence, $l=k-2$ gives $$\label{bk-2 term k-1 divisible by p}
|b_{p-1}|\left |\frac{(k-2)!}{\alpha_1!\alpha_2!}\right
||a_2|^2=\frac{|a_2|^{p}}{|\lambda^{p-2}-1||\lambda^{p-3}-1|\dots
|\lambda-1|}.$$ Note that, since $m=1$ in our case, we have $|1-\lambda
^{p-1}|=|1-\lambda |$. Moreover, by assumption, $|1-\lambda
|>p^{-1}$. Hence, the $b_{k-2}$-term (\[bk-2 term k-1 divisible by p\]) is strictly greater than the $b_{k-1}$-term (\[bk-1 term k-1 divisible by p\]), and all $b_l$-terms for which $l<k-2$. Consequently, $$\label{bk2}
|b_{p+1}|=\frac{|b_{p-1}||a_2|^2}{|\lambda^{p}-1|}=\frac{|a_2|^{p}}{|\lambda^{p}-1||\lambda^{p-2}-1||\lambda^{p-3}-1|\dots
|\lambda-1|}.$$ Note the lack of the factor $|\lambda^{p-1}-1|$. Now, since according to Lemma \[lemma distance 1-lambda m char 0,p\] $$|1-\lambda^p|<|1-\lambda|=|1-\lambda^{p-1}|,$$ we have that $$|1-\lambda^{p}|\prod_{j=1}^{p-2}|1-\lambda^{j}|<|1-\lambda^{p-1}|\prod_{j=1}^{p-2}|1-\lambda^{j}|.$$ Therefore we have for $k=p+2$ that the $b_{k-1}$-term is again strictly greater than all the others in the right hand side of (\[bk-equation\]) so that $$|b_{p+2}|=\frac{|b_{p+1}||a_2|}{|\lambda^{p+1}-1|}=\frac{|a_2|^{p+1}}{|1-\lambda^{p+1}||1-\lambda^p||1-\lambda^{p-2}||1-\lambda^{p-3}|\dots
|1-\lambda |}.$$ The $b_{k-1}$-term will dominate until $p$ divides $k-1$ again, i.e. for $k=2p+1$ (which means that we “loose” the factor $|\lambda^{2p-1}-1|$). Repeated application of these arguments yields that $$\label{bk quadratic}
|b_{k}|=\frac{|a_2|^{k-1}\prod_{i\cdot p\leq
k-1}|1-\lambda^{ip-1}|} {\prod_{n=1}^{k-1}|1-\lambda^n|}.$$ Note that $|1-\lambda^{ip-1}|=|1-\lambda |$, since $m=1$ in this case. Hence, we obtain (\[bk quadratic simplified\]) as required.
It remains to prove that that the corresponding linearization disc is the open disc $D_{\tau}(0)$. Recall that the estimates for the $b_k$:s in the previous sections where based on the estimate (\[b\_k estimate by prod 1-lambda n\]). Moreover, $$\left (|1-\lambda|^{ \lfloor \frac{k-1}{p} \rfloor }\right
)^{1/k}\to |1-\lambda |^{\frac{1}{p}}, \quad k\to \infty.$$ This suggests that $g$ converges on an open disc of radius $$\tau =|1-\lambda |^{-\frac{1}{p}}\sigma.$$ In fact, $g$ diverges on the sphere of radius $\tau$; let $I\geq
s+1$ be an integer, then by Lemma \[lemma case 1\], \[lemma case 2\], \[lemma case 3\] and (\[bk quadratic simplified\]) we have $$\label{bk tau k quadratic}
|b_{p^I+1}|\tau ^{p^I+1}=p^{-\frac{1}{p-1}}
|1-\lambda|^{-\frac{1}{p}}\sigma=p^{-\frac{1}{p-1}}\tau ,$$ which does not approach zero as $I$ goes to infinity. Furthermore, in a similar way, applying Lemma \[lemma case 1\], \[lemma case 2\], and \[lemma case 3\] to the identity (\[bk quadratic simplified\]) we obtain $$|b_{k}|\tau ^{k}\leq p^{-\frac{1}{p-1}}\tau.$$ Consequently, $g$ is one-to-one on $D_{\tau}(0)$. It follows that the linearization disc of the quadratic polynomial $f$ is the disc $\Delta_f=D_{\tau}(0)$.
The theorem implies, in particular, that $f$ can have no periodic points (except the fixed point at the origin) in the disc $D_{\tau}(0)$. However, there may be periodic points on the boundary. We will give sufficient conditions that there is a fixed point on the boundary $S_{\tau}(0)$. Note that $f$ has a fixed point $\hat{x}=(1-\lambda)/a_2$. Solving the equation $$\label{equation fixed point}
\tau=|1-\lambda |/|a_2|,$$ yields that $\hat{x}$ is located on $S_{\tau}(0)$ if $s=1$ and $\alpha =1$ and $|1-\lambda |=p^{-1/2(p-1)}$, or if $s\geq 2$ and $|1-\lambda |=p^{-t(s)}$, where $t(s)=[(s-1)p^{s-1}(p-1)+p^{s-1}-1]/p^{s-1}(p-1)$. Furthermore, $\hat{x}$ cannot be located on $S_{\tau}(0)$ if $s=0$; the only solution to (\[equation fixed point\]) for $s=0$ is $|1-\lambda
|=p^{-p/(p-1)}$, and hence, $\lambda $ does not belong to the annulus $\{z:p^{-1}<|1-z|<1\}$.
As in [@Lindahl:2004 Corollary 2.1] the previous result on quadratic polynomials works also for power series containing a dominating quadratic term in the following sense.
\[theorem quadratic power series\] Let $p$ be an odd prime and let $\lambda\in\mathbb{C}_p$, not a root of unity, belong to the annulus $\{z:p^{-1}<|1-z|<1\}$. Let $f\in \mathbb{C}_p[[x]]$ be a power series of the form $$f(x)=\lambda x + a_2x^2 +\sum_{i\geq 3}a_ix^i,$$ where $$\label{condition quadratic power series}
|1-\lambda |^{1/p}|a_2|>1, \quad |1-\lambda |^{1/p}|a_2|>|a_i|,
\quad i\geq 3.
$$ Then, the coefficients of the conjugacy $g$ satisfy (\[bk quadratic simplified\]). Moreover, the linearization disc about the origin, $\Delta_f=D_{\tau}(0)$, where $\tau=|1-\lambda|^{-1/p}\sigma$.
By the condition (\[condition quadratic power series\]), the same terms as in the proof of Theorem \[theorem quadratic polynomials\] will be stricly larger than all the others in (\[bk-equation\]). The reason for the factor $|1-\lambda|^{1/p}$, is the lack of the factor $|1-\lambda
^{p-1}|$ in (\[bk2\]).
\[corollary quadratic power series\] Let $f$ be a power series satisfying the condions of Theorem \[theorem quadratic power series\], then the radius of the linearization disc $$\tau <p^{-1/(p-1)p^s}a^{-1}.$$ In particular, the linearization disc cannot contain the disc $D_{1/a}(0)\cap K$ for any algebraic extension $K$ of $\mathbb{Q}_p$.
Minimality and ergodicity
=========================
Minimality and conjugation in non-Archimedean fields
----------------------------------------------------
In this section we consider power series defined over an arbitrary complete non-Archimedean field $K$, rather than just over $\mathbb{C}_p$. The notion of transitivity and minimality on subsets of $K$ are defined as follows. Let $X$ be a subset of $K$ and let $f\in K[[x]]$ be a power series which converges on $X$. Suppose that $X$ is invariant under $f$, i.e. $f(X)\subseteq X$. The map $f:X\to X$ is said to be *transitive* if there is an element $x\in X$, such that its forward orbit $\{f^{\circ
n}(x)\}_{n=0}^{\infty}$ is dense in $X$. We say that $f:X\to X$ is *minimal* on $X$ if for every $x\in X$, its forward orbit $\{f^{\circ n}(x)\}_{n=0}^{\infty}$ is dense in $X$.
We will look for dense orbits near indifferent non-resonant fixed points of $f$. By Lemma \[lemma linearization disc isometry\], the dynamics on a linearization disc $\Delta _f(x_0,K)$ is located on invariant spheres, about the fixed point $x_0$. As in previous sections we will assume (without loss of generality) that $f$ has an indifferent fixed point at the origin. Given $\lambda\in K$, let $T_{\lambda}: K \to K$ be the multiplication map, $x
\mapsto \lambda x$. We will prove that transitivity of $f(x)= \lambda x +O(x^2)\in K[[x]]$, on some subset $X$, of the corresponding linearization disc $\Delta_f$ about the origin, is equivalent to transitivity of $T_{\lambda}$ on $g(X)$. Moreover, transitivity and minimality are equivalent if $X$ is compact.
\[theorem-minimalitypreserved\] Let $f(x)=\lambda x +O(x^2)\in K[[x]]$ be analytically conjugate to $T_{\lambda}$, on a linearization disc $\Delta_f$ about the origin, via a conjugacy function $g$, such that $g(0)=0$ and $g'(0)=1$. Suppose that the subset $X\subseteq \Delta_f$ is invariant under $f$. Then, the following statements hold:
1) $f$ is transitive on $X$ if and only if $T_{\lambda}$ is transitive on $g(X)$.
2) If $X$ is compact and $f$ is transitive on $X$, then $f$ is minimal on $X$. Moreover, $f(X)=X$ and $g(X)=T_{\lambda}(g(X))$.
Let $f$ be analytically conjugate to $T_{\lambda}$, with conjugacy function $g$. Suppose that $X\subseteq \Delta_f$ is invariant under $f$. By the conjugacy relation we must have $g(f(X))=T_{\lambda}(g(X))$. It follows that $T_{\lambda}(g(X))\subseteq g(X)$. In other words, $g(X)$ is invariant under $T_{\lambda}$.
Now, suppose that $T_{\lambda}$ is transitive on $g(X)$. Then, there is an $x\in X$ such that the orbit $\{T_{\lambda}^{\circ
n}(x)\}$ is dense in $g(X)$. Recall that by Lemma \[lemma linearization disc isometry\], $g:X\to g(X)$ is bijective and isometric. Hence, given $\epsilon
>0$ and $y\in X$, there is an integer $n\geq 1$ such that in view of the conjugacy relation $$\begin{aligned}
\epsilon &>& |T_{\lambda}^{\circ{n}}\circ g(x)-g(y)|=|g^{-1}\circ T_{\lambda}^{\circ{n}}\circ g(x)-g^{-1}\circ
g(y)|=|f^{\circ n}(x)-y|.
\end{aligned}$$ It follows that the orbit $\{f^{\circ n}(x)\}$ is dense in $X$. Accordingly, $f$ is transitive on $X$. Likewise, transitivity of $f$ implies transitivity of $T_{\lambda}$. Now we consider the second statement of the theorem. Suppose that the subset $X\subset \Delta _f$ is compact, and that $f:X\to X$ is transitive so that $T_{\lambda}$ is transitive on $g(X)$. In view of the conjugacy relation we have $g(f^{\circ
n}(x))=T_{\lambda}^{\circ n}(g(x))$. Hence, transitivity of $T_{\lambda}$ implies that $g(X)$ is dense in $T_{\lambda}(g(X))$. Continuity of $g$ and $T_{\lambda}$, and compactness of $X$, gives $g(X)=T_{\lambda}(g(X))$. Consequently, $f(X)=g^{-1}\circ
T_{\lambda} \circ g(X)=X$. Hence, $f:X\to X$ is not only one-to-one and isometric but also surjective.
It is well known that a transitive bijective isometry is minimal, see e.g. [@Walters:1982]. Accordingly, minimality and transitivity are equivalent on compact subsets of non-Archimedean linearization discs. This completes the proof.
\[remark f isometry X subset sphere\] By Lemma \[lemma linearization disc isometry\], $f:\Delta_f\to
\Delta_f$ is also an isometry. Therefore, if $f$ is minimal on some subset $X\subseteq \Delta_f$ and $X\neq \{0\}$, then $X\subseteq S$ for some sphere $S\subset \Delta_f$.
Minimality in $\mathbb{Q}_p$
----------------------------
It follows from the previous section that transitivity, and hence minimality, on spheres about an indifferent fixed point can be characterized in terms of the multiplier map. By Lemma \[lemma transitivity multiplier\], the multiplier map $T_{\lambda}$ is transitive on a sphere about the origin in $\mathbb{Q}_p$ if and only if $\lambda$ is maximal and $p$ is odd. Moreover, in view of Theorem \[theorem-minimalitypreserved\], minimality and transitivity of $T_{\lambda}$ coincide, as proven earlier by various authors [@Bryk/Silva:2003; @Coelho/Parry:2001; @Gundlach/Khrennikov/Lindahl:2001:a; @Oselies/Zieschang:1975] in the $p$-adic setting.
Recall that, if $\lambda$ is maximal, then the corresponding linearization disc is maximal as stated in Corollary \[corollary p-adic linearization disc\].
\[theorem-minimalityQp\] Let $f\in \mathcal{F}_{\lambda,a}\cap \mathbb{Q}_p[[x]]$ for some prime $p$. Then, $f$ is minimal on each sphere $S\subset\mathbb{Q}_p$ of radius $r<1/a$ about the origin, if and only if $\lambda $ is maximal and $p$ is odd. Moreover, if $\lambda $ is maximal and $p$ is odd, then $g(S)=f(S)=S$.
Suppose that $f$ is minimal on a rational sphere $S\subset \Delta
_f(0,\mathbb{Q}_p)$ about the origin. In view of Theorem \[theorem-minimalitypreserved\], $T_{\lambda}$ is minimal on $g(S)$. Hence, by the conjugacy relation, $f^{\circ
n}(g^{-1}(x))=g^{-1}(\lambda ^n(x))$, the image $g^{-1}(S)$ is dense in $S$. By continuity of $g^{-1}$ and compactness of $S$, $g^{-1}(S)=S$. Accordingly, $g(S)=S$. It follows that $T_{\lambda
}$ is minimal on $S$. Consequently, $\lambda $ is maximal.
On the other hand, suppose that $\lambda $ is maximal. Then, $T_{\lambda}$ is minimal on each sphere $S_r(0)\subset
\mathbb{Q}_p$. In view of Corollary \[corollary p-adic linearization disc\], the semi-conjugacy relation $g(f^{\circ n}(x))=\lambda ^n
g(x)$, holds for all $x\in \Delta_f(0,\mathbb{Q}_p)\supseteq
D_{1/a}(0)$. By similar arguments as above, we conclude that $g(S_r(0))=S_r(0)$ if $r<1/a$. It follows that $T_{\lambda}$ is minimal on $g(S_r(0))$ for $r<1/a$. Consequently, $f\in
\mathcal{F}_{\lambda,a}\cap \mathbb{Q}_p[[x]]$ is minimal on $S_r(0)$ for each $r<1/a$.
As shown in Section \[section non-Archimedean power series\], $f:\Delta_f\to \Delta_f$ is not only one-to-one, but also surjective, in the algebraic closure $\mathbb{C}_p$. By Theorem \[theorem-minimalityQp\], $f$ may also be surjective on $\Delta
_f(\mathbb{Q}_p)=\Delta_f \cap \mathbb{Q}_p$.
\[corollary surjective in Qp\] Let $f\in \mathcal{F}_{\lambda,a}\cap \mathbb{Q}_p[[x]]$ for some odd prime $p$. Suppose that $\lambda $ is maximal, then $f:\Delta_f(0,\mathbb{Q}_p)
\to \Delta_f(0,\mathbb{Q}_p)$ is bijective.
Unique ergodicity
-----------------
Let $K$ be a complete non-Archimedean field. In the following $X$ will be a compact subset of $K$. For example, if $K=\mathbb{C}_p$, $X$ could be any disc or any sphere in a finite extension of $\mathbb{Q}_p$.
A continuous map $T:X\to X$ is *uniquely ergodic* if there exists only one probability measure $\mu$, defined on the Borel $\sigma$-algebra $\mathcal{B}(X)$ of $X$, such that $T$ is measure-preserving (i.e. $\mu (A)=\mu (T^{-1}(A))$ for all $A\in
\mathcal{B}(X)$) with respect to $\mu$. As $\mu $ is unique it follows that $T$ must be ergodic with respect to $\mu$. Recall that $T$ is ergodic if for any $A\in \mathcal{B}$, whenever $T^{-1}(A)=A$, then $\mu(A)\mu(A^{c})=0$. (Here $A^{c}$ denotes the complement of $A$.)
As shown by Oxtoby [@Oxtoby:1952], a bijective isometry of a compact metric space is uniquely ergodic, hence ergodic, if and only if it is minimal. See Bryk and Silva [@Bryk/Silva:2003] for a shorter proof in the $p$-adic setting.
In view of Lemma \[lemma linearization disc isometry\] and Theorem \[theorem-minimalitypreserved\], $f(x)\in\lambda x+
O(x^2)\in K[[x]]$ is certainly bijective and isometric on compact invariant subsets of a linearization disc $\Delta_f$ about the origin in $K$. Consequently, transitivity, minimality, ergodicity and unique ergodicity are all equivalent and preserved under analytical conjugation on compact subsets of $\Delta_f$.
\[theorem minimlity ergodicity subset\] Let $K$ be a complete non-Archimedean field. Let the power series $f(x)=\lambda x
+O(x^2)\in K[[x]]$ be analytically conjugate to $T_{\lambda}$, on a linearization disc $\Delta_f$ about the origin, via a conjugacy function $g$, such that $g(0)=0$ and $g'(0)=1$. Suppose that the subset $X\subset \Delta_f$ is non-empty, compact and invariant under $f$. The following statements are equivalent:
1. $T_{\lambda} : g(X)\to g(X) $ is minimal.
2. $f: X\to X$ is minimal.
3. $f: X\to X$ is uniquely ergodic.
4. $f$ is ergodic for any $f$-invariant measure $\mu$ on $\mathcal{B}(X)$ that is positive on non-empty open sets.
Now, we return to the $p$-adic case. Note that a rational sphere $S\subset\mathbb{C}_p$ is not compact since $\mathbb{C}_p$ is not locally compact. (Recall that a sphere $S\subset K$ is rational if and only if it is non-empty, i.e. the radius is a number in the value group $|K^*|$.) However, in $\mathbb{Q}_p$ (or any finite extension of $\mathbb{Q}_p$), every rational sphere is compact. By Theorem \[theorem-minimalityQp\] we have the following result.
\[theorem generator minimality ergodicity\] Let $f\in \mathcal{F}_{\lambda,a}\cap \mathbb{Q}_p[[x]]$ for some odd prime $p$. Let $S\subset\mathbb{Q}_p$ be a rational sphere of radius $r<1/a$ about the origin. Then, the following statements are equivalent:
1. $\lambda $ is maximal.
2. $f:S\to S$ is minimal.
3. $f: S\to S$ is uniquely ergodic.
4. $f$ is ergodic for any $f$-invariant measure $\mu$ on $\mathcal{B}(S)$ that is positive on non-empty open sets.
As proven in [@Bryk/Silva:2003; @Coelho/Parry:2001; @Oselies/Zieschang:1975], in this case the unique invariant measure $\mu$ is the normalized Haar measure $\mu$ for which the measure of a disc is equal to the radius of the disc.
Note that the estimate of the radius $1/a$ is maximal in the sense that there exist examples of such $f$ which diverges or have a zero on the sphere $S_{1/a}(0)$, see Lemma \[lemma f one-to-one\] and Corollary \[corollary p-adic linearization disc\].
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank Prof. Andrei Yu. Khrennikov for introduction to the theory of $p$-adic dynamical systems,the formulation of the problem on conjugate maps for $p$-adic dynamical systems, and for introducing me to the the work of Bryk and Silva [@Bryk/Silva:2003] that helped me solve the problem on ergodicity. I also thank C.E. Silva for sending me preprints on their work on $p$-adic ergodicity.
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[^1]: Revised version of preprint 04098 MSI, 2004, Växjö University, Sweden and part of the thesis [@Lindahl:2007]
|
---
abstract: |
Background
: X-ray telescopes are powerful tools for the study of neutron stars and black holes. In order to probe such fascinating astrophysical objects future X-ray telescopes will carry superconducting transition-edge sensors. The analysis of the signals produced by X-rays absorption onto transitional-edge sensor is already mature. Several methods have already been developed (e.g., principal component analysis, nonlinear optimal filtering or high-rate processing method) to analyse the pulses that result from X-rays absorbed in these sensors.
Purpose
: Our goal is to develop a lightweight, linear filter that will maximize energy and time resolution when X-ray photons are detected by transition-edge sensors. Such a method could be implemented in the new generation of X-ray space telescopes. Furthermore, we find the minimal sampling rate that will not degrade the energy and time-resolution of these techniques.
Method
: Our method is designed for the widest range of photon energies (from $0.1$ keV to $30$ keV). Transition-edge sensors exhibit a non-linear response that becomes more pronounced with increasing photon energy; therefore, we need to treat high-energy photons differently from low-energy photons. In general, the switching-point energy depends on the properties of the detector and corresponds to the energy of the photon that begins to saturate the superconductor to normal transition of the TES (at about $4.6$ keV here). In order to retrieve the energy and the arrival time of the photon, we fit simulations of the evolution of the current including the typical noise sources in a sensor with simulated theoretical models. The curve-fitting parameters are interpolated to extract the energy and time resolution.
Results
: For energies from $0.1$ keV to $30$ keV and with a sampling rate of $195$ kHz, we obtain a $2\sigma$-energy resolution between $1.67$ eV and $6.43$ eV. Those results hold if the sampling rate decreases by a factor two. About time resolution, with a sampling rate of $195$ kHz we get a $2\sigma$-time resolution between $94$ ns and $0.55$ ns for a sensor with the physical parameters as those used in the HOLMES experiment.
Conclusions
: We have successfully developed a new method that enables to maximize the energy and the arrival time of photons detected by a TES, using a very simple implementation. In order to make this method useful on a larger scale, it will be essential to get a more general description of the noise in a TES, and it will be necessary to develop a robust way to identify pile-up events.
author:
- Paul Ripoche
- Jeremy Heyl
title: |
Maximizing time and energy resolution for photons\
detected by transition-edge sensors
---
Introduction
============
X-ray telescopes enable us to study fascinating compact objects, such as neutron stars and black holes. In the future, several missions, e.g., Athena [@2018SPIE10699E..1GB], Colibrì [@2019BAAS...51g.175H] or Lynx [@2018SPIE10699E..0NG], are planned to carry arrays of superconducting transition-edge sensors (TESs). Consequently, maximizing energy resolution and timing for photons detected with such sensors is a crucial goal in X-ray astronomy [@2015SuScT..28h4003U].
Recently, several techniques to analyze X-ray data from transition-edge sensors have been developed, such as principal component analysis [@2016JLTP..184..382B], optimal filtering of the resistance signal [@2015ApPhL.107v3503L], nonlinear optimal filtering [@2014AIPA....4k7106S] and optimal fitting [@2002AIPC..605..339F; @2004NIMPA.520..555F; @2019JATIS...5b1008S]. Those techniques achieve an energy resolution between $0.7$ keV and $3.4$ keV full width at half maximum (FWHM), for low energies (below $6$ keV). However, those techniques may be difficult to implement easily on an X-ray space telescope.
Therefore, we propose a new and lighter method, in order to get the energy and the arrival time of photons detected by a TES. We aim to use this technique to predict the behavior of the new generation of non-focusing X-ray telescopes. Indeed, such X-ray telescopes will be studying variable X-ray emissions coming from neutron stars and black holes, and probe the region very close to them, where the dynamical timescales of those region are of the order of microseconds. Since accreting neutron stars and black holes shine bright in the $0.5-10$ keV range, we developed a technique that maximizes time and energy resolution over the widest range of photon energies (here, from $0.1$ keV to $30$ keV).
This paper first describes our detector model and how events were simulated. We next address the issue of noise in TESs and present our method that enables to maximize energy resolution and timing. Finally, we discuss the outcomes.
Transition-edge sensors to detect X-rays
========================================
Detector model {#sec:detectModel}
--------------
A transition-edge sensor is made of a superconducting metal film functioning near its transition temperature (typically $0.1$ K). While electrons move freely in a superconducting metal, they encounter some significant resistance when the metal switches to its normal phase. The transition from superconductor to normal metal occurs within about a narrow 1 mK change in the temperature, but results in a large change in resistance. Thus after an X-ray photon deposits energy in the sensor, the superconductor heats up, the resistance increases, the electric current drops and an X-ray photon is detected.
Indeed, when an X-ray photon hits a TES, the provided energy provokes the needed rise in temperature for the transition to happen; therefore measuring how much the intensity of the current diminishes and how long the sensor takes to recover enables us to determine the energy of that photon. Finally, after the absorption of a photon, a TES needs to be cooled down to its initial temperature by using a cooling bath, at temperature $T_\text{bath}$, in order to be able to detect the next photon. In this work, we simulated current pulses in a TES, using differential equations from the Irwin-Hilton model [@Irwin2005]. The temperature $T$ and the current $I$ of the detector evolve as follows $$\diff{I}{t} = - \frac{R(T,I)+R_L}{L}I + \frac{V}{L},
\label{eq:Idyn}$$ $$\diff{T}{t} = \frac{R(T,I)}{C_V}I^{2} - \frac{k}{C_V}(T^{n} - T_\text{bath}^{n}),
\label{eq:Tdyn}$$ where the detector resistance is given by [@2014AIPA....4k7106S] $$R(T,I) = \frac{R_N}{2} \left \{ 1 + \tanh\left[\frac{T-T_C + (I/A)^{2/3}}{2\ln(2)T_W}\right] \right \}.
\label{eq:Rdetect}$$ We assume that the physical parameters in Eq. (\[eq:Idyn\]), Eq. (\[eq:Tdyn\]) and Eq. (\[eq:Rdetect\]) are the same as for detectors being developed at NIST (National Institute of Standards and Technology) for the HOLMES experiment. Those parameters are summarized in [@2016JLTP..184..263A] and are reported in Table \[tab:physparam\].
[|l|l|]{}\
&\
$n=3.25$ & $V=146.9$ nV\
$k=23.3$ nW.K$^{-n}$ & $T_W = 0.565$ mK\
$T_c=0.1$ K & $A = 1.133$ A.K$^{-3/2}$\
$C_V=0.5$ pJ.K$^{-1}$ & $T_0=0.0980$ K\
$R_L=0.3$ m$\Omega$ & $I_0 = 63.85$ $\mu$A\
$T_\text{bath}=0.07$ K &\
$R_N=10$ m$\Omega$ &\
$R_0=2$ m$\Omega$ &\
$L \in \{12, 24, 48\footnote{We use this value of inductance throughout this work, except mentioned otherwise.}\}$ nH &\
In order to test those parameters, we ran our TES simulation without any photon arrivals. We replaced $R(T,I)$ by $R_0$ in Eq. (\[eq:Idyn\]) and Eq. (\[eq:Tdyn\]), and we used the initial temperature $T_0$ and current $I_0$ given in Table \[tab:physparam\]. We obtained a slightly different current at quiescence, $I_0=63.87$ $\mu$A. This new current at quiescence is the one that will be used throughout this work.
Simulations of events {#sec:simulEvents}
---------------------
[|c|c|c|]{}\
\
& &\
& &\
250 & 5.12 & 195[^1]\
125 & 10.2 & 97.7\
62 & 20.6 & 48.4\
31 & 41.3 & 24.2\
13 & 98.5 & 10.2\
In order to develop a method that gives the energy and the arrival time of incoming photons, while maximizing the resolution for those two parameters, we simulated single events. To simulate each photon arrival, we solved Eq. (\[eq:Idyn\]) and Eq. (\[eq:Tdyn\]) letting the initial temperature and current of the TES be: $$T_{0,\gamma}= T_0 + \frac{E_\gamma}{C_V},$$ $$I_{0,\gamma} = I_0,$$ where $E_\gamma$ is the energy of each simulated incoming photon. The current evolution is shown in Fig. \[fig:singleEventCurrent\] for a $7$ keV photon. We observe that current in a TES drops with a relaxation time of about $0.5$ ms, which limits how much the acquisition time can be reduced.
![Change in current for a single event in a TES. The incoming photon has an energy of $7$ keV.[]{data-label="fig:singleEventCurrent"}](theoretical_TES_current_7keV.eps){width="\columnwidth"}
We generated single events at energies in the $0.1-30$ keV range, for the three different values of inductance coming from Table \[tab:physparam\]. We observed that the total drop in the current in the TES increases with the energy of the incoming photon, until it clearly saturates for $E_\gamma \geq 10$ keV. Once the current drop saturates, the relaxation time increases significantly with the energy of the incoming photon. Those two simple observations, are the basis of the method we have developed.
In order to illustrate those two observations, we plot the maximum current drop in the TES as a function of $E_\gamma$ (see Fig. \[fig:maxIdrop\]), and the FWHM of the current drop as a function of $E_\gamma$ (see Fig. \[fig:FWHM\]). The maximum current drop has a linear behavior for $E_\gamma \leq 4.6$ kev, whereas the FWHM of the current drop becomes linear for $E_\gamma \geq 4.6$ kev. Although the magnitude of X-ray pulses as a function of energy has already been used to retrieve the energy of an incoming photon [@2013ITAS...2300705B], the energy resolution obtained at high-energy is poor because TESs have a very non-linear response. Consequently, to maximize time and energy resolution for X-ray photons, we need to treat high-energy and low-energy photons with two different techniques. In general, the switching-point energy depends on the properties of the detector and corresponds to the energy of the photon that begins to saturate the superconductor to normal transition of the TES.
![Maximum current drop in a TES as a function of the energy of the incoming photon.[]{data-label="fig:maxIdrop"}](theoretical_max_TES_drop.eps){width="\columnwidth"}
![FWHM of the current drop in a TES as a function of the energy of the incoming photon.[]{data-label="fig:FWHM"}](theoretical_FWHM.eps){width="\columnwidth"}
Noise in a transition-edge sensor
=================================
Two irreducible sources of noise in a TES are thermal fluctuation noise (also known as phonon noise), and Johnson–Nyquist noise [@McCammon2005]. Thermal fluctuation (TF) noise is the statistical fluctuations that arise from energy exchange between the detector and the heat sink. Johnson–Nyquist (JN) noise is produced by the thermal agitation of electrons in a TES, in other words, it is the electronic noise at equilibrium.
In order to implement our method, we first need to add noise to our simulations. Following the power spectrum of the ARMA-generated noise used in [@2016JLTP..184..263A], we generate a power spectral density (PSD) for the thermal fluctuation noise and for the Johnson-Nyquist noise, at equilibrium: $$% PSD_\text{TF}^\text{eq}(f_j)=\frac{0.15\times10^{-12}}{(4\times10^3)^2+f_j^2}~\mathrm{A^2 Hz^{-1}}
PSD_\text{TF}^\text{eq}(f)=\frac{9.4\times10^{-21}}{1 + \left (\frac{f}{4\times 10^3 \mathrm{Hz}}\right)^2}~\mathrm{A^2 Hz^{-1}}
\label{eq:theoPSDTF}$$
$$PSD_\text{JN}^\text{eq}(f)=0.6\times10^{-21}~\mathrm{A^2 Hz^{-1}},
\label{eq:theoPSDJN}$$
where the variables with indices “eq” are taken at equilibrium. The PSD of the total noise is simply the sum of the two PSDs. Then, we convert these power spectral densities into real noise signals, for each type of noise by performing an inverse Fourier transform using each power spectral density with randomly chosen phases; that is, we assume that the two noise sources are uncorrelated and add them incoherently.
In order to account for non-stationary effects in the noise [@McCammon2005], we scaled both these noises sources at equilibrium, using current, temperature and resistance in a TES at a given moment, $t_j$: $$N_\text{TF}^\text{non-eq}(t_j) = \frac{T}{R}\frac{R_\text{eq}}{T_\text{eq}}\diff{R}{T}\left(\diff{R}{T}\right)_\text{eq}^{-1}N_\text{TF}^\text{eq}(t_j)
\label{eq:expnoiseTF}$$ $$N_\text{JN}^\text{non-eq}(t_j) = \sqrt{\frac{T}{R}\frac{R_\text{eq}}{T_\text{eq}}}N_\text{JN}^\text{eq}(t_j).
\label{eq:expnoiseJN}$$ The total noise is simply the sum of the two noise signals. In order to obtain the resulting equilibrium and non-equilibrium PSDs, we perform a Fourier transform of the noise signals, yielding resulting depicted in Fig. \[fig:comparisonpsd\]. We compare the PSDs of the total non-equilibrium noise (averaged over 1,000 iterations) to the one at equilibrium. They slightly differ for photons with high energies because of the scaling effects in non-stationary noise [@Irwin2005]. As the sensor cools from absorbing a 9.5 keV photon, the thermal fluctuation noise is slightly larger than at equilibrium and conversely the Johnson-Nyquist noise is slightly smaller. Simulating these non-stationary noise signals enables us to test our methods in close-to-real conditions.
![Comparison between the non-equilibrium PSDs (dashed lines) and the ones at equilibrium (solid lines). The Johnson-Nyquist noise has a constant PSD whereas the thermal fluctuation noise has a PSD which drops off at high frequencies. The incoming photon has an energy $E_\gamma = 9.5$ keV.[]{data-label="fig:comparisonpsd"}](psd.eps){width="\columnwidth"}
Measuring the energy and the arrival time of an incoming X-ray photon
=====================================================================
We now present our method to maximize energy and time resolution for photons hitting a TES. We use the physical parameters outlined in Tab. \[tab:physparam\] with $L=48$nH and a sampling interval of 5.12 $\mu$s.
High-energy photons
-------------------
As explained in Sec. \[sec:simulEvents\], we need to treat photons differently according to their energy. In this section, we develop a technique to measure energy and arrival time for high-energy photons. We first describe our method through an example. We simulate a single event at $9.5$ keV, with $L =48$ nH. We work with that inductance so that the onset of the pulse can be resolve even with sampling rates less than 100 khZ. We then use a theoretical model at $7$ keV to fit the noisy pulse. This model was obtained by interpolating current-drop simulations, described in Sec. \[sec:simulEvents\], at a given energy. We split this theoretical model in two parts (see Fig. \[fig:noisyDropHighModel\]), the “onset” part, and the “decay” part: $$I_\text{onset} = \frac{3}{4}I_\text{TES}(t<t_\text{max})$$ $$I_\text{decay} = \frac{3}{4}I_\text{TES}(t>t_\text{max}),$$ where $3/4$ is a numerical factor that enables to have enough points for curve fitting while avoiding effects from the shape of the current-drop maximum, and $t_\text{max}$ is the time at which the current drop is maximum. The “onset” part enables us to obtain the arrival time, whereas the decay part enables us to get the energy of the incoming photon. The theoretical model was chosen for an event at $7$ keV; we however do not expect the energy of the theoretical model to affect the resolution, as long as it has a large enough energy to saturate the transition, that is, $E>4.6$ keV.
![Single event at $E_\gamma = 9.5$ keV, and theoretical model at $E_\gamma=7$ keV. We zoom in (box) to attest the presence of noise in the simulation.[]{data-label="fig:noisyDropHighModel"}](noisy_TES_drop_high_energy_model.eps){width="\columnwidth"}
The curve-fitting model is the following: $$I^\text{fit}_\text{onset}= k_\text{onset} I_\text{onset}(t-t_\text{onset}),
\label{eq:yOnset}$$ $$I^\text{fit}_\text{decay}= I_\text{decay}(t-t_\text{decay}).
\label{eq:yDecay}$$
We then ran simulations for energies between $0.1$ keV and $30$ keV, and retrieved the parameter values for each energy. The response of $k_\text{onset}$ is linear at low energies, which gives a first glimpse of the method used for low-energy photons. On the other hand, the response of $(t_\text{decay}-t_\text{onset})$ is linear at high energies, therefore this parameter is used to obtain the energy of the incoming photon. The parameter $t_\text{onset}$ is used to get the arrival time. In our simulations, the actual arrival time is zero, so the value of $t_\text{onset}$ in the simulations yields the uncertainty in the measurement arrival times.
Finally, for each photon energy, we run 1,000 simulations and retrieve the energy and arrival time uncertainty. Going back to our example where we fit a photon of energy 9.5 keV, we obtained the energy of the incoming photon with an uncertainty of {$+2.35$ eV, $-2.23$ eV} within the 68-percent confidence region. Moreover, the resolution on the arrival time is {$+0.45533$ ns, $-0.48716$ ns} with 68-percent confidence.
Low-energy photons
------------------
In this section, we develop a technique to measure energy and arrival time for low-energy photons. We simulated a single event at $0.5$ keV. We then used a theoretical model at $1$ keV to fit the whole noisy pulse [@2002AIPC..605..339F; @2004NIMPA.520..555F; @2019JATIS...5b1008S]. This is showed in Fig. \[fig:noisyDropLowModel\].
The curve-fitting model is the following: $$I^\text{fit}_\text{shape} = k_\text{shape} I_\text{shape}(t-t_\text{shape}),
\label{eq:yShape}$$ We then ran the simulations for energies between $0.1$ keV and $30$ keV, and retrieved the parameters for each energy. The response of $k_\text{shape}$ is linear at low energies, this is therefore the parameter that we use to obtain the energy of the incoming photon. The parameter $t_\text{shape}$ is used to determine the arrival time.
![Single event at $E_\gamma = 0.5$ keV, and theoretical model at $E_\gamma=1$ keV. We zoomed in (box) to attest the presence of noise in the simulation.[]{data-label="fig:noisyDropLowModel"}](noisy_TES_drop_low_energy_model.eps){width="\columnwidth"}
Finally, for each energy, we run 1,000 simulations and retrieve the energy and arrival time uncertainty. Going back to our example, we obtained the energy of the incoming photon with a resolution of {$+0.87$ eV, $-0.83$ eV}. In addition, the resolution on the arrival time is {$+11.69120$ ns, $-10.84495$ ns}. Both of these results are 68-percent confidence intervals.
Energy and arrival time resolution
----------------------------------
We now summarize the energy resolutions for all photon energies in Fig. \[fig:uncertainty2sigmas10kev\]. As one would expect, $k_\text{shape}$ gives the best energy resolution at low energies and $t_\text{decay}-t_\text{onset}$ gives it at high energies. One can see that the switching point energy is at about $4.6$ keV. Our method enables us to reach an energy resolution at $2 \sigma$ between $1.67$ eV and $6.43$ eV, for $0.1$ keV $< E_\gamma < 30$ keV.
Regarding the arrival time, one can notice that the resolution saturates at high energies. Our method enables us to reach a resolution at $2 \sigma$ between $94$ ns and $0.55$ ns, for $0.1$ keV $< E_\gamma < 30$ keV (see Fig. \[fig:timeuncertainty2sigmas\]).
These resolutions were obtained with a very simple method, which is promising for future analysis of X-ray-telescope data. However, pile-up events are known to alter the obtained resolution [@2014JLTP..176...16F]. Therefore, whenever a consistent photon energy cannot be retrieved with our method, the event can be treated separately as a pile-up event.
![Energy uncertainty range (within $2\sigma$), for the parameters $t_\text{decay}-t_\text{onset}$ and $k_\text{shape}$; as a function of $E_\gamma$.[]{data-label="fig:uncertainty2sigmas10kev"}](energy_uncertainty_2sigmas10keV.eps){width="\columnwidth"}
![Arrival time uncertainty range (within $2\sigma$), for the parameters $t_\text{onset}$ and $t_\text{shape}$, as a function of $E_\gamma$.[]{data-label="fig:timeuncertainty2sigmas"}](time_uncertainty_2sigmas.eps){width="\columnwidth"}
Energy and time resolution for different parameters of a TES
============================================================
The resolution obtained depend on the physical parameters of a detector, such as the sampling rate and the heat capacity. Consequently, we need to change those parameters to see how the resolution is impacted. We also digitized the signal with $16$ bits to see how the resolution would be affected by signal processing.
For simulations at $195$ kHz, we use the better energy resolution given by either ($t_\text{decay}-t_\text{onset}$) or $k_\text{shape}$, and the best time resolution given by either $t_\text{onset}$ or $t_\text{shape}$. However, for lower sampling rates not enough points are present in the onset part of the curve fitting. Therefore, for those lower sampling rates, we use the better energy resolution given by either ($t_\text{decay}-t_\text{shape}$) or $k_\text{shape}$, and the time resolution is given by $t_\text{shape}$.
Energy resolutions for different TES parameters are showed in Fig. \[fig:uncertainty2sigmasDiffParam\]. One notices that digitizing the signal with $16$ bits has no effect on energy resolution and time resolution, which is a promising result for a future use of this method on a real telescope. Increasing the heat capacity however diminishes our energy resolution at low energies. Reducing the sampling rate to $97.7$ kHz doesn’t influence our energy resolution; this enables us to use lower sampling rates than in previous methods [@2016JLTP..184..382B]. One can see that if the sampling rate goes as low as $48.4$ kHz, then the energy resolution decreases by a factor two. To account for the decrease in resolution with lower sampling rates, one could be tempted to increase the heat capacity and the acquisition time. However this would increase the FWHM of the signal and would not let enough time to the TES to cool down for the next incoming photon. Consequently, the sampling rate is the only parameter that can be modified without losing significant resolution.
![Energy uncertainty range (within $2\sigma$) for different parameters of the TES.[]{data-label="fig:uncertainty2sigmasDiffParam"}](energy_uncertainty_2sigmas10keV_diffParam.eps){width="\columnwidth"}
Changing the TES parameters affects the time resolution as well, but in any case it remains far below $1~\mu$s.
Conclusion
==========
TESs are key elements for future X-ray astrophysics telescopes [@2017ITAS...2749839D]. While retrieving the energy and the arrival time of a photon detected by a TES is not new; we have successfully developed a new method that enables to optimize the measurements of the energy and the arrival time of photons detected by a TES, using a very simple linear-filter implementation. Such a simple method is promising for future X-ray analysis. Our method treats high-energy photons and low-energy photons separately. Indeed, since TESs have a very non-linear response, scaling effects in non-stationary noise are more important at high energies. We retrieve the energy and arrival time of low-energy photons by fitting the entire current drop in a TES and by interpolating the magnitude of the drop as a function of the energy. This technique is not new, but gives poor results for high-energy photons. Consequently, for the latter, we developed a new technique; we use the width of the current drop instead of its size, by splitting the curve fitting in two parts. In order to work in closest to real conditions, we generated non-stationary noise in a TES. We noticed that using stationary noise does not change the energy resolution at low-energies, but does for high-energies (where the scaling effects are more important).
We successfully retrieved energy and arrival time for an incoming photon, with resolutions similar to the ones obtained with previous methods at low energies. However, thanks to our new technique, we improve the energy resolution for high-energy photons.
In order to make this method useful on a larger scale, it will be essential to get a more general description of the noise in a TES, and it will be necessary to develop a robust way to identify pile-up events. Eventually, we will determine how to efficiently deal with real-time processing, and therefore enable its implementation in future X-ray telescopes.
This work was supported by the Natural Sciences and Engineering Research Council of Canada, the Canada Foundation for Innovation, the British Columbia Knowledge Development Fund. This research has made use of NASA’s Astrophysics Data System Bibliographic Services.
\[1\]\[1\][\#1]{}
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12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty in [**](https://doi.org/10.1117/12.2312409), , Vol. () p. in @noop [**]{}, Vol. () p. in [**](https://doi.org/10.1117/12.2314149), , Vol. () p. [****, ()](https://doi.org/10.1088/0953-2048/28/8/084003) [****, ()](https://doi.org/10.1007/s10909-015-1357-z) [****, ()](https://doi.org/10.1063/1.4936793) [****, ()](https://doi.org/10.1063/1.4901291) in [**](https://doi.org/10.1063/1.1457659), Vol. , () pp. [****, ()](https://doi.org/10.1016/j.nima.2003.11.313) [****, ()](https://doi.org/10.1117/1.JATIS.5.2.021008) , in [**](https://doi.org/10.1007/10933596_3), (, , ) pp. [****, ()](https://doi.org/10.1007/s10909-015-1402-y) [****, ()](https://doi.org/10.1109/TASC.2013.2238752) , in [**](https://doi.org/10.1007/10933596_1), (, , ) pp. [****, ()](https://doi.org/10.1007/s10909-014-1149-x) [****, ()](https://doi.org/10.1109/TASC.2017.2649839)
[^1]: All plots are at this sampling rate, except where mentioned otherwise.
|
---
author:
- |
[^1]\
Nuclear Physics Institute of Moscow State University\
E-mail:
title: Particle spectra at ZEUS
---
Introduction
============
The HERA ep collider provides a rich field for the study of particle production in a wide range of $\mathit{W}$, the photon-proton centre-of-mass (CMS) energy, and the photon virtuality $Q^2$. The data presented here were obtained with the ZEUS detector at $\sqrt{\mathit{s}} \sim 300\:{\mathrm{GeV}}^2$ and concern the study of the hadronisation and parton fragmentation processes, phenomena which give deep insight into the non-perturbative sector of QCD. Establishing universal features in the properties of the final hadronic system in reactions with different initial particles ($\mathrm{e}^+\mathrm{e}^-$, $\mathrm{ep}$, hadron scattering) helps to elucidate how the partonic cascades evolve into observed hadrons. In particular, recent results on multiplicity distributions in DIS as functions of different energy scales and detailed comparisons with $\mathrm{e}^+\mathrm{e}^-$ are discussed below. The formation of hadron jets was also investigated using the multiplicity and momentum spectra of charged hadrons in the dijet photoproduction events. The measurements verify the validity and consistency of the MLLA approach at energy scale accessed at HERA.
The multipurpose ZEUS detector is described in detail elsewhere [@1]
Multiplicity of charged hadrons
===============================
The average multiplicity and multiplicity distributions are being studied intensively in particle collisions. In previous studies of DIS events only the virtuality of the exchanged photon, $Q$, was used as the energy scale [@2]. A reasonable agreement with $\mathrm{e}^+\mathrm{e}^-$ data was shown except for the region of $Q$ below $6-8\:\mathrm{GeV}$. Recently, the ZEUS collaboration has performed a detailed study of the charged multiplicity in the neutral current deep inelastic scattering (DIS) [@3]. These measurements of the charged hadron multiplicity are performed in the Breit and in the hadronic centre-of-mass (HCM) frames. Due to the restricted detector acceptance only hadrons belonging to the current fragmentation regions in both frames were used in the analysis.
![Comparison of multiplicity distributions in KNO form in bins of $M_{\mathrm{eff}}$ (solid markers) with $\mathrm{e}^+\mathrm{e}^-$ data for the whole event (open circles).[]{data-label="fig:kno_02"}](eps_kno-fig7.eps){width="98.00000%"}
![Comparison of multiplicity distributions in KNO form in bins of $M_{\mathrm{eff}}$ (solid markers) with $\mathrm{e}^+\mathrm{e}^-$ data for the whole event (open circles).[]{data-label="fig:kno_02"}](eps_kno-fig8.eps){width="98.00000%"}
[r]{}[0.5]{}
{width="48.00000%"}
ZEUS investigated charged multiplicity distributions and mean charged multiplicity in terms of different energy scales in order to consistently compare $\mathrm{e}\mathrm{p}$ DIS data with the data from $\mathrm{e}^+\mathrm{e}^-$, $\upnu\mathrm{p}$ and $\upmu\mathrm{p}$ scattering. The following alternatives to the $Q$ energy scale were considered: the energy of the current region of the Breit frame $E^{\mathrm{cr}}_{\mathrm{B}}$, the invariant mass of the produced particles $W$, used in the current region of HCM, and the invariant mass of the hadronic system $M_{\mathrm{eff}}$, used in both frames.
The scaling properties of multiplicity distributions in a commonly used form, proposed by Koba-Nielsen-Olsen (KNO) [@4], were studied in bins of $W$, $2\cdot E^{\mathrm{cr}}_{\mathrm{B}}$ and $M_{\mathrm{eff}}$ and compared with $\mathrm{e}^+\mathrm{e}^-$ data. The multiplicity distributions in the KNO form from ZEUS are shown in Figs. \[fig:kno\_01\], \[fig:kno\_02\]. In these plots the scaled multiplicity distributions, $\Psi(z) = \meannch P(n_{\rm ch})$, are plotted as a function of $n_{\rm ch} / \meannch $, where $P(n_{\rm
ch})$ and $\meannch$ are the multiplicity distribution and average multiplicity respectively. Fig. \[fig:kno\_01\] shows a comparison of the KNO distributions in bins of $2\cdot E^{\mathrm{cr}}_{\mathrm{B}}$ ($12 < 2\cdot E^{\mathrm{cr}}_{\mathrm{B}} < 100\:\mathrm{GeV}$) and in bins of $W$ ($ 70 < W < 225\:\mathrm{GeV}$) with measurements in one hemisphere of $\mathrm{e}^+\mathrm{e}^-$, obtained by the TASSO collaboration in the energy range $14 < \sqrt{s_{\mathrm{ee}}} <
44$ [@5] and by the LEP experiments at $\sqrt{s_{\mathrm{ee}}} =
91.2\:\mathrm{GeV}$ [@b6; @b7]. There is a remarkable agreement between $\mathrm{ep}$ and TASSO data; the LEP data differ somewhat from the present measurement in the peak region and at very low $n_{\rm ch} / \meannch$.
[l]{}[0.5]{}
{width="48.00000%"}
A comparison of the KNO distributions with $\mathrm{e}^+\mathrm{e}^-$ data (both hemispheres) in $M_{\mathrm{eff}}$ bins ($8 < M_{\mathrm{eff}} < 30\:\mathrm{GeV}$) in the current region of HCM and in the current region of the Breit frame ($8<M_{\mathrm{eff}}< 20\:\mathrm{GeV}$) is shown in Fig. \[fig:kno\_02\]. There is good agreement between the ZEUS and both TASSO [@5] and LEP data ($91.2 < \sqrt{s_{\mathrm{ee}}} <
209\:\mathrm{GeV}$) [@b6; @b7; @b8]. The mean charged multiplicities were also investigated using the different energy scales discussed above. In Fig. \[fig:energy\_scale\_1\] the mean charged multiplicities, measured in the current regions of the Breit and HCM frames, are presented as functions of the invariant mass of the corresponding hadronic system and compared with the MC predictions. All three MC models describe the data reasonably well although the Herwig prediction is too high in the last bin of $M_{\mathrm{eff}}$ in the current region of the HCM (Figs. \[fig:energy\_scale\_1\] (a), (b)). As is seen in Fig. \[fig:energy\_scale\_1\] (c), the data in the Breit and HCM frames agree in the region of $M_{\mathrm{eff}} < 10\:\mathrm{GeV}$, while at higher $M_{\mathrm{eff}}$, $\meannch$ rises much faster with $M_{\mathrm{eff}}$ in the current region of HCM than in the current region of the Breit frame. In Fig. \[fig:energy\_scale\_1\] (c) the values of $2\cdot {\meannch}$ as a function of $2\cdot
E^{\mathrm{cr}}_{\mathrm{B}}$ are also plotted. The data follow the same dependence as $\meannch$ vs. $M_{\mathrm{eff}}$ in the Breit frame but differ from those obtained in the current region of the HCM.
Finally, Fig. \[fig:energy\_scale\_2\] shows the comparison of the the mean charged multiplicities in the current region of the Breit and HCM frames as a function of $2\cdot E^{\mathrm{cr}}_{\mathrm{B}}$ and $W$ with the data from $\mathrm{e}^+\mathrm{e}^-$ and fixed-target experiments. The fixed-target data were scaled by a factor 2 (since they were measured in one hemisphere only) and were corrected for the ${\mathrm{K}}^0_{\mathrm{S}}$ and $\Lambda$ decays by a factor 1.08, estimated using the ARIADNE MC model.The ZEUS measurements show good overall agreement with the data from other experiments and exhibit approximately the same dependence on the respective energy scale; only the fixed target DIS data deviate at energies above $15\:\mathrm{GeV}$. The energy scale $2\cdot E^{\mathrm{cr}}_{\mathrm{B}}$ gives better agreement with $\mathrm{e}^+\mathrm{e}^-$ data at low values of energy than $Q$. The measurements of $\meannch$ as a function of $W$ agree, within the uncertainties, with the data from $\mathrm{e}^+\mathrm{e}^-$ collisions.
Scaled momentum distributions
=============================
Recently, momentum spectra of charged hadrons in photoproduction were studied in jet fragmentation processes with the ZEUS detector [@b9]. The results are compared with perturbative QCD calculations carried out in the framework of the Modified Leading Log Approximation, MLLA, [@b10] and the hypothesis of Local Parton-Hadron Duality, LPHD [@b11].The MLLA equations give an analytical description of the parton shower evolution and an effective scale parameter of the QCD calculations, ${\mathit{\Lambda}}_{\mathrm{eff}}$, that is assumed to be universal, i.e. independent of the process considered. The LPHD hypothesis predicts that the observed hadron distributions should be related to the calculated parton distributions by a normalisation parameter $\kappa^{\mathrm{ch}}$. Tests of the MLLA predictions in conjunction with the LPHD hypothesis permit to expand our understanding of the underlying physics of jet fragmentation phenomenon.
[l]{}[0.5]{}
{width="48.00000%"}
Related studies [@b12] have been performed before in $\mathrm{e}^+\mathrm{e}^-$ collisions at LEP and PETRA, in DIS $\mathrm{ep}$ collisions at HERA, (anti)neutrino-nucleon interactions from the NOMAD experiment and $\mathrm{p}\bar{\mathrm{p}}$ collisions at the Tevatron. In the analysis presented here, the momentum spectra of charged hadrons are studied in dijet photoproduction ($\upgamma
\mathrm{p}$) events from $\mathrm{ep}$ collisions. The scaled momentum distributions $\xi=\ln\left(1/x_\mathrm{p}\right)$, where $x_\mathrm{p}$ is the fraction of the jet’s momentum carried by the charged particle, were measured in restricted cones of various opening angles $\theta_c$ around the jet axis. Jets were reconstructed from energy-flow objects [@b13] (EFOs) by applying the $k_\mathrm{T}$ cluster algorithm [@b14]. The reconstructed invariant dijet mass was used as an energy scale. It probes the range 19 to $38\:\mathrm{GeV}$, which spans the energy region between those accessed previously by the ZEUS and CDF collaborations.
To check the validity of the MLLA predictions using the measured $\xi$ distributions, two approaches were adopted. The first was based on the position of the peak of the $\xi$ distributions, $\xi_{\rm peak}$. The second was based on a fit of the full shape of the $\xi$ distributions; the limiting spectra, predicted by MLLA+LPHD theory [@b10], were used in the fit in this method.
In the $\xi_{\rm peak}$ analysis the values of $\xi_{\rm peak}$ were extracted from the $\xi$ distributions using a three-parameter Gaussian fit. At leading order (LO), the peak position is predicted to be at\
$\xi_{\rm peak}=\frac{1}{2}Y + \sqrt{cY}-c$, where $c=0.29$ and Y is a function of the jet energy $E_{\rm jet}$ and $\theta_c$ (see eq.(3) in [@b9]) and depends also on the parameter ${\mathit{\Lambda}}_{\mathrm{eff}}$. Thus the peak position can be directly fit to the data, treating ${\mathit{\Lambda}}_{\mathrm{eff}}$ as a free parameter. The best fit value was found to be ${\mathit{\Lambda}}_{\mathrm{eff}}=275
\pm4$(stat.)$^{+4}_{-8}$(syst.) $\mathrm{MeV}$ for $\theta_c$=0.23. In Fig. \[fig:scalmom-01\] the values of ${\mathit{\Lambda}}_{\mathrm{eff}}$ are shown as a function of the energy scale and compared to the results from different experiments. The data are consistent with the prediction that ${\mathit{\Lambda}}_{\mathrm{eff}}$ is a universal parameter.
[r]{}[0.5]{}
{width="48.00000%"}
The $\xi$ distributions were also fitted using the limiting spectrum predicted by MLLA. The values of ${\mathit{\Lambda}}_{\mathrm{eff}}$ extracted from these MLLA fits are in reasonable agreement with those extracted from the $\xi_{\rm peak}$ data, although the values obtained using the MLLA fit have larger uncertainties due to sensitivity of ${\mathit{\Lambda}}_{\mathrm{eff}}$ to the choice of the fitting range. The values of the LPHD parameters $\kappa^{\mathrm{ch}}$ were extracted also as a function of $E_{\rm jet}$ and $\theta_c$ from the fitted limited momentum spectra and are shown in Fig. 6. The value of $\kappa^{\mathrm{ch}}$, measured with $\theta_c=0.23$ and averaged over $E_{\rm jet}$, was $\kappa_{\rm ch}=0.55
\pm0.01\mathrm{(stat.)}^{+0.03}_{-0.02}\mathrm{(syst.)}^{+0.11}_{-0.09}\mathrm{(theo.)}$ and is in good agreement with that reported by CDF collaboration, $\kappa_{\rm ch}=0.56 \pm 0.05\mathrm{(stat.)}\pm
0.09\mathrm{(syst.)}$. The ZEUS data support the predicted universality of $\kappa_{\rm ch}$.
Summary and conclusions
=======================
The charged multiplicity distributions and the mean charged multiplicity have been investigated in NC DIS $\mathrm{ep}$ scattering in terms of different energy scales. Multiplicity distributions in the scaling KNO form in the current regions of the Breit and HCM frames exhibit the same behaviour as those in one hemisphere of $\mathrm{e}^+\mathrm{e}^-$ collisions when $2\cdot
E^{\mathrm{cr}}_{\mathrm{B}}$ or $W$ are considered. When energy scale $M_{\mathrm{eff}}$ is used, the charged multiplicities exhibit the same KNO-scaling behaviour as those for the whole $\mathrm{e}^+\mathrm{e}^-$ event. The energy scales $2\cdot E^{\mathrm{cr}}_{\mathrm{B}}$ and $W$ give better agreement with $\mathrm{e}^+\mathrm{e}^-$ data than $Q$.
The multiplicity distributions of charged particles in dijet photoproduction events have been measured as a function of $\xi=\ln\left(1/x_\mathrm{p}\right)$. Two methods, the $\xi_{\rm peak}$ analysis and fit of the $\xi$ distributions to the MLLA functions, were used to extract the MLLA scale, ${\mathit{\Lambda}}_{\mathrm{eff}}$, and LPHD parameter, $\kappa^{\mathrm{ch}}$. The data support the assumption that both parameters are universal.
Acknowledments
==============
The author would like to thank my ZEUS collaborators for their efforts to produce the physics results presented at the conference, the ZEUS management for giving her an opportunity to report them here and the organizers for their hospitality.
[99]{} ZEUS Coll., U Holm (ed.), [*[The Zeus Detector]{}*]{}. Status report (unpublished)\
DESY(1993), available on http://www-zeus.desy.de/bluebook/bluebook.html. H1 Collaboration, C. Adloff [*[et al.]{}*]{}, *Nucl. Phys.* [**B 504**]{} (1997) 3;\
ZEUS Collaboration, M. Derrick [*[et al.]{}*]{}, *Z. Phys.* [**C 67**]{} (1995) 93. ZEUS Collaboration, S. Chekanov [*[et al.]{}*]{}, *JHEP* [**06**]{} (2008) 061. Z. Koba, H. B. Nielsen and P. Olesen, *Nucl. Phys.* [**B 40**]{} (1972) 317. TASSO Coll., W. Braunschweig [*[et al.]{}*]{}, *Z. Phys.* [**C 45**]{} (1989) 193. DELPHI Coll., P. Abreu [*[et al.]{}*]{}, *Z. Phys.* [**C 50**]{} (1991) 185. OPAL Coll., P. D. Acton [*[et al.]{}*]{}, *Z. Phys.* [**C 35**]{} (1991) 539. see Chekanov [*[et al.]{}*]{}, *JHEP* [**06**]{} (2008) 061 and ref. \[37\] ibidem. ZEUS Collaboration, S. Chekanov [*[et al.]{}*]{}, *JHEP* [**08**]{} (2009) 077. see S. Chekanov [*[et al.]{}*]{}, *JHEP* [**08**]{} (2009) 077 and ref. \[1\] ibidem. Y. I. Azimov [*[et al.]{}*]{}, *Z. Phys.* [**C 27**]{} (1985) 65. see S. Chekanov [*[et al.]{}*]{}, *JHEP* [**08**]{} (2009) 077 and refs. \[3-9\] ibidem. ZEUS Collaboration, J. Brietweg [*[et al.]{}*]{}, *Eur. Phys. J.* [**C 6**]{} (1999) 43. S. Catani [*[et al.]{}*]{}, *Nucl. Phys.* [**B 406**]{} (1993) 187.
[^1]: for the ZEUS Collaboration
|
---
abstract: 'This supplementary material contains a detailed derivation of the eigenenergies and eigenstates for a system consisting of two atoms in a circular, transversely harmonic waveguide in the multimode regime.'
author:
- 'V. A. Yurovsky'
- 'M. Olshanii'
title: 'Supplementary material to: Restricted Thermalization for Two Interacting Atoms in a Multimode Harmonic Waveguide'
---
The waveguide system [@letter], as well as the Šeba billiard [@seba1990], belongs to the class of problems where an integrable system of Hamiltonian $\hat{H}_0$ is perturbed by a separable rank I interaction $V|{\cal L}\rangle\langle {\cal R}|$ (see [*e.g.*]{} [@albeverio2000]). Eigenstates $|\alpha\rangle$ of the interacting system are solutions of the Schrödinger equation $$[\hat{H}_0+V|{\cal L}\rangle\langle {\cal R}|]|\alpha\rangle=E_{\alpha}|\alpha\rangle \,,
\label{SchrSep}$$ where $E_{\alpha}$ are the corresponding eigenenergies. Let us expand the eigenstate $|\alpha\rangle$ over the eigenstates $|\vec{n}\rangle$ (of energy $E_{\vec{n}}$) of the non-interacting hamiltonian $\hat{H}_0$. The Schrödinger equation leads to the following equations for the expansion coefficients $$(E_{\alpha}-E_{\vec{n}})\langle \vec{n}|\alpha\rangle =
\langle \vec{n} |{\cal L} \rangle\langle {\cal R}|\alpha\rangle
\propto \langle \vec{n} |{\cal L} \rangle ,$$ where the omitted factor in the second equality is independent of $\vec{n}$.
As a result, any eigenfunction $|\alpha\rangle$ functionally coincides with the Green function of the non-interacting Hamiltonian taken at the energy of the eigenstate $E_{\alpha}$, $$|\alpha\rangle \propto \sum\limits_{\vec{n}}
{|\vec{n}\rangle \langle \vec{n} |{\cal L} \rangle
\over E_{\alpha}-E_{\vec{n}}}.
%
\label{psiGreen}
%$$ The omitted factor is determined by the normalization conditions.
Substitution of this expression to the Schrödinger equation (\[SchrSep\]) leads to the following eigenenergy equation: $$%
%
\sum\limits_{\vec{n}}
{\langle {\cal R}|\vec{n}\rangle \langle \vec{n} |{\cal L} \rangle
\over E_{\alpha}-E_{\vec{n}}} =\frac{1}{V}.
%
\label{EnerSep}$$ Similar expressions were obtained in Refs.[@seba1990; @albeverio1991] for the case of the Šeba billiard and its generalizations.
In the case of the relative motion of two short-range-interacting atoms in a circular, transversely harmonic waveguide, the derivation uses ideas of the analogous model involving an infinite waveguide [@olshanii1998; @moore2004; @yurovsky2008b]. The unperturbed Hamiltonian here is given by Eq. (1) in [@letter] and the Fermi-Huang interaction (see Eq. (2) in [@letter]) can be written in the separable form [@albeverio2000] with the interaction strength and formfactors given by $$V={2\pi\hbar^2 a_{s}\over \mu},
\quad
|{\cal L}\rangle=\delta_{3}({\bm r}),
\quad
\langle {\cal R}|=\delta_{3}({\bm r}) {\partial\over\partial r}(r\,\cdot ).$$ In what follows, we will restrict the Hilbert space to the states of zero $z$-component of the angular momentum and even under the $z \leftrightarrow -z$ reflection; the interaction has no effect on the rest of the Hilbert space. The non-interacting eigenstates are products of the transverse and longitudial wavefunctions. The transverse two-dimensional zero-angular-momentum harmonic wavefunctions, $$\langle \rho |n\rangle ={1\over \sqrt{\pi }a{ } _{\perp }}L^{\left(
0\right) }_{n}\left( \xi \right) \exp\left( -\xi /2\right) ,
\label{wf_trans}$$ labeled by the quantum number $n \geq 0$, are expressed in terms of the Legendre polynomials, $ L^{\left(
0\right) }_{n}\left( \xi \right)$, where $\xi =\left( \rho
/a_{\perp }\right) ^{2}$ and $a_{\perp }=\left( \hbar/\mu \omega _{\perp }\right) ^{1/2}$ is the transverse oscillator range. The longitudial wavefunctions, labeled by $l\geq 0$, are the symmetric plane waves satisfying periodic conditions with the period $L$: $$\langle z|k\rangle =\left( 2/L\right) ^{1/2}\cos 2\pi k\zeta ,\qquad
\langle z|0\rangle =L^{-1/2} ,
\label{wf_axial}$$ where $\zeta \equiv z/L-1/2$. The unperturbed spectrum is therefore given by $$%%
E_{nl} = 2 \hbar \omega_{\perp} n + \hbar^2 (2\pi l/L)^2/(2\mu)
\label{E0_2b}
%%$$ Substituting the above non-interacting eigenstates and eigenenergies to Eq. (\[psiGreen\]) and using Eq. (1.445.8) in [@gradshteyn] for summation over $l$, one obtains Eq. (3) in [@letter] for the interacting eigenstates. The normalization factor $C_{\alpha} $ is determined by the condition $$2\pi\int\limits_0^L dz\int\limits_0^{\infty} \rho^2 d\rho \langle \alpha'|\rho,z\rangle
\langle \rho,z|\alpha \rangle =\delta_{\alpha\alpha'}.$$ Using orthogonality of the Legendre polynomials it can be expressed as $$\begin{aligned}
C_{\alpha} ={2\over a_{\perp }\sqrt{\pi L}}
\biggl\lbrack \sum\limits^{\infty }_{n=0}\biggl
({\cot\sqrt{\epsilon_{\alpha} -\lambda n}\over \left( \epsilon_{\alpha} -\lambda
n\right) { } ^{3/2}} \nonumber
\\
+{1\over \left( \epsilon_{\alpha} -\lambda n\right) \sin^{2}\sqrt{\epsilon_{\alpha}
-\lambda n}}\biggr)\biggr\rbrack ^{-1/2} , \label{Ceps}\end{aligned}$$ where the rescaled energy $\epsilon_{\alpha}$ is given by $\epsilon_{\alpha}\equiv\lambda E_{\alpha}/\left(
2\hbar\omega_{\perp }\right)$ and $\lambda \equiv\left( L
/a_{\perp }\right) ^{2}$ is the aspect ratio.
For the Fermi-Huang interaction the eigenenergy equation (\[EnerSep\]) attains the form $${\partial \over \partial r}\left [ r\sum\limits^{}_{n,l}{\langle \rho
|n\rangle \langle z|l\rangle \langle n|0\rangle \langle 0|0\rangle
\over E-E_{nl}}\right ]_{r=0}={1\over V}$$ In the limit $r\to 0$, the eigenstate is spherically symmetric. This allows us to deal with the $z$-axis only (see [@moore2004]). Substitution of the noninteracting wavefunctions (\[wf\_trans\]) and (\[wf\_axial\]) and energies (\[E0\_2b\]) with the subsequent summation over $l$ leads to $$\sqrt{\lambda}{\partial \over \partial z}\left [ z \sum\limits^{\infty
}_{n=0}{\cos\left( 2\sqrt{\epsilon _{\alpha}-\lambda n}(z/L-1/2) \right)
\over \sqrt{\epsilon _{\alpha}-\lambda n}\sin\sqrt{\epsilon _{\alpha}-\lambda
n}}\right ]_{z=0}={a_s \over a_{\perp}}.$$ This sum contains both the regular part and the irregular one, the latter being proportional to $z^{-1}$. The regular part can be extracted using the identity $$\begin{aligned}
\lim_{z\rightarrow 0}\left\lbrack \sum\limits^{\infty }_{n=0}
(\lambda n-\epsilon)^{-1/2}\exp(-2\sqrt{\lambda n-\epsilon}z/L)-
{L\over \lambda z}\right \rbrack \nonumber
\\
=-{1\over \sqrt{\lambda }}\zeta \left( {1\over 2},-{\epsilon\over \lambda}\right)\end{aligned}$$ (see [@moore2004]) where $\zeta \left( \nu ,\alpha \right) $ is the Hurwitz zeta function (see, e. g., [@gradshteyn; @yurovsky2008b]. Finally we arrive at the transcendental equation for the eigenenergies (4) in [@letter]. The summands in the sums Eqs. (3) and (4) in [@letter] and in Eq. (\[Ceps\]) decay exponentially with $n$, leading to the fast converging series. Note that the imaginary parts of the two terms in the left hand side of Eq. (4) in [@letter] cancel each other automatically.
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---
abstract: 'We investigate a semigroup construction related to the two-sided wreath product. It encompasses a range of known constructions and gives a slightly finer version of the decomposition in the Krohn-Rhodes Theorem, in which the three-element flip-flop is replaced by the two-element semilattice. We develop foundations of the theory of our construction, showing in the process that it naturally combines ideas from semigroup theory (wreath products), category theory (Grothendieck construction), and ordered structures (residuated lattices).'
author:
- 'Michal Botur, Tomasz Kowalski'
title: 'Two-sided wreath product done right'
---
Introduction {#intro}
============
The purpose of this article is to introduce and investigate a certain semigroup construction which encompasses a range of known constructions including transformation monoids, semigroup actions, and wreath products. We chose the cheeky (but also tongue-in-cheek) title because our construction is inspired by the standard way of presenting the wreath product, say, of groups, as a direct power $G^X$ together with a group $K$ acting on $X$, that is, a set of bijective maps $X\to X$, indexed by elements of $K$. For semigroups, the restriction to bijections seems artificial: after all, semigroups are representable as semigroups of arbitrary maps. And if the maps do not have to be surjective, there seems to be no reason for having the same set of coordinates for every element of $K$.
A rudimentary construction of this type has been used in [@JM06] to settle some questions about *generalised BL-algebras*, which are a subclass of certain special lattice-ordered monoids known as *residuated lattices*. For the purposes of this article, familiarity with residuated lattices is not necessary, but the interested reader is referred to [@JT02] for a very readable albeit slightly old survey.
The construction was expanded and investigated in [@DK14], under the name of *kites*, still in the context of residuated lattices. A series of applications and further generalisations followed, see, e.g., [@DH14] and [@BD15]. A modification of the kite construction (to be precise, a subsemigroup of a kite) was put to a good use in [@BKLT16]. All these, however, stayed within the area of ordered structures, and the interaction of multiplication with order was the main focus. It was clear from the beginning that the kite construction is closely related to wreath products of ordered structures, for example, from [@JT04], or for a more specific case of lattice-ordered groups, from [@HMC69]. Considering order, however, seems to have obscured the properties of the multiplicative structure to some extent.
Here we depart from order (in the content, not in the organisation) and investigate only the multiplicative structure. As an application, we will show that the decomposition of finite semigroups in the celebrated Krohn-Rhodes Theorem (originally in [@KR62], see also [@RS09]) can be given a slightly finer form, namely, the flip-flop monoids can be replaced by two-element semilattices.
Notation
--------
We use the category-theoretic notation for composition of maps, that is, for maps $f\colon A\longrightarrow B$ and $g\colon B\longrightarrow C$ we denote their composition by $g\circ f\colon A\longrightarrow C$, so that $(g\circ f)(a) = g(f(a))$ for all $a\in A$. The set of all maps from the $A$ to $B$ we denote by the usual $B^A$. For a map $f\colon A\longrightarrow B$ and a set $I$ we write $f^I\colon A^I\to B^I$ for the map defined by $f^I(x)(i)=f(x(i))$. The following easy proposition will be used repeatedly without further ado.
\[P1\] Let $\mathbf{G} = (G;\cdot)$ be a groupoid, and let $I$, $J$ be sets. Then for all $x,y\in G^I$ and any $f\in I^J$ the following equality holds $$(x\circ f)\cdot (y\circ f)= (x\cdot y)\circ f.$$
We will frequently use systems of parameterised maps. In order to distinguish easily between parameters and arguments, we will put the parameters in square brackets, so $f[a,b](x)$ will denote the value of a map $f[a,b]$ on the argument $x$.
We will also frequently pass between algebras (semigroups), categories, and other types of structures (systems of maps), typically related to one another. To help distinguishing between them, we will use different fonts. Typically, boldface will be used for algebras (and italics for their universes), sans serif will be used for categories, and script for other types of structures. A few exceptions to these rules will be natural enough not to cause confusion.
The main construction
---------------------
Let $\mathbf{S} = (S,\cdot)$ be a semigroup, and let $(I[s])_{s\in S}$ be an indexed system of sets. For any $(a,b)\in S\times S$, let $\lambda[a,b]\colon I[ab]\to I[a]$ and $\rho[a,b]\colon I[ab]\to I[b]$ be maps satisfying the following conditions
1. $\lambda[a,b]\circ\lambda[ab,c] = \lambda[a,bc]$
2. $\rho[b,c]\circ\rho[a,bc] = \rho[ab,c]$
3. $\rho[a,b]\circ\lambda[ab,c] = \lambda[b,c]\circ\rho[a,bc]$
which make the diagram in Figure \[l-r-system\] commute.
(I\_abc) at (0,0) [$I[abc]$]{}; (I\_a) at (-4,0) [$I[a]$]{}; (I\_c) at (4,0) [$I[c]$]{}; (I\_ab) at (-2,-2) [$I[ab]$]{}; (I\_bc) at (2,-2) [$I[bc]$]{}; (I\_b) at (0,-4) [$I[b]$]{}; (I\_abc) to node\[swap\] [$\lambda[a,bc]$]{} (I\_a); (I\_abc) to node [$\rho[ab,c]$]{} (I\_c); (I\_abc) to node\[swap\] [$\lambda[ab,c]$]{} (I\_ab); (I\_abc) to node [$\rho[a,bc]$]{} (I\_bc); (I\_ab) to node [$\lambda[a,b]$]{} (I\_a); (I\_ab) to node\[swap\] [$\rho[a,b]$]{} (I\_b); (I\_bc) to node\[swap\] [$\rho[b,c]$]{} (I\_c); (I\_bc) to node [$\lambda[b,c]$]{} (I\_b);
Any triple $(\mathbf{I},{\boldsymbol{\lambda}},{\boldsymbol{\rho}})$ of systems of sets and maps satisfying the above conditions will be called a *$\lambda\rho$-system over* $\mathbf{S}$. A *(general) $\lambda\rho$-system* is then a quadruple $(\mathbf{S},\mathbf{I},{\boldsymbol{\lambda}},{\boldsymbol{\rho}})$, where $\mathbf{S}$ is a semigroup and $(\mathbf{I},{\boldsymbol{\lambda}},{\boldsymbol{\rho}})$ is a $\lambda\rho$-system over $\mathbf{S}$. We will typically use script letters to refer to $\lambda\rho$-systems, together with the convention that a $\lambda\rho$-system over a semigroup will be referred to by the script variant of the letter naming the semigroup. Thus, a $\lambda\rho$-system over $\mathbf{S}$ will be generally called $\mathcal{S}$; subscripts, and occasionally other devices, will be used to distinguish between different $\lambda\rho$-systems over the same semigroup. Where convenient, we will also use a more explicit notation $$\bigl(\langle \lambda[a,b],\rho[a,b]\rangle\colon
I[ab]\longrightarrow I[a]\times I[b]\bigr)_{(a,b)\in S^2}$$ for a $\lambda\rho$-system over a semigroup $\mathbf{S}$.
\[rl-prod\] Let $\mathbf{S} = (S;\cdot)$ be a semigroup and let $\mathcal{S} = (\mathbf{I},{\boldsymbol{\lambda}},{\boldsymbol{\rho}})$ be a $\lambda\rho$-system over $\mathbf{S}$. Let $\mathbf{H}$ be a semigroup. Then, we define a groupoid $\mathbf{H}^{[\mathcal{S}]} = (H ^{[\mathcal{S}]};\star)$, by putting
- $H^{[\mathcal{S}]} = \biguplus_{a\in S} H^{I[a]} = \{(x,a)\colon a\in S,\
x\in H^{I[a]}\}$, and
- $(x,a)\star(y,b) = \bigl((x\circ\lambda[a,b])\cdot(y\circ\rho[a,b]),ab\bigr)$.
We will call $\mathbf{H}^{[\mathcal{S}]}$ a *$\lambda\rho$-product*. As the name suggests, $\lambda\rho$-products are closely related to wreath products. We will explore their relationship more closely in Section \[wreath\].
For any $\lambda\rho$-system $\mathcal{S}$ over a semigroup $\mathbf{S}$, we will call $\mathbf{S}$ the *skeleton* of $\mathcal{S}$. We will extend this terminology to $\lambda\rho$-products, that is, for any semigroup $\mathbf{H}$, we will also call $\mathbf{S}$ the skeleton of $\mathbf{H}^{[\mathcal{S}]}$.
\[main\] Let $\mathbf{S} = (S;\cdot)$ be a semigroup and let $$\mathcal{S} =
\bigl(\langle\lambda[a,b],\rho[a,b]\rangle\colon
I[ab]\to I[a]\times I[b]\bigr)_{(a,b)\in S^2}$$ be a system of sets and maps indexed by the elements of $S$. Then, the following are equivalent.
1. $\mathbf{H}^{[\mathcal{S}]}$ is a semigroup, for any semigroup $\mathbf{H}$.
2. $\bigl(\langle\lambda[a,b],\rho[a,b]\rangle\colon
I[ab]\to I[a]\times I[b]\bigr)_{(a,b)\in S^2}$ is a $\lambda\rho$-system over $\mathbf{S}$.
First, note that associativity of the operation $\star$ is equivalent to the statement that the equality $$\label{assoc-eq}\tag{\dag}
\begin{split}
&\bigl((x\circ\lambda[a,b]\circ\lambda[ab,c])
\cdot(y\circ\rho[a,b]\circ\lambda[ab,c])
\cdot(z\circ\rho[ab,c]),\ abc\bigr) = \\
&\bigl((x\circ\lambda[a,bc])
\cdot(y\circ\lambda[b,c]\circ\rho[a,bc])
\cdot(z\circ\rho[a,b]\circ\rho[a,bc]),\ abc\bigr)
\end{split}$$ holds for arbitrary $(x,a), (y,b), (z,c)\in H^{[\mathcal{S}]}$. To see it, we carry out the following straightforward calculation: $$\begin{aligned}
\bigl((x,a)\star(y,b)\bigr)&\star(z,c) =
\bigl((x\circ\lambda[a,b])\cdot(y\circ\rho[a,b]),\ ab\bigr)\star (z,c) \\
&= \biggl(\Bigl(\bigl((x\circ\lambda[a,b])\cdot(y\circ\rho[a,b])\bigr)
\circ\lambda[ab,c]\Bigr)
\cdot\bigl(z\circ\rho[ab,c]\bigr),\ abc\biggr) \\
&= \bigl((x\circ\lambda[a,b]\circ\lambda[ab,c])
\cdot(y\circ\rho[a,b]\circ\lambda[ab,c])
\cdot(z\circ\rho[ab,c]),\ abc\bigr) \\
&= \bigl((x\circ\lambda[a,bc])
\cdot(y\circ\lambda[b,c]\circ\rho[a,bc])
\cdot(z\circ\rho[b,c]\circ\rho[a,bc]),\ abc\bigr)\\
&= \biggl(\bigl(x\circ\lambda[a,bc]\bigr)
\cdot\Bigl(\bigl((y\circ\lambda[b,c])
\cdot(z\circ\rho[b,c])\bigr)\circ\rho[a,bc]\Bigr),\ abc\biggr) \\
&= (x,a)\star
\bigl((y\circ\lambda[b,c])\cdot(z\circ\rho[b,c]),\ bc\bigr)\\
&= (x,a)\star\bigl((y,b)\star(z,c)\bigr) \end{aligned}$$ where the only non-definitional equality is precisely (\[assoc-eq\]). Now, if $\mathcal{S}$ is a $\lambda\rho$-system, then (\[assoc-eq\]) follows immediately from the equations ($\alpha$), ($\beta$) and ($\gamma$). This proves that (2) implies (1). The converse is clear.
Note that, in general, neither $\mathbf{S}$ nor $\mathbf{H}$ is a subsemigroup of $\mathbf{H}^{[\mathcal{S}]}$. However, it is not difficult to show that if either of them is a monoid then the other one is a subsemigroup of $\mathbf{H}^{[\mathcal{S}]}$.
\[skel\] Let $\mathbf{S}$ be a semigroup, and let $\mathbf{1}$ be the trivial semigroup. Then, for any $\lambda\rho$-system $\mathcal{S}$ over $\mathbf{S}$ we have $\mathbf{1}^{[\mathcal{S}]}\cong \mathbf{S}$. Indeed, $\mathbf{1}^I\cong
\mathbf{1}$ for any $I$, so $\mathbf{1}^{[\mathcal{S}]} = \bigl(\{(1,s)\colon s\in S\},\star\bigr)$, with $(1,a)\star(1,b) = (1,ab)$.
The same effect can be achieved in a more fanciful way.
\[empty\] Let $\mathbf{S}$ be a semigroup, and let $I[s] = \emptyset$ for each $s\in S$. Then, $\mathcal{S} = (\mathbf{I},{\boldsymbol{\lambda}},{\boldsymbol{\rho}})$, where $\lambda[a,b]$, $\rho[a,b]$ are empty functions for each $(a,b)\in S^2$, is a $\lambda\rho$-system over $\mathbf{S}$. For any semigroup $\mathbf{H}$ we then have that $H^{I[s]}$ is a singleton for each $s\in S$ its only element is the empty map $\emptyset\colon \emptyset\to H$. Moreover, $(\emptyset,a)\star(\emptyset,b) = (\emptyset, ab)$, for any $a,b\in S$, and thus $\mathbf{H}^{[\mathcal{S}]}\cong \mathbf{S}$.
In either of these ways, every semigroup $\mathbf{S}$ is isomorphic to a $\lambda\rho$-product whose skeleton is $\mathbf{S}$. One can ask how much freedom there is for making some, but not necessarily all, sets $I[s]$ empty. The answer is due to Dominik Lachman [@Lachman].
Let $\mathcal{S} = (\mathbf{I},{\boldsymbol{\lambda}},{\boldsymbol{\rho}})$ be a $\lambda\rho$-system over a semigroup $\mathbf{S}$. Let $J = \{s\in S\colon I[s] = \emptyset\}$. If $J$ is nonempty, then $J$ is a two-sided ideal of $\mathbf{S}$.
\[prod\] Let $\mathbf{S}$ be a semigroup, and let $I[s] = \{1\}$ for each $s\in S$. Then, $\mathcal{S} = (\mathbf{I},{\boldsymbol{\lambda}},{\boldsymbol{\rho}})$, where $\lambda[a,b]$, $\rho[a,b]$ are constant functions for each $(a,b)\in S^2$, is a $\lambda\rho$-system over $\mathbf{S}$. Then, $H^{I[s]}$ is a copy of $H$, for any semigroup $\mathbf{H}$. Moreover, for any $a,b\in S$ and $x,y\in H$, we have $(x,a)\star(y,b) = (xy, ab)$, and thus $\mathbf{H}^{[\mathcal{S}]}\cong \mathbf{H}\times\mathbf{S}$.
\[lzero\] Let $\mathbf{1}$ be the trivial semigroup, and let $I = \{0,1\}$. Next, let $\lambda\colon I\to I$ be the identity map, and let $\rho\colon I\to I$ be the constant map $\overline{0}$. This defines a $\lambda\rho$-system $\mathcal{I}$ over $\mathbf{1}$. Consider $\mathbb{Z}_2^{[\mathcal{I}]}$, whose universe $\mathbb{Z}_2^I$ we will identify in the obvious way with the set $\{00,01,10,11\}$. Here is the multiplication table of $\mathbb{Z}_2^{[\mathcal{I}]}$: $$
$\star$ $00$ $11$ $01$ $10$
--------- ------ ------ ------ ------
$00$ $00$ $11$ $00$ $11$
$11$ $11$ $00$ $11$ $00$
$01$ $01$ $10$ $01$ $10$
$10$ $10$ $01$ $10$ $01$
$$ Partitioning the universe into $\{00,11\}$ and $\{01,10\}$, we obtain a congruence $\theta$, such that $\mathbb{Z}_2^{[\mathcal{I}]}/\theta$ is isomorphic to the two-element left-zero semigroup.
\[flip-flop\] Let $\mathbf{2}=(\{0,1\},\vee)$ be the two-element join-semilattice, and let $\mathcal{Z}$ be the $\lambda\rho$-system over $\mathbf{2}$, defined by putting
1. $I[0]=\{0\},$ $I[1]=\{0,1\}$,
2. $\lambda[1,0]=\rho[0,1]=\lambda[1,1]=id_{I[1]}$ and $\rho[1,1] = \overline{0}$.
This defines a unique $\lambda\rho$-system, since all the remaining maps all have $\{0\}$ as the range. It is easy to show that the semigroup $\mathbb Z_2^{[\mathcal{Z}]}$ is the following: $$\begin{array}{c|cccccc}
\star &0&1&00&11&01&10\\
\hline
0&0&1&00&11&01&10\\
1&1&0&11&00&10&01\\
00&00&11&00&11&00&11\\
11&11&00&11&00&11&00\\
01&01&10&01&10&01&10\\
10&10&01&10&01&10&01
\end{array}$$ Partitioning the universe into $\{0,1\}$, $\{00,11\}$ and $\{01,10\}$ we obtain a congruence $\theta$, such that $\mathbb Z_2^{[\mathcal{Z}]}/\theta$ is isomorphic to the left flip-flop monoid.
In the commonly used terminology, Examples \[lzero\] and \[flip-flop\] show, respectively, that the two-element left-zero semigroup strongly divides a $\lambda\rho$-product of $\mathbb{Z}_2$ over the trivial semigroup, and the three-element left flip-flop monoid strongly divides a $\lambda\rho$-product of $\mathbb{Z}_2$ over a two-element semilattice. Thus, the flip-flop monoid turns out to be decomposable, in this sense. It will be shown in Section \[wreath\] that wreath product is also a special case of $\lambda\rho$-product. The next two last examples pave the way to wreath products.
\[sgrp-act\] Let a semigroup $\mathbf{S}$ act on a set $X$ on the left. The system of maps $$\bigl(\langle \lambda[a,b],\rho[a,b]\rangle\colon
I[ab]\to I[a]\times I[b]\bigr),$$ where $I[s]=X$ for any $s\in S$, and
1. $\lambda[a,b] = {id}_X$ for any $a,b\in S$,
2. $\rho[a,b] = {\underline{\hspace{1ex}}}\cdot a$ for all $a,b\in S$.
is a $\lambda\rho$-system over $\mathbf{S}$. An analogous $\lambda\rho$-system is induced by $\mathbf{S}$ acting on the right.
Recall that a two-sided action of a semigroup $\mathbf{S}$ on a set $X$ is a pair of maps $\ld\colon S\times X\longrightarrow X$ and $\rd\colon X\times S\longrightarrow X$, satisfying $$a\ld (b\ld x) = (b\cdot a)\ld x\qquad (x\rd a)\rd b = x\rd (b\cdot a)\qquad
(a\ld x)\rd b=a\ld (x\rd b)$$ for any $a,b\in S$ and $x\in X$. The slash notation is not the commonest, but we use it because of the connection with residuation, to come in Example \[rl-example\].
\[sgrp-act-two-sided\] Let $(X,\ld,\rd,\mathbf{S})$ consist of a set $X$ together with a two-sided action of a semigroup $\mathbf{S}$ on $X$. Then the system of maps $$\bigl(\langle \lambda[a,b],\rho[a,b]\rangle\colon
I[ab]\to I[a]\times I[b]\bigr),$$ where $I[s]=X$ for any $s\in S$, and
1. $\lambda[a,b] = {\underline{\hspace{1ex}}}\rd b$ for any $a,b\in S$,
2. $\rho[a,b] = a\ld{\underline{\hspace{1ex}}}$ for all $a,b\in S$.
is a $\lambda\rho$-system over $\mathbf{S}$.
The next example comes from the theory of ordered structures, more precisely, residuated lattices. We present it mainly because it is the closest to the kite construction that motivated the present work. The reader unfamiliar with residuated lattices can safely skip the example.
\[rl-example\] Let $(L;\cdot,\ld,\rd,\wedge,\vee,e)$ be a residuated lattice. For each $x\in L$ we put $I[x] = \{a\in L\colon x\leq a\}$. Next, let $\lambda[a,b] = {\underline{\hspace{1ex}}}\rd b$ and $\rho[a,b] = a\ld{\underline{\hspace{1ex}}}$. Then $(\mathbf{I},{\boldsymbol{\lambda}},{\boldsymbol{\rho}})$ is a $\lambda\rho$-system over the skeleton $(L;\cdot)$. The same holds for the more general case of a partially ordered residuated semigroup $(L;\leq, \cdot,\ld,\rd)$.
The last example in this section is hardly more than a curiosity, but we find it quite illustrative. Let $\star$ be any semigroup operation on a two-element Boolean algebra $\mathbf{B}$, say, meet, join, projection, or addition modulo 2. Then, for any set $X$, on the one hand $\star$ is a pointwise operation in $\mathbf{B}^X$, but on the other hand, it has its *alter ego* in the powerset $2^X$, via characteristic functions. Here is an analogue of this for a $\lambda\rho$-system over $\mathbf{B}$.
Let $\mathcal{S} = (\mathbf{I},{\boldsymbol{\lambda}},{\boldsymbol{\rho}})$ be any $\lambda\rho$-system, with a skeleton $\mathbf{S}$. Let $\star$ be any semigroup operation of the two-element Boolean algebra $\mathbf{B}$. Then, $\mathbf{B}^{[\mathcal{S}]}$ is a semigroup whose universe is the disjoint union of $2^{I[x]}$ for the system $\{I[a]\colon a\in S\}$. The semigroup operation can be explicitly written as $$(U,a)\star (W,b) = \bigl(\lambda[a,b]^{-1}(U)\star\rho[a,b]^{-1}(W), ab\bigr)$$ where $U\subseteq I[a]$ and $W\subseteq I[b]$.
One may think of the preimages $\lambda[a,b]^{-1}(U)$ and $\rho[a,b]^{-1}(W)$ as shadows cast by $U$ and $W$ in a stack of Venn diagrams.
Categorical background {#cats}
======================
In this short section we use some categorical tools to show that general $\lambda\rho$-systems form a category in a very natural way. Of itself, it does not add anything new to the construction, it just provides a conceptualisation which will be useful at least once in Section \[simpli\], but we believe it may also prove useful in developing the theory further. Throughout this section $\mathsf{Cat}$ will stand for the category of all categories (with functors as arrows). For any category $\mathsf{C}$, we will write $\mathrm{obj}(\mathsf{C})$ for the class of objects of $\mathsf{C}$.
\[t-over-S\] Let $\mathcal{S} = (\mathbf{I},{\boldsymbol{\lambda}},{\boldsymbol{\rho}})$ and $\mathcal{S}'=(\mathbf{I}',{\boldsymbol{\lambda}}',{\boldsymbol{\rho}}')$ be $\lambda\rho$-systems over a semigroup $\mathbf S$. By a *slice transformation* $t$ from $\mathcal{S}$ to $\mathcal{S}'$ we mean a system of maps $t = (t[a]\colon I[a]\longrightarrow I'[a])_{a\in S}$ satisfying $\lambda'[ab]\circ t[ab] = t[a]\circ \lambda[a,b]$ and $\rho'[ab]\circ t[ab]= t[b]\circ \rho[a,b]$ for all $a,b\in S$, i.e., such that the diagrams below commute. $$\begin{tikzpicture}[>=stealth,auto]
\node (tab) at (0,0) {$I[ab]$};
\node (ab) at (3,0) {$I'[ab]$};
\node (ta) at (0,-2) {$I[a]$};
\node (a) at (3,-2) {$I'[a]$};
\draw[->] (tab) to node {$t[ab]$} (ab);
\draw[->] (tab) to node[swap] {$\lambda[a,b]$} (ta);
\draw[->] (ab) to node[swap] {$\lambda'[a,b]$} (a);
\draw[->] (ta) to node {$t[a]$} (a);
\end{tikzpicture}
\qquad
\begin{tikzpicture}[>=stealth,auto]
\node (tab) at (0,0) {$I[ab]$};
\node (ab) at (3,0) {$I'[ab]$};
\node (tb) at (0,-2) {$I[b]$};
\node (b) at (3,-2) {$I'[b]$};
\draw[->] (tab) to node {$t[ab]$} (ab);
\draw[->] (tab) to node[swap] {$\rho[a,b]$} (tb);
\draw[->] (ab) to node[swap] {$\rho'[a,b]$} (b);
\draw[->] (tb) to node {$t[b]$} (b);
\end{tikzpicture}$$
\[cat-lr-over-S\] Let $\mathbf{S}$ be a semigroup. We define $\bm{\lambda\rho}(\mathbf{S})$ to be the category whose objects are $\lambda\rho$-systems over a semigroup $\mathbf{S}$, and whose arrows are slice transformations.
It is clear that $\bm{\lambda\rho}(\mathbf{S})$ is a category: composition of slice transformations is a slice transformation and the identity arrow is a system of identity maps. In fact, it resembles a slice category—hence the terminology—but we will not dwell on that. Having defined the category of $\lambda\rho$-systems over a fixed semigroup, in the next step we will upgrade this definition to general $\lambda\rho$-systems over arbitrary semigroups. We will do it by means of Grothendieck construction, whose one version we will now recall.
Let $\mathsf{C}$ be an arbitrary category, and let $F\colon \mathsf{C}^{op}\to\mathsf{Cat}$ be a functor. Then, $\mathsf{\Gamma}(F)$ is the category defined as follows.
1. Objects of $\mathsf{\Gamma}(F)$ are pairs of $(A,X)$ such that $A\in \mathrm{obj}(\mathsf{C})$ and $B\in \mathrm{obj}(F(A))$.
2. Arrows between objects $(A_1,X_1),(A_2,X_2)\in
\mathrm{obj}(\mathsf{\Gamma}(F))$ are pairs $(f,g)$ such that $f\colon A_2\to A_1$ is an arrow in the category $\mathsf{C}$ and $g\colon F(f)(X_1)\to X_2$.
3. Having objects and arrows in $\mathsf{\Gamma}(F)$, given below: $$(A_1,X_1)\overset{(f_1,g_1)}\longrightarrow(A_2,X_2)
\overset{(f_2,g_2)}\longrightarrow(A_3,X_3)$$ the composition of arrows is defined by: $$(f_1,g_1)\circ(f_2,g_2)=(f_2\circ f_1, g_1\circ F(f_1)(g_2)).$$
To apply Grothendieck construction to $\lambda\rho$-systems, we first show the existence of a suitable contravariant functor from (the opposite of the category of) semigroups to categories.
Let $\mathsf{Sg}$ be the category of semigroups with homomorphisms. There exists a functor $\bm{\lambda\rho}\colon \mathsf{Sg}^{op}\to
\mathsf{Cat}$ such that $\mathbf S\mapsto \bm{\lambda\rho}(\mathbf S)$, and for each semigroup homomorphism $f\colon \mathbf S_1\to\mathbf S_2$ we have a functor $$\bm{\lambda\rho}(f)\colon \bm{\lambda\rho}(\mathbf S_2)\longrightarrow
\bm{\lambda\rho}(\mathbf S_1)$$ such that
1. If $\mathcal{S} = (\mathbf{I},{\boldsymbol{\lambda}},{\boldsymbol{\rho}})\in
\bm{\lambda\rho}(\mathbf S_2)$, then $$\bm{\lambda\rho}(f)(\mathcal{S}) =
(\bm{\lambda\rho}(f)\mathbf{I},\bm{\lambda\rho}(f){\boldsymbol{\lambda}},
\bm{\lambda\rho}(f){\boldsymbol{\rho}})$$ where $$\begin{aligned}
\bm{\lambda\rho}(f)\mathbf{I} &=& \bigl(I[f(x)]\bigr)_{x\in S_1},\\
\bm{\lambda\rho}(f){\boldsymbol{\lambda}}&=&
\bigl(\lambda[f(x),f(y)]\colon
I[f(xy)]\to I[f(x)]\bigr)_{(x,y)\in S_1\times S_1},\\
\bm{\lambda\rho}(f){\boldsymbol{\rho}}&=&
\bigl(\rho[f(x),f(y)]\colon
I[f(xy)]\to I[f(y)]\bigr)_{(x,y)\in S_1\times S_1}.\end{aligned}$$
2. For any $\lambda\rho$-systems $\mathcal{S} = (\mathbf{I},{\boldsymbol{\lambda}},{\boldsymbol{\rho}})$ and $\mathcal{S}' = (\mathbf{I}',{\boldsymbol{\lambda}}',{\boldsymbol{\rho}}')$ over a semigroup $\mathbf{S}_2$, and for any slice transformation $t\colon \mathcal{S}\to\mathcal{S}'$, such that $$t = \bigl(f[x]\colon I[x]\longrightarrow I'[x]\bigr)_{x\in S_2}$$ we have a slice transformation $\bm{\lambda\rho}(f)t\colon
\bm{\lambda\rho}(f)(\mathcal{S})\to \bm{\lambda\rho}(f)(\mathcal{S}')$ such that $$\bm{\lambda\rho}(f)t =
\bigl(t[f(x)]\colon I[f(x)]\longrightarrow I'[f(x)]\bigr)_{x\in S_1}.$$
The proof is a series of tedious but straightforward calculations, which we omit. A crucial point is that since $\bm{\lambda\rho}(f)$ acts contravariantly, $\bm{\lambda\rho}(f)\mathbf{I}$, $\bm{\lambda\rho}(f){\boldsymbol{\lambda}}$ and $\bm{\lambda\rho}(f){\boldsymbol{\rho}}$ are well defined. For the proofs that ($\alpha$), ($\beta$) and ($\gamma$) are satisfied, and that $\bm{\lambda\rho}(f)$ behaves properly on slice transformations, we only need the definitions, the fact that $f$ is a homomorphism, and a lot of paper.
Now we are ready to define the notion of a *transformation* between general $\lambda\rho$-systems. Our definition may look a little esoteric, but it is an appropriate notion of a morphism for general $\lambda\rho$-systems. Firstly, it is natural for an application of Grothedieck construction, and secondly, as we will show in the next section, it is functorial for $\lambda\rho$-products as well. Slice transformations, defined previously, are just a particular case of transformations.
\[general-t\] Let $\mathcal{S} = (\mathbf{S}, \mathbf{I}, \boldsymbol{\lambda},
\boldsymbol{\rho})$ and $\mathcal{S}' = (\mathbf{S}', \mathbf{I}',
\boldsymbol{\lambda}', \boldsymbol{\rho}')$ be general $\lambda\rho$-systems. Define $\mathbf{t}\colon \mathcal{S}'\to \mathcal{S}$ to be a pair $(t,h)$ consisting of a homomorphism $h\colon \mathbf{S}\to \mathbf{S}'$ and a system of maps $t = \bigl(t[a]\colon I'[{h(a)}] \to I[a]\bigr)_{a\in S}$, such that the diagrams below commute. $$\begin{tikzpicture}[>=stealth,auto]
\node (tab) at (0,0) {$I'[{h(a)h(b)}] = I'[{h(ab)}]$};
\node (ab) at (3,0) {$I[{ab}]$};
\node (ta) at (0,-2) {$I'[{h(a)}]$};
\node (a) at (3,-2) {$I[a]$};
\draw[->] (tab) to node {$t[ab]$} (ab);
\draw[->] (tab) to node[swap] {$\lambda'[{h(a),h(b)}]$} (ta);
\draw[->] (ab) to node[swap] {$\lambda[{a,b}]$} (a);
\draw[->] (ta) to node {$t[a]$} (a);
\end{tikzpicture}
\qquad
\begin{tikzpicture}[>=stealth,auto]
\node (tab) at (0,0) {$I'[{h(a)h(b)}] = I'[{h(ab)}]$};
\node (ab) at (3,0) {$I[{ab}]$};
\node (tb) at (0,-2) {$I'[{h(a)}]$};
\node (b) at (3,-2) {$I[b]$};
\draw[->] (tab) to node {$t[ab]$} (ab);
\draw[->] (tab) to node[swap] {$\rho'[{h(a),h(b)}]$} (tb);
\draw[->] (ab) to node[swap] {$\rho[{a,b}]$} (b);
\draw[->] (tb) to node {$t[b]$} (b);
\end{tikzpicture}$$ Any such pair $\mathbf{t} = (t,h)$ will be called a *transformation*.
Let $\mathcal{S} = (\mathbf{S}, \mathbf{I}, \boldsymbol{\lambda},
\boldsymbol{\rho})$ and $\mathcal{S}' = (\mathbf{S}', \mathbf{I}',
\boldsymbol{\lambda}', \boldsymbol{\rho}')$ be general $\lambda\rho$-systems. If $\mathbf{S} = \mathbf{S}'$, then for any transformation $\mathbf{t}\colon \mathcal{S}'\to\mathcal{S}$ with $\mathbf{t} = (t, id_S)$, we have that $t$ is a slice transformation.
\[subsystem\] Let $\mathcal{S} = (\mathbf{S}, \mathbf{I}, \boldsymbol{\lambda},
\boldsymbol{\rho})$ be a general $\lambda\rho$-system, and let $\mathbf{T}$ be a subsemigroup of $\mathbf{S}$. Let $\mathbf{I}|_T$, ${\boldsymbol{\lambda}}|_T$ and ${\boldsymbol{\rho}}|_T$ be the restrictions of $\mathbf{I}$, ${\boldsymbol{\lambda}}$ and ${\boldsymbol{\rho}}$ to $T$. Then $\mathcal{T} = (\mathbf{T}, \mathbf{I}|_T,{\boldsymbol{\lambda}}|_T,{\boldsymbol{\rho}}|_T)$ is a $\lambda\rho$-system over $\mathbf{T}$. Moreover, $\mathbf{t}\colon \mathcal{S}\to\mathcal{T}$, defined by taking $h\colon\mathbf{T}\to\mathbf{S}$ to be the identity embedding, together with the system $\bigl(t[a]\colon I[h(a)] \to I[a]\bigr)_{a\in T}$, where $h(a) = a$ and $t[a] = {id}_{I[a]}$ is obviously a transformation.
If $\mathcal{S}$ and $\mathcal{T}$ are $\lambda\rho$-systems related as in Example \[subsystem\], we will call $\mathcal{T}$ a *subsystem* of $\mathcal{S}$. We will sometimes write $\mathcal{S}|_T$ for a subsystem of $\mathcal{S}$ over a semigroup $\mathbf{T}\leq\mathbf{S}$.
\[cat-general-lr\] Applying Grothendieck construction with $\mathsf{C} = \mathsf{Sg}$ and $F = \bm{\lambda\rho}$, we obtain a category $\mathsf{\Gamma}(\bm{\lambda\rho})$ of general $\lambda\rho$-systems with transformations as arrows.
Simplifications {#simpli}
===============
If the semigroup $\mathbf{S}$ is in fact a monoid, any $\lambda\rho$-system constructed over $\mathbf{S}$ will contain a set $I[1]$, and maps $\lambda[a,1]$, $\lambda[1,a]$, $\rho[a,1]$, $\rho[1,a]$ for any $a\in S$. It is immediate from the defining equations ($\alpha$), ($\beta$) and ($\gamma$) that that the maps $\rho[1,a]$ and $\lambda[a,1]$ are commuting retractions, that is, they satisfy
- $\lambda[a,1]\circ\lambda[a,1] = \lambda[a,1]$
- $\rho[1,a]\circ\rho[1,a] = \rho[1,a]$
- $\lambda[a,1]\circ\rho[1,a] = \rho[1,a]\circ\lambda[a,1]$
for each $a\in S$. In fact, for monoids it is reasonable to require something stronger, but instead of stating it for this particular case, we will define a general preservation requirement, whose special case will apply to monoids.
\[P-preserving\] Let $P$ be a property of semigroups, and let $\mathcal{S} = (\mathbf{S},\mathbf{I},{\boldsymbol{\lambda}},{\boldsymbol{\rho}})$ be a $\lambda\rho$-system. We will say that $\mathcal{S}$ *preserves* $P$ or, *is $P$ preserving*, if for every $\mathbf{H}$, whenever $\mathbf{H}$ satisfies $P$, so does $\mathbf{H}^{[\mathcal{S}]}$.
Said concisely, $\mathcal{S}$ is $P$ preserving, if $\forall\mathbf{H}\colon P(\mathbf{H})\Rightarrow P(\mathbf{H}^{[\mathcal{S}]})$. If $P$ is the property of having a unit, then $\mathcal{S}$ is $P$ preserving (*unit-preserving*) if and only if $\mathbf{H}^{[\mathcal{S}]}$ is a monoid, for every monoid $\mathbf{H}$.
\[main-monoid\] Let $\mathcal{S} = (\mathbf{S},\mathbf{I},{\boldsymbol{\lambda}},{\boldsymbol{\rho}})$ be a $\lambda\rho$-system. The following are equivalent:
1. $\mathcal{S}$ is unit-preserving,
2. $\mathbf{S}$ is a monoid and the maps $\lambda[a,1]$ and $\rho[1,a]$ are identity maps on $I[a]$, for each $a\in S$.
Assume $\mathcal{S}$ is unit-preserving. Then, in particular, $\mathbf{1}^{[\mathcal{S}]}$ is a monoid, so since $\mathbf{1}^{[\mathcal{S}]}\cong \mathbf{S}$ (see Example \[skel\]), we get that $\mathbf{S}$ is a monoid. Next, consider $\mathbf{H}^{[\mathcal{S}]}$ for an arbitrary nontrivial monoid $\mathbf{H}$. By assumption, $\mathbf{H}^{[\mathcal{S}]}$ is a monoid, so let $(y,b)$ be its unit element. In particular, $(y,b)\star(x,1) = (x,1)$ for any $x\in H^{I[1]}$, which implies $b\cdot 1 = 1$, so $b = 1$. Next, taking $(1,a)$ for any $a\in S$ (where $1$ is the map from $I[a]$ to $H$ identically equal to $1$), we have $(y,1)\star(1,a) = (1,a)$, from which we get $$\begin{aligned}
(1,a) &= \bigl((y\circ\lambda[1,a])\cdot(1\circ\rho[1,a])\ ,a\bigr)\\
&= \bigl((y\circ\lambda[1,a])\cdot 1,\ a\bigr)\\
&= (y\circ\lambda[1,a],\ a).\end{aligned}$$ This implies that $y\circ\lambda[1,a]$ is also identically $1$. Thus, finally, we obtain $$\begin{aligned}
(x,a) &= (y,1)\star(1,a)\\
&= \bigl((y\circ\lambda[1,a])\cdot(x\circ\rho[1,a])\ ,a\bigr)\\
&= \bigl(1\cdot(x\circ\rho[1,a]),\ a\bigr)\\
&= (x\circ\rho[1,a],\ a),\end{aligned}$$ and therefore $x = x\circ\rho[1,a]$. Since this holds for an arbitrary $x$, $\rho[1,a] = {id}_{I[a]}$ as required. By symmetry, the same holds for $\lambda[a,1]$, finishing the proof of (1) $\Rightarrow$ (2).
For the converse, let $\mathbf{H}$ be any monoid. Since $\mathbf{S}$ is a monoid, the set $I[1]$ exists; since $\mathbf{H}$ is a monoid, the constant function $1$ belongs to $H^{I[1]}$. Then, we have $$\begin{aligned}
(1,1)\star (x,1) &= \bigl((1\circ\lambda[1,a])\cdot (x\circ\rho[1,a]),\ 1\cdot 1\bigr)\\
&= (1\cdot (x\circ {id}_{I[a]}),\ 1\bigr)\\
&= (x, 1)\end{aligned}$$ showing that $(1,1)\in H^{I[1]}$ is a left unit. A completely symmetric argument shows that it is a right unit as well.
If a $\lambda\rho$-system satisfies conditions of Theorem \[main-monoid\](2), we will call it *unital*. This piece of terminology is, strictly speaking, redundant, but we find it conceptually useful as a name for an intrinsic characterisation of being unit-preserving. Note that the $\lambda\rho$-system of Example \[lzero\] is not unital, but the one of Example \[flip-flop\] is.
We will now make a series of observations that will eventually lead to a simplification of $\lambda\rho$-systems to a singly indexed version, at a cost of some relatively mild assumptions.
Before we approach it, we state two lemmas, which are rather obvious but they make the transition between $\lambda\rho$-systems over semigroups and monoids smooth. As usual, we will write $\mathbf{S}^1$ for the semigroup $\mathbf{S}$ with the unit element $1$ adjoined. We adopt the convention that $1$ is always a new element, even if $\mathbf{S}$ already has a unit.
\[add-unit\] Let $\mathcal{S}$ be a $\lambda\rho$-system over a semigroup $\mathbf{S}$. Let $1$ be an element not in $S$. Define $\mathcal{S}^1$ to be the following system of maps $$\bigl(\langle\lambda[a,b],\rho[a,b]\rangle\colon
I[ab]\to I[a]\times I[b]\bigr)_{(a,b)\in S^1\times S^1}$$ where $I[1] = \{1\}$, the maps $\lambda[1,a]$, $\rho[a,1]$ are constant maps from $I[a]$ to $I[1]$, and the maps $\lambda[a,1]$, $\rho[1,a]$ are both equal to ${id}_{I[a]}$.
The $\lambda\rho$-system $\mathcal{S}^1$ defined above will be called the *unital extension* of $\mathcal{S}$. The terminology is justified by the next two lemmas, whose proofs are immediate. An illustration of their application is provided by Examples \[lzero\] and \[flip-flop\] in Section \[intro\].
\[unit-added\] Let $\mathcal{S}$ be a $\lambda\rho$-system over a semigroup $\mathbf{S}$. The system $\mathcal{S}^1$ defined above is a unital $\lambda\rho$-system over $\mathbf{S}^1$. Moreover, $\mathcal{S}$ is a subsystem of $\mathcal{S}^1$.
\[lr-prod-with-unit\] Let $\mathcal{S}$ and $\mathcal{S}^1$ be $\lambda\rho$-systems defined above. Let $\mathbf{H}$ be a semigroup. Then, $\mathbf{H}^{[\mathcal{S}]}$ is a subsemigroup of $\mathbf{H}^{[\mathcal{S}^1]}$.
For any unital $\lambda\rho$-system, in particular for $\mathcal{S}^1$, we can simplify the definitions of the systems of maps $\lambda$ and $\rho$ by removing the double indexing. First, defining $\lambda[a] = \lambda[1,a]$ and $\rho[a] = \rho[a,1]$, we split the diagram in Figure \[l-r-system\] into three parts, substituting $1$ for $a$ in the left part, for $b$ in the middle part, and for $c$ in the right part, as in Figure \[split-l-r-system\]. One should think about these diagrams as three separate copies of the diagram from Figure \[l-r-system\], in which certain inessential fragments were suppressed. For example, in the left part of the diagram the maps $\rho[1,b]\colon I[b]\to I[b]$ and $\rho[1,bc]\colon I[bc]\to I[bc]$ are omitted, and so is the map $\rho[b,c]\colon I[bc]\to I[c]$.
(I\_abc\_l) at (-1,0) [$I[bc]$]{}; (I\_a) at (-5,0) [$I[1]$]{}; (I\_ab\_l) at (-3,-2) [$I[b]$]{}; (I\_abc) at (0,-1) [$I[ac]$]{}; (I\_ab) at (-2,-3) [$I[a]$]{}; (I\_bc) at (2,-3) [$I[c]$]{}; (I\_b) at (0,-5) [$I[1]$]{}; (I\_abc\_r) at (1,0) [$I[ab]$]{}; (I\_c) at (5,0) [$I[1]$]{}; (I\_bc\_r) at (3,-2) [$I[b]$]{}; (I\_abc\_l) to node\[swap\] [$\lambda[bc]$]{} (I\_a); (I\_abc\_r) to node [$\rho[ab]$]{} (I\_c); (I\_abc) to node\[swap\] [$\lambda[a,c]$]{} (I\_ab); (I\_abc) to node [$\rho[a,c]$]{} (I\_bc); (I\_abc\_l) to node\[swap\] [$\lambda[b,c]$]{} (I\_ab\_l); (I\_abc\_r) to node [$\rho[a,b]$]{} (I\_bc\_r); (I\_ab\_l) to node [$\lambda[b]$]{} (I\_a); (I\_ab) to node\[swap\] [$\rho[a]$]{} (I\_b); (I\_bc\_r) to node\[swap\] [$\rho[b]$]{} (I\_c); (I\_bc) to node [$\lambda[c]$]{} (I\_b);
Next, renaming $a$, $b$ and $c$ as necessary, and putting the diagrams from Figure \[split-l-r-system\] together, we obtain the diagram of Figure \[pre-l-r-system\], where the dashed lines denote existence requirements, as usual, and where we write $I$ for $I[1]$.
(I\_abc) at (0,0) [$I[ab]$]{}; (I\_a) at (-4,0) [$I$]{}; (I\_c) at (4,0) [$I$]{}; (I\_ab) at (-2,-2) [$I[a]$]{}; (I\_bc) at (2,-2) [$I[b]$]{}; (I\_b) at (0,-4) [$I$]{}; (I\_abc) to node\[swap\] [$\lambda[ab]$]{} (I\_a); (I\_abc) to node [$\rho[ab]$]{} (I\_c); (I\_abc) to node\[swap\] [$\lambda[a,b]$]{} (I\_ab); (I\_abc) to node [$\rho[a,b]$]{} (I\_bc); (I\_ab) to node [$\lambda[a]$]{} (I\_a); (I\_ab) to node\[swap\] [$\rho[a]$]{} (I\_b); (I\_bc) to node\[swap\] [$\rho[b]$]{} (I\_c); (I\_bc) to node [$\lambda[b]$]{} (I\_b);
Now, let $\bigl(\langle\lambda[a],\rho[a]\rangle\colon I[a]\to I\bigr)_{a\in S^1}$ be a system of maps and sets (with $I = I[1]$) such that for every $(a,b)\in S^1\times S^1$ there exist maps $\lambda[a,b]$ and $\rho[a,b]$ making the required diagrams commute, that is, satisfying the obvious counterparts of the identities ($\alpha$), ($\beta$), and ($\gamma$), below.
1. $\lambda[a]\circ\lambda[a,b] = \lambda[ab]$
2. $\rho[b]\circ\rho[a,b] = \rho[ab]$
3. $\rho[a]\circ\lambda[a,b] = \lambda[b]\circ\rho[a,b]$
We will refer to any such system by a rather unimaginative name of *pre-$\lambda\rho$-system*. The rationale for the prefix ‘pre’ will be given shortly, but before that, let us make a few more observations. Given a pre-$\lambda\rho$-system, we can always define $P[ab] = \{(x,y)\in I[a]\times I[b]\colon \rho[a](x) = \lambda[b](y)\}$, and obtain a pullback diagram $$\begin{tikzpicture}[>=stealth,auto]
\node (P_ab) at (0,0) {$P[ab]$};
\node (I_ab) at (-3,0) {$I[ab]$};
\node (I_a) at (2,2) {$I[a]$};
\node (I_b) at (2,-2) {$I[b]$};
\node (I) at (4,0) {$I$};
\draw[->] (P_ab) to node {$\pi[a]$} (I_a);
\draw[->] (P_ab) to node[swap] {$\pi[b]$} (I_b);
\draw[->] (I_b) to node[swap] {$\lambda[b]$} (I);
\draw[->] (I_a) to node {$\rho[a]$} (I);
\draw[bend left,->] (I_ab) to node {$\rho[a,b]$} (I_a);
\draw[bend right,->] (I_ab) to node[swap] {$\lambda[a,b]$} (I_b);
\draw[->,dashed] (I_ab) to node[swap] {$f[{a,b}]$} (P_ab);
\end{tikzpicture}$$ where $\pi[a]$ and $\pi[b]$ are projections. Thus, from any pre-$\lambda\rho$-system we can obtain a ‘finest’ one by systematically replacing $I[ab]$ by $P[ab]$, and factoring the maps $\lambda$ and $\rho$ through. As this is not an essential issue, we will not go into details.
\[way-down\] Let $\mathbf{M}$ be a monoid, and let $$\mathcal{S} = \bigl(\langle\lambda[a,b],\rho[a,b]\rangle\colon
I[ab]\to I[a]\times I[b]\bigr)_{(a,b)\in M^2}$$ be a $\lambda\rho$-system. Then, $$\mathcal{P} = (\lambda[a],\rho[a]\colon I[a]\to I)_{a\in M}$$ where $\lambda[a] = \lambda[1,a]$ and $\rho[a] = \rho[a,1]$ is a pre-$\lambda\rho$-system.
Obvious.
Going the other way is not completely trivial, but it does work under certain natural conditions. Let $\mathcal{P} =
\bigl(\langle\lambda[a],\rho[a]\rangle\colon I[a]\to I\bigr)_{a\in M}$ be a pre-$\lambda\rho$-system over some monoid $\mathbf{M}$. We say that $\mathcal{P}$ *has natural solutions*, if the maps $\lambda[a,b]$, $\rho[a,b]$ satisfying the equations ($\alpha'$), ($\beta'$), and ($\gamma'$), also satisfy four cancellativity properties, namely
1. $\lambda[a]\circ\lambda[a,b]\circ\lambda[ab,c] =
\lambda[a]\circ\lambda[a,bc] \Rightarrow
\lambda[a,b]\circ\lambda[ab,c] = \lambda[a,bc]$,
2. $\rho[b]\circ \rho[b,c]\circ\rho[a,bc] =
\rho[b]\circ\rho[ab,c] \Rightarrow
\rho[b,c]\circ\rho[a,bc] = \rho[ab,c]$,
3. $\rho[b]\circ \rho[a,b]\circ\lambda[ab,c] =
\rho[b]\circ\lambda[b,c]\circ\rho[a,bc] \Rightarrow
\rho[a,b]\circ\lambda[ab,c] = \lambda[b,c]\circ\rho[a,bc]$,
4. $\lambda[b]\circ\rho[a,b]\circ\lambda[ab,c] =
\lambda[b]\circ\lambda[b,c]\circ\rho[a,bc] \Rightarrow
\rho[a,b]\circ\lambda[ab,c] = \lambda[b,c]\circ\rho[a,bc]$.
These conditions may look somewhat esoteric, but they are just ‘prefixed’ versions of ($\alpha$), ($\beta$) and ($\gamma$). Observe that the left-hand sides of ($\delta_i$) for $i=1,2,3,4$ hold in any $\lambda\rho$-system (with $\lambda[a] = \lambda[1,a]$ and $\rho[b] = \rho[b,1]$). Diagrammatically, they are presented in Figure \[ns-pre-l-r-system\], where all possible commutations are postulated. We are not sure what the diagram resembles more: a parachute or a jellyfish. But we promise we will not have any more complicated diagrams in the article.
Importantly, the conditions above hold in three rather important cases: (a) when $\lambda[a]$ and $\rho[a]$ are injective maps, (b) when $\lambda[a]$ and $\rho[a]$ are actions of a semigroup on itself by left and right multiplication, and (c) in the construction over a free monoid, which will be given in Subsection \[free-constr\].
(I\_abc) at (0,0) [$I[abc]$]{}; (I\_a) at (-4,-4) [$I[a]$]{}; (I\_c) at (4,-4) [$I[c]$]{}; (I\_ab) at (-2,-2) [$I[ab]$]{}; (I\_bc) at (2,-2) [$I[bc]$]{}; (I\_b) at (0,-4) [$I[b]$]{}; (I\_1) at (-6,-6) [$I$]{}; (I\_2) at (-2,-6) [$I$]{}; (I\_3) at (2,-6) [$I$]{}; (I\_4) at (6,-6) [$I$]{}; (I\_abc) to node\[swap,sloped\] [$\lambda[abc]$]{} (I\_1); (I\_abc) to node\[sloped\] [$\rho[abc]$]{} (I\_4); (I\_ab) to node\[swap,sloped\] [$\lambda[ab]$]{} (I\_1); (I\_bc) to node\[pos=0.7,swap,sloped\] [$\lambda[bc]$]{} (I\_2); (I\_ab) to node\[pos=0.7,sloped\] [$\rho[ab]$]{} (I\_3); (I\_bc) to node\[sloped\] [$\rho[bc]$]{} (I\_4); (I\_a) to node\[sloped\] [$\lambda[a]$]{} (I\_1); (I\_a) to node\[swap,sloped\] [$\rho[a]$]{} (I\_2); (I\_b) to node\[sloped\] [$\lambda[b]$]{} (I\_2); (I\_b) to node\[swap,sloped\] [$\rho[b]$]{} (I\_3); (I\_c) to node\[sloped\] [$\lambda[c]$]{} (I\_3); (I\_c) to node\[swap,sloped\] [$\rho[c]$]{} (I\_4); (I\_abc) to node\[pos=0.3,swap,sloped\] [$\lambda[a,bc]$]{} (I\_a); (I\_abc) to node\[pos=0.3,sloped\] [$\rho[ab,c]$]{} (I\_c); (I\_abc) to node\[pos=0.7,sloped\] [$\lambda[ab,c]$]{} (I\_ab); (I\_abc) to node\[swap,pos=0.7,sloped\] [$\rho[a,bc]$]{} (I\_bc); (I\_ab) to node\[pos=0.7,sloped\] [$\lambda[a,b]$]{} (I\_a); (I\_ab) to node\[pos=0.7,swap,sloped\] [$\rho[a,b]$]{} (I\_b); (I\_bc) to node\[pos=0.7,swap,sloped\] [$\rho[b,c]$]{} (I\_c); (I\_bc) to node\[pos=0.7,sloped\] [$\lambda[b,c]$]{} (I\_b);
\[way-up\] Let $\mathbf{M}$ be a monoid, and let $$\mathcal{P} = (\lambda[a],\rho[a]\colon I[a]\to I)_{a\in M}$$ be a pre-$\lambda\rho$-system. If $\mathcal{P}$ has natural solutions, then $$\mathcal{S} = \bigl(\langle\lambda[a,b],\rho[a,b]\rangle\colon
I[ab]\to I[a]\times I[b]\bigr)_{(a,b)\in M^2}$$ where $\lambda[a,b]$ and $\rho[a,b]$ are some solutions to the equations $\alpha'$, $\beta'$, and $\gamma'$, is a unital $\lambda\rho$-system.
To show that ($\alpha$) holds, first calculate, using ($\alpha'$) $$\begin{aligned}
\lambda[a]\circ\lambda[a,b]\circ\lambda[ab,c] &=
\lambda[ab]\circ\lambda[ab,c] \\
&= \lambda[abc]\\
&= \lambda[a]\circ\lambda[a,bc]\end{aligned}$$ and then use ($\delta_1$) to cancel $\lambda[a]$ and obtain $\lambda[a,b]\circ\lambda[ab,c] = \lambda[a,bc]$ as desired. By an analogous argument, ($\beta$) holds.
Now that ($\alpha$) and ($\beta$) have been shown to hold, we calculate, using ($\gamma'$) and ($\beta$) $$\begin{aligned}
\rho[b]\circ\rho[a,b]\circ\lambda[ab,c] &=
\rho[ab]\circ\lambda[ab,c] \\
&= \lambda[c]\circ\rho[ab,c]\\
&= \lambda[c]\circ\rho[b,c]\circ\rho[a,bc]\\
&= \rho[b]\circ\lambda[b,c]\circ\rho[a,bc]\end{aligned}$$ ant then use ($\delta_3$) to cancel $\rho[b]$ and obtain $\rho[a,b]\circ\lambda[ab,c] = \lambda[b,c]\circ\rho[ab,c]$ as desired.
Combining Lemma \[lr-prod-with-unit\] with Lemma \[way-up\] we obtain that any $\lambda\rho$-product is uniquely determined by a pre-$\lambda\rho$-system with natural solutions. Therefore, we can—and will—extend the notation $\mathbf{H}^{[\mathcal{S}]}$ to the situation where $\mathcal{S}$ is such a system.
A free construction {#free-constr}
-------------------
We will now show that that pre-$\lambda\rho$-systems with natural solutions, and thus $\lambda\rho$-systems, exist in abundance. Let $X^*$ be the free monoid, freely generated by some set $X$. For any $x\in X$ we let $I[x]$ be a set, and let $\lambda[x]\colon I[x]\to I$ and $\rho[x]\colon I[x]\to I$ be arbitrary maps. Then, we put $I[\varepsilon] = I$, and for each nonempty word $w = x_1x_2\cdots x_k\in X^*$, we define $I[{x_1x_2\cdots x_k}]$ to be the set of sequences $(v_1,v_2,\dots,v_k)\in I[{x_1}] \times \dots \times I[{x_k}]$ such that $$\begin{aligned}
\rho[{x_1}](v_1) &= \lambda[{x_2}](v_2) \\
\rho[{x_2}](v_2) &= \lambda[{x_3}](v_3) \\
& \vdots \\
\rho[{x_{k-1}}](v_{k-1}) &= \lambda[{x_k}](v_k). \end{aligned}$$ To continue the construction, another piece of notation will be handy. Let $w = x_1x_2\cdots x_k$ be a word over $X$, and let $u = x_mx_{m+1}\cdots x_j$ be a subword of $w$, such that $1\leq m\leq j\leq k$. Given a sequence $s = (v_{1},v_{2},\dots,v_{k})\in I[w]$, we write $s|_u$ for the truncated sequence $(v_{m},\dots,v_{j})$. It s clear that $s|_u$ belongs to $I[u]$.
Now, we proceed inductively. First, we put $\lambda[\varepsilon] =
\rho[\varepsilon] = id_I$. The maps $\lambda[z]$ and $\rho[z]$ for any $z\in X$ have already been defined. Let $w = w_1w_2$, with $w_1$ and $w_2$ nonempty. Assume the maps $\lambda[{w_i}]\colon I[{w_i}]\to I$ and $\rho[{w_i}]\colon I[{w_i}]\to I$ have been defined for $i\in \{1,2\}$. Define $\lambda[{w}]\colon I[{w}]\to I$ and $\rho[{w}]\colon I[{w}]\to I$ by putting $$\begin{aligned}
\lambda[w](s) &= \lambda[{w_1}](s|_{w_1})\\
\rho[w](s) &= \rho[{w_2}](s|_{w_2})\end{aligned}$$ for each $s\in I[w]$.
\[free-is-free\] Let $X^*$ be the free monoid generated by $X$, and for each element $w\in X^*$ let $I[w]$ be defined as above. Further, let $$\mathcal{F} =
\bigl(\langle\lambda[w],\rho[w]\rangle\colon I[w]\to I\bigr)_{w\in X^*}$$ be the system of maps defined as above. Then $\mathcal{F}$ is a pre-$\lambda\rho$-system with natural solutions.
For a word $w\in X^*$ we denote its length by $|w|$. We will first prove, by induction on $|w|$, that the system we have defined is a pre-$\lambda\rho$-system. The base case is $|w| = 2$. Then $w_1$ and $w_2$ are generators of $X^*$, say, $w_1 = a$ and $w_2 = b$. Take an arbitrary element $(v, u)\in I[{ab}]$. Then, by definition, $\lambda[{ab}](v,u) = \lambda[a](v)$ and $\rho[{ab}](v,u) = \rho[b](u)$. Thus, taking $\lambda[{a,b}]\colon I[{ab}]\to I[a]$ to be the first projection, and $\rho[{a,b}]\colon I[{ab}]\to I[b]$ to be the second projection, we have that the identities $(\alpha')$, $(\beta')$, $(\gamma')$ are satisfied, by the conditions imposed on $\lambda[a]$ and $\rho[b]$.
For the inductive step assume $\lambda[{w_1}]\colon I[{w_1}]\to I$ and $\rho[{w_2}]\colon I[{w_2}]\to I$ have been defined, and take an arbitrary element $s\in I[{w}] = I[{w_1w_2}]$. Without loss of generality, we can assume $w_1 = u_1a$ and $w_2 = bu_2$, where $a,b\in H$. Then, by construction of $I[w]$ we have that $\rho[{a}](s|_{a}) = \lambda[{b}](s|_{b})$. Now, $\rho[{w_1}] = \rho[{u_1a}]$, so, by inductive hypothesis, we obtain $\rho[{w_1}](s|_{w_1}) = \rho[{a}](s|_{a})$. Similarly, $\lambda[{w_2}] = \lambda[{bu_2}]$, so we get $\lambda[{w_2}](s|_{w_2}) = \lambda[{b}](s|_{b})$. Therefore, $\rho[{w_1}](s|_{w_1}) = \lambda[{w_2}](s|_{w_2})$. Then, taking $\lambda[{w_1,w_2}]\colon I[{w_1w_2}]\to I[{w_1}]$ to be the projection onto $I[{w_1}]$, and $\rho[{w_1,w_2}]\colon I[{w_1w_2}]\to I[{w_2}]$ to be the the projection on $I[{w_2}]$, we can see that $(\alpha')$, $(\beta')$, $(\gamma')$ are satisfied.
It remains to show that $\mathcal{F}$ has natural solutions. To show that we only need to provide maps $\lambda[{ab,c}]$ and $\rho[{a,bc}]$ which make all diagrams in Figure \[ns-pre-l-r-system\] commute. But these maps have been already given by our construction. Namely, taking $w_1 = ab$ and $w_2 = c$ we have that $\lambda[{ab,c}]$ is the first projection from $I[{abc}]\subseteq I[{ab}]\times I[c]$ to $I[{ab}]$. Similarly, $\rho[{a,bc}]$ is the second projection from $I[{abc}]\subseteq I[a]\times I[{bc}]$ to $I[{bc}]$. Formally, the argument should be again cast in the form of induction of the length of the word $w = abc$, but we are afraid it would then produce clutter rather than provide explanation.
The final part of this section will show that the free construction described above is indeed universal. Recall from Section \[cats\], Definition \[general-t\], the notion of a transformation between $\lambda\rho$-systems.
\[trans\] Let $\mathcal{S} = (\mathbf{S},{\boldsymbol{\lambda}},{\boldsymbol{\rho}})$ and $\mathcal{S}' = (\mathbf{S}',{\boldsymbol{\lambda}}',{\boldsymbol{\rho}}')$ be $\lambda\rho$-systems, and let $\mathbf{t} = (t,h)$ be a transformation, with $\mathbf{t}\colon \mathcal{S}'\to \mathcal{S}$. For an arbitrary semigroup $\mathbf{H}$, let $\mathbf{H}^\mathbf{t}\colon
\mathbf{H}^{[\mathcal{S}]} \to \mathbf{H}^{[\mathcal{S}']}$ be the map defined by $\mathbf{H}^\mathbf{t}(x,a) = (x\circ t[a],\ h(a))$ for every $(x,a)\in\biguplus_{a\in S} H^{I[a]}$. Then, $\mathbf{H}^\mathbf{t}$ is a homomorphism. Moreover, $\mathbf{H}^{-}$ is a contravariant functor from the category $\Gamma({\boldsymbol{\lambda}}{\boldsymbol{\rho}})$ to the category $\mathsf{Sg}$ of semigroups.
It is clear that the map $\mathbf{H}^\mathbf{t}$ is well defined. Let $(x,a), (y,b)\in \biguplus_{a\in S} H^{I[a]}$. Then, we have $$\begin{aligned}
\mathbf{H}^\mathbf{t}(x\star y,\ ab)
&= \bigl((x\star y)\circ t[ab],\ h(ab)\bigr)\\
&= \Bigl(\bigl((x\circ\lambda[{a,b}])(y\circ\rho[{a,b}])\bigr)\circ t[ab],\
h(a)h(b)\Bigr)\\
&= \Bigl((x\circ\lambda[{a,b}]\circ t[ab])(y\circ\rho[{a,b}]\circ t[ab]),\
h(a)h(b)\Bigr) \\
&= \Bigl(\bigl(x\circ t[a]\circ \lambda'[{h(a),h(b)}]\bigr)
\bigl(y\circ t[b]\circ\rho'[{h(a),h(b)}]\bigr),\ h(a)h(b)\Bigr)\\
&= \bigl(x\circ t[a],\ h(a)\bigr)\star\bigl(y\circ t[b],\ h(b)\bigr)\\
&= \mathbf{H}^\mathbf{t}(x,a)\star \mathbf{H}^\mathbf{t}(y,b). \end{aligned}$$ This proves that $\mathbf{H}^\mathbf{t}$ is a homomorphism. The proof of the moreover part is straightforward.
Consider $\lambda\rho$-system $\mathcal{S}$ over some semigroup $\mathbf{S}$. Taking $S$ as the set of free generators, we form the free monoid $S^*$. Then, $\mathbf{S}^1$ is a homomorphic image (in fact, a retract) of $S^*$ via the map extending the identity map on $S$. Let $\mathcal{S}^1$ be the $\lambda\rho$-system over $\mathbf{S}^1$, extending $\mathcal{S}$, as in Definition \[add-unit\]. Next, let $\mathcal{P}^1$ be the pre-$\lambda\rho$-system associated with $\mathcal{S}^1$ as in Lemma \[way-down\]. As $\mathcal{P}^1$ is just a restriction of $\mathcal{S}^1$, it has natural solutions, so by Lemma \[way-up\] it induces a unital $\lambda\rho$-system. We will denote that system by $\mathcal{F}(\mathcal{S}^1)$.
Since every element of $S^*$ is a word $s_1s_2\dots s_n$ over $S$, but on the other hand $s_1s_2\cdots s_n$ also represents a product of $s_1,\dots,s_n$ as an element of $S$, we need some notational device to distinguish the two. We will write $s_1s_2\dots s_n$ for the word, and $\otimes(s_1s_2\dots s_n)$ for the product. Thus, for example, $I[{\otimes(s_1s_2\dots s_n)}]$ will be a set from the original system $\mathcal{S}$, and $I[{s_1s_2\dots s_n}]$ will be a set from the system $\mathcal{F}(\mathcal{S}^1)$. Further, $I[\otimes\varepsilon] = I[1]$ is a singleton set from $\mathcal{S}^1$, and, by construction of $\mathcal{F}(\mathcal{S}^1)$, the same set as a member of $\mathcal{F}(\mathcal{S}^1)$ should be denoted by $I[\varepsilon]$. We will simply write $I$ in either case. Also recall that, by construction of $\mathcal{F}(\mathcal{S}^1)$, we have $I[{s_1s_2\dots s_n}]\subseteq I[{s_1}]\times I[{s_2}]\times\dots I[{s_n}]$.
\[free-t\] Let $\mathcal{S}$, $\mathcal{S}^1$ and $\mathcal{F}(\mathcal{S}^1)$ be as above. We define a system $\mathbf{t}$ of maps as follows. First, we let $t\colon S^*\to \mathbf{S}^1$ be the homomorphism extending the identity map on $S$, and such that $t(\varepsilon) = 1$. Next, for any $s_1,s_2,\dots, s_n \in S$, we define the map $$t[s_1s_2\dots s_n]\colon I[\otimes(s_1s_2\dots s_n)] \longrightarrow
I[{s_1}]\times I[{s_2}]\times\dots\times I[{s_n}]$$ for each $v\in I[\otimes(s_1s_2\cdots s_n)]$, by putting $t[s_1s_2\dots s_n](v) = \langle v_1,\dots, v_n\rangle$, where
- $v_1 = \lambda[s_1, \otimes(s_2s_3\cdots s_n)](v)$,
- $v_j = \rho[\otimes(s_1\cdots s_{j-1}), s_j]\circ
\lambda[\otimes(s_1\cdots s_j), \otimes(s_{j+1}\cdots s_n)](v)$,
- $v_n = \rho[\otimes(s_1\cdots s_{n-1}), s_n](v)$.
Finally, we let $t[\varepsilon]\colon I\to I$ be the (unique) constant map.
\[well-defd\] $\mathcal{S}$, $\mathcal{S}^1$ and $\mathcal{F}(\mathcal{S}^1)$ be as above. Then, the following hold:
1. For each $s\in S$, we have $t[s] = {id}_{I[s]}$.
2. For any $s_1,s_2,\dots, s_n \in S$, we have $$\begin{aligned}
v_j &= \rho[\otimes(s_1\cdots s_{j-1}), s_j]\circ
\lambda[\otimes(s_1\cdots s_j), \otimes(s_{j+1}\cdots s_n)](v)\\
&= \lambda[s_j, \otimes(s_{j+1}\cdots s_n)]\circ
\rho[\otimes(s_1\cdots s_{j-1}),\otimes(s_{j}\cdots s_n)](v)\end{aligned}$$ for each $j\in \{2,\cdots,n-1\}$.
For any $s\in S$, we have $I[s] = I[\otimes s]$ by construction. Without loss of generality, let $s = s_1$. Take a $v\in I[s_1]$. By definition we have $t[s_1](v) = \lambda[s_1,\otimes\varepsilon](v) = \lambda[s_1,1](v) = v$ because $\mathcal{S}^1$ is the unital extension of $\mathcal{S}$. This proves (1). Next, (2) follows easily from the fact that $\lambda$ and $\rho$ come from a $\lambda\rho$-system.
Let $\mathcal{S}$, $\mathcal{S}^1$, $\mathcal{F}(\mathcal{S}^1)$ and $\mathbf{t}$ be as above. Then, $\mathbf{t}\colon\mathcal{S}^1\to \mathcal{F}(\mathcal{S}^1)$ is a transformation.
We need to show: (i) that the range of each map $t[s_1s_2\cdots s_n]$ belongs to $I[{s_1s_2\cdots s_n}]$, and (ii) that the appropriate diagrams commute. To show (i), calculate: $$\begin{aligned}
\rho[{s_1}](v_1) &= \rho[{s_1,1}](v_1) \\
&= \rho[s_1,1]\circ\lambda[s_1,\otimes(s_2s_3\cdots s_n)](v) \\
&= \rho[s_1,1]\circ\lambda[s_1,s_2]\circ
\lambda[\otimes(s_1s_2),\otimes(s_3\cdots s_n)](v)\\
&= \lambda[1,s_2]\circ\rho[s_1,s_2]\circ
\lambda[\otimes(s_1s_2),\otimes(s_3\cdots s_n)](v)\\
&= \lambda[1,s_2](v_2)\\
&= \lambda[s_2](v_2)\end{aligned}$$ where the first and last equalities are respectively the definitions of $\rho[s_1]$ and $\lambda[s_2]$, the second and fifth are the definitions of $v_1$ and $v_2$, and the third and fourth follow from ($\alpha$) and ($\beta$). Similarly, but cutting a few corners now, we calculate: $$\begin{aligned}
\rho[s_j](v_j) &= \rho[s_j,1]\circ\lambda[s_j,\otimes(s_{j+1}\cdots s_n)](v) \\
&= \lambda[1,\otimes(s_{j+1}\cdots s_n)]\circ\rho[s_j,\otimes(s_{j+1}\cdots s_n]
\circ\rho[\otimes(s_1\cdots s_{j-1}),\otimes(s_j\cdots s_n)](v)\\
&= \lambda[1,\otimes(s_{j+1}\cdots s_n)]\circ
\rho[\otimes(s_1\cdots s_j),\otimes(s_{j+1}\cdots s_n)](v)\\
&= \lambda[1,s_{j+1}]\circ\lambda[s_{j+1},\otimes(s_{j+2}\cdots s_n)]\circ
\rho[\otimes(s_1\cdots s_j),\otimes(s_{j+1}\cdots s_n)](v)\\
&= \lambda[1, s_{j+1}](v_{j+1})\\
&= \lambda[s_{j+1}](v_{j+1})\end{aligned}$$ and $$\begin{aligned}
\rho[s_{n-1}](v_{n-1}) &= \rho[s_{n-1},1]\circ\lambda[s_{n-1},s_n]
\circ \rho[\otimes(s_1\cdots s_{n-2}),\otimes(s_{n-1}s_n)](v)\\
&= \lambda[1,s_n]\circ\rho[s_{n-1},s_n]
\circ \rho[\otimes(s_1\cdots s_{n-2}),\otimes(s_{n-1}s_n)](v)\\
&= \lambda[1,s_n]\circ\rho[\otimes(s_1\cdots s_{n-1}),s_n](v)\\
&= \lambda[s_n](v_n).\end{aligned}$$
For (ii), let $v\in I[\otimes(s_1s_2\cdots s_n)]$ and let $u = \lambda[\otimes(s_1\cdots s_k),\otimes(s_{k+1}\cdots s_n)](v)$. With this, commutativity of the relevant diagram amounts to the equality between $\langle v_1,\dots, v_k\rangle$ (the first $k$ coordinates of $\langle v_1,\dots, v_n\rangle$) and $\langle u_1,\dots, u_k\rangle$. To verify that these indeed hold, we calculate $$\begin{aligned}
v_1 &= \lambda[s_1,\otimes(s_2\cdots s_n)](v)\\
&= \lambda[s_1,\otimes(s_2\cdots s_k)]\circ
\lambda[\otimes(s_1\cdots s_k),\otimes(s_{k+1}\cdots s_n)](v)\\
&= \lambda[s_1,\otimes(s_2\cdots s_k)](u)\\
&= u_1\end{aligned}$$ then, for $j\in\{2,\dots,k-1\}$ $$\begin{aligned}
v_j &= \rho[\otimes(s_1\cdots s_{j-1}),s_j]\circ
\lambda[\otimes(s_1\cdots s_j),\otimes(s_{j+1}\cdots s_n)](v)\\
&= \rho[\otimes(s_1\cdots s_{j-1}),s_j]\circ
\lambda[\otimes(s_1\cdots s_j),\otimes(s_1\cdots s_k)] \circ
\lambda[\otimes(s_1\cdots s_k),\otimes(s_{k+1}\cdots s_n)](v)\\
&= \rho[\otimes(s_1\cdots s_{j-1}),s_j]\circ
\lambda[\otimes(s_1\cdots s_j),\otimes(s_1\cdots s_n)](u)\\
&= u_j\end{aligned}$$ and finally $$\begin{aligned}
v_k &= \rho[\otimes(s_1\cdots s_{k-1}), s_k]
\circ \lambda[\otimes(s_1\cdots s_k),\otimes(s_{k+1}\cdots s_n)](v)\\
&= \rho[\otimes(s_1\cdots s_{k-1}),s_k](u)\\
&= u_k\end{aligned}$$ proving (ii).
\[divide\] Let $\mathcal{S}$ be a $\lambda\rho$-system over a semigroup $\mathbf{S}$, and let $\mathbf{H}$ be a semigroup. Let $\mathbf{t}\colon \mathcal{S}^1\to \mathcal{F}(\mathcal{S}^1)$ be the transformation from Definition \[free-t\]. Then, $$\mathbf{H}^{\mathbf{t}}\colon \mathbf{H}^{[\mathcal{F}(\mathcal{S}^1)]}
\longrightarrow \mathbf{H}^{[\mathcal{S}^1]}$$ defined as in Lemma \[trans\], is a surjective homomorphism.
The map $\mathbf{H}^{\mathbf{t}}$ is a homomorphism by Lemma \[free-t\]. Surjectivity follows from Lemma \[well-defd\](1).
Let $\mathcal{S}$ be a $\lambda\rho$-system over a semigroup $\mathbf{S}$, and let $\mathbf{H}$ be a semigroup. Then, $\mathbf{H}^{[\mathcal{S}]}\in SH(\mathbf{H}^{[\mathcal{F}(\mathcal{S}^1)]})$. Therefore, $\mathbf{H}^{[\mathcal{S}]}$ divides $\mathbf{H}^{[\mathcal{F}(\mathcal{S}^1)]}$.
By Lemma \[lr-prod-with-unit\], and the well known universal algebraic fact that $SH\leq HS$.
The free construction we presented could be made even ‘freer’ by taking $I[s_1\dots s_n] = I[s_1]\times\dots\times I[s_n]$. But then Lemma \[well-defd\](1) is no longer true, and consequently we lose surjectivity in Theorem \[divide\].
Wreath products {#wreath}
===============
We will now return to the promise made in the introduction and prove that every wreath product can be realised as a $\lambda\rho$-product. Let $(X,\mathbf G)$ consist of a set $X$ and a group $\mathbf{G}$ acting on $X$ on the left, so that $(x\cdot a)\cdot b = x\cdot ab$, for every $x\in X$ and every $a,b\in G$. For any such $(X,\mathbf G)$ and any group $\mathbf{H}$ recall that their wreath product is defined as $$\mathbf H\wr (X,\mathbf G)=(H^X\times G, *)$$ with multiplication $$(a,x)*(b,y)= (a\cdot (b\circ({\underline{\hspace{1ex}}}\cdot x)),\ xy).$$ It is easy to see, that any $(X,\mathbf G)$ can be viewed by a $\lambda\rho$-system $$\mathcal{S}(X,\mathbf G) =
\bigl(\langle \lambda[a,b],\rho[a,b]\rangle\colon
I[ab]\to I[a]\times I[b]\bigr),$$ where $I[s]=X$ for any $s\in S$, and
1. $\lambda[a,b] = {id}_X$ for any $a,b\in S$,
2. $\rho[a,b] = {\underline{\hspace{1ex}}}\cdot a$ for all $a,b\in S$.
as in Example \[sgrp-act\]. Then $\mathbf H^{[\mathcal S(X,\mathbf G)]}\cong \mathbf{H}\wr (X,\mathbf G)$.
\[wr-prod\] Let $\mathcal{S} = (\mathbf{G},\mathbf{I},{\boldsymbol{\lambda}},{\boldsymbol{\rho}})$ be a $\lambda\rho$-system. The, the following are equivalent:
1. $\mathcal{S}$ is group-preserving,
2. $\mathbf{G}$ is a group and $\mathcal{S}$ is unital,
3. $\mathcal{S}\cong \mathcal{S}(X,\mathbf{G})$ for some group $\mathbf{G}$ acting on some set $X$.
Recall from Definition \[P-preserving\] that group-preserving means $\mathbf{H}^{[\mathcal{S}]}$ is a group for any group $\mathbf{H}$.
\(1) $\Rightarrow$ (2). Group-preserving $\lambda\rho$-systems preserve units, so by Theorem \[main-monoid\] $\mathcal{S}$ is unital. Since $\mathbf{1}$ is a (trivial) group and $\mathcal{S}$ is group-preserving then $\mathbf{1}^{[\mathcal S]}\cong \mathbf{G}$, and so $\mathbf{G}$ is a group.
\(2) $\Rightarrow$ (3). Let $\mathcal{S} = (\mathbf{G},\mathbf{I},{\boldsymbol{\lambda}},{\boldsymbol{\rho}})$ be a unital $\lambda\rho$-system, with $\mathbf{G}$ a group. Since $\mathcal{S}$ is unital, we have $${id}_{I[x]}=\lambda[x,e]=\lambda [x,yy^{-1}] =
\lambda [x,y]\circ \lambda[xy,y^{-1}]$$ for all $x,y\in G$ ($e$ is the unit of $\mathbf{G}$, of course). Consequently, $\lambda [x,y]$ is surjective and $\lambda [xy,y^{-1}]$ is injective for all $x,y\in G$. However, $\lambda[x,y]=\lambda [xyy^{-1},y]$ and thus $\lambda [x,y]$ is a bijection. Analogously we can prove bijectivity of $\rho[x,y]$.
Consider the pair $(I[e],\mathbf G)$, The operation $\cdot\colon I[e]\times
G\longrightarrow I[e]$ defined by $$i\cdot x = (\rho[x,e]\circ\lambda[e,x]^{-1})(i)$$ is a group action.
Substituting equalities $$\begin{aligned}
\lambda[ex,y]&=&\lambda[e,x]^{-1}\circ \lambda[e,xy]\\
\rho[x,ye]&=&\rho[y,e]^{-1}\circ \rho[xy,e]\end{aligned}$$ into the equality $$\lambda[e,y]\circ \rho[x,e\cdot y]=\rho[x,e]\circ\lambda[xe,y]$$ we obtain $$\lambda [e,y]\circ \rho[y,e]^{-1}\circ
\rho[xy,e]=\rho[x,e]\circ\lambda[e,x]^{-1}\circ\lambda[e,xy]$$ and hence $$\rho[y,e]\circ\lambda[e,y]^{-1} \circ\rho[e,x]\circ \lambda
[e,x]^{-1}=\rho[xy,e]\circ \lambda[e,xy]^{-1}.\eqno{(\ddag)}$$ It is easy to see that the last equality implies $(i\cdot x)\cdot y= i\cdot
(x\cdot y)$ for all $i\in I[e]$ and $x,y\in G$.
We will show that the system of bijections $\mathbf{t} = \bigl(\lambda[e,x]\colon I[x]\longrightarrow I[e]\bigr)_{x\in G}$ form a transformation and thus an isomorphism of $\lambda\rho$-systems $\mathcal{S}$ and $\mathcal{S}(I[e],\mathbf{G})$. We need to prove commutativity of the following diagrams: $$\begin{tikzpicture}[>=stealth,auto]
\node (tab) at (0,0) {$S(xy)$};
\node (ab) at (3,0) {$S(e)$};
\node (ta) at (0,-2) {$S(x)$};
\node (a) at (3,-2) {$S(e)$};
\draw[->] (tab) to node {$\lambda[e,xy]$} (ab);
\draw[->] (tab) to node[swap] {$\lambda [x,y]$} (ta);
\draw[->] (ab) to node[swap] {$id_{S(e)}$} (a);
\draw[->] (ta) to node {$\lambda[e,x]$} (a);
\end{tikzpicture}
\qquad
\begin{tikzpicture}[>=stealth,auto]
\node (tab) at (0,0) {$S(xy)$};
\node (ab) at (4,0) {$S(e)$};
\node (tb) at (0,-2) {$S(x)$};
\node (b) at (4,-2) {$S(e).$};
\draw[->] (tab) to node {$\lambda[e,xy]$} (ab);
\draw[->] (tab) to node[swap] {$\rho[x,y]$} (tb);
\draw[->] (ab) to node[swap] {$\rho[x,e]\circ\lambda[e,x]^{-1}$} (b);
\draw[->] (tb) to node {$\lambda[e,y]$} (b);
\end{tikzpicture}$$ Commutativity of the first diagram is clear. Applying $\rho[x,y]=\rho[y,e]^{-1}\circ\rho[xy,e]$ to $(\ddag)$, we obtain $$\lambda[e,y]^{-1}\circ\rho[e,x]\circ\lambda[e,x]^{-1}=\rho[x,y]\circ\lambda [e,xy]^{-1}$$ which proves commutativity of the second diagram.
\(3) $\Rightarrow$ (1). Follows from the fact that wreath product of groups is a group.
Theorem \[wr-prod\] shows that $\lambda\rho$-products of groups over groups coincide with wreath products. We already saw that for semigroups the notion of a $\lambda\rho$-products is more general. In particular, the two-sided wreath product of semigroups (see, e.g., [@RT89]) can be accommodated.
\[wr-two-sided\] Let $\mathcal{S}(X,\ld,\rd,\mathbf{S})$ be the $\lambda\rho$-system of Example \[sgrp-act-two-sided\]. Then, $\mathbf{H}^{\mathcal S(X,\backslash,/,\mathbf S)}$ is (isomorphic to) the two-sided wreath product of $\mathbf{H}$ and $\mathbf{S}$.
Combining Krohn-Rhodes Theorem, Theorem \[wr-prod\], and Example \[flip-flop\], we get our final result.
Every finite semigroup divides an iterated $\lambda\rho$-product whose factors are finite simple groups and a two-element semilattice.
Acknowledgement
===============
This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 689176.
|
---
abstract: 'We quantify the gas-phase abundance of deuterium and fractional contribution of stellar mass loss to the gas in cosmological zoom-in simulations from the Feedback In Realistic Environments project. At low metallicity, our simulations confirm that the deuterium abundance is very close to the primordial value. The chemical evolution of the deuterium abundance that we derive here agrees quantitatively with analytical chemical evolution models. We furthermore find that the relation between the deuterium and oxygen abundance exhibits very little scatter. We compare our simulations to existing high-redshift observations in order to determine a primordial deuterium fraction of $(2.549\pm0.033)\times10^{-5}$ and stress that future observations at higher metallicity can also be used to constrain this value. At fixed metallicity, the deuterium fraction decreases slightly with decreasing redshift, due to the increased importance of mass loss from intermediate-mass stars. We find that the evolution of the average deuterium fraction in a galaxy correlates with its star formation history. Our simulations are consistent with observations of the Milky Way’s interstellar medium: the deuterium fraction at the solar circle is $85-92$ per cent of the primordial deuterium fraction. We use our simulations to make predictions for future observations. In particular, the deuterium abundance is lower at smaller galactocentric radii and in higher mass galaxies, showing that stellar mass loss is more important for fuelling star formation in these regimes (and can even dominate). Gas accreting onto galaxies has a deuterium fraction above that of the galaxies’ interstellar medium, but below the primordial fraction, because it is a mix of gas accreting from the intergalactic medium and gas previously ejected or stripped from galaxies.'
author:
- |
Freeke van de Voort,$^{1,2}$[^1] Eliot Quataert,$^{3}$ Claude-André Faucher-Giguère,$^{4}$ Dušan Kereš,$^{5}$ Philip F. Hopkins,$^{6}$ T. K. Chan,$^5$ Robert Feldmann$^{7}$ and Zachary Hafen$^4$\
$^{1}$Heidelberg Institute for Theoretical Studies, Schloss-Wolfsbrunnenweg 35, 69118, Heidelberg, Germany\
$^{2}$Astronomy Department, Yale University, PO Box 208101, New Haven, CT 06520-8101, USA\
$^{3}$Department of Astronomy and Theoretical Astrophysics Center, University of California, Berkeley, CA 94720-3411, USA\
$^{4}$Department of Physics and Astronomy and CIERA, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA\
$^{5}$Department of Physics, Center for Astrophysics and Space Science, University of California at San Diego, 9500 Gilman Drive,\
La Jolla, CA 92093\
$^{6}$TAPIR, Mailcode 350-17, California Institute of Technology, Pasadena, CA 91125, USA\
$^{7}$Institute for Computational Science, University of Zurich, Zurich CH-8057, Switzerland
bibliography:
- 'deuterium.bib'
date: 'Accepted 2018 February 28. Received 2018 February 23; in original form 2017 April 26'
title: On the deuterium abundance and the importance of stellar mass loss in the interstellar and intergalactic medium
---
\[firstpage\]
nuclear reactions, nucleosynthesis, abundances – stars: mass loss – ISM: abundances – galaxies: star formation – intergalactic medium – cosmology: theory
Introduction {#sec:intro}
============
Deuterium is one of the few stable isotopes produced in astrophysically interesting amounts during Big Bang nucleosynthesis, together with helium and lithium (see @Steigman2007 for a review). Helium and lithium can be produced after this initial phase, in stars and via collisions of cosmic ray nuclei, potentially increasing their gas-phase abundances. However, the gas-phase deuterium abundance can only decrease. All primordial deuterium is burned during the collapse of a protostar and deuterium synthesized in stellar interiors is immediately destroyed, because deuterium fuses at relatively low temperatures, $T\approx10^6~K$, easily reached in the interiors of stars and even brown dwarfs [@Epstein1976; @Stahler1988; @Spiegel2011]. Therefore, mass lost from stars (also referred to as ‘recycled’ gas) is deuterium-free, i.e. $\mathrm{(D/H)}_\mathrm{recycled}=0$.
The primordial deuterium fraction, $\mathrm{(D/H)}_\mathrm{prim}$, is sensitive to cosmological parameters and, in particular, to the baryon–photon ratio and thus to the baryonic density of the Universe. Measurements of the cosmic microwave background (CMB) radiation have pinned down the ratio of the mean density of baryons to the critical density and the Hubble parameter [@Planck2015XIII]. The most recent theoretical models of Big Bang nucleosynthesis have incorporated these and derived, for example, $\mathrm{(D/H)}_\mathrm{prim}=(2.45\pm0.05)\times10^{-5}$ [@Coc2015] and $\mathrm{(D/H)}_\mathrm{prim}=(2.58\pm0.13)\times10^{-5}$ [@Cyburt2016], where the quoted errors are $1\sigma$.
An accurate determination of the primordial deuterium fraction, in conjunction with Big Bang nucleosynthesis reaction rates, gives an independent constraint on the cosmic baryon density. If there is disagreement with the value derived from CMB measurements, this could point to a deviation in the expansion rate of the early universe and to non-standard models of big bang nucleosynthesis. Low-metallicity gas likely has a deuterium fraction close to the primordial value, because it has not been substantially enriched by stellar mass loss. Absorption lines in spectra of background quasars have been used to determine the primordial deuterium fraction observationally, finding e.g. $\mathrm{(D/H)}_\mathrm{prim}=(2.547\pm0.033)\times10^{-5}$ [@Cooke2016]. Modern estimates are thus consistent with each other and there is currently no conflict with the standard model of cosmology [e.g. @Steigman2007; @Cooke2014].
Intermediate-mass and massive stars return material to the interstellar medium (ISM) via stellar winds before and during the asymptotic giant branch (AGB) phase and via supernova explosions, respectively. One well-known effect of this recycling process of baryons that become part of a star and are later returned into space is the release of metals into the ISM and the intergalactic medium (IGM). However, it is also important for the destruction of light elements, such as deuterium. If there is no fresh infall of gas onto galaxies and the ISM of these objects is replenished by stellar mass loss, both the metallicity of the gas and young stars increases and the deuterium fraction in the ISM decreases. The ratio of the deuterium fraction in the ISM or IGM and the primordial deuterium fraction, $\mathrm{(D/H)}/\mathrm{(D/H)}_\mathrm{prim}$, is therefore a measure of the fraction of the gas that has not been processed in stars. The inverse of this, i.e. $\mathrm{(D/H)}_\mathrm{prim}/\mathrm{(D/H)}$, is known as the astration factor. Measurements of the evolution of the deuterium abundance have been used to constrain galactic chemical evolution models [e.g. @Audouze1974; @Vangioni1988; @Vangioni1994; @Scully1997; @Olive2012]. These models predict astration factors higher than observed in the local ISM when they only take into account cosmological inflow [e.g. @Fields1996; @Romano2006; @Lagarde2012]. Models that additionally allow for galactic outflows predict lower astration factors [e.g. @Dvorkin2016; @Weinberg2017]. In this way, measurements of the deuterium fraction (and thus the astration factor) can shed light on the balance between primordial inflow, metal-enriched outflow, and recycling through stellar mass loss, which are all related to the star formation and accretion history of a galaxy [e.g. @Casse1998; @Prantzos2001; @Romano2006; @Dvorkin2016; @Weinberg2017].
The fuelling of star formation by stellar mass loss is likely more important in high-mass galaxies and high-density environments. Massive early-type galaxies and satellite galaxies have specific star formation rates (SFRs) far below those of central late-type galaxies. It is not known which process(es) quench(es) galaxies, but galactic outflows are at least partially responsible for quenching massive galaxies and preventing subsequent gas accretion [e.g. @Faucher2011; @Voort2011a]. However, a substantial fraction of local early-type galaxies still have a detectable molecular or atomic gas reservoir [e.g. @Young2011; @Serra2012]. Some of these exhibit gas kinematics indicating a predominantly external gas supply, such as through minor mergers, whereas others (especially those located in a cluster environment) are consistent with their ISM being fed through stellar mass loss [@Davis2011]. Furthermore, some massive galaxies in the centres of clusters are forming stars at a substantial rate ($1-100$ M$_{\astrosun}$ yr$^{-1}$) and contain a considerable amount of dust [@ODea2008; @Donahue2011]. Dust is produced by stars and destroyed by sputtering in hot gas. Therefore, the gas supply is unlikely to have cooled out of the hot halo gas. This also indicates that stellar mass loss may be an important contributor to the fuel for the observed star formation [@Voit2011].
The amount of mass supplied to the ISM through stellar mass loss could also be sufficient to fuel most of the star formation in present-day star-forming galaxies, including the Milky Way [@Leitner2011]. However, galactic outflows were not included, but are likely required to produce correct stellar masses and metallicities. Hydrodynamical simulations that included feedback from stars and/or black holes have found that stellar mass loss becomes more important for fuelling star formation towards lower redshift, although, in general, it does not become the dominant fuel source for star formation [@Oppenheimer2008; @Segers2016].
The predicted deuterium fraction and the importance of stellar mass loss are the focus of this paper. We present results from a suite of high-resolution, cosmological ‘zoom-in’ simulations from the ‘Feedback In Realistic Environments’ (FIRE) project,[^2] which spans a large range in halo and galaxy mass. The FIRE simulation suite has been shown to successfully reproduce a variety of observations, which is linked to the strong stellar feedback implemented. These galactic winds efficiently redistribute gas from galaxies out to large galactocentric distances (see @Muratov2015 [@Muratov2016]). For the purposes of this paper, we highlight the fact that the simulations match the derived stellar–to–halo mass relationship [@Hopkins2014FIRE; @Feldmann2016], the galaxy mass-metallicity relation and gas-phase metallicity gradients at $z=0-3$ [@Ma2016; @Ma2017], and the dense neutral hydrogen, H<span style="font-variant:small-caps;">i</span>, content of galaxy haloes [@Faucher2015; @Faucher2016; @Hafen2017].
This is the first time cosmological, hydrodynamical simulations are used to study the deuterium abundance in the ISM and IGM. Our simulations self-consistently follow the time-dependent assembly of dark matter haloes, the accretion of gas onto galaxies, the formation of stars, the return of mass in stellar winds, and the generation of large-scale galactic outflows, whereas chemical evolution models - which are often used for similar studies - are based on analytic prescriptions. Specifically, we do not assume instantaneous recycling of stellar mass loss nor instantaneous mixing of metals nor a specific parametrized gas accretion or star formation history. However, our simulations do not consider mixing of the gas between resolution elements.
In Section \[sec:sim\] we describe the suite of simulations used, as well as the way we compute the deuterium abundance and the fractional contribution of stellar mass loss to the gas, i.e. the ‘recycled gas fraction’ or $f_\mathrm{recycled}$ (Section \[sec:Dfrac\]). The deuterium retention fraction and recycled gas fraction are related via $\mathrm{(D/D_{prim})}=1-f_\mathrm{recycled}$. In Section \[sec:results\] we present our results, including comparisons to existing observations. Section \[sec:evol\] describes the evolution of the deuterium fraction (and hence of the recycled gas fraction), while Section \[sec:z3\] focuses on high redshift and Section \[sec:z0\] on low redshift. We discuss our results and conclude in Section \[sec:concl\].
Method {#sec:sim}
======
The simulations used are part of the FIRE-1 sample. These were run with <span style="font-variant:small-caps;">gizmo</span>[^3] [@Hopkins2015] in ‘P-SPH’ mode, which adopts the Lagrangian ‘pressure-energy’ formulation of the smoothed particle hydrodynamics (SPH) equations [@Hopkins2013PSPH]. The gravity solver is a heavily modified version of <span style="font-variant:small-caps;">gadget</span>-2 [@Springel2005], with adaptive gravitational softening following @Price2007. Our implementation of P-SPH also includes substantial improvements in the artificial viscosity, entropy diffusion, adaptive timestepping, smoothing kernel, and gravitational softening algorithm.
The FIRE project consists of a suite of cosmological ‘zoom-in’ simulations of galaxies with a wide range of masses, simulated to $z=0$ (@Hopkins2014FIRE [@Chan2015; @Ma2016; @Hafen2017]; Feldmann et al. in preparation), to $z=1.7$ [@Feldmann2016], and to $z=2$ [@Faucher2015]. The simulation sample used is identical to the one used in @Voortetal2016 and the simulation details are fully described in @Hopkins2014FIRE and references therein. The three Milky Way-mass galaxies that are the focus of Figure \[fig:Devol\] and \[fig:Drad\] are simulations ‘m12i’, ‘m12v’, and ‘m11.9a’ (from highest to lowest stellar mass) from @Hopkins2014FIRE and @Hafen2017. A $\Lambda$CDM cosmology is assumed with parameters consistent with the 9-yr Wilkinson Microwave Anisotropy Probe (WMAP) results [@Hinshaw2013]. The initial particle masses for baryons (dark matter) vary from $2.6\times10^2-4.5\times10^5$ M$_{\astrosun}$ ($1.3\times10^3-2.3\times10^6$ M$_{\astrosun}$) for the 16 simulations that were run to $z=0$ (see also @Voortetal2016 for further details). The 23 simulations that were run to $z\approx2$ are described in @Faucher2015 and @Feldmann2016 and their initial baryonic (dark matter) masses are $(3.3-5.9)\times10^4$ M$_{\astrosun}$ ($(1.7-2.9)\times10^5$ M$_{\astrosun}$).
Star formation is restricted to molecular, self-gravitating gas above a hydrogen number density of $n_\mathrm{H}\approx5-50$ cm$^{-3}$, where the molecular fraction is calculated following @Krumholz2011 and the self-gravitating criterion following @Hopkins2013SelfGrav. The majority of stars form at gas densities significantly higher than this imposed threshold. Stars are formed from gas satisfying these criteria at the rate $\dot\rho_\star=\rho_\mathrm{molecular}/t_\mathrm{ff}$, where $t_\mathrm{ff}$ is the free-fall time. When selected to undergo star formation, the entire gas particle is converted into a star particle.
We obtain stellar evolution results from STARBURST99 [@Leitherer1999] and assume an initial stellar mass function (IMF) from @Kroupa2002. Radiative cooling and heating are computed in the presence of the CMB radiation and the ultraviolet (UV)/X-ray background from @Faucher2009. Self-shielding is accounted for with a local Sobolev/Jeans length approximation. We impose a temperature floor of 10 K or the CMB temperature.
The primordial abundances are $X_\mathrm{prim}=0.76$ and $Y_\mathrm{prim}=0.24$, where $X_\mathrm{prim}$ and $Y_\mathrm{prim}$ are the mass fractions of hydrogen and helium, respectively. The simulations include a metallicity floor at metal mass fraction $Z_\mathrm{prim}\approx10^{-4}$ Z$_{\astrosun}$ or $Z_\mathrm{prim}\approx10^{-3}$ Z$_{\astrosun}$, because yields are very uncertain at lower metallicities and we do not resolve the formation of individual first-generation stars. The abundances of 11 elements (H, He, C, N, O, Ne, Mg, Si, S, Ca and Fe) produced by massive and intermediate-mass stars are computed following @Iwamoto1999, @Woosley1995, and @Izzard2004. The amount of mass and metals ejected in a computational time-step depends on the age of the star particle and our simulations therefore self-consistently follow time-dependent chemical enrichment. Mass ejected through supernovae and stellar winds are modelled by transferring a fraction of the mass of a star particle to its neighbouring gas particles, $j$, within its SPH smoothing kernel as follows: $$f_j = \dfrac{\frac{m_j}{\rho_j} W(r_j,h_\mathrm{sml})}{\Sigma_i \frac{m_i}{\rho_i} W(r_i,h_\mathrm{sml})},$$ where $h_\mathrm{sml}$ is the smoothing length of the star particle (determined in the same manner as for gas particles), $r_i$ is the distance from the star particle to neighbour $i$, $W$ is the quintic SPH kernel, and the summation is over all SPH neighbours of the star particle, 62 on average. There is *no* sub-resolution metal diffusion in these simulations.
The FIRE simulations include an explicit implementation of stellar feedback by supernovae, radiation pressure, stellar winds, and photo-ionization and photo-electric heating (see @Hopkins2014FIRE and references therein for details). Feedback from active galactic nuclei (AGN) is not included. For star-forming galaxies, which constitute the majority of our simulated galaxies, AGN are thought to be unimportant. However, AGN-driven outflows are potentially important for the high-mass end of our simulated mass range.
We measure a galaxy’s stellar mass, $M_\mathrm{star}$, within 20 proper kpc of its centre. The deuterium fraction of a galaxy’s ISM is measured within 20 proper kpc of its centre for gas with a temperature below $10^4$ K, which selects the warm ionized and cold neutral gas in the ISM. These choices have a mild effect on the normalization of some of our results, but not on the trends or on our conclusions.
Deuterium fraction in hydrodynamical simulations {#sec:Dfrac}
------------------------------------------------
Determining $\mathrm{(D/H)}/\mathrm{(D/H)}_\mathrm{prim}$ in our simulations is straightforward. The mass of a gas particle can only increase during the simulation by receiving mass lost from nearby stars (no particle splitting is implemented). Therefore, any mass above the initial particle mass, $m_\mathrm{initial}$, is deuterium-free. This is mixed with the initial particle mass, which has the primordial fraction of deuterium. Therefore, for each gas particle, we calculate $$\label{eqn:Dfrac}
\begin{split}
\frac{\mathrm{(D/H)}}{\mathrm{(D/H)}_\mathrm{prim}} & = \frac{\mathrm{D}}{\mathrm{D_{prim}}}\frac{\mathrm{H_{prim}}}{\mathrm{H}} = \frac{m_\mathrm{initial}}{m_\mathrm{initial}+m_\mathrm{recycled}}\frac{X_\mathrm{prim}}{X_\mathrm{gas}} \\
& = \frac{m_\mathrm{initial}}{m_\mathrm{gas}}\frac{X_\mathrm{prim}}{X_\mathrm{gas}},
\end{split}$$ where $m_\mathrm{recycled}$ is the amount of mass received from evolving stars, i.e. the amount of gas that has been ‘recycled’, $X_\mathrm{gas}$ is the mass fraction of hydrogen, and $m_\mathrm{gas}$ is the mass of the particle at the redshift of interest. We refer to this quantity as the deuterium retention fraction, because it is the fraction of deuterium, produced during Big Bang nucleosynthesis, that is not destroyed. The inverse of Equation \[eqn:Dfrac\] is the astration factor. The value of $\mathrm{(D/H)}_\mathrm{prim}$ is well-constrained, both directly from absorption-line observations of low-metallicity gas and indirectly from CMB measurements coupled with Big Bang nucleosynthesis reaction rates [e.g. @Cooke2016; @Coc2015; @Cyburt2016]. Another way to constrain the primordial value is by comparing observations to cosmological simulations, as done in Section \[sec:z3\].
The deuterium retention fraction in Equation \[eqn:Dfrac\] is directly related to the fractional contribution of stellar mass loss to the gas, i.e. the recycled gas fraction, $$\begin{split}
f_\mathrm{recycled} & = \frac{m_\mathrm{recycled}}{m_\mathrm{gas}} = \frac{m_\mathrm{gas}-m_\mathrm{initial}}{m_\mathrm{gas}} \\
& = 1-\frac{\mathrm{D}}{\mathrm{D_{prim}}} = 1-\frac{\mathrm{(D/H)}}{\mathrm{(D/H)}_\mathrm{prim}}\frac{X_\mathrm{gas}}{X_\mathrm{prim}},
\end{split}$$ which is used to study the importance of stellar mass loss in fuelling the ISM and star formation. Besides destroying all deuterium, a fraction of the hydrogen is fused into helium and metals before the gas is recycled into the ISM, $X_\mathrm{gas}=1-Y_\mathrm{gas}-Z_\mathrm{gas}$. This can be approximated well by $X_\mathrm{gas}\approx X_\mathrm{prim}-3Z_\mathrm{gas}$. The factor $X_\mathrm{gas}/X_\mathrm{prim}$ is close to unity for subsolar metallicities, but becomes more important at supersolar metallicities. Even though the differences do not change our conclusions, we will show and discuss both $\mathrm{(D/H)}/\mathrm{(D/H)}_\mathrm{prim}$ and $\mathrm{D/D_{prim}}$ or $f_\mathrm{recycled}$ when relevant.
Results {#sec:results}
=======
Observations of the deuterium fraction exist at both high and low redshift. We will first discuss the evolution of (D/H) and then discuss predictions and observational comparisons at $z=3$ and $z=0$ separately. Throughout the paper, we use oxygen abundance ratios of gas as compared to those of the Sun, i.e. $\mathrm{[O/H]}= \mathrm{log_{10}}(n_\mathrm{O}/n_\mathrm{H}) - \mathrm{log_{10}}(n_\mathrm{O}/n_\mathrm{H})_{\astrosun}$, where $n_\mathrm{O}$ is the oxygen number density, $n_\mathrm{H}$ the hydrogen number density, and $\mathrm{log_{10}}(n_\mathrm{O}/n_\mathrm{H})_{\astrosun}=-3.31$ is the solar oxygen abundance taken from @Asplund2009.
Evolution of the deuterium fraction {#sec:evol}
-----------------------------------
While the total deuterium content of the Universe decreases with time, its total metallicity increases, leading to an inverse correlation between the deuterium and oxygen abundance [e.g. @Steigman2003; @Romano2006; @Dvorkin2016]. Figure \[fig:DOz\] shows the median deuterium retention fraction (black curves) for all gas particles in our simulations as a function of oxygen metallicity at $z=0$ (top panels) and $z=3$ (bottom panels). The left panels only take into account deuterium and oxygen, whereas the right panels fold in the hydrogen abundance for both axes. $Z_\mathrm{O}$ ($Z_\mathrm{O, \astrosun}$) is the oxygen mass fraction of the gas (for solar abundances). The grey, shaded regions show the $1\sigma$ (dark) and $3\sigma$ (light) scatter around the median. Some of our simulations implemented a relatively high metallicity floor of $\mathrm{[O/H]}_\mathrm{initial}=-2.8$. Here, in order to not be affected by the imposed metallicity floor, we excluded gas with a metallicity within a factor of 2 from its initial oxygen abundance, but this choice does not affect our conclusions. At solar oxygen metallicity, about 90 per cent of the primordial deuterium is not destroyed at $z=3$ and 88 per cent at $z=0$. $\mathrm{(D/H)}/\mathrm{(D/H)}_\mathrm{prim}=0.91$ ($z=0$) and $0.93$ ($z=3$) at $\mathrm{[O/H]}=0$, slightly higher because of the small decrease of the hydrogen fraction.
The $1\sigma$ scatter in this relation is very small, which shows that the destruction of deuterium and the enrichment with oxygen are tightly correlated. However, the scatter increases at $z=0$ at the highest metallicities ($\mathrm{[O/H]}\gtrsim0.5$). Additionally, we find large non-Gaussian tails at all metallicities, which means that even at low metallicity, a small fraction of gas particles have substantially reduced deuterium abundances. Our calculations likely underestimate the mixing of gas, because elements in our simulation are stuck to gas particles and do not diffuse to neighbouring gas particles. Adding turbulent diffusion to our simulations would only decrease the scatter in the correlation between deuterium and oxygen, because it smoothes out variations, and would thus strengthen our conclusions. The dependence of (D/H) on \[O/H\] is very steep at high metallicity, because \[O/H\] is a logarithmic quantity.
A small fraction of the gas (0.5 per cent) reaches extremely high metallicities ($\mathrm{[O/H]}>0.5$), which have not been observed. It is possible that such rare systems exist, outside the Milky Way, but are beyond current observational capabilities. Note, however, that the (average) metallicity in sightlines through our simulation is always $\mathrm{[O/H]}\leq0.5$ (see Figure \[fig:HID\]). Another possibility is that there is not enough mixing in our simulations, since the metals are stuck to particles and cannot diffuse, resulting in small metal-rich pockets. Additionally, the yields are very uncertain at such high metallicities. The real uncertainty is therefore larger than the scatter in this regime. Note that although $\mathrm{D/D_{prim}}\leq1$, $\mathrm{(D/H)}/\mathrm{(D/H)}_\mathrm{prim}$ can be larger than unity in rare cases at high metallicity, because $\mathrm{H/H_{prim}}$ can become very small due to hydrogen fusion.
For comparison, the red, dashed curves (identical in top and bottom left panels) show the relation between the oxygen and deuterium abundances obtained from a one-zone chemical evolution model [@Weinberg2017]. This model assumes that chemical equilibrium is reached in the ISM due to the balance between gas inflow and outflow, enrichment though stellar mass loss, and gas consumption due to star formation. The only parameters in the relation are the recycling fraction, $r$, i.e. the fraction of mass returned to the ISM by a simple stellar population, and the oxygen yield, $m_\mathrm{O}$, i.e. the mass fraction of a simple stellar population released into the ISM in oxygen, $$\label{eqn:W16}
\mathrm{\dfrac{D}{D_{prim}}}=\dfrac{1}{1+rZ_\mathrm{O}/m_\mathrm{O}},$$ where $Z_\mathrm{O}$ is the oxygen mass fraction of the gas. We also compare our findings to analytic results derived from a closed box model, i.e. no gas inflow or outflow [@Tinsley1980]. The resulting relation between the deuterium retention fraction and oxygen abundance is[^4] $$\label{eqn:T80}
\mathrm{\dfrac{D}{D_{prim}}}=e^{\dfrac{-r Z_\mathrm{O}}{m_\mathrm{O}}}$$ as shown by the purple, dotted curves (identical in top and bottom panels). Both chemical evolution models assume instantaneous stellar mass loss and enrichment, with no time dependence (whereas our simulations consistently follow time-dependent mass loss and enrichment as the stellar population ages). The ratio $r/m_\mathrm{O}=26.7$ (using $r=0.4$, and $m_\mathrm{O}=0.015$, the fiducial values from @Weinberg2017) is thus the only free parameter. The models match the relative abundances at $z=0$ surprisingly well.
As can be seen by comparing the two panels of Figure \[fig:DOz\], there is relatively little evolution in the correlation between deuterium and oxygen. At fixed oxygen metallicity, the deuterium abundance is slightly higher at $z=3$ than at $z=0$. This is because most of the oxygen is produced in core-collapse supernovae, which also dominate the stellar mass loss at early times. At late times, AGB stars are responsible for most of the mass loss, adding deuterium-free material, but not substantially enriching the gas with oxygen. This can be tested by dividing the cumulative amount of mass loss added to the gas in the simulations by the total amount of gas-phase oxygen at different redshifts. As mentioned before, the fiducial ratio used is $r/m_\mathrm{O}=26.7$, which is close to, though slightly higher than, the value we find at $z=0$, $r/m_\mathrm{O}=24.4$ (directly computed from and averaged over all our simulations). At $z=3$, however, the average ratio in our simulations is substantially different, $r/m_\mathrm{O}=19.4$. We therefore added extra model curves (cyan, dashed using Equation \[eqn:W16\] and orange, dotted using Equation \[eqn:T80\]) to each panel of Figure \[fig:DOz\], where we changed the value of $r/m_\mathrm{O}$ to match the average value in the simulations.
The level of agreement between these simple models, especially the closed box model from @Tinsley1980, and our cosmological simulation results is remarkable given the very different approaches. This lends credence to both methods and shows that the most important factor in this correlation is the ratio $r/m_\mathrm{O}$, which can be calculated from stellar population synthesis models. The complex processes involved in the formation of galaxies, such as galaxy mergers, time-variable star formation and galactic outflows, as well as the lack of mixing in these simulations are thus likely unimportant where these relative abundances are concerned. The improvement from the small variation in $r/m_\mathrm{O}$ with redshift supports our claim that the evolution is due to the extra (almost oxygen-free) mass loss from AGB stars at late times.
Accurate observations of (D/H) at $\mathrm{[O/H]}>-1$ in combination with an accurate determination of $\mathrm{(D/H)}_\mathrm{prim}$, either from observations at low metallicity or derived from CMB measurements, would be able to determine $r/m_\mathrm{O}$. The recycling fraction is governed by intermediate-mass stars as well as massive stars, whereas the oxygen yield depends only on the latter. Therefore, the relation between $\mathrm{(D/H)/(D/H)}_\mathrm{prim}$ and \[O/H\] can potentially be used to constrain stellar evolution models and/or the variation of the IMF at the high-mass end. Although all our simulations were run with the same stellar evolution model and IMF, numerical chemical evolution models have already demonstrated that the deuterium fraction depends on these choices [e.g. @Tosi1998; @Prantzos2001; @Romano2006].
In the top right panel, observational constraints independently derived by @Linsky2006 (blue cross with error bars) and @Prodanovic2010 (red cross with error bars) for the local ISM have been included, slightly offset from $\mathrm{[O/H]}=0$ for clarity and using $\mathrm{(D/H)}_\mathrm{prim}=2.547\times10^{-5}$, as recently obtained by @Cooke2016. Our solar deuterium value is in excellent agreement with that of @Linsky2006, but higher than that of @Prodanovic2010. The latter, however, is interpreted by the authors as a lower limit on the true value, in which case it is also in agreement with the result from our simulations (see Section \[sec:z0\] for further discussion on these results).
Additionally, it is of interest to compare our results to those from numerical chemical evolution models. We therefore reproduced one of the models, based on a @Scalo1986 IMF and @Schaller1992 stellar lifetimes, from @Romano2006, shown in the top right panel as the purple, dotted curve. Although these results agree qualitatively, there is a clear $2\sigma$ discrepancy at high metallicity. This is potentially caused by the different IMF and stellar evolution models used, by the different star formation histories (and thus different importance of AGB stars), or by the inclusion of galactic outflows. Given the excellent match between our simulations and the simple closed box model, we believe the former explanation is the most plausible.
![\[fig:DFe\] The fraction of deuterium, normalized by the primordial deuterium fraction, as a function of iron metallicity at fixed oxygen metallicity at $z=0$ (top) and $z=3$ (bottom). The coloured curves show the median residual relation in four $Z_\mathrm{O}$ bins, 0.1 dex wide, centred on the value indicated by the legend and increasing from top to bottom. Error bars show the 16th and 84th percentiles of the distribution. Only bins containing at least 100 gas particles are shown. The deuterium fraction (recycled gas fraction) decreases (increases) with iron abundance, because intermediate-mass stars become relatively more important, ejecting iron-rich and deuterium-free material into the ISM. The residual correlation between deuterium and iron is stronger at lower redshift, due to the increased importance of older stars, which increases the scatter in the relation between deuterium and oxygen (Figure \[fig:DOz\]).](figures/DFe_X.eps)
Although the $1\sigma$ scatter in the relation between deuterium and oxygen abundances is small, it is nonzero and slightly larger at lower redshift. As mentioned before, almost all oxygen is produced by massive stars and released in core-collapse supernovae, whereas this is only the case for about half of the iron. The other half is synthesized in intermediate-mass stars and released in type Ia supernova explosions and winds from AGB stars [e.g. @Wiersma2009b]. The iron abundance at fixed oxygen abundance therefore enables us to trace the relative importance of massive (younger) stars and intermediate-mass (older) stars and check whether variations in the contribution of stellar mass loss by AGB stars is responsible for the scatter seen in Figure \[fig:DOz\].
Figure \[fig:DFe\] shows the residual dependence of the normalized deuterium fraction on the iron metallicity at fixed oxygen metallicity at $z=0$ (top panel) and $z=3$ (bottom panel). $Z_\mathrm{Fe}/Z_\mathrm{Fe, \astrosun}$ is the iron mass fraction of the gas, normalized by the solar value. The coloured curves show the median relation between deuterium and iron in four bins, 0.1 dex wide, with (from top to bottom) $\mathrm{Log}_{10} Z_\mathrm{O}/Z_\mathrm{O, \astrosun}\approx-1$ (blue), $\approx-0.5$ (cyan), $\approx0$ (black), and $\approx0.5$ (red). Error bars show the 16th and 84th percentiles of the distribution and only bins containing at least 100 gas particles are included.
A larger iron abundance at fixed $Z_\mathrm{O}/Z_\mathrm{O, \astrosun}$ means that older, intermediate-mass stars have been relatively more important for enriching the ISM. The deuterium fraction is thus expected to decrease with increasing iron abundance. Figure \[fig:DFe\] proves that this is indeed the case, although there is significant scatter in the residual relation between deuterium and iron. We therefore conclude that the (small) scatter in the relation between deuterium and oxygen is at least in part due to the varying importance of AGB stars and thus to the varying age of the stellar population responsible for enriching the gas.
Comparing the $z=0$ and $z=3$ results, it is clear that the residual dependence of the deuterium fraction on the iron abundance is stronger at $z=0$ than at $z=3$. This is likely because the variation in stellar population ages, and thus in the importance of AGB stars, is smaller at higher redshift, when the Universe was much younger. This is also consistent with the fact that the $1\sigma$ scatter in the correlation between the deuterium fraction and oxygen abundance is smaller at higher redshift. Measuring $\mathrm{[Fe/H]}$ besides $\mathrm{[O/H]}$ and (D/H) will provide even better constraints on the stellar IMF and stellar evolution models.
### Milky Way-mass galaxies
To understand the chemical evolution of galaxies like the Milky Way, Figure \[fig:Devol\] shows the deuterium evolution for three of our simulated galaxies with stellar masses close to that of the Milky Way at $z=0$, as indicated in the legend. The mean deuterium retention fraction is calculated for the gas within 20 proper kpc of the galaxies’ centres, with a temperature below $10^4$ K, which selects the warm ionized and cold neutral gas in the ISM. The black curves include the evolution of the hydrogen fraction, as it would be measured in observations. The orange curves show the recycled gas fraction, which is lower, but show the same trends with look-back time. The final $z=0$ values vary between the three galaxies, because they have different stellar masses. The mass dependence will be discussed in Section \[sec:z0\]. Here, we are interested in the evolution of (D/H), that is, in the shape of the curves. Initially, the deuterium fraction is equal to its primordial value, after which it decreases. Two of the galaxies show an approximately linear decrease towards $z=0$ (solid and dashed curves), whereas for the galaxy with $M_\mathrm{star}=10^{10.4}$ M$_{\astrosun}$ (dotted curve; ‘m12v’) the deuterium fraction levels off in the last $\approx5$ Gyr. The former have therefore not reached an equilibrium between the inflow of deuterium-rich gas from the IGM, the addition of deuterium-free gas through stellar mass loss, and the outflow of deuterium-poor gas. The latter galaxy has potentially reached chemical equilibrium in its ISM.
![\[fig:Devol\] Evolution of the mean fraction of deuterium in the ISM of three star-forming, Milky Way-mass galaxies, normalized by the primordial deuterium fraction. The black curves include the evolution of the hydrogen fraction (left axis), whereas the orange curves show the recycled gas fraction (right axis). The deuterium fraction in these galaxies decreases with time. For two of the galaxies (solid and dashed curves; ‘m12i’ and ‘m11.9a’), no equilibrium value has been reached by $z=0$. This means that stellar mass loss becomes steadily more important for fuelling the ISM towards the present day. However, $\mathrm{(D/H)/(D/H)}_\mathrm{prim}$ levels out in the last $\approx5$ Gyr for one of the galaxies (dotted curve; ‘m12v’). This difference is likely related to their star formation history, because the majority of stellar mass loss occurs at young stellar ages. The former reach half of their present-day mass at $z\approx0.4$, whereas the latter already formed half of its stars by $z\approx1.1$.](figures/Devol_X.eps)
Our three galaxies have different stellar masses and gas masses, on top of different star formation histories, and we lack the statistical power to control for this. Despite this limitation, we checked whether or not the low-redshift behaviour of the deuterium fraction is related to the star formation history. The galaxy which has reached deuterium equilibrium ($M_\mathrm{star}=10^{10.4}$ M$_{\astrosun}$; ‘m12v’) has already formed half of its stars by $z\approx1.1$, whereas the other two reach half of their present-day stellar mass only at $z\approx0.4$ and have thus experienced much more low-redshift star formation and thus more low-redshift stellar mass loss. Therefore, the reason for the different low-redshift behaviour may indeed lie in the different star formation histories of our simulated galaxies.
We also checked for a dependence of the deuterium evolution on the mass loading factor, i.e. the gas outflow rate from a galaxy divided by the galaxy’s SFR, as suggested by @Weinberg2017. The average mass loading factor at $z_1>z>z_2$ is calculated in the following way. We select all the gas particles within 20 proper kpc of the galaxies’ centres and a temperature below $10^4$ K. We then divide the total mass of those selected particles that have been turned into stars by $z=z_2$ by the total mass of the selected particles that are still gaseous, but located beyond 20 proper kpc of the galaxies’ centres at $z=z_2$. We take $z_1$ and $z_2$ to be approximately 1.5 Gyr apart, which is similar to the gas consumption time-scale. We find that the mass loading factor is relatively constant in the last 5 Gyr. From the most to least massive of our simulated Milky Way-mass galaxies, their average late-time mass loading factors over the last 5 Gyr are 0.2 (‘m12i’), 0.4 (‘m12v’), and 2.1 (‘m11.9a’). The values for ‘m12i’ and ‘m12v’ are consistent with the upper limits from @Muratov2015, who argue that these low mass loading factors are not driven by galactic winds, but caused by random gas motions and/or close passages of satellite galaxies. We conclude that there is no clear correlation of the mass loading factor with the late-time deuterium evolution.
Knowing the evolution of (D/H) can thus potentially help us understand a galaxy’s star formation history. This could be achieved for the Milky Way with an accurate determination of the deuterium fraction in giant planets in the Solar System, such as Jupiter, in combination with present-day measurements in the local ISM [@Lellouch2001]. The deuterium fraction in the giant planets provides a fossil record of the deuterium fraction in the local ISM during the time the Solar System was formed, about 4.5 Gyr ago. Using $\mathrm{(D/H)}_\mathrm{prim}=2.547\times10^{-5}$ from @Cooke2016, the measurement by @Lellouch2001 implies $\mathrm{(D/H)/(D/H)}_\mathrm{prim}=0.82^{+0.12}_{-0.15}$ in Jupiter, which is consistent with all three of our simulated Milky Way-mass galaxies and is not precise enough to distinguish between a declining or constant deuterium fraction. Future observations with higher accuracy would be well-suited for this purpose.
Deuterium fraction at high redshift {#sec:z3}
-----------------------------------
There has been a large observational effort to measure the deuterium fraction in metal-poor gas through absorption lines in spectra of background quasars. Lyman Limit Systems (LLSs; $10^{17.2}<N_\mathrm{H\,\textsc{i}}<10^{20.3}$ cm$^{-2}$, where $N_\mathrm{H\,\textsc{i}}$ is the $H\,\textsc{i}$ column density) and Damped Lyman-$\alpha$ Systems (DLAs; $N_\mathrm{H\,\textsc{i}}>10^{20.3}$ cm$^{-2}$) are optically thick to Lyman limit photons. To make a fair comparison between our simulations and these observations, we calculate column densities based only on the neutral gas. Because the gas comprising these strong absorbers is partially shielded from the ambient UV radiation, it is more neutral than if it were optically thin. This is taken into account in our simulations by using the fitting formula from @Rahmati2013, which has been shown to capture the effect of self-shielding well.
@Cooke2014 argue that the most precise measurements can be made in absorbers with $N_\mathrm{H\,\textsc{i}}>10^{19}$ cm$^{-2}$. In order to compare to these systems, we also restrict ourselves to sightlines with column densities above this limit. Additionally, we discard the rare systems with $N_\mathrm{H\,\textsc{i}}>10^{21}$ cm$^{-2}$ in order to not be dominated by molecular gas. We note that neither this selection nor the self-shielding correction affects our results. We do not find a dependence of (D/H) on column density at fixed metallicity, so absorption line systems at any column density could be used. The vast majority of the selected high column density absorbers are located in the haloes around galaxies [@Voort2012]. We therefore use a simulated region of 300 by 300 proper kpc centred on the main galaxy in each of our zoom-in simulations. We grid this volume into 1 by 1 proper kpc pixels to calculate the column density of $\mathrm{H}\,\textsc{i}$, $\mathrm{D}\,\textsc{i}$, and $\mathrm{O}\,\textsc{i}$. We assume that the neutral fraction is the same for all three atoms, because their ionization potentials are very similar, as is also done in observations.
![\[fig:HID\] The fraction of deuterium in neutral gas in sightlines with column density $10^{19}<N_\mathrm{H\,\textsc{i}}<10^{21}$ cm$^{-2}$, normalized by the primordial deuterium fraction, as a function of oxygen metallicity at $z=3$. We include all our zoom-in simulations. The black curve shows the median deuterium retention fraction in our simulations and the grey shaded regions show the $1\sigma$ and $3\sigma$ scatter around the median. The red error bars show absorption-line observations, with associated $1\sigma$ measurement errors and assuming that $\mathrm{(D/H)}_\mathrm{prim}=2.547\times10^{-5}$ [@Cooke2016]. Comparing these measurements to our simulations, we determine a best-fit value for the primordial deuterium fraction of $\mathrm{(D/H)}_\mathrm{prim}=(2.549\pm0.033)\times10^{-5}$, consistent with the weighted mean of the measurements assuming no metallicity dependence. Because of the tight correlation between (D/H) and \[O/H\], more metal-rich absorbers could also be used for this purpose, by calibrating to the relation between deuterium and oxygen abundances found here, allowing for the expansion of the observational sample.](figures/HIDz3_X.eps)
The black curve in Figure \[fig:HID\] shows the median fraction of deuterium in neutral gas, divided by its primordial value, in the selected LLSs and DLAs at $z=3$ as a function of their metallicity. The different grey scales show the $1\sigma$ (dark) and $3\sigma$ (light) scatter around the median. Observations of $(\mathrm{D}\,\textsc{i}/\mathrm{H}\,\textsc{i})$ compiled by @Cooke2016 and their associated $1\sigma$ errors are shown as red error bars, where we assumed that $\mathrm{(D/H)}_\mathrm{prim}=2.547\times10^{-5}$, the weighted mean of their measurements.
Our simulations confirm that at $\mathrm{[O/H]}\lesssim-2$ the deuterium abundance is very close to the primordial value (within 0.1 per cent), as seen before in Figure \[fig:DOz\]. These low-metallicity systems are therefore appropriate to use to determine $\mathrm{(D/H)}_\mathrm{prim}$. At $\mathrm{[O/H]}=-1$ the median deuterium abundance is still only 1 per cent below primordial, similar to the $1\sigma$ error in the weighted mean of the observational values from @Cooke2016. The scatter in the relation is even smaller than in Figure \[fig:DOz\], because we are including all (neutral) gas along a particular line-of-sight (rather than individual gas particles), decreasing the importance of small fluctuations. Observations of $N_\mathrm{H\,\textsc{i}}$, $N_\mathrm{D\,\textsc{i}}$, and $N_\mathrm{O\,\textsc{i}}$ are therefore well-suited to determine $\mathrm{(D/H)}_\mathrm{prim}$ and the relation between the deuterium and oxygen abundances.
Instead of assuming no variation as a function of metallicity for the 6 observed systems shown in Figure \[fig:HID\], we can test how well they match our simulations, which show a slight downward trend and minor additional scatter. We select those sightlines in our simulations that have the same metallicity as one of the observed absorbers, within $1\sigma$ errors. We then use least square fitting and calculate $\chi^2$ between our simulated sightlines and the observations as a function of $\mathrm{(D/H)}_\mathrm{prim}$, which sets the relative normalization. The minimum $\chi^2$ is reached for $\mathrm{(D/H)}_\mathrm{prim}=(2.549\pm0.033)\times10^{-5}$, where the errors are $1\sigma$ and calculated from the difference in $\chi^2$. This is consistent with theoretical models of Big Bang nucleosynthesis, based on cosmological parameters [@Coc2015; @Cyburt2016]. Our best estimate is very similar to, though slightly higher than, the weighted mean calculated by @Cooke2016, who assumed no metallicity dependence. For this low-metallicity sample, we do not gain much accuracy from comparing the data to our simulations.
However, given that the scatter in the simulations is much lower than the observational measurement error at all metallicities, more metal-rich absorption-line systems can be used to determine the primordial deuterium fraction. This would allow for the expansion of the observational sample, which would improve the accuracy of $\mathrm{(D/H)}_\mathrm{prim}$. Even absorbers with $\mathrm{[O/H]}\gtrsim-1$ can be used when taking into account the relation between the deuterium and oxygen abundance based on hydrodynamical simulations or on Equation \[eqn:T80\] combined with a prescription for the change of $X_\mathrm{gas}$ with metallicity. For the latter, one should use a slowly evolving ratio of recycling fraction to oxygen yield, $r/m_\mathrm{O}$, increasing with time as the contribution of mass lost by intermediate-mass stars increases. Vice versa, observations of metal-rich absorbers can set constraints on the ratio of the recycling fraction and the oxygen yield, assuming that the primordial abundance of deuterium is known from either CMB measurements or from absorption-line observations at $\mathrm{[O/H]}\lesssim-2$. $r/m_\mathrm{O}$ depends on the relative number of intermediate- and high-mass stars and on their stellar yields and can thus potentially help constrain the high-mass end of the stellar IMF and/or stellar evolution models.
It is important to note that the depletion of deuterium onto dust and preferential incorporation into molecules could cause large scatter in (D/H) between quasar sightlines at fixed metallicity, which are not due to variations in the recycled gas fraction. This is probably seen in the local ISM at solar metallicity [e.g. @Wood2004; @Linsky2006; @Prodanovic2010] and briefly discussed in Section \[sec:z0\]. Unfortunately we cannot address this issue with our current simulations. A relatively large sample of (D/H) measurements in absorption-line systems could quantify the scatter in (D/H) between sightlines at fixed metallicity. This will tell us whether the depletion of deuterium onto dust is important in the intergalactic medium at $\mathrm{[O/H]}>-2$, because our simulations have shown that the scatter due to variations in stellar mass loss at fixed metallicity is negligibly small. If dust depletion turns out to be dominant, these systems cannot be used for determining $\mathrm{(D/H)}_\mathrm{prim}$ or $r/m_\mathrm{O}$. However, there is some evidence that LLSs tend to reside in dust-poor environments [@Fumagalli2016]. Additionally, dust depletion seems to be less important in lower column density absorbers [e.g @Linsky2006; @Prodanovic2010]. It is therefore possible that these systems are well-suited for determining $\mathrm{(D/H)}_\mathrm{prim}$ or $r/m_\mathrm{O}$ even at $\mathrm{[O/H]}>-2$.
Deuterium fraction at low redshift {#sec:z0}
----------------------------------
To compare with observations of (D/H) at $z=0$, we focus on the deuterium fraction in the ISM of our simulated Milky Way-like galaxies. The black curves in Figure \[fig:Drad\] show how the ratio of the present-day abundance of deuterium to the primordial abundance varies with 3D distance from the galactic centre, $R_\mathrm{GC}$, for the same three galaxies as shown in Figure \[fig:Devol\]. For completeness, we show the recycled gas fraction as orange curves. $\mathrm{(D/D_{prim})}$ is similar to, though slightly lower than, $\mathrm{(D/H)/(D/H)}_\mathrm{prim}$, because of the decrease of the hydrogen fraction. One galaxy (dashed curve; ‘m11.9a’) has a central hole in its ISM, created by galactic winds, consistent with its relatively large average mass loading factor (see Section \[sec:evol\]). It therefore has no deuterium measurement at $R_\mathrm{GC}<4$ kpc. This galaxy has the lowest deuterium abundance at large radii, because the gas that was originally in its centre has been moved to larger radii. The deuterium retention fraction is low in the centres of the other two galaxies, where the density of stars is high and most of the star formation takes place. The deuterium fraction for all three galaxies increases with galactocentric radius, as previously shown by chemical evolution models [e.g. @Prantzos1996; @Chiappini2002; @Romano2006]. The importance of stellar mass loss therefore increases towards the galaxy centre and recycled gas accounts for about half of the gas in the central kpc. The steepness of the deuterium abundance gradient could also reveal information on the assembly history of a galaxy [e.g. @Prantzos1996]. In our sample, the galaxy with the flattest deuterium profile has the highest outflow rate. It may therefore depend more strongly on (bursty) galactic outflows than on (smooth) gas accretion.
![\[fig:Drad\] The mean fraction of deuterium in the ISM of three star-forming Milky Way-mass galaxies, normalized by the primordial deuterium fraction, as a function of galactocentric radius at $z=0$ (black curves; left axis). The radial dependence of the recycled gas fraction is similar to that of the deuterium fraction and shown as orange curves (right axis). One galaxy (dashed curve; ‘m11.9a’) has a central hole in its ISM at $R_\mathrm{GC}<4$ kpc, created by a strong outflow. This galaxy also has the lowest deuterium abundance at large radii, because it ejected a large amount of gas from its centre into its surroundings. For all three galaxies, the deuterium fraction increases with galactocentric radius. Our simulations are consistent with observational determinations of the deuterium fraction in the local Milky Way ISM [crosses with error bars, @Linsky2006; @Prodanovic2010]. In the galaxy centres, where most star formation occurs, about half of the gas originates from stellar mass loss.](figures/Drad_X.eps)
Large scatter exists between measurements of the deuterium abundance in the local ISM via absorption-line observations. This scatter could be explained by localized infall of pristine gas, with very little mixing. In this case, the average astration factor is relatively high (and mass lost from stars dominates the ISM). However, if this is the case, the oxygen abundance is also expected to decrease locally as it becomes diluted with the metal-free, infalling gas, resulting in large scatter. The fact that oxygen shows much smaller abundance variation than deuterium argues against such localized infall [@Oliveira2005]. Another, more likely, explanation for the large (D/H) sightline variations is that some of the deuterium is depleted onto dust. The probability of deuterium depletion onto dust grains and incorporation into molecules is high, since the zero-point energies of deuterium-metal bonds are lower than those of the corresponding hydrogen-metal bonds [@Jura1982; @Tielens1983]. When the ISM is heated, dust grains and molecules can be destroyed, returning deuterium to the atomic gas phase. Metals, such as iron, silicon, and titanium, are also depleted onto dust grains and the correlation of their abundances with deuterium supports this theory [@Prochaska2005; @Linsky2006; @Lallement2008]. Based on the assumption that the observational scatter is caused by deuterium depletion onto dust, relatively high deuterium abundances, and low astration factors, are derived for the local Milky Way ISM by @Linsky2006 and @Prodanovic2010.
The deuterium retention percentages in the solar neighbourhood, here defined as $7<R_{GC}<9$ kpc, lie between 85 and 92 per cent for our three simulations of star-forming galaxies with masses similar to that of the Milky Way[^5]. This is consistent with Figure \[fig:DOz\], where we found that 91 per cent of its primordial value is recovered for $\mathrm{[O/H]}=0$ at $z=0$. Using the value from @Cooke2016 for the primordial deuterium fraction as in Section \[sec:z3\], the deuterium abundance derived by @Linsky2006 implies that the local ISM still contains $91^{+9}_{-10}$ per cent of the primordial deuterium abundance. This is consistent with our simulations within $1\sigma$. @Prodanovic2010 use the same data compilation, but a different method, to derive a deuterium retention percentage of $79\pm4$ per cent (again assuming $\mathrm{(D/H)}_\mathrm{prim}=2.547\times10^{-5}$). This is consistent (within $2\sigma$) with our most massive Milky Way-like galaxy, with $M_\mathrm{star}=10^{10.8}$ M$_{\astrosun}$. However, @Prodanovic2010 stress that their measurement can also be interpreted as a lower limit in the event that all available sightlines are affected by dust depletion. In this case, our other galaxies are also consistent with their model. Our simulations exhibit low astration factors and therefore agree with the explanation that the large scatter in local ISM observations is due to dust depletion rather than due to poor mixing of freshly accreted gas.
No known galaxy besides the Milky Way has a measurement of the deuterium fraction in their ISM. Such observations would be interesting, because our simulations predict a strong dependence on stellar mass. Figure \[fig:DMstar\] shows the mean deuterium retention fraction (top panel) and recycled gas fraction (bottom panel) within 20 kpc of the centre of the galaxy for gas with a temperature below $10^4$ K as a function of stellar mass at $z=0$. Due to the depletion of hydrogen at higher metallicity, the differences between the deuterium retention fraction and recycled gas fraction increase with stellar mass, but the trends with mass remain the same. The black crosses show the mass-weighted mean, while the red diamonds show the (instantaneous) SFR-weighted mean. The latter is therefore a better indicator of how important stellar mass loss is for the fuelling of star formation, whereas the former is the value that would be measured, for example, in sightlines through the ISM. The galaxy shown in grey with the highest stellar mass has large uncertainties for its deuterium abundance, because it contains only 12 star-forming gas particles (but more than 1000 gas particles in total). Two other massive galaxies are not included, because they contained no star-forming gas and very little non-star-forming gas. Galaxies with $M_\mathrm{star}<10^8$ M$_{\astrosun}$ are also excluded, because they contained no or very little star-forming gas. The other galaxies (shown in black, red, and blue) have 100 star-forming gas particles or more.
![\[fig:DMstar\] The abundance of deuterium normalized by the primordial deuterium abundance (top panel) and the recycled gas fraction (bottom panel) in each galaxy’s ISM, as a function of stellar mass at $z=0$. The black crosses show the mass-weighted mean deuterium retention fraction and the red diamonds show the SFR-weighted mean values. The blue triangles show the mass-weighted $\mathrm{(D/H)/(D/H)}_\mathrm{prim}$ (or $f_\mathrm{recycled}$) for the gas that accreted, i.e. reached $R_\mathrm{GC}<20$ kpc and $T<10^4$ K, after $z=0.1$. The most massive galaxy (grey symbols) has limited ISM resolution and therefore large uncertainty in its deuterium abundance. In general, the deuterium fraction decreases with increasing stellar mass. This means that stellar mass loss is more important for feeding the ISM of high-mass galaxies. Stellar mass loss is even more critical for fuelling star formation, which preferentially occurs at the highest ISM densities. Accreting gas has a deuterium fraction between that of the ISM and the primordial value, which likely means that it is a combination of pristine infalling gas and gas ejected or stripped from satellites and/or gas reaccreting as part of a galactic fountain.](figures/Dmass_X.eps)
There is a definite trend with stellar mass, where more massive galaxies generally have lower deuterium fractions and thus a larger contribution of stellar mass loss. This is a clear prediction from our simulations. The exception is the lowest mass galaxy which has a low (D/H), as well as a high \[O/H\] (as expected from Figure \[fig:DOz\]). This reflects the variation in star formation, inflow, and outflow history. Our sample of galaxies is limited, but quantifying the scatter in the deuterium fraction between galaxies of similar stellar mass would help us to understand how this correlates with their formation history.
Due to their tight relation, the oxygen abundance could provide very similar information as the deuterium abundance. We expect this to work well at subsolar metallicities. At solar and supersolar metallicities, the dependence of (D/H) on \[O/H\] is very steep, which means that to determine the recycled gas fraction, very precise measurements of \[O/H\] are needed. Additionally, the scatter in the relation increases towards high metallicity at $z=0$, due to varying contributions of AGB stars, impeding the determination of the recycled gas fraction through \[O/H\].
The deuterium fraction is always lower when weighted by star formation rate (red diamonds) rather than by mass (black crosses), because this dense, star-forming gas is close to newly formed stars with high mass loss rates. This means that mass loss is more important for fuelling star formation than it is for replenishing the more diffuse ISM. A measurement of the deuterium fraction (such as those in the local ISM) generally provides information on the latter, but not the former.
In our simulations, the star formation in low-mass galaxies is fuelled predominantly by gas accretion, although there is a small contribution by stellar mass loss. For two of the galaxies with $M_\mathrm{star}\approx10^{10.5}$ M$_{\astrosun}$, mass loss fuels about half of the current star formation, while one other is still dominated by gas accretion. For the massive galaxy (or galaxies, if we include the one with large uncertainties) with $M_\mathrm{star}\approx10^{11}$ M$_{\astrosun}$, mass loss fuels the majority of star formation. As mentioned in Section \[sec:intro\], stellar mass loss has been suggested as fuel for the observed star formation in local, massive, early-type galaxies and in central cluster galaxies. Our simulations are consistent with this interpretation.
Notably, there are also detections of neutral deuterium in quasar sightlines through clouds outside the galactic disc of the Milky Way, in the lower galactic halo [@Sembach2004; @Savage2007]. These clouds could provide the fuel to sustain the Milky Way’s steady star formation. However, the errors associated with these measurements are large and therefore not very constraining. More precise determinations of (D/H) could help constrain the nature of this gas. If the deuterium fraction of the gas around the Milky Way is high, it is likely pristine gas falling in from the IGM. If the deuterium fraction is similar to that of the Milky Way, it is likely part of a galactic fountain [@Shapiro1976]. In intermediate cases, the gas could be a mix of both high and low (D/H) gas or the gas could be ejected or stripped from lower mass satellites [e.g. @Angles2017]. These different theories can potentially be tested via estimates of the scatter in (D/H) in the halo, which would be large if the fountain gas is not fully mixed with the gas falling in from the IGM, but small if the gas was ejected or stripped from satellites.
In order to compare the deuterium fraction in the galaxy to the deuterium fraction of accreting gas, the blue triangles in Figure \[fig:DMstar\] show $\mathrm{(D/H)/(D/H)}_\mathrm{prim}$ at $z=0.1$ of the gas that is accreted at $0<z<0.1$, which means it is located at $R_\mathrm{GC}<20$ kpc and has $T<10^4$ K at $z=0$, but is located at larger radii and/or has higher temperatures at $z=0.1$. Our radial boundary is somewhat arbitrary, but our results are not very sensitive to this (except for the normalization, because the deuterium fraction is generally higher at larger radius). The accreting gas has a higher deuterium retention fraction than the ISM gas. This is consistent with a substantial part of the material accreting for the first time onto a galaxy. The fact that $\mathrm{(D/H)/(D/H)}_\mathrm{prim}<1$, however, shows that another part of the accreting gas has previously been ejected or stripped from a galaxy. Such a combination of different accretion channels has already been found in these and other simulations [e.g. @Oppenheimer2008; @Christensen2016; @Angles2017; @Voort2017]. The difference in (D/H) between accreting gas and the ISM is especially large for the more massive galaxies in our sample. As mentioned above, this balance between accretion from the IGM and reaccretion could potentially be shown observationally by measuring (D/H) in clouds that are accreting onto the Milky Way [e.g. @Sembach2004].
Discussion and conclusions {#sec:concl}
==========================
We have quantified the evolution of the deuterium fraction and its dependence on the oxygen abundance, galactocentric radius, and stellar mass in cosmological zoom-in simulations with strong stellar feedback. Because deuterium is only synthesized in the early Universe, it provides an interesting constraint on cosmology and on galaxy evolution. The normalized deuterium fraction is a measure of the recycled gas fraction, i.e. the fractional contribution of stellar mass loss to the gas, because deuterium is completely destroyed in stars and therefore $f_\mathrm{recycled}=1-\mathrm{(D/D_{prim})}$. Observations, however, measure $\mathrm{(D/H)/(D/H)}_\mathrm{prim}$, which is higher than $\mathrm{(D/D_{prim})}$, especially at high metallicity (because $X_\mathrm{gas}/X_\mathrm{prim}\approx1-3Z_\mathrm{gas}$). Our simulations self-consistently follow gas flows into and out of galaxies and the (metal-rich and deuterium-free) mass loss by supernovae and AGB stars. We have compared our predictions to available observations at low and high redshift and found them to be consistent. Our main conclusions can be summarized as follows:
- The deuterium fraction exhibits a tight correlation with the oxygen abundance, evolving slowly with redshift (Figure \[fig:DOz\]). This is captured well by simple chemical evolution models [@Tinsley1980; @Weinberg2017], which depend only on the ratio of the recycling fraction and the oxygen yield. We find a small increase in $r/m_\mathrm{O}$ with time, because of the increased importance of AGB stars. The variation in the importance of AGB stars at fixed oxygen abundance can be traced by the iron abundance and is responsible for some of the (small) scatter in the deuterium fraction (Figure \[fig:DFe\]).
- The three Milky Way-mass galaxies in our sample exhibit different evolution at low redshift (Figure \[fig:Devol\]). The galaxies that form many of their stars at late times have continually decreasing deuterium fractions in their ISM. The galaxy which forms most of its stars before $z=1$ shows a constant deuterium fraction in the last $\approx5$ Gyr, indicating that it may have reached chemical equilibrium. The evolution of the deuterium fraction may therefore be directly connected to the galaxy’s star formation history.
- The deuterium fraction is very close to primordial at $\mathrm{[O/H]}\lesssim-2$ (within 0.1 per cent). These are the metallicities of LLSs and DLAs typically used to measure the primordial deuterium abundance. Because of the tight correlation with metallicity, deuterium measurements in more metal-rich systems can also be used to constrain the primordial deuterium fraction (Figure \[fig:HID\]) if dust depletion is unimportant.
- We compared our simulations to the observational sample of @Cooke2016 to determine a primordial deuterium fraction of $\mathrm{(D/H)}_\mathrm{prim}=(2.549\pm0.033)\times10^{-5}$, very close to, though slightly higher than, their original estimate, which assumed no dependence of the measured (D/H) on metallicity. Our result is also in agreement with cosmological parameters and Big Bang nucleosynthesis [@Coc2015; @Cyburt2016].
- The deuterium fraction increases with galactocentric radius. Our simulations are consistent with the available estimates from the local Milky Way ISM where the observed scatter between sightlines is assumed to be caused by depletion onto dust (Figure \[fig:Drad\]).
- The deuterium fraction decreases with increasing stellar mass, which means that the importance of stellar mass loss in our simulations increases with stellar mass (Figure \[fig:DMstar\]). Mass loss is more important for fuelling star formation than for replenishing the general ISM (which has, on average, a lower gas density). Accreting gas has a higher deuterium fraction than the ISM of galaxies, but lower than primordial. This is consistent with previous findings that some gas accretes directly from the IGM, but some has been previously been ejected or stripped from a galaxy [e.g. @Angles2017].
Due to the tight correlation of (D/H) and \[O/H\], shown in Figure \[fig:DOz\], measurements of the oxygen abundance could provide the same information as the deuterium abundance if this relation is accurately calibrated by observations at $\mathrm{[O/H]}\gtrsim-1$. The relation evolves slowly, because of the increased importance of stellar mass loss from AGB stars, which should be taken into account (see Section \[sec:evol\]). If the importance of mass loss for fuelling the ISM increases for massive galaxies, as in our simulations, (D/H) will decrease with mass and \[O/H\] (and other metal abundances) will increase. There is observational evidence from gas-phase and stellar metallicities that this is indeed the case, although \[O/H\] may saturate at the highest stellar masses [e.g. @Tremonti2004; @Gallazzi2005; @Mannucci2010; @Peng2015]. Using the median relation derived from our simulations (Figure \[fig:DOz\]) one can immediately estimate the contribution of stellar mass loss given the gas-phase oxygen metallicity. However, this only works in the situation where galactic winds remove mass loss from supernovae and AGB stars (approximately) equally, but may not if ejecta from young stars are removed from a galaxy (through supernovae or AGN, quenching star formation) after which its ISM is replenished by mass loss from old stars alone. This is likely the reason that the scatter in (D/H) increases at supersolar metallicity in our simulations. In observations, however, the scatter at high metallicity is probably dominated by the depletion of deuterium onto dust, an effect which is not included in our calculations.
Our cosmological, hydrodynamical simulations follow time-dependent chemical enrichment and the assembly of galaxies self-consistently, whereas simple, analytical chemical evolution models assume instantaneous recycling and specify a specific star formation history [@Tinsley1980; @Weinberg2017]. We nevertheless compare our results to these models and find a remarkable agreement (especially with @Tinsley1980) when considering the relation between the deuterium and oxygen abundance, despite the very different methods used. Although other galaxy properties are not necessarily well reproduced in these simplified models [e.g. @Binney1998], these do not play a dominant role when relative abundances are concerned. The bursty star formation and strong galactic outflows present in our simulations thus have no major impact on the correlation between (D/H) and \[O/H\]. Although @Weinberg2017 find that a relatively high mass loading factor is necessary to match the $z=0$ deuterium abundance in the local ISM, two of our simulated galaxies show low mass loading factors at late times (but high mass loading factors at early times, see @Muratov2015).
@Leitner2011 use zoom-in simulations with and without stellar mass loss to show that mass loss dominates the fuelling of star formation at late times for galaxies in haloes of similar mass to that of the Milky Way. However, their simulation did not include outflows from either star formation or AGN, which results in the galaxies being too massive at $z=0$. This means that too much baryonic mass is locked up in stars and therefore less gas is available for accretion at late times. Furthermore, the majority of the mass lost by stars is retained by the galaxy and not ejected by a galactic wind. Our simulations show that for Milky-Way mass galaxies with strong stellar feedback, cosmological inflow either dominates over or rivals stellar mass loss for the fuelling of star formation. However, recycled gas dominates the SFR at $M_\mathrm{star}\approx10^{11}$ M$_{\astrosun}$ in our simulations.
@Segers2016 use large-volume simulations with stellar and AGN feedback that match the stellar–to–halo mass relation and find that the contribution of mass loss to the fuelling of star formation is largest for galaxies with $M_\mathrm{star}\approx10^{10.5}$ M$_{\astrosun}$. The deuterium fraction in the ISM is therefore the lowest around this mass and increases for more massive galaxies. Our simulations (without AGN feedback) find a different behaviour at the high-mass end, where mass loss becomes increasingly important and the deuterium fraction decreases. @Segers2016 also show results from simulations without AGN feedback, which do not match the stellar–to–halo mass relation, but agree qualitatively with our simulations. In the absence of AGN feedback, there are no strong galactic outflows, which means that most of the stellar mass loss is retained in the ISM. Different implementations of AGN feedback (ejective or preventive) could also result in different deuterium abundances. Future observations of (D/H) in massive galaxies therefore have the potential to discriminate between these different models.
Numerical chemical evolution models generally find somewhat lower deuterium fractions (higher astration factors) than we do in our cosmological simulations [e.g. @Dearborn1996; @Tosi1998; @Chiappini2002; @Romano2006]. This mild difference could be due to the different assumptions made for the stellar IMF, metal yields, and stellar mass loss. A detailed quantitative comparison between the two different numerical approaches requires using the same IMF and stellar evolution models, which is left for future work. The star formation history of a galaxy determines the relative importance of AGB stars with respect to core-collapse supernovae, which may explain any remaining discrepancy. This could potentially be tested by comparing the oxygen and iron abundances (see Figure \[fig:DFe\]). Another possibility is that the inclusion of galactic outflows in our simulations reduces the importance of stellar mass loss, because a substantial fraction of the recycled gas is ejected from the ISM.
The large observed scatter in the deuterium fraction between different sightlines through the local ISM could be interpreted as evidence for an inhomogeneous ISM due to localized gas accretion, giving rise to a low average deuterium fraction [e.g. @Hebrard2003]. Our work instead favours the interpretation that the ISM is well-mixed and the observational scatter is caused by the depletion of deuterium onto dust, which leads to a local deuterium fraction not much lower than the primordial value [@Linsky2006; @Prodanovic2010]. It is also possible that both dust depletion and localized infall play a roll and the true value is intermediate between those cases [@Steigmanetal2007]. More precise observations of the deuterium and metal abundances would help clarify which of these interpretations is correct.
In summary, we have quantified the deuterium fraction in a suite of zoom-in simulations and found it to be tightly correlated with the oxygen metallicity and consistent with current observational constraints. We conclude that the primordial deuterium fraction (and thus early cosmological expansion and Big Bang nucleosynthesis) can also be constrained by using observations at medium to high metallicity in combination with our simulations. Or, vice versa, if the primordial deuterium fraction is known, these measurements can inform us about the ratio of the recycling fraction to the oxygen yield and thus about the high-mass end of the stellar IMF and stellar evolution models. Our simulations predict that the deuterium fraction is lower at smaller galactocentric radii and for higher mass galaxies. This means that stellar mass loss could provide most of the fuel for star formation in massive early-type galaxies and in the centres of less massive, star-forming galaxies. Grid-based calculations or SPH simulations with explicit diffusion would be useful to determine whether or not small-scale mixing modifies the deuterium and oxygen abundances. Accurate observations of the deuterium fraction provide us with the possibility to understand the fuelling of star formation through stellar mass loss in galaxies in general and the Milky Way in particular.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank the Simons Foundation and the organizers and participants of the Simons Symposium ‘Galactic Superwinds: Beyond Phenomenology’, in particular David Weinberg, for interesting discussions and inspiration for this work. We also thank Thomas Guillet and Joop Schaye for helpful discussions and Tim Davis for useful comments on an earlier version of the manuscript. We would like to thank the referees for valuable comments that helped clarify our results and put them into context. Support for FvdV was provided by the Klaus Tschira Foundation. EQ was supported by NASA ATP grant 12-APT12-0183, a Simons Investigator award from the Simons Foundation, and the David and Lucile Packard Foundation. CAFG was supported by NSF through grants AST-1412836 and AST- 1517491 and by NASA through grant NNX15AB22G. DK was supported by the NSF through grant AST-1412153 and by the Cottrell Scholar Award from the Research Corporation for Science Advancement. Support for PFH was provided by an Alfred P. Sloan Research Fellowship, NASA ATP Grant NNX14AH35G, and NSF Collaborative Research Grant \#1411920 and CAREER grant \#1455342. Numerical calculations were run on the Caltech compute cluster “Zwicky” (NSF MRI award \#PHY-0960291), through allocation TG-AST120025, TG-AST130039 and TG-AST150045 granted by the Extreme Science and Engineering Discovery Environment (XSEDE) supported by the NSF, and through NASA High-End Computing (HEC) allocation SMD-14-5189, SMD-14-5492, SMD-15-5950, and SMD-16-7592 provided by the NASA Advanced Supercomputing (NAS) Division at Ames Research Center.
\[lastpage\]
[^1]: E-mail: freeke.vandevoort@h-its.org
[^2]: http://fire.northwestern.edu/
[^3]: http://www.tapir.caltech.edu/[.17ex]{}phopkins/Site/GIZMO.html
[^4]: Note that in the literature, this equation is usually expressed with a yield defined with respect to the final stellar remnant mass (after mass loss) rather than the initial stellar mass as is the case in our definition of $m_\mathrm{O}$.
[^5]: Although our calculations are done using 3D distance, the results are unchanged when we restrict ourselves to the ISM, because most of the gas mass lies within the star-forming disc.
|
---
author:
- 'V.R. Khalack , F. LeBlanc, B.B. Behr[^1], G.A. Wade, D. Bohlender'
date: 'Received [*date will be inserted by the editor*]{} Accepted [*date will be inserted by the editor*]{} '
title: 'Search for vertical stratification of metals in atmospheres of blue horizontal-branch stars'
---
[The observed abundance peculiarities of many chemical species relative to the expected cluster metallicity in blue horizontal-branch (BHB) stars presumably appear as a result of atomic diffusion in the photosphere. The slow rotation (typically $v\sin{i}<$ 10 km s$^{-1}$) of BHB stars with effective temperatures $T_{\rm eff}>$ 11,500 K supports this idea since the diffusion mechanism is only effective in a stable stellar atmosphere.]{} [In this work we search for observational evidence of vertical chemical stratification in the atmospheres of six hot BHB stars: B84, B267 and B279 in M15 and WF2-2541, WF4-3085 and WF4-3485 in M13.]{} [We undertake an abundance stratification analysis of the stellar atmospheres of the aforementioned stars, based on acquired Keck HIRES spectra.]{} [We have found from our numerical simulations that three stars (B267, B279 and WF2-2541) show clear signatures of the vertical stratification of iron whose abundance increases toward the lower atmosphere, while the other two stars (B84 and WF4-3485) do not. For WF4-3085 the iron stratification results are inconclusive. B267 also shows a signature of titanium stratification. Our estimates for radial velocity, $v\sin{i}$ and overall iron, titanium and phosphorus abundances agree with previously published data for these stars after taking the measurement errors into account. The results support the hypothesis regarding the efficiency of atomic diffusion in the stellar atmospheres of BHB stars with $T_{\rm eff}>$ 11,500 K. ]{}
Introduction
============
According to the current understanding of stellar evolution, the horizontal-branch (HB) stars are post-main sequence stars that burn helium in their core and hydrogen in a shell (e.g. Moehler [@Moehler04]). In this paper we consider the HB stars that are located in the blue part of the HB, to the left of the RR Lyrae instability strip. Most researchers call them blue horizontal-branch (BHB) stars to distinguish them from the red horizontal-branch (RHB) stars, which exhibit different observational properties. Sandage & Wallerstein ([@S+W60]) have found from analysis of the colour-magnitude diagrams of globular clusters[^2] that the HB generally becomes bluer with decreasing metallicity. Derived masses of the cool ($T_{\rm eff}<$11,500 K) BHB stars in the globular cluster NGC 6388 (Moehler & Sweigart [@Moehler+S06]) are in a good agreement with the predictions of canonical HB evolution, except for the hot BHB stars with $T_{\rm eff}>$11,500 K, where the estimated stellar masses seem to be lower than the canonical values.
\[tab1\]
The Hertzsprung-Russell diagrams of some globular clusters show long blue tails (an extension of the HB), populated by very hot BHB stars and extreme horizontal branch (EHB) stars. Published data on the BHB stars argue that the hot BHB stars show remarkable differences in physical properties when compared to the cool BHB stars. Using high-precision photometry of stars in M13, Ferraro et al. ([@Ferraro+98]) have found gaps in the distribution of stars along the blue tail. One of these gaps, labeled as G1, is located at $T_{\rm eff}\sim$11,000-12,000 K. Grundahl et al. ([@Grundahl+98]) used the results of Strömgren $uvby\beta$-photometry finding good agreement between the theoretical prediction of stellar evolution models and the observed location of BHB stars, except for the hot BHB stars, whose $u$-magnitudes are brighter than predicted. It appears that this $u$-jump is observed for the hot BHB stars and coincides with the temperature range of the G1 gap in M13. Similar $u$-jumps have also been found for other globular clusters (Grundahl et al. [@Grundahl+99]). For the hot BHB with effective temperatures up to 20,000 K, the surface gravities derived from the fits of Balmer and helium line profiles appear to be lower than the predictions of stellar evolution models, while the gravities derived for the stars outside this temperature range are in good agreement with theoretical predictions (Moehler et al. [@Moehler+95; @Moehler+97a; @Moehler+97b; @Moehler+03]). The stellar rotation velocity distribution of BHB stars also appears to have a discontinuity at $T_{\rm eff} \simeq $ 11,500 K (Peterson et al. [@Peterson+95]; Behr et al. [@Behr+00a]; Recio-Blanco et al. [@RB+04]), indicating that the hotter stars show modest rotation with $v\sin{i}<$ 10 km s$^{-1}$, while the cooler stars rotate more rapidly on average. Comprehensive surveys of abundances also show that the hot BHB stars have abundance anomalies when compared to the cool BHB stars in the same cluster (Glaspey et al. [@Glaspey+89]; Grundahl et al. [@Grundahl+99]; Behr et al. [@Behr+99; @Behr+00b], Behr [@Behr03a]; Fabbian et al. [@Fabbian+05]; Pace et al. [@Pace+06]).
The observed phenomena such as the low gravity, photometric jumps and gaps, abundance anomalies and slow rotation suggest that atomic diffusion could be important in the stellar atmospheres of hot BHB stars. Atomic diffusion arises from the competition between radiative acceleration and gravitational settling. This can produce a net acceleration on atoms and ions, which results in their diffusion through the atmosphere (Michaud [@Michaud70]). In order for atomic diffusion to produce a vertical stratification of the abundances of particular elements, the stellar atmosphere must be hydrodynamically stable. According to Landstreet ([@Landstreet98]), photospheric convection should be very weak at the effective temperatures of BHB stars. Theoretical atmospheric models of Hui-Bon-Hoa, LeBlanc & Hauschildt ([@Hui-Bon-Hoa+00]) showed that the observed photometric jumps and gaps for hot BHB stars can be explained by elemental diffusion in their atmosphere. Behr ([@Behr03b]) has shown that adoption of a microturbulent velocity of 0 or 1 km s$^{-1}$ provides the best fit to line strengths in the spectra of hot BHB stars. This fact supports the proposal that strong velocity fields are not present in the atmospheres of hot BHB stars.
While synthesizing spectral line profiles, Khalack et al. ([@Khalack+07]) have recently found vertical abundance stratification of sulfur in the atmosphere of the field BHB star HD 135485. In this paper we also attempt to detect signatures of vertical abundance stratification of elements from line profile analyses of several other BHB stars for which we have appropriate data. Together with the data on stratification of the sulfur abundance in HD 135485, new positive results would provide a convincing argument in favour of efficient atomic diffusion in the atmospheres of hot BHB stars. In Sec. \[obs\] we discuss the properties of the acquired spectra, while in Sec. \[mod\] we describe details concerning the simulation routine and adopted atmospheric parameters for the program stars. The evidence for vertical stratification of some chemical species is given in Sec. \[vert\], while the estimation of mean abundances and velocities is described in Sec. \[mean\]. A discussion follows in Sec. \[discuss\].
Observations {#obs}
============
In this paper we have selected hot BHB stars from the list of objects published by Behr ([@Behr03b]) and which have comparatively high signal-to-noise ratio (S/N) spectra available. Spectroscopic observations of the selected stars were undertaken in August 1998 with the Keck I telescope and the HIRES spectrograph. The journal of spectroscopic observations is shown in Table \[tab1\] where individual columns give the object identification, the heliocentric Julian Date of the observation, the exposure time, the S/N per pixel the spectral coverage, the size of the seeing disk (FWHM) and the size of the C1 slit that provides a spectral resolution of $R=\lambda/\delta\lambda=$45,000. For the aforementioned stars, Behr ([@Behr03b]) has found that the underfilling of the slit should not change the estimated spectral resolution by more than 4%–7%.
The package of routines developed by McCarthy ([@McCarthy90]) for the FIGARO data analysis package (Shortridge [@Shortridge93]) was employed to process the spectra. A comprehensive description of the data acquisition and reduction procedure is presented by Behr ([@Behr03b]). This followed a standard prescription with bias subtraction, flat-fielding, order extraction, and wavelength calibration from thorium-argon comparison lamp observations. To minimize the potential distortion of narrow spectra features, cosmic ray hits were identified and removed by hand.
Line profile simulations {#mod}
========================
Stellar atmosphere parameters
-----------------------------
The line profile simulations were performed with the [Zeeman2]{} spectrum synthesis code (Landstreet [@Landstreet88]; Wade et al. [@Wade+01]). The stellar atmosphere models were calculated with the [Phoenix]{} code (Hauschildt et al. [@Hauschildt+97]) assuming LTE (Local Thermodynamic Equilibrium) and using the stellar atmosphere parameters extracted from Behr ([@Behr03b]) and listed in Table \[tab2\]. In calculating the stellar atmosphere models for program stars, we have used solar metallicity with the enhanced iron and depleted helium abundances derived by Behr ([@Behr03b]). The depleted helium abundance has also been taken into account during line profile simulations using [Zeeman2]{}. We used gaussian instrumental profiles with widths derived from the comparison arc spectra.
To simulate the spectra of BHB stars, Behr ([@Behr03b]) initially adopted a microturbulent velocity $\xi=$2 km s$^{-1}$ and subsequently updated it during the simulation routine according to the results of his model fits. He has obtained best fit values of $\xi$ from 0 to 0.9 km s$^{-1}$ for the BHB stars studied here (see Table \[tab2\]). Since we search for possible vertical abundance stratification, our results can be affected by microturbulence. To estimate this influence we have adopted here two different microturbulent velocities $\xi=$0 and 2 km s$^{-1}$ for our simulations.
-------------- --------------- ----------- --------------- ---------------
Cluster/Star $T_{\rm eff}$ $\log{g}$ $\xi$ $V_{\rm r}$
(K) (dex) (km s$^{-1}$) (km s$^{-1}$)
M13/WF2-2541 13000 4.0 0.0 -257.5
M13/WF4-3085 14000 4.0 0.0 -255.7
M13/WF4-3485 13000 4.0 0.0 -246.8
M15/B84 12000 3.5 0.5 -108.2
M15/B267 11000 3.5 0.0 -114.4
M15/B279 11000 3.5 0.9 -104.4
-------------- --------------- ----------- --------------- ---------------
: Parameters of stellar atmospheres for the selected hot BHB stars from Behr ([@Behr03b]).
\[tab2\]
Procedure {#proc}
---------
We examined the spectrum of each star to establish a line list suitable for abundance and stratification analysis. The line identification was performed using the VALD-2 (Kupka et al. [@Kupka+99]; Ryabchikova et al. [@Ryab+99]), Pickering et al. ([@Pickering+01]) and Raassen & Uylings[^3] ([@RU+98]) line databases. For all of the program stars the best represented element (with a number of readily visible line profiles) is iron. Some stars also show Ti[ii]{} or P[ii]{} lines. Therefore iron, titanium and phosphorus were selected for analysis of their possible vertical abundance stratification. For Fe[ii]{} lines, we extracted atomic data from Raassen & Uylings ([@RU+98]), while for P[ii]{} lines we used VALD-2. Atomic data for Ti[ii]{} lines were taken from Pickering et al. ([@Pickering+01]).
$\lambda$, Å $\log gf$ $E_i, cm^{-1}$ $\log \gamma_{\rm rad}$ Ref.
-------------- ----------- ---------------- ------------------------- ------ ------ ------ ----- ----- ----- -----
WF2-2541 WF4-3085 WF2-3585 B84 B267 B279
4122.668 -3.300 20830.58 8.49 R&U ... x ... x ... ...
4128.748 -3.578 20830.58 8.61 R&U ... x ... ... ... x
4173.461 -2.617 20830.58 8.61 R&U x x x x x ...
4177.618 -3.776 73395.93 8.61 R&U ... x ... x ... x
4177.692 -3.449 20516.96 8.61 R&U ... x ... x ... x
4178.862 -2.535 20830.58 8.49 R&U x x x x ... x
4184.261 -1.938 90397.87 8.35 R&U ... x ... ... ... ...
\[tab3\]
The [Zeeman2]{} code has been modified (Khalack & Wade [@Khalack+Wade06]) to allow for an automatic minimization of the model parameters using the [*downhill simplex method*]{} (Press et al. [@press+]). The minimization routine finds the global minimum of the $\chi^\mathrm{2}$ function, which is specified as the measure of differences between simulated and observed line profiles. The relatively poor efficiency of the downhill simplex method, requiring a large number of function evaluations, is a well known problem. However, repeating the minimization routine several times in the vicinity of a supposed minimum in the chosen parameter space allows us to verify if the method converges to the global minimum (for more details about this minimization routine see Khalack et al. [@Khalack+07]).
To search for the presence of abundance stratification, we have estimated the abundance of a chemical element from an independent analysis of each selected line profile. This method operates with three free model parameters (the element’s abundance, the radial velocity $V_{\rm r}$ and $v\sin{i}$) that are derived from each line profile using the aforementioned automatic minimization routine. To analyse the vertical abundance stratification in this method we build the scale of optical depths $\tau_{\rm 5000}$ for the list of selected line profiles. First, we calculate the line optical depth $\tau_{\rm \ell}$ in the line core for every layer of the stellar atmosphere model. Next, we suppose that each analyzed profile is formed mainly at $\tau_{\rm \ell}$=1, which corresponds to a particular layer of the stellar atmosphere with respective continuum optical depth $\tau_{\rm 5000}$. All simulations are performed with stellar atmosphere models that contain 50 layers.
In general, we have selected for our analysis lines found to be free of predicted or inferred blends. However, if a blend is from a line of the same element that forms the main line profile, such a line was also included in our simulation. When a simulated line profile results in a radial velocity which differs significantly (more than 2 km s$^{-1}$) from the average $V_{\rm r}$ for the analysed star, we exclude such a line from further consideration. The difference in radial velocity may be evidence of line misidentifications or inaccurate line wavelengths. Atomic data (and sources) for the final list of iron lines selected for our study is given in Table \[tab3\].
Vertical abundance stratification {#vert}
=================================
Applying the technique described in Subsec. \[proc\], we have attempted to determine if the iron abundance is vertically stratified in the atmospheres of the six selected BHB stars. For comparison, we have also investigated the possible vertical stratification of titanium and phosphorus for some BHB stars, where these elements are represented by a sufficient number of spectral lines.
All of the selected BHB stars are slowly rotating and have sharp absorption lines that result in only 8 to 13 spectral bins per line profile. This fact, together with the S/N and the uncertainties in the atomic data, are the main contributors to errors in the abundance inferred from a single line. This makes the detection of weak vertical abundance variations difficult.
Our abundance analysis for $\xi = 0$ kms$^{-1}$ shows that in the atmospheres of four BHB stars (B267, B279, WF4-3085 and WF2-2541) the iron abundance generally increases towards the deeper atmospheric depths (see Figs. \[B267\] to \[WF2-2541\]). It should be noted that the range of optical depths diagnosed by the iron lines is much smaller in the stars B279 and WF2-2541 than in the other two stars shown here. To estimate the stratification profile of the iron abundance we fit the data with a straight line using a least-square algorithm. We have also checked the dependence of each element’s abundance with respect to the lower level excitation potential $E_{i}$ for the sample of analyzed lines (see as an example Fig. \[excitation\]). Slopes of the linearly approximated dependence of each element’s abundance with respect to $\log{E_{i}}$ and $\log{\tau_{5000}}$ are given in Table \[slope\] for the case of zero microturbulent velocity. Table \[slope\] shows a clear correlation between the slope of the abundance versus $\log{\tau_{5000}}$ and $\log{E_{i}}$ for the elements considered in each star. This correlation strongly suggests that the obtained stratification is physical and not due to uncertainties in the description of the temperature profile of the stellar atmosphere.
The detected stratification of iron, obtained in some of the stars studied here, is far too large ($>$ 2 dex) to be interpreted as the result of measurement errors. Our stellar atmosphere models were calculated taking into account the enhanced iron abundance and helium depletion (Behr [@Behr03b]) and the error in estimating the iron abundances, for a given model atmosphere, is expected to be less than 0.2 dex. Meanwhile, Khan & Shulyak ([@K+S07]) have shown that using atmospheric models with varying Fe abundance (from one to ten times solar) can modify the abundances inferred from observed line profiles by up to $\pm$0.25 dex.
From the results in Table \[slope\] we can conclude that no detectable iron stratification exists in WF4-3485 and B84. Two stars (B267 and B279) show quite large slopes of iron abundance with respect to both $\log{E_{i}}$ and $\log{\tau_{5000}}$. The corresponding slopes for the stars WF2-2541 and WF2-3085 are smaller but still statistically significant. The least significant of these slopes is that of the abundance versus $\log{E_{i}}$ for WF2-3085 which has only approximately a $3\sigma$ value.
Another factor that must be evaluated that could mimic stratification is an error in the effective temperature of the underlying atmospheric model used. To verify the potential importance of this factor, we calculated the Fe stratification with models with an effective temperature higher and lower by 1000 K from the $T_{\rm eff}$ values listed in Table \[tab2\]. We also calculated the stratification with models assuming solar He and Fe abundances to evaluate the effect of this change on the inferred stratification profiles. Our simulations show that models with higher $T_{\rm eff}$ and models with solar abundances usually slightly decrease the slope of the abundance with respect to $\log{E_{i}}$, while models with lower $T_{\rm eff}$ increase this slope for all elements. In the case of iron in WF4-3085, the model with higher $T_{\rm eff}$ and the model with solar abundances result in a negligible slope of its abundance versus $\log{E_{i}}$. Therefore, the results are not sufficient to report unambiguous detection of vertical stratification of iron in WF4-3085. The slopes for iron found for the other three stars (B267, B279 and WF-2541) are still significant in all these models, and stratification is therefore confirmed.
The two BHB stars (B84 in M15 and WF4-3485 in M13) which do not show any signs of stratification of their iron abundances are not different from the other stars with regards to their stellar atmosphere characteristics and rotation (see Table \[tab2\]). However, these two stars possess very small titanium and phosphorus abundances as compared to the other stars studied here. Their spectra have very weak or invisible Ti[ii]{} and P[ii]{} lines. Their iron abundance is also close to the solar value, at least for $\xi=0$kms$^{-1}$ (see Table \[tab5\]).
We observe an upturn in the iron abundance obtained assuming $\xi=0$kms$^{-1}$ at low optical depths for some of the stars studied (e.g. Figs. \[B267\]a, \[B279\]a, \[WF4-3085\]a, \[WF2-2541\]a). The points on the figures corresponding to the upturn feature are the strong Fe[ii]{} lines and are not taken into account during the least-square fit of the data. To estimate the influence of microturbulence we have performed an abundance analysis with $\xi=$ 2kms$^{-1}$, which leads to elimination of the upturn (see, for example, Fig. \[B267\]a). Performing a set of simulations with different microturbulent velocities, we have determined the minimum value of the microturbulence ($\xi_{min}$) for which the upturn disappears for each star studied. These values are reported in Table \[tab5\]. As the inclusion of microturbulent velocity amplifies the iron stratification, the stratification obtained for iron in the first four stars studied with $\xi=$ 0kms$^{-1}$ can be considered to be a lower limit to the possible stratification in these stars. A more detailed study including abundance stratification in the calculation of the synthesized spectra might shed some light on this strange behaviour, but this is outside of the scope of the present paper.
Figure \[B267\]b indicates that B267 shows a possible signature of vertical stratification of the titanium abundance. The respective slopes of Ti abundance versus $\log{E_{i}}$ and $\log{\tau_{5000}}$ are significantly higher than zero for this star and are statistically significant (see Table \[slope\]). For B279, we can not confidently conclude that stratification exists since the slope of the abundance with respect to $\log{E_{i}}$ becomes weak when using either solar abundances or assuming a $T_{\rm eff}$ increase of 1000 K. The slope of Ti abundance with respect to $\log{\tau_{5000}}$ also becomes weak in the model assuming a $T_{\rm eff}$ increase of 1000 K. It should be noted that the Ti lines in B279 do not sample a large portion of the atmosphere and this fact renders the detection of stratification more difficult.
It is clear that phosphorous shows no clear signs of stratification in WF2-2541. For WF4-3085 we cannot make firm conclusions concerning phosphorous stratification since its abundance variation in the range of optical depths under consideration is only approximately 0.4 dex. Also, the slope of its abundance with respect to $\log{E_{i}}$ is not statistically significant.
Mean abundances and velocities {#mean}
==============================
The mean photospheric abundances derived for the selected BHB stars from an analysis of the available iron, titanium and phosphorus lines are reported in Table \[tab5\]. For each chemical element, three columns are shown containing the mean abundance for simulations with microturbulent velocity $\xi$= 0 and 2 km s$^{-1}$ respectively, and the number of analyzed line profiles. The reported uncertainties are equal to the standard deviation calculated from the results of individual line simulations for all lines considered.
The derived heliocentric radial and projected rotation velocities as well as microturbulent velocities (see Table \[tab5\]) are generally in agreement with the data provided by Behr ([@Behr03b]) for these stars. Some inconsistencies are found between $v\sin{i}$ values obtained for WF2-2541 ($v\sin{i}=0.0^{+4.07}_{-0.0}$ km s$^{-1}$) and B279 ($v\sin{i}=5.92^{+1.6}_{-1.69}$ km s$^{-1}$), but they are within the respective error bars given by Behr ([@Behr03b]). The precision of our velocity estimates appears to be comparatively higher because of a more stringent selection of analysed line profiles. Our measurements of the average abundance of iron, titanium and phosphorus also agree with the corresponding values published by Behr ([@Behr03b]), taking into account the error bars and the value of the applied microturbulent velocity. Small differences between our results and abundances from the aforementioned paper are not surprising. Our average abundance is calculated with the individual abundances obtained from each line, while Behr ([@Behr03b]) fitted the whole spectrum of each element to obtain its abundance.
Discussion {#discuss}
==========
In this paper we continue our attempts to detect vertical abundance stratification in the atmospheres of BHB stars. After the report by Bonifacio et al. ([@Bonifacio+95]) of the vertical stratification of helium in the atmosphere of Feige 86, it became clear that the abundances of other chemical species may also be stratified. We devoted special interest to iron because hot BHB stars usually have an enhanced iron abundance (e.g. Behr [@Behr03b]), suggesting that this element may be strongly affected by diffusion. Analysing the spectra of another hot BHB star HD 135485 (Khalack et al. [@Khalack+07]) we did not find direct evidence of iron stratification, but revealed strong signatures of sulfur depletion in the deeper atmospheric layers. However, HD 135485 is different from the other BHB stars in that its spectrum shows evidence of helium enrichment (in comparison with the solar abundance), while in the atmospheres of the other BHB stars helium is depleted.
Therefore, we directed our attention to BHB stars where the iron abundance is near the solar abundance or enhanced, and helium is depleted. The results obtained argue that at least three stars (B267 and B279 in M15 and WF2-2541 in M13) show clear signatures of vertical stratification of their iron abundance, while for WF4-3085 the results are suggestive, but not conclusive. The other two stars studied here (B84 in M15 and WF4-3485 in M13) do not show stratification of iron and their averaged iron abundance is close to solar (but is enhanced in comparison with its cluster value). B267 shows also a signature of vertical stratification of titanium (see Fig. \[B267\]b).
Since our simulations show that the turnup feature observed in the iron stratification profile is strongly dependent on microturbulent velocity, the value of the abundance at low optical depths is uncertain. Of course, if the theoretical framework supposes that these abundance gradients are due to atomic diffusion, microturbulence should be weak since a stable atmosphere is needed for diffusion to be dominant. It should be noted that for corresponding optical depths the abundance profile of iron is similar in the three BHB stars that exhibit stratification. The reason that the other two stars in our study do not show clear signs of iron stratification (B84 and WF4-3485) might be related to evolutionary effects or the presence of other competing hydrodynamical processes. The absence of Ti[ii]{} and P[ii]{} lines in their spectra might be evidence of this.
In conclusion, the results shown here add to the mounting evidence of the existence of vertical abundance stratification, and hence atomic diffusion, in the atmospheres of BHB stars.
This research was partially funded by the Natural Sciences and Engineering Research Council of Canada (NSERC). We thank the Réseau québécois de calcul de haute performance (RQCHP) for computing resources. GAW acknowledges support from the Academic Research Programme (ARP) of the Department of National Defence (Canada). BBB thanks all the dedicated people involved in the construction and operation of the Keck telescopes and HIRES spectrograph. He is also grateful to Judy Cohen, Jim McCarthy, George Djorgovski, and Pat Côté for their contributions of Keck observing time. We are grateful to Dr. T.Ryabchikova and Dr. L.Mashonkina for helpful discussions and suggestions.
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[^1]: Current address: US Naval Observatory, 3450 Massachusetts Avenue NW, Washington DC 20392
[^2]: Most of the known BHB stars are found in globular clusters.
[^3]: ftp://ftp.wins.uva.nl/pub/orth
|
---
address: |
Department of Mathematics\
The Hebrew University of Jerusalem\
Jerusalem, 91904, Israel
author:
- Daniel Lowengrub
bibliography:
- 'ttt-arxiv.bib'
title: 'One Dimensional t.t.t Structures'
---
Introduction
============
The notion of a first order topological structure was introduced by Pillay [@P1] as a generalization of the notion of an o-minimal structure. The idea is to provide a general framework in which model theory can be used to analyze a topological structure whose topology isn’t necessarily induced by a definable order. In the o-minimal case, the topology is generated from a basis where each basis set can be defined by substituting the variables $y_{1}$ and $y_{2}$ by suitable parameters in the following formula$$\phi(x,y_{1},y_{2})=y_{1}<x<y_{2}$$
A first order topological structure generalizes this to the case where $\phi$ is some arbitrary formula with more than one variable.
Pillay also introduced the notion of topologically totally transcendental (t.t.t) structures which have the additional property that any definable set has a finite number of connected components. For example, by definition o-minimal structures are t.t.t.
In the previously mentioned paper, Pillay proved that one dimensional t.t.t structures have some characteristics in common with o-minimal structures such as the exchange property. Furthermore, he showed that if the topology of a one dimensional t.t.t structure is induced by a definable dense linear ordering then the structure is o-minimal.
In this paper we’ll focus on $\omega$-saturated one dimensional t.t.t structures and prove that under a few additional topological assumptions, such structures are composed of o-minimal components in a relatively simple manner.
Our main result which will be proved in section \[sec:Structures-With-Splitting\] will be to show that if we assume that removing any point from the structure splits it into at least two connected components, then the structure must be a one dimensional simplex of a finite number of o-minimal structures:
Let $M$ be a 1-dimensional connected $\omega$-saturated t.t.t structure such that for each point $x\in M$, $M\backslash\{x\}$ has at least two definably connected components. Then there exists a finite set $X\subset M$ such that each of the finite number of connected components of $M\backslash X$ are o-minimal.\[thm:intro-splitting\]
In section \[sec:Structures-Without-Splitting\] we’ll analyze the case where removing a point doesn’t necessarily split the structure, and will find two alternative topological properties which guarantee that the structure is locally o-minimal. This is done by showing that basis sets which are small enough can be split by removing a point.
Let $M$ be a 1-dimensional $\omega$-saturated t.t.t structure such that one of the following holds:\[thm:intro-no-splitting\]
1. There exist a definable continuous function $F:M^{2}\rightarrow M$ and a point $a\in M$ such that for each $x\in M$, $F(x,x)=a$ and $F(x,\cdot)$ is injective.
2. For every basis set $U$, $\vert bd(U)\vert=2$.
Then for all but a finite number of points, for every point $x\in M$ there’s a basis set $U$ containing $x$ such that $U$ is o-minimal.
An immediate corollary of part 1 of theorem \[thm:intro-no-splitting\] is that if an $\omega$-saturated one dimensional t.t.t structure admits a topological group structure then it is locally o-minimal.
Towards the end of the section we’ll prove a version of the monotonicity theorem for locally o-minimal structures. This shows that locally o-minimal structures share many characteristics with standard o-minimal structures.
Theorem \[thm:intro-splitting\] illustrates that the defining characteristic of o-minimal structures within the general setting of $\omega$-saturated one dimensional t.t.t structures isn’t existence of the order itself, but rather the ability to disconnect the structure by removing a point.
Theorem \[thm:intro-no-splitting\] shows that even in the case where an $\omega$-saturated one dimensional t.t.t structure isn’t o-minimal, it will at least be o-minimal on a local scale provided that it has a rudimentary internal structure.
An important step in proving the theorems above will be to show that the relation $a\sim_{x}b$, which says that $a$ and $b$ are in the same connected component of $M\backslash\{x\}$, is definable.
Let $M$ be a 1-dimensional $\omega$-saturated t.t.t structure such that for each point $x\in M$, $M\backslash\{x\}$ has more than one connected component. Then the relation $a\sim_{x}b\subset M^{3}$ is definable.
In section \[sec:Connected-Components-in\] we’ll prove that that the number of connected components in a definable family is uniformly bounded.
Let $(M,\phi)$ be a $1$-dimensional $\omega$-saturated t.t.t structure and $$\alpha(x,y_{1},\dots,y_{l})\in L$$ . Then there exists a constant $C\in\mathbb{N}$ such that for every $l$-tuple $c_{1},\dots,c_{l}\in M$, $\alpha(c_{1},\dots,c_{l})$ has less than $C$ connected components.
This in turn will allow us to prove that elementary extensions of such structures are t.t.t as well.
Preliminaries
==============
In this section we review some of the notions, definitions and results from Pillay [@P1] which will be used heavily throughout the following sections.
Let $M$ be a two sorted $L$ structure with sorts $M_{t}$ and $M_{b}$ and let $\phi(x,y_{1},\dots,y_{k})$ be an $L$ formula such that $\{\phi^{M_{t}}(x,\bar{a})\vert\bar{a}\in M_{b}^{k}\}$ is a basis for a topology on $M_{t}$. Then the pair $(M,\phi)$ will be called a *first order topological structure*. When we talk about the topology of $M_{t}$ we mean the one generated by the basis described above.
In Pillay’s paper, first order topological structures were defined on a one-sorted structure where each element can be both a parameter for a basis set, and a point in the topological space. However, in practice this double meaning isn’t needed, so we’re using the two sorted definition both for clarity and in order to slightly strengthen some of the theorems.
In addition, we consider the following condition on a ** first order topological structure $M$:
\(A) Every definable set $X\subset M_{t}$ is a boolean combination of definable open subsets.
In this paper we assume that $M_{t}$ is Hausdorff and $(M,\phi)$ is a first order topological structure satisfying (A).
The following topological result is also helpful in this context and was proved by Robinson [@R 4.2].
Let $V$ be a topological space, and $W\subset V$ a non-empty subset. Let $A\subset V$ be a boolean combination of open subsets of $V$ and let $B=V\backslash A$. Then either $W\cap A$ or $W\cap B$ has an interior with respect to the induced topology on $W$.\[lem:set-or-comp-int\]
Let $M$ be a first order topological structure satisfying (A) and let $X\subset M_{t}$ be a closed definable subset of $M_{t}$. The ordinal valued $D_{M}(X)$ is defined by:
1. If $X\neq\emptyset$ then $D_{M}(X)\geq0$.
2. If $\delta$ is a limit ordinal and $D_{M}(X)\geq\alpha$ for all $\alpha<\delta$ then $D_{M}(X)\geq\delta$.
3. If there’s a closed definable $Y\subset M_{t}$ such that $Y\subset X$, $Y$ has no interior in $X$ and $D_{M}(Y)\geq\alpha$ then $D_{M}(X)\geq\alpha+1$.
We’ll write $D_{M}(X)=\alpha$ if $D_{M}(X)\geq\alpha$ and $D_{M}(X)\ngeq\alpha+1$. We’ll write $D_{M}(X)=\infty$ if $D_{M}(X)\geq\alpha$ for all $\alpha$.
We say that $M$ *has dimension* if $D_{M}(X)\neq\infty$ for all closed definable subsets $X\subset M_{t}$.
In addition, we define the number of definable connected components for definable subsets of our topology:
Let $X\subset M_{t}$ be definable. Then $d_{M}(X)$ is the maximum number $d<\omega$ such that there are disjoint definable clopen sets $X_{1},\dots,X_{d}\subset X$ with $X=\cup_{i=1}^{d}X_{i}$ , and $\infty$ if no such $d$ exists.
Throughout the paper, when we say “connected” we always mean “definably connected”.
And now for the main definition:
We say that $M$ is *topologically totally transcendental (t.t.t)* if $M$ is a first order topological structure satisfying (A) with dimension such that for every definable set $X\subset M_{t}$, $d_{M}(X)<\infty$. We say that a theory $T$ is t.t.t is every model of $T$ is t.t.t.
The following lemma was proved by Pillay [@P1 6.6] and plays a key role in most of the proofs in this paper.
Let $M$ be a 1-dimensional t.t.t structure. Then:\[lem:Let–be\]
1. For any closed and definable $X\subset M_{t}$, $D(X)=0$ iff $X$ is finite.
2. The set of isolated points of $M_{t}$ is finite.
3. For any definable $X\subset M_{t}$ there are pairwise disjoint definably connected definable open subsets $X_{1},\dots,X_{m}\subset M_{t}$ and a finite set $Y\subset M_{t}$ such that $X=(\cup_{i=1}^{m}X_{i})\cup Y$.
4. For any definable $X\subset M_{t}$, the set of boundary points of $X$ is finite.
One consequence of part $3$ of lemma \[lem:Let–be\] which will be used many times below is that if a set $A\subset M_{t}$ is definable then the statement *“$A$ is infinite”* is expressible in first order logic as it’s equivalent to the statement *“$A$ has no interior”*.
Connected Components in Definable Families\[sec:Connected-Components-in\]
=========================================================================
In this section we’ll show that the number of connected components is uniformly bounded over a definable family. This is used to prove that in 1-dimensional $\omega$-saturated structures, the property of being t.t.t is preserved under elementary equivalence.
Let $(M,\phi)$ be a $1$-dimensional $\omega$-saturated t.t.t structure. Then there exists a number $K\in\mathbb{N}$ such that for each point $b\in M_{b}$, $\vert bd(\phi^{M_{t}}(b)\vert\leq K$.\[lem:bounded-boundary\]
For each $b\in M_{b}$, $\vert bd(\phi^{M_{t}}(b)\vert$ is finite. The lemma then follows from the fact that $M$ is $\omega$-saturated.
Let $(M,\phi)$ be a definably connected $1$-dimensional $\omega$-saturated t.t.t structure, $K\in\mathbb{N}$ a number such that for each point $b\in M_{b}$ we have $\vert bd(\phi^{M_{t}}(b)\vert\leq K$, and $X\subset M_{t}$ a definable subset such that $bd(X)=n$. Then $d_{M}(X)\leq n\cdot K$.\[lem:bounded-connected\]
Let $N=d_{M}(X)$ and let $\{Y_{1},\dots,Y_{N}\}$ be pairwise disjoint clopen (in $X$) subsets of $X$ such that $X=\cup_{i=1}^{N}Y_{i}$. In addition, we denote the elements of $bd(X)$ by $bd(X)=\{a_{1},\dots,a_{n}\}$.
By the Hausdorffness of $M_{t}$, we can find basis sets $\{U_{1},\dots,U_{n}\}$ such that for all $1\leq i\leq n$:
1. $a_{i}\in U_{i}$
2. For all $1\leq j\leq N$, if $Y_{j}\neq\{a_{i}\}$ then $Y_{j}\backslash U_{i}\neq\emptyset$.
For all $1\leq j\leq N$, if $Y_{j}$ isn’t a point then there exists an $1\leq i\leq n$ such that $a_{i}\in\bar{Y_{i}}$ and $bd(U_{i})\cap Y_{j}\neq\emptyset$.
Let $1\leq j\leq N$ be chosen such that $Y_{j}$ isn’t a point. Without loss of generality, $Y_{j}\neq X$ because otherwise $X$ would be connected and the lemma would be trivial. Since $M_{t}$ is definably connected, $bd(Y_{j})\neq\emptyset$. In addition, $Y_{j}$ is clopen in $X$ so $bd(Y_{j})\subset bd(X)$. Therefore, there exists some $1\leq i\leq n$ such that $a_{i}\in\bar{Y_{i}}$.
We’ll now see that $bd(U_{i})\cap Y_{j}\neq\emptyset$.
Assume for contradiction that $bd(U_{i})\cap Y_{j}=\emptyset$. Then both $U_{i}\cap Y_{j}$ and $U_{i}^{c}\cap Y_{j}$ are non-empty clopen subsets of $X$, which is a contradiction to the fact that $Y_{j}$ is a connected component.
This completes the claim.
Without loss of generality, let’s choose an integer $L$ between $1$ and $N$ such that $\{Y_{1},\dots,Y_{L}\}$ are points and $\{Y_{L+1},\dots,Y_{N}\}$ are not points. Furthermore, let’s choose an integer $M$ between $1$ and $n$ such that $\{a_{1},\dots,a_{M}\}$ are isolated and $\{a_{M+1},\dots,a_{n}\}$ are not. It’s clear that $L\leq M$.
According to the claim, for each index $L+1\leq j\leq N$ there exists an integer $1\leq i\leq n$ and a point $y_{j}$ such that $y_{j}\in bd(U_{i})$ and $a_{i}\in\bar{Y_{j}}$. We note that from the fact that $a_{i}\in\bar{Y_{j}}$, it follows that $a_{i}$ is not an isolated point. This gives us a mapping: $$\phi:\{Y_{L+1},\dots,Y_{N}\}\rightarrow\bigcup_{i=M+1}^{n}bd(U_{i})$$ Since $y_{k}\neq y_{l}$ for each $L+1\leq k<l\leq N$ , the map $\phi$ is injective. Furthermore, $$\vert\bigcup_{i=M+1}^{n}bd(U_{i})\vert\leq(n-M)\cdot K\leq(n-L)\cdot K$$ so by the injectivity of $\phi$ we get that $N-L\leq(n-L)\cdot K$. But $K\geq1$ so $N\leq n\cdot K$.
Let $(M,\phi)$ be a $1$-dimensional $\omega$-saturated t.t.t structure and let $\alpha(x,y_{1},\dots,y_{l})\in L$ be a formula. Then there exists a constant $C\in\mathbb{N}$ such that for every $l$-tuple $c_{1},\dots,c_{l}\in M$, $d_{M}(\alpha^{M_{t}}(c_{1},\dots,c_{l}))<C$. \[prop:bounded-connected\]
First of all, let $K\in\mathbb{N}$ a number such that for each point $b\in M_{b}$ we have $\vert bd(\phi^{M_{t}}(b)\vert\leq K$. By lemma \[lem:bounded-boundary\], for each $\bar{c}\in M^{l}$ there exists a number $n_{\bar{c}}\in\mathbb{N}$ such that $bd(\alpha^{M_{t}}(\bar{c}))<n_{c}$. Therefore, since $M$ is $\omega$-saturated, there exists some $n\in\mathbb{N}$ such that for each tuple $\bar{c}\in M^{l}$, $bd(\alpha^{M_{t}}(\bar{c}))<n$.
We’ll show that we can choose $C$ to be $d_{M}(M_{t})\cdot K\cdot n$. Let $m=d_{M}(M_{t})$ and let $\{Y_{1},\dots,Y_{m}\}$ be pairwise disjoint definably connected subsets such that $M_{t}=\cup_{i=1}^{m}Y_{i}$. By lemma \[lem:bounded-connected\], for each $1\leq i\leq m$ and $\bar{c}\in M^{l}$, $d_{M}(\alpha^{M_{t}}(\bar{c})\cap Y_{i})<n\cdot K$. The proposition then follows immediately.
We’ll now use this boundedness result in order to prove that a certain set of first order properties are necessary and sufficient for an $\omega$-saturated first order topological structure to be t.t.t.
Let $(M,\phi)$ be an $\omega$-saturated 1-dimensional t.t.t structure. Then $M$ has the following properties:\[thm:1D-t.t.t-props\]
1. For every formula $\alpha(x,y_{1},\dots,y_{l})\in L$, there exists some $C\in\mathbb{N}$ such that for every tuple $\overline{c}\in M^{l}$, there exist $C$ points $x_{1},\dots,x_{C}$ in $\alpha^{M_{t}}(\overline{c})$ such that $\alpha^{M_{t}}(\overline{c})\backslash\{x_{1},\dots,x_{C}\}$ is open.
2. For every formula $\alpha(x,y_{1},\dots,y_{l})\in L$, there exists a constant $C\in\mathbb{N}$ such that for all $c_{1},\dots,c_{l}\in M$, $d_{M}(\alpha^{M_{t}}(c_{1},\dots,c_{l}))<C$.
3. For any pair of formulas $\alpha(x,y_{1},\dots,y_{s})$ and $\beta(x,y_{1},\dots,y_{t})$ in $L$, and for all $\overline{a}\in M^{s}$ and $\overline{b}\in M^{t}$, if $B=\beta^{M_{t}}(\overline{b})\subset\alpha^{M_{t}}(\overline{a})=A$ is closed and non empty and doesn’t have an interior in $A$, then $A$ has an interior in $M_{t}$.
Furthermore, if $(M,\phi)$ is a first order topological structure which satisfies these three properties and is Hausdorff, then $M$ is a 1-dimensional t.t.t structure.
First we’ll see that the three properties are sufficient. Assume that $(M,\phi)$ is a first order topological structure such that $M_{t}$ is Hausdorff and has the three properties in the theorem.
By property 1, every definable set $X$ is a boolean combination of open sets so $M$ has property (A). By property 2, every definable set has a finite number of definably connected components. Finally, by property 3, $D(M)=1$.
Now we’ll prove the first part of the theorem. Let $(M,\phi)$ be an $\omega$-saturated t.t.t structure. By the definition of t.t.t, $M_{t}$ is Hausdorff. We’ll now prove that $M$ has each one of the required properties.
1. Let $\alpha(x,y_{1},\dots,y_{l})\in L$. Since $M$ is t.t.t, for every $\overline{c}\in M^{l}$, there exist $C$ points $x_{1},\dots,x_{C}$ in $\alpha^{M_{t}}(\overline{c})$ such that $\alpha^{M_{t}}(\overline{c})\backslash\{x_{1},\dots,x_{C}\}$ is open. Since $M$ is $\omega$-saturated, we can choose $C$ uniformly for all $\overline{c}\in M^{l}$.
2. This property is essentially proposition \[prop:bounded-connected\].
3. This follows from the fact that $D(M)=1$.
Let $\phi(x,y_{1},\dots,y_{k})$ be a formula and let $(M,\phi)$ be a $1$-dimensional t.t.t structure which is $\omega$-saturated. In addition, let $N$ be a model such that $N\equiv M$. Then $(N,\phi)$ is a $1$-dimensional t.t.t structure.
This is immediate from the fact that all of the properties in theorem \[thm:1D-t.t.t-props\] can be expressed in first order logic.
Structures With Splitting \[sec:Structures-With-Splitting\]
===========================================================
Introduction
------------
Our main result in this section is that for any 1-dimensional $\omega$-saturated t.t.t structure, if removing any point splits the space into more that one connected component then there exists a finite set $X\subset M_{t}$ such that each connected component of $M_{t}\backslash X$ is o-minimal.
In order to prove this, we first obtain some intermediate results such as the fact that the equivalence relation $y\sim_{x}z$ specifying if $y$ and $z$ are in the same connected component of $M_{t}\backslash\{x\}$ is a definable relation in $M_{t}^{3}$. We also introduce a notion of “local flatness” which is used as a stepping stone between t.t.t structures and o-minimality.
For example, consider the structure $R_{int}=\langle\mathbb{R},I(x,y,z)\rangle$ where $I(x,y,z)$ is true if $z$ lies on the interval between $x$ and $y$. In example \[exa:interval-example\] we’ll show that $R_{int}$ has the property that removing any point splits the space into more that one connected component. We’ll use this fact to show that the relation $y\sim_{x}z$ is indeed definable. In the end of this section we’ll demonstrate how applying the construction of the order to $R_{int}$ gives the standard ordering on the reals.
In this section we’re assuming that $M_{t}$ has no isolated points. This doesn’t pose a problem because $M_{t}$ has at most a finite number of isolated points so we can remove them without affecting any of our results.
A Definable Relation
--------------------
The following equivalence relation is useful for analyzing what happens when a point is removed from a structure.
Let $M$ be a 1-dimensional t.t.t structure. Let $x,a,b\in M_{t}$. Then $a\sim_{x}b$ will be a relation which is true iff $a$ and $b$ are in the same definable connected component of $M_{t}\backslash\{x\}$.
Note that by proposition \[prop:bounded-connected\], there exists an $N\in\mathbb{N}$ such that for each point $x\in M_{t}$, $\sim_{x}$ has less than $N$ equivalence classes.
Our first goal is to show that if for every $x\in M_{t}$ we have $d_{M}(M_{t}\backslash\{x\})\geq2$, then $\sim_{x}\subset M_{t}^{3}$ is definable.
We start by showing that for any $x$ such that $d_{M}(M_{t}\backslash\{x\})>2$, $x\in acl(\emptyset)$.
The following technical lemma will be used many times throughout the proof.
Intuitively, the lemma says that after removing two points, the space is divided into three distinct components. The part “in between” the points we removed and one additional side for each of the points.
Let $M$ be a 1-dimensional $\omega$-saturated t.t.t structure, $C\subset M_{t}$ an open connected definable subset, $a\neq b\in C$ and $2\leq k,l\in\mathbb{N}$ such that $d_{M}(C\backslash\{a\})=k$ and $d_{M}(C\backslash\{b\})=l$. Let $A_{1},\dots,A_{k}$ and $B_{1},\dots,B_{l}$ be the connected components of $C\backslash\{a\}$ and $C\backslash\{b\}$ respectively such that $a\in B_{1}$ and $b\in A_{1}$. Then:\[lem:disjoint-components\]
1. $bd(\cup_{i=2}^{k}A_{i})=\{a\}$
2. $bd(\cup_{j=2}^{l}B_{j})=\{b\}$
3. $bd(A_{1}\cap B_{1})=\{a,b\}$ and for every open set $U$ containing $a$ or $b$, $U\cap(A_{1}\cap B_{1})\neq\emptyset$.
4. The following union is disjoint:$$C=(\cup_{i=2}^{k}A_{i})\cup\{a\}\cup(A_{1}\cap B_{1})\cup\{b\}\cup(\cup_{j=2}^{l}B_{j})$$
First we’ll prove $1$.
Since $C$ is connected, $a\in\overline{A_{i}}$ for each $1\leq i\leq k$. Therefore, since $a\notin A_{i}$ for each $2\leq i\leq k$, $a\in bd(\cup_{i=2}^{k}A_{i})$. In addition, $\cup_{i=2}^{k}A_{i}$ is open which means that $bd(\cup_{i=2}^{k}A_{i})\subset\{a\}\cup A_{1}$. But $A_{1}$ is open as well and disjoint to $\cup_{i=2}^{k}A_{i}$ so $bd(\cup_{i=2}^{k}A_{i})=\{a\}$.
Similarly, $bd(\cup_{j=2}^{l}B_{j})=\{b\}$.
We’ll now show that $\cup_{i=2}^{k}A_{i}\subset B_{1}$. In order to do that we first prove that $(\cup_{i=2}^{k}A_{i})\cup\{a\}$ is connected in $C\backslash\{b\}$. Assume for contradiction that $X_{1}$ and $X_{2}$ form a clopen partition of $(\cup_{i=2}^{k}A_{i})\cup\{a\}$ in $C\backslash\{b\}$. Without loss of generality, $a\in X_{1}$ which means that $a\notin\overline{X_{2}}$. Furthermore, $A_{1}$ is open and $b\in A_{1}$ which means that $b\notin\overline{X_{2}}$. Together this means that $X_{2}\subset(\cup_{i=2}^{k}A_{i})$ and $b\notin bd(X_{2})$. Therefore, $X_{2}$ is clopen in $C$ which is a contradiction to the fact that $C$ is connected.
Now, $B_{1}$ is the connected component of $C\backslash\{b\}$ containing $a$. So from the fact that $(\cup_{i=2}^{k}A_{i})\cup\{a\}$ is connected in $C\backslash\{b\}$ it follows that $\cup_{i=2}^{k}A_{i}\subset B_{1}$.
We’re now ready to prove $4$. It’s immediate that $$(\cup_{i=2}^{k}A_{i})\cap(A_{1}\cap B_{1})=(\cup_{j=2}^{l}B_{j})\cap(A_{1}\cap B_{1})=\emptyset$$ In addition, since $(\cup_{i=2}^{k}A_{i})\subset B_{1}$ it follows that $(\cup_{j=2}^{l}B_{j})\cap(\cup_{i=2}^{k}A_{i})=\emptyset$. This shows that the union is disjoint so all that’s left is to show that it’s equal to $C$.
Let $c\in C$ be a point such that $$c\notin(\cup_{j=2}^{l}B_{j})\cup\{a\}\cup(\cup_{i=2}^{k}A_{i})\cup\{b\}$$ Since $c\in C\backslash\{a\}$ and $c\notin(\cup_{i=2}^{k}A_{i})$ it follows that $c\in A_{1}$. Similarly, $c\in B_{1}$. Therefore, $c\in A_{1}\cap B_{1}$.
We’re now ready to prove $3$.
First of all, assume for contradiction that $A_{1}\cap B_{1}=\emptyset$. Then by parts 1, 2 and 4 of the lemma, the sets $(\cup_{i=2}^{k}A_{i})\cup\{a\}$ and $\{b\}\cup(\cup_{j=2}^{l}B_{j})$ would form a clopen partition of $C$ which is a contradiction to the assumption that $C$ is connected.
We’ll now show that $bd(A_{1}\cap B_{1})=\{a,b\}$ On the one hand, $a\in int(B_{1})$ and $a\in\overline{A_{1}}$ so $a\in bd(A_{1}\cap B_{1})$. Similarly, $b\in A_{1}\cap B_{1}$. On the other hand, $A_{1}\cap B_{1}$ is open so $$bd(A_{1}\cap B_{1})\subset(\cup_{i=2}^{k}A_{i})\cup\{a\}\cup\{b\}\cup(\cup_{j=2}^{l}B_{j})$$ But $\cup_{i=2}^{k}A_{i}$ and $\cup_{j=2}^{l}B_{j}$ are open well so $bd(A_{1}\cap B_{1})\subset\{a\}\cup\{b\}$. Together we get that $bd(A_{1}\cap B_{1})=\{a,b\}$.
Finally, let $U$ be an open set containing $a$ or $b$. Since $bd(A_{1}\cap B_{1})=\{a,b\}$, it follows that $U\cap(A_{1}\cap B_{1})\neq\emptyset$.
Let $M$ be a 1-dimensional $\omega$-saturated connected t.t.t structure. Let $D\subset M_{t}$ be an open definable subset, $E(x,a,b)\subset M_{t}^{3}$ a definable relation and $N\in\mathbb{N}$ such that:\[lem:infinite-sep\]
1. $N\geq2$.
2. For every $x\in D$ and $a,b\in M_{t}$, $a\sim_{x}b\Rightarrow E(x,a,b)$.
3. For every $x\in D$, $E(x,a,b)$ is an equivalence relation with $N$ classes.
Then for each point $a\in D$, there exists a point $b\in D$ such that the definable set $X=\{x\in D\vert\neg E(x,a,b)\}$ is infinite.
Let $a\in D$. Without loss of generality, $D$ is connected. Otherwise, we’ll look at the connected component containing $a$. We’ll now show that there exists a point $b\in D$ such that for an infinite number of points $x\in D$ we have $\neg E(x,a,b)$. In order to do this, we’ll inductively construct a sequence of points $(b_{1},b_{2},\dots)$ in $D$ such that for each $n\in\mathbb{N}$ and each $1\leq j<n$, $a\sim_{b_{n}}b_{j}$ and $\neg E(b_{j},a,b_{n})$.
For $n=1$, we can choose any $b_{1}\in D\backslash\{a\}$.
Let’s assume that we constructed the sequence up to $b_{n}$. Let $X_{1},\dots,X_{c(b_{n})}$ be the connected components of $M_{t}\backslash\{b_{n}\}$ such that $a\in X_{2}$. We choose $b_{n+1}\in D$ to be some point such that $\neg E(b_{n},a,b_{n+1})$. By our assumptions on $E(x,a,b)$, $b_{n+1}\notin X_{2}$. So without loss of generality, in $b_{n+1}\in X_{1}$. Let $Y_{1},\dots,Y_{c(b_{n+1})}$ be the connected components of $M\backslash\{b_{n+1}\}$ such that $b_{n}\in Y_{1}$. By lemma \[lem:disjoint-components\], for all $1<j\leq c(b_{n+1})$, $Y_{j}\cap X_{2}=\emptyset$. By the inductive hypothesis, $b_{j}\in X_{2}$ for all $1\leq j<n$. This means that for all $1\leq j<n$, $b_{j}\in Y_{1}$. Similarly, $a\in Y_{1}$ and we already know that $b_{n}\in Y_{1}$. Together we’ve shown that $a\sim_{b_{n+1}}b_{j}$ for all $1\leq j<n+1$.
We’ll now show that for all $1\leq j<n$, $\neg E(b_{j},a,b_{n+1})$. This will be enough because we already know that $\neg E(b_{n},a,b_{n+1})$.
Let $j$ be an index such that $1\leq j<n$. Let $X_{1},\dots,X_{c(b_{j})}$ be the connected components of $M_{t}\backslash\{b_{j}\}$ such that $a\in X_{2}$ and $b_{n}\in X_{1}$. In addition, let $Y_{1},\dots,Y_{c(b_{n})}$ be the connected components of $M_{t}\backslash\{b_{n}\}$ such that $b_{j},a\in Y_{1}$ and $b_{n+1}\in Y_{2}$. Since $b_{j}\in Y_{1}$ and $b_{n}\in X_{1}$, by lemma \[lem:disjoint-components\] it follows that $Y_{2}\subset X_{1}$ which means that $b_{n}\sim_{b_{j}}b_{n+1}$. By our assumptions on $E$ this implies $E(b_{j},b_{n},b_{n+1})$. Therefore, as $\neg E(b_{j},a,b_{n})$, we can conclude that $\neg E(b_{j},a,b_{n+1})$.
Now, by the $\omega$-saturation, there exists a point $b\in D$ such that $\vert\{x\in D:\neg E(x,a,b)\}\vert=\infty$.
Let $M$ be a 1-dimensional $\omega$-saturated connected t.t.t structure. Let $D\subset M_{t}$ be an open definable subset, $E(x,a,b)\subset M_{t}^{3}$ a definable relation and $N\in\mathbb{N}$ such that:
1. For every $x\in D$ and $a,b\in M_{t}$, $a\sim_{x}b\Rightarrow E(x,a,b)$.
2. For every $x\in D$, $E(x,a,b)$ is an equivalence relation with $N$ classes.
Then $N\leq2$. \[lem:not-all-connected-k\]
Assume for contradiction that $N>2$. For ease of notion, we define $c(x)=d_{M}(M_{t}\backslash\{x\})$ for each $x\in M_{t}$. We note that for all $x\in D$, $c(x)>2$.
By lemma \[lem:infinite-sep\], there exist points $a,b\in M_{t}$ such that for an infinite number of points $x\in D$, $\neg E(x,a,b)$.
We denote the infinite set $\{x\in D:\neg E(x,a,b)\}$ by $X$.
Let $x,y\in X$, let $X_{1},\dots,X_{c(x)}$ be the connected components of $M_{t}\backslash\{x\}$ such that $a\in X_{1}$ and $b\in X_{2}$ and let $Y_{1},\dots,Y_{c(y)}$ be the connected components of $M_{t}\backslash\{y\}$ such that $a\in Y_{1}$ and $b\in Y_{2}$.
First we note that for every $j$ such that $3\leq j\leq c(x)$, $y\notin X_{j}$. Because let $j$ be an index such that $3\leq j\leq n(x)$, let $k$ be an index such that $x\in Y_{k}$ and assume for contradiction that $y\in X_{j}$. Then, since $y\in X_{j}$ and $x\in Y_{k}$, it follows from lemma \[lem:disjoint-components\] that $X_{1},X_{2}\subset Y_{k}$ which means that $a,b\in Y_{k}$. However, this contradicts our assumption that $a\nsim_{y}b$.
In an analogous fashion, for each index $j$ such that $3\leq j\leq c(y)$ we have $x\notin Y_{j}$.
Therefore, $x\in Y_{1}\cup Y_{2}$ and $y\in X_{1}\cup X_{2}$.
Since this is true for any pair of points $x,y\in X$, by lemma \[lem:disjoint-components\] we get that:
1. for all $3\leq i\leq c(x)$ and $3\leq j\leq c(y)$ , $X_{i}\cap Y_{j}=\emptyset$
2. for all $3\leq i\leq c(x)$, $X_{i}\cap X=\emptyset$.
From these two results we’ll show that that $M_{t}\backslash X$ has an infinite number of definable connected components. First of all, by $2$ it follows that for each point $x\in X$, the classes of $E(x,a,b)$ not containing $a$ and $b$ are definable sets which are contained and clopen in $M_{t}\backslash X$. Furthermore, by $1$, all the sets obtained this way are disjoint.
But since $X$ is both infinite and definable, this is a contradiction to the fact that $M$ is t.t.t.
Now, let $p$ be some type in $S(\emptyset)$. We now show that there exist a $\emptyset$-definable relation $R_{p}(x,a,b)\subset M_{t}^{3}$ and an infinite $\emptyset$-definable set $D_{p}\subset M_{t}$ such that:
1. For all elements $x\models p$ and points $a,b\in M_{t}$, $R_{p}(x,a,b)\iff a\sim_{x}b$.
2. For all elements $x\in D_{p}$ and points $a,b\in M_{t}$, $a\sim_{x}b\Rightarrow R_{p}(x,a,b)$.
3. For all elements $x\in D_{p}$, $R_{p}(x,a,b)\subset M_{t}^{2}$ is an equivalence relation with $d_{M}(M_{t}\backslash\{y\})$ equivalence classes where $y$ is some element realizing $p$.
4. For every element $x$ that realizes $p$, $x\in D_{p}$.
We construct $R_{p}$ and $D_{p}$ in the following way. First, let $x$ be some realization of $p$ and define $N$ by $N=d_{M}(M_{t}\backslash\{x\})$. Then there exist $\phi_{1}(x,\bar{y}),\dots,\phi_{N}(x,\bar{y})$ such that:
([\*]{}) for some $\bar{y}$, $\phi_{1}^{M_{t}}[\bar{y}],\dots,\phi_{N}^{M_{t}}[\bar{y}]$ partition $M_{t}\backslash\{x\}$ into $N$ disjoint clopen sets. Furthermore, for any other $\bar{z}$, if $(\phi_{1}^{M_{t}}[\bar{z}],\allowbreak \dots,\allowbreak \phi_{N}^{M_{t}}[\bar{z}])$ is a partition of $M_{t}\backslash\{x\}$ into disjoint clopen sets then it’s the same partition as $(\phi_{1}^{M_{t}}[\bar{y}],\allowbreak \dots,\allowbreak \phi_{N}^{M_{t}}[\bar{y}])$.
Since this is a first order statement, ([\*]{}) holds for all $x\models p$.
Now, we define $D_{p}$ as the set of all the points $x\in M_{t}$ such that ([\*]{}) holds for $x$ with the formulas $\phi_{1}(x,\bar{y}),\dots,\phi_{N}(x,\bar{y})$. We then define $R_{p}(x,a,b)$ as a relation which is true iff for one of the points $\bar{y}$ guaranteed by ([\*]{}) for $x$, the sets $\phi_{1}^{M_{t}}[\bar{y}],\dots,\phi_{N}^{M_{t}}[\bar{y}]$ partition $M_{t}\backslash\{x\}$ into $N$ disjoint clopen sets such that $a$ and $b$ are in the same section of the partition.
Let $M$ be a 1-dimensional $\omega$-saturated connected t.t.t structure and let $x\in M_{t}$ be a point such that $d_{M}(M_{t}\backslash\{x\})>2$. Then $D_{tp(x/\emptyset)}$ is finite and in particular, $x\in acl(\emptyset)$.\[prop:bigger-2-acl\]
Let $N=d_{M}(M_{t}\backslash\{x\})$ and $p=tp(x/\emptyset)$. Since $N>2$, by applying lemma \[lem:not-all-connected-k\] with $D=int(D_{p})$ and $E=R_{p}$, $int(D_{p})$ is finite. Therefore, $D_{p}$ is finite.
We now look at what happens if $d_{M}(M_{t}\backslash\{x\})=2$.
As before, let $p\in S(\emptyset)$ be a type such that for some element $x$ realizing $p$ we have $d_{M}(M_{t}\backslash\{x\})=2$. We define $\tilde{D}_{p}\subset D_{p}$ as the set of points $x\in D_{p}$ such that there exist elements $a,b\in M_{t}$ and a basis set $U\subset M_{t}$ containing $x$ such that for all points $u\in U$, $\neg R_{p}(u,a,b)$.
Let $M$ be a 1-dimensional $\omega$-saturated connected t.t.t structure and let $p\in S_{1}(\emptyset)$ be a complete type in $M_{t}$. In addition, assume that for some (all) elements $x\models p$, $\sim_{x}$ has $2$ equivalence classes. Then, for each point $x$ realizing $p$, one of the following hold:\[prop:equals-2\]
1. There exists a finite $\emptyset$-definable subset of $D_{p}$ containing $x$ and in particular, $x\in acl(\emptyset)$.
2. $int(\tilde{D}_{p})$ is a set containing $x$ such that for every point $y\in int(\tilde{D}_{p})$, $d_{M}(M_{t}\backslash\{y\})=2$.
First of all, if $D_{p}$ is finite then the first case holds for all $x\models p$.
Let’s assume that $D_{p}$ is infinite. Now, suppose that for each point $x$ realizing $p$:
([\*]{}) for all $a,b\in M_{t}$ and for every basis set $U$ containing $x$, there exists a point $u\in U$ such that $R(u,a,b)$.
We define the set $C\subset D_{p}$ as the set of points in $D_{p}$ with property ([\*]{}). $C$ is clearly $\emptyset$-definable. Furthermore, for all $x\models p$, $x\in C$. Assume for contradiction that $C$ is infinite. In that case, by lemma \[lem:infinite-sep\] there exist points $a,b\in M_{t}$ and a basis set $U\subset C$ such that for all $u\in U$, $\neg R(u,a,b)$. This is clearly a contradiction to ([\*]{}). This means that $C$ is finite so again we’re in the first case for every point $x\models p$.
Therefore, we can assume that for all elements $x$ realizing $p$, $x\in\tilde{D}_{p}$.
If $\tilde{D}_{p}$ is finite then again we’re in the first case for every point $x\models p$.
We’ll now see that if $\tilde{D}_{p}$ is infinite then for each point $y\in int(\tilde{D}_{p})$, $d_{M}(M_{t}\backslash\{y\})\leq2$. This will finish the proposition because we already know that for every point $x\in\tilde{D}_{p}$, $d_{M}(M_{t}\backslash\{x\})\geq2$. We also note that if $x\in\tilde{D_{p}}\backslash int(\tilde{D}_{p})$ then clearly we’re in the first case as $\vert\tilde{D}_{p}\backslash int(\tilde{D}_{p})\vert<\infty$.
Let’s assume for contradiction that $y\in int(\tilde{D}_{p})$ and $d_{M}(M_{t}\backslash\{y\})>2$. Since $y\in\tilde{D}_{p}$, there exist points $a,b\in M_{t}$ and a basis set $U\subset\tilde{D}_{p}$ containing $y$ such that for all $u\in U$, $\neg R_{p}(u,a,b)$. Let $Y_{1},Y_{2},Y_{3}$ be three definable disjoint clopen sets partitioning $M_{t}\backslash\{y\}$ such that $a\in Y_{1}$ and $b\in Y_{2}$. Since $M_{t}$ is connected, there exists a $z$ such that $z\in Y_{3}\cap U$. Since $z\in D_{p}$, $k=d_{M}(M_{t}\backslash\{z\})\geq2$. Let $Z_{1},\dots,Z_{k}$ be definable pairwise disjoint clopen sets partitioning $M_{t}\backslash\{z\}$ such that $y\in Z_{1}$. Since $y\in Z_{1}$ and $z\in Y_{3}$, by lemma \[lem:disjoint-components\] it follows that $Y_{1}\cup Y_{2}\subset Z_{1}$ which means that $a,b\in Z_{1}$. However, this is a contradiction to the fact that $\neg R_{p}(z,a,b)$.
We now use the previous two propositions to show that if for each point $x\in M_{t}$ we have $d_{M}(M_{t}\backslash\{x\})>1$, then the relation $a\sim_{x}b\subset M_{t}^{3}$ is $\emptyset$-definable.
Let $M$ be a 1-dimensional connected $\omega$-saturated t.t.t structure such that for each point $x\in M_{t}$, $d_{M}(M_{t}\backslash\{x\})>1$. Then the relation $a\sim_{x}b\subset M_{t}^{3}$ is definable.\[prop:definable-relation\]
We’ll show that both $a\sim_{x}b$ and $a\nsim_{x}b$ are $\bigvee$-definable by formulas without parameters.
$a\nsim_{x}b$ is clearly $\bigvee$-definable by formulas without parameters because $a\nsim_{x}b$ iff there exist two open sets whose boundary is $\{x\}$ such that one contains $a$ and the other contains $b$.
We’ll now prove that $a\sim_{x}b$ is $\bigvee$-definable by formulas without parameters. This is done by showing that for each point $x\in M_{t}$, there exists a set $C_{x}\subset M_{t}^{3}$ which is definable without parameters such that:
1. For all points $y,a,b\in M_{t}$, $(y,a,b)\in C_{x}\Rightarrow a\sim_{y}b$
2. $(x,a,b)\in C_{x}\iff a\sim_{x}b$
Let’s choose some $x\in M_{t}$ and define $p=tp(x/\emptyset)$.
If $d_{M}(M_{t}\backslash\{x\})=N>2$ then, by proposition \[prop:bigger-2-acl\], $D_{p}$ is a finite $\emptyset$-definable set containing $x$. Furthermore, for every $y\in D_{p}$, $d_{M}(M_{t}\backslash\{x\})\geq N$. Let’s denote the points in $D_{p}$ by $D_{p}=\{y_{1},\dots,y_{k}\}$. Without loss of generality, there exists some $0\leq l<k$ such that for all $1\leq i\leq l$, $d_{M}(M_{t}\backslash\{y_{i}\})>N$ and for all $l+1\leq i\leq k$, $d_{M}(M_{t}\backslash\{y_{i}\})=N$. It’s easy to see that for each $1\leq i\leq l$, $x\notin D_{tp(y_{i}/\emptyset)}$. Therefore, we can define:$$C_{x}=((D_{p}\backslash\bigcup_{i=1}^{l}D_{tp(y_{i}/\emptyset)})\times M_{t}^{2})\cap R_{p}$$ . Finally, let’s assume that $d_{M}(M_{t}\backslash\{x\})=2$.
If $D_{p}$ contains a finite $\emptyset$-definable set containing $x$, then we can define $C_{x}$ in the same way as in the previous case. Otherwise, by proposition \[prop:equals-2\], $int(\tilde{D}_{p})\subset D_{p}$ is a set containing $x$ such that for all $y\in int(\tilde{D}_{p})$, $d_{M}(M_{t}\backslash\{y\})=2$. Therefore, we can define:
$$C_{x}=(int(\tilde{D}_{p})\times M_{t}^{2})\cap R_{p}$$ . This finishes the proof of the proposition.
Let’s look at the structure $R_{int}=\langle\mathbb{R},I(x,y,z)\rangle$ where $I(x,y,z)$ is true if $z$ lies on the interval between $x$ and $y$. In other words:\[exa:interval-example\]$$I(x,y,z)=\{(x,y,z)\in\mathbb{R}^{3}\vert(x<z<y)\vee(y<z<x)\}$$
The basis sets will be given by:
$$\{I^{R_{int}}(a,b,z)\vert a,b\in\mathbb{R}\}$$
Since $\langle\mathbb{R},<\rangle$ is an $\omega$-saturated one dimensional t.t.t structure, so is $R_{int}$. We’ll now see that for every point $a\in\mathbb{R}$, $\mathbb{R}\backslash\{a\}$ has two definably connected components in $R_{int}$.
Let $a$ be some point in $\mathbb{R}$. Let $c$ and $b$ be two constants in $\mathbb{R}$ such that $c<a<b$. Then:$$x<a\iff I(x,a,c)\vee I(c,a,x)$$
$$a<x\iff I(a,x,b)\vee I(a,b,x)$$ this shows that $\mathbb{R}\backslash\{a\}$ has two definably connected components in $R_{int}$.
Therefore, by proposition \[prop:definable-relation\], the relation $a\sim_{x}b$ is definable in $R_{int}$. Indeed:$$a\sim_{x}b\iff\neg I(a,b,x)$$
Local and Global Flatness
-------------------------
We’ll now prove that, under the condition that removing any point creates at least two connected components, there exist a finite number of points such that after removing them, the remaining finite number of connected components are o-minimal. This is done by first showing that up to a finite number of points the structure is ”locally o-minimal”, and then showing that local o-minimality implies global o-minimality. In addition, the definability of the relation $a\sim_{x}b\subset M_{t}^{3}$ will play a crucial role.
We start by defining a notion of “local flatness” and then showing that locally flat points have a neighborhood which behaves similarly to an o-minimal one.
Let $M$ be t.t.t structure. We say that the point $x\in M_{t}$ is *locally flat* if there exist points $a,b\in M_{t}$ and a basis set $U$ such that for every point $u\in U$, $a\nsim_{u}b$. We say that a set $D$ is locally flat if all of it’s points are locally flat.
We first show that in the type of structures which we’re currently interested in, all but a finite number of points are locally flat.
Let $M$ be a 1-dimensional $\omega$-saturated connected t.t.t structure such that for each point $x\in M_{t}$, $d_{M}(M_{t}\backslash\{x\})>1$. Then for all but a finite number of points $x\in M_{t}$, $x$ is locally flat.\[prop:finite-not-flat\]
Let $X\subset M_{t}$ be the set of points $x\in M_{t}$ such that for all points $a,b\in M_{t}$ and for every basis set $U$, there exists a point $y\in U$ such that $a\sim_{y}b$. By proposition \[prop:definable-relation\], $X$ is definable.
Assume for contradiction that $X$ is infinite. Then, by lemma \[lem:infinite-sep\], there exists a pair of points $a,b\in M_{t}$ such that the set $\tilde{X}=\{x\in X\vert a\nsim_{x}b\}$ is infinite. In addition, $\tilde{X}$ is definable so there exists a basis set $U$ which is contained in $\tilde{X}\subset X$. This is clearly a contradiction to the definition of $X$.
The next few propositions will show that points which are locally flat have a neighborhood on which we can define a linear order. This motivates the “flatness” in the definition.
Let $M$ be a 1-dimensional $\omega$-saturated connected t.t.t structure such that for all $x\in M_{t}$, $d_{M}(M_{t}\backslash\{x\})>1$. Let $x\in M_{t}$ be locally flat and let $U\subset M_{t}$ be an open connected definable set containing $x$. Then $U\backslash\{x\}$ has two connected components.\[lem:flat-two-components\]
Since $d_{M}(M_{t}\backslash\{x\})>1$ and $M_{t}$ is connected, it’s enough to show that $U\backslash\{x\}$ has no more than two connected components.
Assume for contradiction that $U_{1},\dots,U_{k}$ are the connected components of $U\backslash\{x\}$ with $k>2$. In addition, let $a,b\in M_{t}$ be points and let $V\subset U$ be a basis set containing $x$ such that for all $v\in V$, $a\nsim_{v}b$.
Let the sets $X_{1},X_{2}\subset M_{t}\backslash U$ be part of a clopen partition of $M_{t}\backslash U$ such that $a\in X_{1}$ and $b\in X_{2}$. By the connectedness of $M_{t}$, there exist points $b_1, b_2 \in bd(U)$ such that $b_1 \in X_1$ and $b_2 \in X_2$. Without loss of generality, $b_1 \in bd(U_{1})$ and $b_2 \in bd(U_{2})$. Furthermore, the connectedness of $M_{t}$ implies that $U_{3}\cap V\neq\emptyset$.
Now, let $y$ be a point in $U_{3}\cap V$. By lemma \[lem:disjoint-components\], $U_{1}$ and $U_{2}$ are both subsets of the connected component of $U\backslash\{y\}$ which contains $x$. So since $\overline{U_{1}}\cap X_{1}\neq\emptyset$ and $\overline{U_{2}}\cap X_{2}\neq\emptyset$, both $X_{1}$ and $X_{2}$ are in the same connected component of $M_{t}\backslash\{y\}$. This means that $a\sim_{y}b$ which is a contradiction to the fact that $y\in V$.
One consequence of lemma \[lem:flat-two-components\] which will be used later is that if we remove all of the finite number of points which aren’t locally flat then for each of the remaining connected components $C$, and for each point $x\in C$, $C\backslash\{x\}$ will have exactly two connected components. This will be used to show that $C$ is o-minimal.
Our next goal is to define an order on some neighborhood of each locally flat point. Let $x_{0}\in U$ be locally flat and let $U\subset M_{t}$ be a connected definable neighborhood of $x_{0}$. We define a relation $<_{x_{0},U}$ on $U$ in the following way.
By lemma \[lem:flat-two-components\], $U\backslash\{x_{0}\}$ has two connected components which we’ll denote by $V_{+}$ and $V_{-}$. Let $x$ and $y$ be points in $U$. We’ll say that $x<_{x_{0},U}y$ if one of the following hold:
- $x,y\in V_{+}$ and $x_{0}\nsim_{x}y$
- $x,y\in V_{-}$ and $x_{0}\nsim_{y}x$
- $y\in V_{+}$ and $x\in V_{-}$.
- $y=x_{0}$ and $x\in V_{-}$.
- $x=x_{0}$ and $y\in V_{+}$.
Note that by proposition \[prop:definable-relation\], $x<_{x_{0},U}y$ is definable.
We now show that if $x_{0}$ is locally flat then there exists a neighborhood $x_{0}\in U$ such that $<_{x_{0},U}$ defines a dense linear order on $U$.
Let $M$ be a 1-dimensional connected $\omega$-saturated t.t.t structure such that for all $x\in M_{t}$, $d_{M}(M_{t}\backslash\{x\})>1$. Let $D\subset M_{t}$ be a connected open definable subset such that for every point $x\in D$, $d_{M}(M\backslash\{x\})=2$. Let $x_{0}\in D$ be a locally flat point. Then there exists a connected open neighborhood $U\subset D$ of $x_{0}$ such that $<_{x_{0},U}$ defines a dense linear order on $U$.\[prop:exist-set-with-order\]
Since $x_{0}$ is locally flat, there exist points $a,b\in M_{t}$ and a basis set $V\subset D$ such that $x_{0}\subset V$ and for every point $y\in V$ we have $a\nsim_{y}b$.
We can assume that $V$ is connected because otherwise we can take the connected component containing $x_{0}$. In addition, let $V_{+}$ and $V_{-}$ be the two connected components of $V\backslash\{x_{0}\}$ (by lemma \[lem:flat-two-components\] there are exactly two) and let $C_{a}$ and $C_{b}$ be the connected components of $M_{t}\backslash\{x_{0}\}$ such that $a\in C_{a}$ and $b\in C_{b}$. In addition, without loss of generality $V_{+}\subset C_{a}$ and $V_{-}\subset C_{b}$. This follows from the fact that $C_{a}\cap V$ and $C_{b}\cap V$ partition $V$ into two clopen sets so by lemma \[lem:flat-two-components\], one must equal $V_{+}$ and the other must equal $V_{-}$.
We’ll now show that $<_{x_{0},V}$ is a dense linear order on $V$.
Let $x$, $y$ and $z$ be points in $V_{+}$. In addition, let $X_{a}$ and $X_{b}$ be the connected components of $M_{t}\backslash\{x\}$ such that $a\in X_{a}$ and $b\in X_{b}$. $Y_{a}$, $Y_{b}$, $Z_{a}$ and $Z_{b}$ are defined analogously for $y$ and $z$.
Since $x,y\in V_{+}\subset C_{a}$, it follows from lemma \[lem:disjoint-components\] that $x_{0}\in X_{b}$ and $x_{0}\in Y_{b}$. Because if we assume for contradiction that $x_{0}\in X_{a}$ then by lemma \[lem:disjoint-components\] we get that $C_{b}\cap X_{b}=\emptyset$ which is a contradiction to the fact that $b\in C_{b}\cap X_{b}$. The proof that $x_{0}\in Y_{b}$ is identical.
1. $x\nless_{x_{0},V}y\Rightarrow y<_{x_{0},V}x$:
According to the assumption, $x_{0}\sim_{x}y$ which together with the fact that $x_{0}\in X_{b}$ means that $y\in X_{b}$. Now let’s assume for contradiction that $x\in Y_{b}$. Since $y\in X_{b}$ we get from lemma \[lem:disjoint-components\] that $Y_{a}\cap X_{a}=\emptyset$ which is a contradiction since $a\in X_{a}\cap Y_{a}$. Therefore, $x\in Y_{a}$ which together with $x_{0}\in Y_{b}$ gives $y<x_{o,V}x$.
2. $x<_{x_{0},V}y\Rightarrow y\nless_{x_{0},V}x$:
Since $x<_{x_{0},V}y$ and $x_{0}\in X_{b}$, we get that $y\in X_{a}$. Now, we claim that $x\in Y_{b}$. For otherwise, if $x\in Y_{a}$ then we’d get from lemma \[lem:disjoint-components\] that $Y_{b}\cap X_{b}=\emptyset$ which is a contradiction.
Therefore, since $x_{0}\in Y_{b}$ as well, $y\nless_{x_{0},V}x$.
3. $x<_{x_{0},V}y\wedge y<_{x_{0},V}z\Rightarrow x<_{x_{0},V}z$:
According to the assumptions, $x_{0},x\in Y_{b}$, $z\in Y_{a}$, $x_{0}\in X_{b}$ and $y\in X_{a}$. We have to prove that $z\in X_{a}$ as well. But again by lemma \[lem:disjoint-components\], $y\in X_{a}\wedge x\in Y_{b}\Rightarrow Y_{a}\subset X_{a}$.
The proof of these claims is either trivial or identical when $x$, $y$ and $z$ are distributed differently among $V_{+}$, $V_{-}$ and $\{x_{0}\}$.
This shows that $<_{x_{0},V}$ is indeed a linear order. We’ll now show that if $x<_{x_{0},V}y$ then there exists a point $z\in V$ such that $x<_{x_{0},V}z<_{x_{0},V}y$. Again we’ll assume that $x,y\in V_{+}$. As before, this means that $x\in Y_{b}$ and $y\in X_{a}$ which by lemma \[lem:disjoint-components\] implies that $(X_{a}\cap Y_{b})\cap V\neq\emptyset$. Let $z$ be some point in $(X_{a}\cap Y_{b})\cap V$. By the definition of $<_{x_{0},V}$ and the fact that it’s linear we get that $x<_{x_{0},V}z<_{x_{0},V}y$.
Before extending the order defined to connected components, we introduce the notion of an interval in a t.t.t structure and prove some useful properties.
Let $M$ be a 1-dimensional t.t.t structure and $x,y\in M_{t}$ such that $d_{M}(M_{t}\backslash\{x\})=d_{M}(M_{t}\backslash\{y\})=2$. In addition, let $X_{1}$ and $X_{2}$ be a clopen partition of $M_{t}\backslash\{x\}$ and let $Y_{1}$ and $Y_{2}$ be a clopen partition of $M_{t}\backslash\{y\}$ such that $x\in Y_{1}$ and $y\in X_{1}$. Then *the interval between $x$ and $y$* will be defined as $I(x,y)=X_{1}\cap Y_{1}$. If $x=y$ then $I(x,y)=\emptyset$.
By lemma \[lem:disjoint-components\], if $V\subset M_{t}$ is an open definable connected subset and $x,y\in V$ then $I(x,y)\cap V\neq\emptyset$ and the following union is disjoint: $$V=(X_{2}\cap V)\cup\{x\}\cup(I(x,y)\cap V)\cup\{y\}\cup(Y_{2}\cap V)$$ This motivates us to think of $I(x,y)$ as the set lying “in between” $x$ and $y$.
Let $M$ be a 1-dimensional connected $\omega$-saturated t.t.t structure such that for every point $x\in M_{t}$, $d_{M}(M_{t}\backslash\{x\})>1$. Let $D\subset M_{t}$ be a connected open definable subset which is locally flat. Let $x\neq y$ be points in $D$. Then:\[lem:interval-lemma\]
1. $I(x,y)\cap D$ is a non-empty definable open connected set.
2. $\{x,y\}=bd(I(x,y))$.
3. If $a,b\in D\backslash\{x,y\}$ such that $a\sim_{x}b$ and $a,b\notin I(x,y)$, then $a\sim_{y}b$.
4. If $a,b\in D\backslash\{x,y\}$ such that $a\nsim_{x}b$ and $a,b\notin I(x,y)$, then $a\nsim_{y}b$.
Let $X_{1}$ and $X_{2}$ be the connected components of $M_{t}\backslash\{x\}$ and let $Y_{1}$ and $Y_{2}$ be the connected components of $M_{t}\backslash\{y\}$. Note that by proposition \[lem:flat-two-components\], both $D\backslash\{x\}$ and $D\backslash\{y\}$ have exactly two connected components which are given by $X_{1}\cap D$, $X_{2}\cap D$ and $Y_{1}\cap D$, $Y_{2}\cap D$ respectively. Without loss of generality, $x\in Y_{1}$ and $y\in X_{1}$ so $I(x,y)=X_{1}\cap Y_{1}$, $X_{2}\subset Y_{1}$ and $Y_{2}\subset X_{1}$. By lemma \[lem:disjoint-components\] this means that
([\*]{}) $D=(X_{2}\cap D)\cup\{x\}\cup(I(x,y)\cap D)\cup\{y\}\cup(Y_{2}\cap D)$.
We’ll now prove the four parts of the lemma.
1. First of all, since $X_{1}$ and $Y_{1}$ are definable, $I(x,y)$ is definable as well.
By lemma \[lem:disjoint-components\], $I(x,y)\cap D\neq\emptyset$. $I(x,y)$ is open as the intersection of open sets.
Now, assume for contradiction that $I(x,y)\cap D$ isn’t connected. Let $A_{1}$ and $A_{2}$ be a clopen partition of $I(x,y)\cap D$. By lemma \[lem:disjoint-components\] , the boundaries of $A_{1}$ and $A_{2}$ in $D$ are contained in $\{x,y\}$. Since $D$ is connected, for each $i=1,2$ we have either $x\in bd(a_{i})$ or $y\in bd(A_{i})$. Assume for contradiction that $x\notin\overline{A_{1}}$ and $y\notin\overline{A_{2}}$. By ([\*]{}) this means that the sets $$(D\cap X_{2})\cup\{x\}\cup A_{2},\:(D\cap Y_{2})\cup\{y\}\cup A_{1}$$ form a clopen partition of $D$ which is a contradiction to the assumption that $D$ is connected.
Therefore, without loss of generality, we can assume that $x\in bd(A_{1})$ and $x\in bd(A_{2})$. But then the set $U=D\cap Y_{1}$ is open and $U\backslash\{x\}$ has more than two connected components which is a contradiction to lemma \[lem:flat-two-components\].
2. This follows immediately from lemma \[lem:disjoint-components\].
3. If $a,b\in X_{2}\cap D$ then since $X_{2}\subset Y_{1}$, $a,b\in Y_{1}$ which means that $a\sim_{y}b$. So we can assume that $a,b\in X_{1}\cap D$. By the assumption, $a,b\notin Y_{1}$ which by ([\*]{}) implies that $a,b\in Y_{2}\Rightarrow a\sim_{y}b$.
4. Without loss of generality, $a\in X_{1}$ and $b\in X_{2}$. Therefore, $b\in Y_{1}$. In addition, $Y_{2}\subset X_{1}$ and $a\notin Y_{1}\cap X_{1}$ so $a\in Y_{2}$. This means that $a\nsim_{y}b$.
Let $M$ be a 1-dimensional connected $\omega$-saturated t.t.t structure such that for all $x\in M_{t}$, $d_{M}(M_{t}\backslash\{x\})>1$. Let $D\subset M_{t}$ be a connected open definable subset which is locally flat. Let $x_{0}\in D$. Then there exist points $a,b\in D$ such that:\[lem:exists-interval-with-order\]
1. $x_{0}\in I(a,b)$
2. $<_{x_{0},I(a,b)}$ defines a dense linear order on $I(a,b)$
3. $I(a,b)\subset D$
4. For all $x\in I(a,b)$, $a\nsim_{x}b$
By proposition \[prop:exist-set-with-order\], there exists a definable connected open neighborhood $U\subset D$ of $x_{0}$ such that $<_{x_{0},U}$ defines a dense linear order on $U$. Since $M_{t}$ is Hausdorff and $bd(U)$ is finite, we can assume that $<_{x_{0},U}$ defines a dense linear order on $\overline{U}$. Let $a$ be the point on $bd(U)$ such that $x_{0}<_{x_{0},U}a$ and such that for every point $y\in bd(U)$ with $x_{0}<_{x_{0},U}y$, $a\leq_{x_{0},U}y$. Similarly, Let $b$ be the point on $bd(U)$ such that $b<_{x_{0},U}x_{0}$ and for each point $y\in bd(U)$ with $y<_{x_{0},U}x_{0}$, $y\leq_{x_{0},U}b$.
Let $A_{1}$ and $A_{2}$ be a clopen partition of $M_{t}\backslash\{a\}$ such that $b\in A_{1}$ and let $B_{1}$ and $B_{2}$ be a clopen partition of $M_{t}\backslash\{b\}$ such that $a\in B_{1}$. Furthermore, let $X_{+}$ and $X_{-}$ be a clopen partition of $M_{t}\backslash\{x_{0}\}$ such that $a\in X_{+}$ and $b\in X_{-}$.
First we prove that $x_{0}\in I(a,b)=A_{1}\cap B_{1}$. Assume for contradiction that $x_{0}\in A_{2}$. Since $a\in X_{+}$ it follows from lemma \[lem:disjoint-components\] that $X_{-}\subset A_{2}$ which implies that $b\in A_{2}$ which is a contradiction. Similarly, $x_{0}\in B_{1}$. Together this shows that $x_{0}\in A_{1}\cap B_{1}$.
Next we’ll prove that for every point $y\in I(a,b)\cap\overline{U}$ we have $a\nsim_{y}b$ and $b<_{x_{0},U}y<_{x_{0},U}a$. Let $y$ be some point in $I(a,b)\cap\overline{U}$. Without loss of generality, $y\in X_{+}$. Since in addition $y\in A_{1}$, $a\nless_{x_{0},U}y$ which means that $y<_{x_{0},U}a$. By the definition of the order this implies that $x_{0}\nsim_{y}a$. Let $Y_{1}$ and $Y_{2}$ be a clopen partition of $M_{t}\backslash\{y\}$ such that $x_{0}\in Y_{1}$ and $a\in Y_{2}$. By lemma \[lem:disjoint-components\], $X_{-}\subset Y_{1}$ which means that $b\in Y_{1}$. This proves that $y<_{x_{0},U}a$ and $a\nsim_{y}b$. Similarly, $b<_{x_{0},U}y$.
We’ll now see that $I(a,b)\subset U$. By lemma \[lem:interval-lemma\], $I(x,y)\cap D$ is a non-empty open connected set. Furthermore, as we showed above, $x_{0}\in I(a,b)$. Assume for contradiction that $I(a,b)\backslash U\neq\emptyset$. Then by the connectedness of $I(a,b)\cap D$, $I(a,b)\cap bd(U)\neq\emptyset$. Let $y$ be a point in $I(a,b)\cap bd(U)$. Then as we saw before, $b<_{x_{0},U}y<_{x_{0},U}a$ which is clearly a contradiction to the choice of $a$ and $b$.
Let $M$ be a 1-dimensional connected $\omega$-saturated t.t.t structure such that for every point $x\in M_{t}$, $d_{M}(M_{t}\backslash\{x\})>1$. Let $D\subset M_{t}$ be a connected open definable subset which is locally flat. Let $x_{0}$, $a$ and $b$ be points in $D$ such that $x_{0}\in I(a,b)\subset D$, for every point $y\in I(a,b)$ we have $a\nsim_{y}b$ and $<_{x_{0},I(a,b)}$ defines a dense linear order on $I(a,b)$.
Then for each pair of points $c,d\in I(a,b)$ such that $c<_{x_{0},I(a,b)}d$, \[lem:subinterval-in-interval\]$$I(c,d)=\{y\in I(a,b)\vert c<_{x_{0},I(a,b)}y<_{x_{0},I(a,b)}d\}$$ .
Let $C_{a}$ and $C_{b}$ be a clopen partition of $M_{t}\backslash\{c\}$ such that $a\in C_{a}$ and $b\in C_{b}$ and let $D_{a}$ and $D_{b}$ be a clopen partition of $M_{t}\backslash\{d\}$ such that $a\in D_{a}$ and $b\in D_{b}$. Furthermore, let $A_{1}$ and $A_{2}$ be a clopen partition of $M_{t}\backslash\{a\}$ such that $c,d,b\in A_{1}$ and let $B_{1}$ and $B_{2}$ be a clopen partition of $M_{t}\backslash\{b\}$ such that $c,d,a\in B_{1}$.
Let $X_{+}$ and $X_{-}$ be a clopen partition of $M_{t}\backslash\{x_{0}\}$ such that $a\in X_{-}$ and $b\in X_{+}$. We’ll assume that $c,d\in X_{+}$ since the other cases are either similar or trivial.
Now we’ll show that $x_{0}\in C_{a}$ and $x_{0}\in D_{a}$. Assume for contradiction that $x_{0}\in C_{b}$. Since $c\in X_{+}$, it follows from lemma \[lem:disjoint-components\] that $X_{-}\cap C_{a}=\emptyset$ which is a contradiction to the fact that $a\in X_{-}\cap C_{a}$. The proof that $x_{0}\in D_{a}$ is similar.
Since $c<_{x_{0},I(a,b)}d$ and $x_{0}\in C_{a}\cap D_{a}$, by the definition of $<_{x_{0},I(a,b)}$ it follows that $d\in C_{b}$ and $c\in D_{a}$.
We’re now ready to prove the lemma.
First we note that by lemma \[lem:disjoint-components\], it follows from the assumptions above that $I(a,b)=A_{1}\cap B_{1}$, $C_{b}\subset A_{1}$ and $D_{a}\subset B_{1}$. In addition, since $d\in C_{b}$ and $c\in D_{a}$ it follows from lemma \[lem:disjoint-components\] that $I(c,d)=C_{b}\cap D_{a}$. Together this means that $I(c,d)\subset I(a,b)$.
We’ll now prove that $$I(c,d)=\{y\in I(a,b)\vert c<_{x_{0},I(a,b)}y<_{x_{0},I(a,b)}d\}$$ Let $y$ be some point in $I(a,b)$ such that $c<_{x_{0},I(a,b)}y<_{x_{0},I(a,b)}d$. Since $c<_{x_{0},I(a,b)}y$, $y\in X_{+}$. In addition, $y\in X_{+}\cap C_{a}\Rightarrow c\nless_{x_{0},I(a,b)}y$ so $y\in C_{b}$. In a similar fashion it follows that $y\in D_{a}$. Together this means that $y\in I(c,d)$.
Now, let $y$ be a point in $I(c,d)=C_{b}\cap D_{a}$. Since $C_{b}\subset X_{+}$, $y\in X_{+}$. Furthermore, since $y\in C_{b}$, $c<_{x_{0},I(a,b)}y$. Finally, since $y\in D_{a}$, $d\nless_{x_{0},I(a,b)}y$ which means that $y<_{x_{0},I(a,b)}d$.
Now, let’s assume that $M$ and the set $D\subset M_{t}$ fulfill the assumptions in lemma \[lem:exists-interval-with-order\]. Then for each point $x_{0}\in D$ there exists a pair of points $a$ and $b$ in $D$ such that $x_{0}\in I(a,b)\subset D$, for every point $y\in I(a,b)$ we have $a\nsim_{y}b$ and $<_{x_{0},I(a,b)}$ defines a dense linear order on $I(a,b)$.
By lemma \[lem:subinterval-in-interval\], this means that for all $c,d\in I(a,b)$, $I(c,d)\subset I(a,b)$ and: $$\{y\in I(a,b)\vert c<_{x,V_{x}}y<_{x,V_{x}}d\}=I(c,d)$$ . In other words, in the set $I(a,b)$ guaranteed by lemma \[lem:exists-interval-with-order\], the notion of an interval we defined above coincides with the interval induced by the order $<_{x_{0},I(a,b)}$.
We’ll now prove three lemmas about locally flat points which together will show that the order we defined above can be extended from locally flat points to connected locally flat sets.
Let $M$ be a 1-dimensional connected $\omega$-saturated t.t.t structure such that for every element $x\in M_{t}$, $d_{M}(M_{t}\backslash\{x\})>1$. Let $D\subset M_{t}$ be a connected open definable subset which is locally flat. Let $x$, $a$ and $b$ be points in $D$ such that for every open set $U\subset D$ containing $x$ there exists some point $u\in U$ such that $a\nsim_{u}b$. Then $a\nsim_{x}b$.\[lem:locally-flat-closed-sep\]
Since $M_{t}$ is Hausdorff, there exists an open definable connected set $U\subset D\backslash\{a,b\}$ containing $x$. By lemma \[lem:exists-interval-with-order\], there exists a pair of points $c$ and $d$ in $U$ such that $x\in I(c,d)\subset U$, for every point $y\in I(c,d)$ we have $c\nsim_{y}d$ and $<_{x,I(c,d)}$ defines a dense linear order on $I(c,d)$.
By the assumptions of the lemma, there exists some $y\in I(c,d)$ such that $a\nsim_{y}b$. By lemma \[lem:subinterval-in-interval\], $I(x,y)\subset I(c,d)$ which means that $a,b\notin I(x,y)$. So by lemma \[lem:interval-lemma\], $a\nsim_{x}b$.
Let $M$ be a 1-dimensional connected $\omega$-saturated t.t.t structure such that for all $x\in M_{t}$, $d_{M}(M_{t}\backslash\{x\})>1$. Let $D\subset M_{t}$ be a connected open definable subset which is locally flat. Let $x$, $a$ and $b$ be points in $D$ such that $a\nsim_{x}b$. The there exists a definable open set $U\subset D$ containing $x$ such that for every $u\in U$, $a\nsim_{u}b$.\[lem:locally-flat-open-sep\]
As in the previous lemma, there exists an open definable connected set $U\subset D\backslash\{a,b\}$ containing $x$. By lemma \[lem:exists-interval-with-order\], there exists a pair of points $c$ and $d$ in $U$ such that $x\in I(c,d)\subset U$, for every point $y\in I(c,d)$ we have $c\nsim_{y}d$ and $<_{x,I(c,d)}$ defines a dense linear order on $I(c,d)$.
Let’s choose some point $y\in I(c,d)$. Since $a,b\notin I(c,d)$ it follows that $a,b\notin I(x,y)\subset I(c,d)$. Therefore, by lemma \[lem:interval-lemma\], $a\nsim_{y}b$.
We now use the previous two lemmas to show that in some well defined sense, locally flat sets look like a line.
Let $M$ be a 1-dimensional $\omega$-saturated t.t.t structure such that for every point $x\in M_{t}$, $d_{M}(M_{t}\backslash\{x\})>1$. Let $D\subset M_{t}$ be a connected open definable subset which is locally flat. Then, there doesn’t exist a definable connected closed subset $F\subset D$ such that $bd(F)>2$.\[lem:locally-flat-connected-boundary-not-bigger-2\]
Assume for contradiction that $F\subset D$ is a definable closed connected subset and that $a,b,c\in bd(F)$.
Let $F_{ab}$ denote the set of points $x\in F$ such that $a\nsim_{x}b$.
$F_{ab}\neq\emptyset$.
By lemma \[lem:exists-interval-with-order\], there exists a pair of points $x$ and $y$ in $D$ such that $a\in I(x,y)\subset D$, for every point $z\in I(x,y)$ we have $x\nsim_{z}y$ and $<_{a,I(x,y)}$ defines a dense linear order on $I(x,y)$.
Let $s$ and $t$ be points in $I(x,y)$ such that $s<_{a,I(x,y)}a<_{a,I(x,y)}t$. By lemma \[lem:subinterval-in-interval\], $$\{z\in I(a,b)\vert s<_{a,I(x,y)}z<_{a,I(x,y)}t\}=I(s,t)$$ so we can use the notion $I(s,t)$ to represent the interval given by the order $<_{a,I(x,y)}$. We’ll use the result throughout the proof of the claim.
Since $I(s,t)=I(s,a)\cup\{a\}\cup I(a,t)$ and $a\in bd(U)$, we can assume without a loss of generality that for every point $v\in I(a,t)$ there exists a point $u\in I(a,v)$ such that $u\in F$. Therefore, there exists some point $u\in I(a,t)$ such that $I(a,u)\subset F$. Because assume for contradiction that for every $u\in I(a,t)$ there existed some $v\in I(a,u)$ such that $v\notin F$. In that case, the definable set $I(a,t)\cap F$ would have an infinite number of connected components which is a contradiction to $M$ being t.t.t.
Now, since $a\notin int(F)$, for every point $u\in I(s,a)$ there must be some point $v\in I(u,a)$ such that $v\notin F$. Similarly to above, this means that there exists some point $v\in I(s,a)$ such that $I(v,a)\cap F=\emptyset$.
Together, this means that without loss of generality we can assume that $I(x,a)\cap F=\emptyset$ and $I(a,y)\subset F$.
If $b\in I(a,y)$ then for any $u\in I(a,b)\subset I(a,y)\subset F$, $a\nsim_{u}b$ and so $u\in F_{ab}$.
Let’s assume that $b\notin I(a,y)$.
Let $A_{1}$ and $A_{2}$ be a clopen partition of $M_{t}\backslash\{a\}$ such that $x\in A_{1}$ and let $X_{1}$ and $X_{2}$ be a clopen partition of $M_{t}\backslash\{x\}$ such that $a\in X_{1}$. By the choice of $A_{1}$ and $X_{1}$, $I(x,a)=A_{1}\cap X_{1}$. Furthermore, since $I(x,a)\cap F=\emptyset$ and $U$ is connected, from lemma \[lem:disjoint-components\] it follows that $F\subset\overline{A_{2}}$ or $F\subset\overline{X_{2}}$. But $I(a,y)\subset A_{2}$ and $I(a,y)\subset F$ so it must be that $F\subset\overline{A_{2}}=A_{2}\cup\{a\}$. This means that $b\in A_{2}$.
Now, let $u$ be some point in $I(a,y)\subset F$. Let $U_{1}$ and $U_{2}$ be a clopen partition of $M_{t}\backslash\{u\}$ such that $a\in U_{1}$. Assume for contradiction that $b\in U_{1}$. Since $b\in A_{2}$ this means that $b\in A_{2}\cap U_{1}$. But $a\in U_{1}$ and $u\in A_{2}$ which means that $I(a,u)=A_{2}\cap U_{1}$. This implies that $b\in I(a,u)\subset I(a,y)$ which is a contradiction to our assumption. Therefore, $b\in U_{1}$ and $a\in U_{2}$ which means that $u\in F_{ab}$.
This concludes the proof of the claim.
By lemmas \[lem:locally-flat-open-sep\] and \[lem:locally-flat-closed-sep\], $U_{ab}$ is clopen. Therefore, $U=U_{ab}$.
Similarly, if $U_{ac}$ and $U_{bc}$ are defined in the analogous fashion, $U=U_{ac}=U_{bc}=U_{ab}$. We’ll now show that this is a contradiction.
Let’s choose a point $x\in int(U)$. Let $X_{1}$ and $X_{2}$ be the two connected components of $D\backslash\{x\}$. Either $X_{1}$ or $X_{2}$ will contain two out of $a$, $b$, and $c$. Without loss of generality, $a,b\in X_{1}$. However, since $x\in U_{ab}$, $a\nsim_{x}b$ which is clearly a contradiction.
We’re now ready to show that every locally flat set is o-minimal. In order to do this, we’ll extend our previous notion of order from neighborhoods of locally flat points to locally flat sets.
Let $D$ be a definable open connected locally flat set. Let $a\in D$ be some arbitrary point which we’ll think of as the center. In addition, let $D_{+}$ and $D_{-}$ be the two connected components of $D\backslash\{a\}$ which we’ll think of as the “positive side” and the “negative side”. Finally let $x,y\in D$. We say that $x<_{a,D}y$ if one of the following holds:
- $x,y\in D_{+}$ and $a\nsim_{x}y$
- $x,y\in D_{-}$ and $a\nsim_{y}x$
- $y\in D_{+}$ and $x\in D_{-}$
- $y=a$ and $x\in D_{-}$
- $x=a$ and $y\in D_{+}$
By proposition \[prop:definable-relation\], $<_{a,D}$ is definable.
The next proposition shows that $<_{a,D}$ defines a dense linear order on $D$ such that the induced interval topology is equivalent to the topology induced by $M_{t}$.
Let $M$ be a 1-dimensional connected $\omega$-saturated t.t.t structure such that for each point $x\in M_{t}$, $d_{M}(M_{t}\backslash\{x\})>1$. Let $D\subset M_{t}$ be a connected open definable subset which is locally flat and let $a$ be some point in $D$. Then $<_{a,D}$ defines a dense linear order on $D$ such that the induced interval topology is equivalent to the topology induced by $M_{t}$.\[prop:locally-flat-set-o-minimal\]
Let $D_{+}$, $D_{-}$ be the sets used in the definition of $<_{a,D}$ above.
Let $x,y,z\in D$. In addition, let $X_{1}$ and $X_{2}$ be the connected components of $D\backslash\{x\}$ such that $a\in X_{1}$. $Y_{1}$, $Y_{2}$, $Z_{1}$ and $Z_{2}$ are defined analogously for $y$ and $z$.
1. $x\nless_{a,D}y\Rightarrow y<_{a,D}x$:
We assume that $x,y\in D_{+}$. The other possibilities are either identical or trivial. By the assumption, $y\in X_{1}$. Assume for contradiction that $x\in Y_{1}$. Since $a\in X_{1}\cap Y_{1}$, by lemma \[lem:disjoint-components\] both $X_{2}$ and $Y_{2}$ are subsets of $D_{+}$.
Let’s define $$F=(D_{+}\cap X_{1}\cap Y_{1})\cup\{a\}\cup\{x\}\cup\{y\}$$ . By lemma \[lem:disjoint-components\] $F$ is closed and $bd(F)\subset\{a,x,y\}$. We’ll now show that $F$ is connected and that $bd(F)=\{a,x,y\}$.
Let $C$ be some connected component of $F$. Since $D$ is connected, $bd(C)=\{a,x,y\}$. For assume for contradiction that one of the points in the set $\{a,x,y\}$ was not included in $bd(C)$. Without loss of generality, let’s assume that $bd(C)=\{x,y\}$. Since $bd(C)\subset\{a,x,y\}$, $bd(X_{2})=\{x\}$ and $bd(Y_{2})=\{y\}$, it follows that $C\cup X_{2}\cup X_{1}\cup\{x\}\cup\{y\}$ is a clopen subset of $D$ which is clearly a contradiction. Assuming that $bd(C)$ is equal to some other strict subset of $\{a,x,y\}$ gives a similar contradiction.
In addition, $a$, $x$ and $y$ are boundary points of $D_{-}$, $X_{2}$ and $Y_{2}$ respectively so by lemma \[lem:flat-two-components\], each one of $a$, $x$, and $y$ is the boundary point of at most one connected component of $F$. Therefore, $F$ has only one connected component and $bd(F)=\{a,x,y\}$.
However, this is a contradiction to lemma \[lem:locally-flat-connected-boundary-not-bigger-2\].
2. $x<_{a,D}y\Rightarrow y\nless_{a,D}x$:
Also in this case we’ll assume that $x,y\in D_{+}$. Since $x<_{a,D}y$ and $a\in X_{1}$, we get that $y\in X_{2}$. Now, we claim that $x\in Y_{1}$. For otherwise, if it was true that $x\in Y_{2}$ then we’d get from lemma \[lem:disjoint-components\] that $Y_{1}\cap X_{1}=\emptyset$ which is a contradiction to the fact that $a\in X_{1}\cap Y_{1}$.
3. $x<_{a,D}y\wedge y<_{a,D}z\Rightarrow x<_{a,D}z$:
According to the assumptions, $a,x\in Y_{1}$, $z\in Y_{2}$, $a\in X_{1}$ and $y\in X_{2}$. We have to prove that $z\in X_{2}$ as well. But again by lemma \[lem:disjoint-components\], $$y\in X_{2}\wedge x\in Y_{1}\Rightarrow Y_{2}\subset X_{2}\Rightarrow z\in X_{2}$$ .
This shows that $<_{a,D}$ is a linear order. We’ll now show that $<_{a,D}$ is dense.
Let’s assume that $x<_{a,D}y$. As we showed above, this means that $y\in X_{2}$ and $x\in Y_{1}$. By lemma \[lem:interval-lemma\], $X_{2}\cap Y_{1}=I(x,y)\neq\emptyset$. Let $s$ be some point in $I(x,y)$. Since $s\in X_{2}\cap Y_{1}$, it follows from the definition of $<_{a,D}$ that $x<_{a,D}s<_{a,D}y$.
We’ll now see that the order topology induced on $D$ by $<_{a,D}$ is equivalent to the topology on $D$ induced by $M_{t}$.
As a first step, we note that if $x<_{a,D}y$, then $$I(x,y)\cap D=\{z\in D\vert x<_{a,D}z<_{a,D}y\}$$ . This is immediate from the definitions of $I(x,y)$ and $<_{a,D}$.
Let $U\subset D$ be an open set in $D$ with $x\in U$. By lemma \[lem:exists-interval-with-order\], there exists a pair of points $s,t\in U$ such that $x\in I(s,t)$ and $I(s,t)\subset U$. Without loss of generality, $s<_{a,D}t$.
Therefore, $$x\in\{u\in D\vert s<_{a,D}u<_{a,D}t\}=I(s,t)\cap D\subset U$$ . The other direction is trivial as $I(x,y)$ is open in $D$ for every $x,y\in D$.
We now obtain our primary result as an immediate consequence of propositions \[prop:finite-not-flat\] and \[prop:locally-flat-set-o-minimal\].
Let $M$ be a 1-dimensional connected $\omega$-saturated t.t.t structure such that for every point $x\in M_{t}$, $d_{M}(M_{t}\backslash\{x\})>1$. Then there exists a finite set $X\subset M_{t}$ such that each of the finite number of connected components of $M_{t}\backslash X$ are o-minimal.\[thm:remove-finite-o-minimal\]
Let $X$ be the definable set of points in $M_{t}$ which aren’t locally flat. By proposition \[prop:finite-not-flat\], $X$ is finite. Let $D$ be a connected component of $M_{t}\backslash X$. Since there’re only a finite number of connected components, $D$ is a connected open definable subset which is locally flat. By proposition \[prop:locally-flat-set-o-minimal\], there exists a definable dense linear order which induces the topology on $D$. By [@P1 6.2], this means that $D$ is o-minimal.
Note that even if $M_{t}$ isn’t connected, we can obtain theorem \[thm:remove-finite-o-minimal\] for any open connected definable subset $D\subset M_t$ with the property that removing any point from $D$ splits $D$ into more than one connected component.
Let’s return to the structure $R_{int}=\langle\mathbb{R},I(x,y,z)\rangle$ from example \[exa:interval-example\]. By theorem \[thm:remove-finite-o-minimal\], we should be able to recover the standard order $<$ on $\mathbb{R}$ from $I$.
Let $D=\mathbb{R}$ and let $a$ be some point from $\mathbb{R}$. In addition, let $x$ and $y$ be points in $\mathbb{R}$ such that $a<x<y$. By the construction of, $<_{a,D}$ it’s clear that $a<_{a,D}x<_{a,D}y$. By checking the other possibilities for $x$ and $y$ in a similar fashion it’s easy to see that $<_{a},D$ is equivalent to $<$.
Structures Without Splitting\[sec:Structures-Without-Splitting\]
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In this section we look at structures where removing a point doesn’t split the structure into more than one connected component. One example of such a structure is the unit circle. Our main goal in this section will be to find alternative topological properties which ensure that the structure is at least locally o-minimal as in the case of the unit circle.
Let $M$ be an $\omega$-saturated one dimensional t.t.t structure. Let $A$ be a definable open set, $f:A\rightarrow\mathcal{P}(M_{t})$ a function such that $f(x)$ is finite for each point $x\in M_{t}$ and $\Gamma$ the graph of $f$. Then for each point $x\in A$, the fiber $(\overline{\Gamma})_{x}$ is finite.\[lem:finite-fiber\]
Let’s assume for contradiction that there exists a point $x\in A$ and a sequence of points $(y_{i})_{i<\omega}$ in $M_{t}$ such that for every $i<\omega$, every basis set $U\subset A$ containing $x$ and every basis point $V$ containing $y_{i}$, there exists some $z\in U\backslash\{x\}$ such that $f(z)\in V$.
For every $i<\omega$ and every basis set $V$ containing $y_{i}$ there exists a basis set $U\subset A$ containing $x$ such that for every basis set $W\subset U$ containing $x$ we have $f(bd(W))\cap V\neq\emptyset$.
Let’s take some $i<\omega$. Let $V$ be some basis set containing $y_{i}$. Assume for contradiction that for every basis set $U\subset A$ containing $x$ there exists some basis set $W\subset U$ containing $x$ such that $f(bd(W))\cap V=\emptyset$.
We now define $X=f^{-1}(V)\cap A$. By the definition of $y_{i}$, for every basis set $W$ containing $x$, $W\cap(X\backslash\{x\})\neq\emptyset$. Therefore, by the assumption for contradiction there exists a descending sequence of basis sets $(W_{i})_{i<\omega}$ such that for all $i<\omega$:
- $x\in W_{i}\subset X$
- $\overline{W_{i+1}}\subsetneq W_{i}$
- $bd(W_{i})\cap X=\emptyset$
By the last two properties, for every $i<\omega$ the set $W_{i}\backslash W_{i+1}$ is clopen in $X$. But this means that $X$ can be partitioned into an infinite number of definable clopen subsets which is a contradiction.
Now, by the $\omega$-saturation we can assume that there exists some $N<\omega$ such that for every $z\in A$, $\vert f(z)\vert<N$. Similarly, there exists some $B<\omega$ such that for every basis set $V$, $\vert bd(V)\vert<B$. Let $V_{1},\dots,V_{NB+1}$ be pairwise disjoint basis sets such that for every $1\leq i\leq NB+1$ we have $y_{i}\in V_{i}$. By the claim, there exists a basis set $U\subset A$ containing $x$ such that for every basis set $W\subset U$ containing $x$ and every every $1\leq i\leq NB+1$ we have $f(bd(W))\cap V_{i}\neq\emptyset$.
Let $W\subset U$ be some basis set. By the definitions of $N$ and $B$, $\vert f(bd(W))\vert\leq NB$ which is a contradiction to the fact that $V_{1},\dots,V_{NB+1}$ are pairwise disjoint.
Let $M$ be an $\omega$-saturated one dimensional t.t.t structure such that there exist a definable continuous function $F:M_{t}^{2}\rightarrow M_{t}$ and a point $a\in M_{t}$ such that for each $x\in M_{t}$, $F(x,x)=a$ and $F(x,\cdot)$ is injective. Let $f:M_{t}\rightarrow\mathcal{P}(M_{t})$ be a function such that for every $x\in M_{t}$, $\vert f(x)\vert<\infty$ and $f(x)\neq x$. Let $\Gamma$ be the graph of $f$. Then for every basis set $U\subset M_{t}$, there exists a point $x\in U$ such that $(x,x)\notin\overline{\Gamma}$.\[prop:F-function\]
Assume for contradiction that there exists some basis set $U\subset M_{t}$ such that for every $x\in U$, $(x,x)\in\overline{\Gamma}$. We now define the function $g:U\rightarrow \mathcal{P}(M_{t})$ by$$g(x)=\{F(x,y)\vert y\in f(x)\}$$ .
In addition, we define the function $h:M_{t}\rightarrow\mathcal{P}(U)$ by $$h(y)=g^{-1}(y)\cap U$$ . Let $\Gamma_{g}$ and $\Gamma_{h}$ be the graphs of $g$ and $h$ respectively.
By our assumption on $F$, $F(x,x)=a$ for each $x\in U$. Therefore, by the continuity of $F$ together with the assumption that for every $x\in U$, $(x,x)\in\overline{\Gamma}$, we get that $(x,a)\in\overline{\Gamma_{g}}$ for each $x\in U$.
Furthermore, since for every $x\in U$ we have $f(x)\neq x$ and $F(x,\cdot)$ is injective, $g(x)\neq a$ for all $x\in U$.
We’ll now show that there exists an open set $A$ containing $a$ such that for every $y\in A$, the set $h(y)$ is finite. By our assumptions on $f$, $g(x)$ is finite for every $x\in M_{t}$. Therefore, by the exchange principle there are a finite number of points $y\in M_{t}$ such that $h(y)$ is infinite. Furthermore, $h(a)=\emptyset$ so by the Hausdorffness of $M_{t}$, there exists an open set $A$ containing $a$ such that for every $y\in A$, the set $h(y)$ is finite.
In addition, since $(x,a)\in\overline{\Gamma_{g}}$ for each $x\in U$, the fiber $(\overline{\Gamma_{h}})_{a}$ is infinite. However, this is a contradiction to lemma \[lem:finite-fiber\].
Let $M$ be an $\omega$-saturated one dimensional t.t.t structure such that there exist a definable continuous function $F:M_{t}^{2}\rightarrow M_{t}$ and a point $a\in M_{t}$ such that for each $x\in M_{t}$, $F(x,x)=a$ and $F(x,\cdot)$ is injective.
Then for every basis set $U$ there exists a basis set $V\subset U$ such that for every point $x\in V$ there exists a basis set $W$ such that $bd(W)\cap V=\{x\}$. \[prop:boundary-only-x\]
First of all, without loss of generality we can assume that for every $x\in M_{t}$ there is some basis set $W$ such that $x\in bd(W)$. Because let $X$ be the set of all such points. $X$ is clearly definable. Assume for contradiction that $X^{c}$ has a non empty interior. Let $W$ be a basis set such that $\overline{W}\subset X$. Then $bd(W)\subset X$ which is clearly a contradiction. Therefore, $X^{c}$ is finite.
We define a function $f:U\rightarrow\mathcal{P}(M_{t})$ by$$f(x)=\{y\neq x\vert\textrm{there exists a basis set }W\textrm{ such that }\{x,y\}\subset bd(W)\}$$
Let $N$ be an integer such that for every basis set $U$, $\vert bd(U)\vert<N$. Let $\Gamma$ be the graph of $f$.
We now look at two cases.
For the first case let’s assume that there exists a basis set $V\subset U$ such that for each $x\in V$, $\vert f(x)\vert<\infty$. By proposition \[prop:F-function\], there exists some basis set $W\subset V$ such that $(W\times W)\cap\Gamma=\emptyset$. This means that for every point $x\in W$, there exists a basis set $A$ such that $bd(A)\cap W=\{x\}$. Because let $x$ be some point in $W$ and let $A$ be a basis set such that $x\in bd(A)$. Since $(W\times W)\cap\Gamma=\emptyset$, the rest of the boundary points of $A$ are not contained in $W$ which means that $bd(A)\cap W=\{x\}$.
For the second case, assume that for every basis set $V\subset U$ there exists some point $x\in V$ such that $f(x)$ is infinite. We now assume for contradiction that there doesn’t exist a basis set $V\subset U$ such that for every point $x\in V$, there exists a basis set $W$ such that $bd(W)\cap V=\{x\}$.
In order to get a contradiction, we’ll inductively build a sequence of tuples of points, basis sets and functions $(x_{i},V_{i},f_{i})_{i=1}^{N}$ with the following properties:
- $V_{1}$ is an arbitrary basis set in $U$, $x_{1}$ is a point in $V_{1}$ such that $f(x_{1})$ is infinite and $f_{1}=f$.
- For all $1\leq i\leq N$, $f_{i}:U\rightarrow\mathcal{P}(M_{t})$ is defined by $$f_{i}(x)=\{y\neq x,x_{1},\dots,x_{i-1}\vert\textrm{there exists a basis set }W$$ $$\textrm{ such that }\{x,x_{1},\dots,x_{i-1},y\}\subset bd(W)\}$$
- For all $1\leq i\leq N$, $x_{i}\in V_{i}$
- For all $1\leq i\leq N$, $f_{i}(x_{i})$ is infinite.
- For all $i<j$, $x_{i}\notin V_{j}$
The existence of $(x_{1},V_{1},f_{1})$ follows immediately from our assumptions in the second case.
Let’s assume that we’ve constructed the sequence up to the $i$-th place. Since $f_{i}(x_{i})$ is infinite, there exists some basis set $V_{i+1}\subset f_{i}(x_{i})$ such that $x_{i}\notin V_{i+1}$. We define $f_{i+1}$ as above.
Now, if for all $x\in V_{i+1}$ the set $f_{i+1}(x)$ would be finite then just as in the first case, together with the fact that $i<j\Rightarrow x_{i}\notin V_{j}$, there would exist a basis set $W\subset U$ such that for every $y\in W$, there exists a basis set $A$ such that $bd(A)\cap W=\{y\}$. Therefore, by our assumption for contradiction, there exists some point $x_{i+1}\in V_{i+1}$ such that $f_{i+1}(x_{i+1})$ is infinite. Thus we’ve found a tuple $(x_{i+1},V_{i+1},f_{i+1})$ satisfying the requirements.
However, the existence of the tuple $(x_{N},V_{N},f_{N})$ is clearly a contradiction because on the one hand $f_{N}(x_{N})$ is infinite but by the definition of $N$, for every point $x\in U$ the set $f_{N}(x)$ is empty.
We’ll now prove a similar proposition under the assumption that all of the basis sets have only two points in their boundary.
Let $X$ be a topological space and let $U\subset X$ and $V\subset X$ be connected open sets such that $$bd(U)\cap V=bd(V)\cap U=\emptyset$$ and $U\neq V$. Then $U\cap V=\emptyset$.\[lem:connected-same-boundary\]
Let’s look at the open set $W=U\cap V$. If $W=\emptyset$ then we’re finished.
Let’s assume that $W\neq\emptyset$. If $W\neq U$ then since $U$ is connected, the boundary of $W$ in $U$ must be non-empty. Let $x\in U$ be a point in $bd(W)$. Since $x\notin W$, $x\notin V$. But $x\in\overline{W}\subset\overline{V}$ which means that $x\in bd(V)$. This is a contradiction to the fact that $x\in U$. Therefore, $W=U$. Similarly, $W=V$. Together this means that $U=V$ which is a contradiction to the assumption.
Let $M$ be a t.t.t structure and let $X\subset M_{t}$ be some finite subset. Then there are only a finite number of basis sets $U\subset M_{t}$ such that $bd(U)=X$.\[lem:finite-basis-sets\]
Let $\mathcal{B}$ be the set of basis sets $U$ such that $bd(U)=X$. Assume for contradiction that $\mathcal{B}$ is infinite. Let $\mathcal{C}$ be defined by$$\mathcal{C}=\bigcup_{U\in\mathcal{B}}\{C\subset M_{t}\vert C\textrm{ is a connected component of }U\}$$ For each $C\in\mathcal{C}$, $bd(C)\subset X$. In addition, since $\mathcal{B}$ is infinite, $\mathcal{C}$ is infinite as well. However, by lemma \[lem:connected-same-boundary\], for each pair of connected components $C_{1},C_{2}\in\mathcal{C}$, $C_{1}\cap C_{2}=\emptyset$. Therefore, the definable set $$\bigcup\mathcal{C}=\bigcup\mathcal{B}$$ can be partitioned into an infinite number of clopen sets which is a contradiction to the fact that $M$ is t.t.t.
Let $M$ be an $\omega$-saturated one dimensional t.t.t structure such that for every basis set $U$, $\vert bd(U)\vert=2$.
Then for every basis set $U$ there exists a basis set $V\subset U$ such that for every point $x\in V$ there exists a basis set $W$ such that $bd(W)\cap V=\{x\}$.\[pro:every-basis-boundary-2\]
As before, without loss of generality we can assume that for every $x\in M_{t}$ there is some basis set $W$ such that $x\in bd(W)$.
We also use the function $f:U\rightarrow\mathcal{P}(M_{t})$ defined above by$$f(x)=\{y\neq x\vert\textrm{there exists a basis set }W\textrm{ such that }\{x,y\}\subset bd(W)\}$$
Let $U$ be some basis set. First let’s assume that there exists a point $u\in U$ such that $\vert f(u)\cap U\vert=\infty$. Since $M_{t}$ is Hausdorff, there exists some basis set $V\subset f(u)\cap U$ such that $u\notin V$. $V$ clearly satisfies the requirements of the proposition.
On the other hand, assume that for each point $u\in U$, $\vert f(u)\cap U\vert<\infty$. By lemma \[lem:finite-basis-sets\] this means that there are only a finite number of basis sets $W\subset U$ such that $u\in bd(W)$. By the $\omega$-saturation this means that there exists some number $N\in\mathbb{N}$ such that for each point $u\in U$ there are at most $N$ basis sets $W\subset U$ such that $x\in bd(W)$.
We’ll now show using downward induction that there exists a basis set $V\subset U$ such that for every point $v\in V$, there are no basis sets $W\subset V$ such that $v\in bd(W)$ which is clearly a contradiction.
Assume that we found a basis set $V_{i}$ for $0<i\leq N$ such that for every point $v\in V_{i}$, there are at most $i$ basis sets $W\subset U$ such that $x\in bd(W)$. Let $v_{i}$ be some point in $V_{i}$ and let $X$ be the set of points $x\in V_{i}$ such that there exists a basis set $W$ with $v_{i}\in W$ and $x\in bd(W)$. Again by the fact that $M_{t}$ is Hausdorff it follows that $X$ is infinite. We choose $V_{i-1}$ to be some basis set such that $V_{i-1}\subset X$ and $v_{i}\notin V_{i-1}$.
Now, let $x$ be some point in $V_{i-1}$. Since $x\in V_{i}$, there are at most $i$ basis sets $W\subset V_{i}$ such that $x\in bd(W)$. However, one of these sets contains $v_{i}$ which isn’t an element in $V_{i-1}$. Therefore, there are at most $i-1$ basis sets $W\subset V_{i-1}$ such that $x\in bd(W)$.
This finishes the downward induction and the proposition.
Let $M$ be a 1-dimensional $\omega$-saturated t.t.t structure such that one of the following holds:
1. There exist a definable continuous function $F:M_{t}^{2}\rightarrow M_{t}$ and a point $a\in M_{t}$ such that for each $x\in M_{t}$, $F(x,x)=a$ and $F(x,\cdot)$ is injective.
2. For every basis set $U$, $\vert bd(U)\vert=2$.
Then for all but a finite number of points, for every point $x\in M_{t}$ there’s a basis set $U$ containing $x$ such that $U$ is o-minimal.\[thm:o-min-neighborhood\]
It’s enough to prove that for every basis set $U$ there exists a point $x_{0}\in U$ with an o-minimal neighborhood.
Let $U$ be a basis set. By propositions \[prop:boundary-only-x\] and \[pro:every-basis-boundary-2\], there exists a basis set $V\subset U$ such that for every point $x\in V$ there exists a basis set $W$ such that $bd(W)\cap V=\{x\}$. This means that for every point $x\in V$, $V\backslash\{x\}$ has at least two connected components. Without loss of generality $V$ is connected. By theorem \[thm:remove-finite-o-minimal\] (and the remark immediately after it), this means that after removing a finite number of points from $V$ the remaining connected components are o-minimal. Let $C$ be one of the o-minimal components and let $x_{0}$ be some point in $C$. Clearly $x_{0}$ has an o-minimal neighborhood.
Let $M$ be an $\omega$-saturated one dimensional t.t.t structure which admits a topological group structure . Then all but a finite number of points have an o-minimal neighborhood.
We define a function $F:M_{t}^{2}\rightarrow M_{t}$ by$$F(x,y)=x-y$$ The function $F$ clearly satisfies the conditions of theorem \[thm:o-min-neighborhood\].
Let $M$ be an $\omega$-saturated one dimensional t.t.t structure such that all but a finite number of points have an o-minimal neighborhood. Let $f:M_{t}\rightarrow M_{t}$ be definable function. Then $f$ is continuous for all but a finite number of points.
We’ll show that in every basis set $U$ there is a point $x_{0}\in U$ such that $f$ is continuous at $x_{0}$.
Let $U$ be a basis set and let $\Gamma$ be the graph of $f$.
If the projection of $\Gamma\cap(U\times M_{t})$ onto the second coordinate is finite then there exists a basis set $V\subset U$ such that $f$ is constant on $V$.
On the other hand, if the projection is infinite then there exists some point $y\in M_{t}$ with an o-minimal neighborhood such that for any basis set $V$ containing $y$ there exists some $x\in U$ such that $y\neq f(x)\in V$. Let $V$ be an o-minimal basis set containing $y$. By the choice of $y$, $f^{-1}(V)\cap U$ is infinite so by the assumption there exists some o-minimal basis set $W\subset U$ such that $f(W)\subset V$. Since both $W$ and $V$ are o-minimal, by the monotonicity theorem there exists some $x_{0}\in W\subset U$ such that $f$ is continuous at $x_{0}$.
We conclude this section by giving two examples of theorem \[thm:o-min-neighborhood\].
For the first example we return to the unit circle mentioned in the beginning of the section. We’ll look at the structure $S=\langle S^{1},R(x,y,z)\rangle$ where $R(x,y,z)$ is true if $x$ and $y$ are not opposite each other and $z$ lies on the short arc between $x$ and $y$. If we define the set of basis sets as$$\{R^{S}(a,b,z)\vert a,b\in S\}$$ then $S$ is a 1-dimensional $\omega$-saturated t.t.t structure. In addition, for every basis set $U$, $\vert bd(U)\vert=2$ so by theorem \[thm:o-min-neighborhood\] $S$ is locally o-minimal. This is indeed true as locally $S$ looks like the structure $R_{int}$ from example \[exa:interval-example\] which was shown to be o-minimal.
This example is a slight variant of $R_{int}=\langle\mathbb{R},I(x,y,z)\rangle$. Let’s define the relation $RI(x,y,z)$ in $R=\langle\mathbb{R},+,\cdot,0,1,<\rangle$ by:
$$RI(x,y,z)\iff(x<z<y)\wedge(-1<x-y<1)$$ So $RI$ is a version of $I$ restricted to intervals with a length of less than 1.
Let $R\prec\overline{R}$ be an $\omega$-saturated elementary extension and let $\overline{\mathbb{R}}$ be the universe of $\overline{R}$. We define $\overline{R}_{rint}=\langle\overline{\mathbb{R}},RI\rangle$ to be the restriction of $\overline{R}$ to the language $\{RI\}$.
Clearly $\overline{R}_{rint}$ is $\omega$-saturated. In addition, since $\overline{R}$ is o-minimal, $\overline{R}_{rint}$ is a one dimensional t.t.t structure.
However, for any point $a\in\overline{\mathbb{R}}$, $\overline{\mathbb{R}}\backslash\{a\}$ has only one definably connected component in $R_{rint}$. Because assume for contradiction that the sets $$A_{+}=\{x\in\overline{\mathbb{R}}\vert x>a\}$$ $$A_{-}=\{x\in\overline{\mathbb{R}}\vert x<a\}$$
were definable in $\overline{R}_{rint}$ using the constants $c_{1},\dots,c_{N}$. Let’s define subsets$$\tilde{A}_{+}=\{x\in\overline{\mathbb{R}}\vert(x>a)\wedge(\forall n<N\forall k(x>c_{n}+k)\}$$
$$\tilde{A}_{-}=\{x\in\overline{\mathbb{R}}\vert(x<a)\wedge(\forall n<N\forall k(x<c_{n}+k)\}$$
By the definition of $RI$, an automorphism of $\overline{\mathbb{R}}$ which swaps $\tilde{A}_{+}$ and $\tilde{A}_{-}$ is an automorphism of $\overline{R}_{rint}$ together with the constants $c_{1},\dots,c_{N}$ which is a contradiction.
On the other hand, the basis sets of $R_{rint}$ have two boundary points so by theorem \[thm:o-min-neighborhood\], every point in $R_{rint}$ has an o-minimal neighborhood. This makes sense because for any point $a\in\overline{\mathbb{R}}$ and interval $U$ containing $a$ with a length of less than one, we can define an order on $U$ in the same way that we defined an order on $R_{int}$ in example \[exa:interval-example\].
Acknowledgement {#acknowledgement .unnumbered}
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I’d like to thank Ehud Hrushovski for providing valuable insights and guidance throughout the time spent working on this paper.
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abstract: 'In this work, we report the fabrication of experimental setup for high temperature thermal conductivity ($\kappa$) measurement. It can characterize samples with various dimensions and shapes. Steady state based axial heat flow technique is used for $\kappa$ measurement. Heat loss is measured using parallel thermal conductance technique. Simple design, lightweight and small size sample holder is developed by using a thin heater and limited components. Low heat loss value is achieved by using very low thermal conductive insulator block with small cross sectional area. Power delivered to the heater is measured accurately by using 4-wire technique and for this, heater is developed with 4-wire. This setup is validated by using $Bi_{0.36}Sb_{1.45}Te_3$, polycrystalline bismuth, gadolinium, and alumina samples. The data obtained for these samples are found to be in good agreement with the reported data. The maximum deviation of 6 % in the value $\kappa$ are observed. This maximum deviation is observed with the gadolinium sample. We also report the thermal conductivity of polycrystalline tellurium from 320 K to 550 K and the non monotonous behavior of $\kappa$ with temperature is observed.'
author:
- Ashutosh Patel
- 'Sudhir K. Pandey'
title: Fabrication of setup for high temperature thermal conductivity measurement
---
Introduction
============
In current scenario, finding new sources for power generation are very important to fulfill the demand of electricity. Various renewable energy sources like solar PV cell, wind turbine, etc. are discovered and currently are using at large scale. Various other sources have also been discovered but are in development stage. Power generation through thermolectric (TE) materials is one of them. Research is going on in this field to find good TE materials, specially for high temperature application. A good TE material should have a high Seebeck coefficient ($\alpha$), low thermal conductivity ($\kappa$), and high electrical conductivity ($\sigma$). Combining all three parameters, a common term introduced called figure of merit defined as $Z\overline{T} = \alpha^2\sigma \overline{T}/ \kappa$, where $\overline{T}$ is the mean temperature across TE material. A good TE material should have high Z$\overline{T}$ and for this, the Seebeck coefficient and electrical conductivity should be high, whereas the thermal conductivity should be low. The potential sites for thermoelectric generators are available at high temperature, so characterization of TE materials at high temperature are required. The measurement of all three parameters are required to find Z$\overline{T}$. Among all these three properties, $\kappa$ is most difficult to measure experimentally.
Various available methods to measure $\kappa$ are discussed in Ref. \[1\]. Measurement methods are categorized in two parts, steady-state based and non-steady state based methods. Non steady state based methods like laser pulse method, 3$\omega$-method, etc. require very less time for measurement as compared to steady state based methods. Laser pulse method is used most widely in commercial instruments but it is limited to thin disc sample only.[@laserpulse] 3$\omega$-method is also a popular technique, where time delay between the heating of the sample and the temperature response is used to measure $\kappa$.[@3omega] Steady state based methods, like Axial heat flow technique, comparative technique, guarded heat flow technique, heat-flow meter technique, etc. are also used for measurement.[@method] The benefit of steady state based methods are that, setup cost is low and samples with various shapes and dimensions can be characterized. Steady state based axial heat flow method is a basic technique in which one-dimensional Fourier’s law of thermal conduction equation is used to measure $\kappa$, which can be written as $$\kappa=\frac{\dot{Q}_s}{A}.\frac{l}{\Delta T}$$ Where $\dot{Q}_s$ is the net heat flow through the sample. A, $l$, and $\Delta T$ are cross-sectional area of the sample, its thickness, and temperature difference across it, respectively. The main difficulty with this method is to measure the heat flow through the sample ($\dot{Q}_s$) accurately. This difficulty arises due to the undefined amount of heat loss during the heat transfer process by conduction, convection and radiation, which becomes important at high temperature.[@amatya] Various instruments are reported based on this method for the measurement of $\kappa$. *Muto et al.*[@muto] used a heat flux sensor to measure $\dot{Q}_s$. *Amatya et al.*[@amatya] used comparative axial heat flow technique to measure $\dot{Q}_s$. In this technique, a sample is sandwiched between standard reference materials of known $\kappa$. Radiation losses are taken care by using some physical model. *Zawilski et al.*[@zawilski] discussed a different approach to find the value of $\dot{Q}_s$, called parallel thermal conductance technique. In this technique, heat loss is measured by running the instrument without sample. This heat loss data is act as a baseline for $\kappa$ measurement. Using this technique, they performed measurement from 12 K to room temperature. Further, *Dasgupta et al.*[@dasgupta] used this technique and performed measurement at high temperature. They developed a model in which they run the instrument at constant power without sample and with sample. The value of heat flow through the sample is obtained using heater constant and difference in equilibrium temperatures change in the heater block without sample and with sample. The value of heater constant is evaluated from the plot of equilibrium temperature change in the heater block without sample vs power supplied to the heater. Heater heats the side wall of copper rod and this copper block heat the sample. They put the sample holder in a glass chamber and kept it in a furnace.
In the above discussed instruments, various techniques are used to measure $\dot{Q}_s$. Use of heat flux sensor limit the measurement temperature, as it is not suitable for high temperature applications. Similarly, comparative axial heat flow technique requires standard reference materials and number of temperature sensors. Radiation loss at high temperature is estimated by using some physical model. To get a more accurate result by using the method discussed by *Zawilski et al.* heat loss should be as low as possible. Use of bulk heater causes more heat loss as it heat side walls and also high temperature surface is directly exposed to chamber environment.
In this work, we have developed a low cost setup for high temperature $\kappa$ measurement. The Parallel thermal conductance technique is used to measure heat flow through the sample. Heat loss is minimized by using low thermal conductive gypsum as an insulator block with small cross sectional area. Thin heater is built to heat the sample. Its negligible surfaces are exposed to vacuum environment, which also minimized the heat loss. The use of thin heater and small cross section insulator block minimizes the size of the sample holder assembly. This small size sample holder requires limited components, which make it lightweight. Thin heater is built with 4-probe contacts, by which we can measure accurate power delivered to the heater. This setup can characterize samples with various shapes and dimensions. $Bi_{0.36}Sb_{1.45}Te_3$, polycrystalline bismuth, gadolinium and alumina samples are used to validate the instrument. The data collected on these samples are found to be in good agreement with the reported data. We also report the $\kappa$ of the polycrystalline tellurium.
{width="9cm" height="6.7cm"}
MEASUREMENT METHODOLOGY
=======================
One-dimensional Fourier’s law of thermal conduction equation is used to measure the value of $\kappa$. As discussed in previous section, in this method accuracy of the $\kappa$ measurement mainly depends on the value of heat flow through the sample. It requires the estimation of heat loss by means of conduction, convection, and radiation.
For the measurement setup described in this work, heat generated by the heater is mainly divided into two paths. One is through sample and another one is through the insulator block. Heat flow through the insulator block is evaluated by running the setup without sample. This method to find the heat loss is known as parallel thermal conductance technique.[@zawilski] The schematic diagrams of heat flow without sample and with sample are shown in Fig. 1. The heat loss data obtained by running the instrument without sample includes the conduction loss through the insulator block ($\dot{Q}_{ins}$) and also the radiation losses through the side walls of the insulator block and the surface of hot side copper block ($\dot{Q}_{rad}$). During heat loss measurement, hot side temperature with respect to vacuum chamber temperature ($T_{HR}$) is recorded along with the power supplied to the heater. As the power supply changes, $T_{HR}$ changes. The heat loss measurement is performed till the temperature for which $\kappa$ measurement has to be performed. From Fig. 1, The heat loss equation can be written as $$\dot{Q}_\textit{l}=\dot{Q}_{ins}+\dot{Q}_{rad}$$ Where $\dot{Q}_\textit{l}$ is the net power delivered to the heater during heat loss measurement.
During $\kappa$ measurement, the sample is placed in the holder. As shown in the Fig. 1, heat generated by the heater flows though insulator as well as sample also. The heat loss through side walls of the sample by means of conduction and convection to the air is less than 1 % of the total heat flow through the sample, even at high hot side temperature (above 773 K).[@amatya] This heat loss can be ignored. The heat flow equation with a sample can be written as $$\dot{Q}_{s} \approx \dot{Q}_{in}-\dot{Q}_\textit{l}$$ Where, $\dot{Q}_{in}$ is the net power delivered to the heater during $\kappa$ measurement. This $\dot{Q}_s$ obtained using above equation is used to calculate the $\kappa$ of the sample. In the above equation, equality sign is replaced by approximate symbol, as the value of heat loss through the sidewalls of the sample by means of radiation is ignored. This loss depends on the sample emissivity, geometrical shape and the temperature gradient across it. The whole measurement process are carried out at a constant room temperature (300 K).
MEASUREMENT SETUP
=================
{width="6cm" height="9.77cm"}
The actual photograph of the sample holder assembly is shown in Fig. 2, where different components are represented by numbers. The sample, ***1***, is sandwiched between two copper blocks, ***2*** & ***3***, of $8mm \times 8mm$ cross section and 2 mm thickness. The two K-type PTFE shielded thermocouples, ***4*** & ***5***, of 36 swg are embedded at the center of each copper block. Measured $\Delta T$ includes the temperature gradient generated due to 1 mm of each copper block, the thermal resistance of interface surfaces and the sample. This instrument is used to measure low thermal conductivity (up to 30 W/m.K) sample, where the temperature gradient due to copper block is negligible as it has a very high thermal conductivity ($\sim$400 W/m.K at room temperature). The temperature gradient due to the interface surfaces has been minimized by using GaSn liquid metal eutectic, which has very low thermal contact parasitic (0.05 $Kcm^2/W$).[@amatya] High boiling temperature of GaSn liquid metal eutectic ($\sim$1800 K) make it suitable for high temperature application. 13 $\Omega$ thin heater, ***6***, is used to heat the sample. It is made by winding 40 swg kanthal wire over the alumina sheet of $7mm \times 7mm$ cross section and 0.6 mm of thickness. Each end of kanthal wire is welded with two nickel wires of 35 swg by using gas welding. At each end, one wire is used to supply power and another is used to measure the voltage across kanthal wire. This 4-wire technique measures exact power delivered to the heater. The current carrying nickel wires are kept shorter and exposed to the environment of vacuum chamber just after welded tip. Due to this, the heat generated in the nickel wires will not merge with heat generated through the heater. Also Adding two more nickel wires to achieve 4-wire configuration will not affect the heat loss value considerably, as these wires are very fine. This heater is further coated with high temperature cement to insulate it electrically and fixed over gypsum insulator block, ***7***, by using high temperature cement. Gypsum is used for this purpose as it has a very low thermal conductivity ($\sim$0.017 W/m.K at room temperature) and can be used at high temperature also.[@gypsum] As heat loss is very important and should be as low as possible, we take gypsum insulator block of $8mm \times 8mm$ cross section area and 25 mm thickness. Further, we minimized its effective cross sectional area by making a cylindrical hole across its thickness. This insulator block is supported by a brass plate, ***8***, by using high temperature cement.
{width="9cm" height="6.88cm"}
{width="18cm" height="6.88cm"}
This brass plate is fixed with a brass rod , ***9***, using screw , ***10***. Insulator block, ***11***, is also used at the cold side to insulate sample electrically and optimize $\Delta T$ across it. SS rod, ***12***, of 6 mm diameter and 60 mm length are used to apply pressure over the sample. This rod is having round tip at its end, which allows self aligning of the surfaces at the interface of sample and copper blocks. This self aligning ensures uniform pressure throughout the sample cross section. Another brass plate, ***13***, is used to screw this SS rod. It is fixed over a plane ss rod, ***14***, of 6 mm dia. Use of two different rods provide two independent paths for heat flow. Brass rod is for heat loss and ss rod is for heat flow through the sample. Brass rod and plane SS rod are fixed on a SS flange. On this SS flange, hermetically sealed electrical connector is fixed to make electrical connections. PT-100 RTD is connected to this connector to measure the temperature of the chamber. This sample holder assembly is placed inside the vacuum chamber, which is made by using seamless SS pipe of 10 cm diameter and 30 cm in height. KF25 port is provided over the vacuum chamber at the bottom. This port is used to connect this chamber with vacuum pump. Diffusion vacuum pump is used to create a vacuum inside the chamber upto a level of $\sim3\times10^{-5}$ mbar. At this level of vacuum, the conduction and convection loss through the air can be ignored because of this pressure thermal conductivity of air is 3$\times10^{-6}$ W/m.K . It is four order of magnitude lower than thermal conductivity at atmospheric pressure.[@4order]
Sourcemeter is used to supply power to the heater and digital multimeter with scanner card is used to measure various signals. Sourcemeter is used in constant current mode and based on the resistance of the heater, power value is defined. LabVIEW based program is built to control the whole measurement process.
RESULTS AND DISCUSSIONS
=======================
We calibrated our instrument by using $Bi_{0.36}Sb_{1.45}Te_3$, polycrystalline bismuth, gadolinium, and alumina samples. Samples with various $\kappa$ values (varies from 1.4 W/m.K to 25 W/m.K) have been used to show the flexibility of the instrument. $Bi_{0.36}Sb_{1.45}Te_3$ sample is extracted from commercially available thermoelectric generator (TEC1-12706). The composition of the sample is obtained by performing EDX analysis. We also report the thermal conductivity of polycrystalline tellurium, which is not available to the best of our knowledge. The dimensions of the samples used for measurement are given in the table.
\[1\][D[1]{}[\#1]{} ]{}
-------------------------- --------------- ----------------- ----------- --
Sample Cross section Cross sectional Thickness
name type area ($mm^2$) ($mm$)
$Bi_{0.36}Sb_{1.45}Te_3$ Rectangular 1.96 1.6
Poly. Bismuth Rectangular 63 12
Gadolinium Rectangular 2.56 2.4
Alumina Rectangular 24 20
Poly. Tellurium Rectangular 53 10
-------------------------- --------------- ----------------- ----------- --
\
The reproducibility and repeatability of the instrument is verified by successive measurements on the same sample by remounting the sample for each measurement. The average value of $\kappa$ along with error bar are shown for all the samples.
Heat loss, heat flow, and temperature gradient measurements
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Heat loss values are very important for $\kappa$ measurement. At the start of the measurement, the value of $\dot{Q}_{s}$ is very low. A small error in its value can change the result by considerable magnitude. For better result, $\dot{Q}_\textit{l}$ should be as low as possible. It is achieved by using low thermal conductive gypsum as insulator block. Again, the heat loss value is further minimized by reducing the effective surface area. A plot of heat loss with respect to $T_{HR}$ is shown in Fig. 3. This heat loss data follow cubic degree polynomial, which is obtained by fitting of heat loss data with respect to $T_{HR}$. Initially, $\sim$ 3 mW/K of heat loss is observed. This rate is increasing with increase in temperature, which is due to the increase in thermal conductivity of gypsum and radiation losses.
The value of heat flow through the samples are evaluated by subtracting the heat loss data. The obtained values of $\dot{Q}_{s}$ for the $Bi_{0.36}Sb_{1.45}Te_3$, polycrystalline bismuth, gadolinium and alumina samples with respect to $T_{HR}$ are shown in the Fig. 4(a). The closer values of $\dot{Q}_{s}$ with $T_{HR}$ for $Bi_{0.36}Sb_{1.45}Te_3$ and gadolinium samples are observed. This $\dot{Q}_{s}$ depends on the total thermal resistance of the circuit, which include sample, cold side insulator block, SS rod, brass plate and brass rod. So, sample having low or high thermal resistance does not change the total thermal resistance by large value, which results almost same heat flow. The $\dot{Q}_{s}$, sample dimension and value of $\kappa$ decides the $\Delta T$ across it. Due to large cross sectional area of bismuth and high $\kappa$ of alumina, the values of $\Delta T$ with these samples are very small ($<$1 K) at the start of the measurement. This very small value of $\Delta T$ leads to give large uncertainty, because it also includes very small temperature gradient due to the interface surfaces and copper blocks which we have ignored. To nullify these gradients, the measured value of $\Delta T$ should not be too small. This issue is resolved by removing the insulator block used above the cold side copper block. This minimizes the thermal resistance of the circuit and results in large increment of $\dot{Q}_{s}$. This large increment in $\dot{Q}_{s}$ increases the $\Delta T$ proportionally. The values of $\Delta T$ with respect to $T_{HR}$ for all the four samples are shown in Fig. 4.(b). At the start of the measurement, the value of $\Delta T$ for $Bi_{0.36}Sb_{1.45}Te_3$, polycrystalline bismuth, gadolinium, and alumina samples are 9 K, 1.7 K, 2 K, and 1.4 K, respectively. These values increase with the increase in temperature and reached to 90 K, 26 K, 42 K, and 120 K, respectively.
Thermal conductivity measurement
--------------------------------
Measured and reported values of $\kappa$ for $Bi_{0.36}Sb_{1.45}Te_3$, polycrystalline bismuth, and gadolinium samples at various temperatures are shown in Fig. 5 (a), 5 (b), and 5 (c), respectively. Measured values of $\kappa$ for alumina sample at various temperatures are shown in Fig. 5 (d).
Initially, the value of $\kappa$ for $Bi_{0.36}Sb_{1.45}Te_3$ sample is 1.57 W/m.K. It decreases with increase in temperature with very small rate till $\overline{T}$=350 K and reaches to 1.565 W/m.K. After this temperature, $\kappa$ increases with temperature and reaches to 2.67 W/m.K at $\overline{T}$=502 K. *Ma et al.*[@bisbte] reported $\kappa$, in which they used commercial $BiSbTe$ ingot as a sample. They obtained composition of sample by comparing XRD data of the sample with available XRD data of $Bi_{0.5}Sb_{1.5}Te_3$. Our data show similar behavior compared to the reported data. From 320 K to 375 K, almost constant deviation of 0.22 W/m.K is observed. It increases with increase in temperature and reach to 0.5 W/m.K at 500 K. From the available experimental data in Ref. \[11\], it is observed that the value of $\kappa$ increases with increase in the concentration of bismuth in BiSbTe alloy. The concentration of bismuth in our sample is more compared to the sample used in the reference, which may be the reason that the value of $\kappa$ for our sample is higher than the reported value of $\kappa$.
In Bismuth, the electron and phonon components of $\kappa$ are comparable in magnitude above 120 K. The value of $\kappa$ decreases with temperature due to decrease in the magnitude of electron and phonon components with temperature.[@bismuthphonon] For a polycrystalline bismuth sample, we observed similar behavior. The value of $\kappa$ at $\overline{T}$=318 K is 7.57 W/m.K. It decreases with increase in temperature and reaches to 6.73 W/m.K at $\overline{T}$=440 K. Our data shows similar behavior with the data available in handbook.[@bismuth] A deviation of 0.3 W/m.K is observed at $\overline{T}$=318 K. This deviation decreases with temperature and reach to 0.08 W/m.K. The maximum deviation of 0.3 W/m.K is observed at 318 K, which is 4 % of the value of the $\kappa$.
Gadolinium is a rare earth metal, which shows increasing behavior in thermal conductivity with temperature after ferromagnetic-antiferromagnetic transition. Its ferromagnetic-antiferromagnetic transition occurs near room temperature.[@rare] Our data of gadolinium sample measured from 315 K to 590 K shows increasing behavior. Its thermal conductivity at $\overline{T}$=315 K is 7.76 W/m.K, with temperature $\kappa$ increases and at at $\overline{T}$=590 K, its value is 10.9 W/m.K. Data of gadolinium sample are compared with the reported data available in Ref. \[15\]. In this reference, values of $\kappa$ are obtained by using thermal diffusivity data. A deviation of 0.37 W/m.K is observed at 315 K. This deviation decreases with temperature and match closely with reference data at 340 K. At 475 K and 575 K, deviation of $\sim$0.18 W/m.K is observed. The maximum deviation of 0.37 W/m.K is observed at 315 K, which is the 5 % of the reported value of the $\kappa$.
It is observed that the thermal conductivity of oxide material decreases with temperature.[@oxide] Our data for alumina sample show similar behavior. At $\overline{T}$=310 K, the value of $\kappa$ is 27.1 W/m.K. It decreases with increase in temperature and reaches to 19 W/m.K at $\overline{T}$=578 K. For alumina sample, the values of $\kappa$ from various sources are compared in Ref. \[17\]. Near room temperature its value lies in the range of 24 W/m.K and 30 W/m.K. It decreases with temperature and at 600 K its value lies between 13.5 W/m.K to 17.5 W/m.K. Our data show similar behavior compared to the reported data. At 310 K, our data lies in the range of the reported data and at 578 K, a deviation of 0.5 W/m.K is observed.
{width="9cm" height="6.88cm"}
We also performed measurement over polycrystalline tellurium sample. It is taken from commercially available ingot. The values of $\kappa$ for this sample with respect to $\overline{T}$ are shown in Fig. 5. Variation in $\Delta T$ along with $\overline{T}$ is shown in the inset of Fig. 6. At $\overline{T}$=320 K, $\Delta T$ is 3 K. $\Delta T$ increases and reaches to 90 K at $\overline{T}$=550 K. At $\overline{T}$=320 K, $\kappa$ is 3.14 W/m.K. It increases with increase in temperature with a very small rate till $\overline{T}$=400 K, where its value is 3.3 W/m.K. After this temperature, its value decreases and reached to 2.95 W/m.K at $\overline{T}$=550 K.
Thermoelectric properties of polycrystalline tellurium with different charge carrier concentrations are studied and measured with by *Lin et al.*[@tellurium]. Sample is prepared by melting and followed by quenching in cold water and annealing. charge carrier concentration tuned by using dopants including phosphorus (P), arsenic (As), antimony (Sb) and bismuth (Bi). With different carrier concentration, the value of $\kappa$ changes. Its maximum value near room temperature is $\sim$2.25 W/m.K for highest carrier concentration and with temperature it decreases.
A wide variety of samples are used for measurement, whose $\kappa$ value lies in the range of 1.5 W/m.K to 27 W/m.K. At high temperature, previously only *Dasgupta et al.*[@dasgupta] used parallel conductance technique for $\kappa$ measurement. They performed high temperature measurement with samples only upto a value of 2.5 W/m.K. By comparing Fig. 4 (b) with Fig. 5, the behavior of $\kappa$ for all four samples shown in Fig. 5 can be also obtained, by obtaining change in the slope of $\Delta T$ vs temperature graph (Fig. 4 (b)). The increase in the slope of graph shows decreasing behavior of $\kappa$, while decrease in the slope shows increasing behavior of $\kappa$. We found reproducibility of our data with a maximum error bar of $\pm$3 %. We also observed a maximum deviation of 6 % for gadolinium sample compared to the reported data at high temperature. Since we have ignored the radiation loss through the sidewalls of the sample in the heat flow measurement, may be the possible reason for the observed deviation.
CONCLUSION
==========
In this work, we have developed simple, low cost and user friendly setup for high temperature thermal conductivity measurement. The wide verity of samples with various shapes and dimensions can be characterize using this setup. Good reproducibility of measured data is observed with a maximum error bar of $\pm$3 %. The parallel thermal conductance technique is used for measurement and data measured with better accuracy than previously reported literature using this method. We measured $\kappa$ at high temperature using samples upto a value of 27 W/m.K. This is achieved by low heat loss and accurate power measurement by using 4-wire technique. Thin heater, simple design and limited components, make this setup light weight and small in size. This setup is validated by using $Bi_{0.36}Sb_{1.45}Te_3$, polycrystalline bismuth, gadolinium, and alumina samples. The measured data were found in good agreement with the reported data and observed a maximum deviation of 6 % with the gadolinium sample, which indicate that this instrument is capable to measure $\kappa$ with fairly good accuracy. We also reported the thermal conductivity of polycrystalline tellurium and observed the non monotonous behavior of $\kappa$ with temperature.
ACKNOWLEDGEMENTS
================
The authors acknowledge R S Raghav and other workshop staffs for their support in the fabrication process of vacuum chamber and sample holder parts.
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abstract: 'We develop latent variable models for Bayesian learning based low-rank matrix completion and reconstruction from linear measurements. For under-determined systems, the developed methods are shown to reconstruct low-rank matrices when neither the rank nor the noise power is known a-priori. We derive relations between the latent variable models and several low-rank promoting penalty functions. The relations justify the use of Kronecker structured covariance matrices in a Gaussian based prior. In the methods, we use evidence approximation and expectation-maximization to learn the model parameters. The performance of the methods is evaluated through extensive numerical simulations.'
address: |
ACCESS Linnaeus Center, School of Electrical Engineering\
KTH Royal Institute of Technology, Stockholm, Sweden\
[masundi@kth.se, crro@kth.se, janssonm@kth.se, sach@kth.se]{}
title: |
Bayesian Learning for\
Low-rank Matrix Reconstruction
---
Introduction {#sec:intro}
============
Reconstruction of a high dimensional low-rank matrix from a low dimensional measurement vector is a challenging problem. The low-rank matrix reconstruction (LRMR) problem is inherently under-determined and have been receiving attention [@Candes2010noise; @Candes2009] due to its generality over popular sparse reconstruction problems along with many application scopes [@Candes2009; @Candes2010noise; @Fazel2002; @Zachariah2012; @Fan; @yu]. Here we consider the LRMR system model $$\begin{aligned}
\label{eq:measurements}
\mathbf{y} = \mathbf{A}\mathrm{vec}(\mathbf{X}) + \mathbf{n}\end{aligned}$$ where $\mathbf{y} \in \mathbb{R}^m$ is the measurement vector, $\mathbf{A} \in \mathbb{R}^{m \times pq}$ is the linear measurement matrix, $\mathbf{X}\in \mathbb{R}^{p \times q}$ is the low-rank matrix, $\mathbf{n}\in\mathbb{R}^m$ is additive noise (typically assumed to be zero-mean Gaussian with covariance $\mathrm{Cov}(\mathbf{n}) = \beta^{-1} \mathbf{I}_m$) and $\mathrm{vec}(\cdot)$ is the vectorization operator. With $m < pq$, the setup is underdetermined and the task is the reconstruction (or estimation) of $\mathbf{X}$ from $\mathbf{y}$. To deal with the underdetermined setup, a typical and much used strategy is to use a regularization in the reconstruction cost function. Regularization brings in the information about low rank priors. A typical type I estimator is $$\begin{aligned}
\label{eq:marginal_MAP}
\hat{\mathbf{X}} = \arg \min_{\mathbf{X}} \beta ||\mathbf{y - A}\mathrm{vec}(\mathbf{X})||_2^2 + g(\mathbf{X}) , \end{aligned}$$ where $\beta > 0$ is a regularization parameter and $g(\cdot)$ is a fixed penalty function that promotes low rank in $\hat{\mathbf{X}}$. Common low-rank penalties in the literature [@Fazel2002; @Candes2010noise; @irls] are $$\begin{aligned}
&g(\mathbf{X}) = ||\mathbf{X}||_* = \mathrm{tr}( (\mathbf{X X^\top})^{1/2} ) , \tag{nuclear norm}\\
&g(\mathbf{X}) = \mathrm{tr}( (\mathbf{X X^\top})^{s/2} ) , \tag{Schatten s-norm}\\
&g(\mathbf{X}) = \log |\mathbf{X X^\top} + \epsilon \mathbf{I}_p | , \tag{log-determinant penalty}\end{aligned}$$ where $\mathrm{tr}(\cdot)$ denotes the matrix trace, $|\cdot |$ denotes determinant, and $0< s \leq1$ and $\epsilon >0$. We mention that the nuclear norm penalty is a convex function.
In the literature, LRMR algorithms can be categorized in three types: convex optimization [@Fazel2002; @Candes2009; @Candes2010noise; @Candes10; @Cai2010; @Boyd], greedy solutions [@Admira; @Tang2011; @Zachariah2012; @Johnstone; @irls] and Bayesian learning [@Babacan]. Most of these existing algorithms are highly motivated from analogous algorithms used for standard sparse reconstruction problems, such as compressed sensing where $\mathrm{vec}(\mathbf{X})$ in is replaced by a sparse vector. Using convex optimization we can solve when $g(\mathbf{X})$ is the nuclear norm, which is an analogue of using $\ell_1$-norm in sparse reconstruction problems. Further, greedy algorithms, such as iteratively reweighted least squares [@irls] solves by using algebraic approximations. While convex optimization and greedy solutions are popular they often need more a-priori information than knowledge about structure of the signal under reconstruction; for example, convex optimization algorithms need information about the strength of the measurement noise to fix the parameter $\beta$, and greedy algorithms need information about rank. In absence of such a-priori information, Bayesian learning is a preferred strategy to use. Bayesian learning is capable of estimating the necessary information from measurement data. In Bayesian learning we evaluate the posterior $p(\mathbf{X}|\mathbf{y})$ with the knowledge of prior $p(\mathbf{X})$. If $\mathbf{X}$ has a prior distribution $p(\mathbf{X}) \propto e^{-\frac{1}{2}g(\mathbf{X})}$ and the noise is distributed as $\mathbf{n} \sim \mathcal{N}(\mathbf{0},\beta^{-1} \mathbf{I}_m)$, then the maximum-a-posteriori (MAP) estimate can be interpreted as the type I estimate in . As type I estimation requires more information (such as $\beta$), type II estimators are often more useful. Type II estimation techniques use hyper-parameters in the form of latent variables with prior distributions. While for sparse reconstruction problems, Bayesian learning via type II estimation in the form of relevance vector machine [@Tipping1; @Tipping2] and sparse Bayesian learning [@zhang; @zhang2] have gained significant popularity, the endeavor to design type II estimation algorithms for LRMR is found to be limited. In [@Babacan], direct use of sparse Bayesian learning was used to realize an LRMR reconstruction algorithm. Bayesian approaches were used in [@Carin; @wipf_lowrank] for a problem setup with a combination of low rank and sparse priors, called principal component pursuit [@candes2011robust]. In [@Carin], Gaussian and Bernoulli variables was used and the parameters were estimated using Markov Chain Monte Carlo while in [@Wipf] an empirical Bayesian approach was used. Type II estimation methods are typically iterative where latent variables are usually treated via variational techniques [@wipf_lowrank; @Babacan], evidence approximation [@Tipping1; @Tipping2], expectation maximization [@zhang; @zhang2] and Markov chain Monte Carlo [@Carin; @yu]. Our objective in this paper is to develop new type II estimation methods for LRMR. Borrowing ideas from type II estimation techniques for sparse reconstruction, such as the relevance vector machine and sparse Bayesian learning algorithms, we model a low-rank matrix by a multiplication of precision matrices and an i.i.d. Gaussian matrix. The use of precision matrices helps to realize low-rank structures. The precision matrices are characterized by hyper-parameters which are treated as latent variables. The main contributions of this paper are as follows.
1. We introduce one-sided and two-sided precision matrix based models.
2. We show how the Schatten s-norm and log-determinant penalty functions are related to latent variable models in the sense of MAP estimation via type I estimator .
3. For all new type II estimation methods, we derive update equations for all parameters in iterations. The methods are based on evidence approximation and expectation-maximization.
4. The methods are compared numerically to existing methods, such as the Bayesian learning method of [@Babacan] and nuclear norm based convex optimization method [@Candes2010noise].
We are aware that evidence approximation and expectation-maximization are unable to provide globally optimal solutions. Hence we are unable to provide performance guarantees for our methods. This paper is organized as follows. We discuss the preliminaries of sparse Bayesian learning in section \[sec:preliminaries\]. In section \[sec:one\_sided\] we introduce one-sided precisions for matrices and derive the relations to type I estimators. Two-sided precisions are introduced in section \[sec:two\_sided\] and in section \[sec:algorithms\] we derive the update equations for the parameters. In section \[sec:simulations\] we numerically compare the performance of the algorithms for matrix reconstruction and matrix completion.
Preliminaries {#sec:preliminaries}
-------------
In this section, we explain the relevance vector machine (RVM) [@Tipping1; @Tipping2; @bayesianCS] and sparse Bayesian learning (SBL) methods [@zhang; @zhang2; @Wipf] for a standard sparse reconstruction problem. The setup is $$\begin{aligned}
\mathbf{y = Ax + n},
\label{eq:CS_setup}\end{aligned}$$ where $\mathbf{x}\in\mathbb{R}^n$ is the sparse vector to be reconstructed from the measurement vector $\mathbf{y}\in\mathbb{R}^m$ and $\mathbf{n} \sim \mathcal{N}(\mathbf{0},\beta^{-1} \mathbf{I}_m)$ is the additive measurement noise. The approach is to model the sparse signal $\mathbf{x} = [x_1 \, x_2 \, \hdots \, x_n]^{\top}$ as $$\begin{aligned}
\label{eq:vec_model}
x_i = \gamma_i^{-1/2} u_i , \end{aligned}$$ where $u_i \sim \mathcal{N}(0,1)$ and $\gamma_i > 0$ is the precision of $x_i$. This is equivalent to setting $$\begin{aligned}
p(x_i | \gamma_i) = \sqrt{\frac{\gamma_i}{2\pi}} \exp (-\gamma_i x_i^2 /2) = \mathcal{N}(x_i |0,\gamma_i^{-1}).\end{aligned}$$ The main idea is to use a learning algorithm for which several precisions go to infinity, leading to sparse reconstruction. Alternatively said, the use of precisions allows to inculcate *dominance* of a few components over other components in a sparse vector. Note that $\boldsymbol{\gamma} = [\gamma_1 \, \gamma_2 \, \hdots \, \gamma_n]^{\top}$ and $\beta$ are latent variables that also need to be estimated. We find the posterior $$\begin{aligned}
p(\mathbf{x}|\mathbf{y}) = \int p(\mathbf{x}|\mathbf{y},\boldsymbol{\gamma},\beta) \, p(\boldsymbol{\gamma},\beta | \mathbf{y}) \, d\boldsymbol{\gamma} \, d\beta
\approx p(\mathbf{x}|\mathbf{y},\hat{\boldsymbol{\gamma}},\hat{\beta}), \nonumber
\end{aligned}$$ if $p(\boldsymbol{\gamma},\beta | \mathbf{y})$ is assumed sharply peaked around $\hat{\boldsymbol{\gamma}}$, $\hat{\beta}$ (this is version of the so-called Laplace approximation described in Appendix \[appendix:laplace\]). Assuming the knowledge of $\boldsymbol{\gamma} = \hat{\boldsymbol{\gamma}}$ and $\beta=\hat{\beta}$, the MAP estimate is $$\begin{aligned}
\hat{\mathbf{x}} \triangleq [\hat{x}_1 \, \hat{x}_2 \, \hdots \, \hat{x}_n]^{\top} \leftarrow \arg \max_{\mathbf{x}} p(\mathbf{x}|\mathbf{y}, \boldsymbol{\gamma}, \beta) = \beta \boldsymbol{\Sigma} \mathbf{A}^\top \mathbf{y}
\label{eq:MAP_x_rvm}\end{aligned}$$ where $\boldsymbol{\Sigma} = (\mathrm{diag}(\boldsymbol{\gamma}) + \beta \mathbf{A^\top A} )^{-1}$. In the notion of iterative updates, we use $\leftarrow$ to denote the assignment operator. The precisions $\boldsymbol{\gamma}$ and $\beta$ are estimated by $$\begin{aligned}
(\gamma_i^{new},\beta^{new}) & \leftarrow \arg \max_{\gamma_i,\beta} p(\mathbf{y},\boldsymbol{\gamma},\beta) \\
& = \arg \max_{\gamma_i,\beta} p(\mathbf{y}|\boldsymbol{\gamma},\beta) \, p(\boldsymbol{\gamma}) \, p(\beta), \end{aligned}$$ where $p(\boldsymbol{\gamma}) = \prod_i p(\gamma_i)$ and $p(\mathbf{y}|\boldsymbol{\gamma},\beta) = \mathcal{N}(\mathbf{y}|\mathbf{0},(\mathbf{A}\mathrm{diag}(\boldsymbol{\gamma}) \mathbf{A}^\top + \beta^{-1} \mathbf{I}_m)^{-1})$. Gamma distributions are typically chosen as hyper-priors for $p(\gamma_i)$ and $p(\beta)$ with the form $$\begin{aligned}
p(\beta) = \mathrm{Gamma}(\beta | a+1,b) = \frac{b^{a+1}}{\Gamma (a+1)} \beta^a e^{-b\beta}, \end{aligned}$$ with $a > -1$, $b>0$ and $\beta \geq 0$. The evaluation of $(\gamma_i^{new},\beta^{new})$ leads to coupled equations and are therefore solved approximately as $$\begin{aligned}
\gamma_i \leftarrow \frac{1 - \gamma_i \Sigma_{ii} + 2a}{\hat{x}_i^2 + 2b} \,\, \mathrm{and} \,\,
\beta\leftarrow \frac{\sum_{i=1}^n \gamma_i \Sigma_{ii} + 2a}{||\mathbf{y - A\hat{x}}||_2^2 + 2b},
\label{eq:EdidenceApprox_rvm}\end{aligned}$$ where $\Sigma_{ii}$ is the $i$’th diagonal element of $\boldsymbol{\Sigma}$. The parameters of the Gamma distributions for $p(\boldsymbol{\gamma})$ and $p(\beta)$ are typically chosen to be non-informative, i.e. $a,b \to 0$. The update solutions of and are repeated iteratively until convergence. In sparse Bayesian learning algorithm a standard expectation-maximization framework is used to estimate $\boldsymbol{\gamma}$ and $\beta$.
Finally we mention that the RVM and SBL methods have connection with type I estimation [@Wipf; @Babacan_laplacian]. If the precisions have arbitrary prior distributions $p(\gamma_i)$, then the marginal distribution of $x_i$ becomes $$\begin{aligned}
p(x_i) = \int p(x_i | \gamma_i) \, p(\gamma_i) \, d\gamma_i \propto e^{-h(x_i)/2} ,\end{aligned}$$ for some function $h(\cdot)$. Given and for a known $\beta$, the MAP estimate is $$\begin{aligned}
\hat{\mathbf{x}} = \arg \min_\mathbf{x} \left( \beta || \mathbf{y - Ax} ||_2^2 + \sum_{i=1}^n h(x_i) \right) . \end{aligned}$$ If $p(\gamma_i)$ is a gamma prior then $p(x_i)$ is a Student-$t$ distribution with $h(x_i) = \mathrm{constant} \times \log (x_i^2 + \mathrm{constant}) $. One rule of thumb is that a “more” concave $h(x_i)$ gives a more sparsity promoting model [@Wipf], see some example functions in Figure \[Fig:illustration\]. In the figure, $h(x_i) = |x_i|$ corresponds to a Laplace distributed variable $x_i$, $\log(x_i^2+1)$ to a Student-$t$ and $|x_i|^{1/2}$ to a generalized normal distribution [@general_gaussian]. The relation between the sparsity promoting penalty function $h(x_i)$ and the corresponding prior $p(\gamma_i)$ of the latent variable $\gamma_i$ was discussed in [@Wipf], see also [@Rojas] and [@Stoica2014].
(-3,0) – (3,0) node\[right\] [$x_i$]{}; (0,0) – (0,2); (-2.5,2.5) – (0,0) – (2.5,2.5) node\[right\] [$|x_i|$]{}; plot (,[ln(+ 1)]{}) node\[right\] [$\log (x_i^2 + 1)$]{}; plot (,[sqrt(abs())]{}) node\[right\] [$|x_i|^{1/2}$]{};
One-sided precision based model {#sec:one_sided}
===============================
The structure of a low-rank matrix $\mathbf{X}$ is characterized by the dominant singular vectors and singular values. Like the use of precisions in for the standard sparse reconstruction problem via inculcating dominance, we propose to model the low-rank matrix $\mathbf{X}$ as $$\begin{aligned}
%\label{left_precision}
\mathbf{X} = \boldsymbol{\alpha}^{-1/2} \mathbf{U},
\label{eq:X_oneprior_1}\end{aligned}$$ where the components of $\mathbf{U} \in \mathbb{R}^{p \times q}$ are i.i.d. $\mathcal{N}(0,1)$ and $\boldsymbol{\alpha} \in \mathbb{R}^{p \times p}$ is a positive definite random matrix (which distribution will be described later). This is equivalent to $$\begin{aligned}
p(\mathbf{X} | \boldsymbol{\alpha}) = \frac{|\boldsymbol{\alpha}|^{q/2}}{(2\pi)^{pq/2}} \exp \left( - \frac{1}{2} \mathrm{tr}(\mathbf{X}^\top \boldsymbol{\alpha} \mathbf{X}) \right).
\label{eq:X_oneprior}\end{aligned}$$ Denoting $\mathbf{Z = XX^\top}$ and $\mathrm{tr}(\mathbf{X}^\top \boldsymbol{\alpha} \mathbf{X}) = \mathrm{tr}(\boldsymbol{\alpha} \mathbf{X} \mathbf{X}^\top) = \mathrm{tr}(\boldsymbol{\alpha} \mathbf{Z}) $, we evaluate $$\begin{aligned}
\label{eq:marginal_x}
p(\mathbf{X}) & = \int_{\boldsymbol{\alpha} \succ \mathbf{0}} p(\mathbf{X} | \boldsymbol{\alpha}) \, p(\boldsymbol{\alpha}) \, d \boldsymbol{\alpha} \nonumber \\
& = \int_{\boldsymbol{\alpha} \succ \mathbf{0}} e^{-\frac{1}{2} \mathrm{tr}(\boldsymbol{\alpha}\mathbf{Z})} \frac{|\boldsymbol{\alpha}|^{q/2}}{(2\pi)^{pq/2}} p(\boldsymbol{\alpha}) d \boldsymbol{\alpha} \\
& \propto e^{- \frac{1}{2} \tilde{g}(\mathbf{Z})} \propto e^{-\frac{1}{2} g(\mathbf{X})}. \nonumber\end{aligned}$$ We note that $g(\mathbf{X})$ must have the special form $g(\mathbf{X}) = \tilde{g}(\mathbf{XX^\top})$ for $p(\mathbf{X})$ to hold (as $\boldsymbol{\alpha}$ is integrated out). As $p(\mathbf{X}) \propto e^{-\frac{1}{2} g(\mathbf{X})}$, the resulting MAP estimator can be interpreted as the type I estimator .
Relation between priors
-----------------------
Next we investigate the relation between the priors $p(\mathbf{X})$ and $p(\boldsymbol{\alpha})$. The motivation is that the relations are necessary for designing practical learning algorithms. From , we note that $p(\mathbf{X})$ is the Laplace transform of $|\boldsymbol{\alpha}|^{q/2} p(\boldsymbol{\alpha})/(2\pi)^{pq/2}$ [@Terras], which establishes the relation. Naturally, we can find $p(\boldsymbol{\alpha})$ by the inverse Laplace transform [@Terras] as follows $$\begin{aligned}
\label{eq:inverse_laplace}
%p(\boldsymbol{\alpha}) = \frac{|\boldsymbol{\alpha}|^{-q/2} (2\pi)^{pq/2}}{(4\pi i)^{p(p+1)/2}} \int\limits_{ \substack{ \mathbf{Z} \\ \mathrm{Re} \mathbf{Z} = \boldsymbol{\alpha}_* } } e^{\frac{1}{2} \mathrm{tr}(\boldsymbol{\alpha}\mathbf{Z})} e^{-\frac{1}{2} \tilde{g}(\mathbf{Z})} d\mathbf{Z} ,
%p(\boldsymbol{\alpha}) \propto |\boldsymbol{\alpha}|^{-q/2} \int\limits_{ \substack{ \mathbf{Z} \\ \mathrm{Re} \mathbf{Z} = \boldsymbol{\alpha}_* } } e^{\frac{1}{2} \mathrm{tr}(\boldsymbol{\alpha}\mathbf{Z})} e^{-\frac{1}{2} \tilde{g}(\mathbf{Z})} d\mathbf{Z} ,
p(\boldsymbol{\alpha}) \propto |\boldsymbol{\alpha}|^{-q/2} \int\limits_{\mathrm{Re} \, \mathbf{Z} = \boldsymbol{\alpha}_* } e^{\frac{1}{2} \mathrm{tr}(\boldsymbol{\alpha}\mathbf{Z})} e^{-\frac{1}{2} \tilde{g}(\mathbf{Z})} d\mathbf{Z} , \end{aligned}$$ where the integral is taken over all symmetric matrices $\mathbf{Z} \in \mathbb{C}^{p\times p}$ such that $\mathrm{Re} \, \mathbf{Z} = \boldsymbol{\alpha}_*$ where $\boldsymbol{\alpha}_*$ is a real matrix so that the contour path of integration is in the region of convergence of the integrand. While the Laplace transform characterizes the exact relation between priors, the computation is non-trivial and often analytically intractable. In practice, a standard approach is to use the Laplace approximation [@mackay] where typically the mode of the distribution under approximation is found first and then a Gaussian distribution is modeled around that mode. Let $p(\boldsymbol{\alpha})$ have the form $p(\boldsymbol{\alpha}) \propto e^{-\frac{1}{2} K(\boldsymbol{\alpha})}$; then the Laplace approximation becomes $$\begin{aligned}
\tilde{g}(\mathbf{Z}) = & \min_{\boldsymbol{\alpha} \succ \mathbf{0}} \left\{ \mathrm{tr}(\boldsymbol{\alpha}\mathbf{Z}) - q\log |\boldsymbol{\alpha}| + K(\boldsymbol{\alpha}) \right\} \\
& - \log \left| \mathbf{H} \right| + \mathrm{constant},
%\boldsymbol{\alpha}_0^{-2} - K''(\boldsymbol{\alpha}_0) \right)\end{aligned}$$ where $\mathbf{H}$ is the Hessian of $\mathrm{tr}(\boldsymbol{\alpha}\mathbf{Z}) - q\log |\boldsymbol{\alpha}| + K(\boldsymbol{\alpha})$ evaluated at the minima (which is assumed to exist). The derivation of the Laplace approximation is shown in Appendix \[appendix:laplace\].
Denoting $\tilde{K}(\boldsymbol{\alpha}) = q\log |\boldsymbol{\alpha}| - K(\boldsymbol{\alpha})$ and assuming that the Hessian is constant (independent of $\mathbf{Z}$) we get that $$\begin{aligned}
\tilde{g}(\mathbf{Z}) = \min_{\boldsymbol{\alpha} \succ \mathbf{0}} \left\{ \mathrm{tr}(\boldsymbol{\alpha}\mathbf{Z}) - \tilde{K}(\boldsymbol{\alpha}) \right\}, \end{aligned}$$ where we absorbed the constants terms into the normalization factor of $p(\mathbf{X})$. We find that $\tilde{g}(\boldsymbol{Z})$ is the concave conjugate of $\tilde{K}(\boldsymbol{\alpha})$ [@Boyd]. Hence, for a given $\tilde{g}(\boldsymbol{Z})$ we can recover $\tilde{K}(\boldsymbol{\alpha})$ as $$\begin{aligned}
\label{eq:inverse_conjugate}
\tilde{K}(\boldsymbol{\alpha}) = \min_{\boldsymbol{Z} \succ \mathbf{0}} \left\{ \mathrm{tr}(\boldsymbol{\alpha}\mathbf{Z}) - \tilde{g}(\boldsymbol{Z}) \right\}\end{aligned}$$ if $\tilde{K}(\boldsymbol{\alpha})$ is concave (which holds under the assumption that $K(\boldsymbol{\alpha})$ is convex). Further, we can find $K(\boldsymbol{\alpha})$ from $\tilde{K}(\boldsymbol{\alpha})$ followed by solving the prior $p(\boldsymbol{\alpha}) \propto e^{-\frac{1}{2} K(\boldsymbol{\alpha})}$. Using the concave conjugate relation , we now deal with the task of finding appropriate $K(\boldsymbol{\alpha})$ for two example low-rank promoting penalty functions, as follows.
1. [*For Schatten $s$-norm:*]{} The Schatten $s$-norm based penalty function is $g(\mathbf{X}) = \mathrm{tr}((\mathbf{XX^\top})^{s/2})$. We here use a regularized Schatten $s$-norm based penalty function as $$\begin{aligned}
g(\mathbf{X}) = & \mathrm{tr}((\mathbf{XX^\top + \epsilon \mathbf{I}_p})^{s/2}),
\label{eq:Schatten_reg_penalty}
\end{aligned}$$ where the use of $\epsilon >0$ helps to bring numerical stability to the algorithms in Section \[sec:algorithms\]. For the penalty function , we find the appropriate $K(\boldsymbol{\alpha})$ as $$\begin{aligned}
K(\boldsymbol{\alpha}) = C_s \, \mathrm{tr}(\boldsymbol{\alpha}^{-\frac{s}{2-s}})
+ q\log|\boldsymbol{\alpha}| + \epsilon \, \mathrm{tr}(\boldsymbol{\alpha}), %\nonumber
\label{eq:schatten_latent}
\end{aligned}$$ where $C_s = \frac{2-s}{s} \left( \frac{2}{s}\right)^{-\frac{s}{2-s}}$. The derivation of is given in Appendix \[appendix2\]. Note that, for $s=1$, $g(\mathbf{X})$ becomes the regularized nuclear norm based penalty function $$\begin{aligned}
g(\mathbf{X}) = \mathrm{tr}((\mathbf{XX^\top} + \epsilon \mathbf{I}_p)^{\frac{1}{2}}) = \sum_{i=1}^{\min (p,q)} (\sigma_i(\mathbf{X})^2 + \epsilon)^{\frac{1}{2}}.
\end{aligned}$$
2. [*Log-determinant penalty:*]{} For the log-determinant based penalty function $$\begin{aligned}
g(\mathbf{X}) = \nu \log \left| \mathbf{XX^\top} + \epsilon \mathbf{I}_p \right|,
\end{aligned}$$ where $\nu > q-2$ is a real number, we find $K(\boldsymbol{\alpha})$ as $$\begin{aligned}
K(\boldsymbol{\alpha}) = \epsilon \, \mathrm{tr}( \boldsymbol{\alpha}) + \left(q - \nu \right)\log |\boldsymbol{\alpha}|.
\label{eq:rsvm_latent}
\end{aligned}$$ As $p(\boldsymbol{\alpha}) \propto e^{-\frac{1}{2} K(\boldsymbol{\alpha})}$, we find that the prior $\boldsymbol{\alpha}$ is Wishart distributed (Wishart is a conjugate prior the distribution ). For a scalar instead of a matrix, the prior distribution becomes a Gamma distribution as used in the standard RVM and SBL.
We have discussed a left-sided precision based model in this section, but the same strategy can be easily extended to form a right-sided precision based model. Then a natural question arises, which model to use? Our hypothesis is that the user choice stems from minimizing the number of variables to estimate. If the low-rank matrix is fat then the left-sided model should be used, otherwise the right-sided model. A further question arises on the prospect of developing a two sided precision based model, which is described in the next section.
Two-sided precision based model {#sec:two_sided}
===============================
In this section, we propose to use precision matrices on both sides to model a random low-rank matrix. We call this the two-sided precision based model. Our hypothesis is that the two-sided precision helps to enhance dominance of a few singular vectors. For low-rank modeling, we make the following ansatz $$\begin{aligned}
\mathbf{X} = \boldsymbol{\alpha}_L^{-1/2} \, \mathbf{U} \, \boldsymbol{\alpha}_R^{-1/2}
\label{eq:X_doubleprior_1}\end{aligned}$$ where $\boldsymbol{\alpha}_L \in \mathbb{R}^{p \times p}$ and $\boldsymbol{\alpha}_R \in \mathbb{R}^{q \times q}$ are positive definite random matrices. Using the relation $\mathrm{vec}(\mathbf{X}) = (\boldsymbol{\alpha}_R^{-1/2} \otimes \boldsymbol{\alpha}_L^{-1/2}) \, \mathrm{vec}(\mathbf{U}) =
(\boldsymbol{\alpha}_R \otimes \boldsymbol{\alpha}_L)^{-1/2} \, \mathrm{vec}(\mathbf{U})$, we find $$\begin{aligned}
&p(\mathbf{X} | \boldsymbol{\alpha}_L,\boldsymbol{\alpha}_R) \nonumber\\
&= \frac{| \boldsymbol{\alpha}_R \otimes \boldsymbol{\alpha}_L |^{1/2}}{(2\pi)^{pq/2}} \exp \left( - \mathrm{vec}(\mathbf{X})^\top (\boldsymbol{\alpha}_R \otimes \boldsymbol{\alpha}_L) \mathrm{vec}(\mathbf{X})/2 \right) \nonumber \\
& =\frac{|\boldsymbol{\alpha}_L|^{q/2}|\boldsymbol{\alpha}_R|^{p/2}}{(2\pi)^{pq/2}} \exp \left( - \mathrm{tr}(\mathbf{X}^\top \boldsymbol{\alpha}_L \mathbf{X} \boldsymbol{\alpha}_R)/2\right).
\label{eq:X_doubleprior}\end{aligned}$$ To promote low-rank, we use a prior distribution $p(\boldsymbol{\alpha}_L,\boldsymbol{\alpha}_R) = p(\boldsymbol{\alpha}_L) p(\boldsymbol{\alpha}_R) $. The marginal distribution of $\mathbf{X}$ is $$\begin{aligned}
p(\mathbf{X}) = \int\limits_{ \substack{ \boldsymbol{\alpha}_L \succ \mathbf{0} \\ \boldsymbol{\alpha}_R \succ \mathbf{0} }} p(\mathbf{X} | \boldsymbol{\alpha}_L,\boldsymbol{\alpha}_R) \, p(\boldsymbol{\alpha}_L) \, p(\boldsymbol{\alpha}_R) \, d\boldsymbol{\alpha}_R \, d\boldsymbol{\alpha}_L .
\label{eq:marginal_X_doubleprior}\end{aligned}$$ We have noticed that the use of in evaluating does not bring out suitable connections between the resulting $p(\mathbf{X})$ functions and the usual low-rank promoting $g(\mathbf{X})$ functions (such as nuclear norm, Schatten s-norm and log-determinant). Thus it is non-trivial to establish a direct connection between $p(\mathbf{X} | \boldsymbol{\alpha}_L,\boldsymbol{\alpha}_R)$ of and the type I estimator of .
Instead of a direct connection we can establish an indirect connection by an approximation. For a given $(\boldsymbol{\alpha}_R,\beta)$ and by marginalizing over $\boldsymbol{\alpha}_L$, we have $p(\mathbf{X} | \boldsymbol{\alpha}_R ) \propto e^{- \frac{1}{2} \tilde{g}(\mathbf{X} \boldsymbol{\alpha}_R \mathbf{X}^{\top})}$ and hence the corresponding type I estimator cost function is $$\begin{aligned}
\min_{\mathbf{X}} \beta ||\mathbf{y - A}\mathrm{vec}(\mathbf{X})||_2^2 + \tilde{g}(\mathbf{X} \boldsymbol{\alpha}_R \mathbf{X}^{\top}). \end{aligned}$$ A similar cost function can be found for a given $(\boldsymbol{\alpha}_L,\beta)$ by marginalizing over $\boldsymbol{\alpha}_R$. We discuss the roles of $\boldsymbol{\alpha}_L$ and $\boldsymbol{\alpha}_R$ in the next section.
Interpretation of the precisions
--------------------------------
From , we can see that the column and row vectors of $\mathbf{X}$ are in the range spaces of $\boldsymbol{\alpha}_L^{-1}$ and $\boldsymbol{\alpha}_R^{-1}$, respectively.
Further let us interpret this in a statistical sense with the note that a skewed precision matrix comprises of correlated components. Let us denote the $(i,j)$th component of $\boldsymbol{\alpha}_R^{-1}$ by $[\boldsymbol{\alpha}_R^{-1}]_{ij}$. If $\boldsymbol{\alpha}_R^{-1}$ is highly skewed then $[\boldsymbol{\alpha}_R^{-1}]_{ij}$ and $[\boldsymbol{\alpha}_R^{-1}]_{ii}$ are highly correlated. Suppose $\mathbf{x}_i \in \mathbb{R}^p$ denotes the $i$’th column vector of $\mathbf{X}$. Then following , we can write $$\begin{aligned}
\left[
\begin{array}{c}
\mathbf{x}_i \\
\mathbf{x}_j
\end{array}
\right] \sim
\mathcal{N}
\left(
\left[
\begin{array}{c}
\mathbf{0} \\
\mathbf{0}
\end{array}
\right]
,
\left[
\begin{array}{cc}
[\boldsymbol{\alpha}_R^{-1}]_{ii} \, \boldsymbol{\alpha}_L^{-1} & [\boldsymbol{\alpha}_R^{-1}]_{ij} \, \boldsymbol{\alpha}_L^{-1} \\
{ [\boldsymbol{\alpha}_R^{-1}]_{ji} \, \boldsymbol{\alpha}_L^{-1} } & [\boldsymbol{\alpha}_R^{-1}]_{jj} \, \boldsymbol{\alpha}_L^{-1}
\end{array}
\right]
\right).\end{aligned}$$ The above relation shows that a presence of highly skewed $\boldsymbol{\alpha}_R^{-1}$ leads to the cross-correlation $[\boldsymbol{\alpha}_R^{-1}]_{ij} \, \boldsymbol{\alpha}_L^{-1}$ between $\mathbf{x}_i$ and $\mathbf{x}_j$ that is comparably strong to the auto-correlations $[\boldsymbol{\alpha}_R^{-1}]_{ii} \, \boldsymbol{\alpha}_L^{-1}$. We mention that a low-rank property can be established in a qualitative statistical sense by the presence of columns having strong cross-correlation. The one-sided precision based model can be seen as the two-sided model where $\boldsymbol{\alpha}_R^{-1} = \mathbf{I}_q$. Hence the one-sided precision based model is unable to capture information about cross-correlation between columns of $\mathbf{X}$. A similar argument can be made for the right sided precision based model where $\boldsymbol{\alpha}_L^{-1} = \mathbf{I}_p$.
Practical algorithms {#sec:algorithms}
====================
Considering the potential of two-sided precision matrices, the optimal inference problem is $$\begin{aligned}
\max \, p(\mathbf{X,y}, \boldsymbol{\alpha}_L, \boldsymbol{\alpha}_R, \beta)\end{aligned}$$ which is the MAP estimator for amenable priors and often connected with the type I estimator in . Direct handling of the optimal inference problem is limited due to lack of analytical tractability. Therefore various approximations are used to design practical algorithms which are also type II estimators. This section is dedicated to design new type II estimators via evidence approximation (as used by the RVM) and expectation-maximization (as used in SBL) approaches.
Evidence approximation
----------------------
In the evidence approximation, we iteratively update the parameters as$$\begin{aligned}
\begin{aligned}
\hat{\mathbf{X}} \leftarrow \arg \max_{\mathbf{X}} p(\mathbf{X}| \mathbf{y} , \boldsymbol{\alpha}_L, \boldsymbol{\alpha}_R, \beta) ,
\end{aligned} \label{eq:LMMSE_estimate_X} \\
\begin{aligned}
\beta &\leftarrow \arg \max_{\beta} p(\mathbf{y} , \boldsymbol{\alpha}_L, \boldsymbol{\alpha}_R, \beta) ,
\end{aligned} \label{eq:RVM_like_estimate_beta} \\
\left.
\begin{aligned}
\boldsymbol{\alpha}_L &\leftarrow \arg \max_{\boldsymbol{\alpha}_L} p(\mathbf{y} , \boldsymbol{\alpha}_L, \boldsymbol{\alpha}_R, \beta) , \\
\boldsymbol{\alpha}_R &\leftarrow \arg \max_{\boldsymbol{\alpha}_R} p(\mathbf{y} , \boldsymbol{\alpha}_L, \boldsymbol{\alpha}_R, \beta) ,
\end{aligned}
\right \} \label{eq:RVM_like_estimate_ALPHAs} \end{aligned}$$ The solution of is the standard linear minimum mean square error estimator (LMMSE) as $$\begin{aligned}
&\mathrm{vec}(\hat{\mathbf{X}}) \leftarrow \beta \boldsymbol{\Sigma} \mathbf{A^\top y} , \,\, \mathrm{where} \,\, \boldsymbol{\Sigma} = \left( (\boldsymbol{\alpha}_R \otimes \boldsymbol{\alpha}_L) + \beta \mathbf{A^\top A} \right)^{-1}.
\nonumber%\label{eq:LMMSE_estimate_X}\end{aligned}$$ Using a standard approach (see equations (45) and (46) of [@Tipping1] or (7.88) of [@Bishop]), the solution of can be found as $$\begin{aligned}
\beta \leftarrow \frac{m + 2a}{||\mathbf{y} -\mathbf{A} \mathrm{vec}(\hat{\mathbf{X}})||_2^2 + \mathrm{tr}(\mathbf{A}\boldsymbol{\Sigma}\mathbf{A}^\top) + 2b}.
\label{eq:noise_update1}\end{aligned}$$ The standard RVM in [@Tipping1] uses the different update rule [@mackay1991] $$\begin{aligned}
\beta \leftarrow \frac{\mathrm{tr}(( \boldsymbol{\alpha}_R \otimes \boldsymbol{\alpha}_L)\boldsymbol{\Sigma}) + 2a}{||\mathbf{y} -\mathbf{A} \mathrm{vec}(\hat{\mathbf{X}})||_2^2 + 2b},
\label{eq:noise_update2}\end{aligned}$$ which often improves convergence [@mackay1991]. The update rule has the benefit over of having established convergence properties. In simulations we used the update rule since it improved the estimation accuracy.
Finally we deal with as follows.
1. [*For Schatten $s$-norm:*]{} Using the Schatten $s$-norm prior gives us the update equations $$\begin{aligned}
\begin{aligned}
&\boldsymbol{\alpha}_L \leftarrow c_s \left( \hat{\mathbf{X}}\boldsymbol{\alpha}_R\hat{\mathbf{X}}^\top + \tilde{\boldsymbol{\Sigma}}_L + \epsilon\mathbf{I}_p \right)^{(s-2)/2} ,\\
&\boldsymbol{\alpha}_R \leftarrow c_s \left( \hat{\mathbf{X}}^\top\boldsymbol{\alpha}_L\hat{\mathbf{X}} + \tilde{\boldsymbol{\Sigma}}_R + \epsilon\mathbf{I}_q \right)^{(s-2)/2} ,
\end{aligned}\label{eq:alpha_schatten}\end{aligned}$$ where $c_s = (s/2)^{s/2}$ and the matrices $\tilde{\boldsymbol{\Sigma}}_L$ and $\tilde{\boldsymbol{\Sigma}}_R$ have elements $$\begin{aligned}
&[\tilde{\boldsymbol{\Sigma}}_L ]_{ij} = \mathrm{tr}(\boldsymbol{\Sigma}(\boldsymbol{\alpha}_R \otimes \mathbf{E}_{ij}^{(L)})) , \\
&[\tilde{\boldsymbol{\Sigma}}_R]_{ij} = \mathrm{tr}(\boldsymbol{\Sigma}(\mathbf{E}_{ij}^{(R)} \otimes \boldsymbol{\alpha}_L )) , \end{aligned}$$ and where $\mathbf{E}_{ij}^{(L)} \in \mathbb{R}^{p\times p}$ and $\mathbf{E}_{ij}^{(R)} \in \mathbb{R}^{q\times q}$ are matrices with ones in position $(i,j)$ and zeros otherwise.
2. [*Log-determinant penalty:*]{} For the log-determinant prior the update equations become $$\begin{aligned}
\begin{aligned}
&\boldsymbol{\alpha}_L \leftarrow \nu \left( \hat{\mathbf{X}}\boldsymbol{\alpha}_R\hat{\mathbf{X}}^\top + \tilde{\boldsymbol{\Sigma}}_L + \epsilon\mathbf{I}_p \right)^{-1} ,\\
&\boldsymbol{\alpha}_R \leftarrow \nu \left( \hat{\mathbf{X}}^\top\boldsymbol{\alpha}_L\hat{\mathbf{X}} + \tilde{\boldsymbol{\Sigma}}_R + \epsilon\mathbf{I}_q \right)^{-1}.
\end{aligned} \label{eq:alpha_rsvm}\end{aligned}$$ We see that the update rule for the log-determinant penalty can be interpreted as in the limit $s \to 0$.
The derivations of and are shown in Appendix \[appendix2\] and \[appendix3\]. The corresponding update equations for the one-sided precision based model are obtained by fixing the other precision matrix to be the identity matrix. In the spirit of the evidence approximation based relevance vector machine, we call the developed algorithms in this section as relevance singular vector machine (RSVM). For Schatten-s norm and log-determinant priors, the methods are named as RSVM-SN and RSVM-LD, respectively.
EM
--
In expectation-maximization [@Bishop], the value of the precisions $\theta \triangleq \{ \boldsymbol{\alpha}_L, \, \boldsymbol{\alpha}_R ,\, \beta\}$ are updated in each iteration by maximizing the cost (EM help function in MAP estimation) $$\begin{aligned}
\label{eq:EM_cost}
Q(\theta,\theta') + \log p(\theta)\end{aligned}$$ where $\theta'$ are the parameter values from the previous iteration. The function $Q(\theta,\theta')$ is defined as $$\begin{aligned}
& Q(\theta,\theta') \nonumber = \mathcal{E}_{\mathbf{X}|\mathbf{y},\theta'} [ \log p(\mathbf{y,X}|\theta)] = \text{constant} \nonumber \\
& - \frac{\beta}{2} ||\mathbf{y - A}\mathrm{vec}(\hat{\mathbf{X}}) ||_2^2 - \frac{1}{2} \mathrm{tr}(\boldsymbol{\alpha}_L \hat{\mathbf{X}} \boldsymbol{\alpha}_R \hat{\mathbf{X}}^\top) -\frac{1}{2} \mathrm{tr}(\boldsymbol{\Sigma}^{-1} \boldsymbol{\Sigma}') \nonumber \\
%& \,\,\,\,
& + \frac{q}{2} \log |\boldsymbol{\alpha}_L| + \frac{p}{2} \log |\boldsymbol{\alpha}_R| + \frac{m}{2} \log \beta,
\label{eq:LowRank:EM_help_function}\end{aligned}$$ where $\boldsymbol{\Sigma}' = \left( (\boldsymbol{\alpha}'_R \otimes \boldsymbol{\alpha}'_L) + \beta' \mathbf{A^\top A} \right)^{-1}$, and $\mathcal{E}$ denotes the expectation operator. The maximization of $Q(\theta,\theta') + \log p(\theta)$ leads to update equations which are identical to the update equations of evidence approximation. That means that for the Schatten-s norm, the maximization leads to , and , and for log-determinant penalty, the maximization leads to , and . For the noise precision, EM reproduces the update equation . The derivation of and update equations are shown in Appendix \[appendix:EM\]. Unlike evidence approximation, EM has monotonic convergence properties and hence the derived update equations are bound to improve estimation performance in iterations.
Balancing the precisions
------------------------
We have found that in practical algorithms, there is a chance that one of the two precisions becomes large and the other small over iterations. A small precision results in numerical instability in the Kronecker covariance structure . To prevent the inbalance we rescale the matrix precisions in each iteration such that $1)$ the a-priori and a-posteriori squared Frobeniun norm of $\mathbf{X}$ are equal, $$\begin{aligned}
&\mathcal{E}[||\mathbf{X}||_F^2 | \boldsymbol{\alpha}_L, \boldsymbol{\alpha}_R] = \mathrm{tr}(\boldsymbol{\alpha}_L^{-1}) \mathrm{tr}(\boldsymbol{\alpha}_R^{-1}) \\
&= \mathcal{E}[||\mathbf{X}||_F^2 | \boldsymbol{\alpha}_L, \boldsymbol{\alpha}_R,\beta, \mathbf{y} ] = ||\hat{\mathbf{X}}||_F^2 + \mathrm{tr}(\boldsymbol{\Sigma}),\end{aligned}$$ and $2)$ the contribution of the precisions to the norm is equal, $$\begin{aligned}
\mathrm{tr}(\boldsymbol{\alpha}_L^{-1}) = \mathrm{tr}(\boldsymbol{\alpha}_R^{-1}) .\end{aligned}$$ The rescaling makes the algorithm more stable and often improves estimation performance.
Simulation experiments {#sec:simulations}
======================
In this section we numerically verify our two hypotheses, and compare the new algorithms with relevant existing algorithms. Our objectives are to verify:
- the hypothesis that the left-sided precision is better than the right sided precision for a fat low-rank matrix,
- the hypothesis that the two-sided precision based model performs better than one-sided precision based model.
- the proposed methods perform better than a nuclear-norm minimization based convex algorithm and a variational Bayes algorithm.
In the simulations we considered low-rank matrix reconstruction and also matrix completion as a special case due to its popularity.
Performance measure, experimental setup and competing algorithms
----------------------------------------------------------------
To compare the algorithms, the performance measure is the normalized-mean-square-error $$\begin{aligned}
\text{NMSE} \triangleq \mathcal{E}[||\hat{\mathbf{X}} - \mathbf{X}||_F^2]/\mathcal{E}[||\mathbf{X}||_F^2].\end{aligned}$$
In experiments we varied the value of one parameter while keeping the other parameters fixed. For given parameter values, we evaluated the NMSE as follows.
1. For LRMR, the random measurement matrix $\mathbf{A} \in \mathbb{R}^{m \times pq}$ was generated by independently drawing the elements from $\mathcal{N}(0,1)$ and normalizing the column vectors to unit norm. For low rank matrix completion, each row of $\mathbf{A}$ contains a 1 in a random position and zero otherwise with the constraint that the rows are linearly independent.
2. Matrices $\mathbf{L} \in \mathbb{R}^{p \times r}$ and $\mathbf{R} \in \mathbb{R}^{r \times q}$ with elements drawn from $\mathcal{N}(0,1)$ were randomly generated and the matrix $\mathbf{X}$ was formed as $\mathbf{X = LR}$. Note that $\mathbf{X}$ is of rank $r$ (with probability $1$).
3. Generate the measurement $\mathbf{y} = \mathbf{A} \mathrm{vec}(\mathbf{X}) + \mathbf{n}$, where $\mathbf{n} \sim \mathcal{N}(\mathbf{0},\sigma_n^2 \mathbf{I}_m)$ and $\sigma_n^2$ is chosen such that the signal-to-measurement-noise ratio is $$\begin{aligned}
\mathrm{SMNR} \triangleq \frac{\mathcal{E}[||\mathbf{A} \mathrm{vec}(\mathbf{X})||_2^2]}{\mathcal{E}[||\mathbf{n}||_2^2]} = \frac{rpq}{m\sigma_n^2}.\end{aligned}$$
4. Estimate $\hat{\mathbf{X}}$ using competing algorithms and calculate the error $||\hat{\mathbf{X}} - \mathbf{X}||_F^2$.
5. Repeat steps $2-4$ for each measurement matrix $T_1$ number of times.
6. Repeat steps $1-5$ for the same parameter values $T_2$ number of times.
7. Then compute the NMSE by averaging.
In the simulations we chose $T_1 = T_2 = 25$, which means that the averaging was done over 625 realizations. We normalized the column vectors of $\mathbf{A}$ to make the SMNR expression realization independent.
Finally we describe competing algorithms. For comparison, we used the following nuclear norm based estimator $$\begin{aligned}
\hat{\mathbf{X}} = \arg \min_\mathbf{X} ||\mathbf{X}||_* \text{ , s.t. } ||\mathbf{y - A}\mathrm{vec}(\mathbf{X})||_2 \leq \epsilon,\end{aligned}$$ where we used $\epsilon = \sigma_n \sqrt{m + \sqrt{8m}}$ as proposed in [@candes2006]. The cvx toolbox [@cvx] was used to implement the estimator. For matrix completion we also compared with the Variational Bayesian (VB) developed by Babacan et. al. [@Babacan]. In VB, the matrix $\mathbf{X}$ is factorized as $$\begin{aligned}
\mathbf{X = \underbar{L} \, \underbar{R}},\end{aligned}$$ and (block) sparsity inducing priors are used for the column vectors of $\mathbf{\underbar{L}} \in \mathbb{R}^{p \times \min(p,q)}$ and $\mathbf{\underbar{R}}^\top \in \mathbb{R}^{q \times \min(p,q)}$. The VB algorithm was developed for matrix completion (and robust PCA), but not for matrix reconstruction. We note that, unlike RSVM and VB, the nuclear norm estimator requires a-priori knowledge of the noise power. We also compared the algorithms to the Cram[é]{}r-Rao bound (CRB) from [@Werner; @Tang2011] (as we know the rank a-priori in our experimental setup). We mention that the CRB is not always a valid lower bound in this experimental setup because all technical conditions for computing a valid CRB are not always fulfilled and the estimators are not always unbiased. The choice of CRB is due to absence of any other relevant theoretical bound.
Simulation results
------------------
Our first experiment is for verification of the first two hypotheses. For the experiment, we considered LRMR and fixed $\mathrm{rank}(\mathbf{X}) = r = 3$, $p = 15$, $q = 30$, $\mathrm{SMNR} = 20$ dB and varied $m$. The results are shown in Figure \[fig:single\_double\_rec\] where NMSE is plotted against normalized measurements $m/(pq)$. We note that RSVM-SN with left precision is better than right precision. Same result also hold for RSVM-LD. This verifies the first hypothesis. Further we see that RSVM-SN and RSVM-LD with two sided precisions are better than respective one-sided precisions. This result verifies the second hypothesis. In the experiments we used $s=0.5$ for RSVM-SN as it was found to be the best (empirically). Henceforth we fix $s=0.5$ for RSVM-SN.
![NMSE vs. $m/(pq)$ for low-rank matrix reconstruction.[]{data-label="fig:single_double_rec"}](single_double2nocrb){width="0.9\linewidth"}
The second experiment considers comparison with nuclear-norm based algorithm and the CRB for LRMR. The objective is robustness study by varying number of measurements and measurement noise power. We used $r = 3$, $p = 15$ and $q = 30$. In Figure \[fig:alpha\_snr\_rec\] (a) we show the performance against varying $m/(pq)$; the SMNR = 20 dB was fixed. The performance improvement of RSVM-SN is more pronounced over the nuclear-norm based algorithm in the low measurement region. Now we fix $m/(pq)=0.7$ and vary the SMNR. The results are shown in Figure \[fig:alpha\_snr\_rec\] (b) which confirms robustness against measurement noise.
![NMSE vs. $m/(pq)$ and SMNR for low-rank matrix reconstruction. (a) SMNR = 20 dB and $m/(pq)$ is varied. (b) $m/(pq)=0.7$ and SMNR is varied.[]{data-label="fig:alpha_snr_rec"}](alpha_snr_rec2c){width="0.9\linewidth"}
Next we deal with matrix completion where the measurement matrix $\mathbf{A}$ has a special structure and considered to be inferior to hold information about $\mathbf{X}$ than the same dimensional random measurement matrix used in LRMR. Therefore matrix completion requires more measurements and higher SMNR. We performed similar experiments as in our second experiment and the results are shown in Figure \[fig:alpha\_snr\_comp\]. In the experiments the performance of the VB algorithm is included. It can be seen that RSVM-SN is typically better than the other algorithms. We find that the the VB algorithm is pessimistic.
![NMSE vs. $m/(pq)$ and SMNR for low-rank matrix completion. (a) SMNR = 20 dB and $m/(pq)$ is varied. (b) $m/(pq)=0.7$ and SMNR is varied.[]{data-label="fig:alpha_snr_comp"}](alpha_snr_comp2){width="0.9\linewidth"}
Finally in our last experiment we investigated the VB algorithm to find conditions for its improvement and compared it with RSVM-SN. For this experiment, we fixed $r = 3$, $p = 15$, $m/(pq)=0.7$ and SMNR = 20 dB, and varied $q$. The results are shown in Figure \[fig:q\_completion\] and we see that VB provides good performance when $p=q$. The result may be attributed to an aspect that VB is highly prone to a large number of model parameters which arises in case $\mathbf{X}$ is away from a square matrix.
![NMSE vs. $q$ for low-rank matrix completion.[]{data-label="fig:q_completion"}](q_comp2){width="0.9\linewidth"}
Conclusion
==========
In this paper we developed Bayesian learning algorithms for low-rank matrix reconstruction. The framework relates low-rank penalty functions (type I estimators) to the latent variable models (type II estimators) with either left- or right-sided precisions through the matrix Laplace transform and the concave conjugate formula. The model was further extended to the two-sided precision based model. Using evidence approximation and expectation maximization, we derived the update equations for the parameters. The resulting algorithm was named the Relevance Singular Vector Machine (RSVM) due to its similarity with the Relevance Vector Machine for sparse vectors. Especially we derived the update equations for the estimators corresponding to the log-determinant penalty and the Schatten $s$-norm penalty, we named the algorithms RSVM-LD and RSVM-SN, respectively.
Through simulations, we showed that the two-sided precision based model performs better than the one-sided model for matrix reconstruction. The algorithm also outperformed a nuclear-norm based estimator, even though the nuclear-norm based estimator knew the noise power. The proposed methods also outperformed a variational Bayes method for matrix completion when the matrix is not square.
Derivation of the Laplace Approximation {#appendix:laplace}
---------------------------------------
The Laplace approximation is an approximation of the integral $$\begin{aligned}
I = \int e^{-\frac{1}{2} f(\mathbf{a})} d\mathbf{a},%\limits\end{aligned}$$ where the integral is over $\mathbf{a} \in \mathbb{R}^n$. The function $f(\mathbf{a})$ is approximated by a second order polynomial around its minima $\mathbf{a}_0$ as $$\begin{aligned}
f(\mathbf{a}) \approx f(\mathbf{a}_0) + \frac{1}{2} (\mathbf{a - a}_0)^\top \mathbf{H} (\mathbf{a - a}_0),\end{aligned}$$ where $\mathbf{H} = \nabla^2 f(\mathbf{a}) |_{\mathbf{a = a}_0}$ is the Hessian of $f(\mathbf{a})$ at $\mathbf{a}_0$. The term linear in $\mathbf{a}$ vanishes and $\mathbf{H} \succ \mathbf{0}$ at $\mathbf{a}_0$ since we expand around a minima. With this approximation, the integral becomes $$\begin{aligned}
I \approx \int e^{-\frac{1}{2} f(\mathbf{a}_0) - \frac{1}{4} (\mathbf{a - a}_0)^\top \mathbf{H} (\mathbf{a - a}_0)} d\mathbf{a} = \sqrt{\frac{(4\pi)^{n}}{|\mathbf{H}|}} e^{-\frac{1}{2} f(\mathbf{a}_0)}.\end{aligned}$$ In , the integral is given by $$\begin{aligned}
I = \frac{1}{(2\pi)^{pq/2}} \int_{\boldsymbol{\alpha} \succ \mathbf{0}} e^{-\frac{1}{2} [ \mathrm{tr}(\boldsymbol{\alpha}\mathbf{Z}) -q\log|\boldsymbol{\alpha}| + K(\boldsymbol{\alpha}) ]} d \boldsymbol{\alpha}.\end{aligned}$$ Set $f(\mathbf{a}) = \mathrm{tr}(\boldsymbol{\alpha}\mathbf{Z}) -q\log|\boldsymbol{\alpha}| + K(\boldsymbol{\alpha})$, where $\mathbf{a} = \mathrm{vec}(\boldsymbol{\alpha})$. Let $\boldsymbol{\alpha}_0 \succ \mathbf{0}$ denote the minima of $f(\mathbf{a})$ and $\mathbf{H}$ the Hessian at $\boldsymbol{\alpha}_0$. Assuming that $\boldsymbol{\alpha}_0$ and $\mathbf{H}$ are “large” in the sense that the integral over $\boldsymbol{\alpha} \succ \mathbf{0}$ can be approximated by the integral over $\boldsymbol{\alpha} \in \mathbb{R}^{p \times p}$ we find that $$\begin{aligned}
&I \approx \frac{1}{(2\pi)^{pq/2}} \int e^{-\frac{1}{2} f(\mathbf{a}_0) - \frac{1}{4} (\mathbf{a - a}_0)^\top \mathbf{H} (\mathbf{a - a}_0)} d\mathbf{a} \\
&= \frac{(4\pi)^{p^2/2}}{(2\pi)^{pq/2} |\mathbf{H}|^{1/2}} e^{-\frac{1}{2} f(\mathbf{a}_0)},\end{aligned}$$ where $\mathbf{a}_0 = \mathrm{vec}(\boldsymbol{\alpha}_0)$.
The EM help function {#appendix:EM}
--------------------
The EM help function $Q(\theta,\theta')$ is given by $$\begin{aligned}
&Q(\theta,\theta') = \mathcal{E}_{\mathbf{X}| \mathbf{y},\theta'} [\log \, p(\mathbf{X}|\mathbf{y},\theta)] = c + \frac{m}{2} \log \, \beta \\
&- \frac{\beta}{2} \mathcal{E}[||\mathbf{y - A}\mathrm{vec}(\mathbf{X})||_2^2 - \frac{1}{2} \mathcal{E}[\mathrm{tr}(\boldsymbol{\alpha}_L \mathbf{X} \boldsymbol{\alpha}_R \mathbf{X}^\top)] \\
&+ \frac{q}{2} \log |\boldsymbol{\alpha}_L| + \frac{p}{2} \log |\boldsymbol{\alpha}_R|,\end{aligned}$$ where $c$ is a constant. Using that $$\begin{aligned}
&\mathcal{E}[||\mathbf{y - A}\mathrm{vec}(\mathbf{X})||_2^2] = ||\mathbf{y}||_2^2 - 2\mathbf{y}^\top \mathbf{A} \mathrm{vec}(\hat{\mathbf{X}}) \\
&+ \mathrm{tr}(\mathbf{A^\top A}(\mathrm{vec}(\hat{\mathbf{X}}) \mathrm{vec}(\hat{\mathbf{X}})^\top + \boldsymbol{\Sigma}')) \\
&= ||\mathbf{y - A}\mathrm{vec}(\hat{\mathbf{X}})||_2^2 + \mathrm{tr}(\mathbf{A}^\top\mathbf{A}\boldsymbol{\Sigma}') , \end{aligned}$$ and $$\begin{aligned}
&\mathcal{E}[\mathrm{tr}(\boldsymbol{\alpha}_L \mathbf{X} \boldsymbol{\alpha}_R \mathbf{X}^\top)] \\
&= \mathrm{tr}((\boldsymbol{\alpha}_R \otimes \boldsymbol{\alpha}_L)(\mathrm{vec}(\hat{\mathbf{X}}) \mathrm{vec}(\hat{\mathbf{X}})^\top + \boldsymbol{\Sigma}')) \\
&= \mathrm{tr}(\boldsymbol{\alpha}_L \hat{\mathbf{X}} \boldsymbol{\alpha}_R \hat{\mathbf{X}}^\top) + \mathrm{tr}((\boldsymbol{\alpha}_R \otimes \boldsymbol{\alpha}_L) \boldsymbol{\Sigma}'),\end{aligned}$$ we recover the expression for the EM help function.
Details for the RSVM with the Schatten $s$-norm penalty {#appendix2}
-------------------------------------------------------
We here set $\mathbf{S} = \epsilon \mathbf{I}_q$ to keep the derivation more general. The regularized Schatten $s$-norm penalty is given by $$\begin{aligned}
\tilde{g}(\mathbf{Z}) = \mathrm{tr}((\mathbf{X^\top X} + \mathbf{S})^{s/2}).\end{aligned}$$ For the concave conjugate formula we find that the minimum over $\mathbf{Z}$ occurs when $$\begin{aligned}
\boldsymbol{\alpha} - \frac{s}{2} (\mathbf{Z + S})^{s/2-1} = \mathbf{0}.\end{aligned}$$ Solving for $\mathbf{Z}$ gives us that $$\begin{aligned}
\tilde{K}(\boldsymbol{\alpha}) = - \mathrm{tr}(\boldsymbol{\alpha}\mathbf{S}) - \frac{2-s}{s} \left( \frac{2}{s} \right)^{-2/(2-s)} \mathrm{tr}(\boldsymbol{\alpha}^{-2/(2-s)}),\end{aligned}$$ which results in .
Using , we find that the minimum of for the Schatten $s$-norm occurs when $$\begin{aligned}
\hat{\mathbf{X}}\boldsymbol{\alpha}_R \hat{\mathbf{X}}^\top + \tilde{\boldsymbol{\Sigma}}_R - \left( \frac{2}{s} \right)^{-s/(2-s)} \boldsymbol{\alpha}_L^{-2/(2-s)} = \mathbf{0}\end{aligned}$$ Solving for $\boldsymbol{\alpha}_L$ gives for $\boldsymbol{\alpha}_L$. The update equation for $\boldsymbol{\alpha}_R$ is derived in a similar manner.
Details for the RSVM with the log-determinant penalty {#appendix3}
-----------------------------------------------------
The log-determinant penalty is given by $$\begin{aligned}
g(\mathbf{X}) = \nu \log | \mathbf{Z + S} |.\end{aligned}$$ For the concave conjugate formula we find that the minimum over $\mathbf{Z}$ occurs when $$\begin{aligned}
\boldsymbol{\alpha} - \nu (\mathbf{Z + S})^{-1} = \mathbf{0}.\end{aligned}$$ Solving for $\mathbf{Z}$ gives $$\begin{aligned}
\tilde{K}(\boldsymbol{\alpha}) = - \mathrm{tr}(\boldsymbol{\alpha}\mathbf{S}) + \nu \log |\boldsymbol{\alpha}| + \nu p - \nu \log \nu.\end{aligned}$$ By removing the constants we recover .
Using , we find that the minimum of with respect to $\boldsymbol{\alpha}_L$ for the log-determinant penalty occurs when $$\begin{aligned}
\hat{\mathbf{X}}\boldsymbol{\alpha}_R \hat{\mathbf{X}}^\top + \tilde{\boldsymbol{\Sigma}}_R + \mathbf{S}_L - \nu \boldsymbol{\alpha}_L^{-1} = \mathbf{0}\end{aligned}$$ Solving for $\boldsymbol{\alpha}_L$ gives us for $\boldsymbol{\alpha}_L$. The derivation of the update equation for $\boldsymbol{\alpha}_R$ is found in a similar way.
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|
---
abstract: '[ A special isotropic energy-momentum tensor causes an anisotropic dynamical segregation of contracting extra dimensions in contrast to expanding dimensions, provided initial data put different arrows of time evolution. Features of such the configuration are problematic and in question. ]{}'
author:
- 'V.V.Kiselev'
bibliography:
- 'bib\_ex-seg.bib'
title: Extra dimensions and its segregation during evolution
---
Introduction
============
Trying to formulate a consistent quantum gravity reveals a problem of extra spatial dimensions [@Green:1987sp; @Green:1987mn] since we do not see those dimensions in any experimental phenomena. If so, we have to point to rigorous dynamical reasons or conditions for such the segregation of extra dimensions with respect to ordinary three dimensional space (3D). The most known approaches in this way are the following: i) extra dimensions are compact in contrast to the ordinary infinite space, ii) extra dimensions are large but almost empty, since the matter propagates only in three separate dimensions [@Rubakov:2001kp]. Both statements imply the situations when properties of extra dimensions specifically differ from that of 3D-space. In this paper we consider a mechanism providing such the difference between extra and 3D spaces. Namely, we show that initially isotropic homogeneous sources of evolution in general relativity with extra dimensions can result in an anisotropic solution when some spatial dimensions become exponentially long in comparison with others[^1] if one explicitly sets different directions for the arrow of time evolution in the initial data for the velocities of expansion, i.e. the opposite signs of scale factors derivatives with respect to time in the initail data of evolution.
Evolution
=========
In the isotropic homogeneous case the Einstein tensor, composed of Ricci tensor $R_{\mu\nu}$, metrics $g_{\mu\nu}$ and scalar curvature $R=g^{\mu\nu}R_{\mu\nu}$, $$\mathcal{G}^\mu_\nu=R^\mu_\nu-\frac12\,\delta^\mu_\nu\,R,$$ should have the spatial components in the form of $$\mathcal{G}^\alpha_\beta=-p\,\delta^\alpha_\beta, \qquad\alpha,\,\beta\neq 0,$$ with $\alpha,\,\beta$ running from $1$ to $d$ being the dimension of space and $p$ denoting the pressure of matter with the energy-momentum tensor $T_{\mu\nu}$ in the Einstein equations $$R^\mu_\nu-\frac12\,\delta^\mu_\nu\,R=8\pi G\,T^\mu_\nu,$$ where $G$ is the multidimensional analogue of Newton constant.
With the stationary tensor of matter background we expect that the pressure remains constant in time, that could be easily reached if the metrics has the form of $$\label{g}
\mathrm{d}s^2= \mathrm{d}t^2-a_+^2(t)\,\mathrm{d}\boldsymbol r_+^2-
a_-^2(t)\,\mathrm{d}\boldsymbol r_-^2,$$ where the scale factors have the exponential dependence on the time $$a_\pm(t)=\mathrm{e}^{\pm H_\pm t},$$ while the spatial dimensions of “$+$ and “$-$" components are equal to $d_+$ and $d_-$, respectively, at $d=d_++d_-$. Thus, we study the possibility of evolution with the isotropic and homogeneous initial energy-momentum tensor, while the scale factors are explicitly chosen in the form permitting the break of isotropy by setting the different initial velocities for two kinds of scale factors. Such the difference in the initial impacts to the scales would transfer the anisotropy during the evolution to produce a discrimination of spatial dimensions with essentially different properties dynamically. Indeed, the spatial components of Einstein tensor with metrics (\[g\]) are equal to $$\mathcal{G}^+_+=-d_+H_+^2+d_-H_-H_+-\frac12\, R, \qquad
\mathcal{G}^-_-=-d_-H_-^2+d_+H_-H_+-\frac12\, R,$$ where the scalar curvature equals $$R=-(H_-d_--H_+d_+)^2-H_-^2d_--H_+^2d_+,$$ while $$\mathcal{G}^+_-=\mathcal{G}^-_+\equiv 0.$$ The temporal component is given by the formula $$\mathcal{G}^0_0= -d_+H_+^2-d_-H_-^2-\frac12\, R.$$ The isotropic condition $\mathcal{G}^+_+=\mathcal{G}^-_-$ requires $$d_+H_+^2+(d_+-d_-)H_+H_--d_-H_-^2=0,$$ that has 2 solutions: the first is $H_+=-H_-$ representing de Sitter space-time being the isotropic solution with $T^\mu_\nu=\rho_\Lambda\delta^\mu_\nu$ of vacuum with the energy density $\rho_\Lambda=d(d-1) H_+^2/(16\pi G)$ setting the curvature $R=-d(d+1)H_+^2$, while the second solution gives $d_+H_+=d_-H_-$, that is of our interest[^2].
Let us emphasise that in the case of de Sitter solution all of spatial dimensions do expand ($H_+>0$) or contract ($H_+<0$), and the discrimination between the regimes of expansion or contraction is set by the explicit choice of $H_+$ sign in the initial data. Therefore, we can ascribe this choice of sign to the isotropic arrow of time evolution, say, the forward arrow corresponds to the expansion or the backward arrow does to the contraction.
In this manner, the anisotropic choice for the arrow of time evolution for the ordinary and extra spatial dimensions in the initial data could mean the anisotropic evolution of those dimensions. Indeed, the anisotropic evolution takes place at $$\label{eq7}
\mathcal{G}^0_0= - \mathcal{G}^+_+ =-\mathcal{G}^-_- =\frac12\,R=- \frac12\,d\,H_+H_-.$$ Note, that $H_+$ and $H_-$ have the same sign. Therefore, the solution in (\[eq7\]) corresponds to the stiff matter with $\rho=p<0$.
Two ‘no-go’ theorems
--------------------
The stiff matter with negative energy density could take place for a free isotropic scalar ghost field with the negative kinetic energy, $$\mathcal{L}=-\frac12\,\dot\phi^2,\qquad\dot\phi=\partial_0\phi.$$ That is unphysical case, and we arrive to the first ‘no-go’ theorem for the realisation of conditions in the mechanism described.
The second option is an effective field theory of scalar $\phi (t)$ depending exceptionally on the time if one has the lagrangian $$\mathcal{L}=\frac12\,\left\{1-F(\phi)\right\}\dot\phi^2,\qquad\dot\phi=\partial_0\phi,$$ where $F$ is a function of kinetic self-interaction, that has the limit of large fields $F(\infty)=2$ and the free field limit of $F(0)=0$, say, $$F(\phi)=2\,\frac{\phi^2}{M^2+\phi^2}=2\sum\limits_{n=0}^\infty
(-1)^n\left(\frac{\phi^2}{M^2}\right)^{n+1}.$$ At $\phi^2\gg M^2$ we arrive to the stiff matter with the negative density of energy in the strong field limit.
Then, we would get the following pattern: in the case of a restricted potential energy at large fields (a plateau of potential) the kinetic term can dominate and give the stiff matter providing the anisotropic evolution of space-time, but after the field reaches values with the comparable kinetic and potential energy density the regime of evolution switches to the inflation, i.e. a slow roll of field from the flat plateau to the minimum of potential.
However, such the functions of self-interaction, $F$ changes the sign of the kinetic term, that points to the strong field limit with losing any control for the field dynamics. So, it reveals the second ‘no-go’ theorem for the realisation of conditions in the mechanism described.
Option of fine tuning
---------------------
Nevertheless, one could exhibit a non-trivial situation with a composition of two fine tuned terms:
- the first term is cosmological, $T_\mu^\nu (\Lambda)=\rho_\Lambda\,\delta_\mu^\nu$ with the energy density $\rho_\Lambda>0$ and the pressure $p_\Lambda=-\rho_\Lambda$,
- the second term is composed by a condensate with a tuned negative energy density $\rho_{c}<0$ and zero pressure $p_{c}=0$.
So, the condition $\rho=p$ results in $$\label{very-1}
\rho_\Lambda+\rho_{c}=-\rho_\Lambda,$$ hence, $$\label{very-2}
2\rho_\Lambda=-\rho_{c}>0.$$ The fine tuning in (\[very-2\]) points to a kind of symmetry, which should follow from some dynamical reasons unknown. In this respect, a nature of such the condensate and a mechanism of its tuning remain open, while the option itself should be pointed certainly.
Effects
=======
The anisotropic solution for the evolution with the isotropic energy-momentum tensor reveals two sub-spaces: the large expanding space of $d_+$ dimension and small contracting space of $d_-$ dimension. A co-moving volume $V_0$ corresponds to the physical volume $$V(t)=a_+^{d_+}(t)\,a_-^{d_-}(t)\,V_0 = V_0\,\mathrm{e}^{(d_+H_+-d_-H_-)t} =V_0,$$ hence, the physical volume remains constant but its “$+$” and “$-$” dimensions do scale in different and opposite rates compensating each other as it was caused by the opposite choice for the arrows of time evolution in the initial data for two kinds of dimensions.
Let us consider properties of external matter evolution at fixed metrics. So, we neglect any back reaction of external matter to the metrics, that means a contribution of matter to the total energy balance to be essentially suppressed. For the sake of simplicity we focus on massless particles, i.e. on the radiation.
Soft and hard modes
-------------------
The mass shell condition of radiation with comoving momentum $k$, $g^{\mu\nu}k_\mu k_\nu=0$ transforms to $$k_0^2=\boldsymbol{k}_+^2 \mathrm{e}^{-2H_+ t}+\boldsymbol{k}_-^2 \mathrm{e}^{2H_- t}.$$ Modes moving exclusively along the “$+$” direction, i.e. at $\boldsymbol{k}_-\equiv 0$, are exponentially ultra-soft, $$k_0^2=\boldsymbol{k}_+^2 \mathrm{e}^{-2H_+ t}\to 0,$$ while modes propagating in both the contracting “$-$” sub-space and expanding “$+$” sub-space, are exponentially ultra-hard, $$k_0^2-\boldsymbol{k}_+^2 \mathrm{e}^{-2H_+ t}=\boldsymbol{k}_-^2 \mathrm{e}^{2H_- t}\to \infty.$$ Therefore, an observer living in the large dimensions, considers the ultra-hard modes as super-heavy alike super-Planckian massive particles with masses rising during such the evolution.
We see that the energy of soft modes scales as $$k_0^+\sim\frac{1}{a_+}\sim\mathrm{e}^{-H_+t},$$ while the energy of hard modes scales as $$k_0^-\sim\frac{1}{a_-}\sim\mathrm{e}^{H_-t}.$$ At the constant physical volume $V(t)$ we expect that for the energy densities of soft and hard modes we get $$\label{dot}
\rho_\pm\sim\frac{1}{a_\pm}\sim \mathrm{e}^{\mp H_\pm t},$$ that is valid. Indeed, the energy-momentum conservation $\nabla_\mu T^\mu_\nu=0$ for the temporal component, i.e. at $\nu=0$, gives $$\dot \rho_++\dot\rho_-+(d_+H_+-d_-H_-) (\rho_++\rho_-) + d_+H_+ p_+ -d_-H_-p_-=0,$$ where $p_\pm$ denote pressures of soft and hard modes. Since the sub-spaces are separately isotropic, but the overall isotropy is explicitly broken, we put the relations for radiation in the sub-spaces $$p_\pm=\frac{1}{d_\pm}\,\rho_\pm,$$ that results in $$(\dot \rho_+ +H_+ \rho_+ )+(\dot \rho_- -H_- \rho_-)=0,$$ which means that the evolution of soft and hard modes can be separated and it gives relation in (\[dot\]).
Then, free isotropic soft and hard modes of radiation compose two cold and hot matter components with different temperatures and pressures. However, interactions could mix these components.
Collisions of soft modes with soft modes cannot produce hard modes. i.e. the extra contracting space is decoupled from the soft $d_+$ dimensional space. Collisions of hard modes with hard and soft modes can produce both hard and soft modes, that can cause the cooling of contracted sub-space and warming of expanding sub-space. The balance depends on the rates of evolution and scattering, that determines the scenario of future. For instance, the high density of hard modes can change a geometry of extra sub-space, say, causing its compactification before the cooling, that means further complete decoupling of extra dimensions.
Conclusion
==========
We have described the novel anisotropic solution with extra dimensions under the conditions of isotropy and homogeneity for the energy-momentum tensor with the explicit breaking the isotropy by setting different arrows of time evolution in the initial scale factors for ordinary and extra dimensions. This anisotropy discriminates expanding spatial dimensions and extra contracted dimensions. Modes propagating in sub-spaces are differentiated in the following way: soft modes live exclusively in the expanding sub-space, while other modes look like ultra-hard and heavy for observers in the expanding world.
Such the effects are provided by the specific option for the energy-momentum tensor that can be reached if one introduces the fine tuning for the cosmological term and pressureless condensate with the negative energy density.
A knowledge on a further fate of such the evolution itself requires to take into account an influence of additional matter to the metrics. Moreover, the regime described needs studies of model with deviations of energy-momentum tensor from the values constant in time.
The segregated evolution allows us to conjecture that extra dimensions do decouple from the expanding sub-space. However, the problem of justification for the special state to produce the segregation remains open.
This work is partially supported by Russian Ministry of Science and Education, project \# 3.9911.2017/BasePart.
[^1]: In the similiar aspect, sources of gravity with non-equivalent dimensions were considered in [@Chodos:1979vk], solutions with a cosmological constant and non-equivalent initial data were investigated in [@Appelquist:1983vs], a non-liner sigma model was analysed in [@Ho:2010vv], a generic case of expanding and shrifting dimensions was discussed recently in [@Benisty:2018gzx].
[^2]: The same relation between the dimensions and Hubble rates was obtained in [@Ho:2010vv] under the condition of constant physical volume at $d_+=3$ (see Section III.)
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abstract: 'The general self-adjoint elliptic boundary value problems are considered in a domain $G\subset \Bbb R^{n+1}$ with finitely many cylindrical ends. The coefficients are stabilizing (as $x\to\infty$, $x\in G$) so slowly that we can only describe some “structure” of solutions far from the origin. This problem may be understood as a model of “generalized branching waveguide.” We introduce a notion of the energy flow through the cross-sections of the cylindrical ends and define outgoing and incoming “waves.” An augmented scattering matrix is introduced. Analyzing the spectrum of this matrix one can find the number of linearly independent solutions to the homogeneous problem decreasing at infinity with a given rate. We discuss the statement of problem with so-called radiation conditions and enumerate self-adjoint extensions of the operator of the problem.'
author:
- 'Kalvine V.O.'
title: 'Self-adjoint elliptic problems in domains with cylindrical ends under weak assumptions on the stabilization of coefficients'
---
Introduction
============
In domain $G\subset \Bbb R^{n+1}$ with finitely many cylindrical ends we consider the general formally self-adjoint boundary value problem. The coefficients tend to limits (as $x\to\infty$, $x\in
G$) too slow to allow obtaining an asymptotic of solution at infinity. Using the results of the paper [@1] (see also [@2 Section 8.5]), one can get some “structure” of solution to the problem: far from the origin a solution is represented as a linear combination of some functional series plus a remainder. The coefficients in the linear combination remain unknown. In this paper we develop an approach, which, in particular, allows to derive expressions for the coefficients in the structure of solution to the problem under consideration.
Let $\Pi^r_+=\{(y,t): y\in \Omega^r,
t>0\}$, $r=1,\dots, N$, stand for the cylindrical ends, where $\Omega^r$ is the cross-section. (The domain $G$ coincides with the union $\Pi^1_+\cup \dots \cup \Pi^N_+$ outside a large ball.) With every cylindrical end $\Pi^r_+$ we associate limit and model problems in the cylinder $\Pi^r=\{(y,t): y\in \Omega^r, t\in \Bbb
R\}$.
As the coefficients of limit problem we take the limits of coefficients of the original problem as $t\to +\infty$, $(y,t)\in
\Pi^r_+$. It is assumed that the limit problems are elliptic. Since the operator of the original problem is formally self-adjoint, the operators of the limit problems are formally self-adjoint as well. As is known (see e.g. [@3 Chapter 5]), one can consider every limit problem as a model of “generalized waveguide.” This means that a generalized notion of the energy flow through the cross-section of the cylinder is introduced, the solution to the homogeneous problem is called incoming (outgoing) wave if the energy flow associated with the solution is positive (negative). The amplitudes of such waves may grow with power or even with exponential rate at infinity.
The operator of the model problem is formally self-adjoint and depends on the parameter $T\in \Bbb R$. The coefficients of the model problem coincide with the coefficients of the original problem on the set $\{(y,t)\in P^r_+,t>T+3\}$ and with their limits (as $t\to +\infty$, $(y,t)\in \Pi^r_+$) on the set $\{(y,t)\in \Pi^r,t<T\}$. The coefficients of the model problem tend to the coefficients of the limit one as $T\to+\infty$. Thus a solution to the homogeneous model problem can be obtained in the form of functional series by the method of successive approximations, as the first approximation it is natural to take a wave of the limit problem. On the analogy of the limit problem, for the model problem we introduce a notion of the energy flow through the cross-section of the cylinder. The formula for the energy flow through the cross-section $\{(y,t)\in \Pi^r, t=R\}$, $R<T$, is the same for both (limit and model) problems because the coefficients of the problems coincide on the set $\{(y,t)\in
\Pi^r, t<T\}$. Moreover, it turns out that a wave and the correspondent solution to the homogeneous model problem have equal energy flows through the left infinitely distant cross-section of $\Pi^r$. This allows to calculate the energy flows of obtained solutions to the homogeneous model problem and also allows to separate these solutions into incoming and outgoing waves (of the model problem). Due to the formally self-adjointness of the model problem operator, the energy flows of such waves remain constant along the cylinder. Recall that the coefficients of the model problem coincide with the coefficients of original problem on the set $\{(y,t)\in \Pi^r_+, t>T+3\}$. Owing to this fact, one can consider the domain $G$ as a branching waveguide, where the waves obtained for the model problem in $\Pi^r$ propagate along the cylindrical end $\Pi^r_+$ of $G$, $r=1,\dots,N$. Using a modification of the scheme suggested in [@1 Theorem 6.2], we get the structure of solutions to the problem in $G$: far from the origin a solution is represented as a linear combination of the waves plus a remainder. Some waves properties obtained on the previous step allow us to derive the formulas for the coefficients in the structure of solution. The results are represented in Theorem \[pasv\] and Theorem \[taswG\].
The remaining part of the paper basically contains corollaries of the theorems \[pasv\] and \[taswG\]. We omit the proofs because they almost repeat the proofs of the similar assertions in [@3 Chapther 5] or in [@11], where it is assumed that the coefficients are stabilizing with exponential rate. The changes in the proofs mainly consist in usage of Theorem \[pasv\] or Theorem \[taswG\] instead of asymptotic representations. In the main text we insert the exact references to the needed proofs.
The operator of the problem acts in weighted spaces. We obtain some information about the kernel of the problem (Propositions \[cl2\] and \[cl5\]) and introduce “scattering matrices.” These unitary matrices take into account waves growing at infinity. Analyzing the spectrum of this matrix one can find the number of linearly independent solutions to the homogeneous problem decreasing at infinity with a given rate (cf. Proposition \[crit\]). We discuss the statement of problem with “radiation conditions:” the domain of operator contains only functions with prescribed structure at infinity. This is a way to choose a solution (with a certain arbitrariness) (cf. Propositions \[cl3\] and \[cl4\]). The intrinsic radiation conditions (the solution mainly consists of outgoing waves) can be utilized in every case. To verify whether given radiation conditions can be used, it is required to know the scattering matrix (cf. Proposition \[cl6\]). In Section \[sss\] the self-adjoint extensions of operator of the problem are found.
Some of the results proved in our paper were announced earlier in the work [@4].
Statement of the problem and preliminaries
==========================================
Domain and self-adjoint boundary value problem
----------------------------------------------
Let $G\subset \Bbb R^{n+1}$ be a domain with smooth boundary $\partial G$ coinciding, outside a large ball, with the union $\Pi^1_+\cup \dots \cup \Pi^N_+$ of non-overlapping semicylinders; here $\Pi^r=\{(y^r,t^r): y^r\in\Omega^r, t^r>0 \}$, $(y^r,t^r)$ are local coordinates, and the cross-section $\Omega^r$ is bounded domain in $\Bbb R^n$. In the domain $G$ we introduce a formally self-adjoint $k\times
k$-matrix $\mathcal
L$ of differential operators $\mathcal L_{ij}(x,D_x)$ with smooth coefficients, where ${\rm ord \,} \mathcal L_{ij}=\tau_i+\tau_j$, the numbers $\tau_1,\dots,\tau_k$ are non-negative integers, and $\tau_1+\cdots+\tau_k=m$. Consider the boundary value problem $$\label{1}
\begin{split}
{\mathcal L} (x,D_x)u(x)&=f(x),\qquad x\in G, \\
{\mathcal B} (x,D_x)u(x)&=g(x),\qquad x\in\partial G,
\end{split}$$ where $\mathcal B$ is an $m\times k$-matrix of differential operators. For a given $\mathcal L$ we find a class of boundary conditions such that for an element $\mathcal B$ of the class the self-adjoint Green formula $$\label{1a}
(\mathcal L u, v)_G+(\mathcal B u,\mathcal Q v)_{\partial
G}=(u,\mathcal L v)_G+(\mathcal Q u,\mathcal B v)_{\partial G}$$ holds with some $m\times k$-matrix $\mathcal Q$ of differential operators for all $u,v\in C_c^\infty(\overline G)$. It is supposed that $\mathcal B$ in (\[1\]) is from the mentioned class and the problem is elliptic.
Boundary conditions and self-adjoint Green formula {#s2}
--------------------------------------------------
If necessary changing the enumeration of the rows and columns in $\mathcal
L_{ij}(x,D_x)$, we may always arrange that $\tau=\tau_1\geq\tau_2\geq\dots\geq\tau_k$. In what follows we suppose that this has been done. Denote by $K_s$, $s=1,\dots,\tau$, the number of values $j$ such that $\tau_j\geq\tau-s+1$. Then $K_1+\dots +K_\tau=\tau_1+\dots+\tau_k=m$ and $ K_1\leq\dots\leq K_\tau \leq\ k$. On the boundary $\partial G$ we introduce the $m\times k$-matrix $$\label{D}
\mathcal D=\left(\begin{array}{c}
{\bf D}^1 \\
\vdots \\
{\bf D}^\tau \\
\end{array}\right),$$ where the block ${\bf D}^s$ consists of the rows $$(\delta_{1,h},\dots,\delta_{k,h})D_\nu^{\tau_h-\tau+s-1},\qquad
h=1,\dots,K_s;$$ here $\nu$ is the unit outward normal to $\partial G$ and $D_\nu=-i\partial/\partial \nu$. With $\mathcal L(x,D_x)$ we associate the sesquilinear form $$\label{2}
a(u,v)=\sum_{i,j=1}^k \sum_{|\eta|\leq\tau_h}\sum_{|\mu|\leq
\tau_j}\int_G a_{ij}^{\eta\mu}(x) D^\mu_x u_j(x)\overline{D^\eta_x
v_i(x)}\,dx.$$ The Green formula $$\label{GF}
a(u,v)=(\mathcal L u,v)_G+(\mathcal N u,\mathcal D v)_{\partial G}$$ holds for $u,v\in C_c^\infty (\overline{ G})$ with some $m\times
k$-matrix $\mathcal N (x,D_x)=\|\mathcal N_{qj} (x,D_x)\|$ of differential operators, $\operatorname{ord} \mathcal
N_{qj}+\operatorname{ord}\mathcal D_{qh}\leq\tau_h+\tau_j-1$; by $(\cdot,\cdot)_G$ and $(\cdot,\cdot)_{\partial G}$ we denote the inner products on $L_2(G)$ and $L_2(\partial G)$.
\[ip\][The proof of the Green formula (\[GF\]) is standard. Using “local maps,” we can consider $G$ as the half-space $\Bbb
R^n_-=\{x\in\Bbb R^n, x_n<0 \}$ (cf. [@5], [@6]). Then $D_\nu=D_{x_n}$ and $\mathcal
L_{ij}(x,D_x)=\sum_{|\alpha|+\beta\leq\tau_i}\sum_{|\mu|\leq\tau_j}D_\tau^\alpha
D_\nu^\beta a_{ij}^{(\alpha\beta)\mu}(x)D_x^\mu$, where $D_\tau=D_{x_1}D_{x_2}\dots D_{x_{n-1}}$. Integrating by parts and changing the order of summation, we obtain $$\begin{split}
&(\mathcal L u,
v)_G=\sum_{i,j=1}^k\sum_{|\alpha|+\beta\leq\tau_i}\sum_{|\mu|\leq
\tau_j}\bigl(D_\tau^\alpha D_\nu^\beta
a_{ij}^{(\alpha\beta)\mu}(x)D_x^\mu u_j,
v_i\bigr)_G=a(u,v)\\&+\sum_{s=1}^\tau\sum_{i=1}^{K_s}\sum_{j=1}^k\sum_{|\mu|\leq\tau_j}\sum_{|\alpha|+\beta\leq\tau_j,\beta\geq
s}\bigl(D_\tau^\alpha D_\nu^{\beta-s+\tau-\tau_i}
a_{ij}^{(\alpha\beta)\mu}(x)D_x^\mu u_j, D_\nu^{\tau_i -\tau +s-1}
v_i \bigr)_{\partial G},
\end{split}$$ where $u,v\in C_c^\infty(\overline G)$. This implies (\[GF\]).]{}
\[R\] A matrix $\mathcal P=\mathcal P(x,D_x)$ is called a Dirichlet system on the boundary $\partial G$ if there exists an $m\times
m$-matrix $\mathcal R=\mathcal R(x,D_x)$ satisfying the following conditions.
\(i) $\mathcal P(x,D_x)=\mathcal R(x,D_x)\mathcal D(x,D_x)$, where $\mathcal D$ is given in (\[D\]).
\(ii) The matrix $\mathcal R$ consists of $K_p\times K_s$-blocks ${\mathcal R}_{[p,s]}$ ($p,s=1,\dots,\tau$). The elements of ${\mathcal R}_{[p,s]}$, $s\leq p$, are tangential differential operators with smooth coefficients on $\partial G$ of order not higher than $p-s$, while the elements of ${\mathcal R}_{[p,s]}$, $s>p$, are zeros. The ${\mathcal R}_{[p,p]}(x)$ are nondegenerate matrices, $|\det{\mathcal R}_{[p,p]}(x)|>\varepsilon>0$ for $x\in\partial G$.
If in particular $k=1$, then $\mathcal P(x,D_x)$ is the usual Dirichlet system of order $\tau$ (see [@7],[@8],[@9]).
\[Rr\] The operator $\mathcal R$ from Definition \[R\] has an inverse $\mathcal R^{-1}$, which is a matrix of differential operators of the same structure as $\mathcal R$. The block ${\mathcal R}^{-1}_{[p,p]}(x)$ of $\mathcal R^{-1}(x,D_x)$ is the inverse matrix $({\mathcal R}_{[p,p]}(x))^{-1}$, $p=1,\dots,\tau$. The blocks ${\mathcal R}^{-1}_{[p,s]}$, $s<p$, of sizes $K_p\times K_s$ can be successively found from the relations $${\mathcal R}^{-1}_{[p,s]}(x,D_x)={\mathcal
R}_{[p,p]}^{-1}(x)\sum_{\ell=s}^{p-1}(-{\mathcal
R}_{[p,\ell]}(x,D_x)){\mathcal R}_{[\ell, s]}^{-1}(x,D_x).$$ The blocks ${\mathcal R}^{-1}_{[p,s]}$, $s>p$, consist of zeros.
Let $$\label{p}
\mathcal P(x,D_x)=\mathcal R(x,D_x)\mathcal D(x,D_x)$$ be a Dirichlet system on $\partial G$. We set $$\label{t}
\mathcal T(x,D_x)={\mathcal R}^{-1}_*(x,D_x)\mathcal
N(x,D_x),$$ where ${\mathcal R}^{-1}_*(x,D_x)$ is the formally adjoint differential operator to ${\mathcal R}^{-1}(x,D_x)$ and $\mathcal N(x,D_x)$ is from the Green formula (\[GF\]). Introduce $m\times k$-matrices $\mathcal B$ and $\mathcal Q$ such that $$\label{3}
(\mathcal B_{q1},\dots,\mathcal B_{qk})=(\mathcal
T_{q1},\dots,\mathcal T_{qk}), \quad (\mathcal
Q_{q1},\dots,\mathcal Q_{qk})=(\mathcal P_{q1},\dots,\mathcal
P_{qk})$$ for some numbers $q$, $1\leq q\leq m$, while $$\label{4}
(\mathcal B_{q1},\dots,\mathcal B_{qk})=(\mathcal
P_{q1},\dots,\mathcal P_{qk}), \quad (\mathcal
Q_{q1},\dots,\mathcal Q_{qk})=-(\mathcal T_{q1},\dots,\mathcal
T_{qk})$$ for the remaining rows of $\mathcal B$ and $\mathcal Q$. Therefore, $$\label{i.1}
\begin{split}
(\mathcal N u,\mathcal D v)_{\partial G}-(\mathcal D u,\mathcal N
v)_{\partial G}=(\mathcal N u,\mathcal R^{-1}\mathcal P
v)_{\partial G}-(\mathcal R^{-1}\mathcal P u,\mathcal N
v)_{\partial G}\\=(\mathcal Tu,\mathcal P v)_{\partial
G}-(\mathcal P u,\mathcal T v)_{\mathcal G}=(\mathcal B u,\mathcal
Q v)_{\partial G}-(\mathcal Q u,\mathcal B v)_{\partial G}.
\end{split}$$ Since $\mathcal L$ is formally self-adjoint, the form (\[2\]) is symmetric (i.e. $a(u,v)=\overline {a(v,u)}$). From (\[GF\]) we obtain $$\label{i.2}
(\mathcal L u,v)_G+(\mathcal N u, \mathcal D v)_{\partial
G}=(u,\mathcal L v)_G+(\mathcal D u,\mathcal N v)_{\partial G}.$$ Together with (\[i.1\]) this leads to the Green formula (\[1a\]).
Limit operators {#slo}
---------------
Let $r$, $r=1,\dots,N$, be a fixed number. We write the superscript $r$ at $\mathcal L$, $\mathcal R$, and other operators if they are written in the coordinates $(y,t)$ inside the semicylinder $\overline\Pi^r_+$. Let $\mathcal L^r=\|\mathcal
L^r_{ij}\|$ and let $$\label{i.3a}
\mathcal
L^r_{ij}(y,t,D_y,D_t)=\sum_{|\eta|+\mu\leq\tau_j+\tau_h}\ell_{ij}^{\eta\mu}(y,t)D^\eta_y
D^\mu_t.$$ We set $\psi_T(y,t)\equiv\psi(t-T)$ for $(y,t)\in\overline\Pi^r_+$, where $\psi\in C^\infty(\Bbb R)$ is a cutoff function such that $\psi(t)=0$ for $t<1$ and $\psi(t)=1$ for $t>2$.
We say that $\mathcal L$ is stabilizing in the semicylinder $\Pi^r_+$ if there exist functions ${\bf l}_{hj}^{\eta\mu}$ of $y\in\overline \Omega^r$ ([*limit coefficients*]{}) such that $$\label{i.4}
\lim_{T\to +\infty}\|\psi_T({\ell}_{ij}^{\eta\mu}-{\bf
l}_{ij}^{\eta\mu});C^\infty(\overline\Pi^r_+)\|=0,\quad
|\eta|+\mu\leq\tau_i+\tau_j,\quad i,j=1,\dots,k.$$
Let $\mathcal F(x,D_x)=\|\mathcal F_{qj}(x,D_x)\|$ be an operator given on the boundary $\partial G$. Write down $\mathcal F_{qh}$ in the local coordinates (y,t): $$\mathcal F^r_{q
j}(y,t,D_y,D_t)=\sum_{|\alpha|+\beta\leq\operatorname{ord}\mathcal
F_{qj}} f_{qj}^{\alpha \beta}(y,t)D_y^\alpha D^\beta_t.$$ We say that $\mathcal F$ is stabilizing in $\Pi^r_+$ if there exist functions ${\bf f}_{qj}^{\alpha \beta}$ of $y\in\partial
\Omega^r$ such that $$\lim_{T\to+\infty}\|\psi_T(f_{qj}^{\alpha\beta}-{\bf
f}_{qj}^{\alpha\beta});C^\infty(\partial\Omega^r\times \Bbb
R_+)\|=0$$ for $|\alpha|+\beta\leq\operatorname{ord}\mathcal F_{qj}$ and for all values of $q$ and $j$; here $\Bbb R_+=\{t\in\Bbb R: t>0\}$.
Since the coefficients of $\mathcal D_{qj}^r$ do not depend on $t$, the operator $\mathcal D(x,D_x)$ from (\[D\]) is stabilizing in $\Pi^1_+,\dots,\Pi^N_+$. Assume that $\mathcal L$ is stabilizing in $\Pi^r_+$. The stabilization of $\mathcal
N(x,D_x)$ in $\Pi^r_+$ is guaranteed by (\[i.4\]) (the coefficients of $\mathcal N^r(y,t,D_y,D_t)$ are expressed in terms of $\ell_{ij}^{\eta\mu}$; see Remark \[ip\]). Therefore an operator $\mathcal B(x,D_x)$ constructed of the rows of $\mathcal N (x, D_x)$ and $\mathcal D (x,D_x)$ is stabilizing in $\Pi^r$ as well; this case $\mathcal R(x,D_x)\equiv I$, see (\[p\]), (\[t\]) and (\[3\]), (\[4\]). In the general case we assume the stabilization of $\mathcal R(x,D_x)=\|\mathcal
R_{hq}(x,D_x)\|_{h,q=1}^m$. Then the operator $\mathcal R^{-1}$ is stabilizing (see Remark \[Rr\]), we get the stabilization of the operators $\mathcal P$, $\mathcal T$, $\mathcal B$, and $\mathcal
Q$.
Let the elements ${ L}^r_{ij}(y,D_y,D_t)$ of the limit operator ${ L}^r=\|{ L}^r_{ij}\|$ be given by the right-hand side of (\[i.3a\]) with $\ell_{ij}^{\eta\mu}$ replaced by ${\bf
l}_{ij}^{\eta\mu}$. Likewise, changing the coefficients to the limit ones, we define the limit operators ${ N}^r$, ${ R}^r$, ${
B}^r$ and etc. The relations ${ P}^r={ R}^r{\mathcal D}^r$ and ${ T^r}=({ R}^r)_*^{-1}{ N}^r$ are fulfilled, where $({R}^r)_*^{-1}$ is formally adjoint to $({ R}^r)^{-1}$. From (\[3\]) and (\[4\]) it follows that the matrix ${ B}^r$ and ${ Q}^r$ consist of the rows of ${ P}^r$ and ${ T}^r$. The Green formula $$\label{GFl}
({ L}^r u,v)_{\Pi^r}+({ B}^r u,{ Q}^r v)_{\partial \Pi^r}=(u,{
L}^r v)_{\Pi^r}+({ Q}^r u, { B}^r v)_{\partial \Pi^r}$$ is valid in the cylinder $\Pi^r=\Omega^r\times\Bbb R$, where $u,v\in C_c^\infty(\overline\Pi^r)$. We assume that the limit problem $$\label{lp}
\begin{split}
{ L}^r (y,D_y, D_t) u(y,t)&=F(y,t),\quad (y,t)\in\Pi^r,\\
{ B}^r (y,D_y, D_t) u(y,t)&=G(y,t),\quad (y,t)\in\partial\Pi^r,
\end{split}$$ is elliptic.
Denote by $W^l_\gamma(\Pi^r)$ the space with norm $\|e_\gamma\cdot;H^l(\Pi^r)\|$, where $H^l(\Pi^r)$ is the Sobolev space, $e_\gamma: (y,t)\mapsto \exp(\gamma t)$, and $\gamma\in
\Bbb R$. For $l\geq\tau$ we set $$\label{sp}
\begin{split}
\mathcal D_\gamma^l(\Pi^r) &= \prod_{j=1}^k H_\gamma^{l+\tau_j}(\Pi^r),\\
\mathcal R_\gamma^l(\Pi^r) &=\prod_{i=1}^k
H_\gamma^{l-\tau_i}(\Pi^r)\times \prod_{q=1}^m
H_\gamma^{l-\tau-\sigma_q-1/2}(\partial \Pi^r)
\end{split}$$ with $\sigma_q=\operatorname{ord} B^r_{qj}-\tau-\tau_j$, $\sigma_q<0$. The map $$\label{lo}
A^r_\gamma=\{L^r,B^r\}:\mathcal D_\gamma^\ell(\Pi^r)\to\mathcal
R_\beta^\ell(\Pi^r)$$ is continuous. We introduce the operator pencil $$\label{op}
\Bbb C\ni\lambda\mapsto\mathfrak
A^r(\lambda)=\{L^r(y,D_y,\lambda),B^r(y,D_y,\lambda)\}$$ in the domain $\Omega^r$. The spectrum of $\mathfrak A^r$ is symmetric about the real line and consists of normal eigenvalues. Any strip $\{\lambda\in\Bbb C:|\operatorname{Im}\lambda|\leq
h<\infty\}$ contains at most finitely many points of the spectrum. Denote by $\lambda_{-\nu^0},\dots,\lambda_{\nu^0}$ with $\nu^0\geq 0$ all the real eigenvalues of $\mathfrak A^r$ (if the number of real eigenvalues is even, then $\lambda_0$ is absent). We enumerate the nonreal eigenvalues so that $0<\mbox{Im}\,\lambda_{\nu^0+1}\leq\mbox{Im}\,\lambda_{\nu^0+2}\leq\dots$ and $\lambda_\nu=\overline\lambda_{-\nu}$, where $\nu=\nu^0+1,\nu^0+2,\dots$. Let $\{\varphi_\nu^{(0,j)},\dots,\varphi_\nu^{(\varkappa_{j\nu}-1,j)};j=1,\dots,J_\nu:=\dim\ker\mathfrak
A^r(\lambda_\nu)\}$ be a canonical system of Jordan chains of the pencil $\mathfrak A^r$ corresponding to $\lambda_\nu$, i.e. $\varphi_\nu^{(0,j)}$ is an eigenvector and $\varphi_\nu^{(1,j)},\dots,\varphi_\nu^{(\varkappa_{j\nu}-1,j)}$ are associated vectors (e.g., see [@10]). The functions $$\label{2.11}
u_\nu^{(\sigma, j)}(y,t)=\exp (i\lambda_\nu t)\sum_{\ell=0}^\sigma
\frac 1 {\ell !} (it)^\ell \varphi_\nu^{(\sigma-\ell,j)}(y)$$ with $\sigma =0,\dots,\varkappa_{j\nu}-1$ satisfy the homogeneous problem (\[lp\]). We introduce the form $$\label{2.12}
q^r(u, v):=({L}^r u,v)_{\Pi^r}+({ B}^r u, { Q}^r v)_{\partial \Pi^r}-
(u, { L}^r v)_{\Pi^r}-({ Q}^r u, { B}^r v)_{\partial \Pi^r}.$$ It is obvious that $q^r(u,v)=-\overline {q^r(v,u)}$ and $q^r(u,u)\in i \Bbb R$. The Green formula (\[GFl\]) extends by continuity to the functions $U\in\mathcal D_\gamma^l(\Pi^r)$ and $V\in\mathcal D_{-\gamma}^l(\Pi^r)$, therefore $q^r(U,V)=0$.
\[p2.1\] [(i)]{} Let $\chi \in C^\infty(\mathbb R)$, $\chi (t)=1$ for $t\geq2$ and $\chi (t)=0$ for $t\leq 1$. One can choose Jordan chains $\{\varphi_\nu^{(\sigma,j)}\}$ to satisfy the following conditions: $$\begin{aligned}
\label{2.13}q^r(\chi u_\nu^{(\sigma,j)},\chi u_\mu^{(\tau,p)}) =i\delta_{-\nu,\mu}\delta_{j,p}\delta_{\varkappa_{j\nu}-1-\sigma,\tau},\quad |\nu|>\nu_0, |\mu|>\nu_0,\\
\label{2.14} q^r(\chi u_\nu^{(\sigma,j)},\chi
u_\mu^{(\tau,p)})=\pm
i\delta_{\nu,\mu}\delta_{j,p}\delta_{\varkappa_{j\nu}-1-\sigma,\tau},\quad |\nu|\leq \nu_0,|\mu|\leq \nu_0,\\
\label{2.15}q^r(\chi u_\nu^{(\sigma,j)},\chi u_\mu^{(\tau,p)})
=0,\quad |\nu|\leq \nu_0,|\mu|>\nu_0,\end{aligned}$$ where the functions $u_\nu^{(\sigma,j)}$ are given in (\[2.11\]). In (\[2.14\]) the sign depends on $\nu$ and $j$ (and cannot be taken arbitrarily). The equality (\[2.15\]) remains true for arbitrary choice of Jordan chains and for any superscripts. The conditions (\[2.13\]) – (\[2.15\]) do not depend on the choice of $\chi$.
[(ii)]{} The map (\[lo\]) is an isomorphism if and only if the line $\Bbb R+i\gamma=\{\lambda\in \Bbb
C:\operatorname{Im} \lambda=\gamma\}$ is free of the spectrum of the pencil $\mathfrak A^r$.
Let $\gamma\geq 0$. Denote by $\mathcal W_\gamma(\Pi^r)$ the linear span of the functions $\{u_\nu^{(\sigma,j)}: |\operatorname
{Im}\lambda|\leq\gamma\}$. It is known (see [@3]) that the total algebraic multiplicity of all the eigenvalues of the pencil $\mathfrak A^r$ in the strip $\{\lambda\in \Bbb C:|
\operatorname{Im} \lambda|\leq\gamma\ \}$ is even for any $\gamma\geq 0$; we denote the multiplicity by $2M^r(\equiv 2
M^r_\gamma)$. There is a basis $$\label{bu}
u_1^+,\dots,u_{M^r}^+, u_1^-,\dots, u_{M^r}^-$$ in the space $\mathcal W_\gamma(\Pi^r)$ obeying $$\label{oc}
q^r(\chi u_j^\pm, \chi u_h^\pm)=\mp i\delta_{j,h},\quad
q^r(\chi u_j^\pm,\chi u_h^\mp)=0,\quad j,h=1,\dots,M^r,$$ (see [@3],[@11]); here $\chi$ is the cut-off function from Proposition \[p2.1\]. One can consider the cylinder $\Pi^r$ as a generalized waveguide. The space $\mathcal W_\gamma(\Pi^r)$ is called the space of waves. The quantity $iq^r (\chi u, \chi u)$ represents the energy flow transferred by the wave $u\in\mathcal
W_\gamma(\Pi^r)$ through the cross-section $\Omega^r$ of the cylinder $\Pi^r$. Thus $u_1^+,\dots,u_{M^r}^+$ are incoming waves and $u_1^-,\dots, u_{M^r}^-$ are outgoing waves for the problem (\[lp\]).
The following proposition is a variant of Proposition 3.1.4 and Theorem 3.2.1 from [@3].
\[p2.2\] Assume that $\gamma>0$, the line $\Bbb
R+i\gamma$ is free of the spectrum of the pencil $\mathfrak A^r$, and $\{F,G\}\in\mathcal R^l_\gamma(\Pi^r)\cap \mathcal
R^l_{-\gamma} (\Pi^r)$. Then a solution $u\in\mathcal D_{-\gamma}^\ell (\Pi^r)$ to the problem (\[lp\]) admits the representation $$\label{asu}
u=\sum_{j=1}^{M^r}\{a_j(F,G) u^+_j+b_j(F,G) u^-_j \} +v,$$ where the functions $u^\pm_j$ form a basis in $\mathcal
W_\gamma(\Pi^r)$ and satisfy (\[oc\]), $v$ is a solution to the problem (\[lp\]) in $\mathcal D_\gamma^l(\Pi^r)$. The functionals $a_1,\dots, a_{M^r}$ and $b_1,\dots,b_{M^r}$ are continuous on $\mathcal R^l_\gamma(\Pi^r)\cap \mathcal
R^l_{-\gamma} (\Pi^r)$ and $$\label{c1}
\begin{split}
a_j (F,G) &=i(F,u^+_j)_{\Pi^r}+i(G,Q^r u^+_j)_{\partial\Pi^r},\\
b_j(F,G)&=-i (F,u^-_j)_{\Pi^r}-i(G,Q^r u^-_j)_{\partial\Pi^r},
\end{split}$$ where $Q^r$ is the same as in the Green formula (\[GF\]).
The structure of solutions to the problem (\[1\])
=================================================
Construction of a model problem in $\Pi^r$ {#sGFg}
------------------------------------------
In this subsection we construct a differential operator $\{\mathfrak
L^r_T,\mathfrak B^r_T\}$ in $\Pi^r$ such that the following conditions are satisfied: (i) $\{\mathfrak L^r_T,\mathfrak
B^r_T\}$ coincides with $\{\mathcal L,\mathcal B\}$ on the set $\{(y,t)\in\overline \Pi^r: t>T+3\}$; (ii) $\{\mathfrak
L^r_T,\mathfrak B^r_T\}$ coincides with $\{L^r,B^r\}$ on the set $\{(y,t)\in\overline \Pi^r: t<T\}$; (iii) the norm $\|\Delta_T^r;
\mathcal D^l_\gamma(\Pi^r)\to\mathcal R^l_\gamma (\Pi^r) \|$ of the operator $$\label{Delta}
\Delta_T^r=\{L^r,B^r\}-\{\mathfrak L^r_T,\mathfrak B^r_T\}$$ tends to zero as $T\to +\infty$; (iv) for $u,v\in
C_c^\infty(\overline\Pi^r)$ and sufficiently large $T$ the self-adjoint Green Formula $$\label{GFg}
({\mathfrak L}^r_T u,v)_{\Pi^r}+({\mathfrak B}^r_T u,{\mathfrak
Q}^r_T v)_{\partial \Pi^r}=(u,{\mathfrak L}^r_T
v)_{\Pi^r}+({\mathfrak Q}^r_T u, {\mathfrak B}^r_T v)_{\partial
\Pi^r}$$ holds with some $m\times k$-matrix $\mathfrak Q^r_T$ of differential operators.
Recall that $\psi_T(y,t)$ is a cutoff function, $\psi_T(y,t)\equiv\psi(t-T)$ for $(y,t)\in\overline\Pi^r_+$, where $\psi\in C^\infty(\Bbb R)$, $\psi(t)=0$ for $t<1$ and $\psi(t)=1$ for $t>2$. Let $\mathfrak L^r_T=L^r-\psi_T(L^r-\mathcal L)\psi_T$, where the operator $\psi_T\mathcal L\psi_T$ is extended from $\overline\Pi^r_+$ to the whole cylinder $\overline\Pi^r$ by zero. First we find a Dirichlet system $\EuScript P^r_T$ and an operator $\EuScript T^r_T$ such that $$\label{GFga}
({\mathfrak L}^r_T u,v)_{\Pi^r}+({\EuScript P}^r_T u,{\EuScript
T}^r_T v)_{\partial \Pi^r}=(u,{\mathfrak L}^r_T
v)_{\Pi^r}+({\EuScript T}^r_T u, {\EuScript P}^r_T v)_{\partial
\Pi^r}$$ for $u,v\in C_c^\infty(\overline\Pi^r)$. Then we compose $\mathfrak B^r_T$ and $\mathfrak Q^r_T$ from the rows of $\EuScript P^r_T$ and $\EuScript T^r_T$ (by analogy with (\[3\]), (\[4\] )) and derive (\[GFg\]) from (\[GFga\]).
Denote $ \EuScript N^r_T=N^r-\psi_T(N^r-\mathcal N)\psi_T$ and $
\EuScript D^r_T=\mathcal D^r-\psi_T(\mathcal D^r-\mathcal
D)\Psi_T$ (the operators $\psi_T\mathcal N\psi_T$ and $\psi_T\mathcal
D\psi_T$ are extended to $\overline\Pi^r$ by zero). It is clear that $\EuScript D^r_T\equiv\mathcal D^r$ and $\mathcal D^r_T$ is the Dirichlet system on $\partial\Pi^r$. Since $\mathcal D^r$ consists of normal derivatives, we have $[\mathcal D^r, \psi_T]=0$ on $\partial \Pi^r$; here $[a,b]=ab-ba$. Thus, substituting $\psi_T u$ and $\psi_T v$ for $u$ and $v$ in (\[i.2\]), we obtain $$\label{aa}
\begin{split}
(\psi_T\mathcal L\psi_T u, v)_{\Pi^r}+&(\psi_T\mathcal N\psi_T
u,\mathcal D^r v)_{\partial \Pi^r}\\&=(u,\psi_T\mathcal L\psi_T
v)_{\Pi^r}+(\mathcal D^r u,\psi_T\mathcal N\psi_T v)_{\partial
\Pi^r}.
\end{split}$$ By the same arguments from $$\label{bb}
( L^r u, v)_{\Pi^r}+( N^r u,\mathcal D^r v)_{\partial \Pi^r}=(u,
L^r v)_{\Pi^r}+(\mathcal D^r u, N^r v)_{\partial \Pi^r}$$ we get $$\label{cc}
\begin{split}
(\psi_T L^r\psi_T u, v)_{\Pi^r}+&(\psi_T N^r\psi_T u,\mathcal D^r
v)_{\partial \Pi^r}\\&=(u,\psi_T L^r\psi_T v)_{\Pi^r}+(\mathcal
D^r u,\psi_T N^r \psi_T v)_{\partial \Pi^r}.
\end{split}$$ Adding (\[aa\]) and (\[bb\]) and subtracting (\[cc\]), we arrive at the formula $$\label{dd}
(\mathfrak L^r_T u, v)_{\Pi^r}+(\EuScript N^r_T u,\mathcal D^r_T
v)_{\partial \Pi^r}=(u, \mathfrak L^r v)_{\Pi^r}+(\mathcal D^r_T
u, \EuScript N^r_T v)_{\partial \Pi^r}.$$ Recall that $\mathcal P=\mathcal R\mathcal D$ and $P^r=R^r
\mathcal D^r$. For sufficiently large $T$ we put $\EuScript
R^r_T=R^r+\psi_T(\mathcal R-R^r)\psi_T$. Due to the stabilization of $\mathcal R$ in $\Pi^r_+$, the matrix $\EuScript P^r_T\equiv
\EuScript R^r_T\mathcal D^r$ is a Dirichlet system on $\partial
\Pi^r$, and there exists a differential operator $(\EuScript
R^r_T)^{-1}$ such that $(\EuScript R^r_T)^{-1}\EuScript
R^r_T=\EuScript R^r_T(\EuScript R^r_T)^{-1}=I$; see Remark \[Rr\]. Let $\EuScript T^r_T=(\EuScript
R^r_T)^{-1}_*\EuScript N^r_T$. We have $$\label{ee}
\begin{split}
(\EuScript N^r_T u,\mathcal D^r_T v)_{\partial \Pi^r}-(\mathcal
D^r_T u, \EuScript N^r_T v)_{\partial \Pi^r} &\\
=((\EuScript R^r_T)^{-1}_*\EuScript N^r_T u,\EuScript
R^r_T\mathcal D^r_T v)_{\partial \Pi^r}&-(\EuScript R^r_T\mathcal
D^r_T u, (\EuScript R^r_T)^{-1}_*\EuScript N^r_T v)_{\partial
\Pi^r}\\
&=(\EuScript T^r_T u,\EuScript P^r_T v)_{\partial
\Pi^r}-(\EuScript P^r_T u, \EuScript T^r_T v)_{\partial \Pi^r}.
\end{split}$$ Together with (\[dd\]) this implies (\[GFga\]). Composing the matrices $\mathfrak B^r_T$ and $\mathfrak Q^r_T$ from the rows of $\EuScript P^r_T$ and $\EuScript T^r_T$ by the same rule as in (\[3\]) and (\[4\]), we obtain the Green formula (\[GFg\]).
By the construction of $\{\mathfrak L^r_T,\mathfrak B^r_T\}$ the conditions (i), (ii), and (iv) given in the beginning of this subsection are satisfied. Due to the stabilization of $\mathcal
L(x,D_x)$ and $\mathcal R(x,D_x)$ in $\Pi^r_+$ we have $$\label{lim}
\lim_{T\to+\infty}\|\Delta_T^r;
\mathcal D^l_\gamma(\Pi^r)\to\mathcal R^l_\gamma (\Pi^r) \|=0$$ for all $\gamma\in \Bbb R$, condition (iii) is fulfilled.
In the cylinder $\Pi^r$ we consider the model problem $$\label{pp}
\begin{split}
\mathfrak L^r_T(y,t,D_y,D_t) u(y,t)&=\mathfrak F(y,t), \quad
(y,t)\in\Pi^r,\\
\mathfrak B^r_T (y,t,D_y,D_t) u(y,t)&=\mathfrak G (y,t),\quad
(y,t)\in\partial \Pi^r.
\end{split}$$
The structure of solutions to the model problem (\[pp\])
--------------------------------------------------------
Taking into account (\[lim\]) and the invertibility of the limit operator (\[lo\]) (see Proposition \[p2.1\]), we get the following assertion.
\[p3.1\] Let the operators $\mathcal L(x,D_x)$ and $\mathcal R(x,D_x)$ stabilize in $\Pi^r_+$ and let the line $\Bbb R+ i\gamma$ contain no eigenvalues of the pencil $\mathfrak A^r$. Assume that $T$ is sufficiently large. Then the operator $$\{\mathfrak L^r_T,\mathfrak B^r_T\}:\mathcal
D_\gamma^\ell(\Pi^r)\to\mathcal R_\gamma^\ell(\Pi^r)$$ of the problem (\[pp\]) implements an isomorphism.
We now introduce functions $z_j^\pm$, which play the same role for the problem (\[pp\]) as the waves $u_j^\pm$ play for the limit problem (\[lp\]). Suppose that the assumptions of Proposition \[p3.1\] are fulfilled. We set $$\label{3.5}
z_j^\pm=u_j^\pm+\sum_{q=1}^\infty
((A^r_{-\gamma})^{-1}\Delta_T^r)^q u_j^\pm,\quad j=1,\dots,M^r,$$ where the waves $\{u^\pm_j:j=1,\dots,M^r\}$ form a basis in $\mathcal W_\gamma(\Pi^r)$ obeying (\[oc\]). (Recall that $2
M^r(\equiv 2M^r (\gamma))$ is the total algebraic multiplicity of all the eigenvalues of the pencil $\mathfrak A^r$ in the strip $\{\lambda\in \Bbb C: |\operatorname{Im}\lambda|\leq \gamma\}$.)
Let us discuss the equality (\[3.5\]). Note that $\psi_{T-2}
u_j^\pm\in\mathcal D^l_{-\gamma}(\Pi^r)$ and $\Delta_T^r
u_j^\pm=\Delta_T^r\psi_{T-2}u_j^\pm$ with the same cutoff function $\psi_T$ as in the previous section. By virtue of (\[lim\]) the norm of operator $
(A^r_{-\gamma})^{-1}\Delta_T^r:
\mathcal D^l_{-\gamma}(\Pi^r)\to\mathcal D^l_{-\gamma}(\Pi^r) $ is small; $(A^r_{-\gamma})^{-1}$ is bounded because the spectrum of $\mathfrak A^r$ is symmetric about the real axis, see Proposition \[p2.1\], (ii). The series $\sum_{q=1}^\infty
((A^r_{-\gamma})^{-1}\Delta_T^r)^q u_j^\pm$ converges in the norm of $\mathcal D^l_{-\gamma}(\Pi^r)$. Consequently, $$\label{mod}
z_j^\pm= u_j^\pm\mod \mathcal D^l_{-\gamma}(\Pi^r).$$
\[p3.2\] Let the assumptions of Proposition \[p3.1\] be fulfilled. Then the functions $$\label{zpm}
z_1^+,\dots,z_{M^r}^+,z_1^-,\dots,z_{M^r}^-$$ defined by (\[3.5\]) are linearly independent modulo $\mathcal D_{-\gamma}^l (\Pi^r)$ solutions to the homogeneous problem (\[pp\]).
[**Proof.**]{} It is easy to see that the functions $u_\nu^{(\sigma,j)}$ forming the linear span $\mathcal
W_\gamma(\Pi^r)$ are linearly independent modulo $\mathcal
D_{-\gamma}^l (\Pi^r)$; see (\[2.11\]). Thus the elements of the basis $u_1^+,\dots,u_{M^r}^+,u_1^-,\dots,u_{M^r}^-$ in $\mathcal
W_\gamma(\Pi^r)$ are linearly independent modulo $\mathcal
D_{-\gamma}^l (\Pi^r)$. Together with the relations (\[mod\]) this implies the linear independence of $z_j^\pm$, $j=1,\dots,
M^r$, modulo $\mathcal D_{-\gamma}^l (\Pi^r)$.
Let us show that $z_j^\pm$ satisfy the homogeneous problem (\[pp\]). Consider the equation $$\label{3.2}
\{\mathfrak L^r_T,\mathfrak B^r_T\}w=\Delta^r_T u_j^\pm.$$ The inclusion $\Delta^r_T u_j^\pm\in\mathcal R^l_{-\gamma}(\Pi^r)$ holds for $j=1,\dots, M^r$. The line $\Bbb R-i\gamma$ is free of the spectrum of $\mathfrak A^r$ (because the spectrum of $\mathfrak A^r$ is symmetric about the real line). By Proposition \[p3.1\] there exists a unique solution $w\in\mathcal D_{-\gamma}^l(\Pi^r)$ to the problem (\[3.2\]). Multiplying (\[3.2\]) from left by $(A^r_{-\gamma})^{-1}$ and using (\[Delta\]), we get $$(I-(A^r_{-\gamma})^{-1}\Delta^r_T)w=(A^r_{-\gamma})^{-1}\Delta_T^r
u_j^\pm.$$ Thanks to (\[lim\]) the operator $I-(A^r_{-\gamma})^{-1}\Delta^r_T$ is invertible. We write $(I-(A^r_{-\gamma})^{-1}\Delta^r_T)^{-1}$ as the Neumann series and obtain $$w=\bigl(I-(A^r_{-\gamma})^{-1}\Delta^r_T\bigr)^{-1}(A^r_{-\gamma})^{-1}\Delta_T^r
u_j^\pm=\sum_{q=1}^\infty((A^r_{-\gamma})^{-1}\Delta_T^r)^q
u_j^\pm.$$ Keeping in mind that $\{L^r,B^r\}u_j^\pm=0$ and (\[Delta\]), we deduce from (\[3.2\]) that $\{\mathfrak L^r_T,\mathfrak
B^r_T\}(w+u_j^\pm)=0$. It remains to note that $z_j^\pm\equiv
w+u_j^\pm$.
Introduce the form $$\label{pr}
\begin{split}
p^r_T(u,v)=({\mathfrak L}^r_T u,&v)_{\Pi^r}+({\mathfrak B}^r_T
u,{\mathfrak Q}^r_T v)_{\partial \Pi^r}\\&-(u,{\mathfrak L}^r_T
v)_{\Pi^r}-({\mathfrak Q}^r_T u, {\mathfrak B}^r_T v)_{\partial
\Pi^r}.
\end{split}$$ It is easy to see that $p^r_T(u,v)=0$ for $u\in\mathcal
D^l_{-\gamma}(\Pi^r)$ and $v\in\mathcal D^l_{\gamma}(\Pi^r)$ (indeed, for such functions the Green formula (\[GFg\]) is fulfilled).
\[p3.3\] Let the assumptions of Proposition \[p3.1\] be fulfilled. Then the functions (\[zpm\]) satisfy the conditions $$\label{ocz} p^r_T(\chi z_h^\pm,\chi z_j^\pm)=\mp
i\delta_{h,j},\quad p^r_T(\chi z_h^\pm,\chi z_j^\mp) =0,\quad
h,j=1,\dots,M^r,$$ where $\chi \in C^\infty(\Bbb R)$, $\chi(t)=1$ for $t\geq 2$ and $\chi(t)=0$ for $t\leq 1$.The equalities (\[ocz\]) do not depend on the choice of $\chi$.
[**Proof.**]{} Since the waves $u^\pm_j$ satisfy the homogeneous problem (\[lp\]), we have $q^r(u^\pm_h,u^\pm_j)=0$ and $$\label{3.3}
-q^r(\chi u^\pm_h,u^\pm_j)=q^r((1-\chi)u^\pm_h,u^\pm_j),$$ where $q^r$ is from (\[2.12\]). Note that operator $\{L^r,B^r\}$ coincide with $\{\mathfrak L^r_T,\mathfrak B^r_T\}$ on the support of $(1-\chi)u^\pm_h$. This allows us to write (\[3.3\]) in the form $$\label{+1}
-q^r(\chi u^\pm_h,u^\pm_j)=p_T^r((1-\chi)u^\pm_h,u^\pm_j).$$ Due to (\[mod\]) and $\chi u^\pm_h\in\mathcal
D^l_{\gamma}(\Pi^r)$ the Green formula (\[GFg\]) is valid on the pairs $$\begin{aligned}
& &\{u,v\}=\{(1-\chi)(z^\pm_h-u^\pm_h),z^\pm_j\},\\
& &\{u,v\}=\{(1-\chi)(z^\pm_h-u^\pm_h), z^\pm_j-u^\pm_j\},\\
& &\{u,v\}=\{(1-\chi)z^\pm_h, z^\pm_j-u^\pm_j\}.\end{aligned}$$ Thus $p^r_T(u,v)=0$ on the same pairs, and $$\label{+2}
p_T^r((1-\chi)u^\pm_h,u^\pm_j)=p_T^r((1-\chi) z^\pm_h,z^\pm_j).$$ Thanks to Proposition \[p3.2\] we have $p^r_T(z^\pm_h,z^\pm_j)=0$. Therefore, from (\[+1\]) and (\[+2\]) we get $$-q^r(\chi u^\pm_h,u^\pm_j)=-p_T^r(\chi z^\pm_h,z^\pm_j).$$ Finally we obtain $$q^r(\chi u^\pm_h,\chi u^\pm_j)=q^r(\chi u^\pm_h,
u^\pm_j)=p_T^r(\chi z^\pm_h,z^\pm_j)=p_T^r(\chi z^\pm_h,\chi
z^\pm_j).$$ To establish the first equality in (\[ocz\]) it remains to use (\[oc\]). In a similar way one can prove the second equality in (\[ocz\]).
The first assertion of the following theorem is a variant of Theorem 6.2 from [@1]; see also [@2 Theorem 8.5.7].
\[mp\] Assume that the operators $\mathcal L$ and $\mathcal R$ stabilize in $\Pi^r$. Let $\gamma>0$ and let the line $\Bbb R+i\gamma$ be free of the spectrum of the pencil $\mathfrak A^r$. Then for sufficiently large $T$ the following assertions hold.
[(i)]{} A solution $u\in\mathcal D_{-\gamma}^l (\Pi^r)$ to the problem (\[pp\]) with right-hand side $\{\mathfrak
F,\mathfrak G\}\in \mathcal R^l_\gamma(\Pi^r)\cap \mathcal
R^l_{-\gamma} (\Pi^r)$ admits the representation $$\label{asz}
u=\sum_{j=1}^{M^r} \{a_j(\mathfrak F,\mathfrak
G)\,z_j^++b_j(\mathfrak F,\mathfrak G)\,z_j^-\}+v,$$ where $v$ is a solution to the same problem in $\mathcal
D_\gamma^l(\Pi^r)$, the waves $z_j^\pm$ are defined by (\[3.5\]), and $2 M^r$ is the total algebraic multiplicity of the eigenvalues of the pencil $\mathfrak A^r$ in the strip $\{\lambda\in \Bbb C:|\operatorname{Im}|<\gamma\}$.
[(ii)]{} The functionals $a_1,\dots,a_{M^r}$ and $b_1,\dots,b_{M^r}$ are continuous on $\mathcal
R^l_\gamma(\Pi^r)\cap \mathcal R^l_{-\gamma} (\Pi^r)$ and $$\label{d1}
\begin{split}
a_j (\mathfrak F,\mathfrak G) &=i(\mathfrak F,z^+_j)_{\Pi^r}+i(\mathfrak G,\mathfrak Q^r_T z^+_j)_{\partial\Pi^r},\\
b_j(\mathfrak F,\mathfrak G)&=-i (\mathfrak
F,z^-_j)_{\Pi^r}-i(\mathfrak G,\mathfrak Q^r_T
z^-_j)_{\partial\Pi^r},
\end{split}$$ where $\mathfrak Q^r_T$ is the same as in the Green formula (\[GFg\]).
[**Proof.**]{} (i) Since the spectrum of $\mathfrak A^r$ is symmetric about the real line, the conditions of theorem guaranty that the line $\Bbb R- i\gamma$ is free of the spectrum. From the second assertion of Proposition \[p2.1\] and (\[lim\]) it follows that $$\begin{aligned}
\
& &\|(A_\gamma^r)^{-1}\Delta_T^r;\mathcal
D_\gamma^\ell(\Pi^r)\to\mathcal D_\gamma^\ell(\Pi^r)\|<1,\\
& &\|(A_{-\gamma}^r)^{-1}\Delta_T^r;\mathcal
D_{-\gamma}^\ell(\Pi^r)\to\mathcal D_{-\gamma}^\ell(\Pi^r)\|<1.\end{aligned}$$ Solutions $u\in\mathcal D_{-\gamma}^\ell (\Pi^r)$ and $v\in\mathcal D_\gamma^\ell (\Pi^r)$ to the problem (\[pp\]) satisfy the equations $$u =(A^r_{-\gamma})^{-1}\Delta^r_T
u+(A^r_{-\gamma})^{-1}\{\mathfrak F,\mathfrak G\},\quad v
=(A^r_\gamma)^{-1}\Delta^r_T v+(A^r_\gamma)^{-1}\{\mathfrak
F,\mathfrak G\},
$$ where $\Delta^r_T$ is from (\[Delta\]). Let us solve this equations by the method of successive approximations. We set $$\begin{split}
u_{n+1}&=(A^r_{-\gamma})^{-1}\Delta^r_T u_n+u_0,\quad
u_0=(A^r_{-\gamma})^{-1}\{\mathfrak F,\mathfrak
G\},\\
v_{n+1} &=(A^r_\gamma)^{-1}\Delta^r_T v_n+v_0,\quad
v_0=(A^r_\gamma)^{-1}\{\mathfrak F,\mathfrak G\}.
\end{split}$$ By Proposition \[p2.2\] we have $$\label{3.13}
u_0=i \sum_{j=1}^{M^r} \{a_j(\mathfrak F,\mathfrak G) u_j^+ +b_j
(\mathfrak F,\mathfrak G)u_j^-\}+v_0.$$ Let us write the formulas (\[c1\]) for $a_j(\mathfrak
F,\mathfrak G)$ and $b_j(\mathfrak F,\mathfrak G)$ in the form $$a_j(\mathfrak F,\mathfrak G)=i q^r(v_0,u^+_j), \quad b_j(\mathfrak
F,\mathfrak G)=-i q^r(v_0,u^-_j),$$ where $q^r$ is from (\[2.12\]). We prove by induction that $$\label{3.14}\begin{split}
u_n=v_n+i \sum_{j=1}^{M^r} \sum_{m=0}^n
((A^r_{-\gamma})^{-1}\Delta_T^r)^m\{q^r(v_{n-m},& u^+_j)u_j^+ \\&-
q^r(v_{n-m}, u^-_j)u_j^-\}.
\end{split}$$ If $n=0$ then (\[3.14\]) coincides with (\[3.13\]). We suppose that (\[3.14\]) holds for $n$ and show that it remains valid for $n+1$. From (\[3.14\]) we get $$\label{3.15}
\begin{split}
(&A^r_{-\gamma})^{-1}\Delta_T^r u_n=
(A^r_{-\gamma})^{-1}\Delta_T^r v_n\\&+i\sum_{j=1}^{M^r}
\sum_{m=0}^n ((A^r_{-\gamma})^{-1}\Delta_T^r)^{m+1}\{q^r(v_{n-m},
u^+_j)u_j^+- q^r(v_{n-m}, u^-_j)u_j^-\}.
\end{split}$$ Using Proposition \[p2.2\], we represent $(A^r_{-\gamma})^{-1}\Delta_T^r v_n$ in the form $$\label{3.16}\begin{split}
(&A^r_{-\gamma})^{-1}\Delta_T^r v_n=(A_\gamma^r)^{-1}\Delta^r_T
v_n\\&+i\sum_{j=1}^{M^r}
\bigl\{q^r\bigl((A^r_{-\gamma})^{-1}\Delta_T^r v_{n},
u^+_j\bigr)u_j^+- q^r\bigl((A^r_{\gamma})^{-1}\Delta_T^r v_{n},
u^-_j\bigr)u_j^-\bigr\}.
\end{split}$$ Taking into account (\[3.13\]), (\[3.16\]) and the formulas for $u_{n+1}$ and $v_{n+1}$, we pass from (\[3.15\]) to the equality (\[3.14\]) with $n$ replaced by $n+1$. The formula (\[3.14\]) is proved.
Substituting $v_{n-m}=\sum_{h=0}^{n-m}((A^r_{\gamma})^{-1}\Delta_T^r)^h v_0$ into (\[3.14\]) we obtain $$\label{3.14+}\begin{split}
u_n=v_n+i \sum_{j=1}^{M^r} \sum_{m=0}^n
((A^r_{-\gamma})^{-1}&\Delta_T^r)^m\Bigl\{q^r\bigl(\sum_{h=0}^{n-m}((A^r_{\gamma})^{-1}\Delta_T^r)^h
v_0, u^+_j\bigr)u_j^+ \\&-
q^r\bigl(\sum_{h=0}^{n-m}((A^r_{\gamma})^{-1}\Delta_T^r)^h v_0,
u^-_j\bigr)u_j^-\Bigr\}.
\end{split}$$ The series $ \sum_{h=0}^\infty ((A^r_{\gamma})^{-1}\Delta_T^r)^h
v_0 $ converges in the norm of $\mathcal D_\gamma^\ell(\Pi^r)$, moreover, $\{L^r,B^r\}\sum_{h=0}^\infty((A^r_{\gamma})^{-1}\Delta_T^r)^h
v_0\in \mathcal R_\gamma^\ell(\Pi^r)\cap\mathcal
R_{-\gamma}^\ell(\Pi^r)$. Using the argument given after (\[3.5\]), we justify the passage to the limit in (\[3.14+\]) as $n\to \infty$. As a result we get the representation (\[asz\]), where the functions $u\in\mathcal
D_{-\gamma}^\ell(\Pi^r)$ and $v\in\mathcal D_\gamma^\ell(\Pi^r)$ satisfy the problem (\[pp\]), and $$\begin{aligned}
a_j(\mathfrak F,\mathfrak G)& = &i q^r\bigl(\sum_{h=0}^\infty
((A^r_{\gamma})^{-1}\Delta_T^r)^h v_0, u_j^+), \\b_j(\mathfrak
F,\mathfrak G)&= &-iq^r\bigl(\sum_{h=0}^\infty
((A^r_{\gamma})^{-1}\Delta_T^r)^h v_0, u_j^-).\end{aligned}$$ The assertion (i) is proved.
Let us establish the formulas (\[d1\]). Due to Proposition \[p3.2\] we have $$\label{*}
(\mathfrak F, z_j^\pm)_{\Pi^r}+(\mathfrak G,\mathfrak Q^r_T
z_j^\pm)_{\partial\Pi^r}=p^r_T(u,z_j^\pm),$$ where $u$ is the same as in (\[asz\]). Let $\chi \in
C^\infty(\Bbb R)$, $\chi(t)=1$ for $t\geq 2$ and $\chi(t)=0$ for $t\leq 1$. From (\[mod\]) and $(1-\chi)u_j^\pm \in
D^l_{\gamma}(\Pi^r)$ we obtain $(1-\chi)z_j^\pm\in
D^l_{\gamma}(\Pi^r)$. Then the inclusion $u\in \mathcal
D^l_{\gamma}(\Pi^r)$ implies $p^r_T(u,(1-\chi)z_j^\pm)=0$. Together with (\[asz\]) this allows us to write (\[\*\]) in the form $$\begin{split}
(\mathfrak F,& z_j^\pm)_{\Pi^r}+(\mathfrak G,\mathfrak Q^r_T
z_j^\pm)_{\partial\Pi^r}\\
&= p^r_T(\sum_{h=1}^{M^r} \{a_h(\mathfrak F,\mathfrak
G)\,z_h^++b_h(\mathfrak F,\mathfrak G)\,z_h^-\}+v,\chi z_j^\pm ).
\end{split}$$ Note that $p^r_T(v,\chi z_j^\pm)=0$ as far as $v\in\mathcal
D_{\gamma}^l(\Pi^r)$ and $\chi z_j^\pm\in \mathcal
D_{-\gamma}^l(\Pi^r)$; see (\[mod\]). Finally we have $$\begin{split}
(\mathfrak F,& z_j^\pm)_{\Pi^r}+(\mathfrak G,\mathfrak Q^r_T
z_j^\pm)_{\partial\Pi^r}\\
&= -p^r_T(\sum_{h=1}^{M^r} \{a_h(\mathfrak F,\mathfrak
G)\,z_h^++b_h(\mathfrak F,\mathfrak G)\,z_h^-\},\chi z_j^\pm )\\&=
-p^r_T(\sum_{h=1}^{M^r} \{a_h(\mathfrak F,\mathfrak G)\,\chi
z_h^++b_h(\mathfrak F,\mathfrak G)\,\chi z_h^-\},\chi z_j^\pm).
\end{split}$$ By applying Proposition \[p3.3\], we complete the proof.
Theorem \[mp\] does not allow us to write a structure of $u\in
\mathcal D_\beta^l(\Pi^r)$ with a remainder $v\in \mathcal
D_\gamma^l(\Pi^r)$ if $\beta\ne -\gamma$. Further we correct this trouble.
Let $\alpha_\nu$, $\nu\in\Bbb Z$, be numbers such that every strip $\alpha_\nu\leq\operatorname{Im}\lambda<\operatorname{Im}\lambda_\nu$ is free of the spectrum of $\mathfrak A^r$. For sufficiently large $T$ we set $$\label{w}
w^{(\sigma,j)}_\nu=u^{(\sigma,j)}_\nu+\sum_{q=1}^\infty
((A^r_{\alpha_\nu})^{-1}\Delta^r_T)^q u^{(\sigma,j)}_\nu,$$ where the functions $u^{(\sigma,j)}_\nu$ are given in (\[2.11\]) and satisfy the conditions (\[2.13\])–(\[2.15\]). Repeating the arguments form the proof of Proposition \[p3.2\] one can show that $w^{(\sigma,j)}_\nu$ solves the homogenous model problem (\[pp\]). The functions $w^{(\sigma,j)}_\nu$ do not depend on the choice of $\alpha_\nu$; indeed, $(A^r_{\alpha})^{-1}\{F,G\}=(A^r_{\beta})^{-1}\{F,G\}$ provided that the strip $\alpha\leq\operatorname{Im}\lambda\leq \beta$ is free of the spectrum of the pencil $\mathfrak A^r$ and $\{F,G\}\in\mathcal R^l_\alpha (\Pi^r) \cap \mathcal R^l_\beta
(\Pi^r)$ (see e.g. [@3 Proposition 3.1.4]). From the relations $w^{(\sigma,j)}_\nu = u^{(\sigma,j)}_\nu \mod \mathcal
D_{\alpha_\nu}^l(\Pi^r)$ and the formulas (\[2.11\]) it follows the linear independence of functions $w^{(\sigma,j)}_\nu$.
\[pwz\] Let the assumptions of Theorem \[mp\] be fulfilled and let $\lambda_{-M},\dots,\lambda_M$ be all eigenvalues of $\mathfrak A^r$ from the strip $-\gamma<\operatorname{Im}\lambda<\gamma$. Then for sufficiently large $T$ the relations $$\label{wz}
w^{(\tau,p)}_\mu=\sum_{\nu=1}^M\{a^{(\tau,p)}_{\mu,\nu} z^+_\nu
+b^{(\tau,p)}_{\mu,\nu} z^-_\nu\},\\$$ hold with the coefficients $$\label{ab}
a_{\mu , \nu}^{(\tau, p)}=i p^r_T(\chi w^{(\tau,p)}_\mu,\chi
z_{\nu}^+),\qquad b_{\mu , \nu}^{(\tau, p)}=-i p^r_T(\chi
w^{(\tau,p)}_\mu,\chi z_{\nu}^-),$$ where $\mu=-M,\dots,M$, $p=1,\dots,J_\mu$, and $\tau=0,\dots,\varkappa_{p\mu}-1$; $\chi\in C^\infty (\Bbb R)$, $\chi(t)=1$ for $t>2$ and $\chi(t)=0$ for $t<1$.
[**Proof.**]{} Since $w^{(\tau,p)}_\mu\in
\mathcal D_{\alpha_\mu}^l(\Pi^r)$, where $\gamma>\alpha_\mu\geq
-\gamma$, we have $\chi w^{(\tau,p)}_\mu\in \mathcal
D_{-\gamma}^l(\Pi^r)$ and $(\chi-1)w^{(\tau,p)}_\mu\in \mathcal
D_{\gamma}^l(\Pi^r)$. We put $\{\mathfrak F,\mathfrak G\}
=-\{\mathfrak L^r_T,\mathfrak B^r_T\}\chi w^{(\tau,p)}_\mu$. It is clear that $\{\mathfrak F,\mathfrak G\}\in\mathcal
R_\gamma^l(\Pi^r)\cap\mathcal R_{-\gamma}^l(\Pi^r)$ and $\{\mathfrak L^r_T,\mathfrak B^r_T\}(1-\chi)
w^{(\tau,p)}_\mu=\{\mathfrak F,\mathfrak G\}$. By Proposition \[p3.1\] and Theorem \[mp\] we have $$(1-\chi) w^{(\tau,p)}_\mu=\sum_{\nu=1}^M\{a^{(\tau,p)}_{\mu,\nu}
z^+_\nu +b^{(\tau,p)}_{\mu,\nu} z^-_\nu\}-\chi w^{(\tau,p)}_\mu.$$ This leads to (\[wz\]). The equalities (\[ab\]) are readily apparent from (\[wz\]) and Proposition \[p3.3\].
\[bang-up\] Let $\chi \in C^\infty(\mathbb R)$, $\chi (t)=1$ for $t\geq2$ and $\chi (t)=0$ for $t\leq 1$. The functions $w_\nu^{(\sigma,\nu)}$ given in (\[w\]) satisfy the following conditions: $$\begin{aligned}
\label{w.1}p_T^r(\chi w_\nu^{(\sigma,j)},\chi w_\mu^{(\tau,p)}) =i\delta_{-\nu,\mu}\delta_{j,p}\delta_{\varkappa_{j\nu}-1-\sigma,\tau},\quad |\nu|>\nu_0, |\mu|>\nu_0,\\
\label{w.2} p_T^r(\chi w_\nu^{(\sigma,j)},\chi
w_\mu^{(\tau,p)})=\pm
i\delta_{\nu,\mu}\delta_{j,p}\delta_{\varkappa_{j\nu}-1-\sigma,\tau},\quad |\nu|\leq \nu_0,|\mu|\leq \nu_0,\\
\label{w.3}p_T^r(\chi w_\nu^{(\sigma,j)},\chi w_\mu^{(\tau,p)})
=0,\quad |\nu|\leq \nu_0,|\mu|>\nu_0.\end{aligned}$$ In (\[w.2\]) the sign depends on $\nu$ and $j$ and coincides with the sign in (\[2.14\]). The conditions (\[w.1\]) – (\[w.3\]) do not depend on the choice of $\chi$.
[**Proof.**]{} First we prove that $p_T^r(\chi
w_\nu^{(\sigma,j)},\chi w_\mu^{(\tau,p)})=0$ if $\operatorname
{Im} (\lambda_\nu+\lambda_\mu)\ne 0$.
Let $\operatorname {Im} (\lambda_\nu+\lambda_\mu)> 0$. In this case one can choose $\alpha_\nu$ and $\alpha_\mu$ (see (\[w\])) such that $\alpha_\nu+\alpha_\mu>0$. Then for the functions $u:=\chi
w_\nu^{(\sigma,j)}\in \mathcal D_{\alpha_\nu}^l(\Pi^r)$ and $v:=\chi w_\mu^{(\tau,p)}\in \mathcal D_{\alpha_\mu}^l(\Pi^r)$ the Green formula (\[GFg\]) holds. This implies $p_T^r(\chi
w_\nu^{(\sigma,j)},\chi w_\mu^{(\tau,p)})=0$.
Let us consider the case $\operatorname {Im}
(\lambda_\nu+\lambda_\mu)< 0$. One can choose $\beta_\nu$ and $\beta_\mu$ such that $\beta_\nu>\operatorname {Im}\lambda_\nu$, $\beta_\mu>\operatorname {Im}\lambda_\mu$, and $\beta_\nu+\beta_\mu<0$. Then $(1-\chi) w_\nu^{(\sigma,j)}\in
\mathcal D_{\beta_\nu}^l(\Pi^r)$, $(1-\chi) w_\mu^{(\tau,p)}\in
\mathcal D_{\beta_\mu}^l(\Pi^r)$ and for $u:=(1-\chi)
w_\nu^{(\sigma,j)}$ and $v:=(1-\chi) w_\mu^{(\tau,p)}$ the Green formula (\[GFg\]) holds. This implies $p_T^r((1-\chi)
w_\nu^{(\sigma,j)},(1-\chi) w_\mu^{(\tau,p)})=0$. Since the Green formula (\[GFg\]) holds for $u,v\in C_c^\infty (\overline G)$ and $p^r_T(w_\nu^{(\sigma,j)},w_\mu^{(\tau,p)})=0$, we get $$p_T^r((1-\chi) w_\nu^{(\sigma,j)},(1-\chi)
w_\mu^{(\tau,p)})=p_T^r((1-\chi)
w_\nu^{(\sigma,j)},w_\mu^{(\tau,p)})=-p_T^r(\chi
w_\nu^{(\sigma,j)},\chi w_\mu^{(\tau,p)}).$$ Thus $p_T^r(\chi w_\nu^{(\sigma,j)},\chi w_\mu^{(\tau,p)})=0$ if $\operatorname {Im} (\lambda_\nu+\lambda_\mu)\ne 0$.
Let $\operatorname {Im}
(\lambda_\nu+\lambda_\mu)=0$. Without loss of generality we can assume that $\operatorname{Im}\lambda_\nu\geq 0$. Then $\alpha_\mu<-\operatorname{Im}\lambda_\nu\leq 0$. We set $\{F,G\}=\{L^r, B^r\}(1-\chi)w_\nu^{(\sigma,j)}$. It is clear that $\{L^r, B^r\}(1-\chi)w_\nu^{(\sigma,j)}=\{\mathfrak
L_T^r,\mathfrak B_T^r\}(1-\chi)w_\nu^{(\sigma,j)}\in \mathcal
R^l_{-\alpha_\mu} (\Pi^r)\cap \mathcal R^l_{\alpha_\mu} (\Pi^r)$. Write down the asymptotic of the solution $(1-\chi)w_\nu^{(\sigma,j)}\in \mathcal D_{-\alpha_\mu}^l(\Pi^r)$ to the problem $\{L^r, B^r\}u=\{F,G\}$. We have $$(1-\chi)w_\nu^{(\sigma,j)}=\sum_{h=-M}^M\sum_{s=1}^{J_h}\sum_{\delta=0}^{\varkappa_{sh}-1} c_h^{(\delta,s)}
u_h^{(\delta,s)}\quad \mod\ \mathcal D_{\alpha_\mu}^l(\Pi^r),$$ where $2M$ stands for the total algebraic multiplicity of all eigenvalues of $\mathfrak A^r$ in the strip $-\alpha_\mu>\operatorname{Im}\lambda>\alpha_\mu$; see e.g. [@3 Proposition 3.1.4]. Note that $c_\nu^{(\sigma,j)}=1$ because of the inclusion $(w^{(\sigma,j)}_\nu -
u^{(\sigma,j)}_\nu) \in \mathcal D_{\alpha_\nu}^l(\Pi^r)$. Therefore, $$\begin{aligned}
-p^r_T(\chi w^{(\sigma,j)}_\nu, \chi w^{(\tau,p)}_\mu)=
p^r_T((1-\chi)w^{(\sigma,j)}_\nu,
(1-\chi)w^{(\tau,p)}_\mu)\\=q^r((1-\chi)w^{(\sigma,j)}_\nu,
(1-\chi)w^{(\tau,p)}_\mu)
=q^r(\sum_{h=-M}^M\sum_{s=1}^{J_h}\sum_{\delta=0}^{\varkappa_{sh}-1}
c_h^{(\delta,s)} u_h^{(\delta,s)}, (1-\chi)u^{(\tau,p)}_\mu)\\
=-q^r(\sum_{h=-M}^M\sum_{s=1}^{J_h}\sum_{\delta=0}^{\varkappa_{sh}-1}
c_h^{(\delta,s)} \chi u_h^{(\delta,s)}, \chi u^{(\tau,p)}_\mu)\end{aligned}$$ (in the next-to-last equality we used that $q^r(u,v)=0$ for $u\in\mathcal D^l_{-\alpha_\mu}(\Pi^r)$ and $v\in\mathcal
D^l_{\alpha_\mu}(\Pi^r)$). Taking into account the equality $c_\nu^{(\sigma,j)}=1$ and the relations (\[2.13\])–(\[2.15\]), we complete the proof.
\[tasw\] Assume that $\mathcal L$ and $\mathcal R$ stabilize in $\Pi^r_+$. We also suppose that $\beta>\alpha$ and the lines $\Bbb R+i\alpha$ and $\Bbb R+i\beta$ contain no eigenvalues of the pencil $\mathfrak A^r$. Let $\lambda_K,\dots,\lambda_M$ be all eigenvalues of $\mathfrak A^r$ from the strip $\alpha<\operatorname{Im}\lambda<\beta$ and $\{\mathfrak
F,\mathfrak G\}\in \mathcal R^l_\alpha(\Pi^r)\cap
R^l_\beta(\Pi^r)$. Then for sufficiently large $T$ the following assertions hold.
[(i)]{} A solution $u\in\mathcal D^l_\alpha(\Pi^r)$ to the model problem (\[pp\]) admits the representation $$\label{asw}
u=\sum_{\nu=K}^M\sum_{j=1}^{J_\nu}\sum_{\sigma=0}^{\varkappa_{j\nu}-1}
d_\nu^{(\sigma,j)}(\mathfrak F,\mathfrak G)\,w_\nu^{(\sigma,j)}+v,$$ where $v$ is a solution to the same problem in $\mathcal
D^l_\beta(\Pi^r)$.
[(ii)]{} The coefficients $d_\nu^{(\sigma,j)}(\mathfrak F,\mathfrak G)$ in (\[asw\]) can be found by the formulas $$\label{wd1}
\begin{split}
d_\nu^{(\sigma,j)}(\mathfrak F,\mathfrak G) =i\{(\mathfrak
F,&w_{-\nu}^{(\varkappa_{j\nu}-\sigma-1,j)})_{\Pi^r}\\&+(\mathfrak
G,\mathfrak Q_T^r
w_{-\nu}^{(\varkappa_{j\nu}-\sigma-1,j)})_{\partial\Pi^r}\},\quad
|\nu|>\nu_0,
\end{split}$$ and $$\label{wd2}
\begin{split}
d_\nu^{(\sigma,j)}(\mathfrak F,\mathfrak G)=\pm i\{
(\mathfrak F,&
w_{\nu}^{(\varkappa_{j\nu}-\sigma-1,j)})_{\Pi^r}\\&+ (\mathfrak
G,\mathfrak Q_T^r
w_{\nu}^{(\varkappa_{j\nu}-\sigma-1,j)})_{\partial\Pi^r}\},\quad
|\nu|\leq\nu_0.
\end{split}$$ The sign in (\[wd2\]) is the same as in (\[2.14\]).
[**Proof.**]{} Let $\gamma=\max\{-\alpha, \beta\}$. We again use the cut-off function $\chi\in C^c_\infty (\overline
G)$, $\chi(t)=1$ for $t\geq 2$ and $\chi(t)=0$ for $t\leq 1$.
Let us first consider the case $\gamma=-\alpha$. Let $v\in\mathcal
D_{\beta}^l(\Pi^r)$ satisfy the model problem (\[pp\]). We set $\{F,G\}:=\{\mathfrak L^r_T,\mathfrak B^r_T\}(1-\chi)v\in \mathcal
R^l_{-\gamma}(\Pi^r)\cap \mathcal R^l_{\gamma}(\Pi^r)$. By Theorem \[mp\] we have $$\label{111}
y=\sum_{j=1}^{-K}\{a_j z^+_j+b_j z^-_j\}+(1-\chi)v,$$ where $y\in \mathcal D_{-\gamma}^l (\Pi^r)$. Due to the linear independence of $w^{(\sigma,j)}_\nu$ and (\[wz\]) the waves $z_\nu^\pm$ can be expressed in terms of $w^{(\sigma,j)}_\nu$. From (\[111\]) we get $$\label{112}
u=y+\chi v=\sum_{j=1}^{-K}\{a_j z^+_j+b_j
z^-_j\}+v=\sum_{\nu=K}^{-K}\sum_{j=1}^{J_\nu}\sum_{\sigma=0}^{\varkappa_{j\nu}-1}
d_\nu^{(\sigma,j)}( F, G)\,w_\nu^{(\sigma,j)}+v,$$ where $u=(\chi v+ y)\in \mathcal D^l_{\alpha}(\Pi^r)$. Since $(1-\chi)(u-v)\in\mathcal D^l_{\beta}(\Pi^r)$ and the function $(1-\chi)w_\nu^{(\sigma,j)}$ is in $\mathcal D^l_{\beta}(\Pi^r)$ only if $ \nu\geq K$, we have $d_\nu^{(\sigma,j)}( F, G)=0$ for $\nu=M+1,\dots, -K$. In the case $-\alpha\geq\beta$ the representation (\[asw\]) is proved.
Consider the case $-\alpha<\beta$. Then $\gamma=\beta$. Applying Theorem \[mp\], we get the representation $$\chi u=\sum_{j=1}^{M}\{a_j z^+_j+b_j z^-_j\}+y, \quad
y\in\mathcal D^l_{\gamma}(\Pi^r),$$ for the solution $\chi u\in \mathcal D^l_{-\gamma}(\Pi^r)$ to the model problem (\[pp\]) with right-hand side $\{F,
G\}:=\{\mathfrak L^r_T,\mathfrak B^r_T\}\chi u\in \mathcal
R^l_{-\gamma}(\Pi^r)\cap \mathcal R^l_{\gamma}(\Pi^r)$. Therefore, $$u=\sum_{\nu=-M}^{M}\sum_{j=1}^{J_\nu}\sum_{\sigma=0}^{\varkappa_{j\nu}-1}
d_\nu^{(\sigma,j)}( F, G)\,w_\nu^{(\sigma,j)}+v,$$ where $v=(y+(1-\chi)u)\in \mathcal D^l_{\beta}(\Pi^r)$. Owing to the inclusion $\chi (u-v)\in \mathcal D^l_{\alpha}(\Pi^r)$, we have $d_\nu^{(\sigma,j)}( F, G)=0$ for $\nu=-M,\dots,K-1$. The assertion (i) of the theorem is proved.
Using Proposition \[bang-up\] and the representation (\[asw\]) one can prove the formulas (\[wd1\]) and (\[wd2\]) for the coefficients; see the proof of (\[d1\]) in Theorem \[mp\].
The structure of solutions to the problem (\[1\])
-------------------------------------------------
Let us define the spaces $\mathcal D_\gamma^\ell(G)$ and $\mathcal
R_\gamma^\ell(G)$ by the equalities (\[sp\]) with $\Pi^r$ replaced by $G$; the space $W_\gamma(G)$ is endowed with the norm $\|e_\gamma\cdot;H^\ell(G)\|$, where $e_\gamma$ is smooth positive function in $\overline G$ such that $e_\gamma(y^r,t^r)=\exp\gamma
t^r$ for $(y^r,t^r)\in\bar\Pi^r$.
Assume that the operators $\mathcal L(x,D_x)$ and $\mathcal
R(x,D_x)$ stabilize in $\Pi_+^1,\dots,\Pi_+^N$. As was shown the stabilization in $\Pi_+^r$ implies (\[lim\]). Thus the operator $$\label{o1}
\mathcal A(\gamma)=\{\mathcal L,\mathcal B\}:\mathcal
D_\gamma^\ell(G)\to\mathcal R_\gamma^\ell(G)$$ of the problem (\[1\]) is continuous.
Proposition \[p3.1\] and the well known results of the local theory of elliptic boundary value problems enable one to prove the following proposition in the standard way. The proof is omitted.
\[pF\] Let the operators $\mathcal L(x,D_x)$ and $\mathcal R(x,D_x)$ stabilize in $\Pi_+^1,\dots,\Pi_+^N$ and let the line $\Bbb R+
i\gamma$ be free of the spectrum of the pencils $\mathfrak
A^1,\dots,\mathfrak A^N$. Assume that $T$ is sufficiently large. Then the operator (\[o1\]) of the problem (\[1\]) is Fredholm.
Let us introduce the space of waves $\mathcal W_\gamma (G)$. Suppose that the assumptions of Proposition (\[pF\]) are fulfilled. We extend the functions $\chi z^\pm_j$, $j=1,\dots,M^r$, from the semicylinder $\Pi^r_+$ to the domain $G$ by zero and set $$\label{wv}
v^\pm_h:=\chi z^\pm_j ,\quad h=j+\sum_{p=1}^{r-1}M^p, \
j=1,\dots,M^r,\ r=1,\dots,N;$$ here $z^\pm_j$ are defined by (\[3.5\]), the cut-off function $\chi$ is the same as in Proposition \[p3.3\]. Let $\mathcal W_\gamma (G)$ be the space spanned by functions of the form $v^\pm_h+v$, where $v$ is a function in $\mathcal
D^\ell_{\gamma}(G)$, and $h=1,\dots,M$, $M=\sum_{r=1}^N M^r$. It is clear that $\mathcal W_\gamma (G)\subset \mathcal
D_{-\gamma}^\ell (\Pi^r)$. Note that the elements of $\mathcal
W_\gamma(G)$ do not necessary satisfy the homogeneous problem (\[1\]).
Denote $$\label{q}
q(u,v)=(\mathcal Lu,v)_G+(\mathcal B u, \mathcal Q v)_{\partial
G}- (u,\mathcal L v)_G-(\mathcal Q u, \mathcal B v)_{\partial G}.$$ The quantity $iq ( u, u)$ represents the total energy flow transferred by the wave $u\in\mathcal W_\gamma(G)$ through the infinitely distant cross-sections $\Omega^1, \dots, \Omega^N$ of the cylindrical ends of the domain $G$. It is easy to see that $iq
( u, u)=0$ for an exponentially decreasing function $u\in \mathcal
D_\varepsilon^l (G)$, $\varepsilon>0$.
\[pocv\] Under the circumstances of Proposition \[pF\] the waves $ v^\pm_j$, $j=1,\dots,M$, given by (\[wv\]) satisfy the conditions $$\label{ocv}
q(v^\pm_h,v^\pm_j)=\mp i \delta_{h,j},\quad q(v^\pm_h,v^\mp_j)=0,
\quad j,h=1,\dots,M.$$ Thus $v_1^+,\dots,v_{M}^+$ are incoming waves and $v_1^-,\dots,
v_{M}^-$ are outgoing waves for the problem (\[1\]).
[**Proof.**]{} By Proposition \[p3.3\] the conditions (\[ocz\]) are valid. Due to the Green formula (\[GFg\]) we can replace in (\[ocz\]) the cut-off function $\chi$ by a cut-off function $\zeta_T\in C^\infty (\Bbb R)$, $\zeta_T(t)=1$ for $t>3T$ and $\zeta_T(t)=0$ for $t<2T$. Recall that the operator $\{\mathcal L,\mathcal B\}$ coincides with $\{\mathfrak
L^r_T,\mathfrak B^r_T\}$ on the set $\{(y^r,t^r)\in\overline\Pi^r:t^r>T+3\}$; see section \[sGFg\]. If $v^\pm_j$ and $v^\pm_h$ are related to different semicylinders $\Pi^r_+$ and $\Pi^s_+$ then those supports do not overlap. We have $$q(\zeta_T v^\pm_h,\zeta_T v^\pm_j)=\mp i \delta_{h,j},\quad
q(\zeta_T v^\pm_h,\zeta_T v^\mp_j)=0, \quad j,h=1,\dots,M.$$ Owing to the Green formula (\[GF\]) the cut-off function $\zeta_T$ can be omitted.
\[pasv\] Let $\mathcal L(x,D_x)$ and $\mathcal R(x,D_x)$ stabilize in $\Pi_+^1,\dots,\Pi_+^N$ and let the line $\Bbb R+
i\gamma$ be free of the spectrum of the pencils $\mathfrak
A^1,\dots,\mathfrak A^N$. Then for a solution $u\in \mathcal
D^\ell_{-\gamma}(G)$ to the problem (\[1\]) with right-hand side $\{\mathcal F,\mathcal G\}\in \mathcal R_\gamma^\ell (G)$ the inclusion $$\label{asv}
u-\sum_{j=1}^M\{a_j v_j^++b_j v_j^-\}\in\mathcal
D^\ell_{\gamma}(G)$$ holds. Here $$\label{cv}
a_j=iq(u,v^+_j), \quad b_j=-iq(u,v^-_j), \quad j=1,\dots, M,$$ the waves $v^\pm_j$ are defined by (\[wv\]) and (\[3.5\]), where $T$ is sufficiently large.
[**Proof.**]{} Let $\zeta_T\in C^\infty(\overline G)$, $\zeta_T(t)=1$ for $t>3T$ and $\zeta_T(t)=0$ for $t<2T$. Denote by $\zeta_T^r$ the cut-off function such that $\zeta_T^r$ coincides with $\zeta_T$ inside $\overline\Pi^r_+$ and vanishes on the remaining part of $\overline G$. Theorem \[mp\] implies the representations of the form (\[asz\]) for the solutions $\zeta^r_T u\in \mathcal D_{-\gamma}^l(\Pi^r)$ to the problems (\[pp\]) with the right-hand sides $\{\mathfrak F^r, \mathfrak
G^r\}:=\{\mathfrak L^r_T,\mathfrak B^r_T\}\zeta^r_T u$, $r=1,\dots, N$. To prove (\[asv\]) it remains to note that $\zeta_T\chi=\zeta_T$ and $(1-\sum_{r=1}^N\zeta^r_T)u\in \mathcal
D^\ell_\gamma (G)$.
The equalities (\[cv\]) directly follow from (\[asv\]) and Lemma \[pocv\].
\[taswG\] Assume that $\mathcal L$ and $\mathcal R$ stabilize in $\Pi^r_+$. We also suppose that $\beta>\alpha$ and the lines $\Bbb R+i\alpha$ and $\Bbb R+i\beta$ contain no eigenvalues of the pencil $\mathfrak A^r$. Let $\lambda_K,\dots,\lambda_M$ be all eigenvalues of $\mathfrak A^r$ from the strip $\alpha<\operatorname{Im}\lambda<\beta$ and let $\eta \{\mathfrak
F,\mathfrak G\}\in \mathcal R^l_\beta(G)$, where $\eta\in C^\infty
(\overline G)$, $\operatorname{supp} \eta\in \overline\Pi^r_+$ and $\eta=1$ on the set $\{(y,t)\in\overline\Pi^r_+, t>3\}$. If $u$ is a solution to the problem (\[1\]) such that $\eta u \in \mathcal
D_\alpha^l(G)$ then inside $\Pi^r_+$ the representation $$\label{str}
u=\sum_{\nu=K}^M\sum_{j=1}^{J_\nu}\sum_{\sigma=0}^{\varkappa_{j\nu}-1}
c_\nu^{(\sigma,j)}(\mathfrak F,\mathfrak G)\,w_\nu^{(\sigma,j)} +v$$ holds, where $\eta v\in \mathcal D_\beta^l(G)$ and $$c_\nu^{(\sigma,j)}(\mathfrak F,\mathfrak G) =i q(u, \eta
w_{-\nu}^{(\varkappa_{j\nu}-\sigma-1,j)}),\quad |\nu|>\nu_0,$$ $$c_\nu^{(\sigma,j)}(\mathfrak F,\mathfrak G)=\pm i q(u, \eta
w_{\nu}^{(\varkappa_{j\nu}-\sigma-1,j)}),\quad |\nu|\leq\nu_0.$$ The sign in the last formula is the same as in (\[2.14\]).
[**Proof.**]{} The assertion follows from the item (i) of Theorem \[tasw\] and Proposition \[bang-up\].
Corollaries of Theorems \[pasv\] and \[taswG\]
==============================================
Index properties, Scattering matrices, An existence criterion of exponentially decaying solutions
-------------------------------------------------------------------------------------------------
\[cl1\] Let the assumptions of Theorem \[taswG\] be fulfilled. Then the indexes of operators ${\mathcal A}(\alpha)$ and ${\mathcal A}(\beta)$ are connected by the relation $$\operatorname{Ind}{\mathcal A}(\alpha)-\operatorname{Ind}{\mathcal A}(\beta)=\varkappa,$$ where $\varkappa$ is the total algebraic multiplicity of all eigenvalues of the pencils ${\mathfrak A}^1,\ldots,{\mathfrak
A}^N$ in the strip $\{\lambda\in{\mathbb C}:
\alpha<\operatorname{Im}\lambda<\beta\}$.
The assertion of this proposition follows from the structure (\[str\]) of solution to the problem (\[1\]); see [@3 Section 4.3]. Using Proposition \[cl1\] and the formal self-adjointness of $\{\mathcal L,\mathcal B\}$ one can prove the following proposition; see [@3 Section 5.1.3].
\[cl2\] Let the assumptions of Theorem \[pasv\] be fulfilled. Then $$\operatorname{dim}\operatorname{ker}{\mathcal
A}(-\gamma)-\operatorname{dim}\operatorname{ker}{\mathcal
A}(\gamma) =\operatorname{dim}\operatorname{coker}{\mathcal
A}(\gamma)-\operatorname{dim} \operatorname{coker}{\mathcal
A}(-\gamma)=M,$$ where $2M$ is the total algebraic multiplicity of all eigenvalues of the pencils ${\mathfrak A}^1,\ldots,{\mathfrak A}^N$ in the strip $\{\lambda\in{\mathbb C}:
|\operatorname{Im}\lambda|<\gamma\}$.
The next proposition is a corollary of the formulas (\[cv\]) for the coefficients in the structure (\[asv\]) of solution; see [@3 Propositions 5.3.3, 5.3.4].
\[cl5\] Let the assumptions of Theorem \[pasv\] be fulfilled. Then there exist bases $Z_1,\ldots,Z_M$ and $X_1,\ldots,X_M$ in the space $\operatorname{ker} {\mathcal
A}(-\gamma)$ modulo ${\mathcal
D}_\gamma^\ell W(G)$ such that $$\begin{aligned}
Z_k-\biggl(v_k^++\sum_{j=1}^M {\mathfrak T}_{kj} v_j^-\biggr)\in
{\mathcal D}_\gamma^\ell W(G),\quad k=1,\ldots,M,\label{b1}\\
X_k-\biggl(v_k^-+\sum_{j=1}^M {\mathfrak S}_{kj} v_j^+\biggr)\in
{\mathcal D}_\gamma^\ell W(G),\quad k=1,\ldots,M,\label{b2}\end{aligned}$$ where the scattering matrices ${\mathfrak T}\equiv\|{\mathfrak
T}_{kj}\|$ and ${\mathfrak S}\equiv\|{\mathfrak S}_{kj}\|$ of sizes $M\times M$ are unitary, i.e. ${\mathfrak T}^*={\mathfrak
T}^{-1}$ and ${\mathfrak S}^*={\mathfrak S}^{-1}$; moreover, ${\mathfrak S}={\mathfrak T}^{-1}$.
Before formulating an existence criterion of exponentially decaying solutions to the homogeneous problem (\[1\]) we need to construct a special basis $\{v^\pm_j\}_{j=1}^{M'}$ modulo $\mathcal D_\beta^l (G)$ in the space of waves $\mathcal
W_\beta(G)$, $\beta>\gamma$.
\[lm\] Let $0<\gamma<\beta$, and let $\{v^\pm_j\}_{j=1}^M$ be a basis in the space of waves $\mathcal W_{\gamma}(G)$ modulo $\mathcal D_{\gamma}^l(G)$ subjected to (\[ocv\]). The set $\{v^\pm_j\}_{j=1}^M$ can be supplemented to a basis $\{v^\pm_j\}_{j=1}^{M'}$ in $\mathcal W_{\beta}(G)$ modulo $\mathcal D_{\beta}^l(G)$ so that $v_s^++v_s^-\in {\mathcal
D}_\gamma^l W(G)$ for $s=M+1,\ldots,M'$ and the relations [(\[ocv\])]{} hold for $h,j=1,\ldots,M'$.
[**Proof.**]{} In fact the waves $v^\pm_s$, $s=M+1,\dots,
M'$ can be constructed in the same way as the waves $u^\pm_s$ (see e.g. [@11]), one has to use the functions (\[w\]) instead of functions (\[2.11\]).
For simplicity of description we suppose that domain $G$ has only one cylindrical end $\Pi^1_+$. Assume that the Jordan chains of the pencil $\mathfrak A^1$ are chosen such that the conditions (\[2.13\])–(\[2.15\]) for the functions (\[2.11\]) are valid. To the every Jordan chain $\{\varphi_\nu^{(0,j)},\dots,\varphi_\nu^{(\varkappa_{j\nu}-1,j)}\}$ there correspond the functions $w_\nu^{(0,j)},\dots,w_\nu^{(\varkappa_{j\nu}-1,j)}$ given by (\[w\]). By Proposition \[bang-up\] the conditions (\[w.1\])–(\[w.3\]) are valid. With every eigenvalue $\lambda_\nu$ of $\mathfrak A^1$ such that $\gamma<\operatorname{Im}\lambda_\nu<\beta$ we associate the functions $$\label{002}
w_{\nu,\pm}^{(\sigma,j)}=2^{-1/2}\chi (w_\nu^{(\sigma,j)}\mp
w_{-\nu}^{(\varkappa_{j\nu}-\sigma-1,j)}),\quad j=1,\dots, J_\nu,
\tau=0,1,\dots,\varkappa_{j\nu}-1,$$ where $J_\nu=\dim \ker \mathfrak A^1(\lambda_\nu)$, $\chi$ is the same as in (\[wv\]). Then owing to (\[w.1\]) and (\[w.3\]) we have $p^1_T(w_{\nu,\pm}^{(\sigma,j)},w_{\mu,\pm}^{(\tau,p)})=\mp
i\delta_{\nu,\mu}\delta_{\sigma,\tau}\delta_{j,p}$; the wave $w_{\nu,+}^{(\sigma,j)}$ is incoming and the wave $w_{\nu,-}^{(\sigma,j)}$ is outgoing. Due to the linear independence of the functions $w_\nu^{(\sigma,j)}$ and Lemma \[pwz\], the elements of the basis $\{v^\pm_j\}_{j=1}^M$ can be expressed in terms of functions $\chi w_\nu^{(\sigma,j)}$ corresponding to the eigenvalues of $\mathfrak A^1$ in the strip $\{\lambda\in\Bbb C: |\operatorname{Im}\lambda|<\gamma\}$. Together with (\[w.3\]) this implies $p^1_T(w_{\nu,\pm}^{(\sigma,j)}, v^\pm_s)=0$ for $s=1,\dots,M$. It remains to note that $w_{\nu,+}^{(\sigma,j)}+w_{\nu,-}^{(\sigma,j)}=2\chi
w_\nu^{(\sigma,j)}\in \mathcal D_\gamma^l(G)$. As $v^\pm_j$, $j=M+1,\dots,M'$, we can take the waves $w_{\nu,\pm}^{(\sigma,j)}$.
\[crit\] Let $0<\gamma<\beta$, and let the lines ${\mathbb R}+i\gamma$ and ${\mathbb R}+i\beta$ be free of the spectrum of the pencils ${\mathfrak A}^1,\ldots,{\mathfrak A}^N$. Denote by $\mathfrak
S=\mathfrak S(\beta)$ the scattering matrix corresponding to the basis $\{v_j^\pm\}_{j=1}^{M'}$ from Lemma \[lm\]. Then $$\operatorname{dim} \operatorname{ker} {\mathcal
A}(\gamma)-\operatorname{dim}\operatorname{ ker}{\mathcal
A}(\beta)=\operatorname{dim} \operatorname{ker}({\mathfrak
S}^{2,2}-I),$$ where ${\mathfrak
S}^{2,2}$ is $(M'-M)\times (M'-M)$-block of the $M'\times M'$-matrix $\mathfrak S=\|{
S}^{k,\ell}(\gamma')\|_{k,\ell=1,2}$.
The proof is similar to the proof of Theorem 3.3 from [@11].
Problem with radiation conditions
---------------------------------
As before we suppose that $\mathcal L(x,D_x)$ and $\mathcal
R(x,D_x)$ are stabilizing in $\Pi_+^1,\dots,\Pi_+^N$ and the line $\Bbb R+ i\gamma$ is free of the spectrum of the pencils $\mathfrak A^1,\dots,\mathfrak A^N$. Denote by $\mathcal
W_{out}(G)$ the linear span of the outgoing waves $v^-_1,\dots,v^-_M$ and consider the restriction $A$ of $\mathcal
A(-\gamma)$ to the space $\mathfrak D_{out}(G)=\mathcal
W_{out}(G)[\dot{+}]\mathcal D_\gamma^l(G)$, where by $[\dot{+}]$ we denote the orthogonal with respect to the form (\[q\]) direct sum. The mapping $A:\mathfrak D_{out}(G)\to\mathcal R_\gamma^l(G)$ is continuous.
\[cl3\] Let $z_1, \dots, z_d$ be a basis of $\ker \mathcal A(\gamma)$, and let $\{f,g\}\in\mathcal R_\gamma^l(G)$, $(f,z_j)_G+(g,\mathcal Q
z_j)_{\partial G}=0$, $j=1,\dots,d$.
[(i)]{} There exists a unique up to an arbitrary element of $\operatorname{ker}{\mathcal A}(\gamma)$ solution $u\in\mathfrak D_{out}(G)$ to the problem (\[1\]).
[(ii)]{} The inclusion $$v\equiv u-b_1v^-_1-b_2v^-_2-\ldots-b_M v^-_M\in{\mathcal
D}_\gamma^\ell W(G)$$ holds with the coefficients $$b_j=-i(f,X_j)_G-i(g,{\mathcal Q} X_j)_{\partial G}, \quad
j=1,\dots, M,$$ where $X_1,\dots,X_M$ are elements of $\ker\mathcal A(-\gamma)$ subjected to (\[b2\]).
[(iii)]{} The solution $u$ satisfies the inequality $$\label{e1}
\begin{split}
\|v;{\mathcal D}_\gamma^\ell W(G)\|&+|b_1|+|b_2|+\ldots+|b_M|
\\&\leqslant C(\|\{f,g\};{\mathcal R}^\ell_\gamma W(G)\|+\|e_\gamma
v;L_2(G)\|).
\end{split}$$
[4.]{} The solution $u$ subjected to the additional conditions $(u,z_j)_G=0$, $j=1,\ldots,d$, is unique and satisfies the estimate [(\[e1\])]{} with the right-hand side replaced by $\|\{f,g\}; {\mathcal R}^\ell_\gamma W(G)\|$.
This proposition justifies the statement of the problem (\[1\]) with intrinsic radiation conditions [(]{}only outgoing [“]{}waves[”]{} occur in asymptotic formulas for solutions[)]{}. Up to obvious changes the proof repeats the proof of Theorem 5.3.5 from [@3]. The next two propositions describe the statement of the problem with other radiation conditions. For the proofs we refer to [@3 Theorems 5.5.5, 5.5.6].
\[cl4\] Let $\eta_1,\ldots,\eta_{d}$ be a basis of $\operatorname{ker}
{\mathcal A}(\gamma)$, and let the right-hand side $\{f,g\}\in
{\mathcal R}_\gamma^\ell W(G)$ satisfy the orthogonality conditions $(f,\eta_j)_G+(g, {\mathcal Q} \eta_j)_{\partial G}=0$, $j=1,\ldots,d$. We assume that for the space $\operatorname{ker}
{\mathcal A}(-\gamma)$ one can choose a basis $V_1,\ldots,V_M$ modulo ${\mathcal D}_\gamma^l W(G)$ that compatible with the basis $u_1,\ldots,u_{2M}$ for the quotient space ${\mathcal
W}_\gamma(G)/{\mathcal D}_\gamma^\ell W(G)$ in the following sense[:]{} $$q(u_j,V_k)=-i\delta_{kj}, \quad k,j=1,\ldots,M. \label{ccc}$$ Then the following assertions hold.
[1.]{} There exists a unique up to an arbitrary element of $\operatorname{ker}{\mathcal A}(\gamma)$ solution $u\in {\mathfrak
h[\dot +}]{\mathcal D}_\gamma^\ell W(G)$ to the problem [(\[1\])]{}, where $ {\mathfrak h}$ is the linear span of the functions $u_1, \ldots,u_M$ .
[2.]{} The following inclusion holds[:]{} $$v\equiv u-b_1u_1-b_2u_2-\ldots-b_Mu_M\in{\mathcal D}_\gamma^\ell
W(G),$$ where $b_j=i(f,V_j)_G+i(g,{\mathcal Q} V_j)_{\partial G}$, $j=1,\ldots, M$.
[3.]{} The solution $u$ satisfies the inequality (\[e1\]).
[4.]{} The solution $u$ subjectd to the additional conditions $(u,\eta_j)_G=0$, $j=1,\ldots,d$, is unique and satisfies the estimate [(\[e1\])]{} with the right-hand side replaced by $\|\{f,g\}; {\mathcal R}^\ell_\gamma W(G)\|$.
By Proposition \[cl4\], to enumerate all possible radiation conditions is the same that to enumerate all the bases $u_1,\ldots,u_{2M}$ for the quotient space ${\mathcal
W}_\gamma(G)/ {\mathcal D}_\gamma^\ell W(G)$ and bases $V_1,\ldots,V_M$ modulo ${\mathcal D}_\gamma^\ell W(G)$ for the subspace $\operatorname{ker}{\mathcal A}_*(-\gamma)$ compatible in the sense of (\[ccc\]).
\[cl6\] Let ${\mathcal W}_\gamma(G)$ be the space of waves and let the waves $v^\pm_j$, $j=1,\dots,M$, form a basis of the quotient space ${\mathcal W}_\gamma(G)/{\mathcal D}_\gamma^\ell W(G)$ subjected to (\[ocv\]). Denote by $\{X_1,\ldots,X_M\}$ a set of solutions to the homogeneous problem $\{{\mathcal L},{\mathcal B}\}u=0$ satisfying the inclusions [(\[b2\])]{}. Then the following assertions hold.
[1.]{} If $R$ is arbitrary and $S$ is an invertible operator in ${\mathbb C}^M$, then $$\begin{aligned}
{}&V_k=\sum_{m=1}^M \overline{(S^{-1})_{mk}} X_m,\\
&u_j=\sum_{m=1}^M \biggl(S_{jm}u_m^-+\sum_{p=1}^M
R_{jp}\biggl\{u_p^++\sum_{i=1}^M{\mathfrak S}^*_{pi}u_i^-
\biggr\}\biggr),
\end{aligned}
\label{++++}$$ where $j,k=1,\ldots,M$, satisfy the condition [(\[ccc\])]{}.
[2.]{} If a basis $V_1,\ldots,V_M$ modulo ${\mathcal
D}_\gamma^\ell W(G)$ for the space $\operatorname{ker}{\mathcal
A}(-\gamma)$ and a basis $u_1,\ldots,u_{2M}$ for the quotient space ${\mathcal W}_\gamma(G)/{\mathcal D}_\gamma^\ell W(G)$ satisfy [(\[ccc\])]{}, then there exist operators $R$ and $S$ such that the equalities [(\[++++\])]{} hold.
The extensions of the symmetric operator {#sss}
----------------------------------------
The schemes for the proofs of propositions listed in this section can be found in [@3 Section 5.5], the changes in the proofs consist in usage of Theorem \[pasv\] instead of asymptotic representations.
Here we assume that the elliptic system $\{\mathcal L,\mathcal
B\}$ is homogeneous. In other words, $\tau_1=\tau_2=\ldots=\tau_k\equiv \tau$ and $ {\mathcal
D}_\gamma^\ell W(G):=\prod_{i=1}^k W_\gamma^{2\tau}(G)$. With the problem ([2.1]{}) we associate an operator ${\mathcal M}$ with the domain $${\mathcal D}({\mathcal M})=\{u\in{\mathcal D}_\gamma^\ell W(G):
{\mathcal B}(x,D_x)u(x)=0, x\in\partial G\}$$ that acts in the Hilbert space $$L_2(G;e_{-\gamma})\equiv \prod_{i=1}^k W_{-\gamma}^0 (G)$$ by the formula $$({\mathcal M} u)(x)=e_\gamma(x)^2{\mathcal L}(x,D_x)u(x).$$ We denote by $(\cdot,\cdot)_{-\gamma}$ the inner product $$(u,v)_{-\gamma}=\int\limits_G e_{-\gamma}(x)^2
u(x)\overline{v(x)}\, dx$$ in the space $L_2(G;e_{-\gamma})$.
Suppose that $\mathcal L(x,D_x)$ and $\mathcal R(x,D_x)$ are stabilizing and the line ${\mathbb R}+i\gamma$ is free of the spectrum of the pencils $ {\mathfrak A}^1,\ldots,{\mathfrak A}^N$. Then the operator ${\mathcal M}$ is closed and symmetric. For the $ \operatorname{coker} {{\mathcal M}}$ one can choose a basis $X_1,\ldots,X_M$ modulo ${\mathcal D}_\gamma^l W(G)$ such that the inclusions [(\[b2\])]{} hold.
Let $\mathcal L(x,D_x)$ and $\mathcal R(x,D_x)$ stabilize and let the line ${\mathbb R}+i\gamma$ be free of the spectrum of the pencils $ {\mathfrak A}^1,\ldots,{\mathfrak A}^N$. The operator ${\mathcal M}^*$, adjoint to ${\mathcal M}$ in the space $L_2(G;e_{- \gamma})$, is defined on the set $${\mathcal D}({\mathcal M}^*)=\{v\in{\mathcal W}_\gamma(G):
{\mathcal B}(x,D_x)v(x)=0,x\in\partial G\} \label{5.5}$$ and acts by the formula $({\mathcal M}^*
u)(x)=e_\gamma(x)^2{\mathcal L}(x,D_x)u(x)$.
We extend the operator ${\mathcal M}$ by adjoining representatives of waves to the domain. Let $u\in\mathcal D({\mathcal M}^*)$. Then $u$ is a solution of the problem (\[1\]) with the right-hand side $\{f,g\}=\{\mathcal
L u, 0\}\in L_2(G;e_\gamma)$. In accordance with propositions \[cl4\] and \[cl5\] the function $u$ has the form $$u=u^0+\sum_{k=1}^d c_k \eta_k+\sum_{j=1}^M d_j X_j,$$ where $u^0$ is such that $u^0-b_1 v^-_1-\cdots-b_M v^-_M\in
\mathcal D^l_\gamma (G)$. With every function $u\in\mathcal
D({\mathcal M}^*)$ we associate vectors $d=(d_1,\dots,d_M)$ and $b=(b_1,\dots,b_M)$ of the coefficients.
Suppose that $\mathcal L(x,D_x)$ and $\mathcal R(x,D_x)$ are stabilizing and the line ${\mathbb R}+i\gamma$ is free of the spectrum of the pencils $ {\mathfrak A}^1,\ldots,{\mathfrak A}^N$. Let $\Bbb C^M=\Bbb N_+(+)\Bbb N_0(+)\Bbb N_-$ be an orthogonal sum of subspaces and let $K$ be a self-adjoint operator in $\Bbb N_0$. The operator $\Bbb M$ is a self-adjoint extension of $\mathcal M$ if and only if $\Bbb M$ is defined on the set $$\begin{split}
\mathcal D(\Bbb M)=\{&u\in\mathcal D (\mathcal M^*):
c=\zeta_++\zeta_0,\\&
a=\zeta_--(iK+I)\zeta_0/2-\zeta_+/2,\zeta_\pm\in\Bbb N_\pm,
\zeta_0\in\Bbb N_0\}
\end{split}$$ and acts by the formula $({\Bbb M} u)(x)=e_\gamma(x)^2{\mathcal L}(x,D_x)u(x)$.
[10]{}
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Ja.A. Roĭtberg, Z.G. Sheftel’, “ A theorem of homeomorphisms for elliptic systems and its applications,” Mat. Sb. Vol. [78]{} (1969), 446–472; Math. USSR-Sb. Vol. [7]{} (1969), 439–465.
J.L. Lions, E. Magenes, *Non-Homogeneous Boundary Value Problems and Applications I,* Springer-Verlag, Berlin Heidelberg New York, 1972.
N. Aronszajn, A. N. Milgram, “ Differential operators on Riemannian manifolds”, Rend. Circ. Mat. Palermo (2) 2 (1953), 266–325. M. Schechter, “ General boundary value problems for elliptic partial differential equations,” Comm. Pure Appl. Math. 12 (1959), 457–486. Ju. M. Berezanskii, S.G. Krein, and Ja. A. Roitberg, “ A theorem on homeomorphisms and local increase of smoothness up to the boundery for solutions of elliptic equations,” Dokl. Akad. Nauk SSSR 148 (1963), 745–748. English translation: Soviet Math. Dokl. 4 (1963), 152–155.
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---
abstract: 'The Painlevé transcendents discovered at the turn of the XX century by pure mathematical reasoning, have later made their surprising appearance – much in the way of Wigner’s “miracle of appropriateness” – in various problems of theoretical physics. The notable examples include the two-dimensional Ising model, one-dimensional impenetrable Bose gas, corner and polynuclear growth models, one dimensional directed polymers, string theory, two dimensional quantum gravity, and spectral distributions of random matrices. In the present contribution, ideas of integrability are utilized to advocate emergence of an one-dimensional Toda Lattice and the fifth Painlevé transcendent in the paradigmatic problem of conductance fluctuations in quantum chaotic cavities coupled to the external world via ballistic point contacts. Specifically, the cumulants of the Landauer conductance of a cavity with broken time-reversal symmetry are proven to be furnished by the coefficients of a Taylor-expanded Painlevé V function. Further, the relevance of the fifth Painlevé transcendent for a closely related problem of sample-to-sample fluctuations of the noise power is discussed. Finally, it is demonstrated that inclusion of tunneling effects inherent in realistic point contacts does not destroy the integrability: in this case, conductance fluctuations are shown to be governed by a two-dimensional Toda Lattice.'
address: 'Department of Applied Mathematics, Holon Institute of Technology, Holon 5810201, Israel '
author:
- Eugene Kanzieper
title: Integrable Aspects of Universal Quantum Transport in Chaotic Cavities
---
Introduction {#intro}
============
The low temperature electronic conduction through a cavity exhibiting chaotic classical dynamics is governed by quantum phase-coherence effects (Imry 2002). In chaotic cavities with sufficiently large capacitance, when electron-electron interactions are negligible, the most comprehensive theoretical framework by which the phase coherent electron transport can be explored is provided by the scattering ${\bm {\mathcal S}}$-matrix approach pioneered by Landauer (Landauer 1957, Fisher and Lee 1981, Büttiker 1990). There exist two different, though mutually overlapping, scattering-matrix descriptions (Lewenkopf and Weidenmüller 1991) of quantum transport.
A semiclassical formulation (Richter 2000) of the ${\bm {\mathcal
S}}$-matrix approach is tailor-made to the analysis of energy-averaged charge conduction through an individual cavity. (Energy averaging is performed over such a small energy window near the Fermi energy that keeps the classical dynamics essentially unchanged.) Representing quantum transport observables (such as conductance, shot-noise power, transferred charge etc.) in terms of classical trajectories connecting the leads attached to a cavity, the semiclassical approach efficiently accounts for system-specific features of the quantum transport (Aleiner and Larkin 1996, 1997; Agam, Aleiner and Larkin 2000; Adagideli 2003; Brouwer and Rahav 2006; Whitney and Jacquod 2006; Brouwer 2007). Besides, it also covers the long-time scale [*universal transport regime*]{} (Richter and Sieber 2002, Heusler et al 2006, Braun et al 2006, Müller et al 2007) emerging in the limit $\tau_{\rm
D} \gg \tau_{\rm E}$, where $\tau_{\rm D}$ is the average electron dwell time and $\tau_{\rm E}$ is the Ehrenfest time (the time scale where quantum effects set in). The Ehrenfest time $\tau_{\rm E}\simeq \lambda^{-1} \log (W/\lambda_F)$ is determined by the Lyapunov exponent $\lambda$ of chaotic classical dynamics, the Fermi wavelength $\lambda_F$, and the lead widths $W$. The mean dwell time is estimated as $\tau_{\rm D}\simeq A/(W v_F)$, where $A$ is the area of the cavity and $v_F$ is the Fermi velocity.
The [*universal*]{} regime can alternatively be studied within a stochastic approach based on a random matrix description of electron dynamics in a cavity (Brouwer 1995, Beenakker 1997, Alhassid 2000). Modelling a single electron Hamiltonian by an $M \times M$ random matrix ${\bm {\mathcal H}}$ of proper symmetry, the stochastic approach starts with the Hamiltonian $H_{\rm tot}$ of the total system comprised by the cavity and the leads: $$\begin{aligned}
\label{ham}
H_{\rm tot} = \sum_{k,\ell=1}^M {\bm \psi}_k^\dagger {\mathcal H}_{k\ell}
{\bm \psi}_\ell &+& \sum_{\alpha=1}^{N_{\rm L}+N_{\rm R}}
{\bm \chi}_{\alpha}^\dagger \varepsilon_F {\bm \chi}_{\alpha}\nonumber \\
&+& \sum_{k=1}^M \sum_{\alpha=1}^{N_{\rm L}+N_{\rm R}} \left(
{\bm \psi}_k^\dagger
{\mathcal W}_{k \alpha} {\bm \chi}_{\alpha} +
{\bm \chi}_{\alpha}^\dagger {\mathcal W}_{k \alpha}^* {\bm \psi}_k
\right).\end{aligned}$$ Here, ${\bm \psi}_k$ and ${\bm \chi}_\alpha$ are the annihilation operators of electrons in the cavity and in the leads, respectively. Indices $k$ and $\ell$ enumerate electron states in the cavity: $1\le k,\ell \le M$, with $M\rightarrow \infty$. Index $\alpha$ counts propagating modes in the left ($1\le \alpha \le N_{\rm L}$) and the right ($N_{\rm L}+1 \le \alpha \le N_{\rm R}$) lead. The $M\times N$ matrix ${\bm {\mathcal W}}$ describes the coupling of electron states with the Fermi energy ${\varepsilon_F}$ in the cavity to those in the leads; $N=N_{\rm L}+N_{\rm R}$ is the total number of propagating modes (channels). Since in Landauer-type theories \[see, e.g., a recent review by Lesovik and Sadovskyy (2011)\] the transport observables are expressed in terms of the $N\times N$ scattering matrix $$\begin{aligned}
\label{sm}
{\bm {\mathcal S}}(\varepsilon_F) = \mathds{1}_N - 2i \pi {\bm {\mathcal W}}^\dagger
({\varepsilon}_F \mathds{1}_M - {\bm {\mathcal H}} + i \pi {\bm {\mathcal W}}
{\bm {\mathcal W}}^\dagger)^{-1} {\bm {\mathcal W}},\end{aligned}$$ the knowledge of its distribution is central to the stochastic approach.
For random matrices ${\bm {\mathcal H}}$ drawn from rotationally invariant Gaussian ensembles (Mehta 2004, Forrester 2010), the distribution of ${\bm {\mathcal S}}(\varepsilon_F)$ is described by the Poisson kernel (Hua 1963; Brouwer 1995; Mello and Baranger 1999) $$\begin{aligned}
\label{pk}
P({\bm {\mathcal S}}) \propto \left[
{\rm det}\left(
\mathds{1}_N - {\bm {\mathcal S}_0} {\bm {\mathcal S}}^\dagger\right)
{\rm det}\left(
\mathds{1}_N - {\bm {\mathcal S}} {\bm {\mathcal S}}_0^\dagger\right)
\right]^{ \beta/2 -1 - \beta N/2}.\end{aligned}$$ Here, $\beta$ is the Dyson index accommodating system symmetries ($\beta=1,\, 2, {\rm and} \,\, 4$) whilst ${\bm {\mathcal S}}_0$ is the average scattering matrix (Beenakker 1997), $$\begin{aligned}
\label{s0-1}
{\bm {\mathcal S}}_0 = \frac{M \Delta {\mathds 1}_N - \pi^2 {\bm {\mathcal W}}^\dagger {\bm {\mathcal W}}}
{M \Delta {\mathds 1}_N + \pi^2 {\bm {\mathcal W}}^\dagger {\bm {\mathcal W}}}, \qquad M\gg 1,\end{aligned}$$ whose eigenvalues characterise couplings between the cavity and the leads in terms of tunnel probabilities $\Gamma_j$ of $j$-th mode in the leads ($1\le j \le N$), $$\label{gj}
{\bm {\mathcal S}}_0 = {\bm V}^\dagger \,{\rm diag}
(\sqrt{1-\Gamma_j}
)\, {\bm V}.$$ This relation becomes evident after one realizes that the average scattering matrix is proportional to a reflection matrix of tunnel barriers (Beenakker 1997). Here, $\Delta$ is the mean level spacing; the matrix ${\bm V}$ is ${\bm V}\in G(N)/G(N_{\rm
L}) \times G(N_{\rm R})$ where $G$ stands for orthogonal ($\beta=1$), unitary ($\beta=2$) or symplectic ($\beta=4$) group. The celebrated result Eq. (\[pk\]), that can be viewed as a generalisation of the three Dyson circular ensembles, was alternatively derived through a phenomenological information-theoretic approach reviewed by Mello and Baranger (1999).
Statistical information accommodated in the Poisson kernel is too detailed to make a nonperturbative description of transport observables [*operational*]{}. It turns out, however, that in case of conserving charge transfer through normal chaotic structures, it is sufficient to know a probability measure associated with a set ${\bm T}$ of non-zero transmission eigenvalues $\{T_j \in (0,1)\}$; these are the eigenvalues of the Wishart-type matrix ${\bm t}{\bm t}^\dagger$, where ${\bm t}$ is the transmission sub-block of the reflection-transmission decomposed scattering matrix $$\begin{aligned}
\label{s-block}
\bm{{\mathcal S}} = \left(
\begin{array}{cc}
{\bm r}_{N_{\rm L}\times N_{\rm L}} & {\bm t}_{n_{\rm L}\times N_{\rm R}} \\
{\bm t^\prime}_{N_{\rm R}\times N_{\rm L}} & {\bm r^\prime}_{N_{\rm R}\times N_{\rm R}} \\
\end{array}
\right).\end{aligned}$$ Owing to this observation, the joint probability density function of all transmission eigenvalues emerges as the object of primary interest in the random matrix theories (RMT) of quantum transport which, besides, are particularly suited to the studies of [*integrable aspects*]{} of the universal quantum transport.
The present paper reviews recent advances in the [*integrable theory*]{} of quantum transport in chaotic structures, making particular emphasis on establishing exact relations between experimentally measurable fluctuations of transport observables (such as the Landauer conductance and the noise power) and Painlevé and Toda Lattice equations. It should be stressed that the above relations become manifest only at the level of a moment generating function; this explains, to some extent, why they have been overlooked in the studies of other groups that have merely focussed on a nonperturbative calculation of the moments or cumulants of transport observables \[see, e.g., Savin and Sommers (2006), Savin, Sommers and Wieczorek (2008), Novaes (2008), Khoruzhenko, Savin and Sommers (2009), Mezzadri and Simm (2011)\]. Our exposition is mainly based on the results obtained by Osipov and Kanzieper (2008, 2009) and Vidal and Kanzieper (2012) who have considered the simplest, $\beta=2$ symmetry class referring to chaotic cavities with broken time-reversal symmetry. For discussion of integrability for $\beta=1$ and $\beta=4$ Dyson’s symmetry classes, the reader is referred to the recent paper by Mezzadri and Simm (2013).
In Sections \[SoLC\] and \[itnf\], chaotic cavities probed via two [*ideal leads*]{} \[$\Gamma_j=1$ in Eq. (\[gj\])\] are considered. In this case, stochastic ${\bm {\mathcal S}}$-matrix description becomes particularly simple: the vanishing average scattering matrix ${\bm {\mathcal S}}_0={\bm 0}$ gives rise to the flat measure $P({\bm {\mathcal S}})=1$ which implies that scattering matrices ${\bm {\mathcal S}}$ belong to one of the three Dyson circular ensembles (Blümel and Smilansky 1990, Lewenkopf and Weidenmüller 1991, Brouwer 1995, Mello and Baranger 1999). By combining this observation with the ideas of integrability \[see, e.g., Morozov (1994), Adler and van Moerbeke (2001), Osipov and Kanzieper (2010)\], we show there that the problem of universal quantum transport in chaotic cavities with broken time-reversal symmetry ($\beta=2$) is completely integrable. Although this conclusion is very general (Osipov and Kanzieper 2008, 2009) and applies to a variety of transport observables, the discussion is purposely restricted to two particular problems listed in order of increasing difficulty: (i) statistics of the Landauer conductance (Section \[SoLC\]), and (ii) statistics of current fluctuations quantified by the noise power (Section \[itnf\]). This will help us to keep the presentation as transparent as possible and also show the merits of the integrability based approach. Emergence of (fifth) [*Painlevé transcendents*]{} and [*one-dimensional Toda Lattices*]{} in a nonperturbative description of quantum transport is the main outcome of Sections \[SoLC\] and \[itnf\].
Integrable theory of quantum transport (Vidal and Kanzieper 2012) that accounts for tunneling effects in the point contacts \[[*non-ideal leads*]{}, $0 < \Gamma_j <1$ in Eq. (\[gj\])\] is formulated in Section \[Tunneling\]. There, we show that non-ideality of the leads does not destroy integrability of the problem. In particular, we prove that conductance fluctuations are governed by a [*two-dimensional Toda Lattice*]{} equation.
Integrable theory of conductance fluctuations in chaotic cavities with ideal leads {#SoLC}
==================================================================================
Moment generating function for Landauer conductance
---------------------------------------------------
In what follows, we consider chaotic cavities with broken time-reversal symmetry which are probed, via ballistic point contacts \[no tunneling, $\Gamma_j=1$ in Eq. (\[gj\])\], by two (left and right) leads further attached to outside reservoirs kept at temperature $\theta$; the leads support $N_{\rm L}$ and $N_{\rm R}$ propagating modes, respectively. In this scattering geometry, the Landauer conductance at zero temperature is related the scattering matrix ${\bm {\mathcal S}}$ of the system via Landauer formula $$\begin{aligned}
G = {\rm tr\,} ({\mathcal C}_1 {\bm {\mathcal S}}{\mathcal C}_2 {\bm {\mathcal S}}^\dagger),\end{aligned}$$ where ${\mathcal C}_{1,2}$ are projection matrices of the form $$\begin{aligned}
\label{c1-c2}
{\mathcal C}_1= \left(
\begin{array}{cc}
\mathds{1}_{N_{\rm L}} & 0 \\
0 & 0_{N_{\rm R}} \\
\end{array}
\right),\;\;\;
{\mathcal C}_2= \left(
\begin{array}{cc}
0_{N_{\rm L}} & 0 \\
0 & \mathds{1}_{N_{\rm R}} \\
\end{array}
\right).\end{aligned}$$
In order to describe fluctuations of the conductance $G = {\rm tr\,} ({\mathcal C}_1 {\bm{\mathcal S}}
{\mathcal C}_2 {\bm {\mathcal S}}^\dagger)$ in an adequate way, one needs to know its entire distribution function. To determine the latter, we define the moment generating function (MGF) $$\begin{aligned}
\label{c-iz}
{\mathcal F}_n^{(\nu)}(z) = \left<
\exp\left(
- z G \right)
\right>_{{\bm {\mathcal S}}\in {\rm CUE}(N_{\rm L}+N_{\rm R})}\end{aligned}$$ labeled by the indices $$\label{n-index}
n=\min(N_{\rm L},N_{\rm R}),$$ and $$\label{nu-index}
\nu=|N_{\rm L}-N_{\rm R}|,$$ the latter being the asymmetry parameter (we shall occasionally omit the superscript $(\nu)$ wherever this does not cause a notational confusion). The logarithm, $\log {\mathcal F}_n^{(\nu)}(z)$, Taylor-expanded in a vicinity of $z=0$, supplies the cumulants of Landauer conductance.
While the averaging in Eq. (\[c-iz\]), running over scattering matrices${\bm {\mathcal S}}\in {\rm CUE}(N_{\rm L}+N_{\rm R})$ drawn from the Dyson circular unitary ensemble, can explicitly be performed with the help of the Harish-Chandra-Itzykson-Zuber formula (Harish-Chandra 1957; Itzykson and Zuber 1980), a high spectral degeneracy of the projection matrices ${\mathcal C}_1$ and ${\mathcal C}_2$ makes this calculation quite tedious. To avoid unnecessary technical complications, it is beneficial to decompose the scattering matrix into reflection-transmission submatrices \[see Eq. (\[s-block\])\] to realise that the Landauer conductance $G$ is solely determined by (transmission) eigenvalues $\{ T_j \}$ of ${\bm t}{\bm t}^\dagger$ (Landauer 1957, Fisher and Lee 1981, Büttiker 1990), $$\begin{aligned}
\label{T-cond}
G({\bm T}) = {\rm tr\,} ({\bm t}{\bm t}^\dagger) = \sum_{j=1}^n T_j.\end{aligned}$$ The uniformity of the scattering ${\mathcal S}$-matrix distribution gives rise to a nontrivial joint probability density function of transmission eigenvalues in the form (Baranger and Mello 1994; Jalabert, Pichard and Beenakker 1994; Forrester 2006) $$\begin{aligned}
\label{PnT}
P_n^{(\nu)}({\bm T}) = c_{n,\nu}^{-1} \,\Delta_n^2({\bm T}) \prod_{j=1}^n
T_j^\nu.\end{aligned}$$ Here, $\Delta_n({\bm T})=\prod_{j<k} (T_k-T_j)$ is the Vandermonde determinant and $c_{n,\nu}$ is the normalisation constant (Mehta 2004) $$\begin{aligned}
\label{nc}
c_{n,\nu} = \prod_{j=0}^{n-1} \frac{\Gamma(j+2)\, \Gamma(j+\nu+1) \,\Gamma(j+1)}
{\Gamma(j+\nu+n+1)}.\end{aligned}$$ Let us stress that the description based on Eqs. (\[T-cond\]) and (\[PnT\]) is completely equivalent to the original ${\bm {\mathcal S}}\in {\rm CUE}(N_{\rm L}+N_{\rm R})$ model microscopically justified (Lewenkopf and Weidenmüller 1991; Brouwer 1995) in the universal transport regime we are confined to. Equation (\[PnT\]) is one of the cornerstones of the RMT approach to quantum transport.
Non-perturbative calculation of the moment generating function (easy way) {#Easy-Way}
-------------------------------------------------------------------------
Appearance of the fifth Painlevé transcendent in the Landauer conductance moment generating function (MGF), announced in the abstract to this contribution, can be appreciated in an “easy way” after one realises that the matrix/eigenvalue integral [^1] $$\begin{aligned}
\label{cond-eig-nod}
{\mathcal F}_n^{(\nu)}(z) = c_{n,\nu}^{-1} \int_{(0,1)^n} \prod_{j=1}^n dT_j\, T_j^\nu \exp(-zT_j)\cdot \Delta_n^2({\bm T})\end{aligned}$$ can be described in terms of completely integrable Toda Lattice model (Toda 1989; Teschl 2000). The Darboux theorem (Darboux 1889; Vein and Dale 1999) followed by the Toda-to-Painlevé reduction (Okamoto 1987; Forrester and Witte 2002) are the key ingredients of the calculation to be presented below.
### MGF and the Toda Lattice equation {#TL-Hankel}
A close inspection of the integral Eq. (\[cond-eig-nod\]) reveals that it admits the Hankel determinant representation (Osipov and Kanzieper 2008) $$\begin{aligned}
\label{hd}
{\mathcal F}_n^{(\nu)}(z) = \frac{n!}{c_{n,\nu}} \, {\rm det}}\left[
(-\partial_z)^{j+k}\, {\mathcal F}_1^{(\nu)}(z)
\right]_{(j,k)\in (0,n-1)\end{aligned}$$ with $$\label{f1}
{\mathcal F}_1^{(\nu)}(z) =\int_0^1 dT\, T^{\nu} \,\exp(-zT) = \frac{\Gamma(\nu+1)}{z^{\nu+1}}
\left(
1 - e^{-z} \sum_{\ell=0}^\nu \frac{z^\ell}{\ell!}
\right).$$ In deriving Eqs. (\[hd\]) and (\[f1\]) we have used the Andréief–de Bruijn integration formula (Andréief 1883, de Bruijn 1955).
Equation (\[hd\]), supplemented by the “initial condition” ${\mathcal F}_0^{(\nu)}(z)=1$, has far-reaching consequences. Indeed, by virtue of the Darboux theorem (Darboux 1889; Vein and Dale 1999), the infinite sequence of the moment generating functions $\{{\mathcal F}_1^{(\nu)},{\mathcal
F}_2^{(\nu)},\cdots\}$ obeys the [*one-dimensional Toda Lattice*]{} equation $(n\ge 1)$ $$\label{TL}
\hspace{-1cm}
{\mathcal F}_n^{(\nu)}(z)\,\frac{\partial^2 {\mathcal F}_n^{(\nu)}(z)}{\partial z^2} - \left(
\frac{\partial {\mathcal F}_n^{(\nu)}(z)}{\partial z}\right)^2 = {\rm var}_{n,\nu}(G)\,
{\mathcal F}_{n-1}^{(\nu)}(z)\, {\mathcal F}_{n+1}^{(\nu)}(z),$$ where ${\rm var}_{n,\nu}(G) = n(n+1)^{-1} (c_{n-1,\nu} c_{n+1,\nu}/ c_{n,\nu}^2)$ is, by definition, the conductance variance $$\label{varG}
{\rm var}_{n,\nu}(G) = \frac{n^2 (n+\nu)^2}{(2n+\nu)^2
[(2n+\nu)^2-1]}.$$ (To appreciate this point, project the Toda Lattice equation onto $z=0$.) Importantly, emergence of the Toda Lattice equation for the MGF can be traced back to the broken time-reversal symmetry in the underlying scattering system. For chaotic cavities with conserved time-reversal symmetry ($\beta=1$) and in presence of spin-orbit interactions ($\beta=4$), the statistics of conductance fluctuations is captured by the Pfaff-KP Lattice equation (Mezzadri and Simm 2013). Since ${\mathcal F}_n^{(\nu)}(z)$ is the Laplace transform of conductance probability density$f_n^{(\nu)}(g)=\langle \delta (g - G) \rangle$, the Toda lattice equation provides an exact recursive solution to the problem of conductance distribution in chaotic cavities with an arbitrary number of channels in the leads. Indeed, because of a specific form of ${\mathcal F}_1^{(\nu)}(z)$, a calculation of the inverse Laplace transform of ${\mathcal F}_n^{(\nu)}(z)$ is operationally straightforward. The resulting probability density function $f_n^{(\nu)}(g)$ can be shown to be a nonanalytic function, admitting the representation: $$\begin{aligned}
f_n^{(\nu)}(g) = \sum_{k=0}^n {\rm sgn} (g-k) \, \pi_{k}^{(n,\nu)}(g-k),\qquad g\in(0,n).\end{aligned}$$ Here, $\{\pi_{0}^{(n,\nu)}(x), \pi_{1}^{(n,\nu)}(x),\dots, \pi_{n}^{(n,\nu)}(x)\}$ is a set of hierarchically structured polynomials of degree ${\rm deg\,}\pi_{k}^{(n,\nu)}(x)=n(n+\nu)-1$, satisfying a number of remarkable properties that originate from the structures hiding behind the Toda Lattice Eq. (\[TL\]). For $\nu=0$, explicit formulae for $\pi_{k}^{(n,0)}(x)$ ($1\le n \le 4$) are given in Table \[pol-table\]. Finding explicit expressions for these polynomials for generic $n$ and $\nu$ is a nontrivial problem.
[@\*[2]{}[l]{}]{} $\0\0 n$& $\pi_k^{(n,0)}(x)$
$\0\0 1$& $\pi_0^{(1,0)}(x) = \displaystyle{\frac{1}{2}}$
$\0\0 2$& $\pi_0^{(2,0)}(x) = \phantom{-2}x^3$
$\0\0 {}$& $\pi_1^{(2,0)}(x) = -2 x\, (x^2+3)$
$\0\0 3$& $\pi_0^{(3,0)}(x) = \phantom{-}\displaystyle{\frac{3}{28} x^8}$
$\0\0 {}$& $\pi_1^{(3,0)}(x) = \displaystyle{-\frac{9}{28} \, x^4 \left( x^4 + 56 x^2 -112 x + 140 \right)}$
$\0\0 4$& $\pi_0^{(4,0)}(x) = \phantom{-}\displaystyle{\frac{1}{3003} x^{15}}$
$\0\0 {}$& $\pi_1^{(4,0)}(x) = \displaystyle{-\frac{4}{3003} x^{9}} \left( x^6 + 315 x^4 - 2730 x^3 +15015 x^2 -30030 x + 25025 \right)$
$\0\0 {}$& $\pi_2^{(4,0)}(x) = \displaystyle{-\frac{2}{3003} x^{7}} \left(3 x^8 + 1260 x^6 + 10920 x^4 + 400400 x^2 +900900 \right)$
### MGF and the Painlevé V equation
—While important from conceptual point of view and also operationally useful in generating explicit formulae for the [*distribution*]{} of Landauer conductance for finite $n$, the Toda Lattice equation is of little help in studying the conductance cumulants. Fortunately, the Toda Lattice representation of the MGF can readily be converted into a Painlevé representation since, miraculously, the same Toda Lattice equation governs the behaviour of so-called $\tau$ functions arising in the Hamiltonian formulation of the six Painlevé equations (Clarkson 2003; Noumi 2004), which are yet another fundamental object in the theory of nonlinear integrable lattices. The aforementioned Painlevé reduction (Okamoto 1987; Forrester and Witte 2002) of the Toda Lattice equation (\[TL\]) materialises in the exact representation $$\begin{aligned}
\label{FnP-22}
{\mathcal F}_n^{(\nu)}(z) = \exp\left(
\int_0^z dt \frac{\sigma_{\rm V}(t; n,\nu) - n(n+\nu)}{t}
\right).\end{aligned}$$ Here, $\sigma_{\rm V}(t;n,\nu)=\sigma_{n,\nu}(t)$ satisfies the Jimbo-Miwa-Okamoto form of the [*Painlevé V*]{} equation $$\begin{aligned}
\label{pv-22} \hspace{-1cm}
(t \sigma_{n,\nu}^{\prime\prime})^2 - [\sigma_{n,\nu} - t \sigma_{n,\nu}^\prime
&+& 2 (\sigma_{n,\nu}^\prime)^2 + (2n+\nu)\sigma_{n,\nu}^\prime]^2 \nonumber\\
&+& 4(\sigma_{n,\nu}^\prime)^2 (\sigma_{n,\nu}^\prime + n) (\sigma_{n,\nu}^\prime+n+\nu)=0\end{aligned}$$ subject to the boundary condition $$\begin{aligned}
\label{pv-bc}
\sigma_{n,\nu}(t\rightarrow
0)\simeq n(n+\nu) - \frac{n(n+\nu)}{2n+\nu}t + {\mathcal O}(t^2).\end{aligned}$$
—There exists yet another, “easy” way to derive the Painlevé V representation of the MGF. Spotting that the moment generating function ${\mathcal F}_n^{(\nu)}(z)$ is essentially a gap formation probability $E_{n,\nu}^{{\rm LUE}}(0; (z,\infty))$ within the interval $(z,+\infty)$ in the spectrum of an auxiliary $n \times n$ Laguerre unitary ensemble (Mehta 2004), $$\begin{aligned}
{\mathcal F}_n^{(\nu)}(z) =z^{-n(n+\nu)}\, E_{n,\nu}^{{\rm LUE}}(0; (z,\infty)),\end{aligned}$$ where $$\begin{aligned}
E_{n,\nu}^{{\rm LUE}}(0; (z,\infty)) = c_{n,\nu}^{-1}\int_{(0,z)^n}
\prod_{j=1}^n d\lambda_j \, \lambda_j^\nu \, e^{-\lambda_j}
\cdot \Delta_n^2({\bm \lambda}),\end{aligned}$$ one immediately reproduces (Tracy and Widom 1994) Eqs. (\[FnP-22\]) and (\[pv-22\]). The Painlevé V representation of the MGF (Osipov and Kanzieper 2008) opens a way for an elegant nonperturbative calculation of the Landauer conductance cumulants (see Section \[cum-section\]). As a matter of fact, Eq. (\[FnP-22\]) incorporates all available nonperturbative results for cumulants of the Landauer conductance at $\beta=2$ (see, e.g., Savin and Sommers 2006; Savin, Sommers and Wieczorek 2008; Novaes 2008; Khoruzhenko, Savin and Sommers (2009)).
The $\tau$ function theory of the moment generating function
------------------------------------------------------------
The treatment presented in Section \[Easy-Way\] was largely based on a wealth of “ready-for-use” results (Andréief-de Bruijn formula, Darboux theorem, a connection between the Toda Lattice and Painlevé transcendents, and a well-studied problem of calculating gap formation probabilities) which surprisingly well fitted our goal of a nonperturbative evaluation of the particular moment generating function Eq. (\[cond-eig-nod\]). Since existence of such an “easy way” is clearly the exception rather than the rule (see, e.g., Section \[itnf\] where statistics of thermal-to-shot-noise crossover in chaotic cavities is studied), a regular yet flexible formalism is needed for a nonperturbative description of a wide class of generating functions arising in the context of universal quantum transport in chaotic structures. To outline such a formalism, we first revisit the problem of a nonperturbative evaluation of the Landauer conductance MGF in order to reproduce the main results of the previous Section in a coherent manner.
### The idea
The “deform-and-study” approach (Morozov 1994; Adler, Shiota and van Moerbeke 1995; Adler and van Moerbeke 1995, 2001; Osipov and Kanzieper 2010) borrowed from the theory of integrable systems is central to the nonperturbative calculation of ${\mathcal F}_n^{(\nu)}(z)$. In the present context, the main idea of the method is “embedding” ${\mathcal F}_n^{(\nu)}(z)$ into a more general theory of the $\tau$ function $$\label{tau-def-00}
\tau_n^{(\nu)}({\bm t};z) = \frac{1}{n!}\int_{(0,1)^n} \prod_{j=1}^n dT_j\,T_j^\nu \,\Gamma_{z}(T_j)
\,e^{V({\bm t};T_j)} \Delta_n^2({\bm T})$$ which possesses the infinite-dimensional parameter space ${\bm
t}=(t_1,t_2,\dots)$ arising as the result of the ${\bm t}$ deformation [^2] $$\begin{aligned}
V({\bm t};T) = \sum_{k=1}^\infty t_k T^k\end{aligned}$$ of the MGF of a generic transport observable \[see, e.g., Eq. (\[cond-eig-nod\])\]. In the context of the Landauer conductance fluctuations, the function $\Gamma_{z}(T)$ should be set to $\Gamma_{z}(T)=\exp(-z T)$.
Studying an evolution of the $\tau$ function in the extended $(n,
{\bm t},z)$ space allows us to identify various nonlinear differential hierarchical relations. A projection of these relations onto the hyperplane ${\bm t}={\bm 0}$, $$\begin{aligned}
\label{proj-00}
{\mathcal F}_n^{(\nu)}(z) = \frac{n!}{c_{n,\nu}}\,
\tau_n^{(\nu)}({\bm t};z)\Big|_{{\bm t}={\bm 0}},\end{aligned}$$ generates, among others, a closed nonlinear differential equation for the moment generating function ${\mathcal F}_n^{(\nu)}(z)$.
The two key ingredients of the exact theory of $\tau$ functions are (i) the bilinear identity (Date et al 1983) and (ii) the (linear) Virasoro constraints (Mironov and Morozov 1990).
### Bilinear identity and integrable hierarchies
The bilinear identity encodes an infinite set of hierarchically structured nonlinear differential equations in the variables ${\bm
t}=(t_1, t_2,\dots)$. For the model introduced in Eq. (\[tau-def-00\]), the bilinear identity reads (Adler, Shiota and van Moerbeke 1995; Osipov and Kanzieper 2010): $$\begin{aligned}
\label{bi-id}
\oint_{{\cal C}_\infty} d^2\xi \,e^{a\, v(\bm{t-t^\prime};\xi)}
\tau_{n}^{(\nu)}(\bm{t}-[\bm{\xi}^{-1}])\,
\frac{\tau_{m+1}^{(\nu)}(\bm{t^\prime}+[\bm{\xi}^{-1}])}{\xi^{m+1-n}}\qquad
\nonumber\\
\qquad =\oint_{{\cal C}_\infty} d^2\xi \,e^{(a-1)\, v(\bm{t-t^\prime};\xi)}
\tau_m^{(\nu)}(\bm{t^\prime}-[\bm{\xi}^{-1}]) \frac{\tau_{n+1}^{(\nu)}
(\bm{t}+[\bm{\xi}^{-1}])}{\xi^{n+1-m}}.\end{aligned}$$ Here, $\xi \in {\mathbb C}$ and $a\in {\mathbb R}$ is a free parameter; the integration contour ${\cal C}_\infty$ encompasses the point $\xi=\infty$; the notation ${\bm t} \pm [{\bm \xi}^{-1}]$ stands for the infinite set of parameters $\{t_j\pm \xi^{-j}/j\}$; for brevity, both the Laplace parameter $z$ and the asymmetry index $\nu$ were dropped from the arguments of $\tau$ functions.
Being expanded in terms of $\bm{ t^\prime}-{\bm t}$ and $a$, Eq. (\[bi-id\]) generates a zoo of integrable hierarchies satisfied by the $\tau$ function Eq. (\[tau-def-00\]) in the $(n,{\bm t})$-space. The Toda Lattice (TL) and Kadomtsev-Petviashvili (KP) hierarchies are central to our approach. The first non-trivial members of the TL and KP hierarchies read $$\begin{aligned}
\label{ftl}
\tau_n^{(\nu)}({\bm t})\, \frac{\partial^2 \tau_n^{(\nu)}({\bm t})}{\partial t_1^2} - \left(
\frac{\partial\tau_n^{(\nu)}({\bm t})}{\partial t_1}
\right)^2 = \tau_{n-1}^{(\nu)}({\bm t})\, \tau_{n+1}^{(\nu)}({\bm t})\end{aligned}$$ and $$\begin{aligned}
\fl
\label{fkp}
\left(
\frac{\partial^4}{\partial t_1^4} + 3\,\frac{\partial^2}{\partial t_2^2} -
4\, \frac{\partial^2}{\partial t_1 \partial t_3}
\right)\, \log \tau_n^{(\nu)}({\bm t})
+ \,6\, \left(
\frac{\partial^2}{\partial t_1^2}\, \log \tau_n^{(\nu)}({\bm t})
\right)^2 = 0,\end{aligned}$$ respectively (Adler, Shiota and van Moerbeke 1995; Adler and van Moerbeke 1995). For higher-order equations of the TL and KP hierarchies, as well as a complete list of integrable hierarchies encoded in the bilinear identity Eq. (\[bi-id\]), the reader is referred to Osipov and Kanzieper (2010).
### Virasoro constraints
The projection formula Eq. (\[proj-00\]) makes it tempting to assume that nonlinear integrable hierarchies satisfied by $\tau$ functions in the $(n, {\bm t})$-space should induce similar, hierarchically structured, nonlinear differential equations for the moment generating function ${\mathcal F}_n^{(\nu)}(z)$. To identify them, one has to seek an additional block of the theory that would make a link between the partial $\{t_k\}$ derivatives of $\tau$ functions taken at ${\bm t}={\bm 0}$ and the derivatives of ${\mathcal F}_n^{(\nu)}(z)$ with respect to the Laplace parameter $z$. The study by Adler, Shiota and van Moerbeke (1995) suggests that the missing block is the [*Virasoro constraints*]{} which reflect invariance of the $\tau$ function \[Eq. (\[tau-def-00\])\] under a change of the integration variables.
In the present context, it is useful to demand the invariance under the set of transformations $$\begin{aligned}
\label{v-tr}
T_j \rightarrow \tilde{T}_j +
\epsilon \,\tilde{T}_j^{q+1} (\tilde{T}_j-1),\;\;\; q\ge 0\end{aligned}$$ that leave the integration domain intact. Employing a by now standard procedure (Mironov and Morozov 1990; Adler and van Moerbeke 1995; Osipov and Kanzieper 2010), one readily checks that the transformation (\[v-tr\]) induces the Virasoro constraints in the form $$\label{vc-1}
\big[ \hat{L}_{q+1}({\bm t}) - \hat{L}_{q}({\bm t})\big] \tau_n^{(\nu)}({\bm t};z)
= 0,\;\;\; q\ge 0,$$ where a set of differential operators $$\begin{aligned}
\label{vc-2-00}
\hat{L}_q({\bm t})
=
\hat{\mathcal L}_{q}({\bm t}) - z\, \frac{\partial}{\partial t_{q+1}}
+ \nu \frac{\partial}{\partial t_q}\end{aligned}$$ involves the Virasoro operators $$\begin{aligned}
\label{vc-3}
\hat{{\cal L}}_q({\bm t}) = \sum_{j=1}^\infty jt_j \,\frac{\partial}{\partial t_{q+j}}
+
\sum_{j=0}^q \frac{\partial^2}{\partial t_j \partial t_{q-j}},\end{aligned}$$ satisfying the Virasoro algebra $$\begin{aligned}
\label{va}
[\hat{{\cal L}}_p,\hat{{\cal L}}_q] = (p-q)\hat{{\cal
L}}_{p+q}, \;\;\;\; p,q\ge -1.\end{aligned}$$ Equations (\[vc-2-00\]) and (\[vc-3\]) assume the convention $\partial/\partial t_0 \equiv n$.
### The Toda Lattice equation for MGF
The Toda Lattice equation for the MGF ${\mathcal F}_n^{(\nu)}(z)$ follows from the projection formula Eq. (\[proj-00\]), the TL equation (\[ftl\]) written in the $(n,{\bm t})$-space, and the relation $$\begin{aligned}
\label{t1z}
\frac{\partial}{\partial t_1} \tau_n^{(\nu)}({\bm t};z) = -
\frac{\partial}{\partial z} \tau_n^{(\nu)}({\bm t};z).\end{aligned}$$ Straightforward manipulations bring out Eqs. (\[TL\]) and (\[varG\]), obtained in Section \[TL-Hankel\] by virtue of the Darboux theorem applied to a Hankel determinant form of the MGF.
### The Painlevé V equation for MGF
To derive a differential equation for the MGF ${\mathcal
F}_n(z)$, we combine the projection formula Eq. (\[proj-00\]) with the first KP equation (\[fkp\]). Having in mind the identity Eq. (\[t1z\]) as well as the two lowest Virasoro constraints labeled by $q=0$, $$\begin{aligned}
\fl \label{q0-00}
\left[
\sum_{j=1}^\infty jt_j \left(
\frac{\partial}{\partial t_{j+1}} - \frac{\partial}{\partial t_{j}}\right)
- z \frac{\partial}{\partial t_2} +(2n+\nu+z) \frac{\partial}{\partial t_1}
\right] \log \tau_n^{(\nu)}({\bm t}; z)=n(n+\nu),\end{aligned}$$ and $q=1$ $$\begin{aligned}
\fl \label{q1-00}
\Bigg[
\sum_{j=1}^\infty jt_j \left(
\frac{\partial}{\partial t_{j+2}} - \frac{\partial}{\partial t_{j+1}}\right)
-z \frac{\partial}{\partial t_3} + (2n+\nu+z) \frac{\partial}{\partial t_2}\nonumber\\
- (2n+\nu) \frac{\partial}{\partial t_1}
+ \frac{\partial^2}{\partial t_1^2}\Bigg] \log \tau_n^{(\nu)}({\bm t}; z) + \left(
\frac{\partial}{\partial t_1} \log \tau_n^{(\nu)}({\bm t},z)
\right)^2=0,\end{aligned}$$ we reveal, after lengthy but straightforward manipulations, that the function $$\begin{aligned}
\label{sF-cc}
\sigma_{n,\nu}(z) = n(n+\nu) + z \frac{\partial}{\partial z}\log {\mathcal F}_n^{(\nu)}(z)\end{aligned}$$ satisfies a nonlinear differential equation $$\begin{aligned}
\fl
\label{OK-2008}
z^2 \sigma_{n,\nu}^{\prime\prime\prime} +
z \,\sigma_{n,\nu}^{\prime\prime} +
6 z \left(\sigma_{n,\nu}^\prime
\right)^2
- 4 \sigma_{n,\nu} \sigma_{n,\nu}^\prime \nonumber \\
-
\left[
z^2 -2(2n +\nu)\,z+\nu^2 \right]\,
\sigma_{n,\nu}^\prime -
(2n+\nu -z) \sigma_{n,\nu} = 0\end{aligned}$$ belonging to the Chazy I class (Chazy 1911; Cosgrove and Scoufis 1993). This can be recognised as the Chazy form of the fifth Painlevé transcendent \[Eq. (\[pv-22\])\] written in the Jimbo-Miwa-Okamoto form (Jimbo et al 1980; Okamoto 1987). Equation (\[FnP-22\]) then readily follows. For transition from Eq. (\[OK-2008\]) to Eq. (\[pv-22\]), see Appendix E in the paper by Osipov and Kanzieper (2010).
Cumulants of the Landauer conductance {#cum-section}
-------------------------------------
The Painlevé V representation of the MGF, Eq. (\[FnP-22\]), opens a way for a nonperturbative calculation of the Landauer conductance cumulants.
### Cumulants from Taylor expanded Painlevé V
Perhaps, the most startling consequence of the above nonperturbative calculation is the claim (Osipov and Kanzieper 2008) that the cumulants of the Landauer conductance of a cavity probed via two ideal leads are furnished by the coefficients of a Taylor expanded Painlevé V function. Indeed, since the cumulants $\{\kappa_\ell^{(n,\nu)} (G)\}$ are supplied by the moment generating function $$\begin{aligned}
\log {\mathcal F}_n^{(\nu)}(z) = \sum_{\ell=1}^\infty \frac{(-1)^\ell}{\ell!} \,
\kappa_\ell^{(n,\nu)}(G)\, z^\ell,\end{aligned}$$ one derives, in view of Eq. (\[sF-cc\]), the remarkable identity: $$\begin{aligned}
\label{main-cond-cum}
\sigma_{n,\nu}(z) = n(n+\nu) + \sum_{\ell=1}^\infty \frac{(-1)^\ell}{\Gamma(\ell)}\, \kappa_\ell^{(n,\nu)}(G)\, z^\ell.\end{aligned}$$ Whenever the cumulant dependence on $n$ and $\nu$ is clear from the context, the superscript ${(n,\nu)}$ will be omitted.
### Exact recurrence solution
Alternatively, the cumulants of any finite order can be generated by a recurrence relation. The latter follows upon substitution of the expansion Eq. (\[main-cond-cum\]) into Chazy’s form of the fifth Painlevé transcendent \[Eq. (\[OK-2008\])\]. The resulting recurrence relation reads : $$\begin{aligned}
\label{cumeq}
\fl [(2n+\nu)^2-\ell^2]\,(\ell+1) \kappa_{\ell+1}^{(n,\nu)}(G)
+ (2n+\nu) (2\ell-1)\,\ell \,\kappa_{\ell}^{(n,\nu)}(G) \nonumber\\
\fl \qquad + \ell(\ell-1)(\ell-2)\,
\kappa_{\ell-1}^{(n,\nu)}(G)
- 2 \sum_{j=0}^{\ell-1} (3j+1) (j-\ell)^2 \left({\ell}\atop{j}\right)
\kappa_{j+1}^{(n,\nu)}(G)\, \kappa_{\ell-j}^{(n,\nu)}(G) = 0,\nonumber\\
{}\end{aligned}$$ where $\ell \ge 2$, and $$\begin{aligned}
\left({\ell}\atop{j}\right) = \frac{\Gamma(\ell+1)}{\Gamma(j+1) \Gamma(\ell-j+1)}.\end{aligned}$$ Taken together with the initial conditions provided by the mean conductance $\kappa_1(G)$ and its variance $\kappa_2(G)$ (Baranger and Mello 1994, Jalabert, Pichard and Beenakker 1994), $$\begin{aligned}
\label{ic-cond}
\kappa_1(G) = \frac{n (n+\nu)}{2n + \nu},\quad
\kappa_2(G) = \frac{\kappa_1^2(G)}{(2n+\nu)^2 -1},\end{aligned}$$ this recurrence efficiently generates conductance cumulants of any given order. Since the resulting expressions are very lengthy, we only quote the third cumulant related to the conductance skewness: $$\begin{aligned}
\kappa_{3}(G) = -\frac{\nu^2}{(2n+\nu)}\, \kappa_{1}^2(G).\end{aligned}$$ This reproduces the result first obtained by Savin, Sommers and Wieczorek (2008) who used a Selberg integral technique; for an alternative calculation based on the theory of symmetric functions, see Novaes (2008) and Khoruzhenko, Savin and Sommers (2009). Higher order cumulants can readily be deduced from Eq. (\[cumeq\]) as well.
### Asymptotic analysis of the cumulants of finite order {#cum-fo}
Below, the Painlevé solution for the MGF will be studied in the limit of a large number of propagating modes in both left and right leads. The asymptotic analysis of conductance cumulants to be performed in this Section, refers to chaotic cavities in which the asymmetry parameter $\nu = |N_{\rm L} - N_{\rm R}|$ is kept [*fixed*]{} whilst $n={\rm min}(N_{\rm L},N_{\rm R})$ is assumed to be large, $n \gg 1$. The forthcoming calculation turns out to be particularly simple if one deals with the recurrence equation written in terms of $$\begin{aligned}
\chi_\ell = \frac{(-1)^\ell}{\Gamma(\ell)}\, \kappa_\ell(G).\end{aligned}$$ From Eq. (\[cumeq\]), one derives: $$\begin{aligned}
\fl
\label{rec-sigma}
\phantom{00000000}
(\ell+1)\big[\ell^2 - (2n+\nu)^2\big]\chi_{\ell+1} &+& (2n+\nu) (2\ell-1)\chi_\ell -(\ell-2)\chi_{\ell-1}\nonumber\\
&=&
- 2 \sum_{j=0}^{\ell-1}(\ell -j)(3j+1) \chi_{j+1} \chi_{\ell-j}.\end{aligned}$$ The initial conditions for this equation, that holds down to $\ell=0\;$[^3], read: $$\begin{aligned}
\chi_0 = n(n+\nu),\qquad \chi_1 = -\frac{n(n+\nu)}{2n+\nu}=-\frac{n}{2} - \frac{\nu}{4} +{\mathcal O}(n^{-1}).\end{aligned}$$ To develop a regular $1/n$ expansion for the set $\{\chi_\ell\}$, we separate therein the terms that do not vanish as $n\rightarrow \infty$, $$\begin{aligned}
\label{sigma-ser}
\chi_\ell = n(n+\nu) \delta_{\ell,0} - \frac{2n+\nu}{4} \delta_{\ell,1} +\frac{1}{16} \delta_{\ell,2} + \delta\chi_\ell \,\cdot {\mathds 1}_{\ell\ge 1},\end{aligned}$$ and put forward the ansatz: $$\begin{aligned}
\label{sigma-ans}
\delta\chi_\ell = \frac{a_\ell(\nu)}{(4n)^{\ell}} + \frac{b_\ell(\nu)}{(4n)^{\ell+1}} + \frac{c_\ell(\nu)}{(4n)^{\ell+2}}
+ {\cal O}(n^{-\ell-3}).\end{aligned}$$ In Eq. (\[sigma-ser\]), the fraction $1/16$ originates from a non-vanishing (as $n\rightarrow \infty$) term of the $1/n$ expansion for $$\begin{aligned}
\chi_2 = \frac{n^2 (n+\nu)^2}{(2n+\nu)^2 [(2n+\nu)^2-1]} = \frac{1}{16} +{\mathcal O}(n^{-2}).\end{aligned}$$ Substituting Eqs. (\[sigma-ser\]) and (\[sigma-ans\]) into Eq. (\[rec-sigma\]), the following set of recurrence equations can readily be derived from the resulting $1/n$ expansion: The $\{a_\ell\}$–sequence: $$\begin{aligned}
a_{\ell+1}(\nu) = a_{\ell-1}(\nu),\end{aligned}$$ where $$\begin{aligned}
a_1(\nu)=\frac{\nu^2}{2},\qquad a_2(\nu)=\frac{1-2\nu^2}{4}.\end{aligned}$$ Hence, we deduce the solution $$\begin{aligned}
\label{a-un-fa}
a_\ell(\nu) = \frac{1}{8} \left[
1 + (-1)^\ell (1-4\nu^2)\right].\end{aligned}$$ The $\{b_\ell\}$–sequence: $$\begin{aligned}
b_{\ell+1}(\nu) = b_{\ell-1}(\nu)-4\nu \, a_{\ell+1}(\nu).\end{aligned}$$ This brings, after some effort, $$\begin{aligned}
\label{a-un-fb}
b_\ell(\nu) = -\ell\,\frac{\nu}{4} \left[
1 + (-1)^\ell (1-4\nu^2) \right].\end{aligned}$$ Equivalently, $$\begin{aligned}
\label{a-un-fb-eq}
b_\ell(\nu) = -2\nu\,\ell a_\ell(\nu).\\
{}\nonumber\end{aligned}$$ Combining the above results together, we derive the first few terms in the $1/n$ expansion of $\ell$-th conductance cumulant: $$\begin{aligned}
\fl\label{g-exp-1n}
\kappa_\ell(G) =\frac{2n+\nu}{4}\delta_{\ell,1}+\frac{1}{16}\delta_{\ell,2}+
\frac{1+(-1)^\ell(1 -4\nu^2)}{8}
\, \frac{\Gamma(\ell)}{(4n)^\ell}\,\left(
1-\nu \,\frac{\ell}{2n}
\right) + {\cal O}(n^{-(\ell+2)}).\nonumber\\{}\end{aligned}$$ For $\ell$ kept finite, the next-order terms in $1/n$ can be derived with increasing difficulty.
The $\{c_\ell\}$–sequence: For instance, one may show that the term of the order ${\mathcal O}(n^{-(\ell+2)})$ in Eq. (\[g-exp-1n\]) equals $$\begin{aligned}
(-1)^\ell \Gamma(\ell) \,\frac{c_\ell(\nu)}{(4\nu)^{\ell+2}},\end{aligned}$$ where $$\begin{aligned}
\label{c-addition}
c_\ell(\nu) &=& \frac{\ell}{96} \Big[ \Big(
12\nu^2 (2\ell+3) +3 (3\ell^2-4)\Big)\nonumber\\ &-& (-1)^\ell (1-4\nu^2) \Big( 4\nu^2 (\ell+1)(\ell-7) - 3
(3\ell^2-4)\Big) \Big].\end{aligned}$$ Let us reiterate that the above $1/n$ expansion of $\kappa_\ell(G)$ refers to finite $\ell$.
Equations (\[g-exp-1n\]) – (\[c-addition\]) extend earlier asymptotic results by Brouwer and Beenakker (1996), and Savin, Sommers and Wieczorek (2008) to the case of cumulants of arbitrary finite order. For an extension of the above analysis to other Dyson’s symmetry classes ($\beta=1$ and $4$), the reader is referred to the recent study by Mezzadri and Simm (2013).
Integrable theory of noise fluctuations in chaotic cavities with ideal leads {#itnf}
============================================================================
Charge fluctuations and the noise power {#charge-f}
---------------------------------------
The charge transfer through a phase-coherent cavity exhibiting chaotic classical dynamics is a random process influenced by discreteness of the electron charge $e$ and the quantum nature of electrons (Blanter and Büttiker 2000, Imry 2002). Fluctuations of charge transmitted during a fixed time interval or, equivalently, fluctuations $\delta I(t)$ of current around its mean are quantified by the noise power (Lesovik 1989, Büttiker 1990) $$\begin{aligned}
{\mathcal P} = 2\int_{-\infty}^{+\infty} dt\, \left< \delta I(t+t_0) \delta I (t_0)
\right>_{t_0},\end{aligned}$$ where the brackets $\left<\cdots \right>_{t_0}$ indicate averaging over the reference time $t_0$.
At temperatures $\theta = k_B T$ which are much larger than a bias voltage ${\upsilon}=eV$ applied to the cavity ($\theta \gg
\upsilon$), the current fluctuations are dominated by the equilibrium [*thermal noise*]{}, also known as Johnson-Nyquist noise. Caused by fluctuating occupation numbers in a flow of carriers injected into cavity from electronic reservoirs, thermal noise extends over all frequencies up to the quantum limit $\theta/h$. In the absence of electron-electron interactions, its power at zero bias voltage ($\upsilon =0$) is related to the scattering matrix ${\bm {\mathcal S}}$ of the system composed of the cavity and the leads (Khlus 1987, Lesovik 1989, Büttiker 1990, Martin and Landauer 1992, Büttiker 1992): $$\begin{aligned}
\label{pth}
{\mathcal P}_{\rm th}(\theta) = 4\theta\,G_0\,
{\rm tr} ({\mathcal C}_1 {\bm {\mathcal S}} {\mathcal C}_2 {\bm {\mathcal S}}^\dagger).\end{aligned}$$ Here, $G_0= e^2/h$ is the conductance quantum. The projection matrices ${\mathcal C}_{1,2}$ encoding the information about particular cavity-lead geometry are specified in Eq. (\[c1-c2\]).
In the opposite limit of low temperatures ($\theta \ll \upsilon$), the current fluctuations are still significant even though the flow of incident electrons is essentially noiseless. In this temperature regime, nonequilibrium current fluctuations (known as a [*shot noise*]{}) exist because of (i) the granularity of the electron charge $e$ and (ii) the stochastic nature of electron scattering inside the cavity which splits the electron wave into two or more partial waves leaving the cavity through different exits. It is this “uncertainty of not knowing where the electron came from and where it will go to” (Oberholzer [*et al*]{} 2002) that makes the transmitted charge to fluctuate. At zero temperature, the scattering matrix approach brings the shot noise power in the form $$\label{pshot}
{\mathcal P}_{\rm shot}(\upsilon) = 2 \upsilon\,G_0\left[
{\rm tr} ({\mathcal C}_1 {\bm {\mathcal S}} {\mathcal C}_2 {\bm {\mathcal S}}^\dagger)
- {\rm tr} ({\mathcal C}_1 {\bm {\mathcal S}} {\mathcal C}_2 {\bm {\mathcal S}}^\dagger)^2
\right].$$
At finite temperatures, both sources of noise are operative, the total noise ${\mathcal P}(\theta,\upsilon)$ being a complicated function of temperature and bias voltage [^4]: $$\begin{aligned}
\fl
\label{ptotal}
{\mathcal P}(\theta,\upsilon) = 4 \theta \, G_0\,
\Big(
{\rm tr} ({\mathcal C}_1 {\bm {\mathcal S}}
{\mathcal C}_2 {\bm {\mathcal S}}^\dagger)^2
+ \frac{\upsilon}{2\theta} \, {\rm coth}
\left(
\frac{\upsilon}{2\theta}
\right)
\left[
{\rm tr} ({\mathcal C}_1 {\bm {\mathcal S}} {\mathcal C}_2 {\bm {\mathcal S}}^\dagger)
- {\rm tr} ({\mathcal C}_1 {\bm {\mathcal S}} {\mathcal C}_2 {\bm {\mathcal S}}^\dagger)^2
\right] \Big).\end{aligned}$$ Equation (\[ptotal\]) suggests that the crossover from thermal noise ${\mathcal P}_{\rm th}(\theta)= {\mathcal P}(\theta,0)$ to shot noise ${\mathcal P}_{\rm shot}(\upsilon)= {\mathcal
P}(0,\upsilon)$ depends in a sensitive way on scattering properties of the cavity and the leads incorporated in the scattering matrix ${\bm {\mathcal S}}$. Since chaotic scattering of electrons inside the cavity induces fluctuations of ${\bm {\mathcal S}}$-matrix, the noise power ${\mathcal P}(\theta,\upsilon)$ fluctuates, too.
So far, the thermal to shot noise crossover has only been studied at the level of [*average*]{} noise power. For the two-terminal scattering geometry comprised of the cavity attached to outside reservoirs (kept at temperature $\theta$) via two leads supporting $N_{\rm L}$ and $N_{\rm R}$ propagating modes, respectively, the average noise power equals (Blanter and Sukhorukov 2000, Oberholzer [*et al*]{} 2001, Savin and Sommers 2006) $$\label{paverage}
\left<{\mathcal P}(\theta,\upsilon)\right>_{{\bm {\mathcal S}}} =
\left<{\mathcal P}_{\rm th}\right>_{{\bm {\mathcal S}}}
\left[
1 + \frac{N_{\rm L}N_{\rm R}}{(N_{\rm L}+N_{\rm R})^2-1}\, f_\eta
\right],$$ where $$\label{peq}
\left<{\mathcal P}_{\rm th}\right>_{{\mathcal S}} =
4\theta\, G_0 \frac{N_{\rm L} N_{\rm R}}{N_{\rm L}+N_{\rm R}}$$ is the average equilibrium thermal noise power, and the thermodynamic function $$\begin{aligned}
\label{fbeta}
f_\eta = \eta \coth \,\eta -1\end{aligned}$$ depends on the ratio $$\begin{aligned}
\label{beta-def}
\eta=\frac{\upsilon}{2\theta}\end{aligned}$$ between the bias voltage $\upsilon$ and the temperature $\theta$. Equations (\[paverage\]) and (\[peq\]) hold for cavities with broken time reversal symmetry; the two can readily be extended to other symmetry classes (Savin and Sommers 2006). Derived for the universal transport regime (Beenakker 1997, Richter and Sieber 2002, Müller [*et al*]{} 2007) emerging in the limit $\tau_{\rm D} \gg
\tau_{\rm E}$ (Agam [*et al*]{} 2000), where $\tau_{\rm D}$ is the average electron dwell time and $\tau_{\rm E}$ is the Ehrenfest time (the time scale where quantum effects set in), the above prediction has been confirmed in a remarkable series of experiments (Oberholzer [*et al*]{} 2001, Cron[*et al*]{} 2001, Oberholzer [*et al*]{} 2002).
Below, we examine [*statistics*]{} of the thermal to shot noise crossover. The latter, contained in the distribution function of the noise power ${\mathcal P}(\theta,\upsilon)$ or, equivalently, in its [*cumulants*]{}, can effectively be described within the framework of the formalism exposed in Section 2.
Statistics of shot noise fluctuations and Painlevé V (easy way)
---------------------------------------------------------------
[*Symmetric leads.*]{}—To start with, we shall attempt extending an ‘easy way’ approach outlined in Section 2 to nonperturbatively describe statistics of noise fluctuations at $T=0$ when the thermal noise contribution vanishes. According to Eq. (\[pshot\]), taken in conjunction with reflection-transmission-decomposed scattering matrix \[Eq. (\[s-block\])\], the MGF $$\begin{aligned}
\tilde{{\mathcal F}}_n^{(\nu)}(z) = \left<
\exp \left(z\, P_{\rm shot}\right)
\right>_{\bm{\mathcal{S}} \in {\rm CUE}(N_{\rm L}+N_{\rm R})}\end{aligned}$$ for dimensionless shot-noise power $$\begin{aligned}
P_{\rm shot}({\bm T}) = (2\upsilon G_0)^{-1} {\mathcal P}_{\rm shot}(\upsilon) = \sum_{j=1}^n T_j (1-T_j)\end{aligned}$$ takes the form $$\begin{aligned}
\label{sn-mgf}
\tilde{{\mathcal F}}_n^{(\nu)}(z) = c_{n,\nu}^{-1} \int_{(0,1)^n} \prod_{j=1}^n dT_j \, T_j^\nu \exp \left(
z\, T_j(1-T_j)
\right) \cdot \Delta_n^2({\bm T}).\end{aligned}$$ Similarly to the MGF for Landauer conductance, we make use of the Andréief–de Bruijn integration formula to reduce $\tilde{{\mathcal F}}_n^{(\nu)}(z)$ to the Hankel determinant $$\begin{aligned}
\hspace{-1cm}
\tilde{{\mathcal F}}_n^{(\nu)}(z) = \frac{n!}{c_{n,\nu}} \,{\rm det}
\left[
\int_0^1 dT \, T^{\nu+j+k} \exp \left(
z\, T(1-T)
\right)\right]_{(j,k)\in (0,n-1)}\end{aligned}$$ or, equivalently, $$\begin{aligned}
\label{hd-2}\hspace{-1.5cm}
\tilde{{\mathcal F}}_n^{(\nu)}(z) = \frac{n!}{c_{n,\nu}} \,e^{nz/4} {\rm det}
\left[
\int_{-1/2}^{+1/2} dx \, \left(x+\frac{1}{2}\right)^{\nu} \, x^{j+k} e^{-z x^2}
\right]_{(j,k)\in (0,n-1)}.\end{aligned}$$ Unfortunately, for a generic $\nu$, there is no obvious way to represent the $(j,k)$ entry in Eq. (\[hd-2\]) in an operator form similar to the one appearing in Eq. (\[hd\]). However, for the case of [*symmetric*]{} leads ($\nu=0$), one can readily spot that $\tilde{{\mathcal F}}_n^{(0)}(z)$ reduces to the product of two Hankel determinants, each of them possessing the desired operator structure: $$\begin{aligned}
\label{hd-f}
\tilde{{\mathcal F}}_n^{(0)}(z) =
e^{n z/4} {\tilde {\mathcal F}}_{\lfloor n/2 \rfloor}^{\,+} (z)
{\tilde {\mathcal F}}_{\lceil n/2 \rceil}^{\,-} (z),\end{aligned}$$ where $$\begin{aligned}
\hspace{-1cm}
{\tilde {\mathcal F}}_\ell^{\,\pm} (z) =
c_{\ell,\pm 1/2}^{-1} \, \ell!\, {\rm det} \left[ \int_0^1
d\lambda \, \lambda^{\pm 1/2} e^{-(z/4)\lambda} \lambda^{j+k}
\right]_{(j,k)\in (0,\ell-1)}.\end{aligned}$$ The above factorisation is a consequence of the checkerboard structure of the moment matrix in Eq. (\[hd-2\]) at $\nu=0$, $$\begin{aligned}
\int_{-1/2}^{+1/2} dx \, x^{j+k} e^{-z x^2} \propto \delta_{j+k,{\rm even}}.\end{aligned}$$ In a backward move, we have $$\begin{aligned}
{\tilde {\mathcal F}}_\ell^{\,\pm} (z) =
c_{\ell,\pm 1/2}^{-1} \int_{(0,1)^\ell} \prod_{j=1}^\ell
d\lambda_j \, \lambda_j^{\pm 1/2} e^{-(z/4)\lambda_j} \cdot \Delta_\ell^2({\bm \lambda}).\end{aligned}$$ The latter, being essentially a gap formation probability in the LUE ensemble, can conveniently be expressed in terms of the Landauer conductance MGF \[see Eq. (\[cond-eig-nod\])\] taken at effective asymmetry parameter $\nu = \pm 1/2$, $$\begin{aligned}
{\tilde {\mathcal F}}_\ell^{\,\pm} (z) = {\mathcal F}_\ell^{(\pm 1/2)} \left(\frac{z}{4}\right).\end{aligned}$$ As the result, we observe the relation $$\begin{aligned}
\label{hd-ff}
\tilde{{\mathcal F}}_n^{(0)}(z) =\,
e^{n z/4} {\mathcal F}_{\lfloor n/2 \rfloor}^{(+1/2)} \left(\frac{z}{4}\right)
{\mathcal F}_{\lceil n/2 \rceil}^{(-1/2)} \left(\frac{z}{4}\right)\end{aligned}$$ which, if taken together with the Painlevé V representation of ${\mathcal F}_n^{(\nu)}(z)$ \[Eqs. (\[FnP-22\]), (\[pv-22\]) and (\[pv-bc\])\], yields the shot noise MGF in the form $$\begin{aligned}
\label{FnP-33}\hspace{-1.5cm}
\tilde{{\mathcal F}}_n^{(0)}(z) = \, e^{n z/4} \exp\left(
\int_0^{z/4} dt \frac{\sigma_{\lfloor n/2 \rfloor,+1/2}(t) + \sigma_{\lceil n/2 \rceil,-1/2}(t) - n^2/2}{t}
\right).\end{aligned}$$ We remind that the above result holds for the symmetric chaotic cavity probed via symmetric leads, $N_{\rm L} = N_{\rm R}=n$. Owing to the cumulant expansion $$\begin{aligned}
\log \tilde{{\mathcal F}}_n^{(0)}(z) = \sum_{\ell=1}^\infty \frac{z^\ell}{\ell!}\, \kappa_\ell^{(n,0)}(P_{\rm shot}),\end{aligned}$$ we are now able to express the shot noise cumulants $\kappa_\ell^{(n,0)}(P_{\rm shot})$ through “effective cumulants” $\kappa_\ell ^{(n,\nu)}(G)$ of the Landauer conductance (with $\nu = \pm 1/2$) defined by the Taylor expansion Eq. (\[main-cond-cum\]). Straightforward calculation brings $$\begin{aligned}
\fl
\kappa_\ell^{(n,0)}(P_{\rm shot}) = \frac{n}{4}\delta_{\ell,1}+\frac{(-1)^\ell}{4^\ell} \left(
\kappa_\ell^{(\lceil n/2 \rceil,-1/2)}(G) + \kappa_\ell^{(\lfloor n/2 \rfloor,+1/2)}(G)
\right),\end{aligned}$$ where $\kappa_\ell^{(m,\pm 1/2)}(G)$ are the solutions to the recurrence equation Eq. (\[cumeq\]) taken at $n=m$ and $\nu = \pm 1/2$, and supplemented by appropriate initial conditions \[Eq. (\[ic-cond\])\]. In particular, one has: $$\begin{aligned}
\label{psh-1}
\kappa_1^{(n,0)}(P_{\rm shot}) &=& \frac{n^3}{2(4n^2-1)}, \\
\label{psh-2}
\kappa_2^{(n,0)}(P_{\rm shot}) &=& \frac{n^2 (4n^4 -9n^2+3)}{8(4 n^2-1)^2 (4n^2-9)},\\
\label{psh-3}
\kappa_3^{(n,0)}(P_{\rm shot}) &=& \frac{n^2 (16 n^6 -24 n^4+9 n^2 +1)}{128(4 n^2-1)^4}.\end{aligned}$$ Whenever an overlap exists, Eqs. (\[psh-1\]) – (\[psh-3\]) coincide with the earlier results by Savin and Sommers (2006) and Savin, Sommers and Wieczorek (2008). Nonperturbative formulae for higher order cumulants of the shot noise can readily be generated as well, even though the corresponding expressions become increasingly cumbersome. —Unfortunately, a transparent calculational framework outlined above fails to accommodate statistics of shot noise fluctuations in the cavities probed via [*asymmetric*]{} leads when the factorisation Eq. (\[hd-f\]) is no longer available. Also, an attempt to formulate a $\tau$ function theory \[see Section 2.3\] of the eigenvalue integral Eq. (\[sn-mgf\]) considered at $\nu \neq 0$ brings no fruit: projecting the first KP equation for the associate $\tau$ function onto the hyperplane ${\bm t}={\bm 0}$ with the help of Virasoro constraints turns out to be a nontrivial task since an [*infinitely many*]{} Virasoro constraints appear to be relevant. In the forthcoming Section, we show how this technical obstacle can be overcome by considering a joint distribution of the Landauer conductance and the noise power.
Joint moment generating function of Landauer conductance and the noise power
----------------------------------------------------------------------------
A way out of this difficulty was reported by Osipov and Kanzieper (2009) where it was demonstrated that the shot noise moment generating function can be restored from a two-dimensional lattice equation for the [*joint*]{} moment generating function (JMGF) of the Landauer conductance and the shot noise power. To make a contact with experiments, below we consider a JMGF of the Landauer conductance and the noise power ${\mathcal P}(\theta,\upsilon)$ consisting of two competing contributions – the shot and the thermal noise (see Section \[charge-f\]).
The starting point of our analysis is the JMGF $$\begin{aligned}
\label{cgf-def}
{\mathcal F}_{n}^{(\nu)}(z,w) = \left<
\exp(-z \,G)\exp(- w\, P)
\right>_{{\bm {\mathcal S}}\in {\rm CUE}(N_{\rm L}+N_{\rm R})}\end{aligned}$$ of the dimensionless Landauer conductance $G = {\rm tr} ({\mathcal C}_1
{\bm {\mathcal S}} {\mathcal C}_2 {\bm {\mathcal S}}^\dagger)$ and the dimensionless noise power $$\begin{aligned}
P =
{\rm tr} ({\mathcal C}_1 {\bm {\mathcal S}}
{\mathcal C}_2 {\bm {\mathcal S}}^\dagger)^2
+ \eta \, {\rm coth}\, \eta\,
\left[
{\rm tr} ({\mathcal C}_1 {\bm {\mathcal S}} {\mathcal C}_2 {\bm {\mathcal S}}^\dagger)
- {\rm tr} ({\mathcal C}_1 {\bm {\mathcal S}} {\mathcal C}_2 {\bm {\mathcal S}}^\dagger)^2
\right]\end{aligned}$$ that corresponds to the noise power ${\mathcal P}(\theta,\upsilon)$ measured in the units $4\theta G_0$. The notation ${\bm {\mathcal S}} \in {\rm CUE}(N_{\rm L}+N_{\rm R})$ indicates that averaging runs over scattering matrices ${\bm {\mathcal S}}$ drawn from the Dyson circular unitary ensemble as discussed in Section \[intro\]. The joint dimensionless cumulants $$\begin{aligned}
\label{gp-cum}
\kappa_{\ell,m}^{(n,\nu)}(G,P) =
\langle\!\langle G^\ell P^m
\rangle\!\rangle\end{aligned}$$ can be extracted from the expansion $$\label{c-def}
\log {\mathcal F}_n^{(\nu)}(z,w) = \sum_{\ell,m=0}^\infty (-1)^{\ell+m} \frac{z^\ell w^m}{\ell!\,m!}\,
\kappa_{\ell,m}^{(n,\nu)}(G,P),$$ where $\kappa_{0,0}\equiv 0$ and the subscript $n$ stands for $n=\min(N_{\rm
L},N_{\rm R})$. We notice in passing that a somewhat simpler joint cumulant of Landauer conductance and the [*shot*]{} noise power $$\begin{aligned}
\tilde{\kappa}_{\ell,m}^{(n,\nu)}(G,P_{\rm shot}) =
\langle\!\langle G^\ell P_{\rm shot}^m
\rangle\!\rangle\end{aligned}$$ frequently considered in the literature can be obtained from Eq. (\[gp-cum\]) via the limiting procedure $$\begin{aligned}
\label{lim-rel}
\tilde{\kappa}_{\ell,m}^{(n,\nu)}(G,P_{\rm shot}) = \lim_{\eta\rightarrow \infty} \frac{1}{\eta^m} \, \kappa_{\ell,m}^{(n,\nu)}(G,P).\end{aligned}$$ To perform the averaging in Eq. (\[cgf-def\]) in a most economic way, we employ a decomposition Eq. (\[s-block\]) to bring into play a set of $n$ transmission eigenvalues ${\bm T} = (T_1,\cdots,T_n) \in
(0,1)^n$ distributed in accordance with the joint probability density function given by Eq. (\[PnT\]) and consequently derive the JMGF in the form : $$\label{Fnzw}
{\mathcal F}_n^{(\nu)}(z,w) = c_{n,\nu}^{-1}\int_{(0,1)^n} \prod_{j=1}^n dT_j\,T_j^\nu \,\Gamma_{z,w}(T_j)
\,\Delta_n^2({\bm T}),$$ where $$\begin{aligned}
\label{Gamma}
\Gamma_{z,w}(T) =
\exp\left[
-(z+w)\,T - w\,f_\eta\, T(1-T)
\right].\end{aligned}$$ Here, the exponent contains weighted contributions from the conductance $G({\bm T}) = \sum_{j=1}^n T_j$ and the noise power $$\begin{aligned}
\label{ptotal-T}
P({\bm T})
=
\sum_{j=1}^n T_j + f_\eta \sum_{j=1}^n T_j(1-T_j).\end{aligned}$$ The thermodynamic function $f_\eta$ is defined by Eq. (\[fbeta\]). We also recall that parameter $\nu$ in Eq. (\[Fnzw\]) is a measure of asymmetry between the leads \[Eq. (\[nu-index\])\], the notation $\Delta_n({\bm T})$ stands for the Vandermonde determinant $\Delta_n ({\bm T}) =\prod_{j<k} (T_k-T_j)$, whilst $c_{n,\nu}$ is a normalisation constant specified by Eq. (\[nc\]).
Although the above matrix integral representation of the JMGF ${\mathcal F}_n^{(\nu)}(z,w)$ is by far more complicated than the one appearing in the integrable theory of conductance fluctuations (see Section 2), $$\begin{aligned}
\fl \label{ccgf}
{\mathcal F}_n^{(\nu)}(z,0) =
\left<
\exp(-z \,G)\right>_{{\bm {\mathcal S}}\in {\rm CUE}(N_{\rm L}+N_{\rm R})}
=
c_{n,\nu}^{-1}\int_{(0,1)^n} \prod_{j=1}^n dT_j\,T_j^\nu \,e^{-z T_j}
\,\Delta_n^2({\bm T}), \nonumber\\
&&\end{aligned}$$ it can still be treated nonperturbatively.
The $\tau$ function theory of the joint moment generating function
------------------------------------------------------------------
### The KP equation and Virasoro constraints {#s-223a}
To determine ${\mathcal F}_n^{(\nu)}(z,w)$ nonperturbatively, we define the $\tau$ function $$\label{tau-def}
\tau_n^{(\nu)}({\bm t};z,w) = \frac{1}{n!}\int_{(0,1)^n} \prod_{j=1}^n dT_j\,T_j^\nu \,\Gamma_{z,w}(T_j)
\,e^{V({\bm t};T_j)} \Delta_n^2({\bm T})$$ whose evolution is governed by the bilinear identity Eq. (\[bi-id\]). The latter generates a wealth of nonlinear differential hierarchical relations between various $\tau$ functions in the $(n, {\bm t},z,w)$ space. Of those, we are interested in the first KP equation Eq. (\[fkp\]) because its projection onto the hyperplane ${\bm t}={\bm 0}$, $$\begin{aligned}
\label{proj}
{\mathcal F}^{(\nu)}_n(z,w) = \frac{n!}{c_{n,\nu}}\,
\tau_n^{(\nu)}({\bm t};z,w)\Big|_{{\bm t}={\bm 0}},\end{aligned}$$ will generate a closed nonlinear differential equation for the JMGF ${\mathcal F}_n^{(\nu)}(z,w)$. It is this equation that will further be utilized to determine the noise power cumulants we are aimed at.
Since we are interested in deriving a differential equation for ${\mathcal F}_n^{(\nu)}(z,w)$ in terms of the derivatives with respect to variables $z$ and $w$, we have to use the [*Virasoro constraints*]{}, much in line with our treatment of conductance cumulants in Section 2. Demanding the invariance of $\tau_n^{(\nu)}({\bm t};z,w)$ under the same set of transformations Eq. (\[v-tr\]), one readily derives the following set of Virasoro constraints: $$\label{vc-1}
\big[ \hat{L}_{q+1}({\bm t}) - \hat{L}_{q}({\bm t})\big] \tau_n^{(\nu)}({\bm t};z,w)
= 0,\;\;\; q\ge 0,$$ where $$\begin{aligned}
\fl
\label{vc-2}
\hat{L}_q({\bm t})
=
\hat{\mathcal L}_{q}({\bm t}) + 2f_\eta\, w
\frac{\partial}{\partial t_{q+2}} - [z+(1+f_\eta)\,w] \frac{\partial}{\partial t_{q+1}}
+ \nu \frac{\partial}{\partial t_q}\end{aligned}$$ involves the Virasoro operators Eq. (\[vc-3\]) satisfying the Virasoro algebra Eq. (\[va\]). The usual convention $\partial/\partial t_0 \equiv n$ is everywhere assumed.
### Joint MGF and the Toda Lattice equation {#TL-223}
Similarly to the MGF for Landauer conductance, the joint MGF ${\mathcal F}_n^{(\nu)}(z,w)$ can be shown to satisfy an one-dimensional Toda Lattice equation. For one, it can be derived from the first equation of the TL hierarchy \[Eq. (\[ftl\])\], the identity Eq. (\[id-11\]) below, and the projection relation Eq. (\[proj\]), $$\label{TL-GP}\fl
{\mathcal F}_n^{(\nu)}(z,w)\,\frac{\partial^2 {\mathcal F}_n^{(\nu)}(z,w)}{\partial z^2} - \left(
\frac{\partial {\mathcal F}_n^{(\nu)}(z,w)}{\partial z}\right)^2 = {\rm var}_{n,\nu}(G)\,
{\mathcal F}_{n-1}^{(\nu)}(z,w)\, {\mathcal F}_{n+1}^{(\nu)}(z,w),$$ where ${\rm var}_{n,\nu}(G)$ is the conductance variance. Initial conditions read: $$\begin{aligned}
{\mathcal F}_0^{(\nu)}(z,w) = 1\end{aligned}$$ and $$\begin{aligned}
{\mathcal F}_1^{(\nu)}(z,w) = (\nu+1) \int_0^1 dT\, T^\nu e^{-(z+w)T - w f_\eta T(1-T)}.\end{aligned}$$ Iterating the TL equation $n$ times and calculating a double inverse Laplace transform, one may in principle restore a joint distribution function of the conductance and the the noise power. However, such a procedure is by far more complicated technically as compared to the conductance distribution (see Section \[TL-Hankel\]).
### Nonlinear differential equation for the joint MGF {#s-223}
Aiming at deriving a differential equation for the JMGF ${\mathcal
F}_n^{(\nu)}(z,w)$, we project the first KP equation Eq. (\[fkp\]) onto the hyperplane ${\bm t}={\bm 0}$. Spotting the identities ($f_\eta>0$) $$\begin{aligned}
\label{id-11}
\frac{\partial}{\partial t_1} \tau_n^{(\nu)}({\bm t};z,w)&=& - \frac{\partial}{\partial z} \tau_n^{(\nu)}({\bm t};z,w),\\
f_\eta \frac{\partial}{\partial t_2} \tau_n^{(\nu)}({\bm t};z,w) &=& \frac{\partial}{\partial w} \tau_n^{(\nu)}({\bm t};z,w)
- (1+f_\eta)\frac{\partial}
{\partial z} \tau_n^{(\nu)}({\bm t};z,w),\end{aligned}$$ we combine Eqs. (\[fkp\]) with the Virasoro constraint Eq. (\[vc-1\]) taken at $q=0$, $$\begin{aligned}
\fl
\Bigg[ \sum_{j=1}^\infty jt_j \left(
\frac{\partial}{\partial t_{j+1}}
-
\frac{\partial}{\partial t_{j}}
\right)
+ 2f_\eta\, w \left(
\frac{\partial}{\partial t_{3}} -
\frac{\partial}{\partial t_{2}}
\right) \nonumber \\ \fl
\quad - \left[z+(1+f_\eta)\,w\right] \left(
\frac{\partial}{\partial t_{2}}
-
\frac{\partial}{\partial t_{1}}
\right)
+ (2n+\nu) \frac{\partial}{\partial t_1}
- n(n+\nu) \Bigg]\, \tau_n^{(\nu)}({\bm t};z,w)=0,\end{aligned}$$ to derive: $$\begin{aligned}
\fl
\label{ndeq}
\Bigg[ w f_\eta^2 \frac{\partial^4}{\partial z^4} + \left[2(2n+\nu)f_\eta - 2 z + w\,(1-f_\eta^2)\right]
\frac{\partial^2}{\partial z^2} + 2(z-2w)\frac{\partial^2}{\partial z \partial w}
\nonumber \\ \fl
\quad + 3 w \frac{\partial^2}{\partial w^2} + 2 \left(\frac{\partial}{\partial w} - \frac{\partial}{\partial z}\right) \Bigg]
\,\log {\mathcal F}_n^{(\nu)}(z,w)+ 6 w\,f_\eta^2 \left(
\frac{\partial^2}{\partial z^2} \, \log {\mathcal F}_n^{(\nu)}(z,w)
\right)^2
= 0. \nonumber\\
{}\end{aligned}$$ Owing to Eq. (\[c-def\]), the nonlinear equation Eq. (\[ndeq\]) supplemented by the initial condition $$\begin{aligned}
{\mathcal F}_n^{(\nu)}(z,w=0) = \exp\left(
\int_0^z dt \, \frac{\sigma_{n,\nu}(t) -n(n+\nu)}{t}
\right),\end{aligned}$$ see Eqs. (\[sF-cc\]) and (\[OK-2008\]) \[or (\[pv-22\])\], contains all the information about joint cumulants of the Landauer conductance and the noise power. Although finding explicit formulae expressing ${\mathcal F}_n^{(\nu)}(z,w)$ in terms of the fifth Painlevé transcendent $\sigma_{n,\nu}$ is an unviable task, the joint cumulants $\kappa_{\ell,m}^{(n,\nu)}(G,P)$ can nonetheless be related to the cumulants of Landauer conductance $\kappa_\ell^{(n,\nu)}(G)$. This will be done in the next Section.
Joint cumulants of Landauer conductance and the noise power
-----------------------------------------------------------
### Exact recurrence solution for joint cumulants
Indeed, combining the cumulant expansion Eq. (\[c-def\]) with the differential equation Eq. (\[ndeq\]), we deduce after some algebra a nonlinear recurrence for the joint dimensionless cumulants of conductance and noise power ($\ell,m \ge 0$): $$\begin{aligned}
\label{2drec} \fl
m\,\Big[f_\eta^2\, \kappa_{\ell+4,m-1}^{(n,\nu)}(G,P) + (1-f_\eta^2) \,\kappa_{\ell+2,m-1}^{(n,\nu)}(G,P)\Big] - 2 \,
(2n+\nu)\, f_\eta \,\kappa_{\ell+2,m}^{(n,\nu)}(G,P) \nonumber \\
\fl \qquad\quad
- 2\left(\ell+ 2 m + 1 \right) \, \kappa_{\ell+1,m}^{(n,\nu)}(G,P) + (2\ell+3m+2) \, \kappa_{\ell,m+1}^{(n,\nu)}(G,P)
\nonumber\\
\fl \qquad\quad
+ \, 6m \,f_\eta^2
\sum_{i=0}^{m-1}\left( {m-1}\atop{i} \right) \sum_{j=0}^\ell \left( {\ell}\atop{j} \right)
\kappa_{j+2,i}^{(n,\nu)}(G,P)\, \kappa_{\ell-j+2,m-i-1}^{(n,\nu)}(G,P)=0. \nonumber\\
{}\end{aligned}$$ By virtue of the limiting relation Eq. (\[lim-rel\]), one readily generates a nonlinear recurrence for zero-temperature joint cumulants of Landauer conductance and the shot noise power: $$\begin{aligned}
\label{2drec-shot} \fl
m\,\Big[\tilde{\kappa}_{\ell+4,m-1}^{(n,\nu)}(G,P_{\rm shot}) - \tilde{\kappa}_{\ell+2,m-1}^{(n,\nu)}(G,P_{\rm shot})\Big] - 2 \,
(2n+\nu)\, \tilde{\kappa}_{\ell+2,m}^{(n,\nu)}(G,P_{\rm shot}) \nonumber \\
\fl \qquad\quad
+ (2\ell+3m+2) \, \tilde{\kappa}_{\ell,m+1}^{(n,\nu)}(G,P_{\rm shot})
\nonumber\\
\fl \qquad\quad
+ \, 6m \,
\sum_{i=0}^{m-1}\left( {m-1}\atop{i} \right) \sum_{j=0}^\ell \left( {\ell}\atop{j} \right)
\tilde{\kappa}_{j+2,i}^{(n,\nu)}(G,P_{\rm shot})\, \tilde{\kappa}_{\ell-j+2,m-i-1}^{(n,\nu)}(G,P_{\rm shot})=0. \nonumber\\
{}\end{aligned}$$ To resolve these two-dimensional recursion equations, one must know the cumulants $\kappa_{\ell,0}^{(n,\nu)} = \tilde{\kappa}_{\ell,0}^{(n,\nu)}= \kappa_\ell^{(n,\nu)}(G)$ of dimensionless conductance which play a rôle of boundary conditions. Since these are known \[see Eqs. (\[cumeq\])\], Eqs. (\[2drec\]) and (\[2drec-shot\]) provide a nonperturbative solution to the problem of noise power fluctuations by relating the latter to the cumulants of Landauer conductance. \[For Dyson’s symmetry indices $\beta=1$ and $4$, an analogue of Eq. (\[2drec-shot\]) was recently derived by Mezzadri and Simm (2013).\]
Undoubtedly, the very existence of the above nontrivial relation (which emphasises a fundamental rôle played by Landauer conductance in transport problems) must be well rooted in the mathematical formalism and also have a good physics reason. As far as the former point is concerned, we wish to stress that a naïve attempt to build a theory for the generating function ${\mathcal
F}_n^{(\nu)}(0,w)$ of solely noise power cumulants faces an unsurmountable obstacle: the KP equation \[Eq. (\[fkp\])\] and appropriate Virasoro constraints \[Eq. (\[vc-1\]) at $z=0$\] cannot be resolved jointly in the hyperplane ${\bm t}={\bm 0}$. This justifies the starting point \[Eq. (\[cgf-def\])\] of our analysis. The physics arguments behind the peculiar structure of our solution are yet to be found.
### Noise power cumulants
Some computational effort is required to read off explicit formulae for the noise power cumulants from Eqs. (\[2drec\]) and (\[cumeq\]). Below we provide expressions for two families of joint cumulants expressed in terms of dimensionless cumulants $\kappa_{\ell}^{(n,\nu)}(G) =\langle \! \langle G^\ell \rangle\!\rangle$ of the Landauer conductance. Mean noise power is generated by the lowest order member $(\ell,m)=(0,0)$ of the recurrence Eq. (\[2drec\]); it reads: $$\begin{aligned}
\label{mnp}
\kappa_1^{(n,\nu)}(P) = (2n+\nu) f_\eta\,
\kappa_{2}^{(n,\nu)}(G) + \kappa_{1}^{(n,\nu)}(G).\end{aligned}$$ Being in concert with the known expression Eqs. (\[paverage\]) and (\[peq\]), this result is a particular case of a more general formula $$\begin{aligned}
\label{k-l-1}
\kappa_{\ell,1}^{(n,\nu)}(G,P) = \kappa_{\ell+1}^{(n,\nu)}(G) + (2n+\nu)\frac{f_\eta}{\ell+1}\, \kappa_{\ell+2}^{(n,\nu)}(G).
\end{aligned}$$ Mean shot noise power can be derived by virtue of Eq. (\[lim-rel\]); we deduce: $$\begin{aligned}
\label{mnp-shot}
\kappa_1^{(n,\nu)}(P_{\rm shot}) = (2n+\nu)\, \kappa_{2}^{(n,\nu)}(G) = \frac{n^2 (n+\nu)^2}{(2n+\nu)
[(2n+\nu)^2-1]}\end{aligned}$$ and $$\begin{aligned}
\label{k-l-1-shot}
\kappa_{\ell,1}^{(n,\nu)}(G,P_{\rm shot}) = \frac{2n+\nu}{\ell+1}\, \kappa_{\ell+2}^{(n,\nu)}(G).
\end{aligned}$$ Compare Eq. (\[mnp-shot\]) with Eq. (\[psh-1\]). Noise power variance is supplied by the $(\ell,m)=(0,1)$ member of the recurrence, $$\begin{aligned}
\label{p-cum-2}\fl
\kappa_2^{(n,\nu)}(P) =
\left(
\frac{2}{3}(2n+\nu)^2 -1
\right) \frac{f_\eta^2}{5} \,\kappa_{4}^{(n,\nu)}(G) + (2n+\nu) f_\eta\, \kappa_{3}^{(n,\nu)}(G)
\nonumber\\
\qquad \qquad +\left(
1+ \frac{f_\eta^2}{5}
\right)\, \kappa_{2}^{(n,\nu)}(G) - \frac{6}{5} f_\eta^2 [\kappa_{2}^{(n,\nu)}(G)]^2.\end{aligned}$$ Its generalisation reads: $$\begin{aligned}
\fl \label{k-l-2}
\kappa_{\ell,2}^{(n,\nu)}(G,P) = \left(
\frac{2(2n+\nu)^2}{\ell+3}-1
\right) \frac{f_\eta^2}{2\ell+5}\kappa_{\ell+4}^{(n,\nu)}(G) + 2(2n+\nu) \frac{f_\eta}{\ell+2}\kappa_{\ell+3}^{(n,\nu)}(G)
\nonumber\\
+\left(
1+ \frac{f_\eta^2}{2\ell+5}
\right) \kappa_{\ell+2}^{(n,\nu)}(G)
- 6 \frac{f_\eta^2}{2\ell+5} \sum_{j=0}^\ell \left({\ell}\atop{j}\right) \kappa_{j+2}^{(n,\nu)}(G)\,\kappa_{\ell+2-j}^{(n,\nu)}(G).
\nonumber\\
{}\end{aligned}$$ Shot noise power variance, derived from Eqs. (\[p-cum-2\]) and (\[lim-rel\]), equals $$\begin{aligned}
\label{p-cum-2-shot}\fl
\qquad
\kappa_2^{(n,\nu)}(P_{\rm shot}) =
\frac{1}{5}\Bigg[
\left(
\frac{2}{3}(2n+\nu)^2 -1
\right) \,\kappa_{4}^{(n,\nu)}(G)
+ \kappa_{2}^{(n,\nu)}(G) - 6 [\kappa_{2}^{(n,\nu)}(G)]^2\Bigg]. \nonumber\\
{}\end{aligned}$$ Similarly, $$\begin{aligned}
\fl \label{k-l-2-shot}
\kappa_{\ell,2}^{(n,\nu)}(G,P_{\rm shot}) = \frac{1}{2\ell+5} \Bigg[\left(
\frac{2(2n+\nu)^2}{\ell+3}-1
\right) \,\kappa_{\ell+4}^{(n,\nu)}(G) +
\kappa_{\ell+2}^{(n,\nu)}(G) \nonumber\\
\qquad \qquad \qquad \qquad - 6 \sum_{j=0}^\ell \left({\ell}\atop{j}\right) \kappa_{j+2}^{(n,\nu)}(G)\,\kappa_{\ell+2-j}^{(n,\nu)}(G)\Bigg].\end{aligned}$$ Compare Eq. (\[p-cum-2-shot\]) with Eq. (\[psh-2\]).
The (joint) cumulants $\kappa_{\ell,m}^{(n,\nu)}(G,P)$ of higher order ($m \ge 3$) can be calculated in the same manner albeit explicit expressions become increasingly cumbersome.
### Asymptotic analysis of joint cumulants: Symmetric leads
The nonperturbative solution Eq. (\[2drec\]) has a drawback: it does not supply much desired [*explicit*]{} formula of joint conductance and noise power cumulants $\kappa_{\ell,m}^{(n,\nu)}(G,P)$. To probe the latter, we turn to the large-$n$ limit of the recurrence Eq. (\[2drec\]). In what follows, the asymmetry parameter $\nu$ will be set to zero when the joint cumulants $\kappa_{\ell,m}^{(n,0)}(G,P)$ are solutions to the recurrence equation ($\ell,m
\ge 0$) $$\begin{aligned}
\label{2drec=0} \fl
m\,\Big[f_\eta^2\, \kappa_{\ell+4,m-1}^{(n,0)}(G,P) + (1-f_\eta^2) \,\kappa_{\ell+2,m-1}^{(n,0)}(G,P)\Big] - 4n\,
f_\eta \,\kappa_{\ell+2,m}^{(n,0)}(G,P) \nonumber \\
\fl \qquad\qquad\quad
- 2\left(\ell+ 2 m + 1 \right) \, \kappa_{\ell+1,m}^{(n,0)}(G,P) + (2\ell+3m+2) \, \kappa_{\ell,m+1}^{(n,0)}(G,P)
\nonumber\\
\fl \qquad\qquad\quad
+ \, 6m \,f_\eta^2
\sum_{i=0}^{m-1}\left( {m-1}\atop{i} \right) \sum_{j=0}^\ell \left( {\ell}\atop{j} \right)
\kappa_{j+2,i}^{(n,0)}(G,P)\, \kappa_{\ell-j+2,m-i-1}^{(n,0)}(G,P)=0,\nonumber\\
{}\end{aligned}$$ supplemented by yet another recurrence ($\ell \ge 2$), see Eq. (\[cumeq\]), $$\begin{aligned}
\label{cumeq=0}
\fl (4n^2-\ell^2)\,(\ell+1) \kappa_{\ell+1,0}^{(n,0)}
+ 2n (2\ell-1)\,\ell \kappa_{\ell,0}^{(n,0)} \nonumber\\
\fl \qquad + \ell(\ell-1)(\ell-2)\,
\kappa_{\ell-1,0}^{(n,0)}
- 2 \sum_{j=0}^{\ell-1} (3j+1) (j-\ell)^2\left({\ell}\atop{j}\right)
\kappa_{j+1,0}^{(n,0)}\, \kappa_{\ell-j,0}^{(n,0)} = 0\end{aligned}$$ that brings, in turn, a set of initial conditions $\{\kappa_{\ell,0}^{(n,0)}\}=\{\kappa_\ell^{(n,0)}(G)\}$ to Eq. (\[2drec=0\]).
We start an asymptotic analysis of the recurrence Eq. (\[2drec=0\]) with singling out the large-$n$ Gaussian part from $\kappa_{\ell,m}^{(n,0)}(G,P)$: $$\begin{aligned}
\fl \label{klm-gauss}
\kappa_{\ell,m}^{(n,0)}(G,P) = \frac{n}{2} \left[
\delta_{\ell,1}\delta_{m,0} + \left(1+\frac{f_\eta}{4}\right)\delta_{\ell,0}\delta_{m,1}
\right]\nonumber\\
+ \frac{1}{16}
\left[\delta_{\ell,1}\delta_{m,1} +
\delta_{\ell,2}\delta_{m,0} + \left(
1+ \frac{f_\eta^2}{8}
\right)\delta_{\ell,0}\delta_{m,2}
\right] +\delta\kappa_{\ell,m}.\end{aligned}$$ The Gaussian part was read off from Eqs. (\[k-l-1\]) and (\[k-l-2\]); the term $\delta\kappa_{\ell,m}$ accommodates non-Gaussian corrections to the joint cumulants of Landauer conductance and the noise power. Similarly to the asymptotic analysis of Landauer cumulants (Section \[cum-fo\]), their large-$n$ behavior can be studied by employing the $1/n$ expansion $$\begin{aligned}
\label{ans-joint}
\delta\kappa_{\ell,m} = \frac{1}{n^{\ell+m}}
\left(
\alpha_{\ell,m} + \frac{\beta_{\ell,m}}{n} + \frac{\gamma_{\ell,m}}{n^2} + {\mathcal O}(n^{-3}) \right),\end{aligned}$$ where $\alpha_{1,0}=0$. Substitution of Eqs. (\[klm-gauss\]) and (\[ans-joint\]) into the two-dimensional recurrence Eq. (\[2drec=0\]) brings yet another recurrence equation ($\ell+m>0$): $$\begin{aligned}
\label{rr-a}\fl
m\left(
1 - \frac{f_\eta^2}{4}
\right)\, \alpha_{\ell+2,m-1} - 4 f_\eta \alpha_{\ell+2,m} \nonumber\\
-2 (\ell+1+2m) \alpha_{\ell+1,m} + (2\ell+2+3m)\,\alpha_{\ell,m+1}=0.\end{aligned}$$ Its (unique) solution, subject to the boundary condition (see Eq. (\[a-un-fa\]) in Section \[cum-fo\]) $$\begin{aligned}
\alpha_{\ell,0} =\frac{a_\ell(0)}{2^{2\ell}} =\frac{(\ell-1)!}{2^{2\ell+3}} \left[
1 + (-1)^\ell
\right],\end{aligned}$$ reads ($\ell+m>0$): $$\begin{aligned}
\alpha_{\ell,m} = \frac{(\ell+m-1)!}{2^{2(\ell+m)+3}}
\left[
\left(
\frac{f_\eta}{2}+1
\right)^m + (-1)^\ell \left(
\frac{f_\eta}{2}-1
\right)^m
\right].\end{aligned}$$ Taken together with Eq. (\[ans-joint\]), it yields the leading term in the $1/n$ expansion for joint cumulants of Landauer conductance and the noise power: $$\begin{aligned}
\label{ans-joint-answer}
\delta\kappa_{\ell,m} =\frac{1}{8}
\frac{(\ell+m-1)!}{(4n)^{\ell+m}}
\left[
\left(
\frac{f_\eta}{2}+1
\right)^m + (-1)^\ell \left(
\frac{f_\eta}{2}-1
\right)^m
\right].\end{aligned}$$ Equations (\[klm-gauss\]) and (\[ans-joint-answer\]) are the central result of this Section.
In particular, they bring the following large-$n$ expression for the cumulants of noise power in case of symmetric leads: $$\begin{aligned}
\label{npc} \fl
\kappa_\ell^{(n,0)}(P) =
2n \left(
1+\frac{f_\eta}{4}
\right) \delta_{\ell,1} + \left(
1 + \frac{f_\eta^2}{8}
\right) \delta_{\ell,2} \nonumber \\
+ \frac{(\ell-1)!}{8n^\ell}
\left[
\left( \frac{f_\eta}{2}-1
\right)^\ell
+
\left( \frac{f_\eta}{2}+1
\right)^\ell
\right] + o(n^{-\ell}).\end{aligned}$$ The higher order corrections to the above result can straightforwardly be found as well. An extension of the above results to $\beta=1$ and $4$ symmetry classes can be found in Mezzadri and Simm (2013).
Brief summary
-------------
In the previous sections, we have shown how the ideas of integrability can be utilized to provide a nonperturbative description of universal aspects of quantum transport through chaotic cavities connected to the external world through ideal leads. Focussing on fluctuation properties of the Landauer conductance and the noise power, we revealed that they are remarkably (and quite unexpectedly!) described in terms of fifth Painlevé transcendents and the one-dimensional Toda Lattice. While the formalism presented refers to the cavities with broken time-reversal symmetry ($\beta=2$ Dyson’s symmetry class), an integrable theory of universal quantum transport can equally be extended to other fundamental symmetry classes ($\beta=1$ and $4$) and other transport observables, see a detailed exposition by Mezzadri and Simm (2013).
Importantly, integrable theory of universal quantum transport presented in Sections \[SoLC\] and \[itnf\] assumed that a chaotic cavity is probed via ideal leads attached to the cavity through ballistic point contacts (so that the ${\bm {\mathcal S}}$-matrix is CUE-distributed). Will the integrability persist if one relaxes the ballisticity assumption? Below, this question will be answered in the affirmative.
Quantum transport in chaotic cavities with non-ideal leads and two-dimensional Toda Lattice {#Tunneling}
===========================================================================================
Joint probability density function of reflection eigenvalues {#jpdf-ref-eig}
------------------------------------------------------------
In case of non-ideal leads, the underlying assumption of Sections \[SoLC\] and [\[itnf\]]{} about uniform distribution of the scattering matrix ${\bm {\mathcal S}}$ with respect to the proper Haar measure is no longer justified. Instead, fluctuations of the scattering matrix are captured by the full Poisson kernel \[Eq. (\[pk\])\] which, in turn, induces a nontrivial probability measure for a set ${\bm T}$ of all non-zero transmission eigenvalues.
A first systematic random-matrix-theory study of a largely unexplored territory of non-ideal couplings had recently been undertaken by Vidal and Kanzieper (2012) who managed to lift a point-contact-ballisticity for one (‘left’) of the two leads. To address integrability issues we are concerned with, we shall choose a (reduced) model of [*mode-independent tunneling*]{}, when the average scattering matrix equals \[Eq. (\[gj\])\] $$\begin{aligned}
\label{s0-def}
{\bm {\mathcal S}}_0 = {\bm V}^\dagger
\left(
\begin{array}{cc}
\hat{\bm \gamma}_{\rm L} & 0 \\
0 & 0 \\
\end{array}
\right)
{\bm V},\qquad \hat{\bm \gamma}_L = \mathds{1}_{N_{\rm L}}\sqrt{1 - \Gamma^2}.\end{aligned}$$ Here, $\Gamma = \sqrt{1-\gamma_{\rm L}^2}$ is a tunnel probability for the $j$-th propagating mode ($1\le j \le N_{\rm L}$) in the left, non-ideal lead; the right lead, supporting $N_{\rm R}\ge N_{\rm L}$ open channels, is kept ideal. The joint probability density function of transmission/reflection eigenvalues is the central object of our interest in this Section. It will be derived in three steps. .—Spotting that, in the above setting, the Poisson kernel solely depends on the reflection matrix ${\bm r}$ in Eq. (\[s-block\]), and assuming broken time-reversal symmetry ($\beta=2$), one observes $$\begin{aligned}
P({\bm {\mathcal S}}) \propto {\rm det}^{-N} (\mathds{1}_{N_{\rm L}} - \gamma_{\rm L} {\bm r})
\, {\rm det}^{-N} (\mathds{1}_{N_{\rm L}} - \gamma_{\rm L} {\bm r}^\dagger).\end{aligned}$$ Consequently, the probability density associated with the reflection matrix alone equals $$\begin{aligned}
\label{pr-from-ps}\fl
P({\bm r}) \propto {\rm det}^{-N} (\mathds{1}_{N_{\rm L}} - \gamma_{\rm L} {\bm r})
\, {\rm det}^{-N} (\mathds{1}_{N_{\rm L}} - \gamma_{\rm L} {\bm r}^\dagger)
\int_{{\bm t} \in {\mathbb C}^{N_{\rm L}\times N_{\rm R}}} [d{\bm t}] \,\delta ({\bm t}{\bm t}^\dagger + {\bm r}{\bm r}^\dagger-{\mathds 1}_{N_{\rm L}}),\nonumber\\
{}\end{aligned}$$ where the integral over complex valued transmission matrix ${\bm t} \in {\mathbb C}^{N_{\rm L}\times N_{\rm R}}$ originates from the unitarity of the scattering matrix, ${\bm {\mathcal S}} {\bm {\mathcal S}}^\dagger = \mathds{1}_N$. Performing the integral in Eq. (\[pr-from-ps\]), $$\begin{aligned}
\fl
\int_{{\bm t} \in {\mathbb C}^{N_{\rm L}\times N_{\rm R}}} [d{\bm t}] \,\delta ({\bm t}{\bm t}^\dagger + {\bm r}{\bm r}^\dagger-{\mathds 1}_{N_{\rm L}}) \propto {\rm det}^{N_{\rm R}-N_{\rm L}} (\mathds{1}_{N_{\rm L}} - {\bm r} {\bm r}^\dagger) \, \Theta (\mathds{1}_{N_{\rm L}} - {\bm r} {\bm r}^\dagger),\end{aligned}$$ we obtain: $$\begin{aligned}
\label{pr-int}
P({\bm r}) \propto {\rm det}^{-N} (\mathds{1}_{N_{\rm L}} - \gamma_{\rm L} {\bm r})
\, {\rm det}^{-N} (\mathds{1}_{N_{\rm L}} - \gamma_{\rm L} {\bm r}^\dagger)\, \nonumber\\
\qquad \qquad \times {\rm det}^{N_{\rm R}-N_{\rm L}} (\mathds{1}_{N_{\rm L}} - {\bm r} {\bm r}^\dagger) \, \Theta (\mathds{1}_{N_{\rm L}} - {\bm r} {\bm r}^\dagger).\end{aligned}$$ —Since ${\bm r}$ is a complex valued matrix with no symmetries, it can be pseudo-diagonalized via singular value decomposition, ${\bm r} = {\bm u} {\bm \varrho} {\bm v}^\dagger$, where ${\bm \varrho} = {\rm diag}(\varrho_1,\dots,\varrho_{N_{\rm L}})$ with $\varrho_j>0$ and the unitary matrices ${\bm u}$ and ${\bm v}$ are ${\bm u}\in U(N_{\rm L})$ and ${\bm v}\in U(N_{\rm L})/U(1)^{N_{\rm L}}$. Further, introducing reflection eigenvalues $R_j=\rho_j^2$ (these are the eigenvalues of the matrix ${\bm r}{\bm r}^\dagger$), and observing the relation (see, e.g., Forrester 2006) $$\begin{aligned}
[d{\bm r}] \propto \Delta_{N_{\rm L}}^2({\bm R})\, d\mu({\bm u}) \, d\mu({\bm v})\,\prod_{j=1}^{N_{\rm L}} dR_j\end{aligned}$$ (here, $d\mu$ is the invariant Haar measure on the unitary group), we come down to the joint probability density of reflection eigenvalues in the form: $$\begin{aligned}
\fl \label{PR-01}
P(R_1,\dots,R_{N_{\rm L}};\gamma^2) &\propto& \Delta_{N_{\rm L}}^2({\bm R}) \, \prod_{j=1}^{N_{\rm L}} (1-R_j)^{N_{\rm R}-N_{\rm L}} \Theta(1-R_j)\, dR_j \nonumber\\
&\times& \int_{{\bm U}\in U(N_{\rm L})} d\mu({\bm U})\,
{\rm det}^{-N} (\mathds{1}_{N_{\rm L}} - \gamma_{\rm L} {\bm \varrho}{\bm U})\,
{\rm det}^{-N} (\mathds{1}_{N_{\rm L}} - \gamma_{\rm L} {\bm \varrho} {\bm U}^\dagger).\nonumber\\
{}\end{aligned}$$ Notice, that for any finite $\gamma_{\rm L}$, the $U(N_{\rm L})$ group integral in Eq. (\[PR-01\]) effectively modifies the interaction between reflection eigenvalues, which is no longer logarithmic \[see Eq. (\[PnT\])\]. —The group integral in Eq. (\[PR-01\]) can be evaluated by employing the technique of Schur functions (Macdonald 1995) and the theory of hypergeometric functions of matrix argument (Muirhead 2005, Gross and Richards 1989). (An alternative derivation, based on the theory of $\tau$ functions of matrix argument, was reported by Orlov (2004).) Leaving details of our calculation for a separate publication, we state the final result: $$\begin{aligned}
\label{final-dgi}\fl
\int_{{\bm U}\in U(N_{\rm L})} d\mu({\bm U})\,
{\rm det}^{-N} (\mathds{1}_{N_{\rm L}} - \gamma_{\rm L} {\bm \varrho}{\bm U})\,
{\rm det}^{-N} (\mathds{1}_{N_{\rm L}} - \gamma_{\rm L} {\bm \varrho} {\bm U}^\dagger)
\nonumber\\
= \frac{1}{\Delta_{N_{\rm L}}({\bm R})}\, {\rm det}_{(j,k) \in (1, N_{\rm L})}
\left[
R_j^{k-1} {}_2 F_{1} (N_{\rm R}+k, N_{\rm R}+k; k; \gamma^2 R_j)
\right].\end{aligned}$$ Here, ${}_2 F_1$ is the Gauss hypergeometric function. —Combining the last two equations together, we eventually derive the sought joint probability density function of reflection eigenvalues: $$\begin{aligned}
\fl \label{PR-02}
P(R_1,\dots,R_{N_{\rm L}};\gamma^2) = c(N_{\rm L}, N_{\rm R}) \, (1-\gamma^2)^{N_{\rm L}(N_{\rm L}+N_{\rm R})} \Delta_{N_{\rm L}}({\bm R}) \, \prod_{j=1}^{N_{\rm L}} (1-R_j)^{N_{\rm R}-N_{\rm L}} \nonumber\\
\times
\, {\rm det}_{(j,k) \in (1, N_{\rm L})}
\left[
R_j^{k-1} {}_2 F_{1} (N_{\rm R}+k, N_{\rm R}+k; k; \gamma^2 R_j)
\right].\end{aligned}$$ Here, we reinstated the factor $(1-\gamma^2)^{N_{\rm L}(N_{\rm L}+N_{\rm R})}$ originating from the normalization of the Poisson kernel $\propto {\rm det}^{-N}(\mathds{1}_{N} - {\bm {\mathcal S}}_0 {\bm {\mathcal S}}_0^\dagger)$; the otherwise $\gamma$-independent prefactor $c(N_{\rm L}, N_{\rm R})$ can easily be restored from the $\gamma\rightarrow 0$ limit, where the corresponding joint probability density function of reflection/transmission eigenvalues is known \[Eqs. (\[PnT\]) and (\[nc\])\]. This leads to $$\begin{aligned}
c(N_{\rm L},N_{\rm R}) = \frac{N_{\rm L}! \, N_{\rm R}!}{(N_{\rm L}+N_{\rm R})!}
\prod_{j=1}^{N_{\rm L}} \frac{1}{(j!)^2} \prod_{j=1}^{N_{\rm L}} \frac{(N_{\rm R}+j)!}{(N_{\rm R}-j)!}.\end{aligned}$$ This completes our derivation of the joint probability density of reflection eigenvalues in a chaotic cavity probed via one ideal and one non-ideal lead. For a more general case of mode-dependent tunneling (when $\hat{\bm \gamma}_{\rm L}$ in Eq. (\[s0-def\]) is not proportional to the unity matrix ${\mathds 1}_{N_{\rm L}}$), the reader is referred to Vidal and Kanzieper (2012).
Conductance fluctuations and two-dimensional Toda Lattice
---------------------------------------------------------
To quantify statistics of the Landauer conductance for a cavity probed via both ballistic (right lead) and tunnel (left lead) point contacts, we concentrate on the MGF $$\begin{aligned}
\fl
{\mathcal F}^{(N_{\rm L}, N_{\rm R})} (\gamma^2, z) = \int_{(0,1)^{N_{\rm L}}} \prod_{j=1}^{N_{\rm L}}\left( dR_j \, e^{-z(1-R_j)} \right)\,
P(R_1,\dots,R_{N_{\rm L}};\gamma^2),\end{aligned}$$ where the joint probability density function in the integrand was determined in the previous Section. Having in mind Eq. (\[PR-02\]), one has: $$\begin{aligned}
\fl
{\mathcal F}^{(N_{\rm L}, N_{\rm R})} (\gamma^2, z) = c_\gamma(N_{\rm L}, N_{\rm R})\, e^{-N_{\rm L} z} \int_{(0,1)^{N_{\rm L}}} \prod_{j=1}^{N_{\rm L}}
\left( dR_j \, e^{z R_j} (1-R_j)^{N_{\rm R}-N_{\rm L}}\right)\,\nonumber\\
\times \Delta_{N_{\rm L}}({\bm R})\,
{\rm det}_{(j,k) \in (1, N_{\rm L})}
\left[
R_j^{k-1} {}_2 F_{1} (N_{\rm R}+k, N_{\rm R}+k; k; \gamma^2 R_j)
\right],\end{aligned}$$ where $$\begin{aligned}
c_\gamma(N_{\rm L}, N_{\rm R})=c(N_{\rm L}, N_{\rm R}) \, (1-\gamma^2)^{N_{\rm L}(N_{\rm L}+N_{\rm R})}.\end{aligned}$$ Owing to the Andréief–de Bruijn integration formula, this can further be reduced down to $$\begin{aligned}
\label{mgf-tun-01}\fl
{\mathcal F}^{(N_{\rm L}, N_{\rm R})} (\gamma^2, z) = N_{\rm L}!\, c_\gamma(N_{\rm L}, N_{\rm R})\, e^{-N_{\rm L} z}
\, {\rm det}_{(j,k) \in (1, N_{\rm L})}
\left[ {\mathcal M}_{jk}^{(N_{\rm L},N_{\rm R})}(\gamma^2,z)
\right]\end{aligned}$$ with $$\begin{aligned}
\fl
{\mathcal M}_{jk}^{(N_{\rm L},N_{\rm R})}(\gamma^2,z) =
\int_0^1 dR\, e^{z R} (1-R)^{N_{\rm R}-N_{\rm L}}\, R^{j+k-2}\, {}_2 F_{1} (N_{\rm R}+k, N_{\rm R}+k; k; \gamma^2 R)
\nonumber\\
\fl\qquad
= \left(\frac{\partial}{\partial z}\right)^{j-1}
\,\int_0^1 dR\, e^{z R} (1-R)^{N_{\rm R}-N_{\rm L}}\, R^{k-1}\, {}_2 F_{1} (N_{\rm R}+k, N_{\rm R}+k; k; \gamma^2 R).\end{aligned}$$ The entry ${\mathcal M}_{jk}^{(N_{\rm L},N_{\rm R})}(\gamma^2,z)$ can be written in a more appealing form after one makes use of the identity $$\begin{aligned}
\fl
R^{k-1}\, {}_2 F_{1} (N_{\rm R}+k, N_{\rm R}+k; k; \gamma^2 R) =(k-1)!\nonumber\\
\times \left(\frac{N_{\rm R}!}{(N_{\rm R}+k-1)!} \right)^2
\left( \frac{\partial}{\partial \gamma^2}\right)^{k-1} {}_2 F_1 (N_{\rm R}+1, N_{\rm R}+1; 1; \gamma^2 R)\end{aligned}$$ that yields the representation $$\begin{aligned}
\fl \label{mjk}
{\mathcal M}_{jk}^{(N_{\rm L},N_{\rm R})}(\gamma^2,z) = (k-1)! \left(\frac{N_{\rm R}!}{(N_{\rm R}+k-1)!} \right)^2
\left(\frac{\partial}{\partial z}\right)^{j-1}
\left( \frac{\partial}{\partial \gamma^2}\right)^{k-1}
{\mathcal M}_{11}^{(N_{\rm L},N_{\rm R})}(\gamma^2,z). \nonumber\\
{}\end{aligned}$$ Here, $$\begin{aligned}
\fl
{\mathcal M}_{11}^{(N_{\rm L},N_{\rm R})}(\gamma^2,z)=\int_0^1 dR\, e^{z R} (1-R)^{N_{\rm R}-N_{\rm L}}\,
{}_2 F_{1} (N_{\rm R}+1, N_{\rm R}+1; 1; \gamma^2 R).\end{aligned}$$ With the help of Eqs. (\[mgf-tun-01\]) and (\[mjk\]), we may now put the MGF into a fairly symmetric form: $$\begin{aligned}
\label{mgf-tun-02}\fl
{\mathcal F}^{(N_{\rm L}, N_{\rm R})} (\gamma^2, z) = \tilde{c}_\gamma(N_{\rm L}, N_{\rm R})\, e^{-N_{\rm L} z} \nonumber\\
\times
\, {\rm det}_{(j,k) \in (1, N_{\rm L})}
\left[
\left( \frac{\partial}{\partial z}\right)^{j-1} \left( \frac{\partial}{\partial \gamma^2}\right)^{k-1}
{\mathcal M}_{11}^{(N_{\rm L},N_{\rm R})}(\gamma^2,z) \right],\end{aligned}$$ where $$\begin{aligned}
\fl
\tilde{c}_\gamma(N_{\rm L},N_{\rm R}) = N_{\rm L}! \times (1-\gamma^2)^{N_{\rm L}(N_{\rm L}+N_{\rm R})}
\,\frac{(N_{\rm L}+N_{\rm R})!}{N_{\rm L}! N_{\rm R}!}
\prod_{j=0}^{N_{\rm L}-1} \frac{1}{j!} \prod_{j=1}^{N_{\rm L}} \frac{(N_{\rm R}!)^2}{(N_{\rm R}-j)!\,(N_{\rm R}+j)!}.
\nonumber\\
{}\end{aligned}$$
Equation (\[mgf-tun-02\]) suggests that the conductance MGF ${\mathcal F}_\gamma(N_{\rm L}, N_{\rm R}; z)$ considered as a function of two continuous variables $(z,\gamma^2)$ can be related to a solution of a [*two-dimensional Toda Lattice*]{} equation. To make this statement precise, we need the theorem (see, e.g., Section 6 in: Vein and Dale 1999) going back to Darboux (1889) and stating that for a differentiable function $f(x,y)$ the determinant $$\begin{aligned}
u_n(x,y) = {\rm det}_{(j,k)\in (1,n)} \left[
\left(\frac{\partial}{\partial x}\right)^{j-1} \left(\frac{\partial}{\partial y}\right)^{k-1} f(x,y)
\right]\end{aligned}$$ satisfies the two-dimensional Toda Lattice equation: $$\begin{aligned}
\frac{\partial^2 }{\partial x \partial y} \log u_n(x,y) = \frac{u_{n-1} \, u_{n+1}}{u_n^2}.\end{aligned}$$ It is assumed that $u_0(x,y)=1$.
With the above in mind, we are ready to make a central statement of this Section. Let $\{u_n^{(N_{\rm L}, N_{\rm R})}(\gamma^2, z)\}$ denotes a sequence of determinants $$\begin{aligned}
\fl
u_n^{(N_{\rm L}, N_{\rm R})}(\gamma^2, z) = {\rm det}_{(j,k)\in (1,n)} \left[
\left(\frac{\partial}{\partial z}\right)^{j-1} \left(\frac{\partial}{\partial \gamma^2}\right)^{k-1} {\mathcal M}_{11}^{(N_{\rm L},N_{\rm R})}(\gamma^2,z)
\right]\end{aligned}$$ for $n=1,2,\dots$. Then, $\{u_n^{(N_{\rm L}, N_{\rm R})}(\gamma^2, z)\}$ satisfies the two-dimensional Toda Lattice equation $$\begin{aligned}
\frac{\partial^2 }{\partial z \partial \gamma^2} \log u_n^{(N_{\rm L}, N_{\rm R})}(\gamma^2, z) =
\frac{u_{n-1}^{(N_{\rm L}, N_{\rm R})}(\gamma^2, z)\, u_{n+1}^{(N_{\rm L}, N_{\rm R})}(\gamma^2, z)} {\left(u_{n}^{(N_{\rm L}, N_{\rm R})}
(\gamma^2, z)\right)^2},\end{aligned}$$ and the MGF for Landauer conductance equals the $N_{\rm L}$-th member of the above sequence: $$\begin{aligned}
{\mathcal F}^{(N_{\rm L}, N_{\rm R})} (\gamma^2, z) = u_{n}^{(N_{\rm L}, N_{\rm R})}(\gamma^2, z)\Big|_{n=N_{\rm L}}.\end{aligned}$$ This concludes our derivation of the announced relation between the problem of conductance fluctuations in chaotic cavities with a non-ideal lead and the two-dimensional Toda Lattice equation.
Concluding remarks
==================
In this paper, we reviewed the basics of integrable theory of the universal quantum transport in chaotic structures. (i) Starting with the mathematically simplest case of chaotic cavities with broken time reversal symmetry ($\beta=2$) which are coupled to the outside world through ballistic point contacts (‘ideal leads’), we showed that fluctuations of the Landauer conductance and the noise power are described by a [*one-dimensional Toda Lattice*]{} equation and the [*fifth Painléve transcendent*]{}. This finding, revealed at the level of moment generating functions, was utilized to generate nonlinear recurrence relations between cumulants of transport observables of various orders. Solutions to these relations produced non-perturbative formulae for cumulants of the Landauer conductance and the noise power of any given order. The discovered relation between quantum transport in zero-dimensional chaotic cavities and the theory of integrable lattices appears to be very general and can be carried over to scattering systems with preserved time-reversal symmetry ($\beta=1$ and $4$); there, fluctuations of transport observables are governed by a [*Pfaff-Toda Lattice*]{} (Mezzadri and Simm 2013).
\(ii) Further, we demonstrated that inclusion of tunneling effects inherent in realistic point contacts (‘non-ideal leads’) does not destroy the integrability. Specifically, for $\beta=2$ symmetry class, fluctuations of the Landauer conductance for a cavity probed via both ideal and non-ideal leads were shown to be captured by a [*two-dimensional Toda Lattice*]{} equation. While this result marks quite a progress in understanding integrable aspects of the universal quantum transport, more efforts are required to bring an integrable theory of quantum transport to its culminating point: (i) extending the formalism to other Altland-Zirnbauer symmetry classes (Altland and Zirnbauer 1997) and, even more important, (ii) relaxing a point contact ballisticity for the second lead are the most challenging problems whose solution is very much called for. A progress in solving the former problem will be reported elsewhere (Jarosz, Vidal and Kanzieper 2014).
Acknowledgements {#acknowledgements .unnumbered}
================
I am indebted to A. Jarosz, V. Al. Osipov and P. Vidal for collaboration on the integrability project that led, in part, to the results reported in this contribution. This work was supported by the Israel Science Foundation through the grants No 414/08 and No 647/12.
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[^1]: Essentially the same eigenvalue integral appears in the replica approach to [*statical*]{} correlations in the Calogero-Sutherland model with the interaction parameter $\lambda=1$, see, e.g., Gangardt and Kamenev (2001) and Kanzieper (2002).
[^2]: Here, the notation ${\bm t}$ should not be confused with the one previously used for the transmission sub-block of the scattering matrix ${\bm {\mathcal S}}$, Eqs. (\[s-block\]) and (\[T-cond\]).
[^3]: Both the series $\sum_0^{-1}$ and the coefficient $\chi_{-1}$ should be set to zero.
[^4]: Equation (\[ptotal\]) disregards the low-frequency $1/f$ noise that can efficiently be filtered out in experiments.
|
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abstract: 'The recent extensive availability of “big data” platforms calls for a more widespread adoption by the formal verification community. In fact, formal verification requires high performance data processing software for extracting knowledge from the unprecedented amount of data which come from analyzed systems. Since cloud based computing resources have became easily accessible, there is an opportunity for verification techniques and tools to undergo a deep technological transition to exploit the new available architectures. This has created an increasing interest in parallelizing and distributing verification techniques. In this paper we introduce a distributed approach which exploits techniques typically used by the “big data” community to enable verification of Computation Tree Logic (CTL) formulas on very large state spaces using distributed systems and cloud computing facilities. The outcome of several tests performed on benchmark specifications are presented, thus showing the convenience of the proposed approach.'
author:
-
bibliography:
- 'biblio.bib'
title: Distributed CTL Model Checking in the Cloud
---
Introduction {#sec:intro}
============
Ensuring the correctness of software and hardware products is an issue of great importance. This has led to an increased interest in applying formal methods and verification techniques in order to ensure correctness of developed systems. Among the most successful techniques that are widely used in both research and industry is *model checking*. Model checking of dynamic, concurrent and real-time systems has been the focus of several decades of software engineering research. One of the most challenging task in this context is the development of tools able to cope with the complexity of the models needed in the analysis of real word examples. In fact, the main obstacle that model checking faces is the state explosion problem [@Valmari98]: The number of global states of a concurrent system with multiple processes can be enormous. It increases exponentially in both the number of processes and the number of components per process. The most significant contributions the research has provided in order to cope with this problem are symbolic model checking with ordered binary decision diagrams [@Burch92], partial order reduction techniques [@Alur97], and bounded model checking [@Latvala04].
These breakthrough techniques have enabled the analysis of systems with a fairly big number states. Nevertheless, taking advantage of a distributed environment is still important to cope with real world problems. The idea is to increase the computational power and a larger available memory, by using a cluster of computers. The use of networks of computers can provide the resources required to achieve verification of models representing real world examples. Unfortunately, this last approach requires several skills which—while common in the “big data” community—are still rather rare in the “formal methods” community.
In fact, our recent works were focused on the connection between formal methods in software engineering and big data approaches [@Camilli13; @Camilli12; @Camilli12-2]. The analysis of very complex systems certainly falls in this context, although formal verification has so far poorly explored by big data scientists. We believe, however, the challenges to be tackled in formal verification can benefit a lot from results and tools available for big data access and management. In fact formal verification requires several different skills: On the one hand, one needs an adequate background on formal methods in order to understand specific formalisms and proper abstraction techniques for modeling and interpreting the analysis results; On the other hand, one should also strive to deploy this techniques into software tools able to analyze large amount of data very reliably and efficiently similarly to “big data” projects. Recent approaches have shown the convenience of employing distributed memory and computation to manage large amount of reachable states, but unfortunately exploiting these results requires further skills in developing complex applications with knotty communication and synchronization issues. In particular, adapting an application for exploiting the scalability provided by cloud computing facilities as the Amazon Cloud Computing platform [@amazonElasticMapReduce] might be a daunting task without the proper knowledge of the subtleties of data-intensive and distributed analyses.
In this paper, we try to further reduce the gap between these different areas of expertise by providing a distributed CTL (*Computation Tree Logic*) model checker, which exploits computational models typically used to tackle big data problems. Our software tool is built on top of <span style="font-variant:small-caps;">Hadoop [MapReduce]{}</span> [@Dean08; @hadoop] and can be easily specialized to deal with the verification of CTL formulas on very large state spaces coming from different kinds of formalisms (e.g., different kinds of Petri Nets, Process Algebra etc.), thus it is suitable for simplifying the task of dealing with a large amount of reachable states by exploiting large clusters of machines. The [MapReduce]{} programming model, which has become the *de facto* standard for large scale data-intensive applications, has provided researchers with a powerful tool for tackling big-data problems in different areas [@Lin10; @Camilli13; @Cheng06; @Biswanath09]. We firmly believe that explicit state model checking could benefit from a distributed [MapReduce]{} based approach, but the topic has not been yet explored as far as we know. Exposing this issue to scientists with different backgrounds could stimulate the development of new interesting and more efficient solutions.
Computation Tree Logic {#sec:ctl}
======================
CTL [@Clarke81] is a branching-time logic which models time as a tree-like structure where each moment can be followed by several different possible futures. In CTL each basic temporal operator (*i.e.,* either $X$, $F$, $G$) must be immediately preceded by a path quantifier (*i.e.,* either $A$ or $E$). In particular, CTL formulas are inductively defined as follows: $$\phi ::= p \ | \ \neg \phi \ | \ \phi \vee \phi \ | \ A\psi \ | \ E\psi \ (state \ formulas)$$ $$\psi ::= X\phi \ | \ F\phi \ | \ G\phi \ | \ \phi U \phi \ (path \ formulas)$$
Where $p \in AP$, the set of atomic propositions. The universal path operator $A$ and the existential path operator $E$ express respectively that a property is valid for all paths and for some paths. The temporal operators next $X$ and until $U$ express respectively that a property is valid in the next state, and that a property is valid until another property becomes valid. The interpretation of a CTL formula is defined over a *Kripke structure* (*i.e,* a *state transition system*). A Kripke structure is made up by a finite set of states, a set of transitions (*i.e.,* a relation over the states), and a labeling function which assigns to each state the set of atomic propositions that are true in this state. Such a model describes the system at any point in time represented by states; the transition relation describes how the system evolves from a state to another over one time step. The formal definition is the following.
A Kripke structure $T$ is a quadruple $\langle S, S_0, R, L\rangle$, where:
1. $S$ is a finite set of states.
2. $S_0$ is the set of initial states.
3. $R \subseteq S \times S$ is a a total transition relation, that is: $\forall s \in S \ \exists s' \in S \ \text{such that} \ (s,s') \in R$
4. $L : S \rightarrow 2^{AP}$ labels each state with the set of atomic propositions that hold in that state.
Note that the third point imposes the *seriality* of the transition relation. This means that the system cannot have deadlock states. This condition can be always achieved easily by adding into the system a state of “error” (with one outgoing transition directed to itself) from which the system cannot get out once reached.
A path $\sigma$ in $T$ from a state $s_0$ is an infinite sequence of states $\sigma = s_0s_1s_2\dots$ where $\forall i \geq 0, \ (s_i,s_{i+1}) \in R$.
Given a CTL formula $\phi$ and a state transition system $T$ with $s
\in S$, we say that $T$ satisfy $\phi$ in the state $s$ (written as $T \models_s \phi$) if:
- $T \models_s p$ iff $p \in L(s)$.
- $T \models_s \neg \phi$ iff $T \not\models_s \phi$.
- $T \models_s \phi \wedge \psi$ iff $(T \models_s \phi \wedge T \models_s \psi)$.
- $T \models_s \phi \vee \psi$ iff $(T \models_s \phi \vee T \models_s \psi)$.
- $T \models_s EX\phi$ iff $\exists t$ such that $R(s, t) \wedge T \models_t \phi$.
- $T \models_s EG\phi$ iff $\exists$ a path $s_0s_1s_2\dots$ such that:\
$\forall i\geq0, T\models_{s_i}\phi$.
- $T \models_s E[\phi U \psi]$ iff $\exists$ a path $s_0s_1s_2\dots$ such that:\
$\exists i\geq0, (T \models_{s_i} \psi) \wedge (T \models_{s_j} \phi \ \forall j < i)$.
We can also write $T \models \phi$ which means that $T$ satisfies $\phi$ in all the initial states of the system.
It can be shown that any CTL formula can be written in terms of $\neg,
\vee, EX , EG$, and $EU$, for example $AX\phi$ is $\neg EX \neg \phi$, $EF\phi$ is $E[True \ U \ \phi]$, and so forth. The possible combinations are only eight:
$$AX, EX, AF, EF, AG, EG, AU, EU$$
The semantics of some widely used CTL operators is exemplified in Figure \[fig:trees\].
{width="1.8\columnwidth"}
Let $T$ be a Kripke structure and let $\phi$ be a CTL formula. The model checking problem is to find all the states $s \in S$ such that $T \models_s \phi$.
Fixed-Point Algorithms {#sec:fixedpoint}
======================
One of the existing model-checking algorithms is based on fixed-point characterizations of the basic temporal operators of CTL (similar ideas can be used for LTL model checking) [@Clarke99]. Let $T =
\langle S,S_0,R,L \rangle$ be a Kripke structure. The set $\mathcal{P}(S)$ of all subsets of $S$ forms a lattice under the set inclusion ordering. For convenience, we identify each state formula with the set of states in which it is true. For example, we identify the formula *false* with the empty set of states, and we identify the formula *true* with $S$ (the set of all states). Each element of $\mathcal{P}(S)$ can be viewed both as a set of states and as a state formula (a predicate). Formally, given a CTL formula $\phi$ we can define: $$[\![ \phi ]\!]_T :=\{s \in S \ : \ T \models_s \phi \}$$ This way, we can associate set operators to boolean connectors: $$[\![ \phi_1 \wedge \phi_2 ]\!] = [\![ \phi_1 ]\!] \cup [\![ \phi_2 ]\!], \ [\![ \phi_1 \vee \phi_2 ]\!] = [\![ \phi_1 ]\!] \cap [\![ \phi_2 ]\!],$$ $$[\![ \neg \phi ]\!] = S \ \backslash \ [\![ \phi ]\!]$$ The set of states identified by the temporal operator $EX$, can be defined trivially if we consider the counterimage with respect to the relation $R$. Given $W \in \mathcal{P}(S)$: $$R^-(W) := \{ s \in S \ : \ \exists s' (R(s,s') \wedge s' \in S)\}$$ Thus we can verify easily that the following holds: $$[\![ EX\phi ]\!]_T = R^-([\![ \phi ]\!]_T)$$ Let’s now consider a function $\tau : \mathcal{P}(S) \rightarrow \mathcal{P}(S)$ called *predicate transformer*.
We say that a state formula $X$ is the *least fixed-point* $\mu_X$ (or respectively the *greatest fixed-point* $\nu_X$) of a predicate transformer $\tau$ iff (1) $X = \tau(X)$, and (2) for all state formulas $X'$, if $X' = \tau(X')$, then $X \subseteq X'$ (respectively $X \supseteq X'$).
A predicate transformer $\tau$ is *monotonic* iff for all $X,X' \in \mathcal{P}(S)$ $X \subseteq X'$ implies $\tau(X) \subseteq \tau(X')$.
A monotonic predicate transformer on $\mathcal{P}(S)$ always has a least fixed-point and a greatest fixed-point (by Tarski’s Fixed-Point Theorem [@Tarski55]). The temporal operators $EG$ and $EU$ can each be characterized respectively as the greatest and the least fixed-point of two different monotonic predicate transformers: $$\label{eq:pt1}
[\![ EG\phi ]\!]_T = \nu_X([\![ \phi ]\!]_T \cap R^-(X))$$ $$\label{eq:pt2}
[\![ E[\phi U \psi] ]\!]_T = \mu_X([\![ \psi ]\!]_T \cup ([\![ \phi ]\!]_T \cap R^-(X)))$$
We can calculate the least fixed-point of a monotonic predicate transformer: $\mu_X(\tau(X))$ as follows. We define $X_0 = \emptyset$ and $X_i = \tau(X_{i+1})$ for $i \geq 1$. We first compute $X_1$, then $X_2$, then $X_3$, and so forth, until we find a $k$ such that $X_k = X_{k-1}$. It can be proved that the $X_k$ computed in this manner is the least fixed-point of $\tau$. To compute the greatest fixed-point, we follow a similar procedure but starting from $S$. Pseudocode for this procedure is shown by Algorithm \[alg:lfp\].
$X := \emptyset$ $X := \tau(X)$ $X$
Distributed Model Checking Algorithms {#sec:disalgo}
=====================================
We now recall briefly the [MapReduce]{} computational model (the basis on top of which our application is built) and later on we present our distributed approach in we used the fixed-point algorithms to exploit distributed and “cloud” facilities. The distributed algorithms presented in this section aim just at computing formulas of type $EX$, $EG$, and $EU$ because any CTL formula can be reformulated in terms of these three basic operators (see \[sec:ctl\]).
[MapReduce]{} {#sec:mapred}
-------------
[MapReduce]{} relies on the observation that many information processing activities have the same basic design: a same operation is applied over a large number of records (*e.g.,* database records, or vertices of a graph) to generate partial results, which are then aggregated to compute the final output. The [MapReduce]{} model consists of two functions: The “map” function turns each input element into zero or more key-value pairs. A “key” is not unique, in fact many pairs with a given key could be generated from the Map function; The “reduce” function is applied, for each key, to its associated list of values. The result is a key-value pair consisting of whatever is produced by the Reduce function applied to the list of values. Between these two main phases the system sorts the key-value pairs by key, and groups together values with the same key. This two-step processing structure is presented in Figure \[fig:mapreduce\].
The execution framework handles transparently all non-functional aspects of execution on big clusters. It is responsible, among other things, for scheduling (moving code to data), handling faults, and the large distributed sorting and shuffling needed between the map and reduce phases since intermediate key-value pairs must be grouped by key. The “partitioner” is responsible for dividing up the intermediate key space and assigning intermediate key-value pairs to reducers. The default partitioner computes a hash function on the value of the key modulo the number of reducers.
![The [MapReduce]{} model: the keys are in **bold**.[]{data-label="fig:mapreduce"}](mapreduce.pdf){width="0.9\columnwidth"}
Distributed State Space Generation {#sec:x}
----------------------------------
This task builds the reachability graph $T$ of a given model in a distributed fashion. The idea underlying a distributed algorithm for state space exploration is to use multiple computational units to perform the exploration of different parts of the whole state space in parallel. The task is typically performed by using classical parallel *Worker*s algorithms [@Kristensen04]: States are partitioned among workers by means of a static hash function. The workers explore successor states and assign them to the proper computational units. Communication among different machines is implemented through message passing. Since the partitioning of the state space is a critical issue, different load balancing techniques and compact states representation [@Nicol97; @Kumar05; @Garavel01] were studied. Recent approaches has shown also the convenience of exploiting big data approaches and cloud computing facilities in order to accomplish this task. In particular the *MaRDiGraS* [@Camilli13] framework could be employed to implement distributed state space builders for different formalisms. Given a cluster size of $n$ machines, a *MaRDiGraS* based application generates $n$ files $F_1, F_2, ..., F_n$ containing the whole state space, partitioned into $n$ different sets. The set of states emitted by the $i$th computational unit is $S_i=\{s \in S : \operatorname{\mathrm{Hash}}(f(s)) = i\}$, where $S$ is the set of reachable states, $f$ is a user supplied function and $f(s)$ computes specific features on states such that the equality of the evaluation of these features is a necessary condition for having an inclusion/equality relationship among states. What makes this representation interesting and suitable for further analysis by using our distributed approach is in the transitions management (the $R$ relation). In particular each state stores locally all incoming transitions as a list of state identifiers, therefore, given a set of state $W$, $R^-(W)$ can be easily computed: $$\bigcup_{i=1}^{n}s_i \in S : (id(s_i) \in R^-(s_j), \forall s_j \in W)$$
It is worth noting that the set of predecessor states’ identifiers should be immediately available inside state definition because our [MapReduce]{} based approach exploits the evaluation of $R^-$ as a basic operation without communication among computational units.
In order to apply our distributed fixed-point algorithms the analyzed transition system must preserve the *seriality* of the transition relation (introduced in section \[sec:ctl\]). If not, the *MaRDiGraS* framework can add an output file containing a single “error” state where the list of incoming transitions is made up by itself and all deadlock states.
$EX$ Formulas {#sec:ex}
-------------
To compute $[\![ EX\phi ]\!]_T$, we assume that the set of states satisfying $\phi$ is already computed. Thus $\phi$ can be either a formula locally evaluable or a more complex sub-formula evaluated previously. We can deploy this operation into a single [MapReduce]{} job where the predecessor states of the $[\![ \phi ]\!]_T$ set are evaluated in parallel. The input of this distributed computation is two different sets of files. The first set contains all states belonging to $S \setminus [\![ \phi
]\!]_T$, the second contains all states belonging to $[\![ \phi
]\!]_T$. This way all the mappers can evaluate and emit in parallel the identifiers of the states belonging to $R^-([\![ \phi ]\!]_T)$. As shown by Algorithm \[alg:ex\], the `Map` function emits the identifiers of these states associated with an empty value $\perp$. Then the shuffle phase groups together all the values with the same identifier, so that the `Reduce` function can emit the final result by checking whenever the empty value was passed into the input list.
$emit(e, \perp)$ $emit(k, s)$
$s := s' \in list \ s.t. \ s' \neq \perp$ $emit(k, s)$
$EG$ Formulas {#sec:eg}
-------------
As for the previous formula, to compute $[\![ EG\phi ]\!]_T$, we assume that the set of states satisfying $\phi$ is already computed. The evaluation of the final result is a bit more complex than the previous case. Our approach is based on the greatest fixed-point characterization of the monotonic predicate transformer introduced in (\[eq:pt1\]). Thus we apply an iterative [MapReduce]{} algorithm, where at each iteration we compute the predicate transformer on the output of the previous iteration until we reach the fixed-point. Algorithm \[alg:eg\] shows the `Map` and the `Reduce` functions employed within the job iterations. The input of each [MapReduce]{} job is made up by a set of files containing $[\![ \phi ]\!]_T$ and another set of files $X$ representing the current evaluation of the formula. Since the first iteration should start from $X=S$ and $R^-(S)=S$, we already know the result of the first evaluation of the predicate transformer introduced in (\[eq:pt1\]), thus we start directly from the second iteration by posing $X$ to $[\![ \phi ]\!]_T$. As shown by Algorithm \[alg:eg\], the map phase computes in parallel all the predecessor states and the reduce phase verifies and emits in parallel all predecessors belonging to $[\![ \phi ]\!]_T$. The iterations keep going until the number of key-value pairs given in output by two consecutive jobs becomes equal or we reach the empty set.
$emit(e, \perp)$ $emit(k, s)$
$emit(k, s)$
$EU$ Formulas {#sec:eu}
-------------
As for the previous formulas, to compute $[\![ E[\phi U \psi] ]\!]_T$, we assume that the set of states satisfying the two sub-formulas $\phi$ and $\psi$ are already computed. The approach employed to evaluate this formulas is similar to the previous one, in fact our distributed algorithm is based on the least fixed-point characterization of the monotonic predicate transformer introduced in (\[eq:pt2\]). The iterative map-reduce algorithm, which uses the <span style="font-variant:small-caps;">Map</span> and the <span style="font-variant:small-caps;">Reduce</span> functions presented by the algorithm \[alg:eu\], is employed in order to reach the fixed-point. The input of each iteration is made up by a set of files $X$ containing the current evaluation of the formula and another set of files containing $[\![ \psi ]\!]_T$. Since the first iteration should start from the empty set, but we already know that the predicate transformer (\[eq:pt2\]) computed on the input $X=\emptyset$ is $[\![ \psi ]\!]_T$, we start directly from the second iteration posing $X$ to $[\![ \psi ]\!]_T$. The map phase emits in parallel all predecessor states of $X$ set and forwards all states of $[\![ \psi ]\!]_T$ to reducers. The reduce phase emits in parallel all predecessor states of $[\![ \phi ]\!]_T$ and all states of $[\![ \psi ]\!]_T$.
$emit(e, \perp)$ $emit(k, s)$
$emit(k, s)$
$R^-(X_{i-1}) \subseteq R^-(X_i)$, since Algorithm \[alg:eu\] computes $R^-(X)$ for each iteration and $X_{i-1} \subseteq X_i$. For this reason we implemented an optimized version which computes, for each iteration, just $R^-(X_i \setminus X_{i-1})$.
Experiments {#sec:experiments}
===========
The experiments described in this section were executed using the Amazon Elastic [MapReduce]{} [@amazonElasticMapReduce] on the Amazon Web Service cloud infrastructure. They were supported by an “AWS in Education Grant award” [@grant]. In particular all runs have been performed on clusters of various sizes made up by *m2.2xlarge* computational units [@amazonElasticMapReduce].
As a proof of concept we generated three different state spaces, sized with different order of magnitude. Successively we applied our distributed algorithms in order to verify three different CTL formulas (of type $EX$, $EG$ and $EU$) for each state space. Both models and formulas used during the experiments were introduced in [@mcc2013web]. The models are three Petri Net benchmarks and their state space were generated by means of a *MaRDiGraS* based tool.
Shared Memory {#sec:mem}
-------------
This P/T net models a system composed of 10 processors which compete for the access to a shared memory by using a unique shared bus. The number of reachable states of this model is $1.831\times10^6$. Given the function $m:\mathrm{Place} \rightarrow \mathbb{N}$ which computes the number of tokens for a given place, the three properties verified on this model are: $$EX[A], \ EG[A], \ E[\mathrm{True} \ U \ A]=EF[A]$$ where: $$A:= m(\mathrm{Active}) \neq m(\mathrm{Memory}) \vee m(\mathrm{Queue})=m(\mathrm{Active})$$ Despite the generated state space is relatively small, the benefit gained from our distributed approach grows as the number of states involved in the verification grows (as shown in Table \[tab:report\]): indeed, the verification of the last formula $E[\mathrm{True} \ U A]$ scales better than the previous two.
Dekker {#sec:dekker}
------
This model represents a 1-safe P/T net of a variant of the Dekker’s mutual exclusion algorithm [@Dijkstra02] for $N=20$ processes. The state space generated by this model is an order of magnitude higher than the previous example ($1.153 \times 10^7$ reachable states). The three properties verified on this model are: $$EX[B], \ EG[B], \ E[C \ U \ D]$$ where: $$B:= m(p_{1,18}) \neq m(p_{1,13}) \vee m(p_{0,15})=m(p_{3,18}) \notag$$ $$C:=m(flag_{1,18}) \neq m(p_{0,4}) \wedge m(p_{0,17})=m(flag_{1,11}) \notag$$ $$D:=m(p_{0,17}) = m(flag_{1,11}) \notag$$
In this case, as shown by Table \[tab:report2\] and by the graph shown in Figure \[fig:graphs\](b), the benefits deriving from our distributed approach are clearer. In fact, the evaluation of both the three formulas gets substantially faster by increasing the number of computational units. The graph shown by Figure \[fig:graphs\](b) (and Figure \[fig:graphs\](d) for the next model), plots the function $\operatorname{\mathrm{cheat}}$ defined as follow: $$\operatorname{\mathrm{cheat}}(n) = \frac{\text{exec. time of parallel version with $1$ node}}{\text{exec. time of parallel version with $n$ nodes}}
\label{eq:cheat}$$
Simple Load Balancing {#sec:simple-lbs}
---------------------
This P/T net represents a simple load balancing system composed of 10 clients, 2 servers, and between these, a load balancer process. The reachability graph generated is very large: $4.060 \times 10^8$ states and $3.051 \times 10^9$ arcs for a total size of 120 GB of data. The three properties verified on this model are: $$EX[H], \ EG[J], \ E[K \ U \ H]$$ where: $$\begin{gathered}
H := m(\mathrm{server\_processed}) \neq m(\mathrm{server\_notification}) \wedge \notag \\
m(\mathrm{server\_waiting}) = m(\mathrm{server\_idle}) \notag \\[.3cm]
J := m(\mathrm{client\_idle}) \neq m(\mathrm{client\_waiting}) \notag \\[.3cm]
K := m(\mathrm{client\_idle}) \neq m(\mathrm{client\_waiting}) \wedge \notag \\
m(\mathrm{client\_idle}) = m(\mathrm{client\_request}) \notag \\\end{gathered}$$
As shown by Table \[tab:report3\] and by the graph shown in Figure \[fig:graphs\](d), the benefits deriving from our distributed approach are greater with respect to both previous examples. This points out a clear trend: the major is the complexity of the model to be analyzed, the major is the scalability of our distributed algorithm. In fact, the $\operatorname{\mathrm{cheat}}$ gained during the analysis of this last example greatly overcome the one gained in the analysis of the Dekker model (5.5 using 16 machines to evaluate $EX[B]$). As shown in Figure \[fig:graphs\](d), in this model we reach a super-linear speedup during the evaluation of $EG[J]$.
\
\[fig:graphs\]
Related Work {#sec:rel}
============
The use of distributed and/or parallel processing to tackle the state explosion problem gained interest in recent years. In fact, for very complex models, the state space may not completely fit into the main memory of a single computer and hence model-checking tools becomes very slow or even crash as soon as the memory is exhausted.
[@Lerda99; @Evangelista11; @1240299; @Brim04; @5698463] discuss parallel/distributed verification of *Linear Temporal Logic* (LTL) formulas. They aim at increasing the memory available and reducing the overall time required by LTL formulas verification by employing distributed techniques for searching accepting cycles in Bïchi automata. Distributed and parallel model checking of CTL logic was also proposed. [@Brim05] introduced a CTL model checking technique which works by splitting the given state space into several “partial state spaces”. Each computer involved in the distributed computation owns a partial state space and performs a model checking algorithm on this incomplete structure. To be able to proceed, the border states are augmented by assumptions about truth values of formulas and the computers exchange assumptions about relevant states to compute more precise information. Other approaches were introduced in [@Bell05; @Boukala12].
The main idea of distributed algorithms for both LTL and CTL model checking is in fact similar: the state graph is partitioned among the network nodes, *i.e.,* each network node owns a subset of the state space. The differences are in the way the state space is partitioned (through a *partition function*): this is a crucial issue. In order to increase performance of the parallel model checking, it is key to achieve a good load balancing among machines, meaning that each partition should contain nearly the same number of states. The performance of these algorithms depends also on the number of cross-border transitions of the partitioned state space (*i.e.,* transitions having the source state in a component and the target state in another component). This number should be as small as possible, since it has an effect on the number of messages sent over the network during the analysis [@Bourahla05]. In the context of LTL model checking, probabilistic techniques to partition the state space have been used, for example, in [@Lerda99; @Stern01], and a technique that exploits some structural properties derived from the verified formula has been proposed in [@406260].
Since our distributed algorithms are quite different from message passing approaches, the number of cross-border transitions is not a crucial issue to cope with. The only synchronization point among computational units is the shuffle phase, where key-value pairs are sorted and transferred from map outputs to reducers input. Reducing the number of cross-border transitions may reduce the data exchanged across the network during this phase. Anyway, this phase is partially overlapped with the map phase, which means that the shuffling starts as soon as data become available from mappers without waiting for the entire map output. Furthermore, since we found experimentally that the time required by this phase does not dominate the overall time required by our algorithms, adding a partitioning phase between each [MapReduce]{} iteration could even hurt performances. Nevertheless, we plan to study further this issue in order to understand better how partitioning can impact performances of our [MapReduce]{} based approach.
Our contribution is a set of parallel algorithms designed for distributed memory architectures and cloud computing platform based on a new emerging distributed paradigm. It is worth noting that departing from the current literature on distributed CTL model checking, we considered an important aspect, sometimes understated: we wanted to completely remove the costs of deploying our application into an end-to-end solution, for this reason we developed our software on top of the consolidated <span style="font-variant:small-caps;">Hadoop [MapReduce]{}</span> framework. As far as we now, the effectiveness of a [MapReduce]{} based approach, typically employed to solve big data problems, has been not explored so far by the formal verification community. Thus with our work we aim at further reducing the gap between these two different but related areas of expertise.
Conclusion and Future Work {#sec:conc}
==========================
In this paper we presented a software framework to model check very complex systems by applying iterative [MapReduce]{} algorithms based on fixed-point characterizations of the basic temporal operators of CTL.
Our distributed application exploits techniques typically used by the big data community and so far poorly explored for this kind of problem. Therefore we remark a clear connection between formal verification problems and big data problems conveyed by the recent widespread accessibility of powerful computing resources. Despite model checking software tools are so called “push-button”, the setup phase required by a distributed application, is far from being considered such, especially whenever one wants to exploits general purpose “cloud” computing facilities. Our framework aims at re-enabling a “push-button” mode into the distributed verification context even when these (complex on themselves) computing resources are involved.
Our experiments report that our approach can be used effectively to analyze state spaces of different orders of magnitude. In particular, the major is the complexity of the model to be analyzed, the major is the scalability of our distributed algorithms. In some cases we have shown a potential for a super-linear speedup. We believe that this work could be a further step towards a synergy between two very different, but related communities: the “formal methods” community and the “big data” community. Exposing this issue to scientists with different backgrounds could stimulate the development of new interesting and more efficient solutions.
Acknowledgments {#sec:ack .unnumbered}
===============
The authors would like to thank Amazon.com, Inc. for the “AWS in Education Grant” award which made possible the experiments described in this paper.
|
---
abstract: 'Recent results from CLEO-c, BABAR, and Belle on measurements of absolute branching fractions of $D$ and $D_s$ mesons are reviewed.'
author:
- 'A. Ryd'
title: Determination of Charm Hadronic Branching Ratios and New Modes
---
Introduction
============
Precise measurements of the absolute branching fractions for $D$ and $D_s$ meson decays are important as they serve to normalize most $B$ and $B_s$ decays as well as many charm decays. Recent measurements from CLEO-c, BABAR, and Belle for the measurements of the absolute hadronic branching fractions of $D$ and $D_s$ mesons are presented here.
Results from the CLEO-c experiment at the Cornell Electron Positron Storage Ring based on 281 pb$^{-1}$ recorded at the $\psi(3770)$ are presented here for studies of $D^0$ and $D^+$ decays. In addition, CLEO-c has analyzed 195 pb$^{-1}$ of $e^+e^-$ annihilation data near $E_{\rm cm}=4170$ MeV for studies of $D_s$ decays. These samples provide very clean environments for studying decays of $D$ and $D_s$ mesons. The $\psi(3770)$ produced in the $e^+e^-$ annihilation decays to pairs of $D$ mesons, either $D^+D^-$ or $D^0\bar D^0$. In particular, the produced $D$ mesons can not be accompanied by any additional pions. At $E_{\rm cm}=4170$ MeV $D_s$ mesons are primarily produced as $D_s^{+}D_s^{*-}$ and $D_s^{*+}D_s^{-}$ pairs.
The results from BABAR and Belle use their large samples of $e^+e^-$ data collected by these experiments. The different analyses presented here use integrated luminosities up to 0.55 ab$^{-1}$. For example, Belle has used 0.55 ab$^{-1}$ to study $D_s^+\to K^+K^-\pi^+$ in exclusive production of $e^+e^-\to D_s^*D_{s1}$. BABAR has studied $D_s\to \phi\pi$ using a sample of $B\to D^{(*)}D_{s(J)}^{(*)}$ decays. These examples illustrate that charm produced both in the continuum and in $B$ meson decays are useful for studies of charm at the $B$-factories.
First I will discuss the determination of the absolute $D^0$ and $D^+$ branching fractions. New results from CLEO-c and BABAR are discussed here. Then results for $D_s$ branching fractions from CLEO-c, Belle, and BABAR are presented. Last a few inclusive and rare hadronic decay modes are discussed.
Absolute $D$ hadronic branching fractions at CLEO-c
===================================================
This analysis makes use of a ’double tag’ technique initially used by Mark III [@markiii]. In this technique the yields of single tags, where one $D$ meson is reconstructed per event, and double tags, where both $D$ mesons are reconstructed, are determined. The number of single tags, separately for $D$ and $\bar D$ decays, are given by $N_i=\epsilon_i{\cal B}_i N_{D\bar D}$ and ${\bar N}_j=\bar \epsilon_j{\cal B}_j N_{D\bar D}$ where $\epsilon_i$ and ${\cal B}_i$ are the efficiency and branching fraction for mode $i$. Similarly, the number of double tags reconstructed are given by $N_{ij}=\epsilon_{ij}{\cal B}_i{\cal B}_j N_{D\bar D}$ where $i$ and $j$ label the $D$ and $\bar D$ mode used to reconstruct the event and $\epsilon_{ij}$ is the efficiency for reconstructing the final state. Combining the equations above and solving for $N_{D\bar D}$ gives the number of produced $D\bar D$ events as $$N_{D\bar D}={{N_i}{\bar N_j}\over N_{ij}}{\epsilon_{ij}\over \epsilon_i\bar\epsilon_j}$$ and the branching fractions $${\cal B}_i={N_{ij}\over N_j}{{\epsilon_j}\over \epsilon_{ij}}.$$ In this analysis CLEO-c determine all the single tag and double tag yields in data, determine the efficiencies from Monte Carlo simulations of the detector response, and extract the branching fractions and $D\bar D$ yields from a combined fit to all measured data yields.
This analysis uses three $D^0$ decays ($D^0\to K^-\pi^+$, $D^0\to K^-\pi^+\pi^0$, and $D^0\to K^-\pi^+\pi^-\pi^+$) and six $D^+$ modes ($D^+\to K^-\pi^+\pi^+$, $D^+\to K^-\pi^+\pi^+\pi^0$, $D^+\to K^0_S\pi^+$, $D^+\to K^0_S\pi^+\pi^0$, $D^+\to K^0_S\pi^+\pi^-\pi^+$, and $D^+\to K^-K^+\pi^+$). The single tag yields are shown in Fig. \[fig:dhad\_st\]. The combined double tag yields are shown in Fig. \[fig:dhad\_dt\] for charged and neutral $D$ modes separately. The scale of the statistical errors on the branching fractions are set by the number of double tags and precisions of $\approx 0.8\%$ and $\approx 1.0\%$ are obtained for the neutral and charged modes respectively. The branching fractions obtained are summarized in Table \[tab:dhadresults\].
CLEO-c has presented updated results for these branching fractions[@cleoc_charm07] since these results were presented. The new results, including ${\cal B}(D^0\to K^-\pi^+)=(3.891\pm 0.035\pm 0.059\pm 0.035)\%$, are consistent with the preliminary results presented here. The last error is the uncertainty due to final state radiation.
![The fits for the single tag yields. The background is described by the ARGUS threshold function and the signal shape includes the effects of beam energy spread, momentum resolution, initial state radiation, and the $\psi(3770)$ lineshape. []{data-label="fig:dhad_st"}](3970407-010.eps){width="\linewidth"}
![The fit for the double tag yields combined over all modes for charged and neutral modes separately.[]{data-label="fig:dhad_dt"}](3970707-024.eps "fig:"){width="0.49\linewidth"} ![The fit for the double tag yields combined over all modes for charged and neutral modes separately.[]{data-label="fig:dhad_dt"}](3970707-025.eps "fig:"){width="0.49\linewidth"}
Mode Fitted Value (%) PDG (%)
------------------------- ------------------------ ---------------
${\cal B}(\Dzkpi)$ $3.87\pm 0.04\pm 0.08$ $3.81\pm0.09$
${\cal B}(\Dzkpipiz)$ $14.6\pm 0.1\pm 0.4$ $13.2\pm1.0$
${\cal B}(\Dzkpipipi)$ $8.3\pm 0.1\pm 0.2$ $7.48\pm0.30$
${\cal B}(\Dpkpipi)$ $9.2\pm 0.1\pm 0.2$ $9.2\pm0.6$
${\cal B}(\Dpkpipipiz)$ $6.0\pm 0.1\pm 0.2$ $6.5\pm1.1$
${\cal B}(\Dpkspi)$ $1.55\pm 0.02\pm 0.05$ $1.42\pm0.09$
${\cal B}(\Dpkspipiz)$ $7.2\pm 0.1\pm 0.3$ $5.4\pm1.5$
${\cal B}(\Dpkspipipi)$ $3.13\pm 0.05\pm 0.14$ $3.6\pm0.5$
${\cal B}(\Dpkkpi)$ $0.93\pm 0.02\pm 0.03$ $0.89\pm0.08$
: Preliminary branching fractions from CLEO-c. Uncertainties are statistical and systematic, respectively. []{data-label="tab:dhadresults"}
Measurement of ${\cal B}(D^0\to K^-\pi^+)$ at BABAR
===================================================
![The distribution of the missing mass squared, $M^2_{\nu}$, for (a) right sign events and (b) wrong sign events. The wrong sign events show that the simulation of the background shape is good. (From Ref. [@babar_dtokpi].) []{data-label="fig:babar_m2nu"}](fig1.eps){width="0.9\linewidth"}
BABAR has used a sample of 210 fb$^{-1}$ of $e^+e^-$ data collected at the $\Upsilon(4S)$ resonance to study the decay $D^0\to K^-\pi^+$ decay [@babar_dtokpi]. They use semileptonic $B$ decays, $\bar B^0\to D^{*+}\ell^-\bar\nu$ followed by $D^{*+}\to D^0\pi^+$, where they use the lepton in the $B$ decay and the slow pion from the $D^*$ to tag the signal. As the energy release in the $D^*$ decay is very small the reconstructed slow pion momentum can be used to estimate the four-momentum of the $D^*$ — the slow pion and the $D^*$ have approximately the same velocity. BABAR extracts the number of $\bar B^0\to D^{*+}\ell^-\bar\nu$ decays using the missing mass squared, $M^2_{\nu}$, against the $D^*$ and the lepton. The $M^2_{\nu}$ distribution is shown in Fig. \[fig:babar\_m2nu\]. A clear signal is observed for $M^2_{\nu}>-2.0$ GeV$^2$. However, there are substantial backgrounds that need to be subtracted due to combinatorial backgrounds in $B\bar B$ events and continuum production. Table \[tab:babar\_bkgd\] summarizes the event yields for the inclusive $\bar B^0\to D^{*+}\ell^-\bar\nu$ reconstruction in the column labeled ’Inclusive’. BABAR finds $2,170,640\pm 3,040$ $\bar B^0\to D^{*+}\ell^-\bar\nu$ decays followed by $D^{*+}\to D^0\pi^+$.
Source Inclusive Exclusive
------------------------- ---------------------- -----------------
Data $4,412,390\pm 2100$ $47,270\pm 220$
Continuum $460,030\pm 2090$ $3,090\pm 170$
Combinatorial $B\bar B$ $1,781,720\pm 680$ $8,190\pm 50$
Peaking $1,630\pm 80$
Cabibbo suppressed $550\pm 10$
Signal $2,170,640\pm 3,040$ $33,810\pm 290$
: Event yields for the inclusive $\bar B^0\to D^{*+}\ell^-\bar\nu$ reconstruction and the exclusive analysis where the $D^0\to K^-\pi^+$ final state is reconstructed in the BABAR analysis to determine the branching fraction for $D^0\to K^-\pi^+$ decay. []{data-label="tab:babar_bkgd"}
The next step in this analysis is to use this sample of events and reconstruct the $D^0\to K^-\pi^+$ decay. To extract a clean signal BABAR studies the mass difference $\Delta M\equiv m_{K\pi\pi_s}-m_{K\pi}$ where $\pi_s$ indicate the slow pion from the $D^*$ decay. The mass difference is shown in Fig. \[fig:babar\_dm\]. The yields for this ’Exclusive’ analysis are given in Table \[tab:babar\_bkgd\]. Using simulated events BABAR determine an efficiency of $(39.96\pm0.09)\%$ for reconstructing the $D^0\to K^-\pi^+$ final state. Combining this with the data yields given above BABAR determines $${\cal B}(D^0\to K^-\pi^+)=(4.007\pm0.037\pm0.070)\%.$$ This is slightly larger than the branching fraction CLEO-c obtained, but within errors they are consistent.
![The $\Delta M$ distribution for the reconstructed $D^0\to K^-\pi^+$ candidates in events with a $\bar B^0\to D^{*+}\ell^-\bar\nu$ tag. (From Ref. [@babar_dtokpi].)[]{data-label="fig:babar_dm"}](fig2.eps){width="0.9\linewidth"}
Absolute branching fractions for hadronic $D_s$ decays at CLEO-c
================================================================
This analysis uses a sample of 195 pb$^{-1}$ of data recorded at a center-of-mas energy of 4170 MeV. At this energy $D_s$ mesons are produced, predominantly, as $D_s^+D_s^{*-}$ or $D_s^-D_s^{*+}$ pairs. CLEO-c uses the same tagging technique as for the hadronic $D$ branching fractions; they reconstruct samples of single tags and double tags and use this to extract the branching fractions.
CLEO-c studies six $D_s$ final states ($D^+_s\to K^0_S K^+$, $D^+_s\to K^+K^-\pi^+$, $D^+_s\to K^+K^-\pi^+\pi^0$, $D^+_s\to \pi^+\pi^-\pi^+$, $D^+_s\to \eta\pi^+$, and $D^+_s\to \eta'\pi^+$). The single tag event yields are shown in Fig. \[fig:cleoc\_ds\_st\]. The double tag yields are extracted by a cut-and-count procedure in the plot of the invariant mass of the $D^+_s$ vs. $D^-_s$. This plot is shown in Fig. \[fig:cleoc\_ds\_dt\]. Backgrounds are subtracted from the sidebands indicated in the plot and a total of 471 double tag events are found.
![Single tag yields for $D_s$ modes used in the CLEO-c analysis.[]{data-label="fig:cleoc_ds_st"}](4120706-001.eps "fig:"){width="0.49\linewidth"} ![Single tag yields for $D_s$ modes used in the CLEO-c analysis.[]{data-label="fig:cleoc_ds_st"}](4120706-002.eps "fig:"){width="0.49\linewidth"} ![Single tag yields for $D_s$ modes used in the CLEO-c analysis.[]{data-label="fig:cleoc_ds_st"}](4120706-003.eps "fig:"){width="0.49\linewidth"} ![Single tag yields for $D_s$ modes used in the CLEO-c analysis.[]{data-label="fig:cleoc_ds_st"}](4120706-004.eps "fig:"){width="0.49\linewidth"} ![Single tag yields for $D_s$ modes used in the CLEO-c analysis.[]{data-label="fig:cleoc_ds_st"}](4120706-005.eps "fig:"){width="0.49\linewidth"} ![Single tag yields for $D_s$ modes used in the CLEO-c analysis.[]{data-label="fig:cleoc_ds_st"}](4120706-006.eps "fig:"){width="0.49\linewidth"}
![Double tag yields for $D_s$ modes used in the CLEO-c analysis.[]{data-label="fig:cleoc_ds_dt"}](4120706-010.eps){width="0.7\linewidth"}
From these yields CLEO-c determines the branching fractions listed in Table \[tab:cleoc\_ds\_brfr\]. CLEO-c is not quoting branching fractions for $D_s^+\to \phi\pi^+$ as the $\phi$ signal is not well defined. In particular, the $\phi$ resonance interferes with the $f_0$ resonance. CLEO-c reports preliminary results for partial branching fractions for $D_s^+\to K^+K^-\pi^+$ in restricted invariant mass ranges of $m_{KK}$ near the $\phi$ resonance. In particular, for a 10 MeV cut around the $\phi$ mass the partial branching fraction of $(1.98\pm0.12\pm0.09)\%$ is found while for a 20 MeV cut the corresponding branching fraction is $(2.25\pm0.13\pm0.12)\%$.
Since these results were presented CLEO-c has updated this analysis to include 298 pb$^{-1}$ of data recorded at the $E_{\rm cm}=4170$ MeV [@cleoc_charm07]. In addition to the six mode used in the analysis described above CLEO-c also uses $D_s^+\to K^+\pi^+\pi^-$ and $D^+_s\to K^0_SK^-\pi^+\pi^+$. Among the updated results is the branching fraction ${\cal B}(D_s^+\to K^+K^-\pi^+=(5.67\pm0.24\pm0.18)\%$, in good agreement with the preliminary result presented above.
Belle study of $D_s^+\to K^+K^-\pi^+$
=====================================
Using 0.55 ab$^{-1}$ of $e^+e^-$ data recorded with the Belle detector at KEKB the Belle collaboration has studied the process $e^+e^-\to D_s^{*+}D^-_{s1}$ followed by $D^-_{s1}\to D^{*0}K^-$ and $D_s^{*+}\to D_s^+\gamma$[@belle_dskkpi]. The final state is reconstructed in two ways; either by partially reconstructing the $D_{s1}$ or the $D_s^*$.
Belle obtains the branching fraction ${\cal B}(D^+_s\to K^+K^-\pi^+)=
(4.0\pm0.4\pm0.4)\%$. This is somewhat lower than the CLEO-c result presented in the previous section.
BABAR studies of $D_s\to \phi\pi$
=================================
An earlier BABAR study has used $B\to D^*D_s^*$ decays and a technique of partially reconstructing either the $D^*$ or the $D_s^*$ to measure the $D_s\to \phi\pi$ branching fraction[@babar_dsphipi]. They quote ${\cal B}(D^+_s\to\phi\pi^+)=(4.81\pm0.52\pm0.38)\%$ based on a sample of $123\times 10^6$ $B\bar B$ decays. More recently BABAR[@babar_ds_new] has presented preliminary results based on 210 fb$^{-1}$ of data where they use a tag technique in which one $B$ is fully reconstructed. In events with one fully reconstructed $B$ candidate BABAR reconstructs one additional $D^{(*)}$ or $D_{s(J)}^{(*)}$ meson. Then they look at the recoil mass against this reconstructed candidate. The recoil masses are shown in Figs. \[fig:babar\_d\_recoil\] and \[fig:babar\_ds\_recoil\].
![The recoil mass against a $D$ or $D^*$. (From Ref. [@babar_ds_new].)[]{data-label="fig:babar_d_recoil"}](DATA_FINAL_DSTAR_last_fitresult_rebin2.eps "fig:"){width="0.49\linewidth"} ![The recoil mass against a $D$ or $D^*$. (From Ref. [@babar_ds_new].)[]{data-label="fig:babar_d_recoil"}](DATA_FINAL_DSTAR0_last_fitresult_rebin2.eps "fig:"){width="0.49\linewidth"} ![The recoil mass against a $D$ or $D^*$. (From Ref. [@babar_ds_new].)[]{data-label="fig:babar_d_recoil"}](DATA_FINAL_DC_last_fitresult_rebin2.eps "fig:"){width="0.49\linewidth"} ![The recoil mass against a $D$ or $D^*$. (From Ref. [@babar_ds_new].)[]{data-label="fig:babar_d_recoil"}](DATA_FINAL_D0_last_fitresult_rebin2.eps "fig:"){width="0.49\linewidth"}
![The recoil mass against a $D_s$ or $D_s^*$ (From Ref. [@babar_ds_new].)[]{data-label="fig:babar_ds_recoil"}](DATA_FINAL_DSSTAR_B0_last_fitresult_zoom.eps "fig:"){width="0.49\linewidth"} ![The recoil mass against a $D_s$ or $D_s^*$ (From Ref. [@babar_ds_new].)[]{data-label="fig:babar_ds_recoil"}](DATA_FINAL_DSSTAR_BP_last_fitresult_zoom.eps "fig:"){width="0.49\linewidth"} ![The recoil mass against a $D_s$ or $D_s^*$ (From Ref. [@babar_ds_new].)[]{data-label="fig:babar_ds_recoil"}](DATA_FINAL_DS_B0_last_fitresult_zoom.eps "fig:"){width="0.49\linewidth"} ![The recoil mass against a $D_s$ or $D_s^*$ (From Ref. [@babar_ds_new].)[]{data-label="fig:babar_ds_recoil"}](DATA_FINAL_DS_BP_last_fitresult_zoom.eps "fig:"){width="0.49\linewidth"}
From these modes BABAR extracts ${\cal B}(D_{sJ}(2460)^-\to D_s^{*-}\pi^0)
=(56\pm13\pm9)\%$ and ${\cal B}(D_{sJ}(2460)^-\to D_s^{*-}\gamma)
=(16\pm4\pm3)\%$ in addition to ${\cal B}(D_s^-\to \phi\pi^+)=
(4.62\pm0.36\pm0.50)\%$.
Inclusive measurements of $\eta$, $\eta'$, and $\phi$ production in $D$ and $D_s$ decays
========================================================================================
Using samples of tagged $D$ and $D_s$ decays CLEO-c has measured the inclusive production of $\eta$, $\eta'$, and $\phi$ mesons by looking at the recoil against the tag[@cleoc_inclusive]. The results are summarized in Table \[tab:cleoc\_inclusive\]. The knowledge of inclusive measurements before this CLEO-c measurement was poor, besides limits only ${\cal B}(D^0\to \phi X)=1.7\pm0.8$ was measured. As expected the $\eta$, $\eta'$, and $\phi$ rates are much higher in $D_s$ decays.
Decay ${\cal B}$ (%)
------------------- ----------------------
$D^0\to\eta X$ $9.5\pm0.4\pm0.8$
$D^-\to\eta X$ $6.3\pm0.5\pm0.5$
$D_s^+\to\eta X$ $23.5\pm3.1\pm2.0$
$D^0\to\eta' X$ $2.48\pm0.17\pm0.21$
$D^-\to\eta' X$ $1.04\pm0.16\pm0.09$
$D_s^+\to\eta' X$ $8.7\pm1.9\pm1.1$
$D^0\to\phi X$ $1.05\pm0.08\pm0.07$
$D^-\to\phi X$ $1.03\pm0.10\pm0.07$
$D_s^+\to\phi X$ $16.1\pm1.2\pm1.1$
: Inclusive branching fractions[]{data-label="tab:cleoc_inclusive"}
The doubly Cabibbo suppressed decay $D^+\to K^+\pi^0$
=====================================================
Both CLEO-c and BABAR have studied the doubly Cabibbo suppressed decay $D^+\to K^+\pi^0$. CLEO-c[@cleoc_dcsd] has reconstructed candidates in a 281 pb$^{-1}$ sample of $e^+e^-$ data recorded at the $\psi(3770)$. BABAR[@babar_dcsd] has used a sample of 124 fb$^{-1}$ recorded at the $\Upsilon(4S)$. CLEO-c and BABAR finds branching fractions in good agreement with each other, ${\cal B}(D^+\to K^+\pi^0)=(2.24\pm0.36\pm0.15\pm0.08)\times 10^{-4}$ and ${\cal B}(D^+\to K^+\pi^0)=(2.52\pm0.46\pm0.24\pm0.08)\times 10^{-4}$ respectively.
Modes with $K^0_L$ or $K^0_S$ in the final states
=================================================
It has commonly been assumed that $\Gamma(D\to K^0_S X)=\Gamma(D\to K^0_L X)$. However, as pointed out by Bigi and Yamamoto[@bigi] this is not generally true as for many $D$ decays there are contributions from Cabibbo favored and Cabibbo suppressed decays that interfere and contributes differently to final states with $K^0_S$ and $K^0_L$. As an example consider $D^0\to K^0_{S,L}\pi^0$. Contributions to these final states involve the Cabibbo favored decay $D^0\to \bar K^0\pi^0$ as well as the Cabibbo suppressed decay $D^0\to K^0\pi^0$. However, we don’t observe the $K^0$ and the $\bar K^0$ but rather the $K^0_S$ and the $K^0_L$. As these two amplitudes interfere constructively to form the $K^0_S$ final state we will see a rate asymmetry. Based on factorization Bigi and Yamamoto predicted $$\begin{aligned}
R(D^0)&\equiv&{{\Gamma(D^0\to K^0_S\pi^0)-\Gamma(D^0\to K^0_L\pi^0)}\over
{\Gamma(D^0\to K^0_S\pi^0)+\Gamma(D^0\to K^0_L\pi^0)}}\\
&\approx& 2\tan^2\theta_C\approx 0.11.\\\end{aligned}$$ Using tagged $D$ mesons CLEO-c has measured this asymmetry and obtained $$R(D^0)=0.122\pm0.024\pm0.030$$ which is in good agreement with the prediction.
Similarly, CLEO-c has also measured the corresponding asymmetry in charged $D$ mesons and obtained $$\begin{aligned}
R(D^+)&\equiv&{{\Gamma(D^+\to K^0_S\pi^+)-\Gamma(D^+\to K^0_L\pi^+)}\over
{\Gamma(D^+\to K^0_S\pi^+)+\Gamma(D^+\to K^0_L\pi^+)}}\\
&=&0.030\pm0.023\pm0.025.\\\end{aligned}$$ Prediction of the asymmetry in charged $D$ decays is more involved. D.-N. Gao predicts [@gao] this asymmetry to be in the range 0.035 to 0.044, which is consistent with the observed asymmetry.
Summary
=======
Recently there has been a lot of progress on the determination of absolute hadronic branching fractions of $D$ and $D_s$ mesons. Here recent results from CLEO-c and the B-factory experiments, BABAR and Belle, were reported. CLEO-c uses the extremely clean environment at threshold for these measurements while the B-factory experiments use their very large data samples to explore partial reconstruction techniques to determine the absolute hadronic branching fractions.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the National Science Foundation grant PHY-0202078 and by the Alfred P. Sloan foundation.
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K. Abe [*et al.*]{} (Belle Collab.), arXiv:hep-ex/0701053 (2007).
A. Ryd (on behalf of the CLEO Collab.), “CLEO-c $D$ and $D_s$ Hadronic Decays”, presented at Charm2007, Ithaca NY, August 5-8, 2007.
B. Aubert [*et al.*]{} (BABAR Collab.), arXiv:0704.2080 \[hep-ex\] (2007).
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---
abstract: 'We present the detection of a filament of Ly$\alpha$ emitting galaxies in front of the quasar Q1205-30 at z=3.04 based on deep narrow band imaging and follow-up spectroscopy obtained at the ESO NTT and VLT. We argue that Ly$\alpha$ selection of high redshift galaxies with relatively modest amounts of observing time allows the detection and redshift measurement of galaxies with sufficiently high space densities that we can start to map out the large scale structure at z$\approx$2-3 directly. Even more interesting is it that a 3D map of the filaments will provide a new cosmological test for the value of the cosmological constant, $\Omega_{\Lambda}$.'
author:
- 'Johan P.U. Fynbo'
- Palle Møller
- Bjarne Thomsen
title: 'Detecting filaments at z$\approx$3'
---
Introduction
============
For the past few decades computer simulations have been ahead of the observations when it comes to describing the first structures to form at high redshifts. The present consensus of the model builders is that the gas arranges itself in long string–like structures commonly referred to as filaments (see Fig. \[filament\]). Density variations along the filaments will lead to formation of lumps of cold, self–shielding HI regions and those regions are identified, in the simulations, as regions of starformation. Because of the high column density of neutral Hydrogen a sightline through such a cloud intersects, they are also identified as strong absorbers known as Damped Ly$\alpha$ Absorbers (DLAs). By poking random sightlines through a virtual universe one may simulate observations, and a given model universe will hence predict a specific correlation between DLA systems and the galaxies hosting the DLAs (e.g. Katz et al. 1996). Comparison to real observations of DLA galaxies (Møller & Warren 1998) has shown that there is very good agreement between observations and simulations. This agreement is encouraging, but it would be of great interest if one could observationally map out the actual filaments. Until now this his been done only at low redshifts (e.g. De Lapparent et al. 1991; Bharadwaj et al. 2000), but never at z$>$0.1. Knowing the distribution of scalesizes of filaments at different redshifts will help constrain the allowable parameter space of the simulations.
Unfortunately such a map cannot be constructed directly via absorption studies, because there is currently not a sufficiently tight mesh of background z$>$3 quasars available (e.g. Pichon et al. 2001). The best way to proceed is hence to attempt to find enough centres of starformation to be able to map out filaments by their own light. In order to identify objects to map out filaments one might at first guess that a search for Lyman Break Galaxies (LBGs, Steidel & Hamilton 1992) would be the best procedure. Unfortunately only the very brightest galaxies can be found and have their redshifts measured precisely enough with this technique, and such sparse sampling of the filamentary structure does not allow the structures to be seen. However, it has been shown that both DLA galaxies and galaxies selected for their Ly$\alpha$ emission, are sampling the high redshift galaxy population much further down the Luminosity function than do the LBGs, and one will therefore expect a better sampling of the high redshift structure if one uses DLA galaxies and Ly$\alpha$ galaxies (Fynbo, Møller & Warren 1999; Haehnelt et al. 2000). This has recently been independently confirmed, as deep narrow band Ly$\alpha$ imaging in a known overdensity of LBGs revealed about a factor of 10 more candidate Ly$\alpha$ galaxies than LBGs (Steidel et al. 2000).
Observations
============
In February through March 1998 we obtained deep narrow band imaging in a 21Å wide filter tuned to Ly$\alpha$ at z=3.04. The filter was tuned to the wavelength of a strong Ly$\alpha$ absorption line in the spectrum of the QSO (Fig. \[QSO\]). The data were collected as service observing program at the 3.5-m ESO New Technology Telescope on La Silla, Chile. In total almost 18 hours of narrow band imaging was secured reaching a 5$\sigma$ flux limit of 1.1$\times$10$^{-17}$ erg s$^{-1}$ cm$^{-2}$ (Fynbo, Thomsen & Møller 2000). We detected six good ($>$5$\sigma$) and two marginal ($\sim$4$\sigma$) candidate Ly$\alpha$ emitters in the field of the QSO as well as extended Ly$\alpha$ emission close to the QSO line of sight. In March 2000 we obtained Multi-Object follow-up spectroscopy at the ESO Very Large Telescope using FORS1 on the UT1 unit telescope. We also obtained deep imaging in the B and I bands (reaching B(AB)=26.7 and I(AB)=25.9 at 5$\sigma$).
![The spectrum of Q1205-30 obtained at the VLT in March 2000. The insert in the uper right hand corner shows the region of the spectrum around a strong Ly$\alpha$ absorption line z=3.0322. The narrow filter (transmission curve overplotted) was tuned to Ly$\alpha$ at this redshift. The redshift of the background QSO, as measured from the low ionization OI line, is z=3.0473$\pm$0.0012.[]{data-label="QSO"}](QSO1.ps){width="70.00000%"}
-2cm
Results
=======
The VLT spectroscopy confirmed (by detecting the Ly$\alpha$ line and at the same time excluding the possibility of low redshift interlopers) all six good candidates and one of the two marginal candidates as Ly$\alpha$ emitters at z=3.04 (Fynbo, Møller & Thomsen 2001). The spectral regions around Ly$\alpha$ for all confirmed candidates are shown in Fig. \[spec\]. The spectroscopy of the extended emission close to the QSO line-of-sigt will be presented in a separate paper (Weidinger et al. in preparation).
In Table \[redtab\] we present the redshifts and celestial positions for all confirmed Ly$\alpha$ emitters and for the absorber (from Møller & Fynbo 2001).
Object $\Delta$RA (arcsec) $\Delta$decl. (arcsec) redshift
------------ --------------------- ------------------------ ----------
S7 -143.3$\pm$0.6 41.9$\pm$0.2 3.0402
S8 -141.5$\pm$0.6 59.7$\pm$0.2 3.0398
S9 -124.6$\pm$0.5 63.4$\pm$0.2 3.0350
S10 -119.9$\pm$0.5 59.8$\pm$0.2 3.0353
S11 -77.8$\pm$0.3 0.9$\pm$0.1 3.0312
S12 -43.9$\pm$0.2 54.4$\pm$0.2 3.0333
S13 68.3$\pm$0.3 -52.1$\pm$0.2 3.0228
abs 0.0 0.0 3.0322
\[redtab\]
: Redshifts and positions of seven Ly$\alpha$ emitters and a Ly$\alpha$ absorber in the field of Q1205–30. The positions are given relative to the quasar coordinates: 12:08:12.7, -30:31:06.10 (J2000.0). The uncertainty on the redshifts is 0.0012 (1$\sigma$).
Filamentary structure
---------------------
In Fig. \[filafig\] we show the objects plotted in the box defined by the Field of View of the Camera and the redshift depth of the filter for Ly$\alpha$ at z=3. As seen the 7 Ly$\alpha$ emitters (marked with filled symbols) and the absorber (the open symbol) all align in this diagram. If we assume that the redshifts are all solely due to Hubble flow then this implies a real alignment in 3D space e.g. a filamentary spatial distribution of the objects. However, the measured redshift may not be due to Hubble flow alone for mainly two effects : [*i)*]{} outflows, and [*ii)*]{} peculiar velocities. We now briefly discuss the importance of each of these effects. [*Outflows:*]{} In the nearby starburst galaxy NGC1705 the outflow velocity is estimated to be around 80 km s$^{-1}$ (Heckman et al. 2001 and references therein). Outflows of this strength will cause a shift in the redshift measurement which is of the same order as the combined uncertainty from the wavelength calibration and line centroid measurement (corresponding to 90 km s$^{-1}$ at z=3). Outflows will either produce a systematic blueshift of the emission line redshift if the galaxies are opaque (so that we only see the gas moving towards us) or a broadening with no velocity shift of the lines (if the galaxies are transparent and we also see the gas moving away from us). The fact that the absorber, for which the redshift is detemined from the Ly$\alpha$ absorption line, also follows the alignment is an argument against a significant blueshift due to outflows. [*Peculiar velocities:*]{} In the local universe (v$<$4000 km s$^{-1}$) the $\sigma$(v$_{peculiar}$) of peculiar velocities is of the order 200 km s$^{-1}$ (e.g. Branchini et al. 2001). We do not expect this number to be larger at z=3. Furthermore, any peculiar velocities will tend to smear out any underlying filamentary structure, so the fact that we see alignment is an argument against large peculiar velocities. We therefore conclude that the most likely interpretation of Fig. \[filafig\] is that we see a redshift z=3 filament.
Properties of the filament
--------------------------
We can only determine a lower limit to the length of the filament as it seems to extend beyond the volume mapped by our instrumental setup (CCD and filter). Assuming a Hubble constant of 65 km s$^{-1}$ Mpc$^{-1}$, $\Omega_m=0.3$, and $\Omega_{\Lambda}=0.7$ we find a coming length (defined as the distance between the two outhermost objects) of 4800 proper kpc. The radius of the minimum cylinder containing all objects is 400 proper kpc. Due to the effect of peculiar and outflow velocities this radius should be considered an upper limit.
The derived properties of filaments are strongly dependend on the assumed cosmology. In particular, since filaments are anchored in the Hubble flow, the observed angular distribution of a sample of filaments will be a function of the assumed cosmology. Therefore, it is in principle possible to use a sample of filaments to obtain an independend constrain on the value of the cosmological constant at z$\approx$3 (Weidinger et al. 2001).
-3cm
Summary and outlook
===================
In order to start mapping out the large scale filamentary structure suggested by numerical simulations directly at high redshift we need cosmic sources that are very numerous rather than rare, very bright light houses such as QSOs or Gamma-Ray Bursters. We have here demonstrated that by reaching flux limits below 1$\times$10$^{-17}$ erg s$^{-1}$ cm$^{-2}$ the density of Ly$\alpha$ emitting galaxies is sufficiently high at z=3 to allow a direct mapping of filamentary structure.
The next logical step is to try to map out larger regions of the z$\approx$2–3 universe with Ly$\alpha$ emitters. Therefore we (Møller, Fynbo, Thomsen, Egholm, Weidinger, Haehnelt, Theuns) have initiated a large area survey for Ly$\alpha$ emitters at z=2 with the 2.56-m Nordic Optical Telescope on La Palma. Furthermore, in a pilot project conducted at the ESO VLT we (here Fynbo, Ledoux, Burud, Leibundgut, Møller and Thomsen) have obtained narrow band observations of two fields around QSO absorbers at z$\approx$3. In the field of the z=2.85 absorber towards Q2138-4427, for which our imaging observations are complete, we reach a detection limit of about 7$\times$10$^{-18}$ erg s$^{-1}$ cm$^{-2}$ and detect 34 candidate Ly$\alpha$ emitters in a 45 arcmin$^2$ field over a redshift range of $\Delta$z=0.05. This shows that the density of z=3.04 Ly$\alpha$ emitters in the Q1205-30 field is not unusually high. Follow-up spectroscopic observations of the Q2138-4427 field has not yet been obtained. In the future we hope to map out a large volume with several hundred z=3 Ly$\alpha$ emitters with the VLT.
Acknowledgments {#acknowledgments .unnumbered}
===============
This paper is based on observations collected at the European Southern Observatory, La Silla and Paranal, Chile (ESO project No. 60.B-0843 and 64.O-0187).
[8.]{}
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abstract: 'This paper deals with the approximations of Durrmeyer type generalization of Sz$\acute{\text{a}}$sz-Mirakjan operators. We establish the direct results, quantitative Voronovskaya type theorem, Gr$\ddot{\text{u}}$ss type theorem, $A$-statistical convergence, rate of convergence in terms of the function with derivative of bounded variation. At last, the graphical analysis, comparison study and numerical representations of proposed operators are discussed.'
title: 'Approximation on Durrmeyer modification of generalized Sz$\acute{\text{a}}$sz-Mirakjan operators '
---
[**Rishikesh Yadav$^{1,\dag}$, Ramakanta Meher$^{1,\star}$, Vishnu Narayan Mishra$^{2,\circledast}$**]{}\
$^{1}$Applied Mathematics and Humanities Department, Sardar Vallabhbhai National Institute of Technology Surat, Surat-395 007 (Gujarat), India.\
$^{2}$Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak-484 887, Anuppur, Madhya Pradesh, India\
$^\dag$rishikesh2506@gmail.com, $^\star$meher\_ramakanta@yahoo.com, $^\circledast$vishnunarayanmishra@gmail.com
0.5in
**MSC 2010**: [41A25, 41A35, 41A36]{}.
**Keywords:** Sz$\acute{\text{a}}$sz-Mirakjan operators; modulus of continuity; Lipschitz function; statistical convergence; function of bounded variation.
Introduction
============
In 1950, Sz$\acute{\text{a}}$sz [@OS] studied the approximation properties of generalization of Bernstein’s operators on infinite interval and known as Sz$\acute{\text{a}}$sz-Mirakjan operators. After two decades, in 1977, Jain and Pethe [@JP], introduced new type of Sz$\acute{\text{a}}$sz-Mirakjan operators which are as follows:
$$\begin{aligned}
\label{no1}
\mathcal{LO}_{n}^{[\alpha]}=\sum\limits_{i=0}^{\infty}(1+n\alpha)^{\frac{-x}{\alpha}}\left(\alpha+\frac{1}{n}\right)^{-i}\frac{x^{(i,-\alpha)}}{i!}f\left(\frac{i}{n} \right),\end{aligned}$$
where $x^{(i,-\alpha)}=x(x+\alpha)\cdots(x+(i-1)\alpha$, $x^{(0,-\alpha)}=1$ and the function $f$ is considered to be of exponential type such that $f(x)\leq C e^{Ax},~(x\geq 0$, $A>0)$, with positive constant $C$ and here $\alpha=\alpha_n,~n\in\mathbb{N}$ is as $0\leq\alpha_n\leq\frac{1}{n}$. They determined the approximation properties of the said operators. For $\alpha=\frac{1}{n}$, the above operators (\[no1\]) reduce to the operators which have been defined by Agratini [@AO2]. In 2007, Abel and Ivan [@AI] replaced $\alpha$ by $\frac{1}{nc}$ in the above operators (\[no1\]) and obtained the generalized version operators, which are as:
$$\begin{aligned}
\label{no2}
\mathcal{AO}_{n}^{c}=\sum\limits_{i=0}^{\infty}\left(\frac{c+1}{c}\right)^{-xnc} (1+c)^{-i} \binom{ncx+i-1}{i} f\left(\frac{i}{n} \right),\end{aligned}$$
where $c=c_n$, $n\in\mathbb{N}$ is restricted with a certain constant $\beta>0$ such that $c\geq \beta$. The main purpose to define the above operators (\[no2\]), was to investigate the local approximation properties and to check the asymptotic behavior. Very recently, Dhamiza et al. [@DPD] defined Kantorovich variant of the operators (\[no1\]), for the studying the local approximations properties and rate of convergence. For bounded and integrable function on $[0,\infty)$, the operators are defined by $$\begin{aligned}
\mathcal{MO}_{n}^{\alpha}=\sum\limits_{i=0}^{\infty}(1+n\alpha)^{\frac{-x}{\alpha}}\left(\alpha+\frac{1}{n}\right)^{-i}\frac{x^{(i,-\alpha)}}{i!}\int\limits_{\frac{i}{n}}^{\frac{i+1}{n}} f(t)dt.\end{aligned}$$ A special case, when $\alpha\to 0$, the above operators reduce to Sz$\acute{\text{a}}$sz-Mirakjan-Kantorovich operators defined by Totik [@vt]. The property of Kantorovich type operators are also discussed in [@RYRVN; @RYMVN]. But for the Durremnyer point of view of the Sz$\acute{\text{a}}$sz-Mirakjan operators, in 1985, Mazhar and Totik [@SV], modified the Sz$\acute{\text{a}}$sz-Mirakjan operators into summation integral type operators, which are defined by $$\begin{aligned}
\label{d1}
I_{n}(h;x)=n\sum\limits_{i=0}^{\infty}u_{n,i}(x)\int\limits_0^\infty u_{n,i}(t)h(t)~dt,\end{aligned}$$ where $u_{n,i}(x)=e^{-nx}\frac{(nx)^i}{i!}$, and independently the related aaproximations properties have been discussed by Kasana et al. [@HSG]. In this regard, Gupta and Pant [@VRP] determined the rate of convergence and other approximations properties. Some approximations properties can be seen in [@MKMS]. Their modifications into Durrmeyer version and properties can be seen in various research articles, such as [@GKR; @VGA; @VG; @KAP]. In 2016, Mishra et al. [@VNM] modified Sz$\acute{\text{a}}$sz-Mirakjan Durrmeyer operators using a positive sequence of functions to study the properties like simultaneous approximation, rate of convergence etc. The modified operators are as $$\begin{aligned}
\label{d2}
D_{n}(h;x)=d_n\sum\limits_{i=0}^{\infty}u_{d_n,i}(x)\int\limits_0^\infty u_{d_n,i}(t)h(t)~dt,\end{aligned}$$ where, $d_n\to\infty$ as $n\to\infty$ be a positive sequence of real number which is strictly increasing as well as $d_1\geq 1$ and $u_{d_n,i}(x)=e^{-d_nx}\frac{(d_nx)^i}{i!}$. Moreover, the operators (\[d1\]) can be obtained when $d_n=n$ in the the operators (\[d2\]).
Motivated by above works, we define the Durrmeyer modification of the above operators (\[no1\]) by considering the function as integrable and bounded on the interval $[0,\infty)$ as follows:
$$\begin{aligned}
\label{O1}
\mathcal{U}_{n}^{[\alpha]}(f;x)=n\sum\limits_{i=0}^{\infty}(1+n\alpha)^{\frac{-x}{\alpha}}\left(\alpha+\frac{1}{n}\right)^{-i}\frac{x^{(i,-\alpha)}}{i!}\int\limits_{0}^{\infty}e^{-n u}\frac{(n u)^i}{i!} f(u)~du.\end{aligned}$$
For $\alpha\to 0$, the above operators will be reduced into Sz$\acute{\text{a}}$sz-Mirakjan-Durrmeyer operators, which are defined by equation (\[d1\]).\
The main motive of this article is to investigate the approximation properties of the defined operators (\[O1\]). Therefore, we divide it into sections. The rate of convergence of the defined operators is obtained in the terms of modulus of continuity, second order modulus of continuity with the relations of Peetre’s $K$-functionals in the section \[sec2\]. Section \[sec3\] consists, weighted approximations properties including convergence of the operators using weight function. To determine the properties of the operators (\[O1\]) via quantitatively, quantitative Voronovskaya type theorem and Gr$\ddot{\text{u}}$ss Voronovskaya type theorem are studied in section \[sec4\]. In section \[sec5\], graphical and numerical representations are presented for the support of approximation results. Section \[sec6\] represent $A$-statistical properties of the operators and in section (\[sec7\]), an important property is studied for the rate of the convergence by means of the derivative of bounded variations. Finally, conclusion, result discussion and applications are discussed.
Preliminaries
=============
This section contains, basic lemmas, remark and theorem, which are used to prove our main theorems and study the approximations properties of the proposed operators. Here, we need following lemma.
\[l2\] Following results hold for all $n\in \mathbb{N}$: $$\begin{aligned}
\mathcal{U}_{n}^{[\alpha]}(1;x)&=& 1\\
\mathcal{U}_{n}^{[\alpha]}(t;x)&=& \frac{1+nx}{n}\\
\mathcal{U}_{n}^{[\alpha]}(t^2;x)&=& \frac{2+4nx+n^2x^2+n^2x\alpha}{n^2}\\
\mathcal{U}_{n}^{[\alpha]}(t^3;x)&=& \frac{6+18nx+9n^2x(x+\alpha)+n^3x(x^2+3x\alpha+2\alpha^2)}{n^3}\\
\mathcal{U}_{n}^{[\alpha]}(t^3;x)&=& \frac{24+96nx+72n^2x(x+\alpha)+16n^3x(x^2+3x\alpha+2\alpha^2)+n^4x(x^3+6x^2\alpha+11x\alpha^2+6\alpha^3)}{n^4}.\end{aligned}$$
Here $$\begin{aligned}
\label{s}
\int\limits_{0}^{\infty}e^{-n u}\frac{(n u)^i}{i!} u^m~du=\frac{1}{n^{m+1}}\frac{(i+m)!}{i!},\end{aligned}$$ in particular, if $m=0$ then $$\begin{aligned}
\int\limits_{0}^{\infty}e^{-n u}\frac{(n u)^i}{i!} ~du=\frac{1}{n},\end{aligned}$$ if $m=1$ then $$\begin{aligned}
\int\limits_{0}^{\infty}e^{-n u}\frac{(n u)^i}{i!} u ~du=\frac{(i+1)}{n^2},\end{aligned}$$
and so on...
Notice that $$\begin{aligned}
\label{A}
(1+n\alpha)^{\frac{x}{\alpha}}=\sum\limits_{i=0}^{\infty}\left(\alpha+\frac{1}{n}\right)^{-i}\frac{x^{(i,-\alpha)}}{i!}.\end{aligned}$$
Using the equations (\[s\]) and (\[A\]), we get $$\begin{aligned}
\mathcal{U}_{n}^{[\alpha]}(1;x)&=&\sum\limits_{i=0}^{\infty}(1+n\alpha)^{\frac{-x}{\alpha}}\left(\alpha+\frac{1}{n}\right)^{-i}\frac{x^{(i,-\alpha)}}{i!}\\
&=& (1+n\alpha)^{\frac{-x}{\alpha}}\sum\limits_{i=0}^{\infty} \left(\alpha+\frac{1}{n}\right)^{-i}\frac{x^{(i,-\alpha)}}{i!}\\
&=& (1+n\alpha)^{\frac{-x}{\alpha}} (1+n\alpha)^{\frac{x}{\alpha}}=1\\
\mathcal{U}_{n}^{[\alpha]}(u;x)&=& \sum\limits_{i=0}^{\infty}(1+n\alpha)^{\frac{-x}{\alpha}}\left(\alpha+\frac{1}{n}\right)^{-i}\frac{x^{(i,-\alpha)}}{i!} \frac{i+1}{n}\\
&=&\frac{(1+n\alpha)^{\frac{-x}{\alpha}}}{n}\sum\limits_{i=1}^{\infty}\left(\alpha+\frac{1}{n}\right)^{-i}\frac{x^{(i,-\alpha)}}{i!} i+ \frac{1}{n}\sum\limits_{i=1}^{\infty}(1+n\alpha)^{\frac{-x}{\alpha}}\left(\alpha+\frac{1}{n}\right)^{-i}\frac{x^{(i,-\alpha)}}{i!}\\
&=& x+\frac{1}{n}.\end{aligned}$$
Consider the $\Theta_{n,m}^{[\alpha]}(x)=\mathcal{U}_{n}^{[\alpha]}((t-x)^m;x)$, $m=1,2,3$ are the central moments and here we obtain the following the results.
\[l4\] For each $x\geq 0$ and $n\in\mathbb{N}$, it holds: $$\begin{aligned}
\Theta_{n,1}^{[\alpha]}(x)&=& \frac{1}{n}\\
\Theta_{n,2}^{[\alpha]}(x)&=&\frac{\alpha n^2 x+2 n x+2}{n^2} \\
\Theta_{n,3}^{[\alpha]}(x)&=& \frac{2 \alpha ^2 n^3 x+9 \alpha n^2 x+12 n x+6}{n^3}\\
\Theta_{n,4}^{[\alpha]}(x)&=&\frac{3 \alpha ^2 n^4 x (2 \alpha +x)+4 \alpha n^3 x (8 \alpha +3 x)+12 n^2 x (6 \alpha +x)+72 n x+24}{n^4}\end{aligned}$$
Using the Lemma \[l2\], we can easily prove all parts of the above lemma, so we omit the proof.
For $x\in[0,\infty)$ and for $n\in\mathbb{N}$, we obtain $$\begin{aligned}
\Theta_{n,2}^{[\alpha]}(x)&=& \frac{\alpha n^2 x+2 n x+2}{n^2}=\alpha x+\frac{2x}{n}+\frac{2}{n^2}\\
&\leq & \frac{3x}{n}+\frac{2}{n^2}=\frac{3}{n}\left(x+\frac{1}{n}\right)=\frac{3}{n}\eta_n^2(x)\end{aligned}$$
The above operators \[O1\] can be written as $$\begin{aligned}
\mathcal{U}_{n}^{[\alpha]}(f;x)=\int_0^\infty u_{n}^{[\alpha]} (x,t)f(t)~dt\end{aligned}$$ where $u_{n}^{[\alpha]}(x,t)=n\sum\limits_{i=0}^\infty r_{n,i}^{[\alpha]}(x)p_n(t)$, $r_{n,i}^{[\alpha]}(x)=(1+n\alpha)^{\frac{-x}{\alpha}}\left(\alpha+\frac{1}{n}\right)^{-i}\frac{x^{(i,-\alpha)}}{i!}$ and $p_n(x)=e^{-n x}\frac{(n x)^i}{i!}$.
\[l1\] For every $x\geq 0$ and $\max\alpha=\frac{1}{n}$ then it holds: $$\begin{aligned}
\underset{n\to\infty}\lim\{n\Theta_{n,1}^{\alpha}(x)\}&=& 1,\\
\underset{n\to\infty}\lim\{n\Theta_{n,2}^{\alpha}(x)\}&=& 3x,\\
\underset{n\to\infty}\lim\{n^2\Theta_{n,4}^{\alpha}(x)\}&=& 27x^2,\\
\underset{n\to\infty}\lim\{n^3\Theta_{n,6}^{\alpha}(x)\}&=& 405x^3.\\\end{aligned}$$
If a function $g$ defined on $[0,\infty)$ and bounded with supremum norm $\| f\|=\underset{x\geq0}\sup |f(x)|$ then there is an inequality holds as: $$\begin{aligned}
\left|\mathcal{U}_{n}^{[\alpha]}(f;x) \right|\leq \|f\|.\end{aligned}$$
\[th1\] Consider $g\in C_A[0,\infty)$, $\alpha\in[0,\frac{1}{n}]$ and $\alpha\to 0$ as $n\to\infty$ then we get $$\begin{aligned}
\underset{n\to\infty}\lim\mathcal{U}_{n}^{[\alpha]}(g;x)=g(x),\end{aligned}$$ uniformly on each finite interval of $[0,\infty)$.
Direct Results {#sec2}
==============
This segment, consists the uniform convergence theorem, rate of the convergence of the proposed operators using second order modulus of smoothness and modulus of continuity and a relation exist with Peetre’s $K$-functional.\
To study the approximation properties, we suppose $C_B[0,\infty)$ be the set of all continuous and bounded function $g$ defined on $[0,\infty)$ with supremum norm $\|g\|=\underset{x\geq0} \sup |g(x)|$. And here for any $\xi>0$, the Peetre’s $K$-functional is defined by: $$\begin{aligned}
K_2(g;\xi)=\underset{g_1\in C_B^2[0,\infty)}\inf\{\|g-g_1\|+\xi \|g_1''\|\}, ~~\text{where}~C_B^2[0,\infty)=\{g\in C_B[0,\infty):g',g''\in C_B[0,\infty)\}.\end{aligned}$$ In 1993, an important relation was introduced by De Vore [@q1] by considering Peetre’s $K$-functional and second order modulus of smoothness, which is given below: $$\begin{aligned}
\label{pe1}
K_2(g;\xi)\leq M \omega_2(f;\sqrt{\xi}),\end{aligned}$$ where $ \omega_2(f;\sqrt{\xi})$ is second order modulus of smoothness and is defined by, $$\begin{aligned}
\omega_2(f;\sqrt{\xi})=\underset{h\in[0,\sqrt{\xi}],x\in[0,\infty)}\sup \{|f(x+2h)-2f(x+h)+f(x)|\}, ~f\in C_B[0,\infty)\end{aligned}$$
**Note.** The first order modulus of continuity is $\omega(f;\xi)=\underset{h\in[0,\xi],x\in[0,\infty)}\sup \{|f(x+h)-f(x)|\}, ~f\in C_B[0,\infty)$.\
Consider $f\in C_B[0,\infty)$ and for every $x\in[0,\infty)$, it holds as: $$\begin{aligned}
\left|\mathcal{U}_{n}^{[\alpha]}(f;x)-f(x)\right|\leq M \omega_2(f;\sqrt{\rho_n(x)})+\omega\left(f;\nu_n(x) \right),\end{aligned}$$ where, $\nu_n(x)=\Theta_{n,1}^{[\alpha]}(x)$ and $\rho_n(x)=\Theta_{n,2}^{[\alpha]}(x)+\frac{1}{n^2}$.
For $f\in C_B[0,\infty)$, consider $$\begin{aligned}
\mathfrak{U}_{n}^{[\alpha]}(f;x)=\mathcal{U}_{n}^{[\alpha]}(f;x)+f(x)-f\left(\frac{1+nx}{n}\right).\end{aligned}$$ Here, $\mathfrak{U}_{n}^{[\alpha]}(1;x)=1$ and $\mathfrak{U}_{n}^{[\alpha]}(t;x)=x$, i.e. the auxiliary operators preserve the linear function and constant term. Now, using Taylor’s remainder formula for integral for the function $u\in C_B^2[0,\infty)$, we have $$\begin{aligned}
u(t)-u(x)=(t-x)u'(x)+\int\limits_x^t (t-v)u''(v)~dv.\end{aligned}$$ Applying the operators $\mathfrak{U}_{n}^{[\alpha]}$ to the both sides, we obtain $$\begin{aligned}
\mathfrak{U}_{n}^{[\alpha]}(u(t)-u(x);x)&=&\mathfrak{U}_{n}^{[\alpha]}\left(\int\limits_x^t (t-v)u''(v)~dv \right)\\
&=& \mathcal{U}_{n}^{[\alpha]}\left(\int\limits_x^t (t-v)u''(v)~dv\right)-\left(\int\limits_x^{\left(\frac{1+nx}{n}\right)} \left(\frac{1+nx}{n}-v\right)u''(v)~dv\right).\end{aligned}$$ Here, $$\begin{aligned}
\left|\int\limits_x^t (t-v)u''(v)~dv\right|\leq \int\limits_x^t |t-v||u''(v)|~dv\leq |u''|(t-x)^2.\end{aligned}$$ Similarly, $$\begin{aligned}
\left|\int\limits_x^{\left(\frac{1+nx}{n}\right)} \left(\frac{1+nx}{n}-v\right)u''(v)~dv \right|\leq \frac{|u''|}{n^2}.\end{aligned}$$ Therefore, one has $$\begin{aligned}
\left|\mathcal{U}_{n}^{[\alpha]}(f;x)-f(x)\right| & \leq & |u''| \Theta_{n,2}^{[\alpha]}(x)+\frac{|u''|}{n^2}\\
&=& \rho_n(x)|u''|\end{aligned}$$ Also, $$\begin{aligned}
|\mathfrak{U}_{n}^{[\alpha]}(f;x)|=|\mathcal{U}_{n}^{[\alpha]}(f;x)|+2\|f(x)\|\leq 3\|f(x)\|. \end{aligned}$$ By considering all above inequalities, we obtain $$\begin{aligned}
\left|\mathcal{U}_{n}^{[\alpha]}(f;x)-f(x)\right|&\leq &|\mathfrak{U}_{n}^{[\alpha]}(f-u;x)|+|\mathfrak{U}_{n}^{[\alpha]}(u(t)-u(x);x)|+|u(x)-f(x)|+\left|f\left(\frac{1+nx}{n}\right)-f(x) \right|\\
&\leq & 4\|f-u\|+\rho_n(x)\|u''\|+\left|f\left(\frac{1+nx}{n}\right)-f(x) \right|\\
&\leq & M \{\|f-u\|+\rho_n(x)\|u''\|\}+\omega\left(f;\nu_n(x) \right).\end{aligned}$$ Now, taking minimum overall $u\in C_B^2[0,\infty)$ on the right hand side of above inequality and using Peetre $K$-functional, we obtain $$\begin{aligned}
\left|\mathcal{U}_{n}^{[\alpha]}(f;x)-f(x)\right| &\leq & M K_2(f;\rho_n(x))+\omega\left(f;\nu_n(x) \right)\\
&\leq & M \omega_2(f;\sqrt{\rho_n(x)})+\omega\left(f;\nu_n(x) \right).\end{aligned}$$
Now, we estimate the approximation of the defined operators (\[O1\]), by new type of Lipschit maximal function with order $r\in(0,1]$, defined by Lenze [@LB] as $$\begin{aligned}
\label{eq8}
\kappa_r(f,x)=\underset{x,s\geq 0}\sup \frac{|f(s)-f(x)|}{|s-x|^r},~~x\neq s. \end{aligned}$$ Using Lipschit maximal function, we have an upper bound with the function, given by a theorem.
Consider $f\in C_B[0,\infty)$ with $r\in(0,1]$ then we obtain $$\begin{aligned}
\left|\mathcal{U}_{n}^{[\alpha]}(f;x)-f(x)\right| &\leq \kappa_r(f,x)\left(\nu_n(x)\right)^{\frac{r}{2}}.\end{aligned}$$
By equation (\[eq8\]), we can write $$\begin{aligned}
\left|\mathcal{U}_{n}^{[\alpha]}(f;x)-f(x)\right| &\leq \kappa_r(f,x)\mathcal{U}_{n}^{[\alpha]}(|s-x|^r;x).\end{aligned}$$ Using, H$\ddot{\text{o}}$lder’s inequality with $j=\frac{2}{r}$, $l=\frac{2}{2-r}$, one can get $$\begin{aligned}
\left|\mathcal{U}_{n}^{[\alpha]}(f;x)-f(x)\right| &\leq & \kappa_r(f,x)\left(\mathcal{U}_{n}^{[\alpha]}((s-x)^2;x)\right)^{\frac{r}{2}}=\kappa_r(f,x)\left(\nu_n(x)\right)^{\frac{r}{2}}.\end{aligned}$$ Hence proved.
Next theorem is based on modified Lipschitz type spaces [@OMA] and this spaces is defined by $$\begin{aligned}
Lip_M^{\lambda_1,\lambda_2}(s)=\Bigg\{ f\in C_B[0,\infty):|f(j)-f(k)|\leq M\frac{|j-k|^s}{\left(j+k^2\lambda_1+k\lambda_2\right)^{\frac{s}{2}}},~~\text{where}~j,k\geq0 ~\text{are~variables},~s\in(0,1] \Bigg\}\end{aligned}$$ and $\lambda_1, \lambda_1$ are the fixed numbers.
For $f\in Lip_M^{\lambda_1,\lambda_2}(s)$ and $0<s\leq 1$, we have an inequality holds: $$\begin{aligned}
\left|\mathcal{U}_{n}^{[\alpha]}(f;x)-f(x)\right| &\leq & M\left(\frac{\Theta_{n,2}^{[\alpha]}(x)}{x(x\lambda_1+\lambda_2)}\right)^{\frac{s}{2}}.\end{aligned}$$
To prove the above theorem, we can distribute its proof into two part by considering the case discussion. So here:\
**Case 1.** if $s=1$, proceed ahead, we can observe that $\frac{1}{(y+x^2\lambda_1+x\lambda_2)}\leq \frac{1}{x(x\lambda_1+\lambda_2)}$ then one has $$\begin{aligned}
\left|\mathcal{U}_{n}^{[\alpha]}(f;x)-f(x)\right| &\leq & \mathcal{U}_{n}^{[\alpha]}(|f(t)-f(x)|;x)\\
&\leq & M \mathcal{U}_{n}^{[\alpha]}\left(\frac{|t-x|}{\left(t+x^2\lambda_1+x\lambda_2\right)^{\frac{1}{2}}};x\right)\\
&\leq & \frac{M}{\left(x(x\lambda_1+\lambda_2)\right)^{\frac{1}{2}}}\mathcal{U}_{n}^{[\alpha]}(|t-x|;x) \\
&\leq & \frac{M}{\left(x(x\lambda_1+\lambda_2)\right)^{\frac{1}{2}}}\left(\Theta_{n,2}^{[\alpha]}(x)\right)^{\frac{1}{2}}\\
&\leq & M\left(\frac{\Theta_{n,2}^{[\alpha]}(x)}{x(x\lambda_1+\lambda_2)}\right)^{\frac{1}{2}}.\end{aligned}$$ **Case 2.** if $s\in (0,1)$ then with the help of H$\ddot{\text{o}}$lder inequality by considering $l=\frac{2}{s}, m=\frac{2}{2-s}$, we get $$\begin{aligned}
\left|\mathcal{U}_{n}^{[\alpha]}(f;x)-f(x)\right| &\leq & \left(\mathcal{U}_{n}^{[\alpha]}(|f(t)-f(x)|^{\frac{2}{s}};x)\right)^{\frac{s}{2}}\leq M\mathcal{U}_{n}^{[\alpha]}\left(\frac{|t-x|^{2}}{\left(t+x^2\lambda_1+x\lambda_2\right)};x\right)^{\frac{s}{2}}\\
&\leq & M\mathcal{U}_{n}^{[\alpha]}\left(\frac{|t-x|^{2}}{\left(x(x\lambda_1+\lambda_2)\right)};x\right)^{\frac{s}{2}}\\
&\leq & M\left(\frac{\Theta_{n,2}^{[\alpha]}(x)}{x(x\lambda_1+\lambda_2)}\right)^{\frac{s}{2}}.\end{aligned}$$ Thus, the proof is completed.
Let $g\in C_B[0,\infty)$ and we define Steklov mean function, which is as follows: $$\begin{aligned}
G_h(x)= \frac{1}{h^2}\int\limits_{-\frac{h}{2}}^{\frac{h}{2}}\int\limits_{-\frac{h}{2}}^{\frac{h}{2}} 2(g(x+\kappa+\lambda))-g(x+2(\kappa+\lambda))~d\kappa~d\lambda,~~~\kappa, \lambda\geq0~ \text{and}~h>0.\end{aligned}$$ To approximate continuous functions by smoother functions, Steklov function is used and we appeal to investigate the approximation properties. So, we collect some properties in a next lemma, which are used to prove the main theorem.
Let $g\in C_B[0\infty)$, it is holds following inequalities:
1. $\|G_h-g\|_{C_B[0\infty)} \leq \omega_2(g,h)$
2. $\|G_h' \|_{C_B[0\infty)} \leq \frac{5\omega(g,h)}{h}$ for $G_h'\in C_B[0,\infty)$
3. $\|G_h'' \|_{C_B[0\infty)} \leq \frac{9\omega_2(g,h)}{h^2}$ for $G_h''\in C_B[0,\infty),$
where $\omega(g;h)$ and $\omega_2(g,h)$ are modulus of continuity and second order of modulus of continuity respectively and can be defined as another way, given by $$\begin{aligned}
\omega(f,h)&=&\underset{x,\kappa,\lambda\geq 0}\sup\underset{|\kappa-\lambda|\leq h}\sup|f(x+\kappa)-f(x+\lambda)|,\\
\omega_2(f,h)&=&\underset{x,\kappa,\lambda\geq 0}\sup\underset{|\kappa-\lambda|\leq h}\sup|f(x+2\kappa)-2f(x+\kappa+\lambda)|+f(x+2\lambda).\end{aligned}$$
Consider $f\in C_B[0,\infty)$, for every $x\in [0,\infty)$, we get $$\begin{aligned}
|\mathcal{U}_{n}^{[\alpha]}(f;x)-f(x)|\leq 5\left\{\omega\left(f,\sqrt{\Theta_{n,2}^{[\alpha]}}\right)+\frac{13}{10} \omega_2\left(f,\sqrt{\Theta_{n,2}^{[\alpha]}}\right)\right\},\end{aligned}$$ where $\Theta_{n,2}^{[\alpha]}$ is defined by Lemma \[l4\].
For every $x\geq 0$, using Steklov function, we can write as $$\begin{aligned}
|\mathcal{U}_{n}^{[\alpha]}(f;x)-f(x)|\leq \mathcal{U}_{n}^{[\alpha]}(|f-G_h|;x)+|\mathcal{U}_{n}^{[\alpha]}(G_h-G_h(x);x)|+|G_h(x)-f(x)|.\end{aligned}$$ Since $|\mathcal{U}_{n}^{[\alpha]}(f;x)|\leq \|f(x)\|_{C_B[0,\infty)}$ as $f\in C_B[0,\infty)$ and $x\geq 0$. Then using Steklov mean property, we can have $$\begin{aligned}
\mathcal{U}_{n}^{[\alpha]}(|f-G_h|;x)\leq \|\mathcal{U}_{n}^{[\alpha]}(f-G_h;x)\|_{C_B[0,\infty)}\leq \|f-G_h\|_{C_B[0,\infty)}\leq \omega_2(f,h).\end{aligned}$$ Using the Taylor’s formula and on applying the operators (\[O1\]), we can write $$\begin{aligned}
|\mathcal{U}_{n}^{[\alpha]}(G_h-G_h(x);x)|\leq \|G_h'\|_{C_B[0,\infty)}\sqrt{\Theta_{n,2}^{[\alpha]}}+\frac{\|G_h''\|_{C_B[0,\infty)}}{2!} \Theta_{n,2}^{[\alpha]}.\end{aligned}$$ Using the propery of the Steklov mean, we can write as $$\begin{aligned}
|\mathcal{U}_{n}^{[\alpha]}(G_h-G_h(x);x)|\leq \frac{5\omega(f,h)}{h}\sqrt{\Theta_{n,2}^{[\alpha]}}+\frac{9\omega_2(f,h)}{2h^2}\Theta_{n,2}^{[\alpha]}.\end{aligned}$$ Choosing $h=\sqrt{\Theta_{n,2}^{[\alpha]}}$, we obtain our required result.
If $\Theta_{n,2}^{[\alpha]}\to 0$ as $n\to\infty$ and then $\omega\left(f,\sqrt{\Theta_{n,2}^{[\alpha]}}\right)\to 0,~\omega_2\left(f,\sqrt{\Theta_{n,2}^{[\alpha]}}\right)\to 0$, this imply that $\mathcal{U}_{n}^{[\alpha]}(f;x)$ converge to the function $f(x)$.
Weighted Approximation {#sec3}
======================
For describing the approximation properties, of any sequence of linear positive operators, Gadzhiev [@q5; @q6] introduced the weighted spaces. Recall from there, we consider the functions classes, which are as:
$B_w[0,\infty)=\{f:[0,\infty)\to\mathbb{R} |~~ |f(x)|\leq Mw(x)~~\text{with~the~supremum~norm}~~ \|f\|_w=\underset{x\in [0,\infty)}\sup\frac{f(x)}{w(x)}<+\infty \}$, where $M>0$ is a constant depending on $f$. Also define the spaces $$C_w[0,\infty)=\{f\in B_w[0,\infty), ~f~\text{is~contiuous} \},$$ $$C_w^k[0,\infty)=\{f\in C_w[0,\infty),\underset{x\to\infty}\lim\frac{|f(x)|}{w(x)}=k_f<+\infty\},$$ where $w(x)=1+x^2$ is a weight function.
[@G1; @G2] Let $\mathcal{L}_n:C_w[0,\infty)\to B_w[0,\infty)$ with the the conditions $\underset{n\to\infty}\lim\|L_n(t^r;x)-x^r\|_w=0,~~r=0,1,2$, then for $f\in C_w^k[0,\infty)$, we have $$\underset{n\to\infty}\lim\|L_n(f;x)-f(x)\|_w=0.$$
Let $\{\mathcal{U}_{n}^{[\alpha]}\}$ be a sequence defined by (\[O1\]) then it holds as: $$\begin{aligned}
\underset{n\to\infty}\lim \|\mathcal{U}_{n}^{[\alpha]}(f;x)-f(x)\|_w=0,~~\text{for}~f\in C_w^k[0,\infty).\end{aligned}$$
If we show that $\underset{n\to\infty}\lim \|\mathcal{U}_{n}^{[\alpha]}(t^r;x)-x^r\|_w=0$ holds for $r=0,1,2$ then the above theorem will be proved. Here, it’s obvious
$$\begin{aligned}
\underset{n\to\infty}\lim \|\mathcal{U}_{n}^{[\alpha]}(1;x)-1\|_w=0.\end{aligned}$$
Using the Lemma \[l2\], we have $$\begin{aligned}
\|\mathcal{U}_{n}^{[\alpha]}(t;x)-x\|_w &=&\frac{1}{n}\underset{x\geq 0}\sup \frac{1}{1+x^2}\leq \frac{1}{n}\\
\Rightarrow \|\mathcal{U}_{n}^{[\alpha]}(t;x)-x\|_w\to 0,~~~~\text{as}~n\to\infty.
\end{aligned}$$
Also, $$\begin{aligned}
\|\mathcal{U}_{n}^{[\alpha]}(t^2;x)-x^2\|_w &=& \underset{x\geq 0}\sup \frac{\left|\frac{2+4nx+n^2x^2+n^2x\alpha}{n^2}-x^2\right|}{1+x^2}\\
&\leq & \frac{1}{n}\left(\frac{2}{n}\underset{x\geq 0}\sup\frac{1}{1+x^2}+5\underset{x\geq 0}\sup\frac{x}{1+x^2}\right)\\
&\leq & \frac{2}{n^2}+\frac{5}{2n}\\
\Rightarrow \|\mathcal{U}_{n}^{[\alpha]}(t^2;x)-x^2\|_w\to 0~~~~\text{as}~n\to\infty.
\end{aligned}$$
And hence the proof is completed.
For every $x\geq 0$ and $\alpha=\alpha(n)\to 0$ as $n\to\infty$, let $g\in C_w^k[0,\infty)$ and $l>0$ then we yield $$\begin{aligned}
\underset{n\to\infty}\lim \underset{x\geq0}\sup \frac{\left|\mathcal{U}_{n}^{[\alpha]}(g;x)-g(x)\right|}{(1+x^2)^{1+l}}=0.\end{aligned}$$
Consider $x_0$ be a fixed point, then we can right as $$\begin{aligned}
\label{eq1}
\nonumber\underset{x\geq0}\sup \frac{\left|\mathcal{U}_{n}^{[\alpha]}(g;x)-g(x)\right|}{(1+x^2)^{1+l}}&\leq & \underset{x\leq x_0}\sup \frac{\left|\mathcal{U}_{n}^{[\alpha]}(g;x)-g(x)\right|}{(1+x^2)^{1+l}}+\underset{x>x_0}\sup \frac{\left|\mathcal{U}_{n}^{[\alpha]}(g;x)-f(x)\right|}{(1+x^2)^{1+l}}\\
\nonumber &\leq & \|\mathcal{U}_{n}^{[\alpha]}(g;x)-g(x)\|+\|f\|_{w} \underset{x>x_0}\sup \frac{\left|\mathcal{U}_{n}^{[\alpha]}((1+t^2);x)\right|}{(1+x^2)^{1+l}} +\underset{x>x_0}\sup\frac{|g|}{(1+x^2)^{1+l}}\\
&=& L_1+L_2+L_3~(say). \end{aligned}$$ Here, $$\begin{aligned}
L_3=\underset{x>x_0}\sup\frac{|g|}{(1+x^2)^{1+l}}\leq \frac{\|g\|_{w}}{(1+x_0^2)^l},~~\text{(as~$|g(x)|\leq M(1+x^2))$}\end{aligned}$$ so, for large value of $x_0$, we can consider an arbitrary $\epsilon>0$ such that $$\begin{aligned}
\label{eq2}
L_3=\frac{\|g\|_{w}}{(1+x_0^2)^l}\leq \frac{\epsilon}{3}.\end{aligned}$$ Since, $\underset{n\to\infty}\lim \underset{x>x_0}\sup \frac{\left|\mathcal{U}_{n}^{[\alpha]}((1+t^2);x)\right|}{(1+x^2)}=1$, so let us consider for any arbitrary $\epsilon>0$, there exist $n_1\in\mathbb{N}$, such that $$\begin{aligned}
\label{eq3}
L_2=\| g\|_w\underset{x>x_0}\sup \frac{\left|\mathcal{U}_{n}^{[\alpha]}((1+t^2);x)\right|}{(1+x^2)^{1+l}}\leq \frac{\| g\|_w}{(1+x^2)^l}\leq \frac{\| g\|_w}{(1+x_0^2)^l}<\frac{\epsilon}{3}.\end{aligned}$$ Applying the theorem \[th1\], we can have $$\begin{aligned}
\label{eq4}
L_1=\|\mathcal{U}_{n}^{[\alpha]}(g;x)-g(x)\|_{C[0,x_0]}\|\leq \frac{\epsilon}{3}.\end{aligned}$$ Combining (\[eq2\]-\[eq4\]) and using in (\[eq1\]), we obtain our required result.
For $g\in C_w[0,\infty)$, one can obtain $$\begin{aligned}
|\mathcal{U}_{n}^{[\alpha]}(g;x)-g(x)|\leq 4\mathcal{N}_f(1+x^2) \Theta_{n,2}^{[\alpha]}(x)+2\omega_{l+1}\left(g;\sqrt{\Theta_{n,2}^{[\alpha]}}\right).\end{aligned}$$
From [@IEG], for $0\leq x\leq l$ and $u\geq 0$, it holds $$\begin{aligned}
|g(u)-g(x)|\leq 4\mathcal{N}_f(1+x^2)(u-x)^2+\left(1+\frac{|u-x|}{\theta}\right)\omega_{l+1}(g;\theta),~~\theta>0. \end{aligned}$$ Now, applying the operators defined by (\[O1\]) and with the help of Cauchy-Schwarz inequality, we can obtain $$\begin{aligned}
|\mathcal{U}_{n}^{[\alpha]}(g;x)-g(x)|&\leq & 4\mathcal{N}_f(1+x^2) \Theta_{n,2}^{[\alpha]}+\left(1+\frac{\mathcal{U}_{n}^{[\alpha]}(|u-x|)}{\theta}\right)\omega_{l+1}(g;\theta)\\
&\leq & 4\mathcal{N}_f(1+x^2) \Theta_{n,2}^{[\alpha]}(x)+\left(1+\frac{\sqrt{\Theta_{n,2}^{[\alpha]}} }{\theta}\right)\omega_{l+1}(g;\theta)\\
&\leq & 4\mathcal{N}_f(1+x^2) \Theta_{n,2}^{[\alpha]}(x)+\left(1 +1\right)\omega_{l+1}\left( g;\sqrt{\Theta_{n,2}^{[\alpha]}}\right)\\
&=& 4\mathcal{N}_f(1+x^2) \Theta_{n,2}^{[\alpha]}(x)+2\omega_{l+1}(g;\sqrt{\Theta_{n,2}^{[\alpha]}}).\end{aligned}$$
Quantitative Approximation {#sec4}
==========================
For estimations of the degree of approximation in the weighted space $C_w^k[0,\infty)$, Ispir [@IN1] proposed the weighted modulus of continuity $\Delta(g;\xi)$ for any $\xi>0$, as follows: $$\begin{aligned}
\Delta(g;\xi)=\underset{0\leq h\leq\xi,~0\leq x\leq\infty}\sup \frac{|g(x+h)-g(x)|}{(1+h^2)(1+x^2)},~~~~~~~g\in C_w^k[0,\infty). \end{aligned}$$
For $g\in C_w^k[0,\infty)$ $$\begin{aligned}
\underset{\xi\to 0}\lim\Delta(g;\xi)=0.
\end{aligned}$$
On can obtains as, $\Delta(f;\lambda\xi)\leq2(1+\xi^2)(1+\lambda)\Delta(f;\xi),~~\lambda>0$. Using the weighted modulus of continuity and defined inequality, one can show that $$\begin{aligned}
\nonumber|g(t)-g(x)|&\leq &(1+x^2)(1+(t-x)^2)\Delta(g;|t-x|)\\
&\leq & 2\left(1+\frac{|t-x|}{\xi}\right)(1+\xi^2)(1+(t-x)^2)(1+x^2)\Delta(f;|t-x|),~\text{for every} f\in C_w^k[0,\infty).
\end{aligned}$$ As the consequence of the weighted modulus of continuity, we determine the degree of approximation of the operators $\mathcal{U}_{n}^{[\alpha]}(g;x)$ in the weighted space $C_w^k[0,\infty)$.
Quantitative Voronovskaya type theorem
--------------------------------------
Let $\mathrm{g}', \mathrm{g}''\in C_w^k[0,\infty)$ and for sufficiently large value of $n\in\mathbb{N}$, then for each $x\geq0$, we yield $$\begin{aligned}
n\left |\mathcal{U}_{n}^{[\alpha]}(\mathrm{g};x)-\mathrm{g}(x)-\mathrm{g}'(x)\Theta_{n,1}^{[\alpha]}-\frac{\mathrm{g}''(x)}{2!}\Theta_{n,2}^{[\alpha]}\right|=O(1)\Lambda\left(f;\sqrt{\frac{1}{n}}\right).\end{aligned}$$
By Taylor’s expansion, one can obtain $$\begin{aligned}
\mathrm{g}(t)-\mathrm{g}(x)=\mathrm{g}'(x)(t-x)+\frac{\mathrm{g}''(x)}{2}(t-x)^2+\zeta(t,x),\end{aligned}$$ where $\zeta(t,x)=\frac{\mathrm{g}''(\theta)-\mathrm{g}''(x)}{2!}(\theta-x)^2$ and $\zeta\in (t,x)$. Applying operators (\[O1\]) on both sides to above expansion, then one can obtains $$\begin{aligned}
\label{n1}
n\left|\mathcal{U}_{n}^{[\alpha]}(\mathrm{g};x)-\mathrm{g}(x)-\mathrm{g}'(x)\Theta_{n,1}^{[\alpha]}-\frac{\mathrm{g}''(x)}{2}\Theta_{n,2}^{[\alpha]}\right|\leq n\mathcal{U}_{n}^{[\alpha]}(|\eta(t,x)|;x).\end{aligned}$$ Now using the property of weighted modulus of continuity, we get $$\begin{aligned}
\frac{\mathrm{g}''(\theta)-\mathrm{g}''(x)}{2}
&\leq & \left(1+\frac{|t-x|}{\xi} \right)(1+\xi^2)(1+(t-x)^2)(1+x^2)\Delta(f'',\xi)\end{aligned}$$ and also $$\begin{aligned}
\left|\frac{\mathrm{g}''(\theta)-\mathrm{g}''(x)}{2}\right| &\leq &
\begin{cases}
2(1+\xi^2)^2(1+x^2)\Delta(\mathrm{g}'',\xi),& |t-x|<\xi,\\
2(1+\xi^2)^2(1+x^2)\frac{(t-x)^4}{\xi^4}\Delta(\mathrm{g}'',\xi),& |t-x|\geq\xi.
\end{cases} \end{aligned}$$ Now for $\xi\in(0,1)$, we get $$\begin{aligned}
\left|\frac{\mathrm{g}''(\theta)-\mathrm{g}''(x)}{2}\right| &\leq & 8(1+x^2)\left(1+\frac{(t-x)^4}{\xi^4}\right)\Delta(\mathrm{g}'',\xi). \end{aligned}$$ Hence, $$(|\zeta(t,x)|;x)\leq 8(1+x^2)\left((t-x)^2+\frac{(t-x)^6}{\xi^4}\right)\Delta(\mathrm{g}'',\xi).$$ Thus, applying the Lemma \[l1\] $$\begin{aligned}
\mathcal{U}_{n}^{[\alpha]}(| \zeta(t,x)|;x)&\leq & 8(1+x^2)\Delta(\mathrm{g}'',\xi)\left(\mathcal{U}_{n}^{[\alpha]}((t-x)^2;x)+\frac{\mathcal{U}_{n}^{[\alpha]}((t-x)^6;x)}{\xi^4}\right) \\
&\leq & 8(1+x^2)\Delta(\mathrm{g}'',\xi) \left(O\left(\frac{1}{n} \right)+\frac{1}{\xi^4} O\left(\frac{1}{n^3} \right) \right),~~\text{as}~n\to\infty.\end{aligned}$$ Choose, $\xi=\sqrt{\frac{1}{n}}$, then $$\begin{aligned}
\mathcal{U}_{n}^{[\alpha]}(| \zeta(t,x)|;x)\leq 8 O\left(\sqrt{\frac{1}{n}} \right)\Delta\left(\mathrm{g}'',\sqrt{\frac{1}{n}}\right)(1+x^2).\end{aligned}$$ Hence, we reach on $$\begin{aligned}
\label{n2}
n\mathcal{U}_{n}^{[\alpha]}(|\zeta(t,x)|;x)=O(1)\Delta\left(\mathrm{g}'', \sqrt{\frac{1}{n}}\right).\end{aligned}$$ By (\[n1\]) and (\[n2\]), we obtain the required result.
Gr$\ddot{\text{u}}$ss Voronovskaya type theorem
-----------------------------------------------
\[th2\] Let $\mathsf{f}\in C_w^k[0,\infty)$ then for $\mathsf{f}', \mathsf{f}'', \mathsf{g}', \mathsf{g}''\in C_w^k[0,\infty)$, it holds $$\begin{aligned}
\underset{n\to\infty}\lim n\left(\mathcal{U}_{n}^{[\alpha]}(\mathsf{f}\mathsf{g};x)-\mathcal{U}_{n}^{[\alpha]}(\mathsf{f};x)\mathcal{U}_{n}^{[\alpha]}(\mathsf{g};x) \right)= 2x \mathsf{f}'(x)\mathsf{g}'(x). \end{aligned}$$
By making suitable arrangement and using well known properties of derivative of multiplication of two functions, we get $$\begin{aligned}
n\left(\mathcal{U}_{n}^{[\alpha]}(\mathsf{f}\mathsf{g};x)-\mathcal{U}_{n}^{[\alpha]}(\mathsf{f};x)\mathcal{U}_{n}^{[\alpha]}(\mathsf{g};x) \right)&=& n\Bigg\{\Bigg(\mathcal{U}_{n}^{[\alpha]}(\mathsf{f}\mathsf{g};x)-\mathsf{f}(x)\mathsf{g}(x)-(\mathsf{f}\mathsf{g})'\Theta_{n,1}^{[\alpha]}\\
&&-\frac{(\mathsf{f}\mathsf{g})''}{2!}\Theta_{n,2}^{[\alpha]}\Bigg)-\mathsf{g}(x)\Bigg(\mathcal{U}_{n}^{[\alpha]}(\mathsf{f};x)-\mathsf{f}(x)\\
&&-\mathsf{f}'(x)\Theta_{n,1}^{[\alpha]}-\frac{\mathsf{f}''(x)}{2!}\Theta_{n,2}^{[\alpha]} \Bigg)\\
&&-\mathcal{U}_{n}^{[\alpha]}(\mathsf{f};x)\Bigg(\mathcal{U}_{n}^{[\alpha]}(\mathsf{g};x)-\mathsf{g}(x)-\mathsf{g}'(x)\Theta_{n,1}^{[\alpha]}\\
&&-\frac{\mathsf{g}''(x)}{2!}\Theta_{n,2}^{[\alpha]} \Bigg)+\frac{\mathsf{g}''(x)}{2!}\mathcal{U}_{n}^{[\alpha]}((t-x)^2;x)\\
&&\times \left(\mathsf{f}- \mathcal{U}_{n}^{[\alpha]}(\mathsf{f};x)\right)+\mathsf{f}'(x)\mathsf{g}'(x)\Theta_{n,2}^{[\alpha]}\\
&&+ \mathsf{g}'(x)\Theta_{n,1}^{[\alpha]}\left(\mathsf{f}- \mathcal{U}_{n}^{[\alpha]}(\mathsf{f};x)\right) \Bigg\}.\end{aligned}$$ For sufficiently large value of $n$, i.e. for $n\to\infty$, $\alpha\to 0$. So with the help of Theorem \[th1\] and \[th2\] and talking the limit on both side of the above equation, we obtain the required result.
Graphical and Numerical representation {#sec5}
======================================
This section consists, the graphical approcah and numerical analysis for the convergence of the operators to the function.
Consider the function is $g(x)=x^2\sin{\pi x}$ with $x\in[0,2]$ and choosing $\alpha=\frac{1}{60}$. Then the corresponding operators for $n=15,~35$ are $\mathcal{U}_{15}^{[\alpha]}(g;x)$(green), $\mathcal{U}_{35}^{[\alpha]}(g;x)$(blue) respectively. One can observe that the better rate of convergence can be obtained by graphical representation \[F1\], which is given below.
![The convergence of the operators $\mathcal{U}_{n}^{[\alpha]}(g;x)$ to the function $f(x)(red)$.[]{data-label="F1"}](r1.eps){width=".52\textwidth"}
Let us consider the function $f(x)=te^{-7x}$ (red) for which, the rate of convergence of the defined operators (\[O1\]) is discussed by taking different values of $n\in\mathbb{N}$. Choosing $n=5,10,20,25,35,40,45$, then the corresponding operators are represented by blue, green, cyan, brown, yellow, magenta, purple colors receptively in the given Figure \[F2\]. Here we take $\alpha=\frac{1}{45}$.
![The convergence of the operators $\mathcal{U}_{n}^{[\alpha]}(f;x)$ to the function $f(x)(red)$.[]{data-label="F2"}](r2.eps){width=".52\textwidth"}
For the same function, which has been taken in the above example, we observe by changing the different values of $\alpha$ i.e. choosing $\alpha=\frac{1}{10},\frac{1}{20},\frac{1}{40},\frac{1}{60},\frac{1}{80}$ for which the the corresponding operators for the fixed value of $n=10$ are represented by black, orange, pink, blue, green colors in given Figure \[F3\].
![The convergence of the operators $\mathcal{U}_{n}^{[\alpha]}(f;x)$ to the function $f(x)(red)$.[]{data-label="F3"}](r3.eps){width=".52\textwidth"}
**Concluding Remark:** 1. By the above Figures (\[F1\],\[F2\]), one can observe that as the value of $n$ is increased, the approximation is going to better while on taking the particular value of $\alpha$. i.e. by the suitable choice of $\alpha$, we can show the better approximation by taking large value of $n$. But in Figure \[F3\], convergence can be seen, when the value of $n$ is fixed and the value of $\alpha$ is decreased.\
2. On choosing appropriate values of $\alpha$ and $n$, we can find the better approximation.
Let the function $f=(x^2+1)e^x$, take $$n=5,10,20,25,30,40,50,70,90,130,150,190,240,250,400,500$$ and $\alpha=\frac{1}{5},\frac{1}{10},\frac{1}{20},\frac{1}{30},\frac{1}{50},\frac{1}{100},\frac{1}{150},\frac{1}{200},\frac{1}{250},\frac{1}{500}$ then we obtain the approximation by given table.
$n\downarrow$, $\alpha\to$ $\frac{1}{5}$ $\frac{1}{10}$ $\frac{1}{20}$ $\frac{1}{30}$ $\frac{1}{50}$ $\frac{1}{100}$ $\frac{1}{150}$ $\frac{1}{200}$ $\frac{1}{250}$ $\frac{1}{500}$
---------------------------- --------------- ---------------- ---------------- ---------------- ---------------- ----------------- ----------------- ----------------- ----------------- -----------------
5 2.01244 1.85335 1.7963 1.77966 1.7671 1.75808 1.75515 1.7537 1.75283 1.7511
10 - 1.30625 1.28073 1.2732 1.26749 1.26338 1.26204 1.26137 1.26098 1.26019
20 - - 1.13228 1.12711 1.12318 1.12034 1.11941 1.11896 1.11868 1.11814
25 - - - 1.1034 1.09975 1.09711 1.09625 1.09582 1.09557 1.09506
30 - - - 1.08846 1.08497 1.08246 1.08164 1.08124 1.08099 1.08051
40 - - - - 1.0674 1.06504 1.06427 1.06388 1.06366 1.0632
50 - - - - 1.05732 1.05504 1.05429 1.05392 1.0537 1.05326
70 - - - - - 1.044 1.04328 1.04293 1.04272 1.0423
90 - - - - - 1.03804 1.03734 1.037 1.03679 1.03638
130 - - - - - - 1.03108 1.03074 1.03054 1.03014
150 - - - - - - 1.02923 1.0289 1.0287 1.0283
190 - - - - - - - 1.02639 1.02619 1.02579
240 - - - - - - - - 1.02424 1.02385
250 - - - - - - - - 1.02395 1.02356
400 - - - - - - - - - 1.02093
500 - - - - - - - - - 1.02006
: Effect of $\alpha$ in the convergence of the operators $\mathcal{U}_{n}^{[\alpha]}(f;x)$[]{data-label="t1"}
**Observation:** By the above Table \[t1\], we can observe that as the value of $\alpha$ is decreased, the error is going to least for particular value of $n$. And the same time, if we see the table, we can observe that on increasing the values of $n$, the errors are decreased for particular value of $\alpha$ (excluding the ‘dash-’). These process is running on a particular point of $x$.
A-Statistical Convergence of the defined operators {#sec6}
==================================================
This section contains statistical convergence theorem. We establish some approximation property to study statistical convergence. The basic idea of statistical convergence was first introduced by Fast [@FH] even though the first publication related to statistical convergence was in 1935 (published in Warsaw) and credit goes to Zygmund in his monograph also independently work is seen in paper Steinhaus [@SH] in 1951. In paper [@SIJ], Schoenberg reintroduced the statistical convergence and now a days it has become an area of active research in approximation theory.\
In 2002, it has been seen the use of statistical convergence in Approximation theory by Gadjiev [@GAD1]. We recall the symbol from [@q6], as $A_{ni}=\{a_{ni}\}$ is infinite non-negative infinite summability matrix. Here we denote $A$-transform of matrix $A_{ni}$ for a given sequence $\{x_i\}$ by $$\begin{aligned}
A_{ni} x_i=\sum\limits_{i=0}^\infty a_{ni}x_i,\end{aligned}$$ provided $(A_{ni} x_i)$ converges for each $n\in\mathbb{N}$. Now if $\underset{n\to\infty}\lim(A_{ni} x_i)=\sigma$ whenever $\lim x_i=\sigma$ [@GH] then $A_{ni}$ is said to be regular. And also, $\underset{n\to\infty}\lim a_{ni}=0,~\forall~i\in\mathbb{N}$. In this case, $\{x_i\}$ is said to be $A$-statistically convergent, i.e. for every $\epsilon>0$, $\underset{n\to\infty}\lim \sum_{\{i:||x_i-\sigma|\geq\epsilon\}}a_{ni}=0$ and it is written as $st_A-\lim x_i=0$. For more information, we refer to reader to see [@CJ; @FAS; @FJA; @KE; @MHI].
Consider $A_{ni}=\{a_{ni}\}$ be non-negative regular summability matrix and for each $f\in C_w^k[0,\infty)$ then for every $x\in[0,\infty)$, we have $$\begin{aligned}
st_A-\underset{n\to\infty}\lim \|\mathcal{U}_{n}^{[\alpha]}(f;x)-f(x)\|_{w(x)}=0.\end{aligned}$$
If we show $$\begin{aligned}
\label{eq6}
st_A-\underset{n\to\infty}\lim \|\mathcal{U}_{n}^{[\alpha]}(t^r;x)-x^r\|_{w(x)}=0~~~\text{for}~~r=0,1,2\end{aligned}$$ then we obtain the required result i.e. the proof will be done.
It is clear that $$\begin{aligned}
st_A-\underset{n\to\infty}\lim \|\mathcal{U}_{n}^{[\alpha]}(1;x)-1\|_{w(x)}=0.\end{aligned}$$ Also $$\begin{aligned}
\|\mathcal{U}_{n}^{[\alpha]}(t;x)-x\|_{w(x)}&=&\underset{x\geq0}\sup\frac{1}{1+x^2} \frac{1}{n} \\
&\leq & \frac{1}{n}.\end{aligned}$$ So we define the following sets for given $\epsilon>0$, as $$\begin{aligned}
\mathcal{V}_1&=&\{n:\|\mathcal{U}_{n}^{[\alpha]}(t;x)-x\|\geq \epsilon\}\\
\mathcal{V}_2&=&\left\{n:\frac{1}{n}\geq \epsilon\right\}.\end{aligned}$$ Obviously, $\mathcal{V}_1\subset\mathcal{V}_2$ and hence, $\sum_{i\in \mathcal{V}_1}a_{ni}\leq \sum_{i\in \mathcal{V}_2}a_{ni}$. Therefore $$\begin{aligned}
st_A-\underset{n\to\infty}\lim \|\mathcal{U}_{n}^{[\alpha]}(t;x)-x\|_{w(x)}=0.\end{aligned}$$ Further, $$\begin{aligned}
\|\mathcal{U}_{n}^{[\alpha]}(t^2;x)-x^2\|_{w(x)}&=&\underset{x\geq0}\sup\frac{1}{1+x^2} \left(\frac{2+4nx+n^2x^2+n^2x\alpha}{n^2}-x^2 \right)\\
&\leq & \left(\frac{2}{n^2}+\frac{5}{2n} \right).\end{aligned}$$ Since R.H.S. of above inequality tends to zero as $n\to\infty$. So for given $\epsilon>0$, we can consider following sets, which show as $$\begin{aligned}
\mathcal{W}_1&=&\{n:\|\mathcal{U}_{n}^{[\alpha]}(t^2;x)-x^2\|\geq \epsilon\}\\
\mathcal{W}_2&=& \left\{n:\frac{2}{n^2}\geq \frac{\epsilon}{2}\right\}\\
\mathcal{W}_3&=&\left\{n:\frac{5}{2n}\geq \frac{\epsilon}{2}\right\},\end{aligned}$$ which implies that $\mathcal{W}_1\subset\mathcal{W}_2\cup\mathcal{W}_3 $, and hence $\sum_{i\in \mathcal{W}_1}a_{ni}\leq \sum_{i\in \mathcal{W}_2}a_{ni}+\sum_{i\in \mathcal{W}_3}a_{ni}$. Therefore $$\begin{aligned}
st_A-\underset{n\to\infty}\lim \|\mathcal{U}_{n}^{[\alpha]}(t^2;x)-x^2\|_{w(x)}=0.\end{aligned}$$ Thus the the proof is completed.
Let $w_\zeta(x)\geq 1$ be a continuous function such that $\underset{|x|\to\infty}\lim\frac{w(x)}{w_\zeta(x)}\to 0$ then for each $f\in C_w^k[0,\infty)$ with $\{a_{ni}\}$ be non-negative regular summability matrix, we have $$\begin{aligned}
st_A-\underset{n\to\infty}\lim \|\mathcal{U}_{n}^{[\alpha]}(t^2;x)-x^2\|_{w_\zeta(x)}=0.\end{aligned}$$
By means of Peetre’s $K$-functional, $A$-Statistical convergence is introduced for the operators $\mathcal{U}_{n}^{[\alpha]}$ in next theorem.
Let $g\in C_B^2[0,\infty)$ and for every $x\in[0,\infty)$, we have $$\begin{aligned}
st_A-\underset{n\to\infty}\lim \|\mathcal{U}_{n}^{[\alpha]}(g;x)-g\|_{C_B[0,\infty)}=0,~~~\forall~n\in\mathbb{N}.\end{aligned}$$
Using expansion of Taylor, we have $$\begin{aligned}
g(t)=g(x)+g'(x)(t-x)+\frac{1}{2}g''(\tau)(t-x)^2,\end{aligned}$$ where $\tau\in [t,x]$. Now applying $\mathcal{U}_{n}^{[\alpha]}$, we obtain $$\begin{aligned}
\mathcal{U}_{n}^{[\alpha]}(g(t)-g(x);x)=f'(x)\Theta_{n,1}^{[\alpha]}(x)+\frac{g''(\tau)}{2}\Theta_{n,2}^{[\alpha]}(x).\end{aligned}$$ In this way, $$\begin{aligned}
\label{eq7}
\|\mathcal{U}_{n}^{[\alpha]}(g(t)-g(x);x)\|_{C_B[0,\infty)}&\leq &\|g'(x)\|_{C_B[0,\infty)}\|\Theta_{n,1}^{[\alpha]}(x)\|_{C_B[0,\infty)}+\|g''(\tau)\|_{C_B[0,\infty)}\|\Theta_{n,2}^{[\alpha]}(x)\|_{C_B[0,\infty)}\nonumber\\
&=&\mathcal{E}_1+\mathcal{E}_2, ~(\text{say})\end{aligned}$$ From (\[eq6\]), we can write $$\begin{aligned}
\underset{n\to\infty}\lim\sum_{\{i\in\mathbb{N} \mathcal{E}_1\geq\frac{\epsilon}{2}\}}a_{ni}&=&0,\\
\underset{n\to\infty}\lim\sum_{\{i\in\mathbb{N}: \mathcal{E}_2\geq\frac{\epsilon}{2}\}}a_{ni}&=&0.\end{aligned}$$ So by equation (\[eq7\]), we get $$\begin{aligned}
\underset{n\to\infty}\lim\sum_{\{i\in\mathbb{N}: \|\mathcal{U}_{n}^{[\alpha]}(f(t)-f(x);x)\|_{C_B[0,\infty)}\geq\epsilon\}}a_{ni}\leq \underset{n\to\infty}\lim\sum_{\{i\in\mathbb{N}: \mathcal{E}_1\geq\frac{\epsilon}{2}\}}a_{ni}+\underset{n\to\infty}\lim\sum_{\{i\in\mathbb{N}: \mathcal{E}_2\geq\frac{\epsilon}{2}\}}a_{ni}.\end{aligned}$$ And hence, $$\begin{aligned}
st_A-\underset{n\to\infty}\lim \|\mathcal{U}_{n}^{[\alpha]}(g;x)-g\|=0.\end{aligned}$$
Hence proved.
Consider $f\in C_B[0,\infty)$ and for each $n\in\mathbb{N}$, an inequality holds $$\begin{aligned}
\|\mathcal{U}_{n}^{[\alpha]}(f(t)-f(x);x)\|_{C_B[0,\infty)}\leq \mathcal{C}\omega_2(f;\sqrt{\xi}).\end{aligned}$$
Consider a function $h\in C_B^2[0,\infty)$. We can write $$\begin{aligned}
\|\mathcal{U}_{n}^{[\alpha]}(h(t)-h(x);x)\|_{C_B[0,\infty)}&\leq & \|h'\|_{C_B[0,\infty)} \|\mathcal{U}_{n}^{[\alpha]}((t-x);x)\|_{C_B[0,\infty)}+\frac{1}{2}\|h''\|_{C_B[0,\infty)}\|\Theta_{n,2}^{[\alpha]}\|_{C_B[0,\infty)}\\
&\leq & \eta \|h\|_{C_B^2[0,\infty)}. \end{aligned}$$ Since, $f\in C_B[0,\infty)$ and $h\in C_B^2[0,\infty)$, therefore, by above inequality, one can write $$\begin{aligned}
\|\mathcal{U}_{n}^{[\alpha]}(f(t)-f(x);x)\|_{C_B[0,\infty)}&\leq & \|\mathcal{U}_{n}^{[\alpha]}(f;x)-\mathcal{U}_{n}^{[\alpha]}(h;x)\|_{C_B[0,\infty)} + \|\mathcal{U}_{n}^{[\alpha]}(h(t)-h(x);x)\|_{C_B[0,\infty)}\\
&&\|h-f \|_{C_B[0,\infty)}\\
&\leq & 2\|h-f \|_{C_B[0,\infty)}+\|\mathcal{U}_{n}^{[\alpha]}(h(t)-h(x);x)\|_{C_B[0,\infty)}\\
&\leq & 2\|h-f \|_{C_B[0,\infty)}+\eta \|h\|_{C_B^2[0,\infty)}.\end{aligned}$$ Using the property of Peetre’s $K$-functional (\[pe1\]), one can has $$\begin{aligned}
\|\mathcal{U}_{n}^{[\alpha]}(f(t)-f(x);x)\|_{C_B[0,\infty)}&\leq & \mathcal{C}\{\omega_2(f;\sqrt{\eta})+\min(1,\eta)\|f\|_{C_B[0,\infty)} \}.\end{aligned}$$ Using (\[eq6\]), we obtain $\eta\to 0$ statistically as $n\to\infty$.
In this way, $\omega_2(f;\sqrt{\xi})\to 0$ statistically as $n\to\infty$.
Hence, the rate of convergence via $A$-statistically is obtained of the sequence of linear positive operators defined by \[O1\] to the function $f(x)$.
If $A=I$ then the ordinary rate of convergence is obtained.
Rate of convergence by means of function of bounded variation {#sec7}
=============================================================
In this section, the rate of convergence of the said operators is determined in the space of the functions with derivative of bounded variation. Here, we consider $DBV[0,\infty)$, the set of all continuous function having derivative of bounded variation on every finite sub-interval of the $[0,\infty)$. On observing that for each $g\in DBV[0,\infty)$ and $a>0$, one can write $$\begin{aligned}
g(x)=\int\limits_a^x h(s)~ds+g(a),\end{aligned}$$ where $h$ is a function bounded variation on each finite sub-interval of $[0,\infty)$. Here, we use an auxiliary operators $g_x$ for every $g\in DBV[0,\infty)$ for obtaining the rate of of convergence of the proposed operators.
$$\begin{aligned}
\label{eq5}
g_x(t) &= &
\begin{cases}
g(t)-g(x-),& 0\leq t<x,\\
0,& t=x,\\
g(t)-g(x+), & x<t<\infty.
\end{cases} \end{aligned}$$
Generally, we denote $V_a^b \mathfrak{f}$ is total variation of a real valued function $g$ defined on $[a,b]\subset[0,\infty)$ with the quantity $$\begin{aligned}
V_a^b \mathfrak{f}=\underset{\mathcal{P}}\sup\left(\sum\limits_{k=0}^{n_{P}-1}|g(x_{k+1})-f(x_k)| \right),\end{aligned}$$ where $\mathcal{P}$ is the set of all partition $P=\{a=x_0,\cdots,x_{n_P}=b\}$ of the interval $[a,b]$.
\[l3\] For sufficiently large value of $n$, for every $x\geq 0$ then there exist a positive constant $M>0$ such that
1. $h_n^{[\alpha]}(x,y)= \int\limits_0^y u_{n}^{[\alpha]}(x,t)~dt\leq \frac{3\eta_n^2(x)}{n(x-y)^2},~~0\leq y<x,$
2. $1-h_n^{[\alpha]}(x,z)= \int\limits_z^\infty u_{n}^{[\alpha]}(x,t)~dt\leq \frac{3\eta_n^2(x)}{n(z-x)^2},~~x<z<\infty.$
For $y\in[0,x)$, it is holds $$\begin{aligned}
\int\limits_0^y u_{n}^{[\alpha]}(x,t)~dt
&\leq &\frac{1}{(x-y)^2}\mathcal{U}_{n}^{[\alpha]}((t-x)^2;x)\\
&\leq & \frac{3\eta_n^2(x)}{n(x-y)^2}.\end{aligned}$$ Similarly, other result can be proved.
Let $g\in DBV[0,\infty)$, for sufficiently large value of $n$ and $\max \alpha=\frac{1}{n}$. If $g(t)=O(t^s)$ as $t\to\infty$, then for $x\in[0,\infty)$, we obtain $$\begin{aligned}
|\mathcal{U}_{n}^{[\alpha]}(g;x)-g(x)|&\leq & \frac{3\eta_n^2(x)}{nx^2} |(g(2x)-g(x)-xg'(x+))| +\frac{x}{\sqrt{n}} V_x^t(g_x') + \frac{3\eta_n^2(x)}{n}\sum_{l=1}^{[\sqrt{nx}]} V_x^{x+\frac{x}{l}}(g_x') + \mathcal{M}_{s,x}^\gamma\\
&&+\frac{3|g(x)|\eta_n^2(x)}{nx^2} + |g'(x+)| \sqrt{\frac{3}{n}}\eta_n(x)+\frac{1}{2}\sqrt{\frac{3}{n}}\{g'(x+)-g'(x-)\}\eta_n(x)+ \frac{1}{2n}\{g'(x+)-g'(x-)\},\end{aligned}$$ where $$\begin{aligned}
\mathcal{M}_{s,x}^\gamma=M 2^{\gamma} \left(\int_{0}^\infty (t-x)^{2s} u_{n}^{[\alpha]}(x,t)dt\right)^{\frac{\gamma}{2s}}.\end{aligned}$$
We have $$\begin{aligned}
|\mathcal{U}_{n}^{[\alpha]}(g;x)-g(x)|=\int_0^\infty u_{n}^{[\alpha]} (x,t)(g(t)-g(x)~dt =\int_0^\infty u_{n}^{[\alpha]} (x,t) \left(\int_0^tg'(u)~du\right).\end{aligned}$$ Since $g\in DBV[0,\infty)$, one can write an identity $$\begin{aligned}
\nonumber g'(v)&=&\frac{1}{2}\{g'(x+)+g'(x-)\}+g_x'(v)+\frac{1}{2}\{g'(x+)-g'(x-)\}\text{sgn}(v-x)\nonumber\\
&&+\xi_x(v)\left(g'(v)- \frac{1}{2}(g'(x+)+g'(x-))\right), \end{aligned}$$ where $$\begin{aligned}
\xi_x(v)=
\begin{cases}
1 & v=x\\
0 & v\neq x.
\end{cases}\end{aligned}$$ Now, one can easily obtain as $$\begin{aligned}
\int_0^\infty u_{n}^{[\alpha]}(x,t) \left(\int_0^t \left( \xi_x(v)\left(g'(v)- \frac{1}{2}(g'(x+)+g'(x-))\right)dv\right)\right)dt=0\end{aligned}$$ Also, $$\begin{aligned}
\end{aligned}$$ $$\begin{aligned}
\int_0^\infty u_{n}^{[\alpha]}(x,t) \left(\int_0^t\frac{1}{2}\{g'(x+)-g'(x-)\}\text{sgn}(v-x)dv\right)dt &\leq &\frac{1}{2}\{g'(x+)-g'(x-)\}\mathcal{U}_{n}^{[\alpha]}(|t-x|;x)\nonumber\\
&\leq & \frac{1}{2}\{g'(x+)-g'(x-)\}\left(\mathcal{U}_{n}^{[\alpha]}((t-x)^2;x)\right)^{\frac{1}{2}},\end{aligned}$$ and $$\begin{aligned}
\int_0^\infty u_{n}^{[\alpha]}(x,t) \left(\int_0^t\frac{1}{2}\{g'(x+)-g'(x-)\}dv\right)dt= \frac{1}{2}\{g'(x+)-g'(x-)\}\mathcal{U}_{n}^{[\alpha]}((t-x);x).\end{aligned}$$ So, we have $$\begin{aligned}
\label{in3}
|\mathcal{U}_{n}^{[\alpha]}(g;x)-g(x)| &\leq & \left| \int_0^x\left(\int_0^t g_x'(v) dv\right)u_{n}^{[\alpha]}(x,t) dt+\int_x^\infty \left(\int_0^t g_x'(v) dv\right)u_{n}^{[\alpha]}(x,t) dt \right|\nonumber\\
&& +\frac{1}{2}\{g'(x+)-g'(x-)\}\left(\mathcal{U}_{n}^{[\alpha]}((t-x)^2;x)\right)^{\frac{1}{2}}\nonumber\\
&& + \frac{1}{2}\{g'(x+)-g'(x-)\}\mathcal{U}_{n}^{[\alpha]}((t-x);x)\nonumber\\
&\leq & E_{nx} + F_{nx} +\frac{1}{2}\sqrt{\frac{3}{n}}\{g'(x+)-g'(x-)\}\eta_n(x)+ \frac{1}{2n}\{g'(x+)-g'(x-)\},\end{aligned}$$ where $$\begin{aligned}
E_{nx}=\left| \int_0^x\left(\int_0^t g_x'(v) dv\right)u_{n}^{[\alpha]}(x,t) dt \right|,\end{aligned}$$ and $$\begin{aligned}
F_{nx}=\left| \int_x^\infty \left(\int_0^t g_x'(v) dv\right)u_{n}^{[\alpha]}(x,t) dt \right|.\end{aligned}$$ Now applying Lemma \[l3\], and let $x=\frac{y\sqrt{n}}{\sqrt{n}-1}$, then integrating by part of $$\begin{aligned}
E_{nx}&=&\left| \int_0^x \left(\int_0^t g_x'(v) dv\right) d_t(h_n^{[\alpha]}(x,t)) \right|\\
&=& \left| \int_0^x (h_n^{[\alpha]}(x,t)g_x'(t) dt \right|\\
&\leq & \int_0^y |h_n^{[\alpha]}(x,t)| |g_x'(t)|dt+\int_y^x |h_n^{[\alpha]}(x,t)| |g_x'(t)|dt\\
&=& \int_0^{x-\frac{x}{\sqrt{n}}} h_n^{[\alpha]}(x,t) |g_x'(t)|dt +
\int_{x-\frac{x}{\sqrt{n}}}^x h_n^{[\alpha]}(x,t) |g_x'(t)|dt.\end{aligned}$$ With the help of (\[eq5\]) and using Lemma \[l3\], we get $$\begin{aligned}
\int_{x-\frac{x}{\sqrt{n}}}^x h_n^{[\alpha]}(x,t) |g_x'(t)|dt &\leq & \int_{x-\frac{x}{\sqrt{n}}}^x |g_x'(t)-g_x'(x)|dt \nonumber\\
&\leq & \frac{x}{\sqrt{n}}V_{x-\frac{x}{\sqrt{n}}}^x.\end{aligned}$$ Also, let $v=1+\frac{t}{x-t}$ and using Lemma \[l3\], we obtain $$\begin{aligned}
\int_0^{x-\frac{x}{\sqrt{n}}} h_n^{[\alpha]}(x,t) |g_x'(t)|dt &=& \frac{3\eta_n^2(x)}{n}\int_0^{x-\frac{x}{\sqrt{n}}}\frac{|g_x'(t)|}{(x-t)^2} dt= \frac{3\eta_n^2(x)}{n}\int_1^{\sqrt{n}} V_{x-\frac{x}{v}}^x (g_x') dv \nonumber\\&\leq & \frac{3\eta_n^2(x)}{n}\sum_{l=1}^{[\sqrt{n}]}V_{x-\frac{x}{l}}^x (g_x').\end{aligned}$$ Hence $$\begin{aligned}
E_{nx}\leq \frac{x}{\sqrt{n}}V_{x-\frac{x}{\sqrt{n}}}^x+\frac{3\eta_n^2(x)}{n}\sum_{l=1}^{[\sqrt{n}]}V_{x-\frac{x}{l}}^x (f_x').\end{aligned}$$ Now another part can be written as $$\begin{aligned}
F_{nx}&=& \left| \int_x^\infty \left(\int_0^t g_x'(v) dv\right)u_{n}^{[\alpha]}(x,t) dt \right|\nonumber\\
&\leq & \left|\int_x^{2x} \left(\int_0^t g_x'(v) dv\right) d_t(1-h_n^{[\alpha]}(x,t))\right|+\left| \int_{2x}^\infty \left(\int_0^t g_x'(v) dv\right)u_{n}^{[\alpha]}(x,t) dt \right| \nonumber\\
&=& \left| \int_x^{2x} g_x'(v) dv (1-h_n^{[\alpha]}(x,2x))-\int_x^{2x} g_x'(t)(1-h_n^{[\alpha]}(x,t)) dt \right|\nonumber\\
&&+\left| \int_{2x}^\infty \left(\int_0^t (g'(v)-g'(x+)) dv\right)u_{n}^{[\alpha]}(x,t) dt \right|\nonumber\\
&\leq & \left| \int_x^{2x} (g'(v)-g'(x+)) dv (1-h_n^{[\alpha]}(x,2x))\right|+\left| \int_x^{2x} g_x'(t)(1-h_n^{[\alpha]}(x,t)) dt \right|\nonumber\\
&&+ \int_{2x}^\infty |(g(t)-g(x))| u_{n}^{[\alpha]}(x,t)dt +\int_{2x}^\infty |g'(x+)||(t-x)|u_{n}^{[\alpha]}(x,t)dt \nonumber\\
&\leq & \frac{3\eta_n^2(x)}{nx^2} |(g(2x)-g(x)-xg'(x+))|+\int_{x}^{x+\frac{x}{\sqrt{n}}}|g_x'(t)||(1-h_n^{[\alpha]}(x,t))| dt\nonumber\\
&& + \int_{x+\frac{x}{\sqrt{n}}}^{2x}|g_x'(t)||(1-h_n^{[\alpha]}(x,t))| dt + \int_{2x}^\infty M t^\gamma u_{n}^{[\alpha]}(x,t)dt\nonumber\\
&&+\int_{2x}^\infty |g(x)| u_{n}^{[\alpha]}(x,t)dt+ |g'(x+)| \int_{0}^\infty u_{n}^{[\alpha]}(x,t)|t-x| dt \nonumber\\
&\leq & \frac{3\eta_n^2(x)}{nx^2} |(g(2x)-g(x)-xg'(x+))|+ \int_{x}^{x+\frac{x}{\sqrt{n}}} V_x^t(g_x') dt+ \frac{3\eta_n^2(x)}{n} \int_{x+\frac{x}{\sqrt{n}}}^{2x} \frac{V_x^t(g_x')}{(x-t)^2}dt\nonumber\\
&&+M \int_{2x}^\infty t^\gamma u_{n}^{[\alpha]}(x,t)dt + |g(x)|\int_{2x}^\infty u_{n}^{[\alpha]}(x,t)dt+ |g'(x+)| \left(\int_{0}^\infty u_{n}^{[\alpha]}(x,t)(t-x)^2 dt\right)^{\frac{1}{2}}\nonumber\\
&&\times \left(\int_{0}^\infty u_{n}^{[\alpha]}(x,t) dt\right)^{\frac{1}{2}}\nonumber\\
&\leq & \frac{3\eta_n^2(x)}{nx^2} |(g(2x)-g(x)-xf'(x+))|+ \frac{x}{\sqrt{n}} V_x^t(f_x')+ \frac{3\eta_n^2(x)}{n} \int_{x+\frac{x}{\sqrt{n}}}^{2x} \frac{V_x^t(f_x')}{(x-t)^2}dt\nonumber\\
&&+M \int_{2x}^\infty t^\gamma u_{n}^{[\alpha]}(x,t)dt + |f(x)|\int_{2x}^\infty u_{n}^{[\alpha]}(x,t)dt+ |f'(x+)| \sqrt{\frac{3}{n}}\eta_n(x) \nonumber\\
&\leq & \frac{3\eta_n^2(x)}{nx^2} |(g(2x)-g(x)-xg'(x+))| +\frac{x}{\sqrt{n}} V_x^t(g_x') + \frac{3\eta_n^2(x)}{n}\sum_{l=1}^{[\sqrt{nx}]} V_x^{x+\frac{x}{l}}(g_x')\nonumber\\
&&+M \int_{2x}^\infty t^\gamma u_{n}^{[\alpha]}(x,t)dt + |g(x)|\int_{2x}^\infty u_{n}^{[\alpha]}(x,t)dt+ |g'(x+)| \sqrt{\frac{3}{n}}\eta_n(x).\end{aligned}$$ Here, it is arising a case $t\leq 2(t-x)$ and $x\leq t-x$, when $t\geq 2x$, then by using H$\ddot{\text{o}}$lder inequality, we can obtain $$\begin{aligned}
F_{nx}&\leq &\frac{3\eta_n^2(x)}{nx^2} |(g(2x)-g(x)-xg'(x+))| +\frac{x}{\sqrt{n}} V_x^t(g_x') + \frac{3\eta_n^2(x)}{n}\sum_{l=1}^{[\sqrt{nx}]} V_x^{x+\frac{x}{l}}(g_x') \\
&& + M 2^{\gamma} \left(\int_{0}^\infty (t-x)^{2s} u_{n}^{[\alpha]}(x,t)dt\right)^{\frac{\gamma}{2s}}+ \frac{3|g(x)|\eta_n^2(x)}{nx^2} + |g'(x+)| \sqrt{\frac{3}{n}}\eta_n(x)\\
&=& \frac{3\eta_n^2(x)}{nx^2} |(g(2x)-g(x)-xg'(x+))| +\frac{x}{\sqrt{n}} V_x^t(g_x') + \frac{3\eta_n^2(x)}{n}\sum_{l=1}^{[\sqrt{nx}]} V_x^{x+\frac{x}{l}}(g_x') + \mathcal{M}_{s,x}^\gamma\\
&&+\frac{3|g(x)|\eta_n^2(x)}{nx^2} + |g'(x+)| \sqrt{\frac{3}{n}}\eta_n(x).\end{aligned}$$ Using the values of $E_{nx}$ and $F_{nx}$ in equation (\[in3\]), we get our required result.
Conclusion and result discussion
================================
After whole discussions of the present article, it can be seen that the results are good regarding approximations for the defined operators. We discussed those properties which define the order of approximation in terms of modulus of continuity also by means of modified Lipschitz type space. Steklov function is also one of the best useful function by which the present article deals the approximation property. Some results have been discussed in the weighted spaces which enrich the quality of our works. Quantitative approximation have been studied and asymptotic behavior of the operators, Gr$\ddot{\text{u}}$ss Voronovskaya type theorem have also been discussed quantitatively. As for supporting of the proof of convergence of the said operators, examples took place. An important property which is statistical convergence of the operators, has been established and statistical rate of convergence obtained. At last, the very beautiful property has been discussed and that is the rate of convergence the term of function with derivative of bounded variation. This research article may found to be useful in the area of analysis for the researchers. And results can be applicable in literature on Mathematical Analysis, Applied Mathematics.
Applications
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As an application, the defined operators can be used in Quantum Calculus, Mathematical physics. Moreover, it can be generalized into complex number, also it can be obtained the better rate of convergence by using King’s approach. A consequential research topic is approximation of the function by positive linear operators in general mathematics and it withal provides potent implements to application in areas of CAGD, numerical analysis, and solutions of differential equations. These operators can be generalized by considering hypergeometric function. It can be quite effective for the rate of convergence while using $q$-integer in quantum calculus.
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Gadjiev AD. Theorems of the type of P.P. Korovkin’s theorems, Math. Zametki, 1976, 20 (5), 781-786.(in Russian), Math. Notes, 1976, 20 (5-6), 995-998(Engl. Trans.).
Gadjiev AD. Positive linear operators in weighted spaces of functions of several variables (Russian), Izv. Akad. Nauk Azerbaijan SSR Ser. Fiz. Tehn. Mat. Nauk. 1980;1:32-7. Gadjiev AD, Hacısalihoglu H. Convergence of the sequences of linear positive operators. Ankara University. 1995.
Gadjiev AD, Orhan C. Some approximation theorems via statistical convergence. The Rocky Mountain Journal of Mathematics. 2002 Apr 1:129-38. Gonska H, Kacs[ó]{} D, Ra[ş]{}a I. On genuine Bernstein-Durrmeyer operators. Results in Mathematics. 2007 Aug 1;50(3-4):213-25. Gupta V, Pant RP. Rate of convergence for the modified Sz$\acute{\text{a}}$sz-Mirakjan operators on functions of bounded variation. Journal of Mathematical Analysis and Applications. 1999 May 15;233(2):476-83. Gupta MK, Beniwal MS, Goel P. Rate of convergence for Sz$\acute{\text{a}}$sz-Mirakjan-Durrmeyer operators with derivatives of bounded variation. Applied Mathematics and Computation. 2008 Jun 1;199(2):828-32. Hardy GH. Divergent series Oxford University Press. New York. 1949. Ibikli E, Gadjieva EA. The order of approximation of some unbounded function by the sequences of positive linear operators. Turkish J. Math. 1995;19(3):331-7. Ispir N. Rate of convergence of generalized rational type Baskakov operators. Mathematical and computer modelling. 2007 Sep 1;46(5-6):625-31. Jain GC, Pethe S. On the generalizations of Bernstein and Sz$\acute{\text{a}}$sz-Mirakjan operators. Nanta Math. 1977;10(2):185-93. Kajla A, Agrawal PN. Sz$\acute{\text{a}}$sz-Durrmeyer type operators based on Charlier polynomials. Applied Mathematics and Computation. 2015 Oct 1;268:1001-14. Kasana HS, Prasad G, Agrawal PN, Sahai A., Modified Sz$\acute{\text{a}}$sz operators, (Kfsa Proceedings Series, V. 3) Kuwait) International Conference on Mathematical Analysis and its Applications (1985 \_ Kuwait, A. Hamoui, S. M. Mazhar, N. S. Faour, N. S. Faour - Mathematical Analysis. Kolk E. Matrix summability of statistically convergent sequences. Analysis. 1993;13(1-2):77-84. Lenze B. On Lipschitz-type maximal functions and their smoothness spaces. In Indagationes Mathematicae (Proceedings) 1988 Jan 1 (Vol. 91, No. 1, pp. 53-63). North-Holland. Mazhar S, Totik V. Approximation by modified Sz$\acute{\text{a}}$sz-Mirakjan operators. Acta Scientiarum Mathematicarum. 1985 Jan 1;49(1-4):257-69. Mishra VN, Gandhi RB, Nasaireh F. Simultaneous approximation by Sz$\acute{\text{a}}$sz-Mirakjan-Durrmeyer-type operators. Bollettino dell’Unione Matematica Italiana. 2016 Jan 1;8(4):297-305. Miller HI. A measure theoretical subsequence characterization of statistical convergence. Transactions of the American Mathematical Society. 1995;347(5):1811-9. Mishra VN, Yadav R. Some estimations of summation-integral-type operators. Tbilisi Mathematical Journal. 2018;11(3):175-91. $\ddot{\text{O}}$zarslan MA, Actuǧlu H. Local approximation properties for certain King type operators. Filomat 27 (2013), no. 1, 173-181. Patel P, Mishra VN. A note on simultaneous approximation of some integral generalization of the Lupaş operators. Asian J. Math. Comput. Res. 2015;4(1):28-44. Sz$\acute{\text{a}}$sz O. Generalization of S. Bernstein’s polynomials to the infinite interval. J. Res. Nat. Bur. Standards. 1950 Sep;45:239-45. Steinhaus H. Sur la convergence ordinaire et la convergence asymptotique. In Colloq. Math 1951 (Vol. 2, No. 1, pp. 73-74). Schoenberg IJ. The integrability of certain functions and related summability methods. The American Mathematical Monthly. 1959 May 1;66(5):361-775.
Totik V. Approximation by Sz$\acute{\text{a}}$sz-Mirakjan-Kantorovich operators in $L_p (p> 1)$. Analysis Mathematica. 1983 Jun 1;9(2):147-67. Verma DK, Gupta V, Agrawal PN. Some approximation properties of Baskakov-Durrmeyer-Stancu operators. Applied Mathematics and Computation. 2012 Feb 5;218(11):6549-56.
Yadav R, Meher R, Mishraa VN. Quantitative estimations of bivariate summation-integral-type operators. Mathematical Methods in the Applied Sciences. 2019 Aug 29. 42(18):7172-7191.
|
---
abstract: 'Characters have commonly been regarded as the minimal processing unit in Natural Language Processing (NLP). But many non-latin languages have hieroglyphic writing systems, involving a big alphabet with thousands or millions of characters. Each character is composed of even smaller parts, which are often ignored by the previous work. In this paper, we propose a novel architecture employing two stacked Long Short-Term Memory Networks (LSTMs) to learn sub-character level representation and capture deeper level of semantic meanings. To build a concrete study and substantiate the efficiency of our neural architecture, we take Chinese Word Segmentation as a research case example. Among those languages, Chinese is a typical case, for which every character contains several components called radicals. Our networks employ a shared radical level embedding to solve both Simplified and Traditional Chinese Word Segmentation, without extra Traditional to Simplified Chinese conversion, in such a highly end-to-end way the word segmentation can be significantly simplified compared to the previous work. Radical level embeddings can also capture deeper semantic meaning below character level and improve the system performance of learning. By tying radical and character embeddings together, the parameter count is reduced whereas semantic knowledge is shared and transferred between two levels, boosting the performance largely. On 3 out of 4 Bakeoff 2005 datasets, our method surpassed state-of-the-art results by up to $0.4\%$. Our results are reproducible, source codes and corpora are available on GitHub[^1].'
author:
- |
Han He$^1$, Lei Wu$^2$, Xiaokun Yang$^3$, Hua Yan$^4$\
Zhimin Gao$^5$, Yi Feng$^6$, George Townsend$^7$\
[$^{1,2,3,4}$University of Houston-Clear Lake, U.S.A]{}\
[$^5$University of Houston, U.S.A]{} [$^{6,7}$Alsoma University, Canada]{}\
[$^{1,2,3,4}${heh1996,wul,yangxia,yan}@uhcl.edu]{}\
[$^5$zgao5@uh.edu, $^{6,7}${feng,townsend}@algomau.ca]{}
bibliography:
- 'references.bib'
title: 'Dual Long Short-Term Memory Networks for Sub-Character Representation Learning'
---
AI Algorithms and Applications, Deep Learning, Machine Learning Algorithms, Natural Language Processing, Neural Networks, Pattern Recognition
Introduction
============
Unlike English, the alphabet in many non-latin languages is often big and complex. In those hieroglyphic writing systems, every character can be decomposed into smaller parts or sub-characters, and each part has special meanings. But existing methods often follow common processing steps in latin flavor [@mikolov2010recurrent; @Mikolov:2013wc; @2016arXiv160704606B; @kim2016character; @pinter2017mimicking], and treat character as the minimal processing unit, leading to a neglecting of information inside non-latin characters. Early work exploiting sub-character information usually treat it as a separate level from character [@Sun:2014jn; @Li:2015td; @Shi:2015vx; @Dong:2016bl], ignoring the language phenomenon that some of those sub-characters themselves are often used as normal characters. From this phenomenon, we gained a new motivation to design a novel neural network architecture for learning character and sub-character representation jointly.
[|c|cc|c|cc|]{} SC&Sem.&Pho.&TC&Sem.&Pho.\
[UTF8]{}[gbsn]{}鲤
&
[UTF8]{}[gbsn]{}鱼
&
[UTF8]{}[gbsn]{}里
&
[UTF8]{}[bsmi]{}鯉
&
[UTF8]{}[bsmi]{}魚
&
[UTF8]{}[bsmi]{}里
\
[UTF8]{}[gbsn]{}鲢
&
[UTF8]{}[gbsn]{}鱼
&
[UTF8]{}[gbsn]{}连
&
[UTF8]{}[bsmi]{}鰱
&
[UTF8]{}[bsmi]{}魚
&
[UTF8]{}[bsmi]{}連
\
[UTF8]{}[gbsn]{}河
&
[UTF8]{}[gbsn]{}水
&
[UTF8]{}[gbsn]{}可
&
[UTF8]{}[bsmi]{}河
&
[UTF8]{}[bsmi]{}水
&
[UTF8]{}[bsmi]{}可
\
[UTF8]{}[gbsn]{}沟
&
[UTF8]{}[gbsn]{}水
&
[UTF8]{}[gbsn]{}勾
&
[UTF8]{}[bsmi]{}溝
&
[UTF8]{}[bsmi]{}水
&
[UTF8]{}[bsmi]{}冓
\
[UTF8]{}[gbsn]{}捞
&
[UTF8]{}[gbsn]{}手
&
[UTF8]{}[gbsn]{}劳
&
[UTF8]{}[bsmi]{}撈
&
[UTF8]{}[bsmi]{}手
&
[UTF8]{}[bsmi]{}勞
\
[UTF8]{}[gbsn]{}捡
&
[UTF8]{}[gbsn]{}手
&
[UTF8]{}[gbsn]{}佥
&
[UTF8]{}[bsmi]{}撿
&
[UTF8]{}[bsmi]{}手
&
[UTF8]{}[bsmi]{}僉
\
In linguists’ view, Chinese writing system is such a highly hieroglyphic language, and it has a long history of character compositionality. Every Chinese character has several radicals (
[UTF8]{}[gbsn]{}“部首”
in Chinese), which serves as semantic component for encoding meaning, or phonetic component for representing pronouciation. For instance, we listed radicals of several Simplified and Traditional Chinese characters in Table \[font-table\]. Chinese characters with same semantic component are closely correlated in semantic. As shown above, carp (
[UTF8]{}[gbsn]{}鲤
) and silverfish (
[UTF8]{}[gbsn]{}鲢
) are both fish (
[UTF8]{}[gbsn]{}鱼
). River (
[UTF8]{}[gbsn]{}河
) and gully (
[UTF8]{}[gbsn]{}沟
) are all filled with water (
[UTF8]{}[gbsn]{}水
). To catch (
[UTF8]{}[gbsn]{}捞
) or to pick up (
[UTF8]{}[gbsn]{}捡
) a fish, one needs to use hands (
[UTF8]{}[gbsn]{}手
). To exploit those semantic meanings under character embedding level, radical embedding emerged since 2014 [@Sun:2014jn; @Shi:2015vx; @Mikolov:2013uz; @Dong:2016bl]. These early work treated sub-character and character as two separate levels, omitting that they can actually be unified as single minimal processing unit in language model. Instead of ignoring linguistic knowledge, we respect the divergence of human language, and propose a novel joint learning framework for both character and sub-character representations.
To verify the efficiency of our jointly learnt representations, we conducted extensive experiments on the Chinese Word Segmentation (CWS) task. As those languages often don’t have explicit delimiters between words, making it hard to perform later NLP tasks like Information Retrieval or Question Answering. Chinese language is such a typical non-segmented language, which means unlike English language having spaces between every word, Chinese has no explicit word delimiters. Therefore, Chinese Word Segmentation is a preliminary pre-processing step for later Chinese language process tasks. Recently with the rapid rise of deep learning, neural word segmentation approaches arose to reduce efforts in feature engineering [@Zheng:2013wj; @Collobert:2011tk; @Pei:2014vx; @Chen:2015wa; @Cai:2016tg; @2017arXiv170407047C].
In this paper, we propose a novel model to dive deeper into character embeddings. In our framework, Simplified Chinese and Traditional Chinese corpora are unified via radical embedding, growing an end-to-end model. Every character is converted to a sequence of radicals with its original form. Character embeddings and radical embeddings are pretrained jointly in Bojanowski et al. [@2016arXiv160704606B]’s subword aware method. Finally, we conducted various experiments on corpora from SIGHAN bakeoff 2005. Results showed that our jointly learnt character embedding outperforms conventional character embedding training methods. Our models can improve performance by transfer learning between characters and radicals. The final scores surpassed previous work, and 3 out of 4 even surpassed previous preprocessing-heavy state-of-the-art learning work.
More specifically, the contributions of this paper could be summarized as:
- Explored a novel sub-character aware neural architecture and unified character and sub-character as one same level embedding.
- Released the first full Chinese character-radical conversion corpus along with pre-trained embeddings, which can be easily applied on other NLP tasks. Our codes and corpora are freely available for the public.
Related Work
============
In this section, we review the previous work from 2 directions – radical embedding and Chinese Word Segmentation.
Radical Embedding
-----------------
To leverage the semantic meaning inside Chinese characters, Sun et al.[@Sun:2014jn] inaugurated radical information to enrich character embedding via softmax classification layer. In similar way, Li et al.[@Li:2015td] proposed charCBOW model taking concatenation of the character-level and component-level context embeddings as input. Making networks deeper, Shi et al.[@Shi:2015vx] proposed a deep CNN on top of radical embedding pre-trained via CBOW. Instead of utilizing CNNs, following Lample et al.[@Lample:2016vz], Dong et al.[@Dong:2016bl] used two level LSTMs taking character embedding and radical embedding as input respectively.
Our work is closely related to Dong et al.[@Dong:2016bl], but there are two major differences. In pre-training phase, their character embeddings were pre-trained separately, by utilizing conventional word2vec package, and the radical embeddings are randomly initialized. While we considered radical units as sub-characters (parts of one character) and trained the two level embeddings jointly, following Bojanowski et al. [@2016arXiv160704606B]’s approach. In training and testing phases, our two-level embeddings are tied up and unified as the sole minimal input unit of Chinese language.
Chinese Word Segmentation
-------------------------
Chinese Word Segmentation has been a well-known NLP task for decades[@Huang2007Chinese]. After pioneer Xue et al.[@Xue:2003ti] transformed CWS into a character-based tagging problem, Peng et al. [@peng2004chinese] adopted CRF as the sequence labeling model and showed its effectiveness. Following these pioneers, later sequence labeling based work [@Tseng2005A; @Zhao:2006vi; @Zhao2010A; @sun2012fast] was proposed. Recent neural models [@Zheng:2013wj; @Qi:2014uh; @Pei:2014vx; @Chen:2015wa; @Dong:2016bl; @2017arXiv170407556C] also followed this sequence labeling fashion.
Our model is based on Bi-LSTM with CRF as top layer. Unlike previous approaches, the inputs to our model are both character and radical embeddings. Furthermore, we explored which embedding level is more tailored for Chinese language, either using both embeddings together, or even tying them up.
Joint Learning for Character Embedding and Radical Embedding
============================================================
Previous work treated character and radical as two different levels, used them separately or used one to enhance the other. Although radicals are components of a character (belonging to a lower level), they can actually be learnt jointly. It is linguistically more reasonable to put radical embeddings and character embeddings in exactly the same vector space. We propose to train character vector representation being aware of its internal structure of radicals.
Character Decomposition
-----------------------
Every character can be decomposed into a list of radicals or components. To maintain character information in radical list, we simply add the raw form of character to its radical list. Taking the linguistic knowledge that semantic component contains richest meaning of one character into consideration, we append the semantic component to the end of its radical list, hence to make the semantic component appear more than once.
Formally, denote $c$ as a character, $r$ as a radical, $\mathcal{L}_c = \left[r_1, r_2 \cdots r_n \right]$ as the original radical list of $c$. Let $r_s \in \mathcal{L}_c$ be the semantic component of $c$. Our decomposition of $c$ will be:
$$\mathcal{R}_c=\left[c, r_1, r_2 \cdots r_n, r_s\right]$$
General Continuous Skip-Gram (SG) Model
---------------------------------------
Take a brief review of the continuous skip-gram model introduced by Mikolov et al.[@Mikolov:2013uz], applied in character representation learning.
Given an alphabet, target is to learn a vectorial representation $\mathbf{v}_{c}$ for each character $c$. Let $c_1, ..., c_T$ be a large-scale corpus represented as a sequence of characters, the objective function of the skipgram model is to maximize the log-likelihood of correct prediction. The probability of a context character $c_y$ given $c_x$ is computed by a scoring function $s$ which maps character and context to scores in $\mathbb{R}$.
The general SG model ignores the radical structure of characters, we propose a different scoring function $s$, in order to capture radical information.
Let all radicals form an alphabet of size $R$. Given a character $c$ and the radical list $\mathcal{R}_c \subset \{1, \dots, R \}$ of $c$, a vector representation $\mathbf{z}_r$ is associated to each radical $r$. Then a character is represented by the sum of the vector representations of its radicals. Thus the new scoring function will be: $$s(c_x, c_y) = \sum_{r \in \mathcal{R}_{c_x}} \mathbf{z}_r^\top \mathbf{v}_{c_y}.$$
This simple model allows learning the representations of characters and radicals jointly.
Radical Aware Neural Architectures for General Chinese Word Segmentation
========================================================================
Once character and radical representations are learnt, one evaluation metric is how much it improves a NLP task. We choose the Chinese Word Segmentation task as a standard benchmark to examine their efficiency. One prevailing approach to CWS is casting it to character based sequence tagging problem, where our representations can be applied. A commonly used tagging set is $\mathcal{T} = \{B, M, E, S\}$, representing the **b**egin, **m**iddle, **e**nd of a word, or **s**ingle character forming a word.
Given a sequence $\mathbf{X}$ consisted of $n$ features as $\mathbf{X} = (\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n)$, the goal of sequence tagging based CWS is to find the most possible tags $\mathbf{Y}^* = \{\mathbf{y}_1^*, \dots, \mathbf{y}_n^*\}$: $$\mathbf{Y}^* = \operatorname*{arg\,max}_{\mathbf{Y} \in \mathcal{T}^n} p (\mathbf{Y} | \mathbf{X}), \label{eq:cws-argmax}$$ where $\mathcal{T}=\{B, M, E, S\}$.
Since tagging set restricts the order of adjacent tags, we model them jointly using a conditional random field, mostly following the architecture proposed by Lample et al.[@Lample:2016vz], via stacking two LSTMs with a CRF layer on top of them.
Radical LSTM Layer: Character Composition from Radicals
-------------------------------------------------------
In this section, we’ll review RNN with Bi-LSTM extension briefly, before introducing our character composition network.
#### LSTM
Long Short-Term Memory Networks (LSTMs) [@hochreiter1997long] are extensions of Recurrent Neural Networks (RNNs). They are designed to combat gradient vanishing issue via incorporating a memory-cell which enables long-range dependencies capturing.
#### Bi-LSTM
One LSTM can only produce the representation $\overrightarrow{\mathbf{h}_t}$ of the left context at every character $t$. To incorporate a representation of the right context $\overleftarrow{\mathbf{h}_t}$, a second LSTM which reads the same sequence in reverse order is used. Pair of this forward and backward LSTM is called bidirectional LSTM (Bi-LSTM) [@Graves:2005kt] in literature. By concatenating its left and right context representations, the final representation is produced as $\mathbf{h}_{t} = [\overrightarrow{\mathbf{h}_{t}} ; \overleftarrow{\mathbf{h}_{t}}]$.
We apply a Bi-LSTM to compose character embeddings from radical embeddings in both directions. The raw character is inserted as the first radical, and the semantic component is appended as the last radical. The motivation behind this trick is to make use of LSTM’s bias phenomena. In practice, LSTMs usually tend to be biased towards the most recent inputs of the sequence, thus the first one or last one depends on its direction.
![Radical LSTM Layer – composition of character representation from radicals[]{data-label="radical-lstm"}](radical-lstm.pdf)
As illustrated in Figure \[radical-lstm\], the character
[UTF8]{}[gbsn]{}明
(bright) has the radical list of
[UTF8]{}[gbsn]{}日
(sun) and
[UTF8]{}[gbsn]{}月
(moon) with its raw form and duplicated semantic radical. Its compositional representation $\mathbf{h}_{i}^{r} \in \mathbb{R}^{2k}$ is agglomerated via a Bi-LSTM from these radical embeddings, where $k$ is the dimension of radical embeddings.
Character Bi-LSTM Layer: Context Capturing
------------------------------------------
Once compositional character representation $\mathbf{h}_{i}^{r}$ is synthesized, the contextual representation $\mathbf{h}_{t}^c \in \mathbb{R}^{2d}$ at every character $t$ in input sentence can be agglomerated by a second Bi-LSTM. The dimension $d$ is a flexible hyper-parameter, which will be explored in later experiments.
![Character LSTM Layer – capture contextual representation[]{data-label="character-lstm"}](character-lstm.pdf)
Our architecture for contextual feature capturing is shown in Figure \[character-lstm\]. This contextual feature vector contains the meaning of a character, its radicals and its context.
CRF Layer: Tagging Inference
----------------------------
We employed a Conditional Random Fields(CRF) [@Lafferty2001Conditional] layer as the inference layer. As first order linear chain CRFs only model bigram interactions between output tags, so the maximum of a posteriori sequence $\mathbf{Y}^*$ in Eq. \[eq:cws-argmax\] can be computed using dynamic programming, both in training and decoding phase. The training goal is to maximize the log-probability of the gold tag sequence.
Experiments
===========
We conducted various experiments to verify the following questions:
1. Does radical embedding enhance character embedding in pre-training phase?
2. Whether radical embedding helps character embedding in training phase and test phase (by using character embedding solely or using them both)?
3. Can radical embedding replace character embedding (by using radical embedding only)?
4. Should we tie up two level embeddings?
Datasets
--------
To explore these questions, we experimented on the 4 prevalent CWS benchmark datasets from SIGHAN2005 [@emerson_second_2005]. Following conventions, the last 10% sentences of training set are used as development set.
Radical Decomposition
---------------------
We obtained radical lists of character from the *online Xinhua Dictionary*[^2], which are included in our open-source project.
Pre-training
------------
Previous work have shown that pre-trained embeddings on large unlabeled corpus can improve performance. It usually involves lots of efforts to preprocess those corpus. Here we presented a novel solution.
The corpus used is Chinese Wikipedia of July 2017. Unlike most approaches, we don’t perform Traditional Chinese to Simplified Chinese conversion. Our radical decomposition is sufficient of associate character to its similar variants. Not only traditional-simplified character pairs, those with similar radical decompositions will also share similar vectorial representations.
Further, instead of the commonly used word2vec [@Mikolov:2013wc], we utilized fastText[^3] [@2016arXiv160704606B] to train character embeddings and radical embeddings jointly. We applied SG model, $100$ dimension, and set both maximum and minimal $n$-gram length to $1$, as the radical takes only one token.
Final Results on SIGHAN bakeoff 2005
------------------------------------
Our baseline model is Bi-LSTM-CRF trained on each datasets only with pre-trained character embedding (the conventional word2vec), no sub-character enhancement, no radical embeddings. Then we improved it with sub-character information, adding radical embeddings, tying two level embeddings up. The final results are shown in Table \[bakeoff-result\].
Models PKU MSR CityU AS
---------------------------------------------------- ---------- ---------- ---------- ----------
Tseng et al. [@Tseng2005A] 95.0 96.4 - -
Zhang and Clark [@zhang_chinese_2007] 95.0 96.4 - -
Sun et al. [@sun2009a] 95.2 97.3 - -
Sun et al. [@sun2012fast] 95.4 **97.4** - -
Pei et al. [@Pei:2014vx] 95.2 97.2 - -
Chen et al. [@2017arXiv170407556C] 94.3 96.0 95.6 94.8
Cai et al. [@2017arXiv170407047C]$^{\diamondsuit}$ **95.8** 97.1 95.6 95.3
baseline 94.6 96.0 94.7 94.8
+subchar 95.0 96.0 94.9 94.9
+radical 94.6 96.7 95.3 95.2
+radical -char 94.4 96.5 95.0 95.1
+radical +tie 94.8 96.8 95.3 95.1
+radical +tie +bigram 95.3 **97.4** **95.9** **95.7**
: Comparison with previous state-of-the-art models of results on all four Bakeoff-2005 datasets.
\[bakeoff-result\]
All experiments are conducted with standard Bakeoff scoring program[^4] calculating precision, recall, and $\text{F}_1$-score. Note that results with $\diamondsuit$ expurgated long words in test set.
Model Analysis
--------------
Sub-character information enhances character embeddings. Previous work showed pre-trained character embeddings can improve performance. Our experiment showed with sub-character information (+subchar), performance can be further improved compared to no sub-character enhancement (baseline). By simply replacing the conventional word2vec embeddings to radical aware embeddings, the score can benefit an improvement as much as $0.4\%$.
Radical embeddings collaborate well with character embeddings. By building compositional embeddings from radical level (+radical), performance increased by up to $0.7\%$ in comparison with model (baseline) on MSR dataset. But we also notice that: 1) On small dataset such as PKU, radical embeddings cause tiny performance drop. 2) With the additional bigram feature, performance can be further increased as much as $0.6\%$.
Radical embeddings can’t fully replace character embeddings. Without character embeddings but use radical embeddings solely (+radical -char), performance drops a little ($0.1\%$ to $0.3\%$) compared to the model with character embeddings (+radical).
Tying two level embeddings up is a good idea. By tying radical embeddings and character embeddings together (+radical +tie), the raw feature is unified into the same vector space, knowledge is transferred between two levels, and performance is boosted up to $0.2\%$.
Conclusions and Future Work
===========================
In this paper, we proposed a novel neural network architecture with dedicated pre-training techniques to learn character and sub-character representations jointly. As an concrete application example, we unified Simplified and Traditional Chinese characters through sub-character or radical embeddings. We have utilized a practical way to train radical and character embeddings jointly. Our experiments showed that sub-character information can enhance character representations for a pictographic language like Chinese. By using both level embeddings and tying them up, our model has gained the most benefit and surpassed previous single criterial CWS systems on 3 datasets.
Our radical embeddings framework can be applied to extensive NLP tasks like POS-tagging and Named Entity Recognition (NER) for various hieroglyphic languages. These tasks will benefit from deeper level of semantic representations encoded with more linguistic knowledge.
[^1]: [
https://github.com/hankcs/sub-character-cws](
https://github.com/hankcs/sub-character-cws)
[^2]: <http://tool.httpcn.com/Zi/>
[^3]: <https://github.com/facebookresearch/fastText> With tiny modification to output $n$-gram vectors.
[^4]: <http://www.sighan.org/bakeoff2003/score> This script rounds a score to one digit.
|
---
abstract: '\[abstract\] In this work, we address the problem of estimating sparse communication channels in OFDM systems in the presence of carrier frequency offset (CFO) and unknown noise variance. To this end, we consider a convex optimization problem, including a probability function, accounting for the sparse nature of the communication channel. We use the Expectation-Maximization (EM) algorithm to solve the corresponding Maximum A Posteriori (MAP) estimation problem. We show that, by concentrating the cost function in one variable, namely the CFO, the channel estimate can be obtained in closed form within the EM framework in the maximization step. We present an example where we estimate the communication channel, the CFO, the symbol, the noise variance, and the parameter defining the prior distribution of the estimates. We compare the bit error rate performance of our proposed MAP approach against Maximum Likelihood.'
author:
- Rodrigo Carvajal
- 'Boris I. Godoy'
- 'Juan C. Agüero'
title: '**A Bayesian approach to sparse channel estimation in OFDM systems**'
---
------------------------------------------------------------------------
------------------------------------------------------------------------
------------------------------------------------------------------------
------------------------------------------------------------------------
Introduction {#sec:intro}
============
Sparse channel estimation is an important topic found in many different applications (see e.g [@ref:Cotter2002; @ref:Stojanovic2009; @ref:Berger; @ref:Taubock; @ref:Kim] and the references therein). In fact, in many real-world channels of practical interest, e.g. underwater acoustic channels [@ref:Kilfoyle], digital television channels [@ref:ATSC], and residential ultrawideband channels [@ref:Molisch], the associated impulse response tends to be sparse. To obtain an accurate channel impulse response is crucial since it is used in the decoding stage. Sparsity helps one can obtain better channel estimates. In addition, the most common technique for promoting sparsity is by an $\ell_1-$norm regularization, commonly termed as Lasso [@ref:Tibshirani1996]. However, sparsity can be promoted in different ways. For example, in [@ref:Larsson07], sparsity is promoted by generating a pool of possible models, and then performing model selection.
A special characteristic of OFDM systems is its sensitivity to frequency synchronizations errors (see e.g. [@ref:Carvajal2013]), which is produced (among other causes) by carrier frequency offset (CFO). This adds an extra difficulty to the channel estimation problem, since the CFO must be estimated as well as other channel parameters. To estimate the CFO, Maximum Likelihood (ML) estimation has been successfully utilized (see e.g. [@ref:Moose; @ref:Mo; @ref:Carvajal2013]).
In this work, we combine the following problems: (i) estimation of a sparse channel impulse response (CIR) in OFDM systems, (ii) estimation of CFO, (iii) estimation of the noise variance, (iv) estimation of the transmitted symbol, and (v) estimation of the (hyper) parameter defining the prior probability density function (pdf) of the sparse channel. The estimation problem is solved by utilizing a generalization of the EM algorithm (see e.g. [@ref:Mo; @ref:Carvajal2013; @ref:Carvajal12; @ref:Godoy13] and the references therein) for MAP estimation, based on the $\ell_1-$norm of the CIR. In particular, the same methodology has been applied in [@ref:Godoy13] for the identificaton of a sparse finite impulse response filter with quantized data. Our work generalizes previous work on joint CFO and CIR estimation, see [@ref:Mo] and the generalization [@ref:Carvajal12].
The problem of estimating a sparse channel and the transmitted symbol has been previously addresses in the literature [@ref:Schniter]. In [@ref:Schniter], it is also considered bit interleaved coded modulation (BICM) in OFDM systems. The approach in [@ref:Schniter] corresponds to the utilization of the *generalized approximate message passing* (GAMP) algorithm [@ref:Rangan2], which allows for solving the BICM problem. GAMP corresponds to a generalization of the *approximate message passing* (AMP) algorithm [@ref:Donoho2009], although it does not allow for unknown parameters other than the channel. The AMP and GAMP algorithms are based on belief propagation [@ref:Rangan2; @ref:Donoho2009]. When the system is linear (with respect to the channel response), GAMP and AMP are the same algorithm [@ref:Rangan2]. In addition, for sparsity problems, the AMP algorithm corresponds to an efficient implementation of the Lasso estimator, see[@ref:Rangan2] and the references therein. Hence, under the same setup, the MAP-EM algorithm we propose and the GAMP algorithm utilized in [@ref:Schniter] yield the same results.
OFDM System Model {#section:ofdm_sysid}
=================
We consider the following OFDM system model (see e.g. [@ref:Carvajal2013; @ref:Carvajal12] and the references therein), depiected in Fig. \[fig:signal\] [@ref:Carvajal2013]:
[0cm]{} The channel is modelled as a finite impulse response (FIR) filter $\textbf{h} = [h_0 \,\, h_1 \,\, \ldots \,\, h_{L-1}]^T \in \mathbb{C}^{L}$ with $L$ taps.
CFO is modelled by $\textbf{C}_{\varepsilon}
= \exp \{j\mathrm{diag} \left( \frac{2\pi\varepsilon k}{ N_C}\right
)\}$, with $k = 0,1,\ldots, N_C-1$; $\varepsilon$ is the *normalized* frequency offset ($\vert \varepsilon \vert \leq
1/2$, $N_C$ is the number of subcarriers).
The cyclic prefix (CP) is removed at the receiver. Thus, the received signal is given by $$\label{eq:ofdm_model}
\textbf{r} = \textbf{C}_{\varepsilon}\tilde{\textbf{H}} {\sf P} \textbf{x} + {\boldsymbol{\eta}},
\vspace{-1mm}$$ where the channel matrix $\tilde{\textbf{H}}$ is an $(N_C \times N_C)$ circulant matrix whose first column is given by $ [h_0, \,\, h_1, \,\,
\cdots \,\, $ $ h_{L-1},\,\,0 \,\, \dots \,\, 0]^T $, **x** is the transmitted signal (after the inverse discrete Fourier transform), $\sf P$ is a permutation matrix that shuffles the transmitted symbol samples in any desired fashion [@ref:Carvajal2013], ${\boldsymbol{\eta}}\sim {\mathcal N}(0,\sigma^2 {\bf I}_{N_C} ) $, and ${\bf I}_{N_C}$ is the identity matrix of dimension $N_C$. Notice that the time-domain representation of the multicarrier signals in resembles a single-carrier system. However, the main difference corresponds to the utilization of the cyclic prefix, which yields a circulant channel matrix at the receiver after the cyclic prefix removal.
\[fig:signal\]
The transmitted signal is assumed to have a deterministic part (comprising known training data) and a stochastic part (comprising the unknown data). Thus, the transmitted signal corresponds, after the application of the IDFT, to the time domain multiplexing of a training sequence and data coming from the data terminal equipment. We also need to express the transmitted signal in terms of the known (training) component, $\textbf{x}^{(\mathsf{T})}$, and the unknown component, $\textbf{x}^{(\mathsf{U})}$. Thus, the real representation of the transmitted signal $\textbf{x}$ is given by $$\bar{\textbf{x}} = [{\textbf{x}_{\text{\textscr}}^{\mathsf{(T)}}} ^T \; {\textbf{x}_{\text{\textscr}}^{\mathsf{(U)}}} ^T \;{\textbf{x}_{\text{\textsci}}^{\mathsf{(T)}}} ^T \;{\textbf{x}_{\text{\textsci}}^{\mathsf{(U)}}} ^T ]^T \, \in \mathbb{R}^{2N_C},
\vspace{-1mm}$$ where $(\cdot)_{\text{\textscr}}$, $(\cdot)_{\text{\textsci}}$, $(\cdot)^{\mathsf{(T)}}$ and $(\cdot)^{\mathsf{(U)}}$ represent the real part, imaginary part, training part, and unknown part, respectively.
For estimation purposes, it is possible to express the model in as a real-valued state-space model with sample index $k$: $$\bar{\textbf{y}}_k = \left[
\begin{array}{cc}
\textbf{a}_k & -\textbf{b}_k\\
\textbf{b}_k & \textbf{a}_k
\end{array} \right] {\bar{\textbf{x}}}+ \bar{{\boldsymbol{\eta}}}_k = \bar{\textbf{M}}_k {\bar{\textbf{x}}}+ \bar{{\boldsymbol{\eta}}}_k ,
\label{eq:sss}$$ where $\bar{\textbf{y}}_k = [{\mathfrak{Re} \left \{r_k \right\}} \, {\mathfrak{Im} \left \{r_k \right\}}]^T$, $\bar{{\boldsymbol{\eta}}}_k = [{\mathfrak{Re} \left \{\eta_k \right\}} \, {\mathfrak{Im} \left \{\eta_k \right\}}]^T$, $k = 0,1,...,N_C-1$ is the time sample index of the OFDM symbol, ${\mathfrak{Re} \left \{\cdot \right\}}$ and ${\mathfrak{Im} \left \{\cdot \right\}}$ denote the real and imaginary parts, respectively, $$\begin{aligned}
\textbf{a}_k = & (\cos \psi_k ) \textbf{q}_{k+1}^T {\mathfrak{Re} \left \{\tilde{\textbf{H}} \right\}}
\sf{P} - (\sin \psi_k) \textbf{q}_{k+1}^T {\mathfrak{Im} \left \{\tilde{\textbf{H}} \right\}} \sf{P},
\\
\textbf{b}_k = & (\sin \psi_k) \textbf{q}_{k+1}^T {\mathfrak{Re} \left \{\tilde{\textbf{H}} \right\}}
\sf{P} + (\cos \psi_k) \textbf{e}_{k+1}^T {\mathfrak{Im} \left \{\tilde{\textbf{H}} \right\}} \sf{P},\\
\psi_k = & \frac{2\pi k \varepsilon}{N_C},\end{aligned}$$ and $\textbf{q}_k$ is the $k$th column of the identity matrix. This state-space representation is equivalent to , but it is more convenient for the identification approach used in this work. In addition, and as it will be shown in Section \[section:Q\_ML\], the estimation procedure is based upon expressions in the form of $E[{\bar{\textbf{x}}}^{(\sf U)}|{\bar{\textbf{y}}}]$ and $E[{\bar{\textbf{x}}}^{(\sf U)}{\textbf{x}^{\mathsf{(U)}}} ^T|{\bar{\textbf{y}}}]$, amongst other quantities. The attainment of these two expectations can be achieved, for instance, by applying Bayes’ rule for the *posterior* pdf $$p(\textbf{x}^{\mathsf{(U)}}|{\bar{\textbf{y}}}) = \frac{p({\bar{\textbf{y}}}|\textbf{x}^{\mathsf{(U)}}) p(\textbf{x}^{\mathsf{(U)}})}{p({\bar{\textbf{y}}})},$$ for any given *prior* pdf $p(\textbf{x}^{\mathsf{(U)}})$.
It is possible to extend the state-space model in by including a constant state vector that corresponds to the whole unknown transmitted signal (see e.g [@ref:Soderstrom Chap. 9]). That is, Notice that the subindex $k$ for ${\boldsymbol{\chi}}$ in indicates that ${\boldsymbol{\chi}}$ remains unchanged for every sample index $k = 0,1,...,N_C-1$. This extension allows for the utilization of *filtering techniques* for the attainment of (and consequently and ).
We consider a general state-space model that can be utilized for proper and improper signals[^1] [@ref:Miller]. In this sense, our approach can be applied to all common modulation schemes, such as binary phase shift keying (BPSK) and Gaussian minimum shift keying (GMSK), which are improper (see e.g. [@ref:Buzzi]). [$\bigtriangledown$\
]{}
Regarding the received signal, the conditional pdf of $\textbf{y}=
[\bar{\textbf{y}}^T_{0},\dots ,\bar{\textbf{y}}^T_{N_C-1}]^T$ is given by $p(\textbf{y}\,\vert \, {\boldsymbol{\chi}}, {{\boldsymbol{\theta}}}) \sim {\mathcal{N}}(\textbf{M}{\bar{\textbf{x}}},\boldsymbol{\Sigma}_{y})$ where the vector of parameters ${\boldsymbol{\theta}} = ({\bar{\textbf{h}}} , \varepsilon,
\sigma )$, ${\bar{\textbf{h}}} = [{\mathfrak{Re} \left \{\textbf{h} \right\}} \, {\mathfrak{Im} \left \{\textbf{h} \right\}} ]^T$, and $$\textbf{M} = \begin{bmatrix}
\bar{\textbf{M}}_0^T & \cdots &
\bar{\textbf{M}}_{N_C-1}^T \end{bmatrix}^T, \quad {\boldsymbol{\Sigma}}_y =
\mathbb{E}[\hat{{\boldsymbol{\eta}}} \hat{{\boldsymbol{\eta}}}^T], \vspace{-2mm}$$ with $\hat{{\boldsymbol{\eta}}} = [\bar{{\boldsymbol{\eta}}}_0^T \,
\cdots\,\bar{{\boldsymbol{\eta}}}_{N_C-1}^T]$. For proper additive white noise with variance $\sigma^2$, $\boldsymbol{\Sigma}_{y} = 0.5\sigma^2 {\bf
I}_{2N_C}$. For the unknown part of the transmitted signal, the corresponding pdf is simply expressed as $p({\bar{\textbf{x}}}^{(\sf{U})})$. This allows the transmitted signal to be generated by any modulation scheme, yielding a family of possible pdf’s.
In this work, we consider a time-domain processing for the attainment of the estimates. This approach yields a special structure that closely resembles single-carrier systems [@ref:Wang2004; @ref:Amleh2010]. However, to be consistent with previous works, we present our method in the OFDM framework. Because of the similarity of OFDM systems in time domain and single-carrier systems, the present algorithm can also be modified to cover the latter case. [$\bigtriangledown$\
]{}
MAP estimation in OFDM systems
==============================
To promote sparsity in the parameter ${\bar{\textbf{h}}}$, we include an $\ell_1$ regularization term in the form of a prior distribution, $p({\bar{\textbf{h}}})$, which, in turn, leads to a MAP estimation problem. In general, MAP estimation allows for the inclusion of one or more terms that account for statistical prior knowledge of the parameters ${\boldsymbol{\theta}}$. However, here we are interested in utilizing prior knowledge of the channel impulse response only. On the other hand, in a MAP estimation problem, a good estimate $\hat{\sigma}$ is crucial. Thus, it is important to take into account $\sigma$ in the definition of the problem (see e.g. [@ref:stadler10]). If this is not done, the regularized optimization problem may be non-convex and exhibit numerical difficulties. To address this issue, we can express the prior distribution for ${\bar{\textbf{h}}}$ as $$\begin{aligned}
\label{eq:p_beta}
p({\bar{\textbf{h}}}) &= p({\bar{\textbf{h}}}|\sigma,\varepsilon ) p(\sigma)p(\varepsilon),
\quad \text{with}\\
\label{eq:p_theta_sigma}
p({\bar{\textbf{h}}} | \sigma) &= \left(\frac{1}{2\sigma\tau}\right)^{2L} \exp \left\lbrace- \frac{|
{\bar{\textbf{h}}}|}{\tau \sigma} \right\rbrace.
$$ Since we assume no prior knowledge for the channel noise variance nor the CFO, we choose non-informative marginal prior distributions, for example, $p(\sigma) = p(\epsilon) = 1$ (see [@ref:stadler10]). Then, the maximization problem becomes $$\label{eq:log_map1}
\hat{{\boldsymbol{\theta}}} = \mathrm{arg} \max_{{\boldsymbol{\theta}}} \,\, [\log\,p( \textbf{y} | {\boldsymbol{\theta}})
+ \log\,p({\bar{\textbf{h}}}|\sigma ) ].
$$ where $p(\textbf{y}|{\boldsymbol{\theta}})$ is the *likelihood* function.
To achieve convexity, we now can use the procedure suggested in [@ref:stadler10]. That is, we introduce the following reparametrization: $\textbf{\textscg} = {\bar{\textbf{h}}}/
\sigma$, $\rho = \sigma^{-1}$. We therefore define a new parameter to be estimated: ${\boldsymbol{\gamma}} = (\textbf{\textscg}, \varepsilon, \rho)$. Using the new parametrization, and looking at the second term of the right hand side of , this term can expressed as a function of $\textbf{\textscg}$ (or equivalently of the “individual” terms of $\textbf{{\text{\textscg}}}$, ${\text{\textscg}}_j$ is the $j$th element of $\textbf{\textscg}$, $j = 1,2,...,2L$, as (see e.g. [@ref:Polson]) $$\label{eq:log_pen_sum}
\log\,p(\textbf{\textscg}) = \sum_{j =
1}^{2L}z\left(\frac{{\text{\textscg}}_j}{\tau }\right),\vspace{-2mm}$$ where $z(\cdot)$ is a function specifying the log-prior.
The EM algorithm and MAP estimation
-----------------------------------
The EM algorithm is an iterative method that generates a succession of estimates $\hat{{\boldsymbol{\gamma}}}^{(i)} = ( \hat{ \textbf{\textscg}}^{(i)},
\hat{\varepsilon}^{(i)}, {\hat{\rho} }^{(i)} )$, $i = 1,2, \ldots,$ of the parameters ${\boldsymbol{\gamma}}$, which converges to a local maximum of the log-likelihood function (see e.g. [@ref:Dempster]). The EM algorithm consists of an iterative two-step procedure: i) an expectation step (E-step), and ii) a maximization step (M-step). In our case, we develop an augmented EM algorithm to solve the MAP estimation problem presented in this work. The E-step consists of computing the auxiliary function $$\label{eq:M_mod}
{\mathcal{Q}}({\boldsymbol{\gamma}},\hat{{\boldsymbol{\gamma}}}^{(i)} ) = {\mathcal{Q}}_\text{ML}({\boldsymbol{\gamma}},\hat{{\boldsymbol{\gamma}}}^{(i)} ) + {\mathcal{Q}}_{\text{prior}}(\textbf{\textscg},\hat{\textbf{\textscg}}^{(i)} ) ,\vspace{-2mm}$$ where ${\mathcal{Q}}_{\text{prior}}(\textbf{\textscg},\hat{\textbf{\textscg}}^{(i)}
)$ is the function corresponding to the a priori distribution. The function ${\mathcal{Q}}_\text{ML}({\boldsymbol{\gamma}},\hat{{\boldsymbol{\gamma}}}^{(i)})$ is the typical auxiliary function arising from the related ML estimation problem, given by $$\label{eq:Q}
{\mathcal{Q}}_\text{ML}({\boldsymbol{\gamma}}, {\boldsymbol{\gamma}}^{(i)} ) := \mathbb{E}\left
[\log p({\bar{\textbf{x}}}^{(\sf U)},\textbf{y}\,\vert
\,{\boldsymbol{\gamma}})]\,\vert \textbf{y},\hat{{\boldsymbol{\gamma}}}^{(i)}
\right]. \vspace{-2mm}$$
On the other hand, the M-step consists of maximizing the auxiliary function ${\mathcal{Q}}({\boldsymbol{\gamma}},\hat{{\boldsymbol{\gamma}}}^{(i)} )$, yielding $$\hat{{\boldsymbol{\gamma}}}^{(i+1)} = \mathrm{arg} \max_{\boldsymbol{\gamma}}
\mathcal{Q}({\boldsymbol{\gamma}},\hat{{\boldsymbol{\gamma}}}^{(i)} ).
\label{eq:M}
$$
Evaluation of ${\mathcal{Q}}_{\text{ML}}(\boldsymbol{\gamma},\hat{\boldsymbol{\gamma}}^{(i)})$ and its derivative {#section:Q_ML}
-----------------------------------------------------------------------------------------------------------------
The E-step of the EM algorithm given in can be expressed as $$\begin{aligned}
{\mathcal{Q}}_{\text{ML}}&(\boldsymbol{\gamma},\hat{\boldsymbol{\gamma}}^{(i)}
) = \mathbb{E}\left[\log p({\bar{\textbf{x}}}^{(\sf U)})\vert \textbf{y},
\hat{\boldsymbol{\gamma}}^{(i)}\right]+ K_y \nonumber \\ \label{eq:EM2}
& -\frac{1}{2}\mathbb{E}\left[(\textbf{y}-\textbf{M} {\bar{\textbf{x}}})^T
{\boldsymbol \Sigma}_y^{-1}(\textbf{y}- \textbf{M} {\bar{\textbf{x}}}) \vert \textbf{y}, \hat{\boldsymbol{\gamma}}^{(i)}\right],\vspace{-5mm}\end{aligned}$$ where $\textbf{M}$ is a (matrix) function of the parameters ${\boldsymbol{\gamma}}$, and $K_y = -N_C \log(2\pi) -0.5N_C \log
(0.5) + N_C \log(\rho^{2} )$. In addition, we define $$\begin{aligned}
{\boldsymbol{{\mathcal{M}}}}^T &= \begin{bmatrix}
\bar{{\boldsymbol{{\mathcal{M}}}}}_0^T & \vdots & \bar{{\boldsymbol{{\mathcal{M}}}}}_{N_C-1}^T
\end{bmatrix},\quad
\bar{{\boldsymbol{{\mathcal{M}}}}}_k = \begin{bmatrix}
{\bar{\textbf{a}}}_k & - {\bar{\textbf{b}}}_k\\
{\bar{\textbf{b}}}_k & {\bar{\textbf{a}}}_k
\end{bmatrix}, \\
{\bar{\textbf{a}}}_k= & \,(\cos \psi_k ) \textbf{e}_{k+1}^T {\tilde{\textbf{X}}}_\text{\textscr}{\mathcal{I}} - (\sin \psi_k) \textbf{e}_{k+1}^T{\tilde{\textbf{X}}}_\text{\textsci}{\mathcal{I}},
\\
{\bar{\textbf{b}}}_k = & \,(\sin \psi_k) \textbf{e}_{k+1}^T{\tilde{\textbf{X}}}_\text{\textscr}{\mathcal{I}} + (\cos \psi_k) \textbf{e}_{k+1}^T {\tilde{\textbf{X}}}_\text{\textsci}{\mathcal{I}},\end{aligned}$$ with $\tilde{\textbf{X}}_{\text{\textscr}}$ and $\tilde{\textbf{X}}_{\text{\textsci}}$ being the circulant matrices generated by $\sf{P}\textbf{x}_{\text{\textscr}}$ and $\sf{P}\textbf{x}_{\text{\textsci}}$, respectively, and ${\mathcal{I}} = [
\textbf{I}_{L} \, \textbf{0}]^T$. Then, we can write ${\textbf M}
{\bar{\textbf{x}}} = {\boldsymbol{{\mathcal{M}}}} \textbf{\textscg} \rho^{-1} $. Replacing this equality in , and taking the derivative of ${\mathcal{Q}}_{\text{ML}}({\boldsymbol{\gamma}},\hat{{\boldsymbol{\gamma}}}^{(i)})$ with respect to $\textbf{\textscg}$, and $\rho$, we obtain: $$\begin{aligned}
\label{eq:dQ_dh1}
\frac{\partial {\mathcal{Q}}_{\text{ML}}}{\partial \textbf{\textscg}} & =
2\left( \rho \textbf{y}^T\mathbb{E}[{\boldsymbol{{\mathcal{M}}}}\vert \textbf{y},
\hat{\boldsymbol{\gamma}}^{(i)}] - \mathbb{E}[ {\boldsymbol{{\mathcal{M}}}}^T{\boldsymbol{{\mathcal{M}}}} \vert
\textbf{y}, \hat{\boldsymbol{\gamma}}^{(i)} ] \textbf{\textscg} \right),\\
\frac{\partial {\mathcal{Q}}_{\text{ML}}}{\partial \rho } & = \frac{2
N_C}{\rho} - 2\left( \rho \textbf{y}^T \textbf{y} - \textbf{y}^T
\mathbb{E}\left[({\boldsymbol{{\mathcal{M}}}})\vert \textbf{y},
\hat{\boldsymbol{\gamma}}^{(i)}\right]
\textbf{\textscg}\right).
\label{eq:dQ_dsigma2}\end{aligned}$$
Evaluation of ${\mathcal{Q}}_{\text{prior}}(\textbf{\textscg},\hat{\textbf{\textscg}}^{(i)} )$ and its derivative {#subsec:Qprior}
-----------------------------------------------------------------------------------------------------------------
We express $z(\cdot)$ as a variance-mean Gaussian mixture (VMGM) [@ref:Polson]. When expressed in terms ${\text{\textscg}}_j,\, j = 1,2,...$, a VMGM for the parameters is given by (see e.g [@ref:Polson; @ref:Barndorff-Nielsen]) $$p(\textbf{\textscg}) = \prod_j \int_0^{\infty} p({{\text{\textscg}}_j}|{\lambda_j})p({\lambda_j})d{\lambda_j},\label{eq:mvgm} \vspace{-2mm}$$ where $ {{\text{\textscg}}_j}|{\lambda_j} \sim {\mathcal{N}}_{{\text{\textscg}}_j} (0 ,\lambda_j
\tau^2 )\,, \lambda_j \sim p(\lambda_j)$. In this sense, the random variable $\lambda_j $ ($> 0$) can be considered as a *hidden variable* in the EM algorithm. Hence, given , the auxiliary function ${\mathcal{Q}}_{\text{prior}}(\textbf{{\text{\textscg}}},\hat{\textbf{{\text{\textscg}}}}^{(i)})$, can be expressed as $$\hspace{-10mm} {\mathcal{Q}}_{\text{prior}} (\textbf{\textscg},\hat{\textbf{\textscg}}^{(i)}) =
\sum_{j = 1}^{2L}\int \log [
p({\text{\textscg}}_j,\lambda_j)] p(\lambda_j|\hat{{\text{\textscg}}}_j^{(i)})d(\lambda_j)
\vspace{-3mm}$$ $$= \sum_{j = 1}^{2L}\int \left ( \log
[p({\text{\textscg}}_j|\lambda_j)]+\log
[p(\lambda_j)] \right )p(\lambda_j|\hat{{\text{\textscg}}}_j^{(i)})d(\lambda_j),\vspace{-2mm}
\label{eq:Q_prior1}$$ since $\log p(\textbf{{\text{\textscg}}},{\boldsymbol{\lambda}}) = \sum_{j = 1}^{2L} \log$ $ p({\text{\textscg}}_j,\lambda_j)$.
In the case that is given by , then its derivative is given by where is the expectation obtained from
See [@ref:Carvajal12]. [$\blacksquare$\
]{}
From the M-step, at the $i$th iteration, an estimate ($\hat{{\text{\textscg}}}_j^{(i)}$) of ${\text{\textscg}}_j$ is obtained. This estimate is, then, inserted into , in order to obtain an estimate of $\mathbb{E}_{\lambda_j| \hat{{\text{\textscg}}}^{(i)}_j}
\{\lambda_j^{-1} \}$, which in turn is utilized in the maximization of ${\mathcal{Q}}_\text{prior}$. Once the new estimate $\hat{{\text{\textscg}}}_j^{(i+1)}$ has been obtained, it is inserted into and the iteration continues until convergence has been reached.
In our particular case, we only want to promote sparsity in $\textbf{{\text{\textscg}}}$ (consequently in the CIR ${\bar{\textbf{h}}}$). Thus, our chosen penalty function is $z({\text{\textscg}}_j / \tau ) =
|{\text{\textscg}}_j/ \tau|$. Using , we have that $\mathbb{E}_{\lambda_j|\hat{{\text{\textscg}}}_j}[\lambda_j^{-1}] =
-\tau \text{sign}(\hat{{\text{\textscg}}}^{(i)}_j)/
\hat{{\text{\textscg}}}^{(i)}_j$. Using this value for $\mathbb{E}_{\lambda_j| \hat{{\text{\textscg}}}_j}\{\lambda_j^{-1}\}$, and calculating ${\partial {\mathcal{Q}}_{\text{prior}} /\partial
\textbf{{\text{\textscg}}}} $ we have that $$\begin{aligned}
\frac{\partial {\mathcal{Q}}_{\text{prior}}}{\partial
\textbf{\textscg} } = -\frac{1}{\tau^2}
\textbf{E} \textbf{\textscg},
\label{eq:dQprior_dhbar}
\vspace{-3mm}\end{aligned}$$ where $\textbf{E} = \text{diag}\left(
\mathbb{E}_{\lambda_1|\hat{{\text{\textscg}}}_1^{(i)}}\{ \lambda_1^{-1} \},\,
... ,\,
\mathbb{E}_{\lambda_{2L}|\hat{{\text{\textscg}}}_{2L}^{(i)}} \{
\lambda_{2L}^{-1} \} \right)$.
Combination of ${\mathcal{Q}}_{\text{ML}}({\boldsymbol{\gamma}},\hat{{\boldsymbol{\gamma}}}^{(i)})$ and ${\mathcal{Q}}_{\text{prior}}(\textbf{{\text{\textscg}}},\hat{\textbf{{\text{\textscg}}}}^{(i)})$
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
We are building our strategy on an underlying ML estimation algorithm. Thus, we assume ${\mathcal{Q}}_{\text{ML}}$ and $\partial {\mathcal{Q}}_\text{ML}/\partial \textbf{{\text{\textscg}}}$ known. The strategy is then to derive the augmented E-step considering both ${\mathcal{Q}}_\text{ML}$ and ${\mathcal{Q}}_{\text{prior}}$ with respect to $\textbf{\textscg}$, that is, $$\frac{\partial {\mathcal{Q}}}{\partial \textbf{\textscg} } = \frac{\partial
{\mathcal{Q}}_{\text{ML}}}{\partial \textbf{\textscg} } + \frac{\partial
{\mathcal{Q}}_{\text{prior}}}{\partial \textbf{\textscg} }.
\label{eq:dQ_dh}$$ Using , , , and , and expressing $\textbf{\textscg}$ as a function of $\varepsilon$, we have $$\label{eq:h_bar}
\textbf{\textscg} = \left[\mathbb{E}[{\boldsymbol{{\mathcal{M}}}}^T {\boldsymbol{{\mathcal{M}}}}\vert \textbf{y},
\hat{\boldsymbol{\gamma}}^{(i)} ]
+ \frac{1}{2\tau^2} \textbf{E} \right]^{-1} \! \! \mathbb{E}[{\boldsymbol{{\mathcal{M}}}}\vert \textbf{y},
\hat{\boldsymbol{\gamma}}^{(i)}]^T \rho \textbf{y}.
\vspace{-1mm}$$ Replacing the expression for $\textbf{\textscg}$ in , we can optimize ${\mathcal{Q}}$ in with respect to the parameter $\varepsilon$. Thus, the parameter $\textbf{\textscg}$ (consequently $\bar{\textbf{h}}$) is obtained by replacing the result of the optimization for $\varepsilon$ in . One advantage of our method is that it allows the concentration of the cost in one variable, namely CFO. In addition, we obtain closed form expressions for the optimization of the regularized communication channel, namely CIR, which, in general, is not possible with other methods when applying $\ell_1$-norm regularization.
Estimation of $\tau$ {#subsec:tau_est}
--------------------
So far, the proposed algorithm for sparse channel estimation relies upon knowledge of $\tau$ (or at least a good estimate of it). Knowledge of this variable is important for accurate estimates of ${\bar{\textbf{h}}}$. However, having *a priori* knowledge of this parameter is not always possible. For example, in an urban cellular network, the channel can exhibit different behaviours depending on the location, presenting the possibility of having different values of $\tau$ (at each one of the locations). Thus, we seek an estimate of $\tau$.
Using , the Empirical-Bayes (EB) estimate is given by
We define an auxiliary function $Q(\tau,\tau^{(i)}) =
2L\log(\tau^{-1}/2\sigma) - \tau^{-1} \mathbb{E}_{|{\bar{\textbf{h}}}|/\sigma |
{\bar{\textbf{y}}}, \hat{\tau}^{(i)} }\left[ \frac{|{\bar{\textbf{h}}}| }{\sigma}\right]$, take derivative with respect to $\tau^{-1}$, and then set the result equal to zero. [$\blacksquare$\
]{}
In general, the computation of $\mathbb{E}_{|{\bar{\textbf{h}}}|/\sigma | {\bar{\textbf{y}}},
\hat{\tau}^{(i)} }\left[ \frac{|{\bar{\textbf{h}}}| }{\sigma}\right]$ is computational expensive, requiring, in addition, many observations ${\bar{\textbf{y}}}$. To avoid this problem, we approximate $\mathbb{E}_{|{\bar{\textbf{h}}}|/\sigma | {\bar{\textbf{y}}}, \hat{\tau}^{(i)} }\left[
\frac{|{\bar{\textbf{h}}}| }{\sigma}\right]\approx
| \hat{{\bar{\textbf{h}}}}_{\text{ML}}| /\hat{\sigma}_{\text{ML}}$, where $\hat{{\bar{\textbf{h}}}}_{\text{ML}}$ and $\hat{\sigma}_{\text{ML}}$ are the ML estimates using no regularization term, and where $\bar{\textbf{x}}$ is completely known (100% training).
The solution to the regularized estimation problem (considering the reparametrization) presented in this work can be summarized in the following steps:
- with 100% training, and no regularization term, calculate $\hat{\tau} \approx \hat{{\bar{\textbf{h}}}}_{\text{ML}}/2L\hat{\sigma}_{\text{ML}}$,
- $\hat{{{\boldsymbol{\theta}}}}^{(i)} = ({\hat{{\bar{\textbf{h}}}}}^{(i)} ,
\hat{\varepsilon}^{(i)}, ({\hat{\sigma}^2})^{(i)} )$, and form the new variables: $\textbf{\textscg}^{(i)} = \hat{{\bar{\textbf{h}}}}^{(i)}/\hat{\sigma}^{(i)}$, and $\rho^{(i)} = 1/\hat{\sigma}^{(i)}$,
- with a fixed $(\hat{\sigma}^2)^{(i)}$ from (ii), optimize for $\varepsilon$ after replacing in ,
- with the estimate ($\hat{\textbf{\textscg}}^{(i+1)},
\hat{\varepsilon}^{(i+1)}$) (consequently $\hat{{\bar{\textbf{h}}}}^{(i+1)}$) obtained in (iii), find $\hat{\sigma}^{(i+1)}$ from making zero the right-hand side of , and solve a quadratic equation,
- go back to (ii) until convergence.
Numerical Example
=================
\[table:variance\_u\]
![ ML estimation (upper plot), MAP estimation (lower plot). Continuous line: average value. $\text{SNR} = 10[\text{dB}]$.[]{data-label="fig:ber"}](ber02.pdf){width="0.6\columnwidth"}
In this section, we present a numerical example using our approach for an OFDM system with CFO. We assume that the unknown part of the time-domain transmitted signal is approximately Gaussian distributed (a consequence of the Central Limit Theorem). Thus, $p({\bar{\textbf{x}}}^{(\sf{U})}) \sim {\mathcal{N}} \left( \textbf{0}, {\boldsymbol{\Sigma}}_{\bar{x}^{({\sf U})}} \right)$, where ${\boldsymbol{\Sigma}}_{\bar{x}^{({\sf U})}} = \begin{bmatrix}
{\boldsymbol{\Sigma}}_{\textbf{x}_{\text{\textscr}}^{({\sf U})}} &
{\boldsymbol{\Sigma}}_{\textbf{x}_{\text{\textscr}}^{({\sf U})}
\textbf{x}_{\text{\textsci}}^{({\sf U})}}\\
{\boldsymbol{\Sigma}}_{\textbf{x}_{\text{\textsci}}^{({\sf U})}
\textbf{x}_{\text{\textscr}}^{({\sf U})}} &
{\boldsymbol{\Sigma}}_{\textbf{x}_{\text{\textsci}}^{({\sf U})}}
\end{bmatrix}$, ${\boldsymbol{\Sigma}}_{\textbf{x}_{\text{\textscr}}^{({\sf U})}} = E[\textbf{x}_{\text{\textscr}}^{({\sf U})} {\textbf{x}_{\text{\textscr}}^{({\sf U})}}^T] $, ${\boldsymbol{\Sigma}}_{\textbf{x}_{\text{\textsci}}^{({\sf U})}} = E[\textbf{x}_{\text{\textsci}}^{({\sf U})} {\textbf{x}_{\text{\textsci}}^{({\sf U})}}^T]$, and ${\boldsymbol{\Sigma}}_{\textbf{x}_{\text{\textsci}}^{({\sf U})}
\textbf{x}_{\text{\textscr}}^{({\sf U})}} = E[ \textbf{x}_{\text{\textsci}}^{({\sf U})} {\textbf{x}_{\text{\textscr}}^{({\sf U})}}^T ]$ (known).
The expectations on the right hand side of and can be readily calculated by applying Kalman filtering to the model in . In addition, we consider that channel noise variance is unknown. We consider the following set-up: (i) $N_C = 64$, (ii) a sparse channel impulse response of length $20$, with $14$ taps equal to zero, (iii) the transmitted signal is Gaussian distributed, (iv) the signal to noise ratio is $5[\text{dB}]$ and $10[\text{dB}]$, (v) $\varepsilon = 0.2537$, and (vi) $62.5\%$ of training. As a performance measure, we consider the normalized mean square error, defined as $\text{NMSE}:=
(\textbf{h}-\hat{\textbf{h}})^H (\textbf{h}-\hat{\textbf{h}}
)/(\textbf{h}^H \textbf{h})$. Using 100 different realizations for the noise, the results can be seen in Table \[table:variance\_u\]. We can conclude that regularization only helps if limited amount of data is available. For the case of ${\text{SNR}}= 10[\text{dB}]$, in Fig \[fig:ber\], the average BER for ML estimation is 0.0195, and for MAP estimation is 0.0132.
Conclusions
===========
In this work, we have proposed an algorithm to estimate sparse channels in OFDM systems, the CFO, the variance of the noise, the symbol, and the parameter defining the a priori distribution of the sparse channel. This is achieved in the framework of MAP estimation, using the EM algorithm.
Sparsity has been promoted by using an $\ell_1$-norm regularization, in the form of a prior distribution for the CIR. For that, the EM algorithm has been modified to include this case. In addition, we have concentrated the cost function in the M-step to numerically optimize one single variable ($\varepsilon$).
The numerical examples illustrate the effectiveness of this approach for the partial training case, obtaining, in most cases studied, a lower value for NMSE using regularization compared to the value for NMSE using no regularization. For the full training case, there is no noticeable difference between the estimates obtained with ML and MAP. This confirms that prior knowledge is useful when the amount of data is limited.
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[^1]: This representation also extends to proper and improper CIR and additive noise.
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\#1
Efficient manipulation with optical quanta calls for materials possessing large and lossless optical nonlinearities [@qq1]. Recently it was shown that coherent atomic effects such as electromagnetically induced transparency (EIT) [@EIT] and coherent population trapping (CPT) [@CPT; @sz] are able to suppress the linear absorption of resonant multilevel media while keeping the nonlinear susceptibility at a very high level [@harris90prl; @largenl; @luknature]. Previous experimental and theoretical work has shown that coherent media produce an effective interaction between two electromagnetic fields due to both refractive [@largenl; @luknature; @harris99prl; @lukin00prl] and absorptive [@absorption; @absorption_exp] $\chi^{(3)}$ Kerr nonlinearities.
One method of producing Kerr nonlinearity with vanishing absorption is based on the coherent properties of a three-level $\Lambda$ configuration (see Fig. \[fig1\]a), in which the effect of EIT can be observed. However, since an ideal CPT medium does not interact with the light, it also cannot produce any nonlinear effects [@sz]. To get a nonlinear interaction in such a medium one needs to “disturb” the CPT regime by introducing, for example, the interaction with an additional off-resonant level (the $N$-type scheme, Fig. \[fig1\]b). If the disturbance of CPT is small, i.e., the detuning $\Delta$ is large, the absorption does not increase much, but the nonlinearity can be as strong as in a near-resonant two level system [@largenl; @luknature; @lukin00prl; @zubairy01pra].
In this Letter we propose a new method of resonant enhancement of $\chi^{(3)}$ and higher order nonlinearities without significant optical losses. We consider the $M$-type configuration shown in Fig. \[fig1\]c. There is coherent population trapping in such a scheme which leads to EIT. At the same time, by introducing a small two-photon detuning $\delta$ we may disturb this CPT and produce a strong nonlinear coupling among the electromagnetic fields interacting with the atomic system.
An important advantage of the $M$-type level configuration is that this type of coherence is easily created on the Zeeman sublevels of alkali atoms. In this particular case large self-phase modulation of circularly polarized light is possible. We show here significant change in nonlinear magneto-optical polarization rotation [@faraday] due to $M$-type coherence in Rb vapor. Our experimental results confirm the theoretical predictions.
Our method of creation of a highly nonlinear medium with small absorption has prospects in fundamental as well as applied physics. An advantage of the $M$ configuration is that by increasing the number of the levels it is possible to realize higher orders of nonlinearity. This can be used for construction of non-classical states of light as well as coherent processing of quantum information.
Let us first consider a medium (atomic, molecular, semiconductor, etc.) with an $M$-type energy level structure as in Fig. \[fig1\]c. Here levels $|a_j\rangle$ have natural decays $\gamma_j$ and we assume that the ground state levels $|b_j\rangle$ have no decay. The coherence between levels $|b_i\rangle$ and $|b_j\rangle$ ($i \ne j$) have slow homogeneous decay $\gamma_{0}$. The energy levels are coupled by weak probe electromagnetic fields of Rabi frequency $\alpha_j$ and strong coupling fields of Rabi frequency $\Omega_j \gg |\alpha_j|$. For the sake of simplicity we assume all the fields to be resonant with the corresponding atomic transitions except for the coupling field $\Omega_2$ which has small detuning $\delta \ll \gamma_2$ from the $|b_3\rangle \rightarrow |a_2\rangle$ transition.
We first focus on the case when there is no coherence decay in the system. Then the interaction of the atoms and the fields can be described in the slowly varying amplitude and phase approximations by the Hamiltonian $$\begin{aligned}
\nonumber
H_M = -\hbar \delta |b_3 \rangle \langle b_3| + \hbar \sum
\limits_{j=1}^{2} ( \alpha_j |a_j \rangle \langle b_j| + \Omega_j
|a_j \rangle \langle b_{j+1}| + H.c.),\end{aligned}$$ where $H.c.$ means Hermitian conjugation. If $\delta=0$, there exists the noninteracting eigenstate (“dark eigenstate” [@sz]) of this Hamiltonian, corresponding to the zero eigenvalue $\lambda_{D}=0$: $$\begin{aligned}
\label{d2}
|D\rangle =
\frac{\alpha_1\alpha_2 |b_3\rangle - \Omega_2
\alpha_1 |b_2\rangle + \Omega_1\Omega_2 |b_1 \rangle}{
\sqrt{|\alpha_1|^2 |\alpha_2|^2 + |\Omega_2|^2 |\alpha_1|^2 +
|\Omega_1|^2|\Omega_2|^2}}.\end{aligned}$$
Strictly speaking, there is no “dark state” for finite $\delta$. However when the detuning is small enough $(\delta \ll \gamma_2, |\Omega_2|^2/\gamma_2)$, the disturbance of the “dark” state is small. The disturbed dark state $|\tilde D
\rangle $ and the corresponding eigenvalue of the Hamiltonian $\lambda_{\tilde D}$ in the limit of $|\Omega_j| \gg |\alpha_j|$ are: $$\begin{aligned}
\label{eigenvect}
|\tilde D \rangle \approx \zeta \left [ | D \rangle - \delta
\frac {\alpha_1^*|\alpha_2|^2} {|\Omega_1|^2|\Omega_2|^2} | a_1
\rangle + \delta \frac {\alpha_1 \alpha_2}
{|\Omega_1||\Omega_2|^2} | a_2 \rangle \right ],\end{aligned}$$ $$\begin{aligned}
\label{eigenvalt}
\lambda_{\tilde D} \approx -\hbar \delta \frac
{|\alpha_1|^2|\alpha_2|^2} {|\Omega_1|^2|\Omega_2|^2},\end{aligned}$$ where $\zeta \simeq 1$ is a normalization parameter.
Because the system does not leave the dark state during static or adiabatic interaction with the probe fields we write the Hamiltonian as $H_M \simeq \lambda_{\tilde D} |\tilde D
\rangle \langle \tilde D |$. As $|\tilde D \rangle \langle \tilde
D | \simeq |b_1 \rangle \langle b_1 | \simeq 1$, it is convenient to exclude atomic degrees of freedom from the interaction picture and to rewrite the Hamiltonian in Heisenberg picture with quantized probe fields. The relation between Rabi frequencies of the probe fields and quantum operators describing the corresponding field mode can be written as $$\hat \alpha_i = \sqrt{\frac{2\pi \wp^2_i \nu_i}{\hbar V_i}} \hat
a_i = \xi_i \hat a_i,$$ where $\wp_i$ is the dipole moment of the transition $|a_i\rangle
\rightarrow |b_i\rangle$, $\nu_i$ is the field frequency, $V_i$ is the quantization volume of the mode, $\hat a_i$ and $\hat
a_i^\dag$ are the annihilation and creation operators. Then the Hamiltonian takes the final form $$\begin{aligned}
\label{hamM}
H_M= -\hbar \delta \frac {\xi_1^2\xi_2^2}
{|\Omega_1|^2|\Omega_2|^2} \ \hat a_1^\dag \hat a_1 \hat a_2^\dag
\hat a_2.\end{aligned}$$ We see that the nonlinear coupling between the probe fields increases with an increase in the detuning $\delta$.
To understand the size of the nonlinear interaction we recall the interaction Hamiltonian for an $N$ scheme [@largenl; @zubairy01pra]: $$\begin{aligned}
\label{hamN}
H_N= \hbar \frac {\xi_1^2\xi_2^2} {\Delta |\Omega_1|^2} \ \hat
a_1^\dag \hat a_1 \hat a_2^\dag \hat a_2.\end{aligned}$$ The ratio of coupling constants in (\[hamM\]) and (\[hamN\]) $${\cal R} = \frac{\delta \Delta }{|\Omega_2|^2}$$ determines the relative strength of the nonlinear interaction introduced by the schemes. If $|{\cal R}| > 1$, then the $M$ scheme is more effective than $N$.
However, this comparison is not consistent unless we take into account the absorption of the probe field introduced by spontaneous emission. Simple calculations demonstrate that the ratio of the spontaneous emission probabilities coincide with the ratio of nonlinear susceptibilities for the $M$ and $N$ schemes. Another source of probe absorption is due to the decay of coherence between ground state levels which leads to depopulation of the dark state created in the system and to the absorption of the probe fields independently of each other. It can be shown that the probability of this process is the same for both $M$ and $N$ configurations.
In the usual cells containing gases of alkali atoms the one-photon transitions are Doppler-broadened due to the motion of the atoms. If the condition for EIT is fulfilled ($\Omega_1
\gg W_d \sqrt{\gamma_0/\gamma_1}$, where $W_d$ is the Doppler linewidth $W_d \gg \gamma_1$ [@leearchiv]), then in the $M$-scheme the population of level $|b_2 \rangle$ is approximately equal to $|\alpha_1|^2/|\Omega_1|^2$. The nonlinear interaction results from the refraction and absorption of the second probe field $\alpha_2$, coupled to the second drive field $\Omega_2$. Fields $\alpha_2$ and $\Omega_2$ along with levels $|b_2\rangle$, $|b_3\rangle$, and $|a_2\rangle$ create $\Lambda$ system. Almost all population is in level $|b_2 \rangle$. Therefore, $$\label{sus1}
\chi_M = -i\frac{3}{8\pi^2}{\cal N}\lambda_{\alpha 2}^3
\frac{\gamma_2
(\gamma_0+i\delta)}{(\gamma_0+i\delta)W_d+|\Omega_2|^2}
\frac{|\alpha_1|^2}{|\Omega_1|^2}.$$ where ${\cal N}$ is the atomic density and $\lambda_{\alpha 2}$ is the wavelength of the field $\alpha_2$.
Similar calculations allow us to derive the susceptibility for the field $\alpha_2$ for the Doppler broadened $N$ scheme: $$\label{sus2}
\chi_N = -i\frac{3}{8\pi^2}{\cal N}\lambda_{\alpha 2}^3
\frac{\gamma_2}{W_d+i\Delta} \frac{|\alpha_1|^2}{|\Omega_1|^2},$$ It is easy to see that Eqs. (\[sus1\]) and (\[sus2\]) are interchangeable if $\gamma_0 \rightarrow 0$, and $\Delta
\leftrightarrow \delta/|\Omega_2|^2$. Therefore, ultimately $M$ and $N$ interaction schemes are equally efficient, even though quite different mechanisms are responsible for the nonlinear susceptibility enhancement.
To experimentally demonstrate the enhancement of nonlinearity in the $M$ type level scheme we study the rotation of elliptical polarized light resonant with the $F=2 \rightarrow
F'=1$ transition of the $^{87}$Rb $D_1$ line (Fig. \[fig2\]a). We consider the light as two independent circular components $E_+$ and $E_-$ which generate a coherent superposition of the Zeeman sublevels (a dark state). To disturb this dark state we apply a longitudinal magnetic field $B$ which leads to a splitting $\delta
\propto B$ of the Zeeman sublevels. This atomic transition consists of $\Lambda$ and $M$ level configurations (Fig. \[fig2\]a and \[fig2\]b).
The nonlinear properties of the $M$ level scheme result in significant modification of the nonlinear polarization rotation as a function of the light ellipticity. We write the field amplitudes as $|E_\pm|^2 = (1\pm q)|E_0|^2/2$, where $|q|<1$ characterizes the light ellipticity and find an expression for the rotation angle $\phi$ of the polarization ellipse for small magnetic field. The details of the calculation will be given elsewhere. The enhancement of rotation with regards to the isolated $\Lambda$ system is given by $$\label{rot1}
\frac {\phi(B)}{\phi_\Lambda(B)} \approx \frac 12 +
\frac{2+q^2}{(2-q^2)^2},$$
We have measured the polarization rotation in a cell containing Rb vapor using the technique described in detail in [@novikova'01pra]. The results are shown in Fig. \[fig3\]. The experimental dependence looks slightly different from the theoretical one (Fig. \[fig3\] inset) because of the influence of the Doppler broadening and AC-stark shifts due to light coupling to off-resonant atomic sublevels. However, numerical simulations based on steady state solution of exact density matrix equations give a good agreement with the experiment.
The experimental results pertain to a semiclassical description where the field is classical. An interesting application of the Kerr nonlinearity with quantized field as given in Eq. (\[hamM\]) is the possible implementation of a quantum phase gate. A quantum phase gate, together with a one-bit unitary gate, form the basic building block for quantum computation [@nc]. The transformation properties of a quantum phase gate leave the two qubit states unchanged when one or both input qubits are in the logic state 0 and introduces a phase $\eta
$ only when both the qubits in the input states are 1. For input photon states $|0\rangle$ or $|1\rangle$ for the two qubits, a unitary operator of the form $Q_{\eta}={\rm exp}(i \eta \hat
a_1^\dag \hat a_1 \hat a_2^\dag \hat a_2)$ can lead to such a phase gate, i.e., $Q_{\eta}|0_1,0_2\rangle= |0_1,0_2\rangle$, $Q_{\eta}|0_1,1_2\rangle= |0_1,1_2\rangle$, $Q_{\eta}|1_1,0_2\rangle= |1_1,0_2\rangle$, and $Q_{\eta}|1_1,1_2\rangle={\rm exp}(i \eta) |0_1,0_2\rangle$. It is clear that such a phase gate can be realized via Hamiltonian $H_M$ with the time-evolution unitary operator $exp(-iH_Mt)$ and the corresponding phase $\eta= \hbar \delta \xi_1^2\xi_2^2
t/|\Omega_1|^2|\Omega_2|^2$.
The resonant enhancement of $\chi^{(5)}$ and higher orders of nonlinearity may be achieved using additional $\Lambda$ sections connected to $M$ scheme, similar to the generalized $N$ scheme [@zubairy01pra]. Such $\chi^{(5)}$ nonlinearity may be so high that three-photon phase gates become feasible.
In conclusion, we have proposed a realization of media with resonantly enhanced Kerr nonlinearity where one-photon resonant absorption is suppressed due to coherence effects. Such media have certain advantages over already existing schemes of coherent resonant nonlinearity enhancement and hold promise for the use in the creation of non-classical states of light and in the implementation of quantum computing algorithms.
The authors gratefully acknowledge the support from Air Force Research Laboratory (Rome, NY), DARPA-QuIST, the Office of Naval Research, and the TAMU Telecommunication and Informatics Task Force (TITF) initiative.
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See, for example, M. A. Nielsen and I. L. Chuang, *Quantum Computation and Quantum Information*, (Cambridge University Press, Cambridge 2000).
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---
author:
- 'Hideo <span style="font-variant:small-caps;">Yoshioka</span>$^1$[^1], Hitoshi <span style="font-variant:small-caps;">Seo</span>$^2$ and Hidetoshi <span style="font-variant:small-caps;">Fukuyama</span>$^3$'
---
Most of the conducting molecular crystals are realized by combining two kinds of molecules, $A$ and $B$, with commensurate composition ratios. Typical examples are the well-studied 2:1 compounds, $A_2 B$, which show a variety of phases such as Mott insulator, charge order, superconductivity and so on. [@Ishiguro-Yamaji-Saito; @ChemRev] In the compounds, the molecule $B$ is usually fully ionized either as $-1$ or $+1$ to form a closed shell, and as a consequence the energy band formed by HOMO or LUMO of $A$ is quarter-filled as a whole in terms of holes or electrons, respectively.
Recently, molecular conductors with incommensurate (IC) composition ratios close to 2:1 have been synthesized based on new donor molecules. [@Takimiya; @Kawamoto; @Kawamoto-5] (MDT-TSF)$X_n$ and (MDT-ST)$X_n$ ($X$ = I$_3$, AuI$_2$ or IBr$_2$, $n$ $\simeq$ 0.42 – 0.45) show metallic behavior and undergo superconducting transition at about $T_c$ = 4 K at ambient pressure. [@Takimiya; @Kawamoto] In contrast, in (MDT-TS)(AuI$_2$)$_{0.441}$ a metal-insulator (MI) crossover occurs where the temperature dependence of resistivity displays a minimum at $T_\rho$ = 85 K. [@Kawamoto-5] In addition, an antiferromagnetic transition takes place at $T_{\rm N}$ = 50 K. When pressure is applied to this compound, $T_\rho$ decreases and the superconducting phase appears above $P_c$ = 10.5 kbar ($T_c$ = 3 K). All these compounds are isostructural with alternating donor and anion layers. Since the anions are fully ionized as $X^{-}$ as in the 2:1 compounds, [@Takimiya] the electronic properties can be attributed to the donors with the IC band-filling slightly larger than 3/4 for the HOMO bands. The extended H$\ddot{\rm u}$ckel scheme predicts two-dimensional (2D) Fermi surfaces which are similar to each other. [@Kawamoto; @Kawamoto-5] It should be noticed that the anions in these compounds are not randomly distributed in the layers, but are found by the X-ray scattering experiments to form regular IC lattices with a different periodicity from the donors. [@Takimiya; @Kawamoto]
The metallic state observed in these compounds is naturally expected from the IC band-filling since the system would avoid insulating states due to strong correlation such as Mott insulator or charge order. On the other hand, it is difficult to understand the strong-coupling nature of the insulating ground state in (MDT-TS)(AuI$_2$)$_{0.441}$, deduced from $T_\rho \neq T_{\rm N}$, which is to be explored in this Letter; if it is the weak-coupling spin-density-wave state due to the nesting of the Fermi surface, $T_\rho = T_{\rm N}$ would be expected. [@Ishiguro-Yamaji-Saito]
We consider a one-dimensional (1D) model in order to capture the essence of the MI transition in a more controlled way than considering a 2D model relevant to experiments. Our 1D model consists of $N_L$ donor molecules coupled with $N$ anions both forming regular lattices, as shown in the inset of Fig. \[fig:band\]. The donors are modeled by the 1D extended Hubbard model, known to be relevant for typical 2:1 systems, [@Seo-Hotta-Fukuyama; @Yoshioka] and the small potential from the anions is added, which is crucial for the insulating state to appear. The Hamiltonian is written as follows, $$\begin{aligned}
{\cal H} &=& -t \sum_{j,s}(c_{j+1,s}^\dagger c_{j,s} + h.c.) +
\frac{U}{2} \sum_{j,s} n_{j,s} n_{j,-s} \nonumber \\
&+& V \sum_{j} n_j n_{j+1} +
\sum_{j} v_j n_j,
\label{eqn:H} \end{aligned}$$ where $t$, $U$ and $V$ are respectively the transfer energy between the nearest-neighbor donor sites, the on-site repulsive interaction and the nearest-neighbor repulsion; the creation operator at the $j$-th site with spin $s$=$\pm$ is denoted as $c^\dagger_{j,s}$, $n_{j,s} = c^\dagger_{j,s} c_{j,s}$ and $n_j = \sum_s n_{j,s}$. Since the fully ionized anions form a regular lattice, the anion potential at the $j$-th site, $v_j$, can be expressed as $v_j = N_L^{-1} \sum_{m=-\infty}^{\infty} v(mQ) \e^{\im m Q j a}$ where $Q = 4k_F$, $k_F = \pi n/(2a)$ is the Fermi wavenumber, $n=N/N_L$ is the carrier density (we take the hole picture in the following) in the donor chain and $a$ is the spacing between donor sites. In the following, we consider only the relevant $\pm Q$ component of the potential, $v(\pm Q) \equiv v_0 \e^{\pm \im \chi}$. This can lead to a gap, $2 v_0$, at $\pm 2 \kf$ in the non-interacting band, which we assume to be small compared to the band width. Then the system becomes effectively half-filled in reference to the IC lattices as is seen in Fig.1.
![Energy dispersion in the presence of the anion potential $v_0 = 0.1t$ where the occupied one-particle states are expressed by the thick curve. The figure is written in the case of $n=0.33$ to clarify the characteristics of the present model. Inset: a schematic representation of our model. []{data-label="fig:band"}](chain-II.eps){width="7.0truecm"}
We derive an effective Hamiltonian for low energy scale in terms of phase variables following the just quarter-filling case.[@Yoshioka] To the lowest order of the normalized anion potential $\delta \equiv v_0/\left\{ \epsilon(3k_F) - \epsilon(k_F) \right\}$ with $\epsilon(K) = -2t \cos
Ka$, the phase Hamiltonian is obtained as ${\cal H}_{\rm eff} = {\cal H}_{\rho} + {\cal H}_{\sigma} + {\cal
H}_{\rm \rho \sigma}$. Here ${\cal H}_{\rho}$, ${\cal H}_{\sigma}$ and ${\cal H}_{\rho
\sigma}$ are respectively the charge part, the spin part and the term mixing both degrees of freedom. The spin part, ${\cal H}_{\sigma}$ has the same form as that of the 1D Hubbard model, so the spin excitation becomes gapless. [@Emery] The term ${\cal H}_{\rho \sigma}$ is expressed by the product of the non-linear terms seen in ${\cal H}_\rho$ and ${\cal H}_\sigma$, and then has a larger scaling dimension. Hence we can neglect it [@noteTsuchiizu-Furusaki]. Therefore the properties of the charge degree of freedom are determined by ${\cal H}_{\rho}$, expressed as $$\begin{aligned}
{\cal H}_\rho &=& \frac{v_\rho}{4\pi} \int d x
\left\{ \frac{1}{K_\rho} (\partial_x \theta_\rho)^2 + K_\rho (\partial_x \phi_\rho)^2 \right\} \nonumber \\
&+& \frac{g_{3 \bot}}{(\pi \alpha)^2} \int dx \cos (2 \theta_\rho + 3 \chi) \nonumber \\
&+& \frac{g''_{3 \bot}}{(\pi \alpha)^2} \int dx \cos (2 \theta_\rho + 5 \chi - q_0 x) \nonumber \\
&+& \frac{g_{1/4}}{2 (\pi \alpha)^2} \int dx \cos (4 \theta_\rho + 8 \chi - q_0 x).
\label{eqn:phaseH}\end{aligned}$$ Here $q_0 = 2 \pi/a - 8\kf = 2 \pi (1-2n)/a$ is the misfit parameter, $v = 2 ta \sin \kf a$, $v_\rho = v \sqrt{B_\rho A_\rho}$ and $ K_\rho = \sqrt{B_\rho/A_\rho}$ with $A_\rho = 1+ (g_{4||} + g_{4\bot} + g_{2||} + g_{2\bot} -
g_{1||})/(\pi v)$ and $B_\rho = 1 + (g_{4||} + g_{4\bot} - g_{2||} -
g_{2\bot} + g_{1||})/(\pi v)$. $\alpha^{-1}$ is the ultra-violet cut-off ($\alpha \sim a$). The interaction parameters are written as $$\begin{aligned}
% A_\rho &=& 1 + \frac{g_{4||} + g_{4\bot} + g_{2||} + g_{2\bot} - g_{1||}}{\pi v} \\
%
% B_\rho &=& 1 + \frac{g_{4||} + g_{4\bot} - g_{2||} - g_{2\bot} + g_{1||}}{\pi v} \\
%
% g_{1\bot} &=& \frac{Ua}{2} + Va \cos 2 k_F a - 4 D_1 \left(\frac{Ua}{2} + Va \cos 2 k_F a\right)
%\left(\frac{Ua}{2} + Va \cos 4 k_F a\right) \\
g_{1||} &=& Va \cos 2 k_F a \nonumber \\ &-& 4 D_1 \left(Va \cos 2 k_F a\right) \left(Va \cos 4 k_F a\right), \\
%
g_{2\bot} &=& \frac{Ua}{2} + Va - 2 D_1 \nonumber \\
& & \hspace*{-5em} \times
\left\{
\left( \frac{Ua}{2} + Va \cos 2 k_F a \right)^2
+ \left( \frac{Ua}{2} + Va \cos 4 k_F a \right)^2
\right\}, \nonumber \\ && \\
g_{2||} &=& Va \nonumber \\ &-& 2 D_1
\left\{
\left( Va \cos 2 k_F a \right)^2 + \left( Va \cos 4 k_F a \right)^2
\right\}, \\
%
g_{3 \bot} &=& -4 \delta (\frac{Ua}{2} + Va \cos 2 k_F a ) \nonumber \\
&+& 4 \delta D_2 (\frac{Ua}{2} + Va \cos 4 k_F a) \nonumber \\
& & \times \left(Ua + Va \cos 2 k_F a + Va \cos 6k_F a \right), \\
%g_{3 ||} &=& -4 \delta Va \cos 2 k_F a + 4 \delta D_2 (Va \cos 4 k_F a) (Va\cos 2 k_F a + Va\cos 6k_F a) \\
%
g''_{3 \bot} &=& - 4 \delta D_2 (\frac{Ua}{2} + Va \cos 4 k_F a ) \nonumber \\ && \times (Ua + 2 Va \cos 2 k_F a ), \\
%g''_{3 ||} &=& - 4 \delta D_2 ( Va \cos 4 k_F a ) ( 2 Va \cos 2 k_F a ) \\
%
g_{4 \bot} &=& \frac{Ua}{2} + Va - 2 D_2 \left( \frac{Ua}{2} + Va \cos 4 k_F a \right)^2, \\
g_{4 ||} &=& Va - 2 D_2 \left( Va \cos 4 k_F a \right)^2, \\
%
g_{1/4} &=& \frac{X}{2 (\pi \alpha)^2 (\epsilon(3k_F) - \epsilon(k_F))^2}, \\
X &=& 2 \Big\{
(2 Va \cos 2 k_F a)^2 ( Ua + 2 Va \cos 4 k_F a) \nonumber \\
&+& (Ua + 2Va \cos 2 k_F a)^2 (Ua + 4Va \cos 4k_F a) \nonumber \\
&+& (2Va \cos 2k_F a) (Ua + 2Va \cos 2 k_F a) \nonumber \\
& & \times (Ua + Va \cos 2 k_F a + Va \cos 6 k_F a)
\Big\} \nonumber \\
&+& (Ua + 2Va \cos 2 k_F a)^2 \nonumber \\ & & \times (Va \cos 2 k_F a + Va \cos 6 k_F a), \end{aligned}$$ with $$\begin{aligned}
%
D_1 &=& % \frac{1}{L} \sum_{-k_F < q < k_F} \frac{1}{2(E_+(q) - E_-(0))}
\frac{1}{4 \pi v(3k_F)} \nonumber \\
&& \times \ln \frac{\epsilon(3k_F) - \epsilon(k_F) + v(3k_F)k_F}{\epsilon(3k_F) - \epsilon(k_F)- v(3k_F)k_F}, \\
D_2 &=& %\frac{1}{L} \sum_{-k_F < q < k_F} \frac{1}{E_+(q) + E_+(-q) -2 E_-(0)} \simeq
\frac{1}{4\pi} \frac{2 k_F}{\epsilon(3k_F) - \epsilon(k_F)},\end{aligned}$$ where $v(3k_F) = 2ta \sin 3k_F a$.
In eq.(2), there are three non-linear terms. First, the half-filling Umklapp term, $g_{3 \bot}$, is generated by the anion potential $\delta$ because the band is effectively half-filled as seen in Fig.1. This can lead to a Mott insulator, as we will later show explicitly. We call the state as [*IC Mott insulator*]{} since it has a periodicity not matching with the donors but with the anions. However, it is not trivial whether this IC Mott insulator can be realized, and if so, in which condition it is stabilized, in contrast to the half-filled Hubbard model where infinitesimal on-site repulsion stabilizes the Mott insulator. [@Emery] This is because of the presence of the other two non-linear terms in eq.(2), the “quarter-filling" Umklapp term, $g_{1/4}$, with the misfit which is present even without the anions owing to the proximity to a quarter filling on one hand, and the $g''_{3 \bot}$ term, the combination of both commensurabilities of the donors and the anions on the other hand.
In the present calculation, it is crucial to fix the carrier number at the value determined by the anion density. However, it is to be noted that if one evaluates the quantity based on eq.(2), it results in a deviation of the carrier number from the value in the non-interacting case, $\Delta N_e = (1/L\pi) \int \d x \langle \partial_x \theta_\rho
\rangle$, as $$\begin{aligned}
\Delta N_e / L &=& \frac{2 K_\rho G_{3 \bot}^{''2}}{\pi \alpha} \int_{\alpha}^\infty
\frac{ \d r}{\alpha} \left( \frac{r}{\alpha}\right)^{2 - 4 K_\rho} J_1 (q_0 r) \nonumber \\
& & \hspace*{-5em} + \frac{4 K_\rho G_{1/4}^{2}}{\pi \alpha} \int_{\alpha}^\infty
\frac{ \d r}{\alpha} \left( \frac{r}{\alpha}\right)^{2 - 16 K_\rho} J_1 (q_0 r) \neq 0, \end{aligned}$$ where $G''_{3 \bot} = g''_{3 \bot}/(\pi v_\rho)$, $G_{1/4} = g_{1/4}/(2 \pi v_\rho)$, and $J_n(x)$ is the the Bessel function of the first kind. The origin of the deviation is the existence of the misfit parameter. [@Mori-Fukuyama-Imada] Therefore, we must add the term, $- (\mu/\pi) \int \d x
\partial_x \theta_\rho$ and keep $\Delta N_e$ to zero. To the lowest order of $G''_{3 \bot}$ and $G_{1/4}$, the chemical potential $\mu$ is given as, $$\begin{aligned}
q_\mu \al
&=& - 4 K_\rho G_{3 \bot}''^2 \int_\al^\infty \frac{\d r}{\al} (\frac{r}{\al})^{2 - 4 K_\rho} J_1 (q_0 \al) \nonumber \\
&&- 8 K_\rho G_{1/4}^2 \int_\al^\infty \frac{\d r}{\al} (\frac{r}{\al})^{2 - 16 K_\rho} J_1 (q_0 \al), \end{aligned}$$ where $q_\mu = 4 K_\rho \mu / v_\rho$.
To determine the low energy behavior of this effective Hamiltonian, we derive the renormalization group (RG) equations by rewriting the action $S_\rho$ corresponding to the Hamiltonian, ${\cal H}_{\rho} - (\mu/\pi) \int \d x \partial_x \theta_\rho$, as $$\begin{aligned}
S_\rho &=& \frac{1}{4 \pi K_\rho} \int \d^2 \vr
\left\{ (\delx \ti{\theta}_\rho)^2 + (\dely \ti{\theta}_\rho)^2 \right\} \nonumber \\
&+& \frac{G_{3 \bot}}{\pi \al^2} \int \d^2 \vr \cos \left\{ 2 \ti{\theta}_\rho + 3 \chi - (q_{1/4} - q_3'')x \right\} \nonumber \\
&+& \frac{G''_{3 \bot}}{\pi \al^2} \int \d^2 \vr \cos \left\{ 2 \ti{\theta}_\rho + 5 \chi - q_3''x \right\} \nonumber \\
&+& \frac{G_{1/4}}{\pi \al^2} \int \d^2 \vr \cos \left\{ 4 \ti{\theta}_\rho + 8 \chi - q_{1/4}x \right\}, \end{aligned}$$ where $\ti{\theta}_\rho = \theta_\rho - q_\mu x/2$, $G_{3 \bot}=g_{3
\bot}/(\pi v_\rho)$, $q_{1/4} = q_0 - 2
q_\mu$ and $q''_{3} = q_0 - q_\mu$. The condition, $\Delta N_e = 0$, leads to the following self-consistent equation, $$\begin{aligned}
q_\mu \al &=& -4 K_\rho G_{3 \bot}^2 \int_\al^\infty \frac{\d r}{\al} (\frac{r}{\al})^{2 - 4 K_\rho} \ti{J}_1 ((q_{1/4}-q_3'')\al) \nonumber \\
&&- 4 K_\rho G_{3 \bot}''^2 \int_\al^\infty \frac{\d r}{\al} (\frac{r}{\al})^{2 - 4 K_\rho} \ti{J}_1 (q_3''\al) \nonumber \\
&&- 8 K_\rho G_{1/4}^2 \int_\al^\infty \frac{\d r}{\al} (\frac{r}{\al})^{2 - 16 K_\rho} \ti{J}_1 (q_{1/4}\al), \end{aligned}$$ where $\ti{J}_1 (x) = {\rm sgn}(x) J_1 (|x|)$. Eqs.(16) and (17) lead to the following RG equations, $$\begin{aligned}
\dl K_\rho &=& - 8 K_\rho^2 G_{1/4}^2 J_0(|q_{1/4} \alpha|) \nonumber \\
&&- 2 K_\rho^2 G_{3\bot}^2 J_0(|q_{1/4} \alpha - q_3'' \alpha|) \nonumber \\
&&- 2 K_\rho^2 G_{3\bot}''^2 J_0(|q_3'' \alpha|), \\
%
\dl q_{1/4} \al &=& q_{1/4} \al - 16 K_\rho G_{1/4}^2 \ti{J}_1 (q_{1/4} \al) \nonumber \\
&& - 8 K_\rho G_{3 \bot}^2 \ti{J}_1 (q_{1/4} \alpha - q_3'' \alpha) \nonumber \\
&& - 8 K_\rho G_{3 \bot}''^2 \ti{J}_1 (q_{3}'' \alpha), \\
%
\dl q_{3}'' \al &=& q_{3}'' \al - 8 K_\rho G_{1/4}^2 \ti{J}_1 (q_{1/4} \al) \nonumber \\
&& - 4 K_\rho G_{3 \bot}^2 \ti{J}_1 (q_{1/4} \alpha - q_3'' \alpha) \nonumber \\
&& - 4 K_\rho G_{3 \bot}''^2 \ti{J}_1 (q_{3}'' \alpha), \\
%
\dl G_{3 \bot} &=& (2 - 2 K_\rho) G_{3 \bot} \nonumber \\
&& - G_{3 \bot}'' G_{1/4} J_0 (|(q_3'' \al + q_{1/4} \al)/2|), \\
%
\dl G_{3 \bot}'' &=& (2 - 2 K_\rho) G_{3 \bot}'' \nonumber \\ &&- G_{3 \bot} G_{1/4} J_0 (|q_{1/4} \al - q_3'' \al/2 |), \\
%
\dl G_{1/4} &=& (2 - 8 K_\rho) G_{1/4} \nonumber \\ && - G_{3 \bot} G_{3 \bot}'' J_0 (|q_3'' \al - q_{1/4} \al/2 |), \\
%
\dl q_\mu \al &=& q_\mu \al + 4 K_\rho G_{3 \bot}^2 \ti{J}_1 ((q_{1/4} - q_3'') \al) \nonumber \\
&&+ 4 K_\rho G_{3 \bot}''^2 \ti{J}_1 ( q''_3 \al) \nonumber \\
&&+ 8 K_\rho G_{1/4}^2 \ti{J}_1 ( q_{1/4} \al). \end{aligned}$$ Eqs.(18)-(23) are obtained for the condition of the action, eq.(16), being invariant under RG transformation, whereas eq.(24) is obtained from the condition of the chemical potential, eq.(17). From eqs.(19), (20) and (24), it is shown that the quantity $q_0$, whose dimension is $({\rm length})^{-1}$, is scaled as $q_0(l) = q_0 \e^l$. This shows the fact that the carrier number is indeed conserved without any effects from the interaction.
Typical flows of the RG equations are shown in Fig.2 for $U/t=3.0$ and $V/t=0.0$ where the carrier number is fixed as $n=0.441$ taken from the actual material (MDT-TS)(AuI$_2$)$_{0.441}$.
![ The solutions of the RG equations, $K_\rho$ and $(q_{1/4} - q_3'') \al $ ( the misfit parameter of the $G_{3 \bot}$-term ) for $U/t=3.0$, $V/t=0.0$ and $n=0.441$. The cases of $\delta = 0.001$ and $0.005$ are denoted by the solid and dotted curves, respectively. []{data-label="fig:typica-RG"}](flow-U3-V0-memo-mod-paper.eps){width="7.0truecm"}
In the case of $\delta = 0.005$, $K_\rho$ tends to zero implying that the ground state is an insulator, due to the commensurability in the half-filling, $G_{3\bot}$. This can be seen in the RG equations since the misfit parameter in the $G_{3\bot}$-term, $(q_{1/4} - q_3'') \al = - q_\mu \al$ vanishes (see Fig.2) and then $G_{3\bot}$ affects the renormalization of $K_\rho$ through eq.(18) while those in the $G_{1/4}$- and $G_{3 \bot}''$-terms, $q_{1/4}\al$ and $q_3''\al$, tend to $\infty$ and then these effects become negligible due to the oscillating behavior of the Bessel function. Hence the origin of this insulating state is nothing but the commensurate potential of the effective half-filling generated by the anion potential. Namely, the insulating state is indeed the IC Mott insulator.
On the other hand, in the case of smaller potential due to the anions, $\delta = 0.001$, a metallic state with finite value of $K_\rho$ is realized. Here in eq.(18) the effects of the $G_{3 \bot}$-, $G''_{3 \bot}$- and $G_{1/4}$-terms on $K_\rho$ disappear at the low energy since all the misfit parameters are divergent. This metallic state is [*not*]{} realized if we set $G_{1/4}$ to zero. Therefore we can state that the origin of the MI transition is the interplay between the different kinds of commensurabilities.
Next, we show ground state phase diagrams as a function of the model parameters. First, the phase diagram on the plane of $U/t$ and $\delta$ in the case of $V/t=0$ is shown in Fig.3.
![ The phase diagram on the plane of $\delta$ and $U/t$ in the case of $V/t=0$ and $n = 0.441$. []{data-label="fig:phase-V0"}](phase-dvsU-V0-mod-paper.eps){width="6.5truecm"}
![ The phase diagram on the plane of $\delta$ and $V/t$ in the case of $U/t=3.0$ and $n = 0.441$. []{data-label="fig:phase-U3"}](phase-dvsV-U3-mod-paper.eps){width="6.7truecm"}
Since the quantity $\delta$ is proportional to $v_0/t$, the transition from the metallic state to the IC Mott insulator occurs when the potential from the anions increases and/or the band width decreases. When $U$ $\to$ $\infty$, the present system can be mapped onto a non-interacting spinless Fermion system with the Fermi wavenumber doubled, as $2\kf$.[@Ovchinnikov] In this case the insulating state is realized by an infinitesimal $\delta$ because a gap opens at $\pm 2\kf$ (see Fig.1), consistent with Fig.3.
The role of the $G_{1/4}$-term on the MI transition becomes clearer when $V$ is varied. It is because the coupling constant $G_{1/4}$ changes its sign when $V$ increases.[@Yoshioka] We show the phase diagram on the $V/t$ - $\delta$ plane in Fig. \[fig:phase-U3\] for $U/t = 3.0$. At $V=V_c=0.838 t$ where $G_{1/4} = 0$, the IC Mott insulating state is realized for infinitesimal $\delta$. For $V >V_c$, the absolute value of $G_{1/4}$ increases again and results in a finite metallic region. Therefore, a re-entrant transition, metal $\to$ IC Mott insulator $\to$ metal, occurs when $V$ increases. Note that there is no qualitative difference between the metallic states in the two distinct regions.
Finally let us discuss the relevance of our results to the experiments. The difference of the ground state in metallic MDT-TSF and MDT-ST compounds and that in the MDT-TS compound undergoing MI crossover can be naturally understood as follows. The extended H$\ddot{\rm u}$ckel scheme provides transfer integrals, the band width, of the MDT-TSF families larger than that of the MDT-TS compound [@Kawamoto; @Kawamoto-5], which is consistent with our results that the decrease of the bandwidth lead to an MI transition, as seen in Fig.3. In our 1D model the spin degree of freedom is essentially that of the 1D Heisenberg model showing no magnetic order. However, in the IC Mott insulating state in the actual 2D material we generally expect that antiferromagnetic order appears at low temperature due to the three dimensionality, as in fact observed. [@Kawamoto-5] In this case, the magnetic ordered moment should be large, compared to, e.g., that of the spin-density-wave state due to the nesting of the Fermi surface.
In conclusion, we investigated the electronic state of the one-dimensional extended Hubbard model close to quarter-filling under an incommensurate anion potential. We found that a transition between the metallic state and an incommensurate Mott insulator can occur, whose origin is the interplay between the commensurability energy generated by the anion potential and that in the donor lattice. To the authors’ best knowledge this is the first theoretical study of a “Mott transition" generated by such interplay between different commensurabilities. It would be interesting to investigate the critical properties of this transition in the actual compounds and compare with the “usual" Mott transition seen in the typical 2D $A_2B$ molecular conductors, $\kappa$-(BEDT-TTF)$_2X$, which is recently attracting interests. [@Kagawa]
The authors would like to thank T. Kawamoto for sending them his preprint prior to publication. They also acknowledge G. Baskaran, M. Ogata, K. Kanoda, and J. Kishine for valuable discussions and comments. This work was supported by Grant-in-Aid for Scientific Research on Priority Area of Molecular Conductors (No.15073213) and Grant-in-Aid for Scientific Research (C) (No. 14540302 and 15540343) from MEXT.
[99]{} T. Ishiguro, K. Yamaji and G. Saito: [*Organic Superconductors*]{} (Springer-Verlag, Berlin, 1998) 2nd ed. For various recent reviews, . K. Takimiya : ; ; [*ibid.*]{} 3250. T. Kawamoto : ; ; ; . T. Kawamoto : preprint. H. Seo, C. Hotta and H. Fukuyama: . H. Yoshioka, M. Tsuchiizu and Y. Suzumura: ; . V. J. Emery: in [*Highly Conducting One-Dimensional Solids*]{}, ed. J. Devresse, R. Evrard, and V. Van Doren (Plenum, New York, 1979), p. 247. The spin-charge coupled term plays an essential role when the coefficients of the non-linear terms of both degree of freedom vanish (see M. Tsuchiizu and A. Furusaki: ), which does not happen in the present case. M. Mori, H. Fukuyama and M. Imada: . A.A. Ovchinnikov: ; M. Ogata and H. Shiba: . S. Lefebvre : ; F. Kagawa : .
[^1]: E-mail address: h-yoshi@cc.nara-wu.ac.jp
|
---
abstract: 'A uniaxial strain applied to graphene-like materials moves the Dirac nodes along the boundary of the Brillouin zone. An extreme case is the merging of the Dirac node positions to a single degenerate spectral node which gives rise to a new topological phase. Then isotropic Dirac nodes are replaced by a node with a linear behavior in one and a parabolic behavior in the other direction. This anisotropy influences substantially the optical properties. We propose a method to determine characteristic spectral and transport properties in black phosphorus layers which were recently studied by several groups with angle-resolved photoemission spectroscopy, and discuss how the transmittance, the reflectance and the optical absorption of this material can be tuned.In particular, we demonstrate that the transmittance of linearly polarized incident light varies from nearly 0% to almost 100% in the microwave and far-infrared regime.'
author:
- 'Phusit Nualpijit$^{1,2}$, Andreas Sinner$^2$, and Klaus Ziegler$^2$'
title: Tunable transmittance in anisotropic 2D materials
---
Introduction
============
Since the discovery of graphene \[\], transport properties of different semimetallic materials which crystallize on a hexagonal lattice and have Dirac-like electronic spectra, have been in the focus of intensive research. Because of the subtle issue of Klein tunneling, these materials turn out to have nearly the same conductivity in the broad wave length region ranging from the microwave to the visible spectrum \[\]. An intriguing further development is related to a continuous deformation of the hexagonal lattice, which can result in a topological transition of the spectrum. For the latter it is crucial that the six-fold isotropy of the hexagonal lattice is broken by the deformation. For example, changing the carbon bonds of graphene in one direction (as depicted on the left of Fig. \[fig:CondOm\]) changes the positions of the two Dirac nodes in the perpendicular direction in momentum space. This can even lead to their degeneracy in one point. Then the Dirac cones are also affected, since the resulting spectrum is linear only in the direction of the bond change but it becomes parabolic in the perpendicular direction. This case is topologically different from a single Dirac node, since the corresponding wave function has a zero winding number, contrary to the winding number $\pm 1$ of a single Dirac node. In this sense the system undergoes a topological transition, which has a substantial effect on electronic and optical properties. These ideas have a long history \[\]. More recently, a detailed experimental analysis of black phosphorus layers has revealed that the spectral properties can be tuned by doping \[\]. Using angle-resolved photoemission spectroscopy (ARPES), it was found that there exists a split pair of Dirac nodes that can be moved upon doping towards each other along the zigzag direction of the underlying honeycomb lattice. The detailed mechanism for the movement of the Dirac nodes in black phosphorus is more complex (cf. Ref. \[\]) but is based on the fundamental mechanism of breaking a discrete lattice isotropy, very similar to the case of an anisotropic honeycomb lattice. In the following we propose a method based on light transmittance and absorption which could provide an alternative to ARPES for the observation of the moving Dirac nodes in black phosphorus layers. This could serve as a novel analytic method as well as an application of the moving Dirac nodes for sensors that are sensitive to polarization. The intimate connection between electrical and optical properties in semimetals allows to study them together. Because of the nearly frequency independent conductivity of isotropic graphene its transparency is frequency independent too \[\]. For small uniaxial lattice deformations a variation of the transmittance with respect to the wave plane polarization has been reported in Refs. \[\]. The observed deviation from the isotropic case was small, though, and did not exceed 1%. To extend these studies, we will consider stronger lattice deformations in the regime, where the Dirac nodes are merging. In this case the optical transparency is much more affected and can reach nearly 100% in the microwave regime. It is caused by a divergent optical conductivity in the direction of the stronger bonds and a vanishing optical conductivity in the direction of the weaker bonds for low frequencies \[\]. This property opens a possibility for the development of a new type of single atom thick optical polarization filters.
Model
=====
The tight-binding Hamiltonian for electrons hopping between sites of a hexagonal lattice reads in momentum space $$\label{eq:TBH}
H = -\sum_{j=1}^{3}
\begin{pmatrix}
0 & t_j e^{i {a}_j \cdot k } \\
t_j e^{-i {a}_j \cdot k } & 0
\end{pmatrix},$$ where the positions of the nearest neighbors around an atom at the origin of coordinates are given by $a_1=a(0,-1)$ and $a_{2,3}=a(\pm \sqrt3,1)/2$; $t_j$ denote hopping energies in each direction, cf. Figure \[fig:CondOm\], and $a$ represents the distance between nearest neighbors. The spectrum of (\[eq:TBH\]) has two branches corresponding to positive and negative energies $$E = \pm |t_1 e^{i {a}_1 \cdot {k} } + t_2 e^{i {a}_2 \cdot {k} } + t_3 e^{i {a}_3 \cdot {k} }|.$$ For an isotropic lattice, i.e., in the case where all $t^{}_j$ are equal, both spectral branches touch each other at six corners of the hexagonal Brillouin zone, where they compose two Dirac nodes. An expansion in powers of small momentum deviations around these nodal points yields in leading order a linear spectrum. The situation changes if the isotropy is lifted, i.e., when the hopping integrals are different. Then the Dirac nodes can be continuously moved in momentum space \[\], which also changes the shape of the Brillouin zone, while keeping its area constant. In the following we will study the case where the hopping integrals $t^{}_2$ and $t^{}_3$ are kept fixed at the value of the isotropic lattice $t$, while $t^{}_1$ changes smoothly between $t$ and $2t$ \[\]. Then the nodes (corresponding to Dirac particles with different chirality) start moving in the momentum space towards each other: $$\label{eq:traj}
k^{}_x = \pm\frac{2}{\sqrt{3}} \arccos\left[\frac{t_1}{2t}\right],\;\; k^{}_y = \pm\frac{2\pi}{3}
\ .$$ At the particular value $t^{}_1 = 2t$ the Dirac nodes merge and give rise to an anisotropic spectrum with parabolic dispersion along the direction of the motion and linear perpendicular to it. The effective low-energy Hamiltonian then reads $$\label{eq:EffHam}
H = -\frac{k_x^2}{2m} \hat\sigma_x \pm c k_y \hat\sigma_y
\ ,$$ where $\hat\sigma_{x,y}$ denote Pauli matrices, $m=2/(3ta^2)$ and $c=3at$. The sign ambiguity in the second term is due to different chirality of merging Dirac nodes. The merging point is occupied simultaneously by both copies with opposite chiralities. In our subsequent calculation of the conductivity, which does not depend on the sign in Eq. (\[eq:EffHam\]), we take into account this degeneracy by an additional factor of two and assume the sign to be $+$. The effective Hamiltonian has the eigenvalues $$\epsilon_{k}= \pm ~\sqrt{\displaystyle \frac{k_x^4}{4m^2} + c^2 k_y^2},$$ and the corresponding normalized eigenfunctions $$\psi^{}_{\pm} = \mp \frac{e^{-i k \cdot r}}{\sqrt{2}\epsilon^{}_k}
\begin{bmatrix}
\displaystyle \frac{k_x^2}{2m} + i c k_y \\
\\
\mp\epsilon^{}_k
\end{bmatrix}.$$ The current operators $\displaystyle j_{\mu} = i[H,r_{\mu}]$, corresponding to the anisotropic Hamiltonian in Eq. (\[eq:EffHam\]), read: $$\label{eq:Currents}
j^{}_x = -\frac{k^{}_x}{m}\hat\sigma^{}_x, \;\;\; j^{}_2 = c\hat\sigma^{}_y.$$ Then the interband current matrix elements read: $$\begin{aligned}
\label{eq:MatrElts}
\langle\psi^{}_\pm|j^{}_x|\psi^{}_\mp\rangle &=& \mp i\frac{c}{m}\frac{k^{}_xk^{}_y}{\epsilon^{}_k},\\
\langle\psi^{}_\pm|j^{}_y|\psi^{}_\mp\rangle &=& \pm i\frac{c}{2m}\frac{k^{2}_x}{\epsilon^{}_k},\end{aligned}$$ which will be used to compute the conductivity.
{width="5cm"} {width="8cm"}
Optical conductivity
====================
To calculate the conductivity per valley and spin projection we use the Kubo formula $$\begin{aligned}
{\nonumber}\sigma^{}_{\mu\nu}(\omega) &=& 16i\sigma^{}_0 \sum_{\lambda,\lambda^\prime=\pm}\int\frac{d^2k}{(2\pi)^2} ~
\frac{f_{\beta}(\epsilon^{}_{k,\lambda^\prime})-f_{\beta}(\epsilon^{}_{k,\lambda})}{\epsilon^{}_{k,\lambda}-\epsilon^{}_{k,\lambda^\prime}} \\
\label{eq:kubo}
&&\times~\frac{\langle\psi^{}_\lambda|j^{}_{\mu}|\psi^{}_{\lambda^\prime}\rangle\langle\psi^{}_{\lambda^\prime}|j^{}_\nu|\psi^{}_\lambda\rangle}
{\epsilon^{}_{k,\lambda}-\epsilon^{}_{k,\lambda^\prime} + \omega - i0^+}, \end{aligned}$$ Here, $f_{\beta}(\epsilon)=(1 + \exp[\beta(\epsilon-\epsilon_F)])^{-1}$ denotes the Fermi function at inverse temperature $\beta=1/k_BT$ with Boltzmann constant $k_B$. The conductivity in Eq. (\[eq:kubo\]) is measured in units of universal dc conductivity per valley and spin projection $\sigma_{0} = 1/16 e^2/\hbar$. Because of the preserved time-reversal symmetry of the model, the non-diagonal elements of the conductivity tensor with $\mu\neq\nu$ are zero. In the zero-temperature limit the diagonal elements of the conductivity tensor read $$\label{eq:kubo_t0}
\sigma^{}_{\mu\mu}(\omega) = i\int\frac{d^2k}{(2\pi)^2} \left[ \frac{\Gamma^{}_{\mu\mu}}{2\epsilon_k + \omega+i0^+} -\frac{\Gamma^{}_{\mu\mu}}{2\epsilon_k - \omega - i0^+}\right].$$ Because of the anisotropy, the conductivities in $x$ and $y$ directions are different. We obtain the following matrix elements of the current-current correlator: $$\begin{aligned}
\Gamma^{}_{xx} &=& 8\sigma^{}_0 \left(\frac{c}{m}\right)^2\frac{k_x^2 k_y^2}{\epsilon_k^3} ,\\
\Gamma^{}_{yy} &=& 8\sigma^{}_0 \left(\frac{c}{2m}\right)^2 \frac{k_x^4}{\epsilon_k^3}.\end{aligned}$$ Rewriting momentum integrals as $\int_{-\infty}^{\infty} dk^{}_i \to 2 \int_{0}^{\infty} dk^{}_i$ (which is possible since the integrand function is even under mirroring $k^{}_i\to-k^{}_i$), introducing new integration variables $u=k^2_x/2m$ and $v=ck^{}_y$, and changing into polar coordinates we arrive at $$\begin{aligned}
{\nonumber}\sigma_{xx}(\omega) &=& \frac{i\gamma\sigma^{}_0}{(2\pi)^2} \int_0^{\pi/2}d\phi~\cos^{\frac{1}{2}}\phi~\sin^2\phi \\
&\times&\int^{x^{}_c}_0 dx~\sqrt{x}\left[\frac{1}{x - 1 + i0^+} - \frac{1}{x + 1 - i0^+}\right],
\hspace{0.5cm}\\
{\nonumber}\sigma^{}_{yy}(\omega) &=& \frac{i\gamma\sigma^{}_0}{(2\pi)^2} \int_0^{\pi/2}d\phi~\cos^{\frac{3}{2}}\phi\\
&\times&\int^{x^{}_c}_0\frac{dx}{\sqrt{x}}~\left[\frac{1}{x-1+i0^+} + \frac{1}{x+1-i0^+}\right],
\hspace{0.5cm}\end{aligned}$$ where $\gamma=\sqrt{\omega/mc^2}$ and $x=\sqrt{2\epsilon/\omega}$. Angular parts represent standard elliptic integrals and can be found in the literature: $$\int_0^{\pi/2}d\phi~{\cos^{\frac{1}{2}}\phi}~\sin^2\phi \sim \frac{1}{2},\;\; \int_0^{\pi/2}d\phi~\cos^{\frac{3}{2}}\phi \sim \frac{7}{8}.$$ Separating real and imaginary parts of the fractions under the integral using the Dirac identity $$\frac{1}{x \pm 1 \pm i 0^+} = {\cal P}\frac{1}{x \pm 1} \mp i\pi\delta(x \pm 1),$$ where $\cal P$ denotes the operator of the principal part integration. We ultimately obtain $$\label{eq:ReSigma}
{\rm Re}~\sigma^{}_{xx}(\omega) \sim \frac{2\gamma\sigma^{}_{0}}{\pi} , \;\;
{\rm Re}~\sigma_{yy}(\omega) \sim \frac{7\sigma^{}_{0}}{2 \pi \gamma} ,$$ and $$\begin{aligned}
\label{eq:ImSigmaXX}
{\rm Im}~\sigma^{}_{xx}(\omega) &\sim& \frac{2\gamma\sigma_{0}}{\pi^2}\left[\ln\left|\frac{x^{}_c-1}{x^{}_c+1}\right| + 2\arctan{x^{}_c} \right],
\\
\label{eq:ImSigmaYY}
{\rm Im}~\sigma^{}_{yy}(\omega) &\sim& \frac{7\sigma_{0}}{2\pi^2\gamma}\left[\ln\left|\frac{x^{}_c-1}{x^{}_c+1}\right| - 2\arctan{x^{}_c}\right],\end{aligned}$$ as the real and the imaginary part of the conductivity. $x^{}_c=\sqrt{2\epsilon^{}_{c}/\omega}$ denotes the effective dimensionless band width. At energies $\omega\ll\epsilon^{}_{c}$ we can approximate the expression in the square brackets by $\pi/2$ and obtain the same conductivity amplitudes for both real and imaginary part in each direction. Remarkably, in the low energy regime both conductivities turn out to be functions of the variable $\gamma=\sqrt{\omega/mc^2}$, taken to mutually inverse powers, though. Hence, the product of both conductivities does not depend on the frequency, nor on the parameters of the anisotropic Hamiltonian of Eq. (\[eq:EffHam\]). It is related to the universal conductivity per spin projection $\bar\sigma^{}_0 = 2\sigma^{}_0$, Ref. \[\] $$\label{eq:const}
2\lim_{\omega\to 0}\sqrt{|\sigma_{xx}(\omega)| ~|\sigma^{}_{yy}(\omega)|} \sim 1.2 \bar\sigma_0,$$ where the valley degeneracy is taken into account by the factor $2$. The existence of such a relation is dictated by the current conservation condition which must hold for any smooth lattice deformations which respects parity, time reversal and charge conjugation symmetries. The deviation from unity is due to the low-energy approximation, while numerical evaluations with the full tight-binding spectrum gives a value much closer to unity.
{width="8cm"} {width="8cm"}
Optical properties of strained graphene-like materials
======================================================
To study optical properties we must consider the coupling of the electrons in the hexagonal lattice and an external monochromatic electromagnetic field with frequency $\omega$. For this case we have to solve the classical Maxwell equations for the electromagnetic field together with Ohm’s law due to conductivity obtained above. We consider linearly polarized light propagating perpendicular to the layer along $z$-direction and the interaction with electrons in the graphene-like material at $z=0$. $\phi$ is the angle between the polarization plane and $x$-axis. Then the Maxwell equation of the electromagnetic field reads \[\] $$\label{eq:Maxwell}
\frac{\partial^2 \textbf{E}}{\partial z^2}= \epsilon \frac{\omega^2}{c^2_0}\textbf{E}-4\pi i
\frac{\omega}{c^2_0} \delta(z) ~\textbf{j}$$ with the electronic current in the lattice and $c^{}_0$ denoting the speed of light in vacuum. A unique solution of this equation is obtained if we assume that the tangential component of electric field is continuous at $z=0$: $$\begin{aligned}
\label{sum1}
\tilde{\textbf{E}}^{}_i+\tilde{\textbf{E}}^{}_r = \tilde{\textbf{E}}^{}_t,\end{aligned}$$ with $i,r$ and $t$ denoting [*incident*]{}, [*reflected*]{} and [*transmitted*]{}, respectively, and $\tilde{\textbf E}$ denoting the amplitude of the corresponding field component, along with a discontinuity of its spatial derivative, cf. Appendix $$\label{eq:discont}
-iq \tilde{\textbf{E}}_t + iq (\tilde{\textbf{E}}_i-\tilde{\textbf{E}}_r) =4\pi i \frac{\omega}{c^2_0} ~ \textbf{j}.$$ Exploiting Ohm’s law $\textbf{j}=\sigma \tilde{\textbf{E}}$ and the dispersion relation $\omega=c^{}_0 q$ for the external electromagnetic field, we finally obtain a relationship between the transmitted and the incident electric field: $$\tilde{E}_t^\mu = \frac{\displaystyle \tilde{E}_i^\mu}{\displaystyle 1 + \frac{\pi\alpha}{2} f^{}_{\mu\mu}} ,\;\;\; \mu = x,y
\label{tr.field}
\ ,$$ with $\tilde{E}_{i}^x=|\tilde{E}_{i}|\cos\phi$, $\tilde{E}_{i,t}^y=|\tilde{E}_{i,t}|\sin\phi$. Here $\alpha=e^2/\hbar c^{}_0\sim1/137$ is the fine structure constant, $f^{}_{\mu\mu}=\sigma^{}_{\mu\mu}/\sigma^{}_0$ and the valley and spin degeneracy is taken into account by the factor 4. The transmittance is defined as the ratio of the intensities of the transmitted and the incident field $$\begin{aligned}
{\nonumber}T &\equiv& \frac{I_t}{I_i} = \frac{|E_t^{x}|^2 + |E_t^{y}|^2}{|E_i^{x}|^2 + |E_i^{y}|^2} \\
&&=T_{x} \cos^2{\phi}+T_{y} \sin^2{\phi}\ ,\end{aligned}$$ with $$T^{}_{\mu} = \left|1 + \frac{\pi\alpha}{2}f^{}_{\mu\mu}\right|^{-2}.$$ The reflectance reads according to Eqs. (\[sum1\]) and (\[tr.field\]) $$\begin{aligned}
{\nonumber}R &\equiv& \frac{I_r}{I_i} = \frac{|\tilde{E}_t^{x}-\tilde{E}_i^x|^2 + |\tilde{E}_t^{y}-\tilde{E}_i^y|^2}{|\tilde{E}_i^{x}|^2 + |\tilde{E}_i^{y}|^2} \\
\label{reflectance}
&&= R_{x} \cos^2{\phi}+R_{y} \sin^2{\phi} ,\end{aligned}$$ with $$R^{}_{\mu} = \left(\frac{\pi\alpha}{2}\right)^2|f^{}_{\mu\mu}|^2T_\mu.$$ Both, transmittance and reflectance are plotted versus the polarization angle in Figure \[fig:ReInc\]. Then the ratio of the absorbed and incident intensity $A=I_a/I_i$ is $$\begin{aligned}
{\nonumber}A &=& 1 - R - T \\
\label{eq:abs}
& = & 1-\frac{1+|z_x|^2}{|1+z_x|^2}\cos^2{\phi}-\frac{1+|z_y|^2}{|1+z_y|^2}\sin^2{\phi},\end{aligned}$$ where $z_\mu=\frac{\pi\alpha}{2}f^{}_{\mu\mu}$, which is plotted in Figure \[fig:AbFreqInc\] as function of the polarization angle at fixed light frequency and in Figure \[fig:AbFreqInc\] as function of the light frequency at fixed polarization angle.
Discussion and conclusions
==========================
In sharp contrast to the case of the isotropic hexagonal lattice, lifting the isotropy by a smooth lattice deformation changes substantially the transport properties. Essential for our discussion is that the optical conductivity, which is nearly constant with respect to the frequency of the incident field (at least in the low energy regime) in the isotropic lattice, becomes strongly frequency dependent. In the case studied in this paper, where the armchair oriented hexagonal lattice is compressed along the $y$-axis, the corresponding hopping amplitude $t^{}_1$ increases, as visualized Figure \[fig:CondOm\]. This enhances the conductivity parallel to $y$-axis and suppresses the transport in the perpendicular direction. In the particular case, where $t^{}_1$ becomes twice the hopping energy of isotropic lattice, the dc conductivity in $x$-direction is suppressed and the optical conductivity vanishes like $\sim\gamma$. On the other hand, the dc conductivity in $y$-direction diverges for small $\omega$ like $\sim 1/\gamma$ (cf. Figure \[fig:CondOm\]). Thus, a finite lattice anisotropy ($t_1=2t_{2,3}$) leads to a very strong transport anisotropy at small frequencies. This reflects the degeneracy of the Dirac nodes, which is a topological effect in the spectrum.
{width="8cm"} {width="8cm"}
The strong transport anisotropy has remarkable consequences for the optical properties too, in which the transparency and the reflectivity of the system depend on the orientation of the polarization of the incident light. For instance, if the polarization is parallel to the $x$-axis, the system becomes almost completely transparent in the infrared regime, Figure \[fig:ReInc\]. This behavior is typical for insulators, which is supported by the fact that $\sigma^{}_{xx}\sim 0$ in the infrared. On the other hand, $\sigma^{}_{yy}$ grows towards smaller frequencies and makes the system increasingly more reflective for a polarization parallel to the $y$-axis, cf. Figure \[fig:ReInc\]. Such a behavior is known for conventional metals. Our calculations indicate that much larger transparency oscillations can occur than those reported in Refs. \[\] due to strong deformations with degenerate Dirac nodes.
Another experimentally accessible quantity is the absorption, which is related to the electronic currents induced in the sample. This observable quantity has a non-monotonous behavior as a function of both polarization angle and frequency, cf. Figure \[fig:AbFreqInc\]. In particular, the absorption has a maximum for all frequencies if the polarization is parallel to the $y$-axis. However, in the dc limit the absorption goes to zero for any angle $\phi$, which is in sharp contrast to the isotropic case, where it is given by the constant $\pi\alpha$. This limit is visible in Figure \[fig:AbFreqInc\]. With growing frequency, the absorption initially exhibits a steep growth and reaches a maximum at some specific frequency value.
One could employ the optical method as an alternative to the ARPES studies \[\] of the spectral anisotropy in black phosphorus layers. This would provide a simple and flexible approach either by measuring the surface reflectance, as described in Eq. (\[reflectance\]) and visualized in Figure \[fig:ReInc\], or by measuring the absorption as presented in Figure \[fig:AbFreqInc\]. This method could shed some additional light on the peculiar spectral properties found by ARPES. Moreover, it provides not only information regarding the spectral properties but also information in terms of transport.
Acknowledgments {#acknowledgments .unnumbered}
===============
P.N. acknowledges the financial support by a DPST scholarship of The Institute for the Promotion of Teaching, Science, and Technology (IPST), Thailand. A.S. and K.Z. were supported by a grant of the Julian Schwinger Foundation for Physical Research.
Derivation of the discontinuity condition Eq. (\[eq:discont\]) {#app:details}
==============================================================
Here we obtain the discontinuity condition Eq. (\[eq:discont\]) from the wave equation Eq. (\[eq:Maxwell\]). It is obtained with the plane wave ansatz $$\label{eq:PlaneWave}
\textbf{E}^{}_{i,t} = \tilde{\textbf E}^{}_{i,t}e^{iqz},\;\;\;\textbf{E}^{}_{r} = \tilde{\textbf E}^{}_{r}e^{-iqz}.$$ For this we integrate Eq. (\[eq:Maxwell\]) along $z$–axis within an infinitesimally thin region around $z=0$ $$\lim_{\lambda\to 0} \int^{+\lambda}_{-\lambda}dz\left\{\frac{\partial^2\textbf{E}}{\partial z^2} - \epsilon\frac{\omega^2}{c^2_0}\textbf{E} + 4\pi i\frac{\omega}{c^2_0}\delta(z)\textbf{j} \right\} = 0.$$ Separating the integral in $z<0$ and $z>0$ parts and noticing $$\textbf{E}(z\leqslant0) = \textbf{E}^{}_i + \textbf{E}^{}_{t}, \;\;\; \textbf{E}(z\geqslant0) = \textbf{E}^{}_t,$$ and Eq. (\[sum1\]) precisely at $z=0$ we get $$\begin{aligned}
{\nonumber}\lim_{\lambda\to0} \left\{
\epsilon\frac{\omega^2}{c^2_0}\int^0_{-\lambda} dz~\left({\textbf{E}}^{}_i+{\textbf{E}}^{}_r\right)
-\epsilon\frac{\omega^2}{c^2_0}\int_0^\lambda dz~{\textbf{E}}^{}_t
\right.
\hspace{2mm}
\\
\left.+
\left.\frac{\partial}{\partial z}{\textbf{E}}^{}_t\right|^{}_{z=\lambda} -
\left.\frac{\partial}{\partial z}\left({\textbf{E}}^{}_i+{\textbf{E}}^{}_r\right)\right|^{}_{z=\lambda}
\right\}
= -4\pi i\frac{\omega}{c^2_0}\textbf{j}(0).
\hspace{2mm}\end{aligned}$$ With the plane wave ansatz Eq. (\[eq:PlaneWave\]), terms in the first line of this equation disappear individually in the limit $\lambda\to0$. Remaining terms form the condition Eq. (\[eq:discont\]).
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---
abstract: 'We evaluate the branching ratios for the decays $K\rightarrow \pi\pi\pi e\nu$ at leading order in chiral perturbation theory and give an isospin relation for the decay rates.'
---
hep-ph/9410368\
[$K_{e5}$ decays in chiral perturbation theory]{}\
Stefan Blaser\
Institut für theoretische Physik, Universität Bern\
Sidlerstrasse 5, CH-3012 Bern\
blaser@butp.unibe.ch\
October 1994
[**1**]{}. We discuss the $K_{e5}$ decays $$\begin{aligned}
{K^{+}}&\longrightarrow& {\pi^{+}}{\pi^{-}}{\pi^{0}}e^{+} \nu_{e} ,
\\
{K^{+}}&\longrightarrow& {\pi^{0}}{\pi^{0}}{\pi^{0}}e^{+} \nu_{e} ,
\\
{K^{0}}&\longrightarrow& {\pi^{0}}{\pi^{0}}{\pi^{-}}e^{+} \nu_{e} ,
\\
{K^{0}}&\longrightarrow& {\pi^{+}}{\pi^{-}}{\pi^{-}}e^{+} \nu_{e}\end{aligned}$$ in the framework of chiral perturbation theory (CHPT) [@weinberg; @gasser84]. For low momenta relevant in the present case, the transition amplitude for $ K \rightarrow \pi \pi \pi e^{+} \nu_{e} $ reduces in the standard model to the current times current form $$\label{gl1}
T = \frac{{G_{\mbox{\scriptsize{F}}}}}{\sqrt{2}} V^{\ast}_{us}
\overline{u}(p_{\nu}) \gamma_{\mu} (1-\gamma_{5})
v(p_{e}) (V^{\mu}-A^{\mu}),$$ where $$\label{gl2}
V^{\mu}-A^{\mu} = {\big<}\pi(p_{1}) \pi(p_{2}) \pi(p_{3}) \mbox{ out}
{\big|}{\overline{s}}\gamma^{\mu} (1-\gamma_{5}) u
{\big|}K(p) {\big>}.$$ [**2**]{}. To calculate the hadronic matrix elements $V^\mu$ and $A^\mu$, we use the effective Lagrangian of QCD at leading order, $$\label{gl4}
{{\cal L}}= \frac{F^{2}}{4} \: {\mbox{tr}\,}\left(
\partial_{\mu}U \partial^{\mu}U^{\dagger}
+ \chi U^{\dagger} + \chi^{\dagger} U
\right),$$ where $F=93.2$ MeV is the pion decay constant in the chiral limit. Furthermore, we work in the isospin limit, [*i.e.*]{}, ${m_{u}}= {m_{d}}\doteq \hat{m}$, and set $\chi = 2 B_{0} \, {\mbox{diag}\,}(\hat{m},\hat{m},{m_{s}})$, where $B_{0}$ is related to the quark condensate in the chiral limit [@gasser84]. The unitary $3\times3$ matrix $U$ incorporates the fields of the eight pseudoscalar Goldstone bosons. A convenient parametrization is $U = \exp(i \sqrt{2} \Phi /F)$ with $$\label{gl5}
\Phi = \left(
\begin{array}{ccc}
{\displaystyle}\frac{{\pi^{0}}}{\sqrt{2}} + \frac{{\eta}}{\sqrt{6}} & {\displaystyle}{\pi^{+}}& {\displaystyle}{K^{+}}\\
{\displaystyle}{\pi^{-}}& {\displaystyle}-\frac{{\pi^{0}}}{\sqrt{2}} + \frac{{\eta}}{\sqrt{6}}& {\displaystyle}{K^{0}}\\
{\displaystyle}{K^{-}}& {\displaystyle}{\:\overline{\! K^{0} \!}\:}& {\displaystyle}-\frac{2}{\sqrt{6}} {\eta}\end{array}
\right).$$ In addition, the vector current relevant in the present case reads $$V_{\mu}^{4-i5} = \frac{F^{2}}{4i} \;
{\mbox{tr}\,}\left( [\lambda^{4}-i\lambda^{5}]
[ U\partial_{\mu}U^{\dagger}
+U^{\dagger}\partial_{\mu}U]
\right),$$ where $\lambda^{4}$ and $\lambda^{5}$ denote Gell-Mann matrices. The corresponding axial vector current does not contribute, because it is odd under the transformation $\Phi\rightarrow -\Phi$.
[**3.**]{} The relevant Feynman diagrams at leading order in CHPT are shown in [Fig. \[figur1\]]{}.
(422,117)(119,512) (119,512)
(316,575)[(0,0)\[lb\][\[0pt\]\[0pt\][$\pi$]{}]{}]{} (481,592)[(0,0)\[lb\][\[0pt\]\[0pt\][$K$]{}]{}]{} (431,595)[(0,0)\[lb\][\[0pt\]\[0pt\][$V^{\mu}$]{}]{}]{} (270,595)[(0,0)\[lb\][\[0pt\]\[0pt\][$V^{\mu}$]{}]{}]{} (541,558)[(0,0)\[lb\][\[0pt\]\[0pt\][$\pi$]{}]{}]{} (532,614)[(0,0)\[lb\][\[0pt\]\[0pt\][$\pi$]{}]{}]{} (480,621)[(0,0)\[lb\][\[0pt\]\[0pt\][$\pi$]{}]{}]{} (461,539)[(0,0)\[lb\][\[0pt\]\[0pt\][$K$]{}]{}]{} (384,580)[(0,0)\[lb\][\[0pt\]\[0pt\][$\pi$]{}]{}]{} (256,539)[(0,0)\[lb\][\[0pt\]\[0pt\][$K$]{}]{}]{} (310,512)[(0,0)\[lb\][\[0pt\]\[0pt\][(b)]{}]{}]{} (378,545)[(0,0)\[lb\][\[0pt\]\[0pt\][$\pi$]{}]{}]{} (366,617)[(0,0)\[lb\][\[0pt\]\[0pt\][$\pi$]{}]{}]{} (474,512)[(0,0)\[lb\][\[0pt\]\[0pt\][(c)]{}]{}]{} (165,512)[(0,0)\[lb\][\[0pt\]\[0pt\][(a)]{}]{}]{} (199,553)[(0,0)\[lb\][\[0pt\]\[0pt\][$\pi$]{}]{}]{} (214,585)[(0,0)\[lb\][\[0pt\]\[0pt\][$\pi$]{}]{}]{} (192,613)[(0,0)\[lb\][\[0pt\]\[0pt\][$\pi$]{}]{}]{} (119,539)[(0,0)\[lb\][\[0pt\]\[0pt\][$K$]{}]{}]{} (130,595)[(0,0)\[lb\][\[0pt\]\[0pt\][$V^{\mu}$]{}]{}]{}
Their contribution is $$\begin{aligned}
{\big<}\pi^{+}(p_{1}) \pi^{-}(p_{2}) \pi^{0}(p_{3})\mbox{ out}
{\big|}{\overline{s}}\gamma_{\mu} {u}{\big|}K^{+}(p) {\big>}& = &
- A_\mu(1,2,3) + B_\mu(1,2,3), \label{gl7} \\
{\big<}\pi^{0}(p_{1}) \pi^{0}(p_{2}) \pi^{0}(p_{3})\mbox{ out}
{\big|}{\overline{s}}\gamma_{\mu} {u}{\big|}K^{+}(p) {\big>}& = &
A_\mu(1,2,3) + A_\mu(1,3,2)
+ A_\mu(3,2,1),
\label{gl8} \\
{\big<}\pi^{0}(p_{1}) \pi^{-}(p_{2}) \pi^{0}(p_{3}) \mbox{ out}
{\big|}{\overline{s}}\gamma_{\mu} {u}{\big|}K^{0}(p) {\big>}& = &
\sqrt{2} A_\mu(1,3,2),
\label{gl9} \\
{\big<}\pi^{+}(p_{1}) \pi^{-}(p_{2}) \pi^{-}(p_{3}) \mbox{ out}
{\big|}{\overline{s}}\gamma_{\mu} {u}{\big|}K^{0}(p) {\big>}& = &
-\sqrt{2} \left\{ A_\mu(1,2,3)+A_\mu(1,3,2) \right\},
\label{gl10}
$$ where $$\begin{aligned}
A_\mu(1,2,3) & = & \frac{\sqrt{2}}{4 F^{2}}
\Bigg\{
\frac{p(p_{3}-p_{1})}{M_{K}^{2} - (p_{1}+p_{3}-p)^{2}}
[p_{1}-p_{2}+p_{3}-p]_{\mu} \nonumber \\ &&
\rule{0mm}{8mm}
{} + \frac{p(p_{2}-p_{3})}{M_{K}^{2} - (p_{2}+p_{3}-p)^{2}}
[p_{1}-p_{2}-p_{3}+p]_{\mu} \nonumber \\ &&
\rule{0mm}{8mm}
{} - \frac{p(p_{1}+p_{2})}{M_{K}^{2} - (p_{1}+p_{2}-p)^{2}}
[p_{1}+p_{2}-p_{3}-p]_{\mu} \nonumber \\ && \rule{0mm}{8mm}
{} + \frac{2(M_{\pi}^{2}+2p_{1}p_{2})}
{M_{\pi}^{2} - (p_{1}+p_{2}+p_{3})^{2}}
[p_{1}+p_{2}+p_{3}+p]_{\mu} \nonumber\\ &&
{} + 2 [p_{1}+p_{2}-p_{3}]_{\mu}
\Bigg\}
, \label{gl11}
\end{aligned}$$ $$\begin{aligned}
B_\mu(1,2,3) & = & \frac{\sqrt{2}}{4 F^{2}}
\Bigg\{
\frac{p(p_{2}-p_{1})}{ M_{K}^{2} - (p_{1}+p_{2}-p)^{2}}
[p_{1}+p_{2}-p_{3}-p]_{\mu} \nonumber \\ &&
\rule{0mm}{8mm}
{} + \frac{p(p_{2}-p_{3})}{ M_{K}^{2} -(p_{2}+p_{3}-p)^{2}}
[p_{1}-p_{2}-p_{3}+p]_{\mu} \nonumber \\ &&
\rule{0mm}{8mm}
{} + \frac{p(p_{1}-p_{3})}{ M_{K}^{2} - (p_{1}+p_{3}-p)^{2}}
[p_{1}-p_{2}+p_{3}-p]_{\mu}
\Bigg\}
. \label{gl12}
\end{aligned}$$ [**4.**]{} Defining the Lorentz invariant measure $$\label{gl13}
{d_{\mbox{\scriptsize{LIPS}}}}(p;p_{1},p_{2},\ldots,p_{n}) \doteq
\delta^{4}(p-\sum_{i=1}^{n}p_{i})
\prod_{i=1}^{n} \frac{d^{3} {\bf p}_{i}}{2 p_{i}^{0}},$$ the differential decay rate is given by $$\label{gl14}
d \Gamma = \frac{1}{2 {M_{K}}(2 \pi)^{11}}
\sum_{\mbox{\scriptsize{spins}}} |T|^{2}
{d_{\mbox{\scriptsize{LIPS}}}}(p;p_{e},p_{\nu},p_{1},p_{2},p_{3}).$$ The rates and branching ratios which follow from [Eqs. (\[gl7\])–(\[gl14\])]{} are displayed in [Table \[tafel1\]]{}.
decay rate in s$^{-1}$ branching ratio
----------------------------------------------------------------------------------------- ------------------------ -----------------------
${K^{+}}\to {\pi^{+}}{\pi^{-}}{\pi^{0}}e^{+} \nu_{e}$ $2.4 \cdot 10^{-4} $ $3.0 \cdot 10^{-12}$
${K^{+}}\to {\pi^{0}}{\pi^{0}}{\pi^{0}}e^{+} \nu_{e}$ $2.0 \cdot 10^{-4} $ $2.5 \cdot 10^{-12}$
${{K_{\mbox{\scriptsize{L}}}}^{0}}\to {\pi^{0}}{\pi^{0}}{\pi^{\mp}}e^{\pm} \nu_{e}$ $2.4 \cdot 10^{-4} $ $12 \cdot 10^{-12}$
${{K_{\mbox{\scriptsize{L}}}}^{0}}\to {\pi^{\pm}}{\pi^{\mp}}{\pi^{\mp}}e^{\pm} \nu_{e}$ $6.5 \cdot 10^{-4} $ $33 \cdot 10^{-12}$
: \[tafel1\]Rates and branching ratios of $K_{e5}$ decays, evaluated from the leading order term in CHPT.
The smallness of the decay rates is due to the suppression of phase space. Indeed, consider the ratio of the four- and five-dimensional phase space volumes $$\frac{{M_{K}}^{2}
\int {d_{\mbox{\scriptsize{LIPS}}}}
(p;p_{e},p_{\nu},p_{1},p_{2})}{2! \, (2 \pi)^{12}}
\times
\frac{3! \, (2 \pi)^{15}}{\int {d_{\mbox{\scriptsize{LIPS}}}}
(p;p_{e},p_{\nu},p_{1},p_{2},p_{3})}
\approx 2.3 \cdot 10^{6},$$ where we have inserted ${M_{K}}^{2}$ for dimensional reasons. On the other hand, we find for the ratio of the corresponding rates at tree level in CHPT $$\frac{{\Gamma({K^{+}}\to {\pi^{0}}{\pi^{0}}e^{+} \nu_{e})_{\mbox{\scriptsize{tree}}}}}{{\Gamma({K^{+}}\to {\pi^{0}}{\pi^{0}}{\pi^{0}}e^{+} \nu_{e})_{\mbox{\scriptsize{tree}}}}}
\approx 3.4 \cdot 10^{6},$$ which is of the same order of magnitude.
[**5**]{}. Turning now to the corrections at next-to-leading order, we note that the matrix element of the axial current receives a contribution from the chiral anomaly [@wess; @witten]. Besides the local term of the Wess-Zumino-Witten action, also the nonlocal part, which contains at least five meson fields, gives a contribution. However, an explicit calculation shows that it is suppressed by the factor $m_{e}$ in the matrix element and therefore undetectable in the near future. We expect from experience with other calculations in CHPT that the remaining contributions to the matrix element (\[gl2\]) at this order enhance the tree-level results for the decay rates at most by a factor of two to three.
[**6**]{}. Due to isospin symmetry, the relations which are given in [Eqs. (\[gl7\])–(\[gl10\])]{} are valid to all order in CHPT, with $A_\mu(1,2,3)$ symmetric in $p_{1}$ and $p_{2}$, and $B_\mu(1,2,3)$ totally antisymmetric in $p_{1}$, $p_{2}$, and $p_{3}$. The matrix elements of the axial current $\overline{s}\gamma_\mu\gamma_5u$ have an analogous decomposition, with the same symmetry properties of the reduced matrix elements. From this follows the isospin relation $$\label{gl15}
2 \, \Gamma({K^{+}}\to {\pi^{0}}{\pi^{0}}{\pi^{0}}e^{+} \nu_{e})
=
\Gamma({{K_{\mbox{\scriptsize{L}}}}^{0}}\to {\pi^{\pm}}{\pi^{\mp}}{\pi^{\mp}}e^{\pm} \nu_{e})
-
\Gamma({{K_{\mbox{\scriptsize{L}}}}^{0}}\to {\pi^{0}}{\pi^{0}}{\pi^{\mp}}e^{\pm} \nu_{e}).$$
[**7**]{}. Hitherto, only poor experimental data on $K_{e5}$ decays are available. The Particle Data Group [@pdg] quotes the upper bound $$\frac{\Gamma({K^{+}}\to {\pi^{0}}{\pi^{0}}{\pi^{0}}e^{+} \nu_{e})}{{\Gamma_{\mbox{\scriptsize{total}}}}}
< 3.5 \cdot 10^{-6},$$ which is six orders of magnitude bigger than our result. DA$\Phi$NE, which will produce $K^{\pm}$ and ${{K_{\mbox{\scriptsize{L}}}}^{0}}$ with an annual rate of $9\cdot10^{9}$ and $1.1\cdot10^{9}$, respectively [@dafne], may improve the upper bounds for $K_{e5}$ decays considerably.
To summarize, we have evaluated the rates and branching ratios of $K_{e5}$ decays at leading order in CHPT ([Eqs. (\[gl7\])–(\[gl14\])]{}) and given the isospin relation (\[gl15\]) for the decay modes. We have furthermore seen that $K_{e5}$ decays, in particular any effects from chiral anomaly, will be invisible at DA$\Phi$NE. However, the upper bounds for the branching ratios can be improved significantly.
I thank Jürg Gasser for useful discussions and for a critical reading of the manuscript.
[99]{} S. Weinberg, [*Physica* ]{}[**96A**]{} (1979) 327.
J. Gasser and H. Leutwyler, (a) [*Ann. Phys. *]{}[**158**]{} (1984) 142; (b) [*Nucl. Phys.* ]{}[**B250**]{} (1985) 465.
J. Wess and B. Zumino, [*Phys. Lett. *]{}[**B37**]{} (1971) 95.
E. Witten, [*Nucl. Phys. *]{}[**B223**]{} (1983) 422.
Review of Particle Properties, [*Phys. Rev. *]{}[**D50**]{} (1994) 1173.
The DA$\Phi$NE Physics Handbook, edited by L. Maiani, G. Pancheri and N. Paver (INFN, Frascati, 1992).
|
---
author:
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Jeffrey J. Early, NorthWest Research Associates, USA\
M. Pascale Lelong, NorthWest Research Associates, USA\
K. Shafer Smith, New York University, USA
title: Fast and Accurate Computation of Vertical Modes
---
\
\
This work has not yet been peer-reviewed and is provided by the contributing author(s) as a means to ensure timely dissemination of scholarly and technical work on a noncommercial basis. Copyright and all rights therein are maintained by the author(s) or by other copyright owners. It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author’s copyright. This work may not be reposted without explicit permission of the copyright owner.
Introduction
============
Vertical modes arise as part of the separable solution to both the internal wave problem and quasigeostrophic theory. The eigenvalue problem (EVP) is treated in many introductory physical oceanography textbooks, e.g., @gill1982-book [@cushman2011-book], and the resulting vertical modes describe the vertical structure of the linear solutions for a given density profile. While there are an infinite number of bases that can be used to represent ocean currents and density anomalies satisfying certain boundary conditions, the vertical modes correspond to $O(1)$ dynamical solutions of the equations of motion, and are therefore both diagnostic and prognostic. For this reason, vertical modes are the standard basis with which to represent the vertical structure of ocean currents, and it would be hard to overstate their usefulness for describing and modeling the ocean.
There are two primary uses for the vertical modes: (1) projecting a given flow field onto the vertical modes to determine its spectrum (a forward transformation), or (2) creating a dynamically consistent linear flow field from a given spectrum (an inverse transformation). For example, projection onto the vertical modes was used to construct the Garrett-Munk internal wave spectrum [@munk1981-book; @polzin2011-rg], as well as to describe the vertical structure of balanced flow in the ocean [@wunsch1997-jpo; @wortham2014-jpo].
This study was motivated by two situations where current techniques for computing vertical modes were found to have significant errors for reasonable computation times. In the first situation a numerical model needed to be initialized with a full spectrum of internal wave—a task which requires solving an eigenvalue problem at each resolved wavenumber in the model. In the second situation, we sought to compare an observed horizontal velocity spectrum of internal waves to the Garrett-Munk spectrum near a very strong pycnocline. This requires computing vertical modes at high frequencies—which requires appropriately resolving the mode variability near the pycnocline.
There are a number of sources of error that arise in performing either the forward or inverse transformation. These include the following:
1. a poorly defined mean density function;
2. measurement noise and uncertainty in the density function and, in the case of a forward transform, the dynamical variables;
3. aliasing error, due to the location of the grid points of the dynamical variables (relevant only for forward transform);
4. interpolation error, due to the location of (and lack of) data points specifying the density function;
5. numerical truncation error of the modes generated from the vertical eigenvalue problem.
Whether performing a forward or inverse transformation, these sources of error will compound in some fashion to create error in either the resulting flow field or the resulting spectrum.
The first source of error, a poorly defined mean density function, can result from lack of data, or often just uncertainty in what qualifies as a mean (e.g., questions of what time and length scales to average over, or even whether averaging is the correct approach for a nonlinear system). Measurement noise and uncertainty in data is a topic unto itself, so the methods here proceed without concern for measurement noise when possible. The aliasing error arises when the grid points are placed such that higher modes project onto the lower modes. This source of error is relatively easily controlled by computing the condition number of the projection matrix, which provides a fairly precise cutoff for the number of resolvable modes. On the rare occasion that a density function can be specified analytically, the interpolation error can be minimized to numerical precision. However, in the usual case where the density is given on some grid with uneven spacing, the density function must be interpolated in between those grid points. Our work suggests density interpolation error does not dominate the error for most cases. This manuscript is therefore largely concerned with the final source of errors—arising from the numerical representation of the modes in the vertical eigenvalue problem.
The standard method for solving the vertical eigenvalue problem is to discretize the problem and construct the matrices using second-order finite difference matrices, e.g., @cushman2011-book. However, this approach produces unacceptable errors for all but the very lowest modes in the simplest stratifications. This is problematic because numerical algorithms for solving eigenvalue problems scale as $O(n^3)$, where $n$ is the number of discretization points in the vertical, so small increases in resolution come at a large computational cost. Instead of using finite difference methods, @kelly2016-jpo solves the hydrostatic form of the eigenvalue problem spectrally using Galerkin’s method, at a fraction of the computational cost and with much higher accuracy. @dunphy2009-thesis uses a Chebyshev collocation method to solve the non-hydrostatic case with fixed frequency. Here, we extend the ideas of @kelly2016-jpo and @dunphy2009-thesis to the more general non-hydrostatic cases where the EVP must be solved for each frequency in the spectrum or each resolved wavenumber in a numerical model. Our approach solves the eigenvalue problem spectrally with Chebyshev polynomials to produce high quality vertical modes, even with relatively small $n$. The same techniques are then applied to solve for surface quasigeostrophic modes, used to describe the effect of density anomalies at the ocean boundaries.
This paper begins with a derivation of the two most relevant forms of the vertical eigenvalue problem that arise from the linearized equations of motion in section \[sec:linear\_iw\]. This provides the necessary context for orthogonality relations that form the basis of the normalization of the vertical modes, which in turn shows limitations of using certain vertical modes as a basis. Section \[sec:prob\_with\_fd\] demonstrates some of the problems associated with finite differencing, while section \[sec:z\_coord\] shows how these can be remedied using spectral methods. Details of the numerical implementation are described in section \[sec:implementation\] and some of the other sources of error are examined in section \[sec:error\]. Section \[sec:discussion\] discusses some best practices and potential pitfalls. Finally, the appendices include the exact analytical solutions for constant and exponential stratification that are employed for unit testing, the WKB (Wentzel-Kramers-Brillouin) approximated solution used for the adaptive grid method, and the class inheritance tree of the publicly available Matlab implementation of these methods.
Background {#sec:linear_iw}
==========
The linearized equations of motion for the fluid velocity $u(x,y,z,t)$, $v(x,y,z,t)$, $w(x,y,z,t)$, on the $f$-plane are $$\begin{aligned}
\label{x-momentum}
\partial_t u - f_0 v =& - \frac{1}{\rho_0} \partial_x p \\ \label{y-momentum}
\partial_t v + f_0 u =& - \frac{1}{\rho_0} \partial_y p \\ \label {z-momentum}
\partial_t w =& - \frac{1}{\rho_0} \partial_z p - g \frac{\rho}{\rho_0} \\ \label{continuity}
\partial_x u + \partial_y v + \partial_z w =& 0 \\ \label{thermodynamic}
\partial_t \rho + w \partial_z \bar{\rho} =& 0\end{aligned}$$ where $p(x,y,z,t)$ and $\rho(x,y,z,t)$ are the perturbation pressure and density, respectively. These are defined such that the total pressure $p_{\textrm{tot}}(x,y,z,t) = p(x,y,z,t) + p_0(z)$ and the total density $\rho_{\textrm{tot}}(x,y,z,t) = \rho_0 + \bar{\rho}(z) + \rho(x,y,z,t) $ where $\partial_z p_0(z) = -g \bar{\rho}(z)$. All variables in the equations of motion are functions of $x$, $y$, $z$ and $t$, except for $\bar{\rho}$ which is strictly a function of $z$. The operator $\partial_z$ is understood to reduce to $\frac{d}{dz}$ when applied to univariate functions. We use the usual definition of the buoyancy frequency $N^2(z) \equiv -\frac{g}{\rho_0} \partial_z \bar{\rho}$.
There are three linearly independent solutions to equations \[x-momentum\]-\[thermodynamic\], assuming periodic horizontal boundary conditions and a flat bottom: two wave solutions and the geostrophic solution.
Wave solutions {#sec:wave_soln}
--------------
The positive frequency wave solution is given by, $$\label{positive_wave_solution}
\left[\begin{array}{c} p_+ \\ u_+ \\ v_+ \\ w_+ \\ \rho_+ \end{array}\right] =
A \left[\begin{array}{c}
- \rho_0 g \frac{K h}{\omega} \cos \theta_+ F(z)\\
\left(\cos \alpha \cos \theta_+ + \frac{f_0}{\omega} \sin \alpha \sin \theta_+\right)F(z) \\
\left(\sin \alpha \cos \theta_+ - \frac{f_0}{\omega} \cos \alpha \sin \theta_+\right)F(z) \\
K h \sin \theta_+ G(z) \\
\frac{d \bar{\rho}}{d z} \frac{K h}{\omega} \cos \theta_+G(z)
\end{array}\right]$$ where the functions $F(z)$ and $G(z)$ are the eigenfunctions with corresponding eigenvalue $h$, to be discussed in detail below. The frequency is determined through the dispersion relation, $$\omega = \sqrt{g h (k^2 + l^2) + f_0^2}$$ and the negative rotating wave solution is found by flipping the sign on the frequency, $\omega \mapsto -\omega$. In this notation the phase angle of the wave is given by $\alpha = \tan^{-1} \left(\frac{l}{k}\right)$ and $K = \sqrt{k^2 + l^2}$ is the total horizontal wavenumber. The horizontal phase is given by $\theta_\pm=k x + l y \pm \omega t$. The eigenvalue $h$ is referred to as the equivalent depth and is related to the wave group velocity, $c_g = \sqrt{gh}$. The value $h$ can be replaced in favor of eigenfrequency $\omega$ using the dispersion relation, but here we include both to avoid singularities at $K=0$ and for notational compactness. Applying this solution to equations \[x-momentum\]-\[thermodynamic\] leads to two coupled equations for the vertical structure functions, $$(N^2-\omega^2)G = -g \partial_z F \quad\text{and}\quad
F = h \partial_z G,$$ which can be combined into various second-order eigenvalue problems (see section \[sec:vertical\_evp\]).
Geostrophic solution {#sec:geostrophic_soln}
--------------------
The geostrophic solution is given by, $$\label{geostrophic_solution}
\left[\begin{array}{c} p_g \\ u_g \\ v_g \\ w_g \\ \rho_g \end{array}\right] =
B \left[\begin{array}{c}
\rho_0 g \cos \theta_0 F(z) \\
\frac{g}{f_0} l \sin \theta_0 F(z) \\
-\frac{g}{f_0} k \sin \theta_0 F(z) \\
0 \\
-\frac{d \bar{\rho}}{d z} \cos \theta_0 G(z)
\end{array}\right]$$ where $\theta_0=k x + l y$. The solution can also be written entirely in terms of streamfunction $\psi(x,y,z) = \frac{g}{f_0} \cos \theta_0 F(z)$ where $(u_g,v_g,\rho_g)=(-\partial_y \psi, \partial_x \psi, -\rho_0 f_0 \partial_z \psi/g)$. Satisfying the vertical momentum equation requires that $N^2 G = -g \partial_z F$ but, unlike the wave solutions, geostrophic solutions already satisfy continuity. For a given wavenumber $(k,l)$ the geostrophic solution is entirely specified by a vertical profile of any one of the variables, from which all others are immediately deduced. For example, $\hat{\rho}(k,l,z)$ determines $G(z)$, from which $F(z)$ is determined by integration—the thermal wind balance. There is no preferred basis for the geostrophic solution.
Although the scalings that lead to equations \[x-momentum\]-\[thermodynamic\] result in the linear geostrophic solution where $w_g=0$, near-geostrophic theories with a different choice of scalings, such as quasi-geostrophy [@pedlosky1987-book], have nonzero vertical velocities $w_g \neq 0$ and therefore require full continuity, i.e., that $F = h \partial_z G$. As with the wave solution, this requirement combined with $N^2 G = -g \partial_z F$ results in an eigenvalue problem, detailed in section \[sec:vertical\_evp\].
An eigenbasis constructed with the rigid lid boundary condition ($w(0)=G(0)=0$) precludes non-zero density anomalies at the surface. One workaround to this limitation is to further decompose the geostrophic solution into three parts: two parts resulting from the density anomaly at the boundaries and one part from the remaining density anomaly in the interior. Following @lapeyre2006-jpo, the idea is then to let $$\psi = \psi^{\textrm{int}} + \psi^{\textrm{sur}} + \psi^{\textrm{bot}}$$ where both the surface and bottom components of the flow are required to have no potential vorticity in the interior, $$\label{sqg_eqn}
\nabla^2 \psi^{\textrm{sur/bot}} + \partial_z \left( \frac{f_0^2}{N^2} \partial_z \psi^{\textrm{sur/bot}} \right) = 0,$$ but account for the density anomaly at the boundaries, e.g., $$f_0 \partial_z \psi^{\textrm{sur}} = -\frac{g}{\rho_0} \rho \bigr\rvert_{z=0}, \quad f_0 \partial_z \psi^{\textrm{bot}} = 0 \bigr\rvert_{z=-D}.$$ The resulting modes can be solved for a given wavenumber and are referred to as the surface quasi-geostrophic (SQG) modes. This methodology has been used to construct the interior velocity field from sea-surface height and temperature data @wang2013-jpo.
@smith2013-jpo formulate a new eigenvalue problem that results in modes capable of capturing surface density anomalies for quasigeostrophic flows. Taking equation 9 from @smith2013-jpo and writing it in the present notation we have, $$\label{vertical-eigenvalue-SV}
\tag{SV EVP}
\partial_{zz}G_j - \frac{K^2 N^2}{f_0^2} G_j = -\frac{N^2}{g h_j }G_j$$ with surface boundary condition $G(0)=\frac{D}{\alpha}\partial_z G$ where $\alpha$ is a tunable parameter. Importantly, these modes remain orthogonal, unlike the combined set of SQG modes and interior modes described above.
An alternative to both the SQG and the @smith2013-jpo approaches is to use the internal wave eigenbasis constructed with the free surface boundary condition (detailed below) which also results in orthogonal modes capable of capturing nonzero density anomaly at the ocean surface.
Vertical eigenvalue problem {#sec:vertical_evp}
---------------------------
The vertical eigenvalue problem is formed using the two coupled equations from section \[sec:wave\_soln\] for the vertical structure functions, $(N^2-\omega^2)G = -g \partial_z F$ (vertical momentum) and $F = h \partial_z G$ (continuity). In combination with the dispersion relation, one of the eigenfunctions can be eliminated, resulting in various single second-order eigenvalue problems for the vertical structure functions $F$ or $G$. The two most practical eigenvalue problems to solve are for $G(z)$ with constant $\omega$, $$\label{vertical-eigenvalue-G-with-omega}
\tag{$\omega$-constant EVP}
\partial_{zz} G_j = -\frac{N^2-\omega^2}{g h_j }G_j$$ and for $G(z)$ with constant $K$, $$\label{vertical-eigenvalue-G-with-K}
\tag{$K$-constant EVP}
\partial_{zz}G_j - K^2 G_j = -\frac{N^2 - f_0^2}{g h_j }G_j.$$ The near geostrophic eigenvalue problem is found by combining the vertical momentum condition $N^2G = -g \partial_z F$ with the continuity condition $F = h \partial_z G$, which can be treated as the \[vertical-eigenvalue-G-with-omega\] with $\omega=0$. Note that this is equivalent to making the hydrostatic approximation [@kelly2016-jpo], and removes all dependence on frequency $\omega$ and wavenumber $K$.
For the cases considered here we take the lower boundary condition at $z=-D$ to be either free-slip, where $w(-D)=0$, or no-slip, where $u(-D)=0$. These correspond to $G(-D)=0$ and $F(-D)=0$, respectively. These conditions can be seen as the limiting cases of sloped bottom topography [@lacasce2017-grl]. The surface boundary condition at $z=0$ is taken to be either a rigid lid with $w(0)=0$, $G(0)=0$, or a free surface approximated as $p(x,y,0)=\rho_0 g \eta(x,y,0)$ where $\eta\equiv-(\partial_z \bar{\rho})^{-1} \rho$ is the linear approximation of the isopycnal displacement. In terms of the vertical modes, the free surface boundary condition becomes $ h \partial_z G(0) = G(0)$. Finally, there are many cases where the density profile does not extend to the full depth of the ocean and no boundary conditions (beyond the EVP itself) should be added.
Solving either EVP yields a set of eigenvalues $h_j$ that can be ordered such that $h_1>h_2>h_3>..>h_n$, each with corresponding eigenfunction $G_j$. This means that solving \[vertical-eigenvalue-G-with-omega\] results in wave solutions with the same frequency $\omega$, but different wavenumbers $K_j$, and similarly solving \[vertical-eigenvalue-G-with-K\] results in wave solutions with the same wavenumber $K$ and different frequencies $\omega_j$. Although we do not implement this numerically, note that rearranging the \[vertical-eigenvalue-G-with-K\] poses the EVP for fixed group velocity, $gh$, with eigenvalue $K^2_j$.
The equations for \[vertical-eigenvalue-G-with-omega\] and \[vertical-eigenvalue-G-with-K\] are both Sturm-Liouville problems and share the property that their eigenmodes are orthogonal. Following the procedure in @kelly2016-jpo, the eigenmodes found with the \[vertical-eigenvalue-G-with-omega\] satisfy $$\label{omega_const_ortho}
G_i(0)G_j(0)+\frac{1}{g} \int_{-D}^{0} (N^2(z)-\omega^2) G_i G_j \, dz = \beta \delta_{ij}$$ and $$\int_{-D}^0 F_i F_j \, dz = \beta h_i \delta_{ij}$$ while the eigenmodes found with \[vertical-eigenvalue-G-with-K\] satisfy, $$\label{k_const_ortho}
G_i(0)G_j(0)+\frac{1}{g} \int_{-D}^{0} (N^2(z)-f_0^2) G_i G_j \, dz = \gamma \delta_{ij}$$ and $$\int_{-D}^0 \left( F_i F_j + h_i h_j K^2 G_i G_j \right) \, dz = \gamma h_i \delta_{ij}$$ where $\beta$ and $\gamma$ are unspecified constants that depend on the chosen normalization, as discussed below. It is important to note that these orthogonality conditions only apply for a particular choice of $\omega$ or $K$. For example, an eigenmode $G_j(z,k_1)$ found using $K=k_1$ is not orthogonal to an eigenmode $G_j(z,k_2)$ found using $K=k_2$ if $k_1 \neq k_2$.
The most significant difference between the two EVPs is that eigenmodes from the \[vertical-eigenvalue-G-with-K\] often form a complete basis set for typical ocean stratification profiles, while the eigenmodes from the \[vertical-eigenvalue-G-with-omega\] do not. The \[vertical-eigenvalue-G-with-K\] is a *regular* Sturm-Liouville problem when the weighting function $w_K(z) \equiv N^2 - f_0^2 >0$ for all $z$, a condition typically met in the ocean. We note that although it is fair to say that stratification with $N>f_0$ is typical of the world oceans, after examining 30,000 CTD profiles @kunze2017-jpo finds that 10% of the data have $N<2f$ and a full 30% of the data suggest $N<2f$ within 380 meters of the bottom. In contrast, the weighting function $w_\omega(z) \equiv N^2-\omega^2$ in the \[vertical-eigenvalue-G-with-omega\] switches sign at turning points $z_T$, where $N(z_T)=\omega$. Consequently, the norm of an arbitrary function defined on the domain $[-D,0]$ and satisfying the boundary conditions is not guaranteed to be positive using the norm implied by equation \[omega\_const\_ortho\], a necessary condition for completeness. Intuitively this can be seen in figure \[PycnoclineMode\], which shows that the high frequency modes have no variance beyond the turning points, and are therefore incapable of representing arbitrary functions on the domain.
Normalization {#sec:normalization}
-------------
The amplitude of each vertical mode can be scaled by an arbitrary constant, so one must choose a normalization appropriate for the problem being considered. The four most common scenarios are setting the total energy, a horizontal velocity ($U$), a vertical velocity ($W$), and the sea surface height (SSH) of a given wave.
To set the total energy of the internal wave solution in equation \[positive\_wave\_solution\], we use the modes from the \[vertical-eigenvalue-G-with-K\] and therefore use the norm implied by equation \[k\_const\_ortho\], $$\label{k_const_norm}
G_i^2(0)+\frac{1}{g} \int_{-D}^{0} (N^2(z)-f_0^2) G_i^2 \, dz = 1 \tag{$K$-constant norm}$$ where we have chosen $\gamma=1$ as the normalization constant in order to keep the vertical modes unitless.[^1] Taking the total energy of the wave, $$E(x,y,z,t) = \frac{1}{2}\left(u^2 + v^2 + w^2 + \frac{g^2}{\rho_0^2}\frac{\rho^2}{N^2}\right)$$ and then depth integrating and averaging over space and time produces a wave with energy $P^2/2$ if we set the coefficient $A=P/\sqrt{h_j - (\omega^2 - f_0^2)G^2(0)/\omega^2}$ in equation \[positive\_wave\_solution\].
Setting the maximum initial eastward velocity to $U$ can be accomplished by imposing $\max{F_j}=1$ and $A=U$. The maximum vertical velocity $W$ is set using the norm $\max{G_j}=1$ where $A=W/(Kh_j)$, but is clearly singular for inertial waves which have no vertical velocity. The sea surface height is set using the pressure at the surface with $F(0)=1$ and $A=\textrm{SSH} \cdot \frac{\omega}{K h}$.
To set the total energy of the interior geostrophic solution in equation \[geostrophic\_solution\], we assume the solution uses the typical geostrophic modes from equation \[vertical-eigenvalue-G-with-omega\] with $\omega=0$ and therefore use the norm implied by equation \[omega\_const\_ortho\], $$\label{omega_const_norm}
\frac{1}{D} \int_{-D}^{0} F_i^2 \, dz = 1
\tag{$\omega$-constant norm}$$ where we have taken $\beta = D/h_i$. This produces a mode with energy $P^2$ if we let, $$B^2 = \frac{4 P^2 f^2 h}{g D} \frac{1}{ghK^2 + f^2(1+h G^2(0)/D)}.$$ Setting the maximum eastward velocity requires $B = U \frac{f_0}{g l}$ using $\max{F_j}=1$. The sea-surface height is set using the pressure at the surface by setting $F(0)=1$ and $B=\textrm{SSH}$.
The problem with finite differencing {#sec:prob_with_fd}
====================================
Computing the lowest vertical modes with finite differencing methods can be relatively fast and accurate when considering a single wavenumber or frequency. Although one can encounter problems with the higher modes, this can usually be ameliorated by increasing resolution. The primary issues with finite differencing arise when needing to compute many modes across a broad range of frequencies and wavenumbers—the two scenarios that motivated the present study. To compute a complete internal wave frequency spectrum requires solving the \[vertical-eigenvalue-G-with-omega\] at each resolved frequency between the Coriolis frequency and the maximum buoyancy frequency, roughly $O(10^2)$ EVPs. This is especially challenging near the buoyancy frequency, where all oscillations occur in the narrow region where $N(z)<\omega$. On the other hand, initializing a numerical model with an internal wave spectrum involves solving the \[vertical-eigenvalue-G-with-K\] for each resolved wavenumber in the model, which easily requires $O(10^4)$ computations or more.
![Relative error as a function of vertical mode number using 64 evenly spaced grid points in exponential stratification $N(z)=N_0 e^{z/b}$ where $N_0=3$ cph and $b=1300$ m at latitude 33$^\circ$ in a 5000 m deep ocean for $K=0.0$ (top panel) and $K=\frac{2\pi}{500\textrm{ m}}$ (bottom panel). Shown are two 2nd-order finite differencing methods with (1) 64 grid points (blue), (2) 631 grid points (blue dotted), and three spectral methods using Chebyshev polynomials with coordinates in (3) depth (red), (4) WKB scaled (purple), and (5) density (orange).[]{data-label="InternalModeError"}](InternalModeError){width="19pc"}
A prerequisite to initializing a numerical model with a given internal waves spectrum, is that the modes must be computed for each unique horizontal wavenumber $K$ resolved by the model using the \[vertical-eigenvalue-G-with-K\]. If the numerical model has $(N_x,N_y)$ horizontal grid points, approximately $N_x N_y/2$ unique eigenvalue problems must be solved (up to another factor of 2 can be eliminated with isotropic horizontal resolution). Unfortunately, eigenvalue algorithms scale as $O(n^3)$ for $n$ by $n$ matrices. This means that initialization of an internal wave spectrum scales as $O(N_x N_y N_z^3)$ and thus, with any reasonable vertical resolution, this will quickly become a rate limiting step to a model run. In practical terms, the computation time of these eigenvalue problems takes approximately 1s, 10s, and 100s, for $N_z$ of 512, 1024, and 2048, respectively, on consumer hardware from 2015.
The problem is further exacerbated by the poor performance of finite difference methods. To demonstrate, we compare different numerical methods against an analytical solution. Consider an exponential density profile—the canonical deep ocean stratification profile which has known analytical solutions for both the non-hydrostatic internal modes [@garrett1972-gfd], as well as the SQG modes [@lacasce2012-jpo]. Using a numerical method to solve the \[vertical-eigenvalue-G-with-K\], we can compute the relative error of the numerical solution to the analytical solution. We define the relative error as $$\textrm{rel. err.} = \frac{\max(|f_i - f(z_i)|)}{\max(f(z_i))}$$ where $f(z_i)$ is the true solution evaluated at the grid points $z_i$, and $f_i$ is the numerical approximation. Figure \[InternalModeError\] shows the maximum relative error of the eigenmodes $F$, $G$ and eigenvalue $h$ found by solving the \[vertical-eigenvalue-G-with-K\] using 2nd order finite difference methods for standard exponential stratification with 64 vertical grid points (blue). Details of the numerical implementation of the analytical solutions are given in \[sec:im\_exponential\_strat\].
Reasonable error magnitudes are $O(10^{-2})$ (see \[sec:mode\_error\] for justification), however, the top panel of figure \[InternalModeError\] shows that no modes computed with finite differencing satisfy this condition. The situation is even worse for the $K=\frac{2\pi}{500\textrm{ m}}$ case shown in the bottom panel, where even the lowest mode has an $O(0.1)$ error. The blue dotted line in figure \[InternalModeError\] shows that increasing the resolution tenfold for 2nd-order finite differencing decreases the error by a factor of 100, as would be expected. However, this comes at 1000 times the computational cost, and still barely produces any usable modes.
![The top panel shows relative error as a function of resolution for the 10th mode in exponential stratification with $K=\frac{2\pi}{500\textrm{ m}}$. The density function is specified on an evenly spaced grid (solid lines) or passed directly as an analytical function (dashed lines). The bottom panel shows the number of usable modes as a function of resolution, defined as the number of modes with truncation errors less than $10^{-2}$. The convergence rate of the 2nd order and 6th order finite difference methods are found to be $(\Delta z)^{2.0}$ and $(\Delta z)^{5.8}$, respectively. []{data-label="TruncationErrorK2pi500Mode10"}](TruncationErrorK2pi500Mode10){width="19pc"}
The accuracy of finite differencing can be increased by going to higher orders [@fornberg1998-siam], since the truncation error at order $s$ scales as $(\Delta z)^s$. The truncation error of the 10th mode in exponential stratification is shown in the top panel of figure \[TruncationErrorK2pi500Mode10\] where the blue (solid, dotted) line shows the (2nd, 6th) order finite difference method converging at its predicted rate. The bottom panel shows that even with 1024 grid points, there are only 8 usable modes for the 2nd order finite differencing method, while 6th order gives up to 50 modes. However, while increasing the order of the method does provide some gains in accuracy, the most efficient way to proceed is simply to use spectral methods, which promise exponentially decreasing truncation error, rather than the polynomial truncation errors offered by finite differencing. When using an analytical density function (dashed lines, figure \[TruncationErrorK2pi500Mode10\]) rather than gridded data, there is no interpolation error and the spectral methods truncation errors reach a noise floor somewhere between 64 and 128 grid points. Furthermore, the number of usable modes is an order of magnitude higher than even the 6th order finite difference method. In practical terms, the 2nd order finite difference method is producing about 10 good modes in 100 seconds, while the spectral methods are producing about 100 good modes in 1 second. The increase in truncation error at higher resolution is likely due to increasingly compounded errors of the eigenvalue solvers.
Chebyshev polynomials {#sec:z_coord}
=====================
Written in matrix form the \[vertical-eigenvalue-G-with-K\] is, $$\label{EVPforAB}
\mathsf{A} {\boldsymbol{\mathrm{v}}} = \frac{1}{h} \mathsf{B} {\boldsymbol{\mathrm{v}}}$$ where ${\boldsymbol{\mathrm{v}}}$ is the vector representation of the normal mode $G$ at grid points in $z$, $\mathsf{A}=\partial_{zz} - K^2$ and $\mathsf{B} = (f_0^2 - N^2)/g$. For finite differencing, $\mathsf{A} = \mathsf{D}_{zz} - K^2 \mathsf{I}$, where $\mathsf{D}_{zz}$ is the $N_z \times N_z$ differentiation matrix and $\mathsf{I}$ is the $N_z$ dimensional identity matrix. To use Chebyshev polynomials, we project vector ${\boldsymbol{\mathrm{v}}}$ onto a Chebyshev basis using $\hat{{\boldsymbol{\mathrm{v}}}} = \mathsf{T}^{-1}{\boldsymbol{\mathrm{v}}}$ where $\mathsf{T}$ is the matrix that transforms a vector from a Chebyshev basis to the coordinate basis. In a practical sense, the columns of $\mathsf{T}$ are the Chebyshev polynomials. Then the eigenvalue problem becomes, $$\mathsf{A} \mathsf{T} \mathsf{T}^{-1}{\boldsymbol{\mathrm{v}}} = \frac{1}{h} \mathsf{B} \mathsf{T} \mathsf{T}^{-1}{\boldsymbol{\mathrm{v}}}$$ or simply, $$\label{EVPforT}
(\partial_{zz} \mathsf{T} - K^2 \mathsf{T}) \hat{{\boldsymbol{\mathrm{v}}}} = \frac{1}{h} \left(\frac{f_0^2 -N^2(z)}{g} \right) \mathsf{T} \hat{{\boldsymbol{\mathrm{v}}}}.$$ The vector $\hat{{\boldsymbol{\mathrm{v}}}}$ contains the coefficients needed to reconstruct eigenfunctions and $\partial_{zz} \mathsf{T}$ are the second derivatives of the Chebyshev poynomials.
The optimal choice of grid for Chebyshev polynomials is a Gauss-Lobatto grid [@boyd2001-book; @canuto2006-book], e.g. equation \[lobatto\] below, and thus the eigenmatrices and eignfunctions are always created on a Gauss-Lobatto grid for any chosen coordinate. Because the basis functions are continuous functions of $z$, the resulting vertical modes can be interpolated onto any grid at any resolution by evaluating the functions at the points of interest.
It is, however, rarely the case that density is given as an analytical function, or that observations are made on a Gauss-Lobatto grid, which means that typically the density needs to be interpolated on the appropriate grid. Interpolation is performed using B-splines implemented with the numerical framework described in @early2019-arxiv. The advantage to using B-splines to represent gridded density data is that it is easy to accommodate arbitrary grids. Despite being a low order method, this is generally not a limitation (see section \[sec:error\]). In the cases shown in figure \[InternalModeError\], the algorithms are given the density ($\rho$) on a uniform grid in $z$ of 64 points and the resulting modes are returned on the same grid (except where noted for the high resolution finite differencing case). This is, of course, suboptimal for the spectral cases which use a Gauss-Lobatto grid on various coordinates to compute the eigenvalue problem. When given an analytical function for density, these methods perform even better, as can be seen in figure \[TruncationErrorK2pi500Mode10\].
Despite the potential limitations imposed by interpolating the density with B-splines onto a Gauss-Lobatto grid, the red line in figure \[InternalModeError\] shows that the Chebyshev method performs extremely well, even while interpolating from an evenly-spaced grid, and outputting to the same grid. The first 20 and 14 modes have error less than $O(10^{-2})$ for the $K=0$ and $K=2\pi/500 \textrm{ m}^{-1}$ cases, respectively. However, at higher horizontal resolution (larger wavenumbers $K$), even the spectral method’s errors grow large, because the points at which the functions are evaluated do not sufficiently capture the oscillations of the modes. This can be remedied by using a stretched coordinate, $s$.
Stretched coordinates {#sec:stretched_coord}
---------------------
In order to find an independent coordinate better suited to capturing the structure of the eigenmodes, we rewrite the eigenvalue problems in terms of a generic coordinate $s$ and then consider two concrete examples. Taking $z$ to be a function of $s$ and applying the chain rule leads to $$\partial_z = (\partial_s z)^{-1} \partial_s$$ and $$\partial_{zz} = - (\partial_{ss}z) (\partial_s z)^{-3} \partial_s + (\partial_s z)^{-2} \partial_{ss}.$$ For example, the \[vertical-eigenvalue-G-with-K\] becomes, $$\label{evp_stretched}
\left( - (\partial_{ss}z) (\partial_s z)^{-3} \partial_s + (\partial_s z)^{-2} \partial_{ss} \right) G - K^2 G = -\frac{N^2 - f_0^2}{gh}G$$ where now $F = h (\partial_s z)^{-1} \partial_s G$. The free surface boundary condition in these coordinates becomes $(\partial_s z)^{-1} \partial_s G = G/h$, while the normalization conditions are now, $$\label{k_const_norm_stretched}
\frac{1}{g} \int_{s(-D)}^{s(0)} (N^2(s)-f_0^2) G_i G_j \partial_s z \, ds = \delta_{ij}$$ and $$\label{omega_const_norm_stretched}
\frac{1}{D} \int_{s(-D)}^{s(0)} F_i F_j \partial_s z \, ds = \delta_{ij}.$$ A necessary condition for using stretched coordinates is that the function $s(z)$ must be strictly monotonic.
Density coordinates {#sec:density_coord}
-------------------
For density coordinates $s = -g \bar{\rho}/\rho_0$, equation \[evp\_stretched\] can be written as $$N^4 \partial_{ss}G + \partial_z\left(N^2\right) \partial_s G - K^2 G = -\frac{N^2 - f_0^2}{gh}G$$ where $F = h N^2 \partial_s G$. Note that the derivatives of the density are computed on the $z$ coordinate, then projected onto the $s$ coordinate in the eigenvalue problem. This avoids using inverses of functions that tend towards zero, and therefore has greater numerical stability. While the method does well for the high wavenumber case (figure \[InternalModeError\], lower panel), it performs somewhat poorly with a uniform relative error of $O(0.03)$ for all modes in the low wavenumber (upper panel), as shown by the orange line. Evidently, density coordinates cluster points inefficiently in this case.
![The left panel shows a stratification profile with pycnocline taken from @cushman2011-book. The vertical dashed line represents a frequency with two turning points in the pycnocline. The right panel shows the fourth vertical $F$ mode at that frequency.[]{data-label="PycnoclineMode"}](PycnoclineMode){width="19pc"}
WKB stretched coordinates {#sec:wkb_coord}
-------------------------
A compromise between depth ($z$) and density ($\bar{\rho}$) coordinates is the WKB stretched coordinate, $s = \int_D^z N(z^\prime) \, dz^\prime$. In this case the eigenvalue problem becomes, $$\label{evp_wkb}
\left( (\partial_z N) \partial_s + N^2 \partial_{ss} \right) G - K^2 G = \frac{f_0^2 - N^2}{gh}G$$ where $F=h N \partial_s G$.
The purple line in figure \[InternalModeError\] shows that the vertical modes computed on WKB coordinates have uniform accuracy of $O(10^{-3})$ for $K=0$, outperforming the density coordinate case, and also performing nearly as well as density coordinates in the high wavenumber case.
Adaptive grid for $\omega$-constant EVP {#sec:adaptive_grid}
---------------------------------------
![Relative error as a function of vertical mode number using 64 evenly spaced grid points for the frequency and stratification shown in figure \[PycnoclineMode\]. Shown is the WKB scaled spectral method (purple) as in figure \[InternalModeError\], but also the adaptive grid method (green) that clusters points near the regions of oscillation.Only the first 16 modes are shown.[]{data-label="InternalModeErrorFrequency"}](InternalModeErrorFrequency){width="19pc"}
Solving the \[vertical-eigenvalue-G-with-omega\] suffers from the additional challenge that as the frequency increases and the distance between turning points decreases, the grid spacing necessary to capture the mode structure becomes ever smaller. As noted in the introduction, this issue arises when considering internal waves near a pycnocline. An example stratification profile and vertical mode found at a frequency approaching the maximum frequency in the pycnocline is shown in figure \[PycnoclineMode\]. The relative error as a function of mode for this example is shown in figure \[InternalModeErrorFrequency\], from which it is clear that even the WKB stretched coordinate that performed so well for the \[vertical-eigenvalue-G-with-K\], does relatively poorly in this scenario. Sturm-Liouville theory tells us that the $n$-th $F$ mode will have $n$ zero crossings in the oscillatory region where $N^2(z)>\omega^2$ [@arfken1970-book]. Thus, in order to resolve these oscillations, one would require at least $2 n$ optimally placed points in that region, as well as additional points to capture the variance in the decay regions. Simply increasing resolution of the Chebyshev grid cannot efficiently solve the problem, as grid points will continue to be poorly placed.
To resolve this issue we devise an ad hoc method for clustering points in regions of interest. Our approach is to partition the domain into regions where the modes are hypothesized to be nonzero, formulate the EVP for each region (using WKB stretched coordinates), then couple the equations at the region boundaries. This enables us to assign most of the points to the regions where the solution is assumed to be nonzero, and allocate a few remaining points in the other regions. A comparison of this adaptive method and the standard WKB stretched coordinates is shown in figure \[InternalModeErrorFrequency\], where the adaptive method is able to capture a few more usable modes than the standard WKB stretched coordinate method with one EVP. The value of this method becomes more pronounced as the maximum frequency is reached. The number of usable modes (error $< 10^{-2}$) drops to zero as the maximum buoyancy frequency is approached when using the single EVP, as shown in figure \[GoodModesVsFrequency\]. However, using the adaptive grid algorithm, we are able to guarantee a minimum number of usable modes as points cluster around the turning frequencies.
The equation boundaries are established by using the WKB approximated solution to identify the regions where the modes are expected to be nonzero. Specifically, the equation boundaries are the points where WKB solution (\[Fwkb\]) decays to $10^{-5}$ of its value from the turning point. This is an adjustable tolerance, chosen to be small enough that only a few points are needed in the to capture the nearly zero function, but large enough that the nonzero regions aren’t unnecessarily large. The gray vertical lines in figure \[GoodModesVsFrequency\] show the number of coupled equations being used to solve the EVP. At the lowest frequencies only one equation is used and the method is identical to the WKB stretched coordinate method described in section \[sec:wkb\_coord\]. As the frequency increases, the algorithm eventually separates into two coupled equations: one for the top boundary and pycnocline, and another for the deep region where no mode variance is expected (refer to figure \[PycnoclineMode\]). At high enough frequency the region above the pycnocline is decoupled as well, and three coupled EVP problems are solved.
The EVPs are coupled by requiring that the function and its first derivative are continuous at the equation boundaries, following the procedure described in section 22.3 of @boyd2001-book. The ‘eigenvalue rule-of-thumb’ as discussed in @boyd2001-book states that roughly $n/2$ modes will be accurate when using $n+1$ Chebyshev polynomials away from boundary layers or critical levels. Solving the EVP with turning points near the maximum buoyancy certainly does not satisfy this criterion, but the rule-of-thumb can be modified to use half of the modes *with eigenvalues greater than zero*. Although we make no attempt at proving the general validity of this modification, the dashed line in figure \[GoodModesVsFrequency\] indicates that the rule-of-thumb generally does well at predicting how many modes are good quality.
![The number of usable modes (error $< 10^{-2}$) versus frequency for the stratification in figure \[PycnoclineMode\] using 64 points. The adaptive grid method and standard WKB method are shown in green and purple, respectively. The dashed green line is the rule-of-thumb number of good modes by the adaptive grid method. The two vertical gray lines separate the regions where the adaptive algorithm used 1, 2 or 3 coupled equations. []{data-label="GoodModesVsFrequency"}](GoodModesVsFrequency){width="19pc"}
Numerical implementation {#sec:implementation}
========================
One of the primary products of this paper is the implementation of these methods as classes in Matlab (see \[sec:class\_hierarchy\] for more details). Figure \[InitializationAlgorithm\] shows the flowchart followed by the initialization algorithm for the `InternalModes` class, described in this section.
![Algorithm flowchart for the initialization of `InternalModes` class.[]{data-label="InitializationAlgorithm"}](InitializationAlgorithm){width="19pc"}
The two methods for initializing the classes are both called using,
im = InternalModes(rho,z,zOut,latitude);
where the arguments `rho,z` are either a gridded density field at locations `z`, or function handle valid in the domain spanned by `[min(z) max(z)]`. When the function handle is given, the density function is projected onto Chebyshev polynomials. If gridded data is provided, then the density is interpolated using B-splines. The argument `zOut` specifies the grid on which all output is given, which need not span the full depth.
After initialization, all classes support setting the upper and lower boundary conditions as well as setting the normalization to any of the choices discussed in section \[sec:normalization\].
The two primary functions for computing internal modes are
[F,G,h,omega] = im.ModesAtWavenumber(k);
for the \[vertical-eigenvalue-G-with-K\] and
[F,G,h,k] = im.ModesAtFrequency(omega);
for the \[vertical-eigenvalue-G-with-omega\].
The implementation of these methods for finite differencing is straightforward—the eigenvalue problem is either solved on the gridded input data as given, or on a grid that matches the output grid if specified as a function. The differentiation matrices are created using the algorithms described in @fornberg1998-siam. However, the spectral implementations require additional choices.
The eigenvalue problem being solved is $$\label{generic_evp}
\left[a(s) \mathsf{T}_{ss} + b(s) \mathsf{T}_s + c(s) \mathsf{T}\right]{\boldsymbol{\mathrm{v}}} = \frac{1}{h} d(s) \mathsf{T} \hat{{\boldsymbol{\mathrm{v}}}}$$ where $s$ is a generic stretched coordinate, $a(s)$, $b(s)$, $c(s)$, and $d(s)$ are referred to here as *coefficient functions*.
The algorithm can be separated into the three parts. First we compute the *coefficient functions* for each eigenvalue problem, e.g., $N^2$ and $\partial_z N$ for equation \[evp\_wkb\]. Second, the eigenvalue problem is solved on the appropriate coordinate with $n_\textrm{evp}$ points. Finally, the resulting modes are normalized and projected onto the output grid with an arbitrary number of points.
Initialization with an analytical function
------------------------------------------
If the `InternalModes` class is initialized with a function handle for the density, it is projected onto Chebyshev polynomials which are then used to compute the coefficient functions and, if necessary, the stretched coordinate.
To project onto Chebyshev polynomials we define a grid with $n_\textrm{grid}$ points using $$\label{lobatto}
z^i_{\textrm{lobatto}} \equiv \frac{z_\textrm{max}-z_\textrm{min}}{2}\left( \cos \left(\frac{i\pi}{n_\textrm{grid}-1} \right) + 1 \right) + z_\textrm{min}$$ where $i$ is an integer index ranging from $0$ to $n_\textrm{grid}-1$. We evaluate the density function on that grid, $\bar{\rho}(z^i_{\textrm{lobatto}})$. The density function is then expanded in a Chebyshev polynomial basis such that, $$\bar{\rho}(z^i_{\textrm{lobatto}}) = \sum_{k=0}^{n_\textrm{grid}} \hat{\rho}^k T^k(z^i_\textrm{lobatto})$$ where $\hat{\rho}^k$ indicates the $k$-th coefficient for Chebyshev polynomial defined on $z$ coordinates. The coefficients for the derivative of the function, denoted $\hat{\rho}_z^k$, are then computed using a recursion formula, $$\label{cheb_deriv_formula}
c_k \hat{\rho}_z^k = \hat{\rho}_z^{k+2} + 2(k+1)\hat{\rho}^{k+1}$$ where $c_k=2$ for $k=0$, and $c_k=1$ otherwise. Because a Gauss-Lobatto grid was used for the $z$-coordinate, the Chebyshev transformation is performed with a rescaled fast Fourier transformation in $O(n_\textrm{grid} \log n_\textrm{grid})$ operations. The differentiation requires only $O(n_\textrm{grid})$ operations, which means that all of the coefficient functions for the eigenvalue problem can be computed on a relatively fine grid. For example, $n_\textrm{grid} = O(2^{14})$ takes a fraction of a second on commodity hardware from 2015.
The stretched coordinates implemented here are either $s=z$, $s = -g\bar{\rho}/\rho_0$, or $s = \int_D^z N(z^\prime) \, dz^\prime$, where the latter two cases require density to be strictly monotonic. For those two cases if $\bar{\rho}_z \geq 0$ anywhere in the domain, then an error is thrown. For the WKB coordinate, $s = \int_D^z N(z^\prime) \, dz^\prime$, the integral is computed spectrally using equation \[cheb\_deriv\_formula\].
Initialization with gridded data
--------------------------------
In the more typical scenario where a user initializes the `InternalModes` class with gridded data from observations or a numerical model, B-splines are used to interpolate the data and compute the coefficient functions. The primary advantage to using B-splines in this scenario is that B-splines can be created for arbitrary grids without suffering from Runge’s phenomena at lower orders. We fit the data to 6th order interpolating spline (with 5 nonzero derivatives) using the methodology and numerical implementation described in @early2019-arxiv.
If the method requires that density remain monotonic (e.g. for WKB and density coordinates) , then the B-spline fits are constrained to be monotonic following @pya2015-sc. If the data are not monotonic, then this implicitly smooths to find the nearest monotonic fit in a least-squares sense.
Computing the stretched coordinate and the coefficient functions from the spline interpolant requires derivatives and integrals of the B-splines, which are relatively straightforward to compute because they are just piecewise polynomials [@deboor1978-book]. The WKB method requires computing the square root of the B-spline interpolant. The approach taken here is to build a new interpolating spline of the same order that interpolates between the square root of the data points.
Eigenvalue problem
------------------
All three Chebyshev methods solve their respective eigenvalue problems on a Gauss-Lobatto grid of their respective coordinate, i.e., $s=z$, $s = -g\bar{\rho}/\rho_0$, or $s = \int_D^z N(z^\prime) \, dz^\prime$. The Gauss-Lobatto grid in $s$ is defined in the usual way as, $$s^i_{\textrm{lobatto}} \equiv \frac{s_\textrm{max} - s_\textrm{min}}{2}\left( \cos \left(\frac{i\pi}{n_\textrm{evp}-1} \right) + 1 \right) + s_\textrm{min}$$ where the number of points $n_\textrm{evp}$ reflects the size of the eigenvalue problem, and is therefore also an absolute upper bound to the number of modes that may be computed.
Once the Gauss-Lobatto grid $s^i_{\textrm{lobatto}}$ is created, the corresponding value $z(s^i_{\textrm{lobatto}})$ is computed using the bisection method as implemented in @chebfun2014-book. The method is set to terminate with a relative error of $O(10^{-12})$.
The coefficient functions in equation \[generic\_evp\] are now simply evaluated onto the $s^i_{\textrm{lobatto}}$ grid using the interpolant (either a B-spline or Chebyshev). The Chebyshev polynomials and their derivatives ($T$,$T_s$, and $T_{ss}$) are computed using the standard recursion formulas, and then multiplied by the coefficient functions to create eigenvalue matrices. The boundary conditions are implemented by replacing the first and last rows of the matrices $\mathsf{A}$ and $\mathsf{B}$ in .
The eigenvalue problem is then solved using the standard generalized eigenvalue problem solver. This is typically the rate limiting step in the process, taking $O(n_\textrm{evp}^3)$ operations. The resulting eigenvectors now contain coefficients to the Chebyshev polynomials *defined on the stretched coordinate.*
Adaptive grid
-------------
The adaptive grid is created by locating regions in the domain where the solution is expected to be small, and then allocating fewer points (and therefore fewer Chebyshev polynomials) to those regions. After identifying the turning points $z_T$ where $N^2(z_T)=\omega^2$, the WKB solution \[Fwkb\] is used to identify the equation boundaries $z_{\textrm{bnd}}$, the points where $F^{\textrm{WKB}}_j(z_{\textrm{bnd}},\omega)/F^{\textrm{WKB}}_j(z_{T},\omega)=10^{-5}$. The WKB solution is assumed to work locally in the stratification, and is therefore applied at the turning point, $z_T$, in the direction of decaying variance. The eigenvalue \[wkb\_eigenvalue\] is assumed to be set globally by integration over all oscillatory regions. If $m$ equation boundaries $z_{\textrm{bnd}}$ are identified, they delineate $m+1$ regions: the ‘decay’ regions, where the solution is anticipated to be small and governed by the decaying exponential, and ‘oscillatory’ regions where the solution is expected to dominate and include the oscillatory solution. These decaying and oscillating regions are necessarily alternating.
The $m+1$ regions are coupled using the same technique described in section 22.3 of @boyd2001-book by requiring continuity across boundaries for $G$ and $\partial_z G$. The key benefit to this algorithm is that the decay regions are allocated as few as 6 points each, while oscillatory regions are apportioned the remaining points relative to their WKB length, $L^m = \int N(z^\prime) dz^\prime$. While we find that 6 points appears to be sufficient for the decay regions, in practice, we do in fact apportion 1/16th of the total points evenly between the decay regions as a hedge that this ad-hoc method will fail for some unforeseen cases.
The adaptive grid algorithms use low-order interpolation to identify $z_T$ and $z_{\textrm{bnd}}$ because high accuracy is not required and this keeps the number of computations $O(n_\textrm{grid})$.
Normalization {#normalization}
-------------
The final step is to normalize the resulting eigenvectors and project them onto $z_\textrm{out}$. Normalization using the two integral conditions can be performed exactly invoking the fact that the integral of each Chebyshev polynomial $T^k(z)$ is exactly known, $$w^k \equiv \int_{-1}^{1} T^k(z) \, dz = \begin{cases}
\frac{(-1)^k + 1}{1-k^2} & k \neq 1 \\
0 & k=1
\end{cases}.$$ We have defined $w^k$ such that it can be summed with a vector of Chebyshev coefficients to produce the integral. In other words, if $\hat{v}^k$ is a Chebyshev coefficient vector, then $I = \sum \hat{v}^k w^k$ is the definite integral. The integrands in equations \[k\_const\_norm\_stretched\] and \[omega\_const\_norm\_stretched\] are computed pseudospectrally before integrating (by transforming to the spatial domain, multiplying, then transforming back Chebyshev coefficients).
Computing the max U and and max W norm is more problematic because the function extrema do not necessarily lie on the grid points. For the implementation, we simply take the maximum at the resolved grid points, but if higher accuracy is required, one could locate the extrema using the methods in @boyd2014-book.
Finally, the normalized eigenmodes are projected onto the output grid using the slow Chebyshev transforms, $$\label{output_projection}
v(z^i_{\textrm{out}}) = \sum_{k=0}^{n_\textrm{evp}} \hat{v}^k T^k(s(z^i_\textrm{out})).$$ If a large number of output points are requested, this operation could dominate the total computation time.
The algorithm flowchart for the mode computation is shown in figure \[ModesAlgorithm\]
![Algorithm flowchart for the mode computation.[]{data-label="ModesAlgorithm"}](ModesAlgorithm){width="19pc"}
SQG modes
---------
The two functions for computing the SQG modes are
psi = im.SurfaceModesAtWavenumber(k);
and
psi = im.BottomModesAtWavenumber(k);
where `k` is an array of wavenumbers.
The SQG modes are found from equation \[sqg\_cheb\_eqn\] using a linear solver after replacing the top and bottom points of the matrix with the boundary conditions. As noted in @tulloch2009-jas-note, the SQG modes require a high density of grid points near the boundaries, a task well suited to the Gauss-Lobatto grid in equation \[lobatto\]. The number of Chebyshev polynomials is chosen so that the Gauss-Lobatto grid captures at least 10 points over the e-folding scale. The e-fold scale for constant stratification (see appendix \[sec:sqg\_constant\_strat\]) is $\Delta z_{\textrm{efold}} = \frac{f_0}{K N_0}$ and the distance between the first two points in a Lobatto grid, equation \[lobatto\], is $$\Delta z_{\textrm{boundary}} = \frac{D}{4} \left( \frac{\pi}{(n_{\textrm{grid}}-1)} \right)^2.$$ where we’ve defined $D=z_\textrm{max}-z_\textrm{min}$ as the depth of the domain. Setting $\Delta z_{\textrm{boundary}} = \frac{1}{10} \Delta z_{\textrm{efold}}$ we find that we need $$n_{\textrm{grid}} = 1+\frac{\pi}{2} \sqrt{ \frac{10 D K N_0}{ f_0} }$$ points (and therefore also $n_{\textrm{grid}}$ polynomials) to sufficiently capture the SQG mode. The resulting SQG modes are projected onto the output grid using equation \[output\_projection\].
Unit testing
------------
In order to ensure that each of the algorithm implementations is correct, the numerically generated eigenmodes and eigenvalues are compared against the analytical solutions for constant stratification and exponential stratification shown in \[sec:analytical\_im\_solutions\]. The comparison is performed across a range of wavenumbers and frequencies for both surface boundary conditions and all four norms.
The computed SQG modes are also compared against analytical solutions for constant and exponential stratifications, shown in \[sec:analytical\_sqg\_solutions\].
Other sources of error {#sec:error}
======================
In the introduction we noted five sources of error that contribute to the total error when computing the forward or inverse transformation, but this manuscript has primarily focused on one source of error: the numerics of accurately representing the vertical modes in the EVP. We now discuss these other sources of error and how they are dealt with in the numerical implementation.
Aliasing error {#sec:aliasing_error}
--------------
When performing a forward transformation, where a given dynamical field is projected onto the vertical modes, the data grid will determine how many modes are resolvable. As with a Fourier transformation, higher frequency modes alias into the lower frequency modes. Unlike the Fourier transformation, however, the optimal grid for performing a transformation is not an evenly spaced grid, but depends on the stratification profile, and therefore the eigenvalue problem being solved. Here we show that there is a relatively easy method for determining the number of resolvable modes for a given grid using the condition number of the resulting matrix.
To show the effect of different grid choices on the forward transformation, we use the analytical solution of the vertical modes in exponential stratification in combination with an imposed spectrum to generate stochastic isopycnal displacement profiles typical of the world oceans. In particular, we use the Garrett-Munk spectrum, $$H(j)= \frac{H_0}{(j + j_\ast)^p}$$ where $j_\ast$ is the roll-off mode, usually set to 3 but possibly as high as 20, $p$ is the slope which is usually very nearly $5/2$, and $H_0$ normalizes sum over $j=1..\infty$ to unity. For each stochastically generated set of coefficients, $m^j$, a profile $\eta(z) = \sum_{j=1}^N G^j(z) m^j$ is created. The profile is then evaluated on three different grids: $z^i_\textrm{even}$, $z^i_\textrm{lobatto}$, and $z^i_\textrm{quadrature}$ where $z^i_\textrm{quadrature}$ is the grid of Gaussian quadrature points, determined by the roots of a mode one higher than is trying to be used [@press1992-book; @boyd2001-book]. Using the first $n$ modes, we then attempt to recover the coefficients using least squares—in practice this is Matlab’s `mldivide` (`\`) operator. For example the first $n$ coefficients of the evenly spaced grid are determined by, $$\tilde{m}^j_\textrm{even} = G^{j}(z^i_\textrm{even}) \backslash \eta(z^i_\textrm{even})$$ where $j=1..n$. The root-mean square error (rmse) is defined as the error in the sum of recovered and missing coefficients, $$\textrm{rmse}^2 \equiv \frac{\sum_{j=1}^n \left(m^j-\tilde{m}^j\right)^2 + \sum_{j=n+1}^N \left(m^j\right)^2}{\sum_{j=1}^N \left(m^j\right)^2}$$
![The top panel shows the root mean square error as a function of total modes used to recover the coefficients with an inverse transformation. Vertical dashed lines are the predicted cutoffs for the different grids, based on the matrix condition number at which the modes are no longer resolvable. The condition number as a function of total modes is shown in the bottom panel.[]{data-label="ModeRecoverabilityForDifferentGrids"}](ModeRecoverabilityForDifferentGrids){width="19pc"}
Figure \[ModeRecoverabilityForDifferentGrids\] shows the result of trying to recover the mode coefficients, $\tilde{m}^j$ using successively more modes for the three different grids. In all three cases the rmse decreases as modes are added until a dramatic increase occurs, correlating with a similarly dramatic increase in the condition number of the matrix. The quadrature grid, defined as the roots of the $G^{N-1}$ mode plus the boundaries, performs best, as expected. Including the boundaries in this definition means that the top and bottom boundary points provide no useful information, and therefore only $N-2$ modes are recoverable for $G$, and $N-1$ for $F$ with a rigid lid. This definition is chosen so that the forward and inverse transform for constant stratification coincides with the discrete sine and cosine transform (and their associated grids).
While the condition number of the matrix is clearly controlled by the grid being used, the choice of norm also affects the condition number. Generally speaking, the \[k\_const\_norm\] performs well for transformations with the $G$ modes and the \[omega\_const\_norm\] with the $F$ modes. The exception to this is when the free-surface boundary condition is used, the barotropic mode has a substantially different $L^2$ norm and should be rescaled.
Mode error {#sec:mode_error}
----------
Here, we test whether or not the truncation error in the vertical modes is a limitation in recovering the coefficients of the spectrum, $m_j$. To this end, we created profiles on a quadrature grid with variance distributed using $H(j)$ for a range of parameters including white noise (large $j_\star$), and very smooth (large $p$), as described above, and recorded how many mode coefficients were recoverable for some error tolerance across all the different numerical methods in this paper. We found that discarding modes with truncation errors exceeding the requested error tolerance of the modes guarantees coefficients recovered with the requested error tolerance. In other words, if one wants relative errors in coefficients of less than $10^{-2}$, one needs modes with errors less than $10^{-2}$. The only exception is for very steep spectra (e.g., $p=-10$), where the coefficients of the higher modes are indistinguishable from zero. The reverse of this is not true—in fact, including modes with truncation errors exceeding the error tolerance, can often return coefficients within the bounds of the error tolerance. Evidently, the errors in these ordered, orthogonal bases work systematically in our favor.
Interpolation error
-------------------
Another source of error may arise from interpolation of the density function. This issue is treated separately from a poorly defined or noisy mean density function and therefore assumes that the data given is gridded and without error. For gridded data, our method uses a relatively low-order B-spline to interpolate between grid points which is then evaluated for the coefficient functions where needed. Does this low-order interpolation method limit the mode recoverability described in the previous section?
To address this question we compute the vertical modes for exponential stratification from an evenly spaced density function with variable number of grid points. These modes are then used to recover the coefficients $\tilde{m}^j$, as above, on a high resolution quadrature grid. We find that with as few as 16 grid points for the density function, we are able to recover 90 mode coefficients with less than 1% error.
Poorly defined or noisy mean density
------------------------------------
It is often the case that the mean density function, $\bar{\rho}(z)$, is not easily defined. Averaging over a mooring time series of density will not necessarily result in a monotonic density function—and averaging often removes the sharp gradients that exist in individual profiles, which may not be desirable. Even output from numerical models can suffer from these same issues, depending on the boundary conditions. Noisy data, where errors in the observed value of $\bar{\rho}(z)$ stem from instrument errors, also effectively constitute a poorly defined mean. One way to frame these issues is to ask how a misspecified mean density affects our ability to infer the vertical spectrum of a given flow.
First we note that all methods, as implemented, will proceed without error for noisy data. However, the most notable difference between methods is that the WKB and density coordinate methods use a density function constrained to the nearest monotonic spline fit, as previously described. It is not clear that this implicit smoothing is necessarily ‘better’ than using the unaltered density function with the z-coordinate method for noisy data. This decision, and how to deal with measurement noise in general, is beyond the scope of this manuscript. Additional smoothing of the density data can be done using many techniques, including the constrained smoothing splines described in @early2019-arxiv.
In order to test the effects of misspecifying a mean density profile, we generate density profiles in exponential stratification that follow the GM spectrum, as above, but we attempt to recover the coefficients using vertical modes computed from a noisy mean density profile. The results are consistent with section \[sec:mode\_error\]. For example, as long as the modes from the noisy profile have errors less than $10^{-2}$ relative to the modes generated from the true profile, the recovered coefficients $\tilde{m}_j$ will have errors less than $10^{-2}$ relative to $m_j$. The relative error of the vertical modes increases as a function of mode number until eventually the mode errors and therefore coefficient errors reach $O(1)$. Exactly where mode errors reach $O(1)$ depends on the details of the noise, or misspecification, of the mean density profile.
While accurately recovering the coefficients $m_j$ of the spectrum becomes impossible with a noisy mean density profile, we are able to accurately infer the spectrum from which $m_j$ was generated by either ensemble averaging over additional synthetic profiles, or bin averaging nearby modes. One way to see why this might be true is to note that the coefficient errors never exceed $O(1)$—so although the exact coefficient is incorrect, the magnitude is correct on average. In practice, using modes from the misspecified mean density profile causes variance that should be associated with mode $j=33$, for example, to be assigned to the variance of nearby modes.
That the spectrum is recoverable despite a noisy mean is consistent with previous analysis methods. In @polzin2011-rg, internal wave spectra are found using WKB approximated modes and a WKB stretched grid. It is also important to note that computing the spectrum with a misspecified mean density function still requires an orthogonal set of modes, and therefore fast and accurate mode computation is still helpful.
Discussion {#sec:discussion}
==========
The spectral methods presented here solve the most relevant forms of the vertical eigenvalue and surface quasigeostrophic mode problems efficiently and accurately. The methods also include an algorithm for computing modes in challenging stratification profiles at high frequencies near turning points. The algorithms are implemented in a publicly available Matlab suite using the class hierarchy described in \[sec:class\_hierarchy\] and the implementations are validated against known analytical solutions, under a wide range of conditions. However, the methods do not always perform well under all conditions.
Poorly resolved features and discontinuities in the density profile will produce the Gibbs phenomenon, where ‘ringing’ occurs in the vicinity of the discontinuity. In one example, we found that a narrow 5-meter-wide pycnocline in a 5000-meter-deep ocean produced strong spurious oscillations unless the pycnocline was sufficiently well resolved with enough grid points. In another example, we defined an analytical profile with a discontinuity in $\bar{\rho}_{zz}$ (the highest derivative used in the eigenvalue problem) and this also produced the Gibbs phenomenon. Interestingly, in these cases lower-order finite differencing produced better modes for the same vertical resolution, because these methods implicitly smooth the derivatives. A logical extension of this work would be to apply the ‘splitting’ algorithms in `chebfun` [@chebfun2014-book] to handle such discontinuities.
Our recommendations for projecting observed or modeled fields onto the modes are as follows,
- When possible, use a quadrature grid for the fields and modes, e.g. the function `GaussQuadraturePointsForModesAtWavenumber` in the `InternalModesSpectral` class computes the Gauss quadrature points for $G$. This is a relatively expensive operation (it involves solving the EVP), but provides near- optimal point placement.
- In the usual case where there is no freedom to choose grid points, compute the condition number as a function of mode as described in section \[sec:aliasing\_error\], and limit the number of modes to a low condition number. Alternatively, if computation time is not a limitation, one can perform the least-squares fit of the fields to successively more modes until the coefficients are no longer stable.
Many of the errors described in section \[sec:error\] may still be a concern, but may be quantified with some of the techniques in section \[sec:error\], by using the density function and spectrum specific to the problem.
Internal mode solutions {#sec:analytical_im_solutions}
=======================
We present analytical solutions for the internal mode eigenvalue problem in three different scenarios: constant stratification, exponential stratification, and the WKB approximated solution for arbitrary stratification. These solutions are used to validate the numerical implementations.
Constant stratification {#sec:im_constant_strat}
-----------------------
trigonometric linear hyperbolic
------------------- --------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------- ----------------------------------------------------------------------------------------------------------------
$K$-constant $(-1)^j\sqrt{\sin^2\left(m_jD\right)+\frac{(N_0^2-f_0^2)D}{2g} \left( 1 - \frac{\sin \left( 2 m_j D \right)}{2 m_j D} \right)}$ $D\sqrt{1+\frac{(N_0^2 - f_0^2)D}{3 g}} $ $\sqrt{\sinh^2\left(m_0 D\right)+\frac{(N_0^2 - f_0^2)D}{2g}\left(\frac{\sinh(2 m_0 D)}{2m_0 D} - 1 \right) }$
$\omega$-constant $(-1)^jh_j m_j \sqrt{ \frac{1}{2} + \frac{\sin \left( 2 m_j D \right)}{4 m_j D} }$ $D$ $ h_0 m_0 \sqrt{ \frac{\sinh \left( 2 m_0 D \right)}{4 m_0 D} + \frac{1}{2} }$
$U_\textrm{max}$ $(-1)^jh_j m_j$ $D$ $h_0 m_0 \cosh( m_0 D)$
$W_\textrm{max}$ $(-1)^j, \sin(m_0 D)\textrm{ for }j=0$ $D$ $\sinh(m_0 D)$
\[normalizationtable\]
The internal baroclinic modes in constant stratification are given as, $$\begin{aligned}
\label{baroclinic_g_mode}
G^{\textrm{const}}_j(z) =& A \sin \left( m_j ( z + D) \right)\\ \label{baroclinic_f_mode}
F^{\textrm{const}}_j(z) =& A h_j m_j \cos \left( m_j ( z + D) \right).\end{aligned}$$ with eigendepth $h_j$ and vertical wavenumber $m_j$ given by $$m_j = \frac{j\pi}{D} + \frac{\xi}{D}$$ where we have assumed that $w=0$ at the lower boundary $z=-D$. In the case of a rigid lid ($w=0$ at $z=0$), the correction to the vertical wavenumber is $\xi=0$. However, if the linearly approximated free surface boundary condition is used, $h_j G_j(0)=G_j(0)$, then $\xi$ is nonzero. The equations for $\xi$ are transcendental and are therefore solved with a numerical root finding algorithm. The equations for $\xi$ are written in a form conducive for finding the desired root.
- For fixed wavenumber, $k$, the vertical wavenumber correction $\xi$ is found by solving, $$\label{baroclinic_fixed_k}
(\xi + j \pi) \left(N_0^2 - f_0^2\right) D \cos(\xi) - g \left( k^2 D^2 + (\xi + j \pi)^2 \right) \sin(\xi) = 0$$ near $\xi=0$ and the eigendepth $h_j$ is given by, $$h_j = \frac{1}{g} \frac{N_0^2 - f_0^2}{k^2+m_j^2}.$$
- For fixed frequency, $\omega$, the vertical wavenumber correction $\xi$ is found by solving, $$\label{baroclinic_fixed_omega}
D(N_0^2 - \omega^2) - g (\xi + j \pi)\tan(\xi) = 0$$ near $\xi=0$ and the eigendepth $h_j$ is given by, $$h_j = \frac{1}{g} \frac{N_0^2 - \omega^2}{m_j^2}.$$
The normalization for these modes is given in the first column of table \[normalizationtable\].
In the case of the free surface boundary condition, there also exists a barotropic mode ($j=0$), the solution of which changes from trigonometric to hyperbolic at $\omega = N_0$, or $k=k_\ast$ where $$k_\ast \equiv \sqrt{\frac{N_0^2 - f_0^2}{g D}}.$$ In these cases then, the vertical wavenumber reduces to $m_0=\xi/D$.
- Trigonometric case, $k<k_\ast$ or $\omega < N_0$
The solution is exactly the same as the baroclinic solutions given above in equation \[baroclinic\_g\_mode\] and \[baroclinic\_f\_mode\], but now equation \[baroclinic\_fixed\_k\] is solved when $j=0$. As a practical matter, it is numerically more stable to solve $$\left(N_0^2 - f_0^2\right) D - g \xi^{-1} \left( k^2 D^2 + \xi^2 \right) \tan(\xi) = 0$$ for the positive root near $\xi = \sqrt{(N_0^2-f_0^2)D/g - k^2 D^2}$. For fixed frequency, equation \[baroclinic\_fixed\_omega\] can be used to find the root near $\xi=\sqrt{D(N_0^2 - \omega^2)/g}$.
- Linear case, $k=k_\ast$ or $\omega = N_0$
The solution is given by $$\begin{aligned}
G^{\textrm{const}}_0(z) =& A ( z + D)\\
F^{\textrm{const}}_0(z) =& A D\end{aligned}$$ where $h_0=D$.
- Hyperbolic case, $k>k_\ast$ or $\omega > N_0$
The solution is given by $$\begin{aligned}
G_j(z) =& A \sinh \left( m_0 ( z + D) \right)\\
F_j(z) =& A h_0 m_0 \cosh \left( m_0 ( z + D) \right).\end{aligned}$$
- For fixed wavenumber, $k$, the vertical wavenumber correction $\xi$ is found by solving $$\left(N_0^2 - f_0^2\right) D - g \xi^{-1} \left( k^2 D^2 - \xi^2 \right) \tanh(\xi) = 0$$ for the root near $\xi = \sqrt{k^2 D^2 - (N_0^2-f_0^2)D/g}$ and the eigendepth $h_0$ is given by, $$h_0 = \frac{1}{g} \frac{N_0^2 - f_0^2}{k^2 + m_0^2}.$$
- For fixed frequency, $\omega$, the vertical wavenumber correction $\xi$ is found by solving $$\left(\omega^2 - N_0^2\right) D - g \xi \tanh(\xi) = 0$$ for the root near $\xi = \sqrt{D(\omega^2 - N_0^2)/g}$ and the eigendepth $h_0$ is given by, $$h_0 = \frac{1}{g} \frac{\omega^2 - N_0^2}{m_0^2}.$$
Exponential stratification {#sec:im_exponential_strat}
--------------------------
Exponential stratification serves as the $O(1)$ representation of the ocean stratification away from the poles and continental boundaries. As formulated and first solved in @garrett1972-gfd, we take the stratification to be $N^2 = N_0^2 e^{2z/b}$ where $N_0$ is buoyancy frequency and $b$ is the thermocline depth (with canonical values of 3 cycles per hour and 1300 meters).
Letting the stretched coordinate $s = N_0 e^{z/b}$ as in section \[sec:stretched\_coord\], the \[vertical-eigenvalue-G-with-omega\] becomes, $$\begin{aligned}
s^2 G_{ss} + s G_s + \frac{b^2}{c^2} \left( s^2 - \omega^2 \right)G = 0\end{aligned}$$ which has solution $$G^\textrm{exp}_j (z) = J_\nu \left( \frac{b N_0}{c_j}e^{z/b} \right) - \alpha_j Y_\nu \left( \frac{b N_0}{c_j}e^{z/b} \right)$$ where $$\alpha_j \equiv J_\nu \left( \frac{b N_0}{c_j}e^{-D/b} \right) / Y_\nu \left( \frac{b N_0}{c_j}e^{-D/b} \right)$$ is chosen to satisfy the lower boundary condition $G(-D)=0$. The Bessel function $Y_\nu(s)$ has a singularity at $s=0$, so for many choices of $\omega$ and $c_j$, $\alpha_j \ll 1$ and the second term needs to be neglected for stable numerical evaluation. The order of the Bessel function is set by the frequency $\nu=\frac{b \omega}{c}$, or, using the dispersion relation, wavenumber $\nu = \sqrt{ \frac{b^2 f_0^2}{c^2} + b^2 k^2}$. The $F$ modes are found by taking the derivative, $$\begin{split}
F^\textrm{exp}_j(z) = \frac{N_0}{2g}e^{z/b} \Bigg[ J_{\nu-1} \left( \frac{b N_0}{c_j}e^{z/b} \right) - J_{\nu+1} \left( \frac{b N_0}{c_j}e^{z/b} \right) \\ + \alpha_j \left(Y_{\nu-1} \left( \frac{b N_0}{c_j}e^{z/b} \right) - Y_{\nu+1} \left( \frac{b N_0}{c_j}e^{z/b} \right) \right) \Bigg]
\end{split}$$
The discretization into modes is a result of applying the second boundary condition, in this case either a rigid lid $G(0)=0$ or a free-surface $G(0) = F(0)$, and then finding the eigenmode speeds $c_i = \sqrt{gh_i}$ that satisfy those conditions. The $c_i$’s are therefore determined by the roots of Bessel functions, for which there is no general closed form solution. The challenge then becomes finding bounds for the roots and writing the equation in a form suitable for a root finding algorithm.
In practice it’s easiest to find the inverse of the $c_i$’s, so we let $x=\frac{b N_0}{c}$ and then write the root equation, $f(x)=0$, in terms of parameter function $\nu(x)$ and $s(x)$. Again, stable numerical evaluation requires that we work around the singularity of $Y_\nu$, so if $\nu(x)<s(x) e^{-D/b}$, then we take $$f_{\nu_\textrm{small}}(x) = A(x) J_{\nu(x)}\left( e^{-D/b} s(x) \right) + B(x) Y_{\nu(x)}\left( e^{-D/b} s(x) \right)$$ and when $\nu(x)>s(x) e^{-D/b}$ we take $$f_{\nu_\textrm{big}}(x) = A(x) J_{\nu(x)}\left( e^{-D/b} s(x) \right) / Y_{\nu(x)}\left( e^{-D/b} s(x) \right) + B(x)$$ where in either case the rigid-lid condition requires $$\begin{aligned}
A(x) =& Y_{\nu(x)}\left( s(x) \right) \\
B(x) =& -J_{\nu(x)}\left( s(x) \right) \end{aligned}$$ and the free-surface requires, $$\begin{aligned}
A(x) =& Y_{\nu(x)}\left( s(x) \right) - \frac{\alpha}{s(x)} \left( Y_{\nu(x)-1}\left( s(x) \right) - Y_{\nu(x)+1}\left(s(x)\right) \right) \\
B(x) =& -J_{\nu(x)}\left( s(x) \right) + \frac{\alpha}{s(x)} \left( Y_{\nu(x)-1}\left( s(x) \right) - Y_{\nu(x)+1}\left(s(x)\right) \right).\end{aligned}$$
- $\omega$-constant solution
The parameter functions are defined as, $$\begin{aligned}
\nu(x) =& \frac{\omega x}{\eta} \\
s(x)=& \frac{N_0 x}{\eta}\end{aligned}$$ where the scaling factor $\eta$ is chosen from the WKB approximated solution [@desaubies1973-gfd],
- $\omega > N_0 e^{-D/b}$
The root equation must be taken to be $f_{\nu_\textrm{big}}(x)$ and, $$\eta \pi = \sqrt{N_0^2 - \omega^2} - \omega \cos^{-1}\left( \frac{\omega}{N_0} \right)$$ (from @desaubies1973-gfd eqn 2.18) the first $n$ modes are found within $[0.5, n+1]$.
- otherwise
The root equation must be taken to be $f_{\nu_\textrm{small}}(x)$ and, $$\begin{split}
\eta \pi = \sqrt{N_0^2 - \omega^2} - \omega \cos^{-1}\left( \frac{\omega}{N_0} \right) \\ - \sqrt{N_0^2 e^{-2D/b} - \omega^2} + \omega \cos^{-1}\left( \frac{\omega}{N_0} e^{D/b} \right)
\end{split}$$ (from @desaubies1973-gfd eqn 2.19) the first $n$ modes are found within $[0.5, n+1]$.
- $K$-constant solution
Unlike the $\omega$-constant solution, the order of the Bessel function changes for each root, and therefore the root equation being used may have to change during evaluation.
The parameter functions are defined as, $$\begin{aligned}
\nu(x) =& \sqrt{\epsilon^2 x^2 + \lambda^2} \\
s(x)=& x\end{aligned}$$ where $\epsilon=\frac{f_0}{N_0}$ and $\lambda = b k$. @desaubies1973-gfd provides low-frequency (lf) and high-frequency (hf) bounds for the eigenvalues, which we can be written in terms of wavenumber, $$\begin{aligned}
x_{\textrm{lf}}(j) =& \left( j - \frac{1}{4} \right) \pi + \lambda \frac{\pi}{2} \\
x_{\textrm{hf}}(j) =& \lambda \left[ 1 + \frac{1}{2} \left( \frac{3\pi(4j-1)}{\lambda 8 \sqrt{2}} \right)^\frac{2}{3} \right]\end{aligned}$$ We transition at $$x_\nu = \frac{\lambda}{\sqrt{5^2 e^{-2D/b} - \epsilon^2}}$$ To set the search bounds for the root algorithm, we want $$x_\textrm{lower} =
\begin{cases}
x_{\textrm{lf}}(1) & \lambda < 2(1-\frac{1}{4})\cdot 10^{-1} \\
x_{\textrm{hf}}(1) & \textrm{otherwise}
\end{cases}$$ and, $$x_\textrm{upper} =
\begin{cases}
x_{\textrm{lf}}(1.1n) & \lambda < n-\frac{1}{4} \\
x_{\textrm{hf}}(5n) & \textrm{otherwise}
\end{cases}$$ so the first $n$ modes are found within $[x_\textrm{lower}, x_\textrm{upper}]$.
Normalization is performed by direct evaluation of the mode functions for the $U_\textrm{max}$ and $W_\textrm{max}$, and by numerical integration for the \[k\_const\_norm\] and \[omega\_const\_norm\] norms.
WKB solution {#sec:im_wkb_solution}
------------
@desaubies1973-gfd found the WKB solution for stratification with at most one turning point $z_T$. Simplified and using the notation of this manuscript, the WKB solution with single turning point $z_T$ is, $$\begin{aligned}
\label{Gwkb}
G_j^{\textrm{WKB}}(z,\omega) &= A \sqrt{\pi} \left( \frac{\xi}{\omega^2 - N^2}\right)^{\frac{1}{4}} \operatorname{Ai}(\xi)\\ \label{Fwkb}
F_j^{\textrm{WKB}}(z,\omega) &= A \sqrt{\pi} \left( \frac{\xi}{\omega^2 - N^2}\right)^{\frac{1}{4}} \xi_z \operatorname{Ai}^\prime(\xi)\end{aligned}$$ where $\xi = \operatorname{sgn}(\omega^2-N^2) \left(\frac{3}{2} q \right)^{\frac{2}{3}}$ and we’ve defined $$\label{q_eqn}
q(z) = \frac{1}{\sqrt{gh_j}} \left| \int_{z_T}^z \sqrt{|N^2-\omega^2|}\,dz \right|$$ so that it goes to zero at the turning point, but is positive everywhere else. The eigenvalue is proportional to the integral of the stratification over the oscillatory region, $$\label{wkb_eigenvalue}
\sqrt{gh_j} = \frac{1}{\left(j-\frac{1}{4}\right)\pi}\left| \int_{z_T}^0 \sqrt{|N^2-\omega^2|}\,dz \right|$$ while normalization coefficient for the \[k\_const\_norm\] is $$A(\omega) = (-1)^j \sqrt{2 g} \left[ \int_{z_T}^0 \frac{N^2 - f_0^2}{\sqrt{N^2 - \omega^2} } dz \right]^{-\frac{1}{2}}.$$ It is useful to note that away from the turning point, the Airy function $\operatorname{Ai}(\xi)$ can be approximated as sinusoidal above the turning point and exponentially decaying below the turning point, $$\begin{gathered}
G_j^{\textrm{WKB}}(z,\omega) = \\
\frac{1}{\sqrt{2}} \frac{A(\omega)}{\sqrt[4]{|N^2 - \omega^2|}} \cdot
\begin{cases}
\frac{1}{\sqrt{2}} e^{-q}, & \text{for } z>z_T\\
\sin q + \cos q, & \text{for } z<z_T
\end{cases}\end{gathered}$$
In the case of no turning point ($\omega < N(z), \forall z \in [-D,0]$) the WKB solution is $$\begin{aligned}
G_j^{\textrm{WKB}}(z,\omega) =& \frac{A(\omega)}{\sqrt[4]{N^2 - \omega^2}} \sin q \\
F_j^{\textrm{WKB}}(z,\omega) =& -\frac{A(\omega) N^2_z h_j}{4(N^2 - \omega^2)^{5/4}} \sin q \\
&+ A(\omega) \sqrt{\frac{h_j}{g}} \sqrt[4]{N^2 - \omega^2} \cos q\end{aligned}$$ where we’ve taken $z_T = -D$ in \[q\_eqn\]. The eigenvalue is now $$\sqrt{gh_j} = \frac{1}{j \pi}\left| \int_{-D}^0 \sqrt{|N^2-\omega^2|}\,dz \right|$$ Note that this solution does reduce to the exact solution for constant stratification.
SQG mode solutions {#sec:analytical_sqg_solutions}
==================
The SQG modes are computed by taking the Fourier transform of $\psi$ in equation \[sqg\_eqn\] and scaling by the Fourier transformed boundary condition $\hat{\rho}(k,l,z=0)$ such that $\psi(x,y,z) = \int \int \left(-\frac{g}{\rho_0} \hat{\rho} \bigr\rvert_{z=0}\right)\phi(k,l,z) e^{i(kx+ly)} \, dk \, dl$. Equation \[sqg\_eqn\] then becomes, $$\label{sqg_wavenumber_eqn}
0 = -K^2 \phi + \frac{\partial}{\partial z} \left( \frac{f_0^2}{N^2} \frac{\partial \phi }{\partial z} \right)$$ with surface boundary condition $f_0 \partial_z \phi = 1$ and bottom boundary boundary condition $\partial_z \phi = 0$ (and vice-versa for the bottom boundary modes).
To solve this problem numerically, we project $\phi$ onto a Chebyshev basis where ${\boldsymbol{\mathrm{v}}} = \mathsf{T} {\boldsymbol{\mathrm{\hat{v}}}}$ is the vector representation of $\phi$. The equation for the SQG modes becomes $$\label{sqg_cheb_eqn}
\left[ N^2 \partial_{zz}\mathsf{T} - \partial_z ( N^2 ) \partial_z \mathsf{T}- \frac{K^2 N^4}{f_0^2} \mathsf{T} \right] {\boldsymbol{\mathrm{\hat{v}}}} = 0$$ in the interior with boundary conditions $f_0 \partial_z \mathsf{T} {\boldsymbol{\mathrm{\hat{v}}}} = 1$ and $f_0 \partial_z \mathsf{T} {\boldsymbol{\mathrm{\hat{v}}}} = 0$ for the surface mode.
The numerical implementation of equation \[sqg\_cheb\_eqn\] is validated against the known analytical solutions.
Constant stratification {#sec:sqg_constant_strat}
-----------------------
The surface quasigeostrophic modes for constant stratification can be found in [@tulloch2009-jas]. For the upper boundary this is, $$f_0 \phi_{\textrm{sur}}^{\textrm{const}}(z) = \frac{1}{\lambda} \frac{\cosh\left( \lambda (z+D) \right) }{\sinh\left( \lambda D \right)}$$ and the lower boundary, $$f_0 \phi_{\textrm{bot}}^{\textrm{const}}(z) = -\frac{1}{\lambda} \frac{\cosh\left( \lambda z \right) }{\sinh\left( \lambda D \right)}$$ where $\lambda = \frac{KN_0}{f_0}$. These solutions cannot be evaluated numerically for all wavenumbers and depths because the $\sinh$ function may overflow. Instead, we use $$f_0 \phi_{\textrm{sur}}^{\textrm{const}}(z) = \frac{1}{\lambda} \frac{e^{\lambda z} + e^{-\lambda(z+2D)}}{1 - e^{-2\lambda D}}$$ and $$f_0 \phi_{\textrm{bot}}^{\textrm{const}}(z) = -\frac{1}{\lambda} \frac{e^{\lambda (z-D)} + e^{-\lambda(z+D)}}{1 - e^{-2\lambda D}}$$ for reliable numerical evaluation.
Exponential stratification {#sec:sqg_exponential_strat}
--------------------------
The surface SQG modes for exponential stratification are first solved by @lacasce2012-jpo. Defining the scale $\eta \equiv \frac{N_0 K b}{2 f_0}$, the surface mode is given by $$\phi_{\textrm{sur}}^{\textrm{exp}}=\frac{e^\frac{z}{b}}{N_0 K} \frac{K_0\left(2\eta e^{-\frac{D}{b}}\right)I_1\left(2\eta e^\frac{z}{b}\right) + I_0\left(2\eta e^{-\frac{D}{b}}\right) K_1\left(2\eta e^\frac{z}{b}\right)}{I_0(2\eta)K_0\left(2\eta e^{-\frac{D}{b}}\right) - K_0(2\eta) I_0\left(2\eta e^{-\frac{D}{b}}\right)}$$ while the bottom mode can be shown to be, $$\phi_{\textrm{bot}}^{\textrm{exp}}=\frac{e^\frac{z+2D}{b}}{N_0 K} \frac{K_0(2\eta)I_1\left(2\eta e^\frac{z}{b}\right) + I_0(2\eta) K_1\left(2\eta e^\frac{z}{b}\right)}{K_0(2\eta)I_0\left(2\eta e^{-\frac{D}{b}}\right) - I_0(2\eta) K_0\left(2\eta e^{-\frac{D}{b}}\right)}.$$ Bessel functions $K_0$ and $K_1$ should not be confused with the wavenumber $K$.
WKB solution {#sec:wkb_sqg_solution}
------------
The WKB solution for the SQG mode does not appear to have been previously derived. Assuming a solution of the form $\phi = e^{q(z)}$ and inserting this into equation \[sqg\_eqn\], we have that, $$q^{\prime\prime} + {q^\prime}^2 - 2 \left( \ln N \right)^\prime q^\prime - N^2 K^2 = 0.$$ We then assume a series expansion $q=q_0 + \epsilon q_1 + ...$ combined with the assumption that both $q^{\prime\prime}$ and $N^\prime$ are $O(\epsilon)$. The two lowest order equations are therefore, $$\begin{aligned}
{q_0^\prime}^2 - N^2 K^2 =& 0 & O(1) \\
q_0^{\prime\prime} + 2 q_0^\prime q_1^\prime - 2 \left( \ln N \right)^\prime q_0^\prime &= 0 & O(\epsilon)\end{aligned}$$ which has solution, $$\phi(z) \approx \sqrt{ \frac{N(z)}{N_0} } \left[ A e^{\frac{K}{f_0} \int_{-D}^z N(z)} + B e^{-\frac{K}{f_0} \int_{-D}^z N(z)} \right].$$ Applying boundary condition $f_0 \phi_z(-D) = 0$ and $f_0 \phi_z(0) = 1$, the surface mode is $$\begin{aligned}
\phi_{\textrm{sur}}^{\textrm{WKB}}(z) \approx& \frac{1}{K N_0} \sqrt{ \frac{N(z)}{N_0} } \left[\frac{\alpha e^{s(z)} + e^{\left( -s(z) + 2 s(-D)\right)} }{ \alpha \left( b + 1 \right) + \left(b - 1 \right)e^{2s(-D)} }\right]\end{aligned}$$ where $s(z) \equiv \frac{K}{f_0} \int_{0}^z N(z) \, dz$ in the integral from the surface to depth, while $a=\frac{f_0 N_z(-D)}{2K N^2(-D)}$, $b=\frac{f_0 N_z(0)}{2K N^2_0}$, and $\alpha = \frac{1-a}{1+a}$.
For constant stratification this reduces to the result in appendix \[sec:sqg\_constant\_strat\].
Class hierarchy {#sec:class_hierarchy}
===============
![Class hierarchy for the Matlab implementation of the algorithms in this manuscript. `InternalModesBase` is an abstract class.[]{data-label="ClassHierarchy"}](ClassHierarchy){width="19pc"}
The algorithms in this manuscript are implemented as classes in Matlab in order to take advantage of class-based inheritance. The class hierarchy is shown in figure \[ClassHierarchy\], where `InternalModeBase` is the abstract superclass which defines the primary interface. A class cluster `InternalModes` (not part of the hierarchy) is included to provide a single interface from which to initialize all the concrete subclasses.
The primary `Spectral` class uses depth ($z$) coordinates from section \[sec:z\_coord\] to compute the eigenvalue problem, while the `DensitySpectral` and `WKBSpectral` subclasses use the stretched coordinates described in sections \[sec:density\_coord\] and \[sec:wkb\_coord\], respectively. The `AdaptiveSpectral` class overrides its superclass when the frequency is high enough, as described in section \[sec:adaptive\_grid\]. The `ConstantStratification` class implements the analytical solution from \[sec:im\_constant\_strat\] and \[sec:sqg\_constant\_strat\], while the `ExponentialStratification` class implements the analytical solution from \[sec:im\_exponential\_strat\] and \[sec:sqg\_exponential\_strat\]. The WKB class implements the WKB approximated solution from \[sec:im\_wkb\_solution\] and \[sec:wkb\_sqg\_solution\] and inherits from the `Spectral` class in order to use the spectrally computed stratification function, $N^2$. The `FiniteDifference` class uses finite difference matrices of arbitrary order, on arbitrary grids using the algorithm described in @fornberg1998-siam.
Acknowledgments {#acknowledgments .unnumbered}
===============
All classes available at `https://github.com/JeffreyEarly/GLOceanKit/`.
Thanks to Jonathan Lilly for his careful comments on an early draft of this manuscript. This work was supported by Office of Navy Research grant N00014-15-1-2465, and National Science Foundation award numbers 1658564 and 1536747.
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[^1]: Another reasonable choice is to take $\gamma = N_0^2 D/g$, but using $\gamma=1$ keeps the norm universal, rather than problem specific.
|
---
abstract: 'In this note, we investigate the regularity of the extremal solution $u^*$ for the semilinear elliptic equation $-\triangle u+c(x)\cdot\nabla u=\lambda f(u)$ on a bounded smooth domain of $\mathbb{R}^n$ with Dirichlet boundary condition. Here $f$ is a positive nondecreasing convex function, exploding at a finite value $a\in (0, \infty)$. We show that the extremal solution is regular in the low dimensional case. In particular, we prove that for the radial case, all extremal solutions are regular in dimension two.'
address: |
$^1$Department of Mathematics, East China Normal University, 200241 Shanghai, P.R. China. E-mail: luoxue0327@163.com\
$^2$LMAM, UMR 7122, Université Paul Verlaine de Metz, 57045 Metz Cedex 1, France. E-mail: dong.ye@univ-metz.fr\
$^3$Department of Mathematics, East China Normal University, 200241 Shanghai, P.R. China. E-mail: fzhou@math.ecnu.edu.cn
author:
- 'Xue Luo$^1$, Dong Ye$^2$, Feng Zhou$^3$'
title: Regularity of the extremal solution for some elliptic problems with singular nonlinearity and advection
---
singular nonlinearity, advection, extremal solution, regularity[^1].
Introduction
============
We consider the elliptic problem $$\tag{$P_\lambda$}
\left\{ \begin{aligned}
-\triangle u + c(x)\cdot\nabla u & =\lambda f(u) && \textup{in}\
\Omega,\\
u& > 0 && \textup{in}\ \Omega,\\
u&=0 && \textup{on}\ \partial\Omega,
\end{aligned} \right.$$ where $\l > 0$, $\Omega$ is a smooth bounded domain in $\mathbb{R}^n$ ($n \geq 2$), $c(x)$ is a smooth vector field over $\overline{\Omega}$ and $f: [0, a) \to \mathbb{R}_+$ with fixed $a \in (0, \infty)$ satisfies the following condition $(H)$: $$\mbox{$f$ is $C^2$, positive, nondecreasing and convex in $[0,a)$ with $\displaystyle \lim_{t\to
a^-}f(t)=\infty$.}$$ In the literature, $f$ is refered as a [*singular nonlinearity*]{}. We say that $u$ is a regular solution if $u \in C^2(\overline\O)$, and we also deal with solutions in the following weak sense.
We say that $u$ is a weak solution of ($P_\lambda$) if $0\leq
u\leq a$ a.e. in $\Omega$ such that $f(u)d(x, \p\O)\in L^1(\Omega)$ and $$-\int_\Omega u\Delta\phi - \int_\Omega u{\rm div}(\phi c)
=\lambda\int_\Omega f(u)\phi,
\quad\forall\; \phi\in C^2(\overline{\Omega})\cap H_0^1(\Omega).$$ Moreover, $u$ is a weak super-solution of ($P_\lambda$) if ${``="}$ is replaced by ${``\geq"}$ for all nonnegative functions $\phi\in C^2(\overline{\Omega})\cap
H_0^1(\Omega)$.
Clearly, a weak solution is regular if $\sup_\O u < a$. For regular solutions, we introduce a notion of stability.
\[regularity and stability\] A regular solution $u$ of ($P_\lambda$) is said to be stable if the principal eigenvalue of the linearized operator $L_{u,\lambda,c}: =-\triangle + c\cdot\nabla - \lambda f'(u)$ is nonnegative in $H_0^1(\O)$.
Exploiting some ideas in [@m; @gg], the solvability of $(P_\l)$ is characterized by a parameter $\l^*$:
\[general\] There exists $\lambda^*\in(0,\infty)$ such that
- For $0<\lambda<\lambda^*$, the problem ($P_\lambda$) has a minimal solution $u_\lambda$, $u_\l$ is regular and the map $\l \mapsto u_\l$ is increasing. Moreover, $u_\l$ is the unique stable solution of $(P_\l)$.
- For $\l = \l^*$, $(P_{\l^*})$ admits a unique weak solution $u^* := \lim_{\l\to\l^*} u_\l$, called the extremal solution.
- For $\l > \l^*$, $(P_\l)$ admits no weak solution.
Here the minimal solution means that $u_\l \leq v$ for any solution $v$ of $(P_\l)$. We remark immediately a close similarity between $(P_\l)$ and the Emden-Fowler equation with superlinear [*regular nonlinearity*]{}, that is $$\begin{aligned}
\label{EF}
-\Delta u = \l g(u) \;\mbox{ in } \;\O \subset {\mathbb R}^n; \quad u = 0 \; \mbox{ on }\; \p\O,\end{aligned}$$ with $\l > 0$ and $g: [0, \infty) \rightarrow (0, \infty)$ satisfies $$\begin{aligned}
\label{super}
\mbox{$g$ is $C^2$, nondecreasing, convex and }\; \lim_{t\to \infty} \frac{g(t)}{t} = \infty.\end{aligned}$$ In fact, there exists also a critical parameter $\overline\l \in (0,
\infty)$ for such that all conclusions in the above proposition hold true by replacing $\l^*$ by $\overline\l$ (see [@bcmr; @m]). It is well known by classical examples as $g(u) = (1+u)^p$ with $p > 1$ or $g(u) = e^u$, the extremal solution $u^*$ can be either a regular solution or a real weak solution in the distribution sense with $\sup_\O u = \infty$.
For general nonlinearity $g$ satisfying , the regularity of the extremal solution $u^*$ to is obtained by Nedev [@n] for any bounded smooth domain $\Omega \subset \mathbb{R}^n$ if $n = 2, 3$; by Cabré [@c1] for convex domains in ${\mathbb R}^4$; and for radial symmetry case in ${\mathbb R}^n$ with $n \leq 9$ by Cabré & Capella [@cc]. In [@yz1], it is proved that, under mild condition on $g$, the extremal solution $u^*$ is regular for any smooth bounded domain $\O \subset {\mathbb R}^n$ if $n \leq 9$.
We can ask the same question about the problem ($P_\lambda$): For $f$ verifying $(H)$, is it true that the extremal solution to ($P_\lambda$) is regular for general vector field $c$ and general domain $\Omega \subset\mathbb{R}^n$ with low dimensions $n$? We will partly answer this question. It is worthy to mention that for studying the explosion phenomena in a flow, Berestycki [*et al.*]{} [@bknr] have considered the problem $(P_\l)$ with a general source $f$ verifying .
Without loss of generality, fix $a = 1$ in the sequel. The problem $(P_\l)$ can be linked to equation up to the transformation $v = -\ln{(1-u)}$. In fact, let $u$ solve $(P_\l)$, $v$ verifies then $$\tag{$Q_\lambda$}
\left\{ \begin{aligned}
-\triangle v+|\nabla v|^2+c(x)\cdot\nabla v&=\lambda e^vf(1-e^{-v}):=\lambda g(v) && \textup{in}\ \Omega,\\
v&=0 && \textup{on}\ \partial\Omega.
\end{aligned} \right.$$ Therefore $g$ verifies and $v^*=-\ln(1-u^*)$ is the extremal solution for the problem ($Q_\lambda$). Thus the regularity of $u^*$ is equivalent to the boundedness of $v^*$, however the situation could be very different with the presence of advection terms (see [@cg; @wy]). In last decade, a model describing the steady state of MEMS (Micro-Electro-Mechanical Systems) device given by Pelesko and Bernstein in [@pb], has drawn many attentions (see [@egg] and the references therein). $$-\Delta u = \frac{\l}{(1 - u)^2} \;\mbox{ in } \;\O \subset {\mathbb R}^n; \quad u = 0 \; \mbox{ on }\; \p\O.$$ More generally, many precise studies have been done for the singular nonlinearities with negative exponent $f(u) = (1 -u)^{-p}$ ($p > 0$) in the advection-free situation, i.e. $c \equiv 0$. In that case, when $\O $ is moreover the unit ball in ${\mathbb R}^n$, it is known that $u^*$ is regular if and only if (see [@mp; @gg]) $$\begin{aligned}
\label{np}
n < n_p:= 2 + \frac{4p}{p+1} + 4\sqrt{\frac{p}{p+1}}.\end{aligned}$$ Tending $p \to 0^+$ in , we see that $n_p \to 2$. Therefore we cannot expect in general better than dimension two to claim the regularity of $u^*$.
For the radial case of $(P_\l)$, equally when $\O$ is a ball and $c(x)$ is the gradient of a smooth radial function, $u_\l$ is radial by uniqueness of the minimal solution. We obtain the following optimal results which are new even for the advection-free case.
\[reguradial\] Assume that $n = 2$, $\O = B_1$. Let $\gamma$ is a smooth radial function and $c = \nabla\gamma$, then the extremal solution $u^*$ is regular for any $f$ satisfying $(H)$.
\[radialu’\] For any $f$ satisfying $(H)$, $\O = B_1$ and smooth radial function $\gamma$, there exists $C > 0$ such that for all $\l \in (0, \l^*]$ $$|u_\l'(r)| \leq \left\{\begin{array}{ll}
Cr^{-1} & \mbox{if } n \geq 10;\\
Cr^{-\frac{n}{2}+1+\sqrt{n-1}} & \mbox{if } 3 \leq n \leq 9;
\end{array}
\right. \quad \forall \; r = {|x|} \in (0, 1]$$ where ${|\cdot|}$ is the Euclidean norm in ${\mathbb R}^n$.
\[1.1\] The above estimates are optimal. In fact, when $f(u) = (1 - u)^{-p}$, $p > 0$, $\O = B_1$ and $c \equiv 0$, it is well known that $u^*(x) = 1 - r^{\frac{2}{p+1}}$ if $n \geq n_p$ with $n_p$ given in , and we have $$n \geq n_p \quad \mbox{ iff }\quad n \geq 10 \;\mbox{ or }\; 3\leq n \leq 9, \; \frac{2}{p+1} \leq -\frac{n}{2} + 2 + \sqrt{n - 1}.$$
But is the extremal solution $u^*$ of $(P_\l)$ regular with general singular nonlinearity $f$ verifying $(H)$, vector field $c$ and smooth bounded domains in ${\mathbb R}^2$? The answer is affirmative under some additional mild condition on $f$.
\[main theorem\] Assume that $f$ satisfies conditions $(H)$ and the additional conditions, $$\limsup_{t\rightarrow1^-}\frac {f(t)}{f'(t)(1-t)\ln^2(1-t)}<1 \leqno{(H1)}$$ and $$\liminf_{t\rightarrow1^-} \frac
{f(t)f''(t)}{f'^2(t)} > 0.\leqno{(H2)}$$ Then $u^*$ is regular solution to ($P_{\l^*}$) if $n = 2$, i.e. $\Omega\subset\mathbb{R}^2$.
Under more precise conditions on the growth of $f$, the extremal solution can be showed to be regular in some higher dimensions.
\[main result\] Let $f$ verify (H) and $g(v) = e^vf(1 - e^{-v})$. Assume that $g$ satisfies $$\liminf_{t\rightarrow\infty}
\frac{g'(t)}{g(t)} = 1+\delta > 1\leqno{(H3)}$$ and $$\liminf_{t\rightarrow\infty} \frac {g''(t)g(t)}
{g'^2(t)}=\mu>\frac 1 {1+\delta}. \leqno{(\widetilde{H2})}$$ Then $v^* = -\ln(1 - u^*)$ is bounded (so $u^*$ is regular) when $$\begin{aligned}
\label{ndelta}
n < 2+\frac {4\delta}{1+\delta}+\frac
{4\sqrt{\delta(\mu + \mu\delta -1)}}{1+\delta}.
\end{aligned}$$
Consequently, if $\mu\delta > 1$, $u^*$ is regular for all $n\leq 6$. Furthermore, if we can tend $\delta$ to $\infty$, which means $g = o(g')$ near $\infty$, then $u^*$ is regular for $n < 6 + 4\sqrt{\mu}$ with any $\mu > 0$. However, we can never have $\mu > 1$, since otherwise $g$ blows up at finite value and contradicts , so the best result we can expect is for $n \leq 9$. For example, if $f(u) = e^{\frac{1}{1 - u}}$, then $g(v) = e^{v + e^v}$ verifies $\delta = \infty$ and $\mu = 1$.
\[fast increasing nonlinearity\] Let $f$ verify (H) and $g(v) = e^vf(1 - e^{-v})$. Assume that $g = o(g')$ near $\infty$. Rewrite $g(t) = g(0) + te^{h(t)}$ in $(0, \infty)$, suppose there exists $t_0>0$ such that $t^2h'(t)$ is nondecreasing for $t \geq t_0$, then for any bounded smooth domain $\Omega\subset \mathbb{R}^n$ with $n\leq 9$, $u^*$ is a regular solution.
Furthermore, when $g = o(g')$ near $\infty$, the condition $(\widetilde{H2})$ is just equivalent to $(H2)$, since $$\frac{f''(t)f(t)}{f'^2(t)} = \frac{\left(g'' - g'\right)g}{\left(g' - g\right)^2}(s) = \left(\frac{g''g}{g'^2} - \frac{g}{g'}\right)\times \left(1 - \frac{g}{g'}\right)^{-2}(s), \qquad \forall\; t = 1 - e^{-s}.$$ It is also easy to see that $(H3)$ is equivalent to the condition $$\liminf_{t\rightarrow 1^-}
\frac{f'(t)(1-t)}{f(t)} = \delta > 0.$$ If the equality holds for the whole limit, we have the following optimal result. The case $f(u) = (1 - u)^{-2}$ was obtained in [@cg] with a different argument.
\[negative p\] Assume that $$\begin{aligned}
\label{H2'}
\lim_{u\to 1^-}\frac{f'(u)(1 - u)}{f(u)} = p > 0.\end{aligned}$$ Then $u^*$ is a regular solution if $n < n_p$ where $n_p$ is defined in .
One of the main difficulties here is due to the vector field $c(x)$. When $c \ne 0$, the operator $-\Delta + c\cdot\nabla$ is not self-adjoint, we use ideas from [@cg] to get some energy estimates. However if $c$ is a gradient, say $c=-\nabla\gamma$ in $\O$, then $-\Delta +c\cdot\nabla$ can be rewritten as $e^{-\gamma}L_\gamma$ where $L_\gamma = -{\rm div}(e^\gamma\nabla)$ is a self-adjoint operator. In that case, $(P_\l)$ admits a variational structure and we can expect more precise estimates of minimal solutions $u_\l$, as in the radial case.
The paper is organized as follows: In section 2, we prove quickly Proposition \[general\] and show some general consequences of the stability of $u_\l$. The section 3 is devoted to the proof of Theorems \[main theorem\] to \[negative p\] for general domains. In section 4, we discuss the radial case. The norm $\|\cdot\|_q$ denotes always the standard $L^q$ norm for any $q \in [1, \infty]$. The capital letter $C$ denotes a generic positive constant independent of $\l$, it could be changed from one line to another.
Preliminaries
=============
As mentioned above, $-\Delta + c\cdot\nabla$ is not a self-adjoint operator for general vector field $c$. However using Lemma 1 in [@cg], we have a kind of Hodge decomposition, which tells us that for any vector field $c \in C^\infty(\overline{\Omega},\mathbb{R}^n)$, there exist a smooth scalar function $\gamma$ and a vector field $b \in C^\infty(\overline{\Omega},\mathbb{R}^n)$ such that $$\begin{aligned}
\label{decomp-c}
c = -\nabla\gamma+b \; \mbox{ and }\; {\rm div}(e^\gamma b)= 0 \quad \mbox{in }\; \overline\O.\end{aligned}$$ Therefore the problem ($P_\lambda$) can be rewritten as $$\tag{$P_\lambda'$}
-{\rm div}(e^\gamma\nabla u)+ e^\gamma b\cdot\nabla u=\lambda
e^\gamma f(u)\quad\textup{in}\ \Omega.$$ On the other hand, we don’t have a suitable variational characterization in general to use the stability assumption. Fortunately, we can adopt an energy inequality as in [@cg], which is derived from a generalized Hardy inequality of [@co].
Let $u_\l$ be minimal solution of $(P_\l)$. For any $1\leq\beta<2$, we have $$\begin{aligned}
\label{Hardy ineqn}
\lambda\int_\Omega e^\gamma f'(u_\lambda)\psi^2
\leq \frac2\beta\int_\Omega
e^\gamma|\nabla\psi|^2+\frac{\|b\|_\infty^2}{2(2-\beta)}\int_\Omega
e^\gamma\psi^2, \quad \forall\; \psi\in H_0^1(\Omega).
\end{aligned}$$ where $b$ is the vector field in , $\|b\|_\infty = \max_{\overline\Omega}|b(x)|$.
[**Proof.**]{} We use a Hardy type inequality given by Theorem 2 in [@cg], which says that for a positive principal eigenfunction $\varphi$ of $L_{u_\lambda, \l, c}$, for $\beta \in [1, 2)$ and any $\psi \in H_0^1(\O)$, $$\lambda\int_\Omega e^\gamma f'(u_\lambda)\psi^2\leq\frac2\beta\int_\Omega e^\gamma|\nabla\psi|^2
+\int_\Omega \left[-\frac{2-\beta}2\frac{|\nabla\varphi|^2}{\varphi^2}+
\frac{b\cdot\nabla\varphi}\varphi\right]e^\gamma\psi^2.$$ By Cauchy-Schwarz inequality, it is easy to see $$-\frac{2-\beta}2\frac{|\nabla\varphi|^2}{\varphi^2}+\frac{b\cdot\nabla\varphi}\varphi \leq \frac{|b(x)|^2}{2(2-\beta)} \leq \frac{\|b\|_\infty^2}{2(2-\beta)},$$ so we are done.
Another main ingredient of our approach is just the transformation $v = -\ln(1 - u)$. Let $\phi$ and $\xi$ be nonnegative $C^1$ functions satisfying $\phi(0) = \xi(0) = 0$ and $\xi'=\phi'^2$. Define $v_\l = -\ln(1 - u_\l)$ and $g(v_\l) = e^{v_\l}f(1 - e^{-{v_\l}})$. Using $(Q_\l)$, we get $-{\rm div}(e^\gamma\nabla v_\l) + e^\gamma b\cdot\nabla v_\l \leq \l e^\gamma g(v_\l)$ in $\O$. Let $\psi = \phi(v_\l)$ in (\[Hardy ineqn\]), $\forall\; \l \in (0, \l^*)$, $$\begin{aligned}
&\; \lambda\int_\Omega e^\gamma
f'(u_\lambda)\phi^2(v_\lambda)\\
\leq &\;\frac2\beta\int_\Omega e^\gamma|\nabla\phi(v_\lambda)|^2
+\frac{\|b\|_\infty^2}{2(2-\beta)}\int_\Omega e^\gamma\phi^2(v_\lambda)\\
= &\; \frac2\beta\int_\Omega e^\gamma\nabla \xi(v_\lambda) \nabla
v_\lambda + C_\beta\int_\Omega e^\gamma\phi^2(v_\lambda)\\
= &\; -\frac2\beta\int_\Omega {\rm div}(e^\gamma\nabla
v_\lambda)\xi(v_\lambda)+ C_\beta\int_\Omega e^\gamma\phi^2(v_\lambda)\\
\leq &\; \frac{2\lambda}\beta\int_\Omega e^\gamma g(v_\lambda)\xi(v_\lambda) - \frac{2}\beta\int_\Omega e^\gamma b\cdot \xi(v_\l)\nabla v_\l + C_\beta\int_\Omega
e^\gamma\phi^2(v_\lambda)\\
= &\; \frac{2\lambda}\beta\int_\Omega e^\gamma g(v_\lambda)\xi(v_\lambda) + C_\beta\int_\Omega
e^\gamma\phi^2(v_\lambda).\end{aligned}$$ The last line is due to ${\rm div}(e^\gamma b) = 0$. We claim then
Let $1\leq \beta < 2$. For any $\l \in (0, \l^*)$ and any nonnegative $C^1$ test functions $\phi$, $\xi$ verifying $\phi(0) = \xi(0) = 0$ and $\xi'=\phi'^2$, there hold $$\begin{aligned}
\label{psi and xi}
\lambda\int_\Omega e^\gamma f'(u_\lambda)\phi^2(v_\lambda)
\leq \frac{2\lambda}\beta\int_\Omega e^\gamma g(v_\lambda)\xi(v_\lambda) + C_\beta\int_\Omega
e^\gamma\phi^2(v_\lambda)\end{aligned}$$ and $$\begin{aligned}
\label{est1} \lambda\int_\Omega e^\gamma f'(u_\l)\phi^2(u_\l) \leq
\frac{2\lambda}{\beta} \int_\Omega e^\gamma f(u_\l)\xi(u_\l) + C_\beta
\int_\Omega e^\gamma \phi^2(u_\l).\end{aligned}$$
The proof of is completely similar to but using $(P'_\l)$ instead of $(Q_\l)$.
We also make use the following behavior of $f$ proved in [@yz2].
\[ff’\] For any $f$ verifying $(H)$, we have $\lim_{t\to 1} f(t)/f'(t) = 0$.
Choose first $\phi(u) = e^u - 1$ in , then $\xi(u) = \frac{e^{2u} - 1}{2}$ and $$\begin{aligned}
\lambda\int_\Omega e^\gamma f'(u_\l)\left(e^{u_\l} - 1\right)^2 \leq
\frac{\lambda}{\beta} \int_\Omega e^\gamma f(u_\l)\left(e^{2u_\l} - 1\right) + C_\beta
\int_\Omega e^\gamma \left(e^{u_\l} - 1\right)^2.\end{aligned}$$ Fix $\beta \in (1, 2)$. By Lemma \[ff’\], $$\lambda\int_\Omega e^\gamma f'(u_\l)e^{2u_\l} \leq C.$$ Consequently $\|f'(u_\l)\|_1$ is uniformly bounded, so is $\|f(u_\l)\|_1$. Multiplying $(P_\l)$ by $u_\l$, $$\int_\O |\nabla u_\l|^2 = \int_\O \frac{{\rm div}(c)}{2}u_\l^2 + \l\int_\O f(u_\l)u_\l \leq C,$$ which gives
\[H01\] The family of minimal solutions $\{u_\l\}_{0 < \l < \l^*}$ is uniformly bounded in $H^1_0(\O)$.
As far as we know, it is always an open question whether the similar $H^1$ energy estimation holds for minimal solutions of with general regular nonlinearity satisfying and general domain $\O$ when $n \geq 6$ (see [@n] for $n\leq 5$). For the advection-free case $c = 0$, it was proved in [@yz2] that $u^* \in H^2\cap H_0^1(\O)$ under the condition $(H)$, it is also true for the gradient case $c = \nabla \gamma$ (see Lemma \[gradientH2\]).
[**Sketches of proof of Proposition \[general\].**]{} We follow the ideas coming from [@bknr; @m; @gg]. The main argument is the maximum principle for operators $-\Delta + c\cdot\nabla$ and $L_\gamma$ under the Dirichlet boundary condition, we use also the super-sub solution method and monotone iteration.
Let $w \in H_0^1(\O)$ be the regular solution of $-\Delta w + c\cdot\nabla w = 1$ in $\O$ and fix $\alpha > 0$ such that $\alpha\max_\O w < 1$. It is easy to verify that $\alpha w$ is a supersolution of $(P_\l)$ for $\l > 0$ small enough. As $0$ is a subsolution and $\alpha w > 0$ in $\O$, $(P_\l)$ admits a regular solution for $\l > 0$ small enough. As any regular solution $u$ of $(P_\l)$ is also a supersolution for $(P_\mu)$ if $\mu \in (0, \l)$, the set of $\l$ for which $(P_\l)$ admits a regular solution is just an interval. Moreover, for these $\l$, using $(H)$ and the monotone iteration $v_0 = 0$; $-\Delta v_{n+1} + c\cdot\nabla v_{n+1} = \l f(v_n)$ in $\O$ with $v_{n+1} = 0$ on $\p\O$ for $n \in {\mathbb N}$, we get the minimal solution $u_\l = \lim_{n\to\infty} v_n$.
If we suppose that the principal eigenvalue of $L_{u_\l, \l, c}$ is negative, we can construct, as in [@bknr] another solution $v \leq u_\l$ using the associated first eigenfunction, this is just impossible by the definition of $u_\l$, hence $u_\l$ is stable. The uniqueness of stable solution comes from Lemmas 2.16 and 2.17 in [@CR].
Take a positive first eigenfunction $\varphi$ of $L_\gamma$ with the Dirichlet boundary condition, by $(P_\l')$, $$\l f(0)\int_\O e^\gamma \varphi \leq \int_\O\l e^\gamma f(u)\varphi = \int_\O {\l_1(L_\gamma)}u\varphi - \int_\O {\rm div}(e^\gamma b \varphi)u \leq C.$$ So $\l$ is upper bounded. Define the critical threshold $\lambda^*$ as the supermum of $\lambda > 0$ for which ($P_\lambda$) admits a regular solution, as $u^*$ is the monotone limit of $u_\l$ when $\l \to \l^*$, we deduce that $u^* \in H^1_0(\O)$ is a weak solution of $(P_\l)$ by Proposition \[H01\].
Suppose that $u$ is a weak solution to $(P_\l)$. By the monotonicity of $f$, it is easy to verify that for any $\delta > 1$, the function $v = \delta^{-1}u$ is a weak supersolution for $(P_{\l/\delta})$, then the monotone iteration will enable us a weak solution $w$ of $(P_{\l/\delta})$ satisfying $0 \leq w \leq v \leq \delta^{-1} < 1$. The regularity theory implies then $w$ is a regular solution of $(P_{\l/\delta})$. This means that $\l/\delta \leq \l^*$. Let $\delta$ tend to 1, we get $\l \leq \l^*$. Therefore, no weak solution exists for $\l > \l^*$.
The uniqueness of the weak solution can be proved in the very similar way as in [@m] using the monotonicity and convexity of $f$, with the strong maximum principle for the operator $-\Delta + c\cdot\nabla$ associated to Dirichlet boundary condition, so we omit the details.
Regularity of $u^*$ for general $c$ and $\O$
============================================
For proving our results, we will choose suitable functions $\phi$ to apply or . We need also
\[comp\] For any $q > n/2$, there exists $C > 0$ such that the solution $v$ of $(Q_\l)$ satisfies $0 \leq v \leq C\|g(v)\|_q$ in $\O$.
Indeed, let $w$ be the solution of $L(w) := -\Delta w + c\cdot\nabla w = \lambda g(v)$ in $\O$ with $w = 0$ on $\p\O$. By regularity theory and Sobolev embedding, $\|w\|_\infty \leq C\|w\|_{W^{2, q}(\O)} \leq C'\lambda^*\|g(v)\|_q$ because $q > n/2 \geq 1$. Morover, as $L(w - v) \geq 0$, the maximum principle implies then $0 \leq v \leq w \leq C\|g(v)\|_q$.
Proof of Theorem 1.3
--------------------
For simplicity, we omit the index $\l$ for $u_\l$ or $v_\l$. Let $\phi(u) = v = -\ln(1 - u)$ in , so $\xi(u) = (1 - u)^{-1} - 1$. Fix $\beta \in (1, 2)$ but very close to 2. Repeating the proof of Theorem 2 in [@yz2] with the assumption $(H1)$, there exists $C > 0$ such that $$\lambda\int_\Omega e^\gamma \frac{f(u)}{1 - u} < C + C C_\beta\int_\Omega e^\gamma \phi^2(u).$$ As $\phi^2(u) = o(\xi(u)) = o(f\xi)$ when $u \to 1^-$, $$\lambda\int_\Omega e^\gamma \frac{f(u)}{1 - u} \leq C.$$ Using the equation $(Q_\l)$ and $\p_\nu v \leq 0$ on $\p\O$, $$\begin{aligned}
\int_\Omega |\nabla v|^2 = \lambda\int e^v f(1 - e^{-v}) +
\int_{\partial\Omega} \frac{\partial v}{\partial \nu} d\sigma
-\int_\Omega c\cdot\nabla v &\leq \lambda\int_\Omega \frac{f(u)}{1 - u} + C\|\nabla v\|_2\\
& \leq C + C\|\nabla v\|_2.\end{aligned}$$ Therefore $\|\nabla v\|_2 \leq C$, the classical Moser-Trudinger inequality enables us, as $n = 2$ $$\begin{aligned}
\label{mt}
\int_\Omega e^{qv} \leq C_q, \quad \forall\; q \geq 1.\end{aligned}$$ Take now $\phi(u) = f(u) - f(0)$ in , we need to estimate $$\begin{aligned}
\zeta(u) := f'(u)\phi(u) - \frac{2}{\beta}\xi(u) &= f'(u)\phi(u) - \frac{2}{\beta}\int_0^u f'^2(s) ds\\
&= f'(u)f(u) -
\frac{2}{\beta}\int_0^u f'^2(s) ds - Cf'(u)\\
&:= I(u) -
\frac{2}{\beta}J(u) - Cf'(u).\end{aligned}$$ By $(H2)$, there exists $\delta
> 0$ such that $$\begin{aligned}
I(u) - I(0) = \int_0^u \left[f'^2(s) + f''(s)f(s)\right] ds \geq (1 +
\delta)J(u) - Cf'(u), \;\; \forall\; u \in [0, 1)\end{aligned}$$ Let $\frac{4}{2 + \delta} < \beta < 2$, we get $\zeta(u) \geq C I(u) - C$. Asserting this in , $$\lambda\int_\Omega e^\gamma f'(u)f^2(u) \leq C\int_\Omega e^\gamma f^2(u) + C.$$ Consequently, $\|f'(u)f^2(u)\|_1 \leq C$. By Lemma \[ff’\], we deduce $\|f(u)\|_3 \leq C$. Combining with , $\|g(v)\|_p \leq C$ for any $p < 3$. The proof is completed by Lemma \[comp\] as $n = 2$.
Proof of Theorem 1.4
--------------------
Without loss of generality, we can assume that $g(0)=1$. Let $\phi(t)= g^\alpha(t)-1$ where $\alpha > 0$ is a constant to be determined later. Then $$\begin{aligned}
\label{3.5}
\begin{split}
\xi(t) & =\int_0^t \phi'^2(s)ds\\
& =\alpha^2\int_0^t g^{2\alpha-2}(s)g'^2(s)ds\\
&=\frac {\alpha^2}{2\alpha-1}g^{2\alpha-1}(t)g'(t)-\frac {\alpha^2}{2\alpha-1}\int_0^t
g^{2\alpha-1}(s)g''(s)ds - C_\alpha.
\end{split}\end{aligned}$$ The condition $(\widetilde{H2})$ yields: Given any $\epsilon \in \left(0, \mu-\frac 1{1+\delta}\right)$, there exists $C\geq 0$ such that $g(t)g''(t)\geq(\mu-\epsilon)g'^2(t)-C$ in $[0, \infty)$. Therefore $$\begin{aligned}
\label{3.6}
\begin{split}
-\int_0^t g^{2\alpha-1}(s)g''(s)ds
&\leq -(\mu-\epsilon)\int_0^t
g^{2\alpha-2}(s)g'^2(s)ds+C\\
& \leq -\frac {\mu-\epsilon}{\alpha^2}\xi(t)+C.
\end{split}\end{aligned}$$ We divide the proof into two cases.
[*Case 1*]{}: $\delta > 1$ and $\mu > \frac{1}{1 + \delta}$; or $\delta \leq 1$ with $\mu > \frac{1+\delta}{4\delta}$.
Take $\alpha > \frac{1}{2}$. Combine and , $$\left(1+\frac {\mu-\epsilon}{2\alpha-1}\right)\xi(t)\leq\frac
{\alpha^2}{2\alpha-1}g^{2\alpha-1}(t)g'(t)+C,$$ consequently $$\begin{aligned}
\label{xi-g}
\xi(t)\leq \frac
{\alpha^2}{2\alpha-1+\mu-\epsilon}g^{2\alpha-1}(t)g'(t)+C, \quad \mbox{for any }\; t\geq0.\end{aligned}$$ According to $(H3)$, for any $0 < \delta' < \delta$, there exists $C > 0$ such that $g'(t) \geq (1 + \delta')g(t) - C$ in $[0, \infty)$. Setting these estimates in (\[psi and xi\]), omitting the index $\l$ and recalling that $f'(u) = g'(v) - g(v)$, $$\begin{aligned}
& \frac{\d'\l}{1+\delta'}\int_\Omega e^\gamma g'(v)(g^\alpha(v)-1)^2 - C\l\int_\Omega
e^\gamma(g^\alpha(v)-1)^2\\
\leq & \; \l\int_\Omega e^\gamma f'(u)(g^\alpha(v)-1)^2\\
\leq & \;\frac {2\alpha^2\l}{\beta(2\alpha-1+\mu-\epsilon)}\int_\Omega
e^\gamma g^{2\alpha}(v)g'(v)+ C\l\int_\Omega
e^\gamma g(v) + C\int_\Omega e^\gamma (g^\alpha(v)-1)^2.\end{aligned}$$ Consequently, $$\begin{aligned}
& \left[\frac{\d'}{1+\d'}-\frac
{2\alpha^2}{\beta(2\alpha-1+\mu-\epsilon)}\right]\l\int_\Omega
e^\gamma g'(v)g^{2\alpha}(v)\\
\leq & \;\frac {2\d'C}{1+\d'}\int_\Omega
e^\gamma g'(v)g^\alpha(v) +C\int_\Omega
e^\gamma g(v) + C\int_\Omega e^\gamma (g^\alpha(v)-1)^2.\end{aligned}$$ Choose $\d'$ near $\d$ such that $$\mbox{either } \; \d' > 1 \;\mbox{ and }\; \mu > \frac{1}{1 + \d'} \quad \mbox{or} \quad \d' < \delta \leq 1 \;\mbox{ with } \;\mu > \frac{1+\d'}{4\d'}.$$ Through direct computations, for ${\varepsilon}> 0$ sufficiently small and $\beta = 2 - {\varepsilon}$, there exists $$\alpha \in \left(\frac12,
\frac{\d'}{1+\d'} + \frac{\sqrt{\d'(1+\d')(\mu-\epsilon)-\d'}}{1+\d'}\right)$$ such that $$\begin{aligned}
\label{da}\left[\frac{\d'}{1+\d'}-\frac
{2\alpha^2}{\beta(2\alpha-1+\mu-\epsilon)}\right] > 0.\end{aligned}$$ For such $\alpha$, we obtain $$\label{uniform control}
\l\int_\Omega e^\gamma g^{2\alpha}(v)g'(v)\leq C, \quad \forall\; \l \in (0, \l^*).$$ Tending now $\d'$ to $\d$ and ${\varepsilon}$ to $0$, holds true provided that $$\begin{aligned}
\label{esta}
\alpha < \frac
\delta{1+\delta}+\frac
{\sqrt{\delta\mu(1+\delta)-\d}}{1+\delta}.\end{aligned}$$ Therefore $$\begin{aligned}
\int_\Omega e^\gamma g^{2\alpha+1}(v)\leq C\int_\Omega
e^\gamma g^{2\alpha}(v)g'(v) + C \leq \widetilde C,\end{aligned}$$ which implies that $\|g(v)\|_{2\alpha+1} \leq C$ for $\alpha$ verifying . Applying Lemma \[comp\], we conclude that for $n < 2 + 4\alpha$ with $\alpha$ verifying , $v_\l$ is uniformly bounded, hence $u^*$ is a regular solution if $n$ satisfies .
[*Case 2*]{}: $\d \leq 1$ and $\frac{1}{1 + \d} < \mu \leq \frac{1+\delta}{4\delta}$.
Now we take $\alpha\in \left(\frac 1 2(1-\mu+\epsilon), \frac{1}{2}\right)$, the formulas and imply then $$\left(1+\frac{\mu-\epsilon}{2\alpha-1}\right)\xi(t)\geq\frac
{\alpha^2}{2\alpha-1}g^{2\alpha-1}(t)g'(t)+C.$$ The inequality still holds true. Proceeding as for Case 1, we see that for $\d' < \d$ but nearby, ${\varepsilon}> 0$ small and $\beta = 2 - {\varepsilon}$, there exists $$\alpha \in \left(\frac{1-\mu+\epsilon}2, \frac{\d'}{1+\d'} + \frac{\sqrt{\d'(1+\d')(\mu-\epsilon)-\d'}}{1+\d'}\right) \subset \left(\frac{1-\mu+\epsilon}2, \frac12\right)$$ such that is satisfied. Hence we conclude exactly as in [*Case 1*]{}.
Proof of Theorem 1.5
--------------------
Without loss of generality, assume again $g(0)=1$. Take now $\phi(t)=te^{\alpha h(t)}$, where $\alpha>0$ is a constant to be determined, then $$\begin{aligned}
\xi(t)& =\int_0^t \left[1+s\alpha h'(s)\right]^2e^{2\alpha h(s)}ds\\
& =\int_0^t \left[1+ 2s\alpha h'(s)\right] e^{2\alpha h(s)}ds+\int_0^t
\alpha^2s^2h'^2(s)e^{2\alpha h(s)}ds\\
& =te^{2\alpha h(t)}+K(t).\end{aligned}$$ Thus, for $t \geq t_0$, $$\begin{aligned}
\frac {2K(t)}{\alpha} =2\alpha\int_0^t s^2h'^2(s)e^{2\alpha
h(s)}ds &=C+\int_{t_0}^t s^2h'(s)d\left(e^{2\alpha
h(s)}\right)\\
&\leq C+t^2h'(t)e^{2\alpha
h(t)}-\int_{t_0}^t e^{2\alpha
h(s)}d\left(s^2h'(s)\right),\end{aligned}$$ where the last integration is considered in the sense of Stieltjes. The monotonicity of $s^2h'$ in $[t_0,\infty)$ yields $$K(t)\leq \frac \alpha 2 t^2h'(t)e^{2\alpha h(t)}+C, \quad \forall\; t \geq t_0.$$ So we get $$\xi(t)\leq C+ \left[t+\frac \alpha 2 t^2h'(t)\right] e^{2\alpha
h(t)},\quad \forall\; t \geq 0.$$ Using (\[psi and xi\]) (we drop the index $\lambda$), $$\begin{aligned}
& \int_\Omega e^\gamma\left[e^{h(v)}+vh'(v)e^{h(v)}-ve^{h(v)}-1\right] v^2e^{2\alpha
h(v)}\\
\leq & \; \frac2\beta\int_\Omega e^\gamma\left( 1+ve^{h(v)}\right) \xi(v)
+ C \int_\Omega e^\gamma v^2e^{2\alpha h(v)}\\
\leq & \; \frac2\beta\int_\Omega e^\gamma \left(1+ve^{h(v)}\right)\left[C+ve^{2\alpha h(v)}+\frac \alpha 2 v^2h'(v)e^{2\alpha
h(v)}\right] + C\int_\Omega e^\gamma v^2e^{2\alpha h(v)},\end{aligned}$$ By Young’s inequality, $$\begin{aligned}
\label{estalpha}
\begin{split}
& \left(1-\frac\alpha\beta\right)\int_\Omega e^\gamma v^3h'(v)e^{(2\alpha+1)h(v)}\\ \leq & \; C\int_\Omega e^\gamma \left[1+ v^2h'(v)e^{2\alpha h(v)}+v^3e^{(2\alpha+1)h(v)}\right].
\end{split}\end{aligned}$$ Moreover, $g = o(g')$ at infinity yields $\lim_{t \to \infty} h'(t) = \infty$, hence $$\begin{aligned}
\frac {t^2h'(t)e^{2\alpha
h(t)}+t^3e^{(2\alpha+1)h(t)}}{t^3h'(t)e^{(2\alpha+1)h(t)}} = \frac 1 {g(t)-1}+\frac 1 {h'(t)} \rightarrow 0 \; \mbox{ as } \;t \rightarrow \infty.\end{aligned}$$ Fix $\beta \in (\alpha, 2)$, the inequality implies $$\int_\Omega \frac {[g(v)-1]^{2\alpha+1}}{v^{2\alpha}}=\int_\Omega v
e^{(2\alpha+1)h(v)} \leq C + \int_\Omega v^3h'(v)e^{(2\alpha+1)h(v)}\leq
C.$$ Recall that $g$ is superlinear, we obtain $\|g(v)\|_1 \leq C$. Consider again $w$ satisfying $L(w) = \l g(v)$ in $\O$ and $w = 0$ on $\p\O$, as $v \leq w$ in $\Omega$ by maximum principle, $$\int_\Omega\frac{(g(v)-1)^{2\alpha+1}}{w^{2\alpha}}\leq
C.$$ Following the proof of Lemma 2.1 in [@yz1] (we just need a minor adjustment, say define $\Omega_1 =\{x\in\Omega: g(v)>w^T\}$ instead, here $T > 0$ is a suitable constant), we can obtain that if $2\alpha+1> n/2$, $w$ is uniformly bounded in $L^\infty(\Omega)$, so does $v$. Taking $2 > \beta > \alpha > 7/4$, the result holds for $n \leq 9$.
Proof of Theorem 1.6
--------------------
Here we choose $\phi(u) = (1 - u)^{-\alpha} - 1$ in . For $2\l >
\l^*$ and ${\varepsilon}> 0$, $$\left(p - \frac{2\alpha^2}{\beta(2\alpha + 1)} - 2\epsilon\right)\int_\O \frac{e^\gamma}{(1 - u)^{p+2\alpha +1}} \leq C, \quad \forall\;\beta \in [1, 2).$$ We have used $f'(u)(1 - u) \geq (p - \epsilon)f(u) - C$ in $[0, 1)$ by . As ${\varepsilon}> 0$ is arbitrary, $$\int_\O \frac{1}{(1 - u)^{p+2\alpha +1}} \leq C$$ provided that $$p > \frac{\alpha^2}{2\alpha + 1},\quad \mbox{i.e. when }\;\alpha < p + \sqrt{p(p+1)}.$$ Therefore $\|(1 -
u)^{-1}\|_q \leq C$ if $q < 1 + 3p + 2\sqrt{p(p+1)}$. For any ${\varepsilon}> 0$, as $f'(u)(1-u) \leq (p + {\varepsilon})f(u) + C_{\varepsilon}$ in $[0, 1)$ by , we have $f(u) \leq C(1 - u)^{-p - {\varepsilon}}$, consequently $$g(v) = e^vf(1 - e^{-v}) = \frac{f(u)}{1 - u} \leq C(1 - u)^{-1 -p - {\varepsilon}},$$ hence $\|g(v)\|_r \leq C$ when $$r < \frac{1 + 3p + 2\sqrt{p(p+1)}}{p+1+{\varepsilon}}.$$ According to Lemma \[comp\], the proof is done by taking ${\varepsilon}\to 0^+$.
Radial case
===========
As we have mentioned, when $c = -\nabla\gamma$, the equation $(P_\l)$ is rewritten as $$\begin{aligned}
\label{selfadjoint}
-{\rm div}(e^\gamma\nabla u) = \l e^\gamma f(u).\end{aligned}$$ With the variational structure, the stability of minimal solutions $u_\l$ is equivalent to $$\begin{aligned}
\label{stable2}
\int_\O e^\gamma |\nabla \psi|^2 \geq \l \int_\O e^\gamma f'(u_\l)\psi^2, \quad \forall\; \psi\in H_0^1(\O).\end{aligned}$$ Moreover, for any $C^1$ functions $\phi$ and $\xi$ satisfying $\phi(0) = \xi(0) = 0$ and $\xi'=\phi'^2$, the estimate is replaced by $$\begin{aligned}
\int_\Omega e^\gamma f'(u_\l)\phi^2(u_\l) \leq
\int_\Omega e^\gamma f(u_\l)\xi(u_\l).\end{aligned}$$ Taking now $\phi(t) = f(t) - f(0)$ and working as for Theorem 1 in [@yz2], we have
\[gradientH2\] When $c = \nabla\gamma$, the extremal solution $u^* \in H^2\cap H_0^1(\O)$. More precisely, $$\begin{aligned}
\label{H2}
\int_\O f'(u_\l)f(u_\l) \leq C, \quad \forall\; \l \in (0, \l^*].\end{aligned}$$
When $\O = B_1$ is the unit ball, $\gamma(x) = \gamma(r)$ with $r = |x|$, $u_\l$ is radial by uniqueness of the minimal solution and satisfies $$\begin{aligned}
\label{radial}
-u'' - \frac{n-1}{r}u' - \gamma'u' = \l f(u)\quad \mbox{in } (0, 1],\end{aligned}$$ with $u'(0) = 0$ and $u(1) = 0$. Our main result in this section is the regularity of the extremal solution $u^*$ for any $f$ satisfying $(H)$ provided $n = 2$ and the optimal estimate for $u'$ claimed in Theorem \[radialu’\].
The method we use is similar to [@cc; @v], but the uniform boundedness of $\|u_\l\|_{C^1}$ is not enough to claim the regularity of $u^*$, because a singular $u^*$ could be Lipschitz in many cases (see Remark \[1.1\]). In fact, the estimate is crucial for our proof.
As in [@cc; @v], since $u_\l'(r) \leq 0$ by maximum principle or equation , the boundedness of $\|u_\l\|_{H_0^1}$ implies that for any $k\in {\mathbb N}$, $r > 0$, $\|u_\l\|_{C^k\left(\overline B_1\setminus B_r\right)} \leq C_{k, r}$, $\forall\; \l \in (0, \l^*]$. So we concentrate our attention near the origin. Derivating the equation or with respect to $r$, $$\begin{aligned}
-{\rm div}\left(e^\gamma\nabla u'\right) = e^\gamma u'\left[\l f'(u)- \frac{n-1}{r^2} + \gamma''\right] \; \mbox{ in } (0, 1].\end{aligned}$$ Using $\psi = r\eta(r) u_\l'(r)$ as test function in with $\eta \in H_0^1(B_1)\cap C(\overline B_1)$, by similar calculation as for Lemma 2.1 in [@cc], we obtain $$\begin{aligned}
\label{radialest1}
\int_{B_1} e^\gamma \Big[|\nabla(r\eta)|^2 - (n-1)\eta^2 + \gamma''r^2\eta^2\Big]u_\l'^2 \geq 0, \quad \forall\; \l \in (0, \l^*].\end{aligned}$$
Proof of Theorem 1.1
--------------------
For simplicity, we drop the index $\l$. All estimates below hold uniformly for $\l$. First as $u_\l$ is radial, by maximum principle, we see that $u$ is decreasing in $r$. Since $f$ and $f'$ are nondecreasing functions according to $(H)$, the estimate implies (as $n = 2$) $$\begin{aligned}
\pi r^2 f'(u(r))f(u(r)) \leq \int_{B_r} f'(u)f(u) \leq C, \quad \forall\; r \in (0, 1].\end{aligned}$$ By Lemma \[ff’\], we have $$\begin{aligned}
\label{f}
f(u(r)) \leq \frac{C}{r} \quad \mbox{for all } r \in (0, 1].\end{aligned}$$ Let $r_0 \in (0, \frac{1}{2}]$. Let $\eta$ be a radial function in $H_0^1(B_1)\cap C^0(\overline B_1)$ such that $$\eta(r) = \left\{\begin{array}{ll}
r_0^{-1} & \mbox{if } r < r_0;\\
r^{-1} & \mbox{if } r_0 \leq r \leq \frac{1}{2},
\end{array}
\right.$$ and $\eta$ be a fixed $C^1$ function in $\overline B_1\setminus B_{1/2}$, independent of $r_0$. The direct calculation yields $$|\nabla(r\eta)|^2 - \eta^2 + \gamma''r^2\eta^2 = \left\{\begin{array}{ll}
\gamma''r^2r_0^{-2} & \mbox{if } r < r_0;\\
\gamma'' - r^{-2} & \mbox{if } r_0 < r \leq \frac{1}{2}.
\end{array}
\right.$$ Using , as $u$ is uniformly bounded in $H^1(B_1)$ by Proposition \[H01\] and $r^2r_0^{-2} \leq 1$ in $[0, r_0]$, we get $$\int_{r_0}^{\frac{1}{2}} \frac{u'(r)^2}{r} dr \leq C.$$ Tending $r_0$ to $0$, there holds $$\begin{aligned}
\label{radialest2}
\int_0^1 \frac{u'(r)^2}{r} dr \leq C.\end{aligned}$$ Consider the following test function used in [@v]: For any $r \leq \frac{1}{2}$ and $0 < r_0 < r$, $$\eta(s) = \left\{\begin{array}{ll}
(rr_0)^{-1} & \mbox{if } s < r_0;\\
(rs)^{-1} & \mbox{if } r_0 \leq s < r;\\
s^{-2} & \mbox{if } r \leq s \leq \frac{1}{2}.
\end{array}
\right.$$ Applying again and combining with , we obtain finally (with $r_0 \rightarrow 0$) $$\begin{aligned}
\label{radialest3}
\int_0^r \frac{u'(s)^2}{s} ds \leq Cr^2, \qquad \forall\; r \leq 1.\end{aligned}$$ As $\left(e^\gamma ru'\right)' = -\l e^\gamma rf(u)$ with $n = 2$, so $e^\gamma ru'$ is nonincreasing in $r$. Then $u'(s) \leq Cru'(r)/s$ for $s \in [r, 1]$, hence $u'(s) \leq Cu'(r) \leq 0$ for any $s\in [r, 2r]$ if $r \leq \frac{1}{2}$. By , for any $0 < r \leq \frac{1}{2}$, $$\begin{aligned}
C_1r^2 \geq \int_0^{2r} \frac{u'(s)^2}{s} ds \geq \int_r^{2r} \frac{u'(s)^2}{s} ds \geq \frac{C_2}{r}\int_r^{2r} u'(r)^2ds = C_3u'(r)^2.\end{aligned}$$ That means $$\begin{aligned}
\label{estu'}
|u'(r)| \leq Cr \quad \mbox{in } \;[0, 1].\end{aligned}$$ However, we need to consider also $u''(r)$ as explained above. Let $$G(r) = e^\gamma r u' \quad \mbox{and}\quad \Psi(r) = -2G(\sqrt{r}) - M\int_0^r (r-s)f\left(u(\sqrt{s})\right)ds$$ where $M$ is a constant to be chosen. Using $G' = - \l e^\gamma rf(u)$, $$\begin{aligned}
\Psi''(r) & = \left[\l e^{\gamma(s)}f'\left(u(s)\right)\frac{u'(s)}{2s} + \l e^{\gamma(s)} f\left(u(s)\right)\frac{\gamma'(s)}{2s} - Mf\left(u(s)\right)\right]\Big|_{s = \sqrt{r}}\\
& \leq \left[\l e^{\gamma(s)} f\left(u(s)\right)\frac{\gamma'(s)}{2s} - Mf\left(u(s)\right)\right]\Big|_{s = \sqrt{r}}\\
& \leq C_0f\left(u(\sqrt{r})\right) - M f\left(u(\sqrt{r})\right).\end{aligned}$$ For the last line, we used $|\gamma'(s)|/s \leq C$ in $[0, 1]$ since $\gamma$ is a smooth function (so $\gamma'(0) = 0$). Fix $M > C_0 + 1$, $\Psi$ is then concave in $[0, 1]$. On the other hand, by $$\begin{aligned}
\Psi'(r) = \l e^{\gamma(\sqrt{r})}f\left(u(\sqrt{r})\right) - M\int_0^r f\left(u(\sqrt{s})\right) ds \geq C\l f(0) - CM\sqrt{r}.\end{aligned}$$ There exists $r_1 > 0$ small enough such that $\Psi' \geq 0$ in $[0, r_1]$ with $\l \geq \frac{\l^*}{2}$. Using , and , for $\l \geq \frac{\l^*}{2}$ and $r \leq r_1$, $$\begin{aligned}
& -e^{\gamma(\sqrt{r})}\left[u''(\sqrt{r}) + \frac{u'(\sqrt{r})}{\sqrt{r}} + \gamma'u'(\sqrt{r})\right] - CM\sqrt{r}\\ \leq & \; \Psi'(r) \leq \frac{\Psi(r)}{r} \leq -2e^{\gamma(\sqrt{r})}\frac{u'(\sqrt{r})}{\sqrt{r}} \leq C.\end{aligned}$$ Applying one more time , we see that $u''(\sqrt{r}) \geq -C$ for any $\l \geq \frac{\l^*}{2}$ and $r \leq r_1$. Otherwise, by and , $u''(r) \leq -u'(r)r^{-1} - \gamma'(r)u'(r) \leq C$, we claim then $$\begin{aligned}
\|u''\|_\infty \leq C, \quad \forall\; \l \geq \frac{\l^*}{2}.\end{aligned}$$ Combining with and , it means $\|\l f(u)\|_\infty \leq C$, no singularity will occur.
Proof of Theorem 1.2
--------------------
As above, we drop the index $\l$ and all estimations hold uniformly for $\l$. First, repeating the proof of Theorem 1.8, c) in [@cc], we obtain $f'(u(r)) \leq Cr^{-2}$ in $(0, 1]$. Using Lemma \[ff’\] with , $f(u(r)) \leq Cr^{-2}$ in $(0, 1]$. Consequently, by , for $n \geq 3$, $$\begin{aligned}
0 \leq -e^\gamma r^{n-1}u'(r) = \int_0^r e^{\gamma(s)} s^{n-1}f(u(s))ds \leq C\int_0^r s^{n-3}ds \leq Cr^{n-2}.\end{aligned}$$ Hence $$\begin{aligned}
\label{generalu'}
|u'(r)| \leq \frac{C}{r}.\end{aligned}$$ Let $\eta$ be a radial function in $H_0^1(B_1)\cap C^0(\overline B_1)$ such that $$\eta(r) = \left\{\begin{array}{ll}
r_0^{-\sqrt{n-1}} & \mbox{if } r < r_0;\\
r^{-\sqrt{n-1}} & \mbox{if } r_0 \leq r \leq r_1.
\end{array}
\right.$$ in $\overline B_{r_1}$ and be a fixed $C^1$ function in $\overline B_1\setminus B_{r_1}$, here $r_0$ is any constant in $(0, r_1)$, $r_1 > 0$ is a small constant to be determined. Therefore $$|\nabla(r\eta)|^2 - (n-1)\eta^2 + \gamma''r^2\eta^2 = \left\{\begin{array}{ll}
\left(\gamma''r^2 + 2 - n\right)r_0^{-2\sqrt{n-1}} & \mbox{if } r < r_0;\\
\left(\gamma''r^2 - 2\sqrt{n-1} + 1\right) r^{-2\sqrt{n-1}} & \mbox{if }r \in [r_0, r_1].
\end{array}
\right.$$ We fix $r_1 > 0$ small enough such that $$\max_{r \in [0, r_1]}\left\{\gamma''r^2 \right\} < \min\left(n - 2, 2\sqrt{n-1} - 1\right).$$ By , as $|\nabla(r\eta)|^2 - (n-1)\eta^2 + \gamma''r^2\eta^2 \leq 0$ for $r\in [0, r_0]$, $$\begin{aligned}
\int_{r_0}^{r_1} u'^2(r) r^{n - 1-2\sqrt{n-1}}dr \leq C.\end{aligned}$$ Tending $r_0$ to $0$, we have $$\begin{aligned}
\label{radialest4}
\int_0^{r_1} u'^2(r) r^{n - 1-2\sqrt{n-1}}dr \leq C.\end{aligned}$$ Now we take another test function used in [@v], $$\eta(r) = \left\{\begin{array}{ll}
r_0^{-\sqrt{n-1}-1} & \mbox{if } r < r_0;\\
r^{-\sqrt{n-1}-1} & \mbox{if } r_0 \leq r \leq r_1.
\end{array}
\right.$$ Combining and , we conclude then $$\begin{aligned}
\int_0^{r_0} u'^2(r) r^{n - 1}dr \leq Cr_0^{2+ 2\sqrt{n-1}}, \quad \forall\; r_0 \in [0, r_1].\end{aligned}$$ By the monotonicity of $e^\gamma r^{n-1}u'$, similarly as for , it holds $$\begin{aligned}
|u'(r)| \leq C r^{-\frac{n}{2}+1 + \sqrt{n-1}}, \quad \forall\; r \in [0, 1].\end{aligned}$$ Finally, combining with , we are done (in fact, $-\frac{n}{2}+1 + \sqrt{n-1} \leq -1$ for $n \geq 10$).
[**Acknowledgments**]{} [*Part of the work was completed during X.L.’s visit to University of Connecticut (Uconn) with the financial support of CSC. She would like to thank the Department of Mathematics of Uconn for its warm hospitality. She thanks also Prof. Ryzhik for useful discussion. D.Y. is supported by the French ANR project referenced ANR-08-BLAN-0335-01. F.Z. is supported in part by NSFC No. 10971067, the “basic research project of China" No. 2006CB805902 and Shanghai project 09XD1401600.*]{}
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[^1]: MSC: 35B65, 35B45, 35J60
|
---
abstract: 'Network Intrusion Detection Systems (NIDS) play an important role as tools for identifying potential network threats. In the context of ever-increasing traffic volume on computer networks, flow-based NIDS arise as good solutions for real-time traffic classification. In recent years, different flow-based classifiers have been proposed based on both shallow and deep learning. Nevertheless, these classical machine learning algorithms have some limitations. For instance, they require large amounts of labeled data, which might be difficult to obtain. Additionally, most machine learning models are not general enough to be applied in different contexts. To overcome these limitations, we propose a new flow-based classifier, called Energy-based Flow Classifier (EFC). This anomaly-based classifier uses inverse statistics to infer a model based on labeled benign examples. We show that EFC is capable to accurately perform a two-class flow classification and is resilient to context change. Given the positive results obtained, we consider EFC o be a promising algorithm to perform flow-based traffic classification.'
author:
- 'Camila Pontes, João Gondim, Matt Bishop and Marcelo Marotta [^1] [^2]'
bibliography:
- 'bibliography.bib'
title: 'A new method for flow-based network intrusion detection using inverse statistical physics'
---
=1
Flow-based Network Intrusion Detection, Anomaly-based Network Intrusion Detection, Network Flow Classification, Network Intrusion Detection Systems, Energy-based Flow Classifier.
Introduction
============
Internet Security Threat Report [@symantec_internet_2019] points out a 56% increase in the number of web attacks in 2019. Network scans, denial of service, and brute force attacks are among the most common threats. Such malicious activities threaten not only individuals, but also some collective organizations such as public health, financial, and government institutions. In this context, play an important role as tools for identifying potential threats.
There are two different approaches for regarding the kind of data analyzed: packet-based and flow-based. In the former, deep packet inspection is performed taking into account individual packet payloads as well as header information. In the latter, flows, as packet collections, are analyzed regarding their properties, duration, number of packets, number of bytes, and source/destination port. In order to perform traffic classification in real time, a huge volume of data must be analyzed preventing more accurate and complex mechanisms to be used, such as deep packet inspection. Since flow-based approaches can classify the whole traffic inspecting an equivalent to 0.1% of the total volume [@Sperotto2010], based on flow analysis arise as good solutions for real-time traffic classification.
In recent years, different flow-based classifiers have been proposed based on both shallow and deep learning [@Umer2017]. According to the report in [@Umer2017], the best flow-based classifiers achieve around 99% accuracy. Although quite accurate, classical based classifiers require labeled malicious traffic samples to perform training. However, real traffic samples might be difficult to label as malicious. based classifiers training process, in turn, is specific for the context considered making them unable to be easily generalized. Moreover, algorithms are well-known to be black box mechanisms difficult to be readjusted in detail. In this regard, there is a clear need of a new flow-based classifier for able to be readjusted in details (white box) without training, based on malicious examples, and resilient to context change.
We propose a new flow-based classifier called . EFC is an anomaly-based classifier, it learns from labeled data characteristics typical of normal traffic and use this information to classify unlabeled flows. In this process, flow features are analyzed, both individually and jointly, to identify frequent values associated to normal traffic behavior. Feature values which seldom occur are, in turn, regarded as abnormal. This procedure, as described, resembles problems involving energy landscapes in statistical physics, where the energy of a given system state is determined by the configuration assumed by each individual particle in the system. Particles on lower energy configurations are more frequent and tend to be present in a system in “normal” state. Following this analogy, each specific single or joint flow feature occurrence contribute to raise or lower the “energy” of a given flow. So, a certain “energy” value may be associated with each unlabeled flow and compared to a given normality threshold. Therefore, the idea and rationale rely on mapping network flow anomaly detection to a similar problem already successfully addressed by statistical physics. This is the motivation that drives the method supporting .
is a white box flow classifier based on inverse statistical physics. Since is based on a multivariate statistical model, it can be analyzed in detail regarding individual parameter values. Moreover, as long as normal traffic can be characterized in flows, will be able to detect the opposite, malicious traffic. We compared the performance of against a variety of classifiers trained within different scenarios, a simulated and a real network from the flow dataset CIDDS-001 [@ring2017flow]. Our results show that classifiers based on shallow and deep learning are highly sensitive to context change. Meanwhile, was the only classifier to achieve high accuracy when subjected to context change regarding the two different scenarios from the dataset. Our main contributions are as follows:
- The mapping of flow features to serve as input for an inverse statistical physics based model.
- The proposal of a flow classifier for based on inverse statistical physics.
- A comparison among and the best classifiers found on the literature of flow-based .
The rest of this article is structured as follows. In Section II, we briefly present the state-of-the-art in flow-based . In Section III, we describe the structure of a network flow and present a preliminary analysis of the flow dataset used here. In Section IV, we introduce the statistical model proposed and the classifier implementation. In Section V, we present results obtained regarding the analysis of the statistical model and the classification experiments performed. Finally, in Section VI, we present our conclusions and future work.
Related Work
============
In this section, we briefly review the state-of-the-art in flow-based network intrusion detection. We show some early work in the field, as well as the recent advances. In the end, some previous work performed on CIDDS-001 dataset are shown.
In 2011, Winter [@winter2011inductive] presented an innovative approach for flow-based network traffic classification. They propose a classifier that uses a one-class with learning based on malicious traffic examples. Their algorithm achieves around 98% accuracy with a very low false positive rate. We revisit this idea proposing a new one-class classier with training, however, based solely on benign traffic examples instead of malicious ones.
Apart from one-class-based approaches, which are not easily found in literature, several -based and statistical-based flow classifiers were proposed in recent years. In 2017, Umer wrote a comprehensive survey on flow-based [@Umer2017], in which all recently proposed flow-based classifiers were reviewed. In this work, we deploy most of the -classifiers covered in [@Umer2017] to serve as baselines, against which we compare our classifier.
Recently, flow-based intrusion detection has been explored in modern contexts, networks [@moustafa2018ensemble; @tama2017attack] and cloud environments [@idhammad2018distributed; @idhammad2018detection]. The proposed solutions for intrusion detection in and cloud environments achieved satisfactory classification accuracy and feasible running times. However, their capability to be reused in different scenarios without retraining is still a matter of investigation. In fact, most of the proposed solutions from literature assume that there will be available training sets to be used in all contexts, which is not necessarily true. In this regard, we propose a flow-classifier solution able to be adapted to different contexts without retraining.
To assess performance, we selected a network flow dataset: CIDDS-001. This dataset was used by Verma and Ranga [@verma2018statistical] to assess the performance of and k-means clustering algorithms when classifying traffic. Both algorithms achieved over 99% accuracy. Also, Ring [@ring2018detection] explored slow port scans detection using CIDDS-001. The approach proposed by them is capable of accurately recognizing the attacks with low false alarm rate. Finally, Abdulhammed [@Abdulhammed2019] also performed flow-based classification on CIDDS-001, and proposed an approach that is robust considering imbalanced network traffic. In summary CIDDS-001 is an up-to-date and relevant dataset to be used for network flow-classification solutions, being our dataset choice for assessing the performance of EFC.
Preliminaries
=============
For a given network flow to be considered malicious, it must present at least one observable abnormality. In other words, from the perspective of an anomaly-based classifier, a combination of anomalies is what characterizes a malicious flow. In this regard, as a preliminary analysis, we are going to characterize flows considering two distinct classes, normal and malicious. If it is observed that flows labeled as malicious present abnormal characteristics, as expected, then must be able to correctly classify unlabeled flows after learning normal traffic characteristics. As it will be later seen, learning here corresponds to deriving a model for normal traffic from normal traffic characteristics. Therefore, next we analyze network flows regarding their features. We also describe the statistical distribution of flows within CIDDS-001. And finally, we investigate if there is significant feature variation between different labeled flow classes to consider them malicious or normal.
Network flow profiling
----------------------
A network flow is a set of packets that traverses intermediary nodes between end points within a given time interval. Under the perspective of an intermediary node, an observation point, all packets belonging to a given flow have a set of common features called flow keys. It means that flow keys do not change for packets belonging to the same flow, while the remaining features might vary. FlowScan [@plonka2000flowscan] is an example of tool capable of collecting data from a set of packets and extracting flow features to be later exported in different formats, such as NetFlow and IPFIX. Since NetFlow is the most commonly used format, its main features are listed bellow:
- Source/Destination IP (flow key) - determine the origin and destination of a given flow in the network;
- Source/Destination port (flow key) - characterize different kinds of network services port 22 is used to access an ssh service;
- Protocol (flow key) - characterizes flows regarding the transport protocol used TCP, UDP, ICMP.
- Number of packets (feature) - total number of packets captured in a flow;
- Number of bytes (feature) - total number of bytes in a flow;
- Duration (feature) - total duration of a flow in seconds;
- Initial timestamp (feature) - system time when a flow started to be captured;
Other features such as TCP Flags and Type of Service might also be exported in some cases. The combination of different flow keys and features characterize a flow and determine its particular behavior.
It is important to note that individual packets usually have more features than a flow. The number of features vary according to the collecting and sampling methodologies applied to the network traffic. For instance, packets from NSL-KDD dataset have 42 features, while flows from CIDDS-001 dataset have 10. This information gap makes flow classification inherently less accurate than packet classification. Nevertheless, flow-based approaches are seen as good alternatives to precede packet inspection in real-time . The idea is to deeply inspect only the packets belonging to flows considered to be suspicious by the flow-based classifier. A two-step approach would notably reduce the amount of data analyzed, while maintaining a high classification accuracy [@Sperotto2010]. However, there are still some open questions in the field regarding how many and which features should be taken into account for classification. Trying to address these questions, in the next section we are going to characterize some features distribution within different flow classes.
{width="95.00000%"}
Feature characterization
------------------------
In order to perform a comprehensive feature characterization, a network trace containing different flow classes is necessary. The chosen flow dataset, CIDDS-001 [@ring2017flow], is a relatively recent dataset composed of a set of flow samples captured within a simulated OpenStack environment and another set of flow samples captured in an external server. The former contains only simulated traffic, while the latter contains both real and simulated traffic.
Each sample collected within the simulated environment has one of the labels described in the following:
- *normal* - normal traffic;
- *dos* - attack traffic;
- *portScan* - port scan attack traffic;
- *pingScan* - ping scan attack traffic;
- *bruteForce* - brute force attack traffic.
Flows labeled as *dos*, *portScan*, *pingScan* or *bruteForce* are malicious flows. Each malicious flow is labeled regarding its origin as either *attacker* or *victim*. Flows sampled within the external server environment may have two extra labels, *suspicious* and *unknown*. Traffic was sampled in both the simulated and the external environment during a four week period. For the simulated environment, we consider only traffic captured in the second week to reduce the amount of data to be analyzed. Similarly, only external traffic captured within the third week was assessed. CIDDS-001 dataset flow features are shown in Table \[tab:features\]. All features within Table \[tab:features\] were taken into account for characterization and classification except for *Src IP*, *Dest IP* and *Date first seen*.
**\#** **Name** **Description**
-------- ----------------- ----------------------------------------
1 Src IP Source IP Address
2 Src Port Source Port
3 Dest IP Destination IP Address
4 Dest Port Destination Port
5 Proto Transport Protocol (ICMP, TCP, or UDP)
6 Date first seen Start time flow first seen
7 Duration Duration of the flow
8 Bytes Number of transmitted bytes
9 Packets Number of transmitted packets
10 Flags OR concatenation of all TCP Flags
: Features within CIDDS-001 dataset[]{data-label="tab:features"}
In order to evaluate the relationship between flow features and traffic classification, we characterized how normal/malicious traffic fractions change in response to constraining a given feature. To perform this analysis, 2,000 flows with each of the following labels: *normal*, *dos*, *portScan*, *pingScan*, and *bruteForce* were sampled from the simulated traffic within dataset CIDDS-001. Afterwards, the number of flows with a given feature value smaller than or equal to a given upper limit in each class was counted. Similarly, the same was done for decreasing upper limits of the same feature.
Figure \[fig:features\] A) shows the normal/malicious traffic fractions for different upper limits considering the source port feature. It is possible to observe some anomalous patterns since malicious traffic fraction does not decrease in the same rate as normal traffic fraction. For instance, normal traffic fraction presents a near linear decrease. Since there is no TCP or UDP port number associated to ICMP protocol, ping scan traffic within CIDDS-001 dataset has source port value equal to 0. Given that, ping scan attack is perceived as coming mainly from a source “port” smaller than 50, and, consequently, its traffic fraction remains constant. For DoS, port scan and brute force attacks, their traffic fractions present an abrupt decrease between ports 50,000 and 10,000, showing that traffic of these types comes mostly from higher ports.
Normal/malicious traffic fractions for different upper limits of destination port are shown in Figure \[fig:features\] B). It is possible to see that the anomalous patterns observed in B) change considerably in comparison to A), because malicious traffic comes from higher ports and is mainly aimed at smaller ports. While normal fraction decreases almost linearly, for DoS and brute force there is an abrupt decrease, since DoS traffic is mainly aimed at port 80 and brute force traffic at port 22. Here again it is important to notice that there is no TCP or UDP port number associated to ICMP protocol, so ping scan traffic within CIDDS-001 dataset has destination port value equal to 8. Destination port feature for port scan attack cannot be used to detect an anomalous behavior considering its nearly proportional fraction change.
Figure \[fig:features\] C) shows the normal/malicious traffic fractions for different upper limits of feature number of bytes. Anomalous patterns can be observed considering the number of bytes for all attacks, since normal/malicious traffic fractions decrease at different rates. For instance, traffic presents an abrupt decrease after 500, showing that all flows have 500 bytes or more. This is characteristic of volumetric DoS, which intends to flood the victim with traffic. Port/ping scan traffic fractions have slower decreases, since these types of flows tend to have number of bytes below 60. Meanwhile, like in previous cases, normal traffic fraction decreases almost linearly.
Finally, normal/malicious traffic fractions for different upper limits of duration are shown in Figure \[fig:features\] D). Regarding feature duration, the normal/malicious traffic fractions decrease is proportional for most cases. The only exception is traffic, which presents an abrupt decrease on 0.005 meaning that most flows belonging to this class have duration above 0.001.
Noticeably, the decremental change of the flow features enabled the detection of malicious traffic in a not very effective way. Our main contribution is to provide a mechanism able to combine the different features to perform sharper detection of malicious traffic considering abnormal behavior by combining multiple features. We construct a model based on flow features patterns present in normal traffic using inverse statistical physics to perform traffic classification, such as presented in the next section.
Statistical model
=================
The main task of inverse statistical physics is to infer a statistical distribution based on a sample of it [@Cocco2018]. Methods using inverse statistical physics have been successfully applied to problems in other disciplines, the problem of predicting protein contacts in Biophysics [@Cocco2018; @Morcos2011]. Here, the statistical inference is based on the Potts model [@Wu1982]. This model provides a mathematical description of interacting spins on a crystalline lattice. Within the model framework, interacting spins are mapped into a graph $\func$ (see Figure \[fig:model\] A)), where each node $\nodei \in \node = \{1, ..., N\}$ has an associated spin $\spin_\nodei$, which can assume one value from a set $\alphabet$ that contains all possible individual quantum states. Each node $\nodei$ has also an associated local field $\field_\nodei (\spin_\nodei)$ that is a function of $\spin_\nodei$’s state. Meanwhile, each edge $(\nodei,\nodej) \in \edge$, $\nodei, \nodej \in \node$, has an associated coupling value $\coup_{\nodei\nodej} (\spin_\nodei,\spin_{\nodej})$ that is a function of the states of spins $\spin_\nodei$ and $\spin_\nodej$ associated to nodes $\nodei$ and $\nodej$. A specific system configuration has an associated total energy, determined by the Hamiltonian function $\mathcal{H}(a_1...a_N)$, which depends on all spin states.
In this work, we adapt the Potts model to characterize network flows (see Figure \[fig:model\] B)). An individual flow $\flowid$ is represented by a specific graph configuration $\funck$. Instead of spins, each node represents a selected feature $\featurei \in \node = \{Src Port, ..., Flags\}$. Within a given flow $\flowid$, each feature $\featurei$ assumes one value $\feature_{\flowid\featurei}$ from the set $\featureset_{\featurei}$ that contains all possible values for this feature. As in the Potts Model, each feature $\featurei$ has an associated local field $\field_\featurei(\feature_{\flowid\featurei})$. Meanwhile, $\edge = \{(\featurei,\featurej) | \featurei, \featurej \in \node ; \featurei \neq \featurej\}$ is the set of edges determined by all possible pairs of features. Each edge has an associated coupling value determined by the function $\coup_{\featurei\featurej}(\feature_{\flowid\featurei}, \feature_{\flowid\featurej})$.
![A) Interacting spins on a crystalline lattice. B) Network flow mapped into a graph structure.[]{data-label="fig:model"}](flow_model.pdf){width="0.95\columnwidth"}
Since the values of local fields and couplings depend on the values assumed by features within a given flow, each distinct flow will have a different combination of these quantities. As in the Potts Model, local fields and couplings determine the total “energy” $\mathcal{H}(\feature_{\flowid1}...\feature_{\flowid \featureids})$ of each flow. For instance, in Figure \[fig:model\] B), the total “energy” of the flow is obtained by summing up all values associated to the edges and to the nodes, resulting in a total of -3. Note that what we call energy is analogous to the notion of Hamiltonian in Quantum Mechanics.
It is important to note that the model described here is discrete, ergo continuous features must be discretized. The classes for continuous features discretization were defined based on the analysis performed in the last section and are shown on Appendix \[appendix:classes\]. In the following, we present the framework applied to perform the statistical model inference and subsequent energy-based flow classification.
Model inference
---------------
In this section, coupling and local field values are going to be inferred from a statistical model to perform energy-based flow classification. The main idea consists in extracting a statistical model from benign flow samples to infer coupling and local field values that characterize this type of traffic. When calculating the energies of unlabeled flows using the inferred values, it is expected that benign flows will have lower energies than malicious flows.
Let $( A_{1}...A_{\featureids})$ be a $\featureids$-uple of features, which can be instantiated for flow $\flowid$ as $(\feature_{\flowid 1}...\feature_{\flowid\featureids})$, with $\feature_{\flowid 1} \in \alphabet_1, ..., \feature_{\flowid\featureids} \in \alphabet_\featureids$. Each feature value $\feature_{\flowid\featurei}$ is encoded by an integer from the set $\alphabet = \{ 1, 2, ..., Q \}$, all feature alphabets are the same $\alphabet_\featurei = \alphabet$ of size $Q$. If a given feature can only assume $M$ values and $M < Q$, it is considered that values $M+1, ..., Q$ are possible, but will never be observed empirically. For instance, if the only possible values for feature *protocol* are {*’TCP’*, *’UDP’*}, and given $Q = 4$. In this case, we would have the mapping {*’TCP’*:1, *’UDP’*:2, *’ ’*:3, *’ ’*:4 } and feature values 3 and 4 would never occur.
Now, let $\flowidset$ be the set of all possible flows, all possible combinations of feature values ($\flowidset = \alphabet^N$), and let $\flowset \subset \flowidset$ be a sample of flows. We can use inverse statistical physics to infer a statistical model associating a probability $P(\feature_{\flowid 1}...\feature_{\flowid \featureids})$ to each flow $\flowid \in \flowidset$ based on sample $\flowset$. The global statistical model $P$ is inferred following the Entropy Maximization Principle [@Jaynes1957]:
$$\begin{aligned}
\max_{P} & \ \ \ - \sum_{\flowid \in \flowidset} \ \ \ \ \ \ \ P(\feature_{\flowid1}...\feature_{\flowid \featureids}) log (P(\feature_{\flowid 1}...\feature_{\flowid \featureids})) \label{eq:max}\\
s.t.\nonumber\\
& \ \ \ \sum_{\flowid \in \flowidset | \feature_{\flowid \featurei} = \feature_{\featurei}} \ \ \ \ P(\feature_{\flowid 1} ... \feature_{\flowid \featureids}) = f_\featurei(\feature_\featurei) \label{eq:3} \\
&\qquad\qquad\forall \featurei \in \node;\ \forall \feature_{\featurei} \in \Omega; \nonumber \\
&\sum_{\flowid \in \flowidset | \feature_{\flowid \featurei} = \feature_{\featurei}, \feature_{\flowid \featurej} = \feature_{\featurej}} P(\feature_{\flowid 1} ... \feature_{\flowid \featureids}) = f_{\featurei \featurej} (\feature_\featurei, \feature_{\featurej}) \label{eq:4} \\
&\qquad\qquad\forall (\featurei,\featurej) \in \node^2\ |\ \featurei \neq \featurej;\ \forall (\feature_{\featurei}, \feature_{\featurej}) \in \Omega^2;\nonumber\end{aligned}$$
where $f_\featurei(\feature_\featurei)$ is the empirical frequency of value $\feature_\featurei$ on feature $\featurei$ and $f_{\featurei\featurej}(\feature_\featurei,\feature_{\featurej})$ is the empirical joint frequency of the pair of values $(\feature_\featurei,\feature_{\featurej})$ of features $\featurei$ and $\featurej$. Note that constraints \[eq:3\] and \[eq:4\] force model $P$ to generate single as well as joint empirical frequency counts as marginals. This way, the model is sure to be coherent with empirical data.
Single and joint empirical frequencies $f_\featurei(\feature_\featurei)$ and $f_{\featurei\featurej}(\feature_\featurei,\feature_{\featurej})$ are obtained from set $\flowset$ by counting occurrences of a given feature value $\feature_\featurei$ or feature value pair $(\feature_\featurei, \feature_\featurej)$, respectively, and dividing by the total number of flows in $\flowset$. Since the set $\flowset$ is finite and much smaller than $\flowidset$, inferences based on $\flowset$ are subjected to undersampling effects. Following the theoretical framework proposed in [@Morcos2011], we add pseudocounts to empirical frequencies to limit undersampling effects by performing the following operations: $$f_\featurei(\feature_\featurei) \leftarrow (1 - \alpha ) f_\featurei(\feature_\featurei) + \frac{\alpha}{Q}\label{eq:single_site}$$ $$f_{\featurei \featurej}(\feature_\featurei,\feature_{\featurej}) \leftarrow (1 - \alpha ) f_{\featurei \featurej}(\feature_\featurei,\feature_{\featurej}) + \frac{\alpha}{Q^2}\label{eq:pair_freq}$$ where $(\feature_\featurei, \feature_{\featurej}) \in \alphabet^2$ and $0 \leq \alpha \leq 1$ is a parameter defining the weight of the pseudocounts. The introduction of pseudocounts is equivalent to assuming that $\flowset$ is extended with a fraction of flows with uniformly sampled features.
The proposed maximization can be solved using a Lagrangian function such as presented in [@Jaynes1957], yielding the following Boltzmann-like distribution: $$P^*(\feature_{\flowid1}...\feature_{\flowid\featureids}) = \frac{e^{- \mathcal{H}(\feature_{\flowid1}...\feature_{\flowid\featureids})}}{Z} \label{eq:opt_dist}
$$ where $$\mathcal{H}(\feature_{\flowid1}...\feature_{\flowid\featureids}) = - \sum_{\featurei,\featurej \mid \featurei<\featurej} \coup_{\featurei\featurej}(\feature_{\flowid\featurei},\feature_{\flowid\featurej}) - \sum_\featurei \field_\featurei(\feature_{\flowid\featurei}) \label{eq:hamil}$$ is the Hamiltonian of flow $\flowid$. In eq. (\[eq:opt\_dist\]), $Z$ is the partition function that normalizes the distribution. Since in this work we are not interested in obtaining individual flow probabilities, $Z$ is not required and, as a consequence, its calculation is omitted. Our objective is to calculate individual flows energies, individual Hamiltonians as determined in eq. (\[eq:hamil\]).
Note that the Hamiltonian, as presented above, is fully determined in terms of the Lagrange multipliers $ \coup_{\featurei \featurej}(\cdot)$ and $\field_\featurei(\cdot)$ associated to constraints (\[eq:3\]) and (\[eq:4\]), respectively. Within the Potts Model framework, the Lagrange multipliers have a special meaning, with the set $\{\coup_{\featurei \featurej}(\feature_\featurei,\feature_{\featurej}) | (\feature_\featurei, \feature_\featurej) \in \alphabet^2\}$ being the set of all possible coupling values between features $i$ and $j$ and $\{\field_\featurei(\feature_\featurei) | \feature_\featurei \in \alphabet\}$ the set of possible local fields associated to feature $\featurei$.
Inferring the local fields and pairwise couplings is difficult since the number of parameters exceeds the number of independent constraints. Due to the physical properties of interacting spins, it is possible to infer pairwise coupling values $\coup_{\featurei \featurej}(\feature_\featurei,\feature_{\featurej})$ using a Gaussian approximation. Assuming that the same properties apply for flow features, we infer coupling values as follows: $$\begin{aligned}
\coup_{\featurei \featurej}(\feature_\featurei,\feature_{\featurej}) = -(C^{-1})_{\featurei \featurej} (\feature_\featurei,\feature_{\featurej}),\label{eq:coupling}\\ \forall (\featurei, \featurej) \in \node^2, \forall (\feature_\featurei, \feature_\featurej) \in \alphabet^2, \feature_\featurei, \feature_\featurej \neq Q \nonumber
\end{aligned}$$ where $$C_{\featurei \featurej}(\feature_\featurei,\feature_{\featurej}) = f_{\featurei\featurej}(\feature_\featurei,\feature_{\featurej}) - f_\featurei(\feature_\featurei)f_{\featurej}(\feature_{\featurej})$$ is the covariance matrix obtained from single and joint empirical frequencies. Taking the inverse of the covariance matrix is a well known procedure in statistics to remove the effect of indirect correlation in data [@giraud1999superadditive]. Here, it is important to clarify that, the number of independent constraints in eq. (\[eq:3\]) and eq. (\[eq:4\]) is actually $\frac{N(N-1)}{2}(Q-1)^2 + N(Q-1)$, even though the model in eq. (\[eq:opt\_dist\]) has $\frac{N(N-1)}{2}Q^2 + NQ$ parameters. So, without loss of generality, we set: $$\begin{aligned}
\coup_{\featurei, \featurej}(\feature_\featurei, Q) = \coup_{\featurei, \featurej}(Q, \feature_\featurej) = \field_\featurei (Q) = 0 \end{aligned}$$ Thus, in eq. (\[eq:coupling\]) there is no need to calculate $\coup_{\featurei, \featurej}(\feature_\featurei, \feature_\featurej)$in case $\feature_\featurei$ or $\feature_\featurej$ is equal to $Q$ [@Morcos2011]. Afterwards, local fields $\field_\featurei(\feature_\featurei)$ can be inferred using a mean-field approximation [@georges1991expand]: $$\begin{aligned}
\frac{f_\featurei(\feature_\featurei)}{f_\featurei(Q)} = exp \left ( \field_\featurei(\feature_\featurei) + \sum_{\featurej,\feature_{\featurej}} \coup_{\featurei \featurej}(\feature_\featurei,\feature_{\featurej}) f_{\featurej}(\feature_{\featurej}) \right ), \label{eq:mf}\\ \forall \featurei \in \node, \feature_\featurei \in \alphabet, \feature_\featurei \neq Q \nonumber
\end{aligned}$$ where $f_\featurei(Q)$ is the frequency of the last element $\feature_\featurei = Q$ for any feature $\featurei$ used for normalization. It is also worth mentioning that the element Q is arbitrarily selected and could be replaced by any other value in {1$\dots$Q} as long as the selected element is kept the same for calculations of the local fields of every feature $\featurei \in \node$. Note that in eq. (\[eq:mf\]) the empirical single frequencies $f_\featurei(\feature_\featurei)$ and the coupling values $\coup_{\featurei\featurej}(\feature_\featurei,\feature_{\featurej})$ are known, yielding: $$\begin{aligned}
\field_\featurei(\feature_\featurei) = ln \left (\frac{f_\featurei(\feature_\featurei)}{f_\featurei(Q)} \right ) - \sum_{\featurej, \feature_\featurej}\coup_{\featurei\featurej}(\feature_\featurei, \feature_\featurej)f_\featurej(\feature_\featurej) \label{eq:mf2}
\end{aligned}$$ In the mean-field approximation presented above, the interaction of a feature with its neighbors is replaced by an approximate interaction with an averaged feature, yielding an approximated value for the local field associated to it.
For further details about this calculations please refer to [@Cocco2018]. Now that all model parameters are known, it is possible to calculate a given flow energy according to eq. (\[eq:hamil\]). In the following, we are going to present the theoretical framework implementation to perform a two-class, normal and malicious, flow classification.
Energy-based flow classification {#sec:3b}
--------------------------------
The energy of a given flow can be calculated according to eq. (\[eq:hamil\]) based on the values of its features and the parameters from the statistical model inferred in the last section. In simple terms, a given flow energy is the negative sum of couplings and local fields associated to its features, according to a given statistical model. It means that, if a flow resembles the ones used to infer the model, it is likely to be low in energy. This happens because features that were strongly coupled in sample $\flowset$ that generated the model, are going to be present in that flow.
Since is an anomaly-based classifier, the statistical model used for classification is inferred based on normal flow samples. We would then expect energies of normal samples to be lower than energies of malicious samples. In this sense, it is possible to classify flow samples as normal or malicious based on a chosen energy threshold. The classification is performed by stating that samples with energy smaller than the threshold are normal and samples with energy greater than or equal to the threshold are malicious. Note that the threshold for classification can be chosen in different ways and it can be static or dynamic. In this work we will consider a static threshold. Algorithm \[algo:sitefreq\] implements the first step for model inference, considering equation (\[eq:single\_site\]) to calculate empirical single frequencies. On lines 3-7 the frequencies of each feature value in each column are counted. On line 8, frequency values are reweighted using pseudocount parameter $\alpha$. In the end, this function produces a two-dimensional array with shape $(N, Q)$, where $N$ is the number of features considered and $Q$ is the cardinality of the alphabet. This array contains the empirical frequencies of each possible value for all features.
matrix $f\_i_{(N \times Q)}$ $f\_i[i,a\_i] \leftarrow count(a\_i, flows[:,i]) / K$ $f\_i \leftarrow (1 - \alpha)f\_i + \alpha / Q$
Joint frequencies from equation (\[eq:pair\_freq\]), in turn, are calculated as described in Algorithm \[algo:pairfreq\]. On lines 3-8, the occurrences of each possible feature value pair are counted. Afterwards, on line 9, pairwise frequency values are reweighted using pseudocount parameter $\alpha$. On lines 10-18, to correct inconsistencies caused by the pseudocounts, elements in the diagonal are set to $0.0$ in case $\feature_\featurei \neq \feature_\featurej$ and to $f_\featurei(\feature_\featurei)$ otherwise, since there is no meaning in pairing one feature with itself. The output of this function is an array with shape $(N, Q, N, Q)$ containing the joint empirical frequencies of each possible pair of values ($Q^2$ possibilities) for each pair of features ($N^2$ possibilities).
matrix $f\_ij_{(N \times Q \times N \times Q)}$ $is\_pair \leftarrow [1$ for $k$ in $[1..K]$ if ($flows[k,i] = a\_i$ and $flows[k,j] = a\_j$) $]$ $f\_ij[i,a\_i,j,a\_j] \leftarrow count(1, is\_pair) / K$ $f\_ij \leftarrow (1 - \alpha)f\_ij + \alpha / Q^2$ $f\_ij[i,a\_i,i,a\_j] \leftarrow f\_i[i,a\_i]$ $f\_ij[i,a\_i,i,a\_j] \leftarrow 0.0$
matrix $cov_{(N \times Q-1 \times N \times Q-1)}$ $cov[i,a\_i,j,a\_j] \leftarrow f\_ij[i,a\_i,j,a\_j] - f\_i[i,a\_i] f\_i[j,a\_j]$ $e\_ij \leftarrow - cov^{-1}$
After single and joint empirical frequencies are computed according to equation (\[eq:coupling\]), coupling values are obtained as described in Algorithm \[algo:couplings\]. On lines 3-7, the covariance matrix is build, and on line 8 the negative of its inverse is computed. In the end, this functions returns an array with shape $(N \times (Q-1), N \times (Q-1))$ containing the negative inverse of the covariance matrix, the values of couplings $\coup_{\featurei\featurej}(\feature_\featurei, \feature_\featurej), \forall (\featurei,\featurej) \in \node^2, (\feature_\featurei, \feature_\featurej) \in \alphabet^2$.
matrix $h\_i_{(N \times Q-1)}$ $h\_i[i,a\_i] \leftarrow ln(f\_i[i,a\_i] / f\_i[i,Q])$ $h\_i[i,a\_i] \leftarrow h\_i[i,a\_i] - e\_ij[i,a\_i,j,a\_j] f\_i[j,a\_j]$
Afterwards, local fields from equation (\[eq:mf2\]) are computed as described in Algorithm \[algo:localfields\]. On line 5, local field $\field_\featurei(\feature_\featurei)$ is initialized as $ln \left (\frac{f_\featurei(\feature_\featurei)}{f_\featurei(Q)} \right )$. After that $\field_i(\feature_\featurei)$ is decremented by the values of the couplings $\coup_{\featurei\featurej}(\feature_\featurei, \feature_\featurej)f_\featurej(\feature_\featurej)$ for all possible values of $j$ and $\feature_\featurej$. The output of this function is an array with shape $(N, Q-1)$, containing the values of the local fields $\field_i(\feature_\featurei), \forall \featurei \in \node, \feature_\featurei \in \alphabet$.
**Input:** $normal\_flows_{(K \times N)}$, $Q$, $\alpha$, $cutoff$
import all model inference functions $f\_i \leftarrow SiteFreq(normal\_flows, Q, \alpha)$ $f\_ij \leftarrow PairFreq(normal\_flows, f\_i, Q, \alpha)$ $e\_ij \leftarrow Couplings(f\_i, f\_ij, Q)$ $h\_i \leftarrow LocalFields(e\_ij, f\_i, Q)$
$flow \leftarrow$ wait\_for\_incoming\_flow() e $\leftarrow 0$ $a\_i \leftarrow flow[i]$ $a\_j \leftarrow flow[j]$ $e \leftarrow e - e\_ij[i,a\_i,j,a\_j]$ $e \leftarrow e - h\_i[i,a\_i]$ stop\_flow() forward\_to\_DPI() release\_flow()
Finally, Algorithm \[algo:efc\] shows the implementation of EFC. On lines 2-5, the statistical model for the sampled flows is inferred using the functions described in Algorithms \[algo:sitefreq\]-\[algo:localfields\]. Afterwards, on lines 6-27, the classifier monitors the network waiting for a captured flow. When a flow is captured, its energy is calculated on lines 9-20, according to the Hamiltonian involving the received flow features in equation (\[eq:hamil\]). The computed flow energy is compared to a known threshold (*cutoff*) value on line 21. In case the energy falls bellow the threshold, the flow is classified as malicious and should be forwarded to deep packet inspection (line 23) for assessment. Otherwise, the flow is released and the classifier waits for another flow. Considering this implementation, next, we present the results obtained when EFC is used to perform flow classification.
Results
=======
In this section, we describe and discuss the results obtained regarding the statistical model properties and the classification tests performed on dataset CIDDS-001. First, we show that the statistical model recovers important traffic properties from the flow set used in the model building phase. Then it is shown that the statistical model built based on normal traffic samples is able to separate normal from malicious flows based on their energies. Finally, we present the classification results obtained for and compare them to results obtained for other classifiers in two different test scenarios.
Model analysis
--------------
Tables \[tab:couplings\_normal\] - \[tab:couplings\_bruteforce\] show the most strongly coupled variables in the statistical models inferred based on traffic labeled as *normal*, *dos*, *portScan*, *pingScan* and *bruteForce*, respectively. Entries present in Tables \[tab:couplings\_dos\] - \[tab:couplings\_bruteforce\] but not in Table \[tab:couplings\_normal\] are in bold. By looking at the top 5 couplings in different statistical models, it is possible to see that each model is characterized by different coupling patterns.
Feature $i$ Value $a_i$ Feature $j$ Value $a_j$ $\coup_{ij}(a_i,a_j)$
------------- ------------- ------------- ------------- -----------------------
Protocol UDP Flags ...... 19.5
Src Pt 100-400 Dst Pt 100-400 12.6
Duration 0.000 Packets 1 11.8
Protocol TCP ToS 32 11.4
Bytes 60-70 Flags .A.... 11.0
: Top 5 coupling values obtained from statistical model inferred based on flows labeled as normal[]{data-label="tab:couplings_normal"}
Feature $i$ Value $a_i$ Feature $j$ Value $a_j$ $\coup_{ij}(a_i,a_j)$
------------- ------------- ------------- ------------- -----------------------
Protocol UDP Flags ...... 17.9
**Packets** **5** **Bytes** **400-500** **15.3**
**Packets** **6** **Bytes** **500-700** **14.5**
Duration 0.000 Packets 1 13.4
Bytes 60-70 Flags .A.... 12.0
: Top 5 coupling values obtained from statistical model inferred based on flows labeled as dos[]{data-label="tab:couplings_dos"}
Feature $i$ Value $a_i$ Feature $j$ Value $a_j$ $\coup_{ij}(a_i,a_j)$
-------------- ------------- ------------- ------------- -----------------------
Protocol UDP Flags ...... 15.8
Duration 0.000 Packets 1 14.3
**Bytes** **50-60** **Flags** **....S.** **12.9**
**Protocol** **ICMP** **Src Pt** **0-50** **11.8**
**Src Pt** **0-50** **Bytes** **0-50** **11.6**
: Top 5 coupling values obtained from statistical model inferred based on flows labeled as port scan[]{data-label="tab:couplings_portscan"}
Feature $i$ Value $a_i$ Feature $j$ Value $a_j$ $\coup_{ij}(a_i,a_j)$
-------------- ------------- ------------- ------------- -----------------------
Duration 0.000 Packets 1 14.7
Protocol UDP Flags ...... 14.6
**Protocol** **ICMP** **Src Pt** **0-50** **14.3**
**Protocol** **ICMP** **Dst Pt** **0-50** **14.0**
**Src Pt** **0-50** **Dst Pt** **0-50** **14.0**
: Top 5 coupling values obtained from statistical model inferred based on flows labeled as ping scan[]{data-label="tab:couplings_pingscan"}
Feature $i$ Value $a_i$ Feature $j$ Value $a_j$ $\coup_{ij}(a_i,a_j)$
------------- ------------- ------------- ------------- -----------------------
Protocol UDP Flags ...... 18.7
Src Pt 100-400 Dst Pt 100-400 16.7
Duration 0.000 Packets 1 14.0
Bytes 60-70 Flags .A.... 12.8
**Dst Pt** **0-50** **Flags** **....S.** **12.7**
: Top 5 coupling values obtained from statistical model inferred based on flows labeled as brute force[]{data-label="tab:couplings_bruteforce"}
It is important to note that feature anomalies which characterize different classes of malicious flows in figure \[fig:features\] are strongly coupled in the respective statistical models presented in this section. These results reinforce the conclusion that malicious flows present, in fact, statistically meaningful anomalies in their features. Moreover, Tables \[tab:couplings\_normal\] - \[tab:couplings\_bruteforce\] show that the statistical approach proposed here is capable to capture these anomalies.
In Figure \[fig:energies\_simulated\], flow energies calculated using a statistical model based on normal traffic are shown. The histogram shows energy values of 5,000 randomly sampled flows labeled as *normal* and 5,000 randomly sampled flows labeled as *malicious* from the simulated traffic contained in CIDDS-001. The statistical model used to calculated the energies was inferred based on 4,500 flows randomly sampled from the simulated traffic dataset. The separation between the two flow classes considered, normal and malicious, is clear. Normal flow energy distribution is clearly shifted to the left in relation to malicious flow energy distribution. A chosen energy threshold value used to separate classes is show in red.
![Energy histogram for flow samples in test set I (simulated).[]{data-label="fig:energies_simulated"}](energies_w2_w2.pdf){width="0.95\columnwidth"}
Figure \[fig:energies\_external\] shows flow energies calculated based on the same statistical model, however, coming from the external server traffic scenario. The histogram shows energy values of 5,000 randomly sampled flows labeled as *unknown* and 5,000 randomly sampled flows labeled as *suspicious* from the external sever traffic in CIDDS-001. Traffic labeled as *unknown* is traffic coming from external users with destination port 80 or 443, expected traffic. In this sense, here we consider this traffic to be analogous to normal traffic. Traffic labeled as *suspicious*, on the other hand, is traffic coming from external users aimed at ports other than 80 and 443, unexpected traffic. Hence, this traffic is considered here analogous to malicious traffic. Note that the separation between this two classes, unknown and suspicious, is also clear. In Figure \[fig:energies\_external\] we can see that a portion of unknown traffic is mixed up with suspicious traffic, which might be an indication that this traffic, even with expected destination ports, is actually malicious.
![Energy histogram for flow samples in test set II (external).[]{data-label="fig:energies_external"}](energies_w2_extreal.pdf){width="0.95\columnwidth"}
It is important to note that we can apply the same energy threshold (around 150) to separate the two classes in both the simulated and the external traffic contexts. Because the threshold learned in the simulated traffic scenario can be applied to separate the traffic in the external server scenario, we understand that the statistical model learned is resilient to context change. In the following section, classification results are shown for different classifiers and compared with the results obtained with .
Classification results
----------------------
In this subsection we describe how the test sets were constructed and how the training and testing processes were performed. Next, we present the classification results obtained for and compare them to results obtained for other classifiers.
To evaluate the performance of different algorithms and EFC, we constructed test sets using a subset of CIDDS-001 dataset, as described in the following. The first 100,000 non-redundant flows labeled as *normal* were extracted from week 2 simulated traffic to compose a normal traffic sampling pool. Week 2 was chosen because attacks from different types are more equally distributed on this week than on the other weeks. Similarly, a malicious traffic sampling pool was composed. All flows labeled as *pingScan* and *bruteForce* were extracted from within week 2 simulated traffic. In addition, the first 90,000 non-redundant DoS flows and the first 9,000 non-redundant port scan flows were selected to compose this pool. Dataset undersampling was performed in order to obtain a more homogeneous distribution of the different malicious traffic subclasses. Also, because we want to focus on incoming traffic classification, malicious traffic labeled as *victim* was not selected to compose the sampling pool. Afterwards, 5,000 normal flows and 5,000 malicious flows were sampled from the normal traffic pool and the malicious traffic pool, respectively to compose a set of size 10,000 which was validated using a 10-fold cross validation (see Appendix \[appendix:banks\]). Since flows labeled as *pingScan* and *bruteForce* were rare, all samples with these labels were included in the test set.
Similarly, the first non-redundant 16,900 unknown flows and the first non-redundant 47,700 suspicious flows were extracted from CIDDS-001 week 3 external server traffic. Week 3 was chosen because suspicious and unknown flows are more equally distributed on this week than on the other weeks. A set with 5,000 suspicious and 5,000 unknown flows randomly sampled from the suspicious traffic pool and the unknown traffic pool, respectively, was constructed and validated using a 10-fold cross validation (see Appendix \[appendix:banks\]). Data within the test set was only discretized for EFC, since discretization would probably impair the performance of most algorithms.
To assess the performance of EFC, we deployed eight classifiers found in [@vinayakumar2017evaluating] that are available online at GitHub[^3]. The deployed classifiers are: , , Adaboost, , , , , and polynomial . All classifiers were deployed with their default scikit-learn configurations. Table \[tab:train\_gen\] shows classification results for all test sets considering that all classifiers as well as EFC were trained with the same training set. The only difference regarding the training of the classifiers is that EFC is only trained with normal traffic. The metric used to compare the results is the accuracy, calculated as follows: $$\begin{aligned}
Acc = \frac{TPR + TNR}{2}\end{aligned}$$ where TPR is the true positive rate, percentage of normal traffic classified as normal, and TNR stands for true negative rate, percentage of malicious traffic classified as malicious.
It is clear from Table \[tab:train\_gen\] that most classifiers achieve high accuracy processing test set I (simulated traffic). For instance, , , Adaboost and achieved over 0.980 accuracy. did also obtain good results (Acc = 0.965). , and polynomial , on the other hand, did not obtain very good results. On test set II (external traffic), only EFC achieves accuracy over 0.900, showing that EFC is resilient to context change regarding both the simulated and external environments from dataset CIDDS-001. Such context resilience is a highly desirable trait in a flow-based classifier, since classifiers are trained usually in a synthetic environment to be later applied in real context. Moreover, EFC is the only considered classifier that is independent of malicious flow samples to be trained and could easily learn from real data instead, achieving almost the same performance. In the following, we present our conclusions and future work.
Conclusion
==========
The results presented in the last section show three important and distinguishing characteristics of EFC . Fist, EFC is capable of separating benign from malicious flows based on energy values using a model learned from benign traffic alone. Second, EFC was shown to be resilient to context change within dataset CIDDS-001, while all other classifiers were not. And third, EFC is easy to implement and does not perform costly operation, as shown in Section \[sec:3b\].
Considering the advantages presented, we believe EFC to be a promising algorithm to perform flow-based traffic classification. Nevertheless, despite the promising results achieved, there is still room for further testing and improvement. For instance, in order to obtain a more homogeneous distribution of different attack types, we performed a dataset undersampling, which might have had some effect on the results. Hence, in future work we aim at performing a more comprehensive investigation of EFC applicability using different datasets (CICIDS [@sharafaldin2018toward]) and traffic captured in real networks to thoroughly test it.
{#appendix:classes}
Table \[tab:classes\] shows the classes used for feature discretization. Within CIDDS-001 dataset, TCP Flags is the discrete feature with more possible values (32 possibilities), so the alphabet size $Q$ was set to 32. Each continuous feature values were clustered in a certain number of classes (or bins), up to $Q$ classes. Classes were determined in such a way that the number of values within each class was similar for all classes.
{#appendix:banks}
Table \[tab:class\] shows the composition of two non-redundant sampling pools, which are a subset of CIDDS-001 dataset used to construct the even smaller sets considered to evaluate classification accuracy. The composition of the sets used to assess classification performance is shown in Table \[tab:subclass\]. These sets were validated using a 10-fold cross validation. Dataset undersampling was performed in order to obtain a more homogeneous distribution of the different classes of malicious traffic.
**Class** **Simulated (week 2)** **External (week 3)**
-------------- ------------------------ -----------------------
*normal* 100,000 0
*attack* 100,000 0
*unknown* 0 16,900
*suspicious* 0 47,700
: Structure of sampling pool from CIDDS-001 used to compose 10-fold cross validation sets[]{data-label="tab:class"}
**Label** **Simulated (week2)** **External (week 3)**
-------------- ----------------------- -----------------------
*normal* 5,000 0
*dos* 1,919 0
*portScan* 1,919 0
*pingScan* 152 0
*bruteForce* 910 0
*suspicious* 0 5,000
*unknown* 0 5,000
*all* 10,000 10,000
: Flows labels within 10-fold cross validation sets []{data-label="tab:subclass"}
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors would like to thank Luís Paulo Faina Garcia for helping with dataset analysis. Matt Bishop was supported by the National Science Foundation under Grant Number OAC-1739025. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. João Gondim gratefully acknowledges the support from Project “EAGER: USBRCCR: Collaborative: Securing Networks in the Programmable Data Plane Era” funded by NSF and RNP (Brazilian National Research Network).
\
[Camila F. T. Pontes]{}
is a student at University of Brasilia (UnB), Brasilia, DF, Brazil. She received her M.Sc. degree in Molecular Biology in 2016 from UnB and is currently an undergrad student at the Department of Computer Science (CIC/UnB). Her research interests are Computational and Theoretical Biology, and Network Security.
[João J. C. Gondim]{} was awarded an M.Sc. in Computing Science at Imperial College, University of London, in 1987 and a Ph.D. in Electrical Engineering at UnB (University of Brasilia, 2017). He is an adjunct professor at Department of Computing Science (CIC) at UnB where he is a tenured member of faculty. His research interests are network, information and cyber security.
[Matt Bishop]{} received his Ph.D. in computer science from Purdue University, where he specialized in computer security, in 1984. His main research area is the analysis of vulnerabilities in computer systems. The second edition of his textbook, Computer Security: Art and Science, was published in 2002 by Addison-Wesley Professional. He is currently a co-director of the Computer Security Laboratory at the University of California Davis.
\[[{width="1in" height="1.25in"}]{}\][Marcelo Antonio Marotta]{} is an assistant professor at University of Brasilia, Brasilia, DF, Brazil. He received his Ph.D. degree in Computer Science in 2019 from the Institute of Informatics (INF) of the Federal University of Rio Grande do Sul (UFRGS), Brazil. His research involves Heterogeneous Cloud Radio Access Networks, Internet of Things, Software Defined Radio, Cognitive Radio Networks, and Network Security.
[^1]: C. Pontes, J. Gondim and M. A. Marotta are with the University of Brasilia, Brazil, emails: cftpontes@gmail.com, gondim@unb.br, marcelo.marotta@unb.br;
[^2]: M. Bishop is with the University of California at Davis, Davis, USA, email: mabishop@ucdavis.edu
[^3]: <https://github.com/vinayakumarr/Network-Intrusion-Detection>
|
---
abstract: 'In this paper we study a key exchange protocol similar to the Diffie-Hellman key exchange protocol, using abelian subgroups of the automorphism group of a non-abelian nilpotent group. We also generalize group no.92 of the Hall-Senior table [@halltable] to an arbitrary prime $p$ and show that, for those groups, the group of central automorphisms is commutative. We use these for the key exchange we are studying.'
address: 'Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, NJ 07030, U.S.A.'
author:
- Ayan Mahalanobis
bibliography:
- 'Implementation.bib'
nocite: '[@*]'
title: 'The Diffie-Hellman key exchange protocol and non-abelian nilpotent groups'
---
**MSC**: 94A62, 20D15.\
**Keyword**: Diffie-Hellman key exchange, public-key cryptography, $p$-group, Miller group.
Introduction
============
In this paper we generalize the Diffie-Hellman key exchange protocol from a cyclic group to a finitely presented non-abelian nilpotent group of class $2$. Similar efforts were made in [@braid3; @braid4; @braid1] to use braid groups, a family of finitely presented non-commutative groups [@links; @braid5], in key exchange. We also refer to [@sh1 Section 3] for a formal description of a key exchange protocol similar to ours
[The author expresses his gratitude to the referee for this reference.]{}
. Our efforts are not solely directed to construct an efficient and fast key exchange protocol. We also try to understand the conjecture, *the discrete logarithm problem is equivalent to the Diffie-Hellman problem in a cyclic group*. We develop and study protocols where, at least theoretically, non-abelian groups can be used to share a secret or exchange private keys between two people over an insecure channel. This development is significant because nilpotent or, more specifically, $p$-groups have nice presentations and computation in those groups is fast and easy [@sims Chapter 9]. So our work can be seen as a nice application of the advanced and developed subject of $p$-groups and computation with $p$-groups.
The frequently used public key cryptosystems are slow and use mainly number theoretic complexity. The specific cryptographic primitive that we have in mind is *the discrete logarithm problem*, DLP for short. DLP is general enough to be defined in an arbitrary cyclic group as follows: let $G=\langle g\rangle$ be a cyclic group generated by $g$ and let $g^n=h$, where $n\in\mathbb{N}$. Given $g$ and $h$, DLP is to find $n$ [@stinson Chapter 6]. The security of the discrete logarithm problem depends on the representation of the group. It is trivial in $\mathbb{Z}_n$, but is much harder (no polynomial time algorithm known) in the multiplicative group of a finite field and even harder (no sub-exponential time algorithm known) in the group of elliptic curves which are not supersingular [@blakeCurve]. But with the invention of sub-exponential algorithms for breaking the discrete logarithm problem, like the index calculus and Coppersmith’s algorithm, multiplicative groups of finite fields are no longer that attractive especially the ones of characteristic $2$.
The discrete logarithm problem can be used in many other groups like the group of elliptic curves, in which case a cyclic group or a big enough cyclic component of an abelian group is used. In this article we propose a generalization of DLP or more specifically the Diffie-Hellman key exchange protocol in situations where the group has more than one generator, i.e., in a finitely presented non-abelian group. Let $f$ be an automorphism of a finitely presented group $G$ generated by $\{a_1,a_2,\ldots ,a_n\}$. If one knows the action of $f$ on $a\in G$, i.e., $f(a)$, then it is difficult for him to tell the action of $f$ on any other $b\in G$ i.e., $f(b)$. We describe this in detail later under the name “the general discrete logarithm problem”. In this paper we work with finitely presented groups in terms of generators and relations and do not consider any representation of that group. Though that seems to be a good idea for future research.
Now suppose for a moment that $G=\langle g\rangle $ is a cyclic group and that we are given $g$ and $g^n$ where $\gcd(n,|G|)=1$. DLP is to find $n$. Notice that in this case the map $x\mapsto x^n$ is an automorphism. If we conjecture that finding the automorphism is finding $n$ then one way to see DLP, in terms of group theory, is to find the automorphism from its image on one element. This is the central idea that we want to generalize to non-abelian finitely presented groups, especially to a family of $p$-groups of class $2$. This explains our choice of the name the general discrete logarithm problem.
To work with a finitely presented group and its automorphisms the following properties of the group are needed.
- A consistent and natural representation of the elements in the group.
- Computation in the group should be fast and easy.
- The automorphism group should be known and the automorphisms should have a nice enough presentation so that images can be computed quickly.
We note at this point that for a $p$-group the first two requirements are satisfied [@sims Chapter 9].
Our Contribution in this article
================================
The central idea behind this article is to study a generalization of the discrete logarithm problem (DLP) that we call the general discrete logarithm problem (GDLP). As a cryptographic primitive the concept of GDLP seems to be secure (see Section 4.1).
To use GDLP we use a Diffie-Hellman like key exchange protocol using finitely presented $p$-groups with an abelian central automorphism group. In this case the security depends not only on GDLP but also on GDHP (see Section 4.2) which turns out to be insecure in the specific case we are studying.
Section 8 of this paper contains a brief survey of all the group theoretic results necessary for a reader to understand the later part of this paper. However, a knowledgeable reader might choose to ignore Section 8 altogether and come back to it when required. In Section 10 we survey the existing literature for groups with abelian automorphism group and show that none of them are adequate for the key exchange we are studying.
We found no groups readily available in the literature, hence we had to develop a family of groups $G_n(m,p)$ with abelian central automorphism group (Section 10). This is a significant contribution to the theory of finite groups because $G_n(m,p)$ is a generalization of group no. 92 of the Hall-Senior table. We describe the group of automorphisms for this group and further prove that this group is Miller if and only if $p=2$.
We do not claim that the key exchange protocol is secure. Rather, we show that the key exchange protocol is insecure for the particular family of groups that we picked. Our study raises two important questions which are of interest both mathematically as well as cryptographically.
a
: Are there groups different from $G_n(m,p)$, with an abelian central automorphism group, for which the key exchange protocol is secure?
b
: Does there exist any cryptographic protocol with reductionist security proof, where the security of the protocol depends only on the discrete logarithm problem? If one can find such a protocol using cyclic groups then that could be generalized using GDLP, and since we claim that GDLP is a secure primitive, this will give rise to a secure cryptosystem using non-abelian groups.
Some notations and Definitions
==============================
We now describe some of the definitions and notations that will be used in this paper. The notations used are standard:
- $G$ will denote a finite group. $Z=Z(G)$ denotes the center of the group $G$ and will be denoted by $Z$ if no confusion can arise.
- $G^\prime=[G,G]$ is the commutator subgroup of $G$.
- $\text{Aut}(G)$ and $\text{Aut}_c(G)$ are the group of automorphisms and the group of central automorphisms of $G$, respectively.
- $\Phi(G)$ is the Frattini subgroup of $G$, which is the intersection of all maximal subgroups of $G$.
- We denote the commutator of $a,b$ by $[a,b]$ where $[a,b]=a^{-1}b^{-1}ab$.
- The exponent of a $p$-group $G$, denoted by exp$(G)$, is the largest power of $p$ that is the order of an element in $G$.
The following commutator formulas hold for any element $a$,$b$ and $c$ in any group $G$.
(a)
: $a^b=a[a,b]$
(b)
: $[ab,c]=[a,c]^b[b,c]=[a,c][a,c,b][b,c]$ it follows that in a nilpotent group of class $2$, $[ab,c]=[a,c][b,c]$
(c)
: $[a,bc]=[a,c][a,b]^c=[a,c][a,b][a,b,c]$ it follows that in a nilpotent group of class $2$, $[a,bc]=[a,b][a,c]$
(d)
: $[a,b]^{-1}=[b,a]$
The proof of these formulas follow from direct computation or can be found in [@khukhro].
A group $G$ is called a Miller group if it has an abelian automorphism group, in other words, if $\text{Aut}(G)$ is commutative then the group $G$ is Miller.
Let $G$ be a group, then $\phi\in\text{Aut}(G)$ is called a central automorphism if $g^{-1}\phi(g)\in Z(G)$ for all $g\in G$. Alternately, one might say that $\phi$ is a central automorphism if $\phi(g)=gz_{\phi,g}$ where $z_{\phi,g}\in Z(G)$ depends on $g$ and $\phi$. If $\phi$ is clear from the context then we can simplify the notation as $\phi(g)=gz_g$.
Apart from inner automorphisms, central automorphisms are second best in terms of nice description. They are very attractive for cryptographic purposes, since it is easy to describe the automorphisms and compute the image of an arbitrary element.
The centralizer of the group of inner automorphisms is the group of central automorphisms. Moreover a central automorphism fixes the commutator elementwise.
This theorem first appears in [@fournelle] which refers to [@hall] and [@zeanhaus].
Let $G$ be a group, a finite series of subgroups in $G$ $$G=G_0\trianglerighteq G_1\trianglerighteq G_2\trianglerighteq
G_3\trianglerighteq\ldots\trianglerighteq G_n=1$$ is a polycyclic series if $G_i/G_{i+1}$ is cyclic and $G_{i+1}$ is a normal subgroup of $G_i$. Any group with polycyclic series is a polycyclic group.
It is easy to prove that finitely generated nilpotent groups are polycyclic, hence any finitely generated $p$-group is polycyclic. Let $a_i$ be an element in $G_i$ whose image generates $G_i/G_{i+1}$. Then the sequence $\{a_1,a_2,\ldots ,a_n\}$ is called a polycyclic generating set. It is easy to see that $g\in G$ can be written as $g=a_1^{\alpha_1}a_2^{\alpha_2}\ldots
a_n^{\alpha_n}$, where $\alpha_i$ are integers. If $g=a_1^{\alpha_1}a_2^{\alpha_2}\ldots a_n^{\alpha_n}$ where $0\leq\alpha_i<
m_i$, $m_i=\left|G_i:G_{i+1}\right|$ then the expression is a collected word. Each element $g\in G$ can be expressed by a unique collected word. Computation with these collected words is easy and implementable in computer, for more information on this topic see [@sims Section 9.4] and also [@GAP4 polycyclic package].
Key Exchange
============
We want to follow the Diffie-Hellman Key exchange protocol using a commutative subgroup of the automorphism group of a finitely presented group $G$. The security of the Diffie-Hellman key exchange protocol in a cyclic group rests on the following three factors:
DLP
: The discrete logarithm problem.
DHP
: The Diffie-Hellman problem.
DDH
: The decision Diffie-Hellman problem [@blake1; @boneh; @galb1; @igor; @gonzalez].
We have already described the discrete logarithm problem. The Diffie-Hellman problem is the following: let $G=\langle g\rangle$ be a cyclic group of order $n$. One knows $g$, $g^a$ and $g^b$, and the problem is to compute $g^{ab}$. It is not known if DLP is equivalent to DHP. The decision Diffie-Hellman problem is more subtle. Suppose that DHP is a hard problem, so it is impossible to compute $g^{ab}$ from $g^a$, $g^b$ and $g$. But what happens if someone can compute or predict $80\%$ of the binary bits of $g^{ab}$ from $g^a$, $g^b$ and $g$, then the adversary will have $80\%$ of the shared secret or the private key; that is most of the private key. This is clearly unacceptable. It is often hard to formalize DDH in exact mathematical terms ([@boneh Section 3]); the best formalism offered is a randomness criterion for the bits of the key. In DDH we ask the question, given the triple $g^a, g^b$ and $g^c$ is $c=ab\mod n$? But there is no known link between DDH and any mathematically hard problem for the Diffie-Hellman key exchange protocol in cyclic groups.
Clearly, solving the discrete logarithm problem solves the Diffie-Hellman problem and solving the Diffie-Hellman problem solves the decision Diffie-Hellman problem.
As is usual, we denote by Alice and Bob, two people trying to set up a private key over an insecure channel to communicate securely and Oscar an eavesdropping adversary. In this paper the shared secret or the private key is an element of a finitely presented group $G$.
General Discrete Logarithm Problem
----------------------------------
Let $G=\langle
a_1,a_2,\ldots,a_n\rangle$ and $f:G\rightarrow G$ be a non-identity automorphism. Suppose one knows $f(a)$ and $a\in G$ then GDLP is to find $f(b)$ for any $b$ in $G$. Assuming the word problem is easy or presentation of the group is by means of generators, GDLP is equivalent to finding $f(a_i)$ for all $i$ which in terms gives us a complete knowledge of the automorphism. So in other words the cryptographic primitive GDLP is equivalent to, “*finding the automorphism $f$ from the action of $f$ on only one element*”.
General Diffie-Hellman Problem
------------------------------
Let $\phi,\psi:G\rightarrow G$ be arbitrary automorphisms such that $\phi\psi=\psi\phi$, and assume one knows $a$, $\phi(a)$ and $\psi(a)$. Then GDHP is to find $\phi(\psi(a))$. Notice that GDHP is a restricted form of GDLP, because in case of GDHP one has to compute $\phi(\psi(a))$ for some fixed $a$, not $\phi(b)$ for an arbitrary $b$ in $G$. There is an interesting GDHP attack due to Vladimir Shpilrain. To mount this attack one need not find $\phi$ but finds another automorphism $\phi^\prime$ such that $\phi^\prime\psi=\psi\phi^\prime$ and $\phi^\prime(a)=\phi(a)$. Since $\phi(\psi(a))=\psi(\phi^\prime(a))=\phi^\prime(\psi(a))$, the knowledge of the $\phi^\prime$ breaks the system. We will refer to this attack as the Shpilrain’s attack.
We now describe two key exchange protocols and do some cryptanalysis. We denote by $G$ a finitely presented group and $S$ an abelian subgroup of $\text{Aut}(G)$.
Key Exchange Protocol I {#keyexchange1}
=======================
Alice and Bob want to set up a private key. They select a group $G$ and an element $a\in G\setminus Z(G)$ over an insecure channel. Then Alice picks a random automorphism $\phi_A\in S$ and sends Bob $\phi_A(a)$. Bob similarly picks a random automorphism $\phi_B\in S$ and sends Alice $\phi_B(a)$. Both of them can now compute $\phi_A(\phi_B(a))=\phi_B(\phi_A(a))$ which is their private key for a symmetric transmission.
Step 1
: Alice and Bob selects the group $G$ and an element $a\in
G\setminus Z(G)$ in public. Notice that $G$ and $a$ are public information.
Step 2
: Alice and Bob picks, at random, two automorphisms $\phi_A$ and $\phi_B$ from $S$ respectively. Notice that $\phi_A$ and $\phi_B$ are private information.
Step 3
: Alice and Bob compute $\phi_A(a)$ and $\phi_B(a)$ respectively and exchanges them. Notice that $\phi_A(a)$ and $\phi_B(a)$ are public information.
Step 4
: Both of them compute $\phi_A\left(\phi_B(a)\right)=\phi_B\left(\phi_A(a)\right)$ from their private information; which is their private key.
Comments on Key Exchange Protocol I
-----------------------------------
Though initially it might seem that we do not have enough information to know the automorphisms $\phi_A$ and $\phi_B$, it turns out that if we are using automorphisms which fix conjugacy classes, like inner automorphisms, then the security of the above scheme actually rests on the conjugacy problem.
Let $\phi_A(a)=x^{-1}ax$ and $\phi_B(a)=y^{-1}ay$ for some $x$ and $y$. Then $\phi_A(\phi_B(a))\\=(yx)^{-1}a(yx)$. Since $a$, $\phi_A(a)$ and $\phi_B(a)$ are known, if the conjugacy problem is easy in the group then anyone can find $x$ and $y$ and break the system.
In the above scheme Oscar knows $G$ and $a$. If the automorphisms are central automorphisms, then he also sees $\phi_A(a)=az_{\phi_A ,a}$ and $\phi_B(a)=az_{\phi_B ,a}$. Oscar can compute $z_{\phi_A, a}$ and $z_{\phi_B ,a}$. Now if $G$ is a special $p$-group ($G^\prime=Z(G)=\Phi(G)$) then $Z(G)$ is fixed elementwise by both $\phi_A$ and $\phi_B$. Then $$\label{center attack}
\phi_A(\phi_B(a))=\phi_A(az_{\phi_B ,a})=az_{\phi_A ,a}z_{\phi_B ,a}.$$ Oscar knows $a$ and can compute $z_{\phi_A ,a}$ and $z_{\phi_B ,a}$ and can find the private key $\phi_A(\phi_B(a))$. In the literature all examples of Miller $p$-group with odd prime $p$ are special and the above key exchange is fatally flawed for those groups.
Key Exchange Protocol II {#keyexchange2}
========================
In this case Alice and Bob want to set up a private key and they set up a group $G$ over an insecure channel. Alice chooses a random non-central element $g$ and a random automorphism $\phi_A\in S$ and sends Bob $\phi_A(g)$. Bob picks another automorphism $\phi_B\in S$ and computes $\phi_B(\phi_A(g))$ and sends it back to Alice. Alice, knowing $\phi_A$, computes $\phi_A^{-1}$ which gives her $\phi_B(g)$ and picks another random automorphism $\phi_H\in S$ and computes $\phi_H(\phi_B(g))$ and sends it back to Bob. Bob, knowing $\phi_B$ computes $\phi^{-1}_B$ which gives him $\phi_H(g)$ which is their private key. Notice that Alice never reveals $g$ in public.
Step 1
: Alice and Bob set up the group $G$. Notice that $G$ is public information.
Step 2
: Alice picks $g\in
G\setminus Z(G)$ and a random $\phi_A\in
S$. Then she computes $\phi_A(g)$ and sends that to Bob. Notice that $g$ and $\phi_A$ are private but $\phi_A(g)$ is public.
Step 3
: Bob picks $\phi_B\in S$ at random and computes $\phi_B\left(\phi_A(g)\right)$ and sends that back to Alice. Notice that $\phi_B$ is private but $\phi_B\left(\phi_A(g)\right)$ is public.
Step 4
: Alice computes $\phi_A^{-1}$ and then computing $\phi_A^{-1}\left(\phi_B\left(\phi_A(g)\right)\right)$ she gets $\phi_B(g)$.
Step 5
: Alice now picks another random automorphism $\phi_H\in
S$ and computes $\phi_H\left(\phi_B(g)\right)$ and $\phi_H(g)$. She then sends $\phi_H\left(\phi_B(g)\right)$ to Bob but keeps $\phi_H(g)$ private.
Step 6
: Similar to Step 4, Bob computes $\phi_H(g)$. Now both Alice and Bob knows $\phi_H(g)$ and it is their common key.
Comments on Key Exchange Protocol II
------------------------------------
Notice that for central automorphisms, $\phi_A$ and $\phi_B$, $\phi_A(g)=gz_{\phi_A ,g}$; since $g$ is not known Oscar doesn’t know $z_{\phi_A ,g}$ but if $G$ is special ($Z(G)=G^\prime=\Phi(G)$) then $\phi_B(gz_{\phi_A ,g})\\=gz_{\phi_B ,g}z_{\phi_A ,g}$ from which $z_{\phi_B ,g}$ can be computed. Now $\phi_H(\phi_B(g))=gz_{\phi_B,g}z_{\phi_H ,g}$ is a public information; so using $z_{\phi_B ,g}$ one can compute $gz_{\phi_H ,g}$, which is $\phi_H(g)$ and the scheme is broken. As one clearly sees, this attack is not possible if the group is not special.
The reader might have noticed at this point that all the attacks are GDHP. So certainly in some groups GDHP is easy, even though GDLP is hard.
As we know, any automorphism in $G$ can be seen as a restriction of an inner automorphism in $\text{Hol}(G)$ (see [@kurosh; @maria] for further details on the holomorph of a group). Solving the conjugacy problem in $\text{Hol}(G)$ will break the key exchange protocols for any automorphism. On the other hand, operation in $\text{Hol}(G)$ is twisted so it is possible that the conjugacy problem in $\text{Hol}(G)$ is difficult even though it is easy in $G$. Since any cyclic group is a Miller group, success of the holomorph attack would prove insecurity in DLP. Therefore we believe that the holomorph attack will not be successful in many cases. Though more work needs to be done on this.
Key Exchange using Braid Groups
===============================
In [@braid1] a similar key exchange protocol was defined, in this section we mention some similarities of their approach to ours. We also mention how our system generalizes their system which uses braid groups. See also [@polybraid].
We define braid group as a finitely presented group, though there are fancy pictorial ways to look at braids and multiplication of braids. An interested reader can look in [@links; @braid5]. The braid group $B_n$ with $n$-strands is defined as: $$\begin{aligned}
B_n=\left\langle
\sigma_1,\ldots,\sigma_{n-1}:\sigma_i\sigma_j\sigma_i=\sigma_j
\sigma_i\sigma_j\,\text{if}\,|i-j|=1,
\sigma_i\sigma_j=\sigma_j\sigma_i \,\text{if}\, |i-j|\geq 2
\right\rangle \end{aligned}$$ In [@braid1], the authors found two subgroups $A$ and $B$ of the group of inner automorphisms of $B_n$, $\text{Inn}(B_n)$, such that, if $\phi\in A$ and $\psi\in B$, then $\phi(\psi(g))=\psi(\phi(g))$ for $g\in B_n$. Then the key exchange proceeds similar to the Key Exchange Protocol I above; with the restriction that Alice chooses automorphisms from A and Bob chooses automorphisms from B. There is also a different approach to key exchange using braid groups as in [@braid3; @braid4].
In the same spirit as [@braid1] we can develop a key exchange protocol similar to the key exchange protocol I, where we take two subgroups $A$ and $B$ in $\text{Aut}(G)$ such that for $\phi\in A$ and $\psi\in B$, $\phi(\psi(g))=\psi(\phi(g))$ for all $g\in G$. The use of inner automorphisms is only possible when the conjugacy or the generalized conjugacy problem (conjugator search problem) is known to be hard.
There are significant differences in our approach to that of the approach in [@braid1]. In [@braid1], the authors choose a group and then try to use that group in cryptography. On the other hand, we take the fundamental concept as the discrete logarithm problem, generalize it using automorphisms of a non-abelian group and then look for groups favorable to us. The fact that the central idea in braid group key exchange turns out to be similar to ours is encouraging.
It is intuitively clear at this point that we should start looking for groups with abelian automorphism group, i.e., Miller groups.
Some useful facts from group theory
===================================
The term Miller Group is not that common in the literature. It was introduced by Earnley in [@earnley]. Miller was the first to study groups with abelian automorphism group in [@miller]. Cyclic groups are good examples of Miller groups. G.A. Miller also proved that no non-cyclic abelian group is Miller.
Charles Hopkins began a list of necessary conditions for a Miller group in 1927 [@hopkins]. He complained that very little is known about those groups. The same is true today. Except for some sporadic examples of groups with abelian automorphism groups, there is no sufficient condition known for a group to be Miller.
We now state some known facts about Miller groups which are available in the literature and which we shall need later. For proof of these theorems which we present in a rapid fire fashion, the reader can look in any standard text books, like [@khukhro; @rotman], or the references there.
If $G$ is a non-abelian Miller group, then $G$ is nilpotent and of class $2$.
It follows from the fact that the group of inner automorphisms commute and $G/Z(G)\cong\text{Inn}(G)$.
Since a nilpotent group is a direct product of its Sylow $p$-subgroups $S_p$, and $\text{Aut}(A\times B)=\text{Aut}(A)\times\text{Aut}(B)$ whenever $A$ and $B$ are of relatively prime order, it is enough to study Miller $p$-groups for prime $p$.
If $G$ is a $p$-group of class $2$, then $\text{exp}(G^\prime)=\text{exp}(G/Z(G))$.
In a $p$-group of class $2$, $(xy)^n=x^ny^n[y,x]^{\frac{n(n-1)}{2}}$. Furthermore if $\text{exp}(G^\prime)=n$ is odd, then $(xy)^n=x^ny^n$.
By definition, in a Miller group all automorphisms commute. Since central automorphisms are the centralizer of the group of inner automorphisms, we have proved the following theorem.
In a Miller group $G$, all automorphisms are central.
It follows that to show a group is not Miller, all we have to do is to produce a non-central automorphism.
If the commutator and the center coincide then every pair of central automorphisms commute.
Let $G$ be a group such that $G^\prime=Z(G)$. Then let $\phi$ and $\psi$ be central automorphisms given by $\phi(x)=xz_{\phi ,x}$ and $\psi(x)=xz_{\psi
,x}$ where $z_{\phi ,x},z_{\psi ,x}\in G^\prime$. Then $$\psi(\phi(x))=\psi(xz_{\phi ,x})=\psi(x)z_{\phi ,x}=xz_{\psi ,x}z_{\phi
,x}=xz_{\phi ,x}z_{\psi ,x}=\phi(\psi(x)).$$
A group $G$ is said to be a purely non-abelian group (PN group for short) if whenever $G=A\times B$ where $A$ and $B$ are subgroups of $G$ with $A$ abelian, then $A=1$. Equivalently $G$ has no non-trivial abelian direct factor.
Let $\sigma:G\rightarrow G$ be a central automorphism. Then we define a map $f_{\sigma}:G\rightarrow Z(G)$ as follows: $f_{\sigma}(g)=g^{-1}\sigma(g)$. Clearly this map defines a homomorphism. The map $\sigma\mapsto f_{\sigma}$ is clearly a one-one map. Conversely, if $f\in\text{Hom}(G,Z(G))$ then we define a map $\sigma_f(g)=gf(g)$, $x\in G$. Clearly $\sigma_f$ is an endomorphism. It is easy to see that $$\text{Ker}(\sigma_f)=\{x\in G:\;f(x)=x^{-1}\}.$$ Hence it follows that $\sigma_f$ is an automorphism if and only if $f(x)\neq
x^{-1}$ for all $x\in G$ with $x\neq 1$.
\[correspondence\] In a purely non-abelian group $G$, the correspondence $\sigma\rightarrow
f_{\sigma}$ is a one-one map of $\text{Aut}_c(G)$ onto $\text{Hom}(G,Z(G))$
See [@adney].
For any $f\in\text{Hom}(G,Z(G))$ there is a map $f^\prime\in\text{Hom}(G/G^\prime,Z(G))$ since $f(G^\prime)=1$. Furthermore, corresponding to $f^\prime\in\text{Hom}(G/G^\prime,Z(G))$ there is a map $f:G\rightarrow Z(G)$ explained in the following diagram $$\begin{CD}
G @>\eta>> G/G^\prime @>f^\prime>> Z(G)
\end{CD}$$ where $\eta$ is the natural epimorphism.
Let $G$ be a $p$-group of class $2$, such that exp$(Z(G))=a$, exp$(G^\prime)=b$ and exp$(G/G^\prime)=c$ and let $d=\min(a,c)$. Now from the fundamental theorem of abelian groups, let $$G/G^\prime =A_1\oplus A_2\oplus\ldots A_r\;\; \text{where}\;\; A_i=\langle
a_i\rangle$$ $$Z(G)=B_1\oplus B_2\oplus\ldots B_s\;\; \text{where}\;\; B_i=\langle
b_i\rangle$$ $r,s\in\mathbb{N}$ be the direct decomposition of $G/G^\prime$ and $Z(G)$. If the cyclic component $A_k=\langle a_k\rangle$ has exponent greater or equal to the exponent of $B_j=\langle b_j\rangle$, then one can define a homomorphisms $f:G/G^\prime\rightarrow Z(G)$ as follows $$f(a_i)=\left\{
\begin{array}{lll}
b_j\;\;\text{where}\;\;i=k\\
1 \;\;\;\text{where}\;\;i\neq k
\end{array}
\right.$$ From this discussion it is clear that for $f\in\text{Hom}(G,Z(G))$, $f(G)$ generates the subgroup $$\mathcal{R}=\{z\in Z(G):\; |z|\leq p^d,\; d=\min(a,c)\}.$$
In any abelian $p$-group $A$ written additively, there is a descending sequence of subgroups $$A\supset pA\supset p^2A\supset \ldots \supset p^nA\supset
p^{n+1}A\supset\ldots$$ Then $x\in A$ is of height $n$ if $x\in p^nA$ but not in $p^{n+1}A$. In other words the elements of height $n$ are those that drop out of the chain in the $(n+1)^\text{th}$ inclusion.
For further information on height see [@kaplansky].
Since for a class $2$ group we have $$\text{exp}(G/G^\prime)\geq\text{exp}(G/Z(G))=\text{exp}(G^\prime)$$ it follows that $c\geq b$. Hence if $d=\min(a,c)$ then either $d=b$ or $d>b$.
Let height$(xG^\prime)\geq b$, then $xG^\prime=y^{p^b}G^\prime$ for some $y\in G$. Then for any $F\in\text{Hom}(G,G^\prime)$, $F(yG^\prime)^{p^b}=1$ implying $xG^\prime\in F^{-1}(1)$. Conversely, let height$(xG^\prime)<b$. Then from the previous discussion it is clear that there is a $F^\prime\in\text{Hom}(G/G^\prime,G^\prime)$ such that $xG^\prime$ is not in the kernel, consequently there is a $F\in\text{Hom}(G,G^\prime)$ such that $x\notin\text{ker}(F)$. Combining these two facts we see that: $$\mathcal{K}=\bigcap\limits_{F\in\text{Hom}(G,G^\prime)}F^{-1}(1)=\left\{x\in
G:\;\text{height}(xG^\prime)\geq b\right\}$$
$\mathcal{K}\subseteq\mathcal{R}$
In a class $2$ group, if $x\in\mathcal{K}$ then $xG^\prime=y^{p^b}G^\prime$ for some $y\in G$ and exp$(G/Z)=b$ and $G^\prime\subseteq Z(G)$, hence $x\in
Z(G)$.
Let $x\in\mathcal{K}$, then height$(xG^\prime)\geq b$, hence there is a $y\in G$ such that $y^{p^b}G^\prime=xG^\prime$ i.e., $x=y^{p^b}z$ where $z\in G^\prime$ and $y^{p^c}\in G^\prime$ and $c\geq b$. We have $$x^{p^c}=(y^{p^b})^{p^c}z^{p^c}=(y^{p^c})^{p^b}=1$$ Hence $|x|\leq\min(p^a,p^c)$ which implies that $x\in\mathcal{R}$.
For a PN group $G$ of class $2$, if $\text{Aut}_c(G)$ is abelian then $\mathcal{R}\subseteq\mathcal{K}$.
In a PN group, using Theorem \[correspondence\] and the notation there, two central automorphisms $\sigma$ and $\tau$ commute if and only if $f_{\sigma},f_{\tau}\in\text{Hom}(G,Z(G))$ commute. Then for any $f\in\text{Hom}(G,Z(G))$ and $F\in\text{Hom}(G,G^\prime)$ we have that $f\circ
F=F\circ f=1$. Since $f(G^\prime)=1$, clearly $F\circ f(G)=1$ proving that $\mathcal{R}\subseteq\mathcal{K}$.
Combining the above two propositions, we just proved that in a PN group $G$ of class $2$, if $\text{Aut}_c(G)$ is abelian then $\mathcal{R}=\mathcal{K}$. As discussed earlier there are two cases $d=b$ and $d>b$. Adney and Yen proves that:
If $G$ is a non-abelian $p$ group of class $2$, and $\text{Aut}_c(G)$ is abelian with $d>b$, then $\mathcal{R}/G^\prime$ is cyclic.
See [@adney Theorem 3].
\[adney\] Let $G$ be a purely non-abelian group of class $2$, $p$ odd, let $G/G^\prime=\prod\limits_{i=1}^n\{x_iG^\prime\}$. Then the group $\text{Aut}_c(G)$ is abelian if and only if
- $\mathcal{R}=\mathcal{K}$
- either $d=b$ or $d>b$ and $\mathcal{R}/G^\prime=\{x_1^{p^b}G^\prime\}$
See [@adney Theorem 4].
From the proof of Proposition 8.5 it follows that in a group $G$ with $Z(G)\leq G^\prime$, the central automorphisms commute.
\[sanders\] The group of central automorphisms of a $p$-group $G$, where $p$ is odd, is a $p$-group if and only if $G$ has no non-trivial abelian direct factor.
See [@sanders Theorem B] and its corollary.
At this point we concentrate on building a cryptosystem. We note that Miller groups in particular have no advantage over groups with abelian central automorphism group. It is hard to construct Miller groups and there is no known Miller group for an odd prime, which is not special. So we now turn towards a group $G$ such that $\text{Aut}(G)$ is not abelian but $\text{Aut}_c(G)$ is abelian. We propose to use $\text{Aut}_c(G)$ rather than $\text{Aut}(G)$ in the key exchange protocols described earlier.
Signature Scheme based on conjugacy problem
===========================================
Assume that we are working with a group $G$ with commuting inner automorphisms.
Alice publishes $\alpha$ and $\beta$ where $\beta=a^{-1}\alpha a$ and keeps $a$ a secret. To sign a text $x\in G$ she picks an arbitrary element $k\in G$ and computes $\gamma=k\alpha k^{-1}$ and then computes $\delta$ such that $x=(\delta k)(a\gamma)^{-1}$. Now notice that $$\begin{aligned}
x\alpha x^{-1}=&(\delta k)(a\gamma)^{-1}\alpha((\delta
k)(a\gamma)^{-1})^{-1}&\\
=&(\delta k)\gamma^{-1}a^{-1}\alpha a\gamma k^{-1}\delta^{-1}&\\
=&\delta\gamma^{-1}a^{-1}k\alpha k^{-1}a\gamma\delta^{-1}&\text{Inner
automorphisms commute}\\
=&\delta\gamma^{-1}a^{-1}\gamma a\gamma\delta^{-1}\\
=&\delta a^{-1}\gamma a\delta^{-1}&\\
=&\delta (k\beta k^{-1})\delta^{-1}& \gamma=k\alpha k^{-1}\Rightarrow
a^{-1}\gamma a=k\beta k^{-1}\\\end{aligned}$$ So to sign a message $x\in G$ Alice computes $\delta$ as mentioned and sends $x,(k\delta)$. To verify the message one computes $L=x\alpha x^{-1}$ and $R=\delta k\beta(\delta k)^{-1}$. If $L=R$ then the message is authentic otherwise not.
There is a similar signature scheme in [@braid2], where they exploit the gap between the computational version (conjugacy problem) and the decision version of the conjugacy problem (conjugator search problem) in braid groups. We followed the El-Gamal signature scheme closely [@stinson Chapter 7].
Comments on the above Signature Scheme
--------------------------------------
If one can solve conjugacy problem in the group then from the public information $\alpha$ and $\beta$ he can find out $a$ and our scheme is broken. Conjugacy problem is known to be hard in some groups and hence it seems to be a reasonable assumption at this moment. There is another worry: if Alice sends $k$ and $\delta$ separately then one can find $a$ from the equation $x=(\delta
k)(a\gamma)^{-1}$, since $\gamma$ is computable. However, this is circumvented easily by sending the product $\delta k$ not $\delta$ and $k$ individually and keeping $k$ random.
An interesting family of $p$-groups
===================================
It is well known that cyclic groups have abelian automorphism groups. The first person to give an example of a non-abelian group with an abelian automorphism groups is G.A. Miller in [@miller] which was generalized by Struik in [@struik]. There are three non-abelian groups with abelian automorphism group in the Hall-Senior table [@halltable], they are nos. 91, 92 and 99. Miller’s example is no. 99. In [@jamali], Jamali generalized nos. 91 and 92. His generalization of no. 91 is in one direction, it increases the exponent of the group.
Jamali in the same paper generalizes group no. 92 in two directions, the size of the exponent and the number of generators. His generalization was restrictive in that it works only for the prime $2$. There are other examples of families of Miller $p$-groups in the literature, the most notable one is the family of $p$-groups, for an arbitrary prime $p$, given by Jonah and Konisver in [@jonah]. This was generalized to an arbitrary number of generators by Earnley in [@earnley]. There are other examples by Martha Morigi in [@morigi] and Heineken and Liebeck in [@hl]. All these examples of Miller groups given in [@earnley; @hl; @jonah; @morigi] are special groups, i.e., the commutator and the center are the same. For special groups the key exchange protocols do not work as noted earlier. So there is no Miller $p$-group, readily available in the literature, for arbitrary prime $p$ which can be used right away in construction of the protocol. The only other source are groups nos. 91, 92 and 99 in the Hall Senior table [@halltable] and their generalizations, notice that these groups are not special but are $2$-groups. Of the three generalizations, the generalization of no. 92 best fits our criterion because it is generalized in two directions, *viz*. number of generators and exponent of the center and moreover it is not special; $Z(G)=A\times G^\prime$ where $A$ is a cyclic group. So once we generalize it for arbitrary primes, it has “three degrees of freedom”, the number of generators, exponent of center and the prime; which makes it attractive for cryptographic purposes.
In the rest of the section we use Jamali’s definition in [@jamali] to define a family of $p$-groups for arbitrary prime. So this family is a generalization of Jamali’s example and assuming transitivity of generalizations, ultimately a generalization of group no. 92 in the Hall-Senior table [@halltable]. We study automorphisms of this group and show that the group is Miller if and only if $p=2$, but this family of groups always have an abelian central automorphism group which is fairly large. We then attempt to build a key exchange protocol as described earlier using the central automorphisms. We start with the definition of the group $G_n(m,p)$.
Let $G_n(m,p)$ be a group generated by $n+1$ elements
$\{a_0,a_1,a_2,\ldots ,a_n\}$ where $p$ is a prime number and $m\geq 2$ and $n\geq 3$ are integers. The group is defined by the following relations: $$a_1^p=1,\;\;\;a_2^{p^m}=1,\;\;\;a_i^{p^2}=1\;\;\; \text{for}\;\;\;
3\leq i\leq n,\;\;\;a_{n-1}^p=a_0^p.$$ $$[a_1,a_0]=1,\;\;\;[a_n,a_0]=a_1,\;\;\;[a_{i-1},a_0]=a_i^p
\;\;\text{for}\;\;\;3\leq i\leq n.$$ $$[a_i,a_j]=1 \;\;\;\text{for}\;\;\;1\leq i<j\leq n.$$
We state some facts about the group $G_n(m,p)$ whose proof is by direct computation (see [@ayan1 Section 2.9]).
a
: $G_n(m,p)^\prime$ the derived subgroup of $G_n(m,p)$ is an elementary abelian group $\langle a_1,a_3^p,\ldots a_n^p\rangle\simeq\mathbb{Z}_p^{n-1}$.
b
: $Z(G_n(m,p))=\langle a_2^p\rangle \times G^\prime$.
c
: $G_n(m,p)$ is a $p$-group of class $2$.
d
: $G_n(m,p)$ is a PN group.
$G_n(m,p)$ is a polycyclic group and every element of $g\in G_n(m,p)$ can be uniquely expressed in the form $g=a_0^{\alpha_0}a_1^{\alpha_1}a_2^{\alpha_2}a_3^{\alpha_3}\ldots
a_n^{\alpha_n}$, where\
$0\leq\alpha_i <p$ for $i=0,1$; $0\leq\alpha_2 <p^m$, $0\leq\alpha_i <p^2$ for $i=3,4,\ldots ,n$.
Define $G_0=G_n(m,p)=\langle a_0,a_1,a_2,\ldots ,a_n\rangle$, $G_1=\langle a_1,a_2,\ldots a_n\rangle$ and similarly $G_k=\langle
a_k,a_{k+1},\ldots ,a_n\rangle$ for $k\leq n$. Since $G_1$ is a finitely generated abelian group, it is a polycyclic group [@sims Proposition 3.2]. It is fairly straightforward to show that $$G_1\triangleright
G_2\triangleright \ldots\triangleright G_n\triangleright\langle 1\rangle$$ is a polycyclic series and $\{a_1,\ldots ,a_n\}$ a polycyclic generating sequence of $G_1$.
It is easy to see from the relations of the group that $G_1$ is normal in $G_0$ and $G_0/G_1$ is cyclic. It is also straightforward to show that $\langle
a_iG_{i+1}\rangle=G_i/G_{i+1}$ and $|a_iG_{i+1}|=|a_i|$ and hence any element of the group has a unique representation of the above form. We would call an element represented in the above form a *collected word*. See also [@sims Chapter 9, Proposition 4.1].
**Computation with $G_n(m,p)$**: Our group $G_n(m,p)$ is of class 2, i.e., commutators of weight 3 are identity, computations become real nice and easy. Let us demonstrate the product of two collected words $g=a_0^{\alpha_0}a_1^{\alpha_1}a_2^{\alpha_2}a_3^{\alpha_3}a_4^{\alpha_4}$ and $h=a_0^{\beta_0}a_1^{\beta_1}a_2^{\beta_2}a_3^{\beta_3}a_4^{\beta_4}$. To compute $gh$ we use concatenation and form the word $a_0^{\alpha_0}a_1^{\alpha_1}a_2^{\alpha_2}a_3^{\alpha_3}a_4^{\alpha_4}a_0^{\beta_0}a_1^{\beta_1}a_2^{\beta_2}a_3^{\beta_3}a_4^{\beta_4}$ and note that $a_i$’s commute except for $a_0$ hence one tries to move $a_0$ towards the left using the identity $$a_ia_0=a_0a_i[a_i,a_0]=\left\{
\begin{array}{ccc}
a_0a_ia_{i+1}^p &\text{for}& 1\leq i<n\\
a_0a_ia_1 &\text{for}& i=n.
\end{array}
\right.$$ Further note, since commutators are in the center of the group, $a_{i+1}^p$ or $a_1$ can be moved anywhere. Once $a_0$ is moved to the extreme left the word formed is the collected word of $gh$. This process is often referred to in the literature as *collection*. Computing the inverse of an element can be similarly done.
We now prove that the group of central automorphisms of the group $G_n(m,p)$ for an arbitrary prime $p$ is abelian. For sake of simplicity we denote $G_n(m,p)$ by $G$ for the rest of the article, and use notations from Theorem \[adney\].
In $G$, $\mathcal{R}=Z(G)=\mathcal{K}$.
Using the notation from Theorem \[adney\], we see that in $G$, $a=m-1$, $b=1$ and $c=m$ hence $d=m-1$. Clearly, $\mathcal{R}=Z(G)$ hence $\mathcal{K}\subseteq Z(G)$.
Let $x\in Z(G)$, if $x\in G^\prime$ then height$(xG^\prime)=\infty$ and we are done. If not, then $x=z_1z_2$ where $z_1\in\langle a_2^p\rangle$ and $z_2\in G^\prime$. Then $xG^\prime=z_1G^\prime$ and hence height$(xG^\prime)\geq 1$.
It is easy to see that $\mathcal{R}/G^\prime=Z(G)/G^\prime=\langle
a_2^pG^\prime\rangle$ and hence from Theorem \[adney\] we prove the following theorem:
$\text{Aut}_c(G)$ is abelian.
Automorphisms of $G_n(m,p)$
---------------------------
In this section we describe the automorphisms of groups of this kind. The discussion is, in more than one way, an adaptation of the work of Jamali [@jamali] and generalizes his main theorem.
Let $x=a_0^{\beta_0} a_1^{\beta_1}a_2^{\beta_2}\ldots a_n^{\beta_n}$, where $\beta_i$, $i=0,1,2\ldots ,n$ are integers be an element of $G$. If $p=2$ then ${\beta_0}$ is $1$ and
- $x^2=a_1^{\beta_n} a_2^{2\beta_2}a_3^{\gamma_3}\ldots
a_{n-2}^{\gamma_{n-2}}a_{n-1}^{\gamma_{n-1}+2}a_n^{\gamma_n}$ for $p=2$. Where $\gamma_i=2(\beta_{i-1}+\beta_i)$.
- $x^p=a_2^{p\beta_2}a_3^{p\beta_3}\ldots
a_{n-2}^{p\beta_{n-2}}a_{n-1}^{p\beta_{n-1}+p{\beta_0}}a_n^{p\beta_n}$ for $p\neq 2$.
For the case $p=2$ we just collect terms and use the relation $a_{n-1}^2=a_0^2$.
For $p\neq 2$ using Proposition 8.3 we have $$\begin{aligned}
\lefteqn{x^p=(a_0^{\beta_0} a_1^{\beta_1}a_2^{\beta_2}\ldots
a_{n-1}^{\beta_{n-1}}a_n^{\beta_n})^p}\\
&=&(a_0^{\beta_0})^p(a_1^{\beta_1}a_2^{\beta_2}\ldots
a_{n-1}^{\beta_{n-1}}a_n^{\beta_n})^p\\
&=&a_0^{p{\beta_0}}a_2^{p\beta_2}a_3^{p\beta_3}\ldots a_n^{p\beta_n}\end{aligned}$$ Using the relation $a_{n-1}^p=a_0^p$ we have $$a_0^{p{\beta_0}}a_2^{p\beta_2}a_3^{p\beta_3}\ldots
a_n^{p\beta_n}=a_2^{p\beta_2}a_3^{p\beta_3}\ldots a_{n-2}^{p\beta_{n-2}}a_{n-1}^{p\beta_{n-1}+p{\beta_0}}a_n^{p\beta_n}$$
For the group $G$ we note that $H=\langle a_1,a_2,a_3,\ldots
a_n\rangle $ is the maximal abelian normal subgroup of $G$ and is characteristic. It follows that the $H^p$ is also characteristic. Following [@jamali], we define two decreasing sequences of characteristic subgroups $\{K_i\}_{i=0}^{n-1}$ such that $$K_0=H \;\text{and}\;
K_i/K_{i-1}^p=Z(G/K_{i-1}^p)\;\;(1\leq i\leq n-1)$$ and $\{L_i\}$ such that $$L_0=H\;\text{and}\;L_i=\{h:\;h\in H,\;h^p\in[G,L_{i-1}]\}\;(1\leq i\leq
n-1)$$ It follows easily that
$$K_i=\langle a_1,a_2,\ldots ,a_{n-i},a^p_{n-i+1},\ldots ,a_n^p\rangle \;
1\leq i\leq n-1$$ $$L_1=\langle a_1,v,a_3,\ldots ,a_n\rangle$$ $$L_i=\langle a_1,v,a_3^p,\ldots ,a^p_{i+1},a_{i+2},\ldots
,a_n\rangle\;\;\;2\leq i\leq n-1$$ where $v=a_2^{p^{m-1}}$. For $3\leq i\leq n$ we have $$K_{n-i}\cap L_{i-2}=\langle a_1,v,a_3^p,\ldots
a_{i-1}^p,a_i,a_{i+1}^p,\ldots a_n^p\rangle =\langle v,a_i,G^\prime\rangle .$$ Also $K_{n-2}\cap L_0=\langle a_2,G^\prime\rangle $.
Since $\langle v,a_i,G^\prime\rangle$ and $\langle a_2,G^\prime\rangle$ are characteristic, for any $\theta\in\text{Aut}(G)$, $$\begin{aligned}
\theta(a_2)&=a_2^{k_2}z&\text{where}\;\;z\in
G^\prime\;\;\text{and}\;\;k_2\in\mathbb{N}\\
\theta(a_i)&=a_i^{k_i}v^{r_i}z&\text{where}\;\;z\in G^\prime;\;
k_i\in\mathbb{N};\;\;i=3,4,\ldots,n;\;\;0\leq r_i<p. \end{aligned}$$ It is clear that not all $k_2$ and $k_i$ will make $\theta$ an automorphism. To begin with, if $\theta$ is an automorphism then $\gcd(k_i,p)=1$ for all $k_i$, and we may choose $k_i$, such that $0<k_i<p$ for $i=3,4,\ldots,n$.
Let $\theta(a_0)=a_0^{\beta_0}a_1^{\beta_1}a_2^{\beta_2}\ldots
a_n^{\beta_n}$. Since $\theta(a_0^p)=\theta(a_{n-1}^p)=\theta
(a_{n-1})^p=a^{pk_{n-1}}_{n-1}$, from Lemma 10.4 $$a_{n-1}^{pk_{n-1}}=a_2^{p\beta_2}a_3^{p\beta_3}\ldots
a_{n-2}^{p\beta_{n-2}}a_{n-1}^{p\beta_{n-1}+p{\beta_0}}a_n^{p\beta_n}\;\;\text{for}\;\;p\neq
2$$ implying $\beta_0+\beta_{n-1}\equiv k_{n-1}\mod p$, $p^{m-1}|\beta_2$ and $p|\beta_i$ for $i=3,4,\ldots,n-2,n$. Hence $\theta(a_0)=a_0^{k_0}a_{n-1}^{\beta_{n-1}}v^rz$ where $0\leq r<p$. We changed $\beta_0$ to $k_0$ to maintain uniformity in notations.
Notice the relation $[a_i,a_0]=a_{i+1}^p$ for $i=2,3,\ldots,n$ implies that $$[\theta(a_i),\theta(a_0)]=\theta(a_{i+1})^p=a_{i+1}^{pk_{i+1}}.$$ It follows that $[a_i^{k_i},a_0^{k_0}a_{n-1}^{\beta_{n-1}}]=a_{i+1}^{pk_{i+1}}$ which is the same as $[a_i^{k_i},a_0^{k_0}]=a_{i+1}^{pk_{i+1}}$, which implies that $[a_i,a_0]^{k_0k_i}=a_{i+1}^{pk_{i+1}}$. Recall that $G$ is a $p$-group of class $2$ and $a_{n-1}$ commutes with $a_i$ for $i\geq 2$. From these we have a recursive formula for $k_i$, (also see [@ayan1 Theorem 2.9.7]): choose $k_0$ such that $0<k_0<p$ and $k_2$ such that $0<k_2<p^m$ and $\gcd(k_2,p)=1$ and then define $k_{i+1}=k_0k_{i}\mod p$ for $i=2,3,4,\ldots,(n-1)$; and $k_1=k_0k_n\mod p$. In [@jamali Proposition 2.3] Jamali proves that for $p=2$, all automorphisms of $G$ are central. We have just proved that for $p\neq 2$ there is a non central automorphism, take $k_0>1$; the following theorem follows from Theorem 8.4.
The group $G_n(m,p)$ is Miller if and only if $p=2$.
Description of the Central Automorphisms
----------------------------------------
Notice that $G$ is a PN group, so there is a one-one correspondence between $\text{Aut}_c(G)$ and $\text{Hom}(G,Z(G))$. Since it is known from our earlier discussion that $Z(G)=\langle
a_2^p\rangle\times G^\prime$, $\text{Hom}(G,Z(G))=\text{Hom}(G,\langle
a_2^p\rangle)\times\text{Hom}(G,G^\prime)$. It follows: $\text{Aut}_c(G)=A\times B$ where $$A=\{\sigma\in\text{Aut}_c(G):x^{-1}\sigma(x)\in\langle a_2^p\rangle\}$$ $$B=\{\sigma\in\text{Aut}_c(G):x^{-1}\sigma(x)\in G^\prime\}$$ Elements of $A$ can be explained in a very nice way. Pick a random integer $k$ such that $k=lp+1$ where $0\leq l<p^{m-1}$ and a random subset $R$ (could be empty) of $\{0,3,4,\ldots n\}$, and then an arbitrary automorphism in $A$ is $$\begin{aligned}
\label{oneeqn}
\nonumber &\sigma(a_1)=a_1\\
\nonumber&\sigma(a_2)=a_2^k\\
\nonumber&\sigma(a_i)=\left\{
\begin{array}{cccc}
\nonumber a_i &\text{if}\;\;i \not\in R\\
\nonumber a_i\left(a_2^{p^{m-1}}\right)^{r_i}&\text{if}\;\; i\in R
\end{array}\right.\\\end{aligned}$$ We use indexing in $\{0,3,4,\ldots ,n\}$ to order $R$ and $0<r_i<p$ is an integer corresponding to $i\in R$. Conversely, any element in $A$ can be described this way. It follows from the definition of $A$ that $$|A|=p^{m-1}\times p^{n-1}=p^{m+n-2}.$$
The automorphism $\phi\in B$ is of the form $$\begin{aligned}
\label{twoeqn}
\phi(x)=\left\{
\begin{array}{llll}
a_1&\text{if}& x=a_1\\
a_iz&\text{if}& x=a_i,\;\;i\in\{0,2,3,\ldots ,n\}
\end{array}\right.\end{aligned}$$ where $z\in G^\prime$.
We note that $\dfrac{G}{Z(G)}$ is an abelian group and hence $\text{Inn}(G)$ is abelian and hence $\text{Inn}(G)\subseteq\text{Aut}_c(G)$. We further note from the commutator relations in $G$ that $\text{Inn}(G)\subseteq B$.
Using these automorphisms in key-exchange protocol I
----------------------------------------------------
Let us briefly recall the key-exchange protocol described before. Alice and Bob decide on a group $G$ and a non-central element $g\in G\setminus Z(G)$ in public. Alice then chooses an arbitrary automorphism $\phi_A$ and sends Bob $\phi_A(g)$. Similarly Bob picks an arbitrary automorphism $\phi_B$ and sends Alice $\phi_B(g)$. Since the automorphisms commute, both of them can compute $\phi_A(\phi_B(g))$, which is their private key. The most devastating attack on the system is the one in which Oscar, looking at $g$, $\phi_A(g)$ and $\phi_B(g)$, can predict what $\phi_A(\phi_B(g))$ will look like, i.e., a GDHP attack.
If $g=a_0^{\beta_0} a_1^{\beta_1}a_2^{\beta_2}a_3^{\beta_3}\ldots
a_n^{\beta_n}$ is an arbitrary element of $G$, i.e., $0\leq{\beta_0} <p$, $0\leq
\beta_1<p$, $0\leq\beta_2 <p^m$ and $0\leq \beta_i <p^2$ for $3\leq i\leq
n$. Then the vector $v:=({\beta_0},\beta_3,\beta_4,\ldots ,\beta_n)$ is called the parity of $g$. Two elements $g$ and $g^\prime$ are said to be of the same parity condition if $v=v^\prime\mod p$, where $v^\prime$ is the parity of $g^\prime$.
If $\phi:G\rightarrow G$ is any central automorphism then $g$ and $\phi(g)$ have the same parity condition for any $g\in G$.
Notice that an automorphism $\phi$ either belongs to $A$ or $B$ or is of the form $\phi(g)=gf_{\phi}(g)g_{\phi}(g)$ where $f_{\phi}\in\text{Hom}(G,Z(G))$ and $g_{\phi}\in\text{Hom}(G,G^\prime)$. So we might safely ignore elements from $A$, since they only affect the exponent of $a_2$. Also note that $a_1$ being in the commutator remains fixed under any central automorphism.
So we need to be concerned with elements of $B$, from the description of $B$, and each commutator is a word in $p$-powers of the generators and from the fact that $G^\prime\subset Z(G)$, the lemma follows.
Now let us understand what an element in $A$ does to an element $g\in G$. We use notations from Equation \[oneeqn\].
Let $g=a_0^{\beta_0}a_1^{\beta_1}a_2^{\beta_2}a_3^{\beta_3}\ldots
a_n^{\beta_n}$, $\phi\in A$ and if
$\phi(g)=a_0^{\beta_0^\prime}a_1^{\beta_1^\prime}a_2^{\beta_2^\prime}a_3^{\beta_3^\prime}\ldots
a_n^{\beta_n^\prime}$ then $\beta_i=\beta_i^\prime$ for $i\neq 2$ and
$\beta_2^\prime=k\beta_2+p^{m-1}\sum\limits_{i\in R}r_i\beta_i \mod p^m$.
Notice that from Equation \[oneeqn\], it is clear that elements of $A$ only affect the exponent of $a_2$, so $\beta_i^\prime =\beta_i$ for $i\neq 2$ follows trivially. From the definition of $A$ and simple computation it follows that $\beta_2^\prime=k\beta_2+p^{m-1}\sum\limits_{i\in R}r_i\beta_i\mod p^m$.
In the key exchange protocol I, we will only use automorphisms from[^1] $A$. As noted earlier there are two kinds of attack, GDLP (the discrete logarithm problem in automorphisms) and GDHP (the Diffie-Hellman problem in automorphisms). We have earlier stated that GDLP is equivalent to finding the automorphism from the action of the automorphism on one element. It seems that for one to find the automorphism discussed in the previous lemma, one has to find $k$, $R$ and $r_i$. Notice that $\beta_2^\prime=k\beta_2+p^{m-1}\sum\limits_{i\in R}r_i\beta_i\mod p^m$, is a knapsack in $\beta_2$ and $p^{m-1}$. Solving that knapsack is not enough to compute the image of any element, because $R$ is not known so $\beta_i$’s are not known. We shall show in a moment that the security of the key exchange protocol depends on the difficulty of this knapsack, but solving this knapsack does not help Oscar to find the automorphism, just partial information about the automorphism comes out.
Next we show that though it seems to be secure under GDLP, but if the knapsack is solved then the system is broken by GDHP. This proves that GDHP is a weaker problem than GDLP in $G_n(m,p)$. Let $g=a_0^{\beta_0}a_1^{\beta_1}a_2^{\beta_2}a_3^{\beta_3}\hspace*{\fill}\\
\ldots a_n^{\beta_n}$, then as discussed before for $\phi,\psi\in\text{Aut}_c(G)$, with notation from Equation 2 and $k_i\in\mathbb{N}$ for $i=3,4,\ldots,n$: $$\label{eqn1}
\phi(g)=a_0^{\beta_0}a_1^{\beta_1}a_2^{k_2\beta_2+p^{m-1}\sum\limits_{i\in
R}r_i\beta_i}a_3^{\beta_3+k_3p}\ldots a_n^{\beta_4+k_4p}$$ $$\label{eqn2}
\psi(g)=a_0^{\beta_0}a_1^{\beta_1}a_2^{k_2^\prime\beta_2+p^{m-1}\sum\limits_{i\in R^\prime}r_i^\prime\beta_i}a_3^{\beta_3+k_3^\prime p}\ldots a_n^{\beta_4+k_4^\prime
p}$$ From direct computation it follows that the exponent of $a_2$ in $\phi(\psi(g))$ is $$\label{eqn3}
k_2\left(k_2^\prime\beta_2+p^{m-1}\sum\limits_{i\in
R^\prime}r_i^\prime\beta_i\right)+p^{m-1}\sum\limits_{i\in
R}r_i\beta_i$$ where $k_2=lp+1$ and $k_2^\prime=l^\prime p+1$, $0\leq
l,l^\prime <p^{m-1}$. The exponent of $a_0, a_1$ stays the same and the exponent of $a_i$ will be $\beta_i+(k_i+k_i^\prime)p\mod p^2$ for $3\leq i\leq n$. As mentioned before since we are using only automorphisms from $A$, i.e., $\phi$ and $\psi$ are in $A$ hence $k_i=k_i^\prime=0$ for $i=3,4,\ldots,n$.
Notice that $g$, Equations \[eqn1\] and \[eqn2\] are public, so Oscar sees those. Since the exponents of $a_0,a_1,a_3,\ldots,a_n$ are predictable, the key Alice and Bob want to establish is the exponent of $a_2$ in $\phi\left(\psi(g)\right)$, which is given by Equation 6. Since Oscar sees Equations \[eqn1\] and \[eqn2\], if he can compute $k_2$ from $k_2\beta_2+p^{m-1}\sum\limits_{i\in R}r_i\beta_i\mod p^m$, then he can compute $p^{m-1}\sum\limits_{i\in R}r_i\beta_i$ and the scheme is broken. But, $k_2=lp+1$ for some $l\in[0,p^{m-1})$ hence $$k_2\beta_2+p^{m-1}\sum\limits_{i\in R}r_i\beta_i\mod p^m$$ reduces to $$\beta_2+lp\beta_2+p^{m-1}\sum\limits_{i\in R}r_i\beta_i\mod p^m.$$ Since $\beta_2$ is public, Oscar can compute $lp\beta_2+p^{m-1}\sum\limits_{i\in R}r_i\beta_i\mod p^m$. Notice that finding $k_2$ is equivalent to finding $l$, hence one of the security assumptions is that there is no polynomial time algorithm to find $l$ from $$lp\beta_2+p^{m-1}\sum\limits_{i\in R}r_i\beta_i\mod
p^m.$$ Let us write $$\label{eqn4}
M=lp\beta_2+p^{m-1}\sum\limits_{i\in R}r_i\beta_i\mod p^m,$$ then $$M=lp\beta_2\mod p^{m-1}.$$ Now, if $lp<p^{m-1}$ and $\gcd(\beta_2,p)=1$, then one can find $lp$ from the above equation and the scheme is broken. So the only hope of making a secure cryptosystem out of key exchange protocol I and the group $G_n(m,p)$ is to take $l=kp^{m-2}$ where $k=0,1,2,\ldots,(p-1)$. In this case, if we set $l=lp^{m-2}$ and $l^\prime=l^\prime p^{m-2}$ in Equation \[eqn3\], then the key will be $$\begin{aligned}
\lefteqn{\left(1+lp^{m-1}\right)\left((1+l^\prime
p^{m-1})\beta_2+p^{m-1}\sum\limits_{i\in
R^\prime}r_i^\prime\beta_i\right)+p^{m-1}\sum\limits_{i\in
R}r_i\beta_i}\\
&=\left(1+lp^{m-1}+l^\prime p^{m-1}\right)\beta_2+p^{m-1}\sum\limits_{i\in R^\prime}r_i^\prime\beta_i+p^{m-1}\sum\limits_{i\in
R}r_i\beta_i\mod p^m\\
&=\left(\left(1+lp^{m-1}\right)\beta_2+p^{m-1}\sum\limits_{i\in
R}r_i\beta_i\right)+l^\prime p^{m-1}\beta_2+p^{m-1}\sum\limits_{i\in
R^\prime}r_i^\prime\beta_i\mod p^m\end{aligned}$$
Now the information in the last equation is easy to compute from the public information, Equations \[eqn1\] and \[eqn2\]; so the Key Exchange Protocol I is broken for automorphisms from $A$ of $G_n(m,p)$ when $\gcd(\beta_2,p)=1$.
Now if $\gcd(p,\beta_2)\neq 1$, i.e., $\beta_2=kp^i$ for some $i\in[1,m)$ and $1\leq k <p$, then an attack similar to the above breaks the system. The insight behind these attacks is that any solution to Equation \[eqn4\] can be thought of as the image of $g$ under an automorphism $\phi^\prime\in A$. We are talking about a solution to Equation \[eqn4\], which is easy to find, for which $\phi^\prime(g)=M$ and then Shpilrain’s attack breaks the system.
Implementation
==============
There is not much reason left to go into the details of an implementation. We briefly mention that this cryptosystem can be implemented without any reference to the group $G_n(m,p)$. Once the element $g=a_0^{\beta_0}a_1^{\beta_1}a_2^{\beta_2}\ldots a_n^{\beta_n}$ is fixed, Alice can send Bob $k_2\beta_2+p^{m-1}\sum\limits_{i\in R}r_i\beta_i
\mod p^m$ and similarly Bob can send Alice $k_2^\prime\beta_2+p^{m-1}\sum\limits_{i\in R^\prime}r_i^\prime\beta_i \mod
p^m$. Since Alice and Bob know their own $k_2$, $\sum\limits_{i\in
R}r_i\beta_i$ and $k_2^\prime$, $\sum\limits_{i\in
R^\prime}r_i^\prime\beta_i$ respectively, they can both compute the private key or the shared secret. Since the only operation involved in computing the private key is multiplication and addition$\mod p^m$, there can be a very fast implementation of this cryptosystem.
Conclusion
==========
In this paper we studied a key exchange protocol using commuting automorphisms in a non-abelian $p$-group. Since any nilpotent group is a direct product of its Sylow subgroups, the study of nilpotent groups can be reduced to the study of $p$-groups. We argued that our study is a generalization of the Diffie-Hellman key exchange and is a generalization of the discrete log problem. Other public key systems like the El-Gamal cryptosystem which uses the discrete logarithm problem is adaptable to our methods. This is the first attempt to generalize the discrete logarithm problem in the way we did.
We should try to find other groups and try our system in terms of GDLP and GDHP. As we noted earlier, GDHP is a subproblem of the GDLP, and we saw in $G_n(m,p)$, GDHP is a much easier problem than GDLP. Our example was of the form $d>b$ in Theorem \[adney\]. The next step is to look at groups where $d=b$. We note from Theorem \[sanders\], if a $p$-group $G$ is a PN group then $\text{Aut}_c(G)$ is a $p$-group and since $p$-groups have nontrivial centers; one can work in that center with our scheme. In this case we would be generalizing to arbitrary nilpotentcy class while still working with central automorphisms.
Lastly we note that, if we were using some representation for this finitely presented group $G$, for example, matrix representation of the group over a finite field $\mathbb{F}_q$; then security of the system in $G_n(m,p)$ becomes the discrete logarithm problem in a matrix algebra [@menezes1; @menezes2]. Since the discrete logarithm problem in matrices is only as secure as the discrete logarithm problem in finite fields, there is no known advantage to go for matrix representation, but there might be other representations of interest.
There is one conjecture that comes out of this work and we end with that.
If $G$ is a Miller $p$-group for an odd prime $p$, then $G$ is special.
**Acknowledgment**: The author wishes to thank Fred Richman and Rustam Stolkin; they read the whole manuscript and made valuable suggestions. The author is indebted to the referee for his kind comments.
[^1]: In light of Lemma 10.6, we believe that adding automorphisms from $B$ is not going to add to the security of the system.
|
---
abstract: 'As 3D scanning solutions become increasingly popular, several deep learning setups have been developed for the task of scan completion, i.e., plausibly filling in regions that were missed in the raw scans. These methods, however, largely rely on supervision in the form of paired training data, i.e., partial scans with corresponding desired completed scans. While these methods have been successfully demonstrated on synthetic data, the approaches cannot be directly used on real scans in the absence of suitable paired training data. We develop a first approach that works directly on input point clouds, does not require paired training data, and hence can directly be applied to real scans for scan completion. We evaluate the approach qualitatively on several real-world datasets (ScanNet, Matterport3D, [KITTI]{}), quantitatively on 3D-EPN shape completion dataset, and demonstrate realistic completions under varying levels of incompleteness.'
author:
- |
**Xuelin Chen**\
Shandong University\
University College London\
- |
**Baoquan Chen**\
Peking University\
- |
**Niloy J. Mitra**\
University College London\
Adobe Research London\
bibliography:
- 'egbib.bib'
title: Unpaired Point Cloud Completion on Real Scans using Adversarial Training
---
Conclusion
==========
We presented a point-based unpaired shape completion framework that can be applied directly on raw partial scans to obtain clean and complete point clouds. At the core of the algorithm is an adaptation network acting as a generator that transforms latent code encodings of the raw point scans, and maps them to latent code encodings of clean and complete object scans. The two latent spaces regularize the problem by restricting the transfer problem to respective data manifolds. We extensively evaluated our method on real scans and virtual scans, demonstrating that our approach consistently leads to plausible completions and perform superior to other methods. The work opens up the possibility of generalizing our approach to scene-level scan completions, rather than object-specific completions. [Our method shares the same limitations as many of the supervised counterparts: does not produce fine-scale details and assumes input to be canonically oriented.]{} Another interesting future direction will be to combine point- and image-features to apply the completion setup to both geometry and texture details.
Acknowledgements
================
We thank all the anonymous reviewers for their insightful comments and feedback. This work is supported in part by grants from National Key R&D Program of China (2019YFF0302900), China Scholarship Council, National Natural Science Foundation of China (No.61602273), ERC Starting Grant, ERC PoC Grant, Google Faculty Award, Royal Society Advanced Newton Fellowship, and gifts from Adobe.
|
---
address: |
Inter-University Centre for Astronomy and Astrophysics,\
Post Bag 4, Ganeshkhind, Pune 411 007, India.
author:
- Tarun Souradeep
title: Summary of ICGC04 Cosmology Workshop
---
\#1[= 0.6]{}
\#1[= 1.75]{}
\#1\#2\#3[=.6 =.75 =.6]{} \#1\#2\#3\#4[=.4 =.4 =.4 =.4]{} \#1\#2[=.3 =.3]{} \#1\#2\#3\#4\#5\#6\#7
to\#2
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Introduction
============
Recent developments in Cosmology has been largely driven by huge improvement in quality, quantity and the scope of cosmological observations. While on one hand, the observations have constrained theoretical scenarios and models more precisely, some of these observations have thrown up new challenges to theoretical understanding and others that have brought issues from the realm of theoretical speculation to observational verification. The contribution to the workshop on cosmology at the ICGC-04 reflects this vibrancy in the field.
In this article, I summarize both the oral and poster presentations made at the workshop. The two invited talks at the cosmology workshop by [**S. Majumdar**]{} and [**M. Kaplinghat**]{} are included as separate articles in this issue. The contributions have been divided into five broad themes. A brief introduction to each of the themes in the context of the ICGC04 contributions is given in this section. Sections 2 to 6, summarize the contributions in each of the following five themes:
[***1. Structure formation:***]{} The formation of large scale structure in universe has been a very important aspect of modern cosmology. The ‘standard’ model of cosmology must not only explain the dynamics of homogeneous background universe, but also (eventually) satisfactorily describe the perturbed universe – the generation, evolution and finally, the formation of large scale structures in the universe. Recently, large redshift surveys such as the Las Campanas Redshift survey (LCRS), $2$degree field (2dF) and SDSS have mapped out the distribution of matter in the universe [@lah_sut03]. In conjunction, with other observations these surveys have contributed to the community wide effort in cosmological parameter estimation [@sper_wmap03; @max_sdss04].
There are lot of systematic effects that arise in extracting cosmological information from redshift surveys. Contributions to ICGC-04 dealt with some such issues. The distances to distant galaxies are measured in terms of redshifts. It is important to understand in finer detail the mapping from the redshift space to real space known as the redshift space distortions. The galaxies are just tracers of the underlying distribution of dark matter in the universe. Understanding the ‘bias’ in translating from distribution of galaxies to the dark matter remains an important area of research. Finally, to compare observed large scale structures to theory, it is important to devise statistics that can distinguish true morphological features from chance alignments in terms of robust statistics.
[***2. Cosmic Microwave background anisotropy:*** ]{} The cosmic microwave background has played a crucial role in cosmology. In the past decade, the measured anisotropy in the temperature of the Cosmic Microwave background has spearheaded the ongoing transition of cosmology into a precision science [@hu_dod02]. In particular, the exquisite measurements of the angular power spectrum of CMB fluctuations have played a crucial role in identifying an emerging concordance cosmological model. The high quality of data, the good theoretical understanding of CMB anisotropy and polarization, and the relatively unambiguous connection between the two has encouraged a number of researchers to use the CMB data to probe the universe beyond estimating a set of cosmological parameters. Contributions in ICGC-04 have observationally addressed theoretical assumptions such as the statistical isotropy of CMB fluctuations, scale invariance in the primordial power spectrum and parity conservation in electromagnetic interactions. Efforts to mine the CMB deeper pose data analysis challenges in particular to account for finer systematic effects in observations, such as accounting for the non-circularity of the experimental beam response function presented in the meeting.
[***3. Dark energy:***]{} Observations of the luminosity distance using high redshift Supernova Ia have indicated that the present expansion rate of the universe may be accelerating. Within the Friedman-Robertson-Walker cosmology ( and general relativity), this implies that the present matter content of the universe is dominated by dark energy – a yet unidentified, exotic matter with negative pressure. Modeled as an ideal hydrodynamic fluid with equation of state $w$, ($p=w \rho$) the acceleration of universe implies that $w
<-1/3$. Phenomenologically, the simplest model of dark energy is the cosmological constant (or, vacuum energy) where $w\simeq -1$ is consistent with data. Concordant results are also obtained from the formation of large scale structures in the universe by combining the exquisite measurements of the angular spectrum of CMB anisotropy with recent measurements of power spectrum of density perturbation from large redshift survey. Various combinations of the CMB, High-$z$ SN1a and galaxy redshift survey data constrain $w\lsim -0.8$. If the strong energy condition $w \geq -1$ is not imposed as prior then the likelihood appears to spill over and peak in the $w<-1$ regime. Dark energy with $w <-1$, is rather unusual, if interpreted in terms of a hydrodynamic fluid or scalar field energy density. A number of contributions in the workshop are theoretical models and scenarios attempting to explain dark energy with $w < -1$. An interesting possibility which is explored in a contribution to this meeting is is to recover the evolution of the equation of state, $w$, with redshift from the data without imposing the strong energy condition prior.
[***4. Early Universe & extra dimensions:***]{} In the hot Big Bang scenario, the present universe is inescapably linked to the ultra high energy physics of the early universe. It is fair to say that cosmology will remain incomplete without adequate understanding of the early universe. The initial singularity of big bang remains an enigma that time and again attracts imaginative solutions. The recent proposal to ’cap’ the initial FLRW model with a static Einstein universe is explored in the context of higher derivative extensions of gravitation in one of the workshop contributions. The primordial perturbations believed to be generated during inflation is one of most promising probe of physics at ultra-high energies, possibly, even up to trans-Planckian energy scales. A contribution here has explored the signature of trans-Planckian physics that respects Lorentz symmetry. Besides scalar density perturbations, the gravity wave background from inflation is an important clue to the early universe physics. For example, overproduction of gravity waves can severely constrain some braneworld inflation models with steep inflaton potential, unless, as shown one of the contributions, the mechanism of reheating is reworked. Although cosmic defects produced during phase transition in the early universe seem unlikely to be the dominant source of primordial perturbations, the possibility that they play a sub-dominant role in structure formation is still an open possibility. A formalism for studying perturbations from defects was presented in the workshop.
In the past few years, the possibility of large extra dimensions has caught the fancy of both theoretical high energy physicists as well as cosmologists. These braneworld scenarios, where the observed $3+1$ dimensions and all interactions other than gravity reside on a brane embedded in a higher dimension, are usually motivated by string theory. These scenarios, initially invoked to address the hierarchy problem of disparity between the SUSY breaking and the Planck scale, have been used and studied in a variety of other contexts. The observed accelerating universe and dark energy have been linked to braneworld scenario. Construction of stable brane configurations that may do this was discussed in the meeting. Braneworld scenarios can have interesting consequences for inflation as shown by some contributions in the meeting.
[***5. Alternative approaches:*** ]{} Despite the emergence of concordant cosmological model, there is enough width left in cosmology for exploring radically different ideas. The reason is that recent advances in cosmology have been more on the phenomenological, rather than conceptual aspects. Alternative ideas may be the key to some of the intriguing puzzles of contemporary cosmology. While many of these ideas face increasingly difficult challenges to match observations, it is important to allow them to be judged by observations, and not theoretical prejudices.
Large scale structures in the universe
=======================================
The morphology of the structures in the large scale distribution of matter in the universe seen in the recent redshift surveys has to be quantified with a statistically robust measures. One of the most striking visual features in redshift surveys is that galaxies appear to be distributed along filaments that are interconnected to form a network which extends across the entire survey like a cosmic web. Figure \[fig:som\] shows the distribution of galaxies in one of the slices of the Las Campanas redshift Survey (LCRS) and highlights the filamentary structures that appear within it. [**S. Bharadwaj**]{} reported a novel analysis (with S. Bhavsar and J. Sheth) of the Las Campanas redshift Survey (LCRS) to determine the extent to which the filaments are genuine, statistically significant features as against the possibility of their arising from chance alignments [@bhar04]. They find that one of the LCRS slices has statistically significant filamentary features spanning scales as large as $70$ to $80\,h^{-1}{\rm Mpc}$, whereas filaments spanning scales larger than this are not statistically significant (see fig. \[fig:som\]). For the five other LCRS slices, filaments of lengths $50$ to $70\,h^{-1} {\rm Mpc}$ are statistically significant, but not beyond. [*The reality of the $80\,h^{-1}{\rm Mpc}$ filamentary features in the LCRS slice make them the longest coherent features presently known.*]{}
\[fig:som\]
The galaxy two point correlation function measured from redshift surveys exhibits deviations from the predictions of linear theory on scales as large $20\,h^{-1} {\rm Mpc}$ where we expect linear theory to hold. Any attempt at analyzing the anisotropies in the redshift correlation function and determining the linear redshift distortion parameter requires these effects to be taken into account. Usually these non-linear effects are attributed to galaxy random motions, and a heuristic model where the linear redshift correlation is convolved with the random pairwise velocity distribution function is used. [**S. Bharadwaj**]{} reported investigations (with B. Pandey) of a different model which is derived under the assumption that linear theory holds in real space, and which takes into account all non-linear effects that are introduced by the mapping from real to redshift space [@bhar_pan04]. They test this model using N-body simulations and find that the pairwise velocity dispersion predicted by all the models considered are in excess of the values determined directly from the N-body simulations. [*This indicates a shortfall in the understanding of the statistical properties of peculiar velocities and their relation to redshift distortion.*]{}
The distribution of galaxies is believed to trace the underlying distribution of dark matter quantified by a bias factor. [**J. Bagla**]{} presented results from a study (with S. Ray) of the moments of counts in cells for a set of models in real space and redshift space. A comparison of the moments for the entire distribution as well as for regions with over-density above a certain threshold offers insight into the differences between clustering of galaxies and dark matter. They focus mainly on non-linear scales. Tree-PM simulations were used to generate the distribution of particles. They find that at non-linear scales the bias for the second moment of the distribution is scale dependent. The scale dependence is such that the variation of $\sigma$ with scale is the same for all the models studied here. The amplitude of $\sigma$ is very different for these models, and models with a more negative index have a larger linear bias. Skewness for different models studied here is very different for the entire distribution and has expected values for the power law models in the linear as well as the extreme non-linear regime. [*Skewness for over-dense regions in all of these models is the same in redshift space in the non-linear regime*]{}. This startling result implies that in the non-linear regime the skewness of the distribution of particles in over-dense regions is not sensitive to initial conditions.
[**S. Ray**]{} reported on the development of a parallel version of the Tree-PM code optimized for cluster computing. The Tree-PM method is a hybrid method for cosmological N-Body simulations that uses both the tree and mesh based methods to compute force. The force is divided into two parts and mesh method is used to compute the long range force and the tree method is used for computing the short range force [@bag_ray03]. They use domain decomposition for distributing the computation of short range force on a set of processors. Functional decomposition is used to assign the tasks for computing long range and short range forces. They also introduce optimizations to reduce the communication overheads. [*The present version of the code scales almost linearly up to $34$ processors for simulations with $1.6 \times 10^7$ particles. Time taken per step per particle for these simulations is about $18\mu$s.*]{}
[**J. Prasad**]{} reported initial results on a study (with S. Ray and J. Bagla) of the interaction of fluctuations at very different scales by simulating collapse of a plane wave with varying amount of substructure. The substructure is modeled as power in a narrow range of scales at $k_s \gg k_l$, where $k_l$ is the wave number for the long range wave mode. The substructure is along three directions whereas the plane wave collapses along one direction. In absence of substructure they find the usual structure where a pancake forms and has an increasing number of streams as one nears the center. In presence of small scale fluctuations, they find that the size of the multi-stream region shrinks by a small amount as the amplitude of substructure is increased. They believe that this is due to transfer of kinetic energy in directions transverse to the plane wave due to interaction of clumps. The plane wave influences evolution of small scale fluctuations strongly. [*Formation of clumps is suppressed in the regions made under-dense by the plane wave, whereas merging in the over-dense regions leads to a rapid growth of fluctuations at small scale.*]{}
A study of density perturbations of a cosmological scalar field addressing the possibility of using the instability mechanism of Jeans theory, to form self gravitating configurations from a real scalar field scalar field approach to Jeans mass calculation was presented by [**M. Joy**]{} and V. C. Kuriakose [@mjoy]. They consider a massive scalar field arbitrarily coupled to a gravitational background, with the stress-energy tensor expectation values of the quantum field fluctuations computed in a coherent state. [*It is shown that the self-interaction of the scalar field influences the character of instability and the value of the Jeans wave number is altered by the effects of self-interaction.*]{}
The observations of clustering in the distribution of HI can be used to study large scale structures at high redshift. [**S. Bharadwaj**]{} presented a study of the possibility of using Giant Meter-wave Radio Telescope (GMRT) to probe large scale structures in the universe at high redshift by studying fluctuations in the redshifted 1420 MHz emission from the neutral hydrogen (HI) [@bhar_srik04]. The study focuses on the cross-correlations between the visibility signal measured at different baselines and frequencies in radio interferometric observations. They show that the visibility correlations directly probe the power spectrum of HI fluctuation, and present analytic estimates of the signal expected in two of the GMRT bands centered at 325 and 610 MHz. They also simulate GMRT observations including the expected HI signal, galactic and extragalactic foregrounds and system noise. [*The preliminary results indicate that it may be possible to detect the HI signal in around 1000 hours of observations.*]{}
Cosmic Microwave Background anisotropy
=======================================
\[fig:haj\]
The statistical expectation values of the temperature fluctuations of cosmic microwave background (CMB) are assumed to be preserved under rotations of the sky. The assumption of statistical isotropy (SI) of the CMB anisotropy should be observationally verified since detection of violation of SI could have profound implications for cosmology. The Bipolar power spectrum (BiPS) has been recently proposed as a measure of violation of statistical isotropy in the CMB anisotropy map[@haj_sour03]. [**A. Hajian**]{} reported results from a BiPS analysis (with T. Souradeep) statistical isotropy of the CMB anisotropy maps obtained from the first year of data from the WMAP satellite. The CMB maps were smoothed by a family of window functions to isolate and test the SI in the different regions of the multipole space. Figure \[fig:haj\] also shows the BiPS of three different CMB maps obtained from the WMAP data filtered to retain power with $20\lsim \ell \lsim 45$. [*Preliminary results indicate that the CMB anisotropy maps from WMAP do not strongly violate statistical isotropy.*]{}
The non-circularity of the experimental beam has become progressively important as CMB experiments strive to attain higher angular resolution and sensitivity. Recent CMB experiments such as ARCHEOPS, MAXIMA, WMAP have significantly non-circular beams. Future experiments like Planck are expected to be even more seriously affected by non-circular beams. [**S. Mitra**]{} reported a study of the effect of a non-circular beam on CMB power spectrum estimation (done with A. Sengupta and T. Souradeep) [@mit04]. They compute the bias introduced estimated power spectrum. They construct an unbiased estimator using the bias matrix. The covariance matrix of the unbiased estimator is computed for non-rotating smooth beams. The WMAP beams maps are fitted and shown to significantly non-circular. The effect of a non-circular beam on power spectrum estimate is calculated for a CMB map made by an experiment with a beam which is non-circular at a level comparable to the WMAP beam.
Cosmological parameters estimated from CMB anisotropy assume a specific form for the spectrum of primordial perturbations believed to have seeded the large scale structure in the universe. Accurate measurements of the angular power spectrum $C_l$ over a wide range of multipoles from the Wilkinson Microwave Anisotropy Probe (WMAP) [@ben_wmap03] has opened up the possibility to deconvolve the primordial power spectrum for a given set of cosmological parameters. [**A. Shafieloo**]{} presented results from a work (with T. Souradeep) on the direct estimation of the primordial power spectrum from WMAP measured angular power spectrum of CMB anisotropy using an improved Richardson-Lucy deconvolution algorithm [@arm_sour04]. The most prominent feature of the recovered $P(k)$, shown as the solid curve in figure \[pkrec\], is a sharp, infra-red cut off on the horizon scale. It also has a localized excess just above the cut-off which leads to great improvement of likelihood over the simple monotonic forms of model infra-red cut-off spectra considered in the post WMAP literature. The form of infra-red cut-off is robust to small changes in cosmological parameters. Remarkably similar forms of infra-red cutoff is known to arise in very reasonable extensions and refinements of the predictions from simple inflationary scenarios. In figure \[pkrec\], the curve labeled ‘staro’ is the primordial spectrum when the inflaton potential has a kink– a sharp, but rounded, change in slope [@star92] and two curves labeled ‘VF’ are the modification to the power spectrum from a pre-inflationary radiation dominated epoch [@vilfor82].
\[pkrec\]
In addition to temperature fluctuations, the CMB photons coming from different directions have a random, linear polarization. The polarization of CMB can be decomposed into $E$ part with even parity and $B$ part with odd parity. Besides the angular spectrum $C_l^{TT}$, the CMB polarization provides three additional spectra, $C_l^{TE}$, $C_l^{EE}$ and $C_l^{BB}$ which are invariant under parity transformations. The level of polarization of the CMB being about a tenth of the temperature fluctuation, it is only very recently that the angular power spectrum of CMB polarization field has been detected. The Degree Angular Scale Interferometer (DASI) has measured the CMB polarization spectrum over limited band of angular scales in late 2002 [@kov_dasi02]. The WMAP mission has also detected CMB polarization [@kog_wmap03]. WMAP is expected to release the CMB polarization maps very soon.
Parity violating interactions open up the possibility for measuring non-zero $C_l^{TB}$ and $C_l^{EB}$ power spectra. One such possibility is a parity-violating interaction of the antisymmetric tensor Kalb-Ramond (KR) gauge field and the electromagnetic field [@maj03]. [**P. Majumdar**]{} presented results that show that parity forbidden $C_l^{TB}$ spectra can arise due to such interactions [@maj04]. The coupling also leads to an effective time-dependent fine-structure constant in the current cosmological epoch, pointing thereby to possible correlations between these two disparate phenomena.
Accelerating universe and Dark energy
=====================================
Improvements in the measurements of luminosity distance as a function of redshift from High redshift SN1a may eventually allow direct recovery of the evolution of the equation of state of the dark energy component in the universe at low redshifts. [**U. Alam**]{} reported results from a new study (with V. Sahni, T. D. Saini, A. A. Starobinsky) where dark energy parameters are reconstructed from the latest data set of 194 supernovae [@ton03; @bar03] without any priors on the equation of state $w$ [@alam]. They find that dark energy evolves rapidly and metamorphoses from dust-like behavior at high redshift ($w \simeq 0$ at $z \sim 1$) to a strongly negative equation of state at present ($w \lsim -1$ at $z \simeq 0$), as shown in the figure \[fig:var\_w\]. Dark energy metamorphosis appears to be a robust phenomenon which manifests for a large variety of supernova data samples provided one does not invoke the weak energy prior $\rho
+ p \geq 0$. [*These results indicate that dark energy with a rapidly evolving equation of state may provide a compelling alternative to a cosmological constant if data are analyzed in a prior-free manner.*]{}
Typical scalar field models do not provide a scenario with $w<-1$. A radical possibility is to consider a phantom scalar field which has negative kinetic energy and violates null dominant energy condition – now popularly referred to as a ‘phantom field’. [**P. Singh**]{}, reported work (with M. Sami, and N. Dadhich) on a model of phantom field motivated S-brane dynamics using the Supernova Ia observations to constrain the parameters of the model [@param]. [*They find that the model fits High redshift Supernova data fairly well for a large range of parameters and favors a $w < -1$.*]{}
The negative kinetic energy term in the Lagrangian of a phantom field is not very well motivated and also suffers from severe instability. [**S. Das**]{} reported work on explaining dark energy with $w<-1$ in the Brans-Dicke theory of gravity. [*The results indicate that $w<-1$ could be obtained if the gravitational constant is slowly varying with time in a canonical Brans-Dicke theory of gravity without conflict with the solar system constraints.*]{}
The de Broglie-Bohm approach to quantum mechanics helps to demonstrate that the Wheeler-De Witt equation is equivalent to the corresponding classical equation for a special potential; i.e., the universe is both quantum and classical at the same time. [**M. John**]{} presented a comparison of prediction of this approach with observations of High-redshift Supernova data. [*They claim that the observational level, this special solution to cosmology is as good as the conventional matter-$\Lambda$ cosmologies.*]{}
Early universe and Brane world
===============================
Curing the initial singularity of standard FLRW models has been a long standing endeavor of cosmologists. Recently, Ellis and Maartens studied a class of cosmological models in which there is no singularity, no beginning of time and no horizon problem [@ell_maar04]. The universe starts out as an almost static universe and expands slowly, eventually evolving into a hot big-bang era. An example of this scenario is a closed model with a minimally coupled scalar field, with a special self-interaction potential. This potential may be obtained after a conformal transformation of the metric of a higher derivative theory. [**S. Mukherjee**]{} reported a detailed study (with B. Paul, S. Maharaj and A. Beesham) of higher derivative theories, including a cosmological constant and a quadratic term ($R^2$) in the Lagrangian density, where $R$ is the scalar curvature. The field equations are analyzed to determine the general characteristics of the evolution and some quantities of cosmological interest are calculated. The results are compared with those of the proposed emergent universe. [*The stability of the results have been studied by considering an $R^3$ term as a perturbation of the quadratic Lagrangian density. The possibility of a quantum creation of the emergent universe in quantum cosmology has also been considered.*]{}
[**B. Modak**]{} presented application of Noether symmetry as powerful tool to find the solution of the field equations for scalar tensor theory including curvature quadratic term. [*A few physically reasonable solutions like power law inflation were presented.*]{}
Quintessential inflation describes a scenario in which both inflation and dark energy are described by the same scalar field. In conventional brane world models of quintessential inflation gravitational particle production is used to reheat the universe. This reheating mechanism is very inefficient and results in an excessive production of gravity waves which can even violate nucleosynthesis constraints and invalidate the model [@sami]. [**M. Sami**]{} described a new method of realizing quintessential inflation on the brane in which inflation is followed by ‘instant preheating’. [*The larger reheating temperature in this model is shown to result in a smaller amplitude of relic gravity waves which is consistent with nucleosynthesis bounds.*]{} The relic gravity wave background has a ‘blue’ spectrum at high frequencies and is a generic byproduct of successful quintessential inflation on the brane.
[**S. Biswas**]{} presented a study of fermion particle production in early universe using the complex trajectory WKB method developed earlier. They study the particle production in periodic potential, generally used in inflationary cosmology. [*Using this method, they recover results obtained in literature earlier, such as [@kof_green00].*]{}
Due to the tremendous red-shift that occurs during the inflationary epoch in the early universe, it has been realized that trans-Planckian physics may manifest itself at energies much lower than the Planck energy. The presence of a fundamental scale suggests that local Lorentz invariance may be violated at sufficiently high energies. However, certain astrophysical observations seem to indicate that local Lorentz invariance may be preserved to extremely high energies. This suggests considering models of trans-Planckian effects that preserve local Lorentz invariance. [**L. Sriramkumar**]{} (with S. Shankaranarayanan) presented one such model and evaluated the spectrum of density perturbations during inflation in the model [@srir_sank04]. [*They find that, in the case of exponential, as well as, power-law inflation, the corrections to the standard scale-invariant perturbation spectrum (in the Bunch-Davies vacuum) turn out to be small.*]{}
Several promising models (e.g. D-brane inflation) of high energy physics inspired inflationary scenarios terminate with the production of topological defects. In order to investigate the observational implications of such models, [**G. Amery**]{} described a complete 4-d synchronous gauge formalism that may include non-zero curvature and cosmological constants, multiple scalar fields, and topological defects or other active sources [@amery]. [*The formalism provides a concise and geometrically sound description of energy-momentum conservation on all scales, from which appropriate initial conditions may be obtained*]{}. They present preliminary investigations of the possible contributions by active sources to the CMB data.
[**B. C. Paul**]{} presented a study of chaotic inflationary universe in a Brane world model [@paul]. They study the evolution of the universe with a minimally coupled self interacting scalar field when the kinetic energy dominates the potential energy and vice versa and obtain cosmological solutions which permits inflation. In four dimensional gravitation (GTR), the initial value of the inflaton field $\phi_{i} \gsim {\rm few}\, M_{4}$ required for a sufficient inflation is physically unrealistic. [*In contrast, they show that in brane world model sufficient inflation may be obtained even with an initial scalar field having value less than the usual four dimensional Planck scale.*]{}
[**H. K. Jassal**]{} reported on some cosmological consequences of the five dimensional, two brane Randall-Sundrum brane world scenario. The radius of the compact extra dimension is taken to be time dependent. Integrating over extra dimensions, the four dimensional action reduces to that of scalar tensor gravity. The radius of the extra dimension stabilizes to a nonzero separation of branes very quickly. A simple quadratic potential with minimum at zero leads to stabilization at comparable level but also allows for accelerated expansion. After stabilizations the potential does not play any other role except contributing the dark energy component at late times. [*It is shown that requirements for solving the hierarchy problem and getting an effective dark energy can be satisfied simultaneously.*]{}
Alternate views and ideas in Cosmology
======================================
If confirmed, the often discussed periodicity in the redshift distribution of quasars may not be readily explained in the standard Big Bang model. [**P. K. Das**]{} presented results that indicate that the Variable Mass Hypothesis scenario of Hoyle- Narlikar Theory redshift quantization can be invoked to explain any observed periodicity or quantization of quasar redshift distribution.
By considering thermodynamics of open systems in cosmology, Prigogine has proposed an irreversible matter creation process accompanied with large- scale entropy production. [**P. Gopakumar**]{} and G. V. Vijayagovindan discussed the application of this scenario in 3- brane world cosmology in 5-dimensions. The matter creation rate is found to affect the evolution of the scale factor both at high and low energy densities. In the standard brane world scenario, cooling is much slower than in the FLRW case, at high energy densities. With the matter creation in the brane world, the standard FLRW evolution is regained. As a consequence the temperature at the freezing-out of neutrons to protons ratio is the same as in the standard scenario.
[**M. Govender**]{} presented a simple method which generalizes the static isothermal universe first studied by Saslaw [*et al*]{} [@sas]. The cosmic fluid in the static model obeys a barotropic equation of state of the form $p= \alpha
\rho$ [@waghetal]. It has been argued that the isothermal cosmological model of Saslaw [*et al*]{} could represent the asymptotic state of the Einstein-de Sitter model. The generalized model could describe an isothermal sphere of galaxies in quasi-hydrostatic equilibrium with heat dissipation driving the system to equilibrium. A thermodynamical treatment within the framework of extended irreversible thermodynamics of the model is carried out [@maar97].
Acknowledgment {#acknowledgment .unnumbered}
==============
I thank the Scientific Organizing Committee of ICGC-04 for the opportunity to chair the ICGC-04 cosmology workshop. I would like to thank the local organizing committee for ensuring the smooth running of the workshop activities and a wonderful time at Kochi.
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---
abstract: 'Quantum enhanced sensing provides a powerful tool for the precise measurement of physical parameters that is applicable in many areas of science and technology. The achievable gain in sensitivity is largely limited by the influence of noise and decoherence. Here, we propose a paradigm of adiabatic quantum parameter amplification to overcome this limitation. We demonstrate that it allows to achieve generic robust quantum sensing, namely it is robust against noise that may even acting on the same degree of freedom as the field. Furthermore, the proposal achieves entanglement-free Heisenberg limit sensitivity that surpasses the limit of classical statistics.'
author:
- Yu Liu
- Zijun Shu
- 'Martin B. Plenio'
- Jianming Cai
title: Adiabatic quantum parameter amplification for generic robust quantum sensing
---
[*Introduction.—*]{} The emergent fields of quantum information science and quantum technologies are promising devices that make use of quantum properties to achieve performances that exceed what is possible in the realm of classical physics and promise considerable impact in physics, material science and biology. In this context, quantum sensing and metrology has attracted increasing interest because a relatively modest number of quantum systems under experimental control may already achieve a very useful enhancement of performance [@Gio2011]. Indeed, by allowing increasingly precise measurement of physical parameters and sensitive detection, this may enable new potential applications in a wide range of different areas. A great deal of effort has been invested over the last two decades to develop high precision quantum sensing and metrology protocols [@Wineland1992; @Choi08; @Roy08; @Napo11; @Boi07; @Gold11; @Cap12], to study their limitations imposed by environmental noise [@Esc11; @Huel97; @Boixo2008; @Kok12; @chin2012quantum; @Smir16; @Dur16] and to develop means to mitigate the deleterious effects of noise [@Preskill2000; @Macchiavello2002; @Chaves13; @Kotler11; @Dur14; @Kess14; @Arad14; @Lu14; @Kotler14; @Bau16; @unden2016quantum; @Dorner12; @Genes13; @Sza14; @Sun16].
A typical procedure of quantum metrology, e.g., the Ramsey method, uses a quantum particle that is subjected to a time evolution under a Hamiltonian that depends on a parameter that we wish to determine. This involves a sequence of interrogation cycles with the interrogation time duration $T$. As the measurement sensitivity scales as $1/\sqrt{T}$, it is beneficial to extend $T$ to its maximally possible value. This value is limited by the presence of noise. Alternatively, the sensitivity of metrology may be enhanced using N quantum probes that are prepared in a multipartite entangled states and then each subjected to the same time evolution [@Wineland1992; @Gio2011; @Boi07] thereby achieving an $N$ times more rapid accumulation of phase. Nevertheless, quantum entanglement particularly in maximally entangled state is quite fragile under environmental noise. It has been shown that noise would degrade or even completely eliminate the improvement in the scaling of precision [@Huel97], and thus hinder the implementation of Heisenberg limited quantum sensing. The application of active methods such as dynamical decoupling and quantum error correction in order to improve the sensitivity of quantum metrology against noise [@Macchiavello2002; @Kotler11; @Chaves13; @Dur14; @Kess14; @Arad14; @Lu14; @Kotler14; @unden2016quantum; @Sun16; @Bau16] aiming to reinstate the Heisenberg limit has received considerable attention over the years. Nevertheless, both dynamical decoupling and quantum error correction have limitations of their own. Dynamical decoupling based techniques work efficiently only for oscillating fields, while quantum error correction can only improve measurement sensitivity only under limited types of noise [@Sekatski2016]. Most importantly, it remains an open question how to achieve robust quantum parameter estimation of local Hamiltonians in the presence of parallel phase noise [@Macchiavello2002; @Dur14], e.g. for local Hamiltonian $H_s=b \sigma_z$ with noise $H_{noise}=\delta(t) \sigma_z $.
In this work, we propose a paradigm of adiabatic quantum metrology which consists of sensing and probe systems to determine a local parameter e.g. $H_s=b \sigma_z$. The sensing systems are adiabatically prepared into the ground state of a parameter-dependent local driven Hamiltonian. The energy gap protection makes it robust against noise that may even be acting along the $\hat{z}$-direction. By engineering an effective state-dependent interaction between the sensing systems and a probe system, the ground state encoded parameter information can be extracted. The interrogation time is mainly limited by the coherence time of the probe system which itself does not need to interact with the field that is to be measured. Hence it can be assumed to be unaffected by noise, e.g. originating from nuclear spin environments, may thus have a significantly longer coherence time. We demonstrate that it is possible to achieve Heisenberg limit scaling without involving entanglement as a resource in the sensing systems, and provide a new perspective concerning the role of entanglement in connection with quantum metrology [@Caves08; @Cze15; @Aug16]. The present idea of adiabatic quantum metrology is readily implementable in current state-of-art experiment, e.g. trapped ion and superconducting qubits, and may find application in a broad range of scenarios, ranging from the measurement of magnetic field and electric fields to that of forces. [*Adiabatic quantum parameter amplification.—*]{} We start from $N$ two-level quantum sensing systems whose eigenstates are denoted as $\{{\left\vert 0,1 \right\rangle}\}$. The $k$-th sensing system, in an interaction picture with $\omega_0 \sigma_z^{(k)}$ is governed by the following Hamiltonian as $$H_k(t) = b \sigma_z^{(k)} + \lambda_k(t)\sigma_x^{(k)}, \label{eq:local_ham}$$ where $\sigma_z^{(k)}={{\left\vert 0_k \right\rangle}{\left\langle 0_k \right\vert} } - {{\left\vert 1_k \right\rangle}{\left\langle 1_k \right\vert} }$, $\sigma_x^{(k)} = {{\left\vert 0_k \right\rangle}{\left\langle 1_k \right\vert} } + {{\left\vert 1_k \right\rangle}{\left\langle 0_k \right\vert} }$ are the corresponding Pauli operators, $\lambda_k(t)$ quantifies the strength of a time-dependent field (e.g. a laser acting on an atom/ion or a microwave driving field on a spin) applied to the k-th system via $\sigma_x^{(k)}$, and $b$ is the physical parameter that we would like to estimate. We remark that the parameter $b$ may also arise from the interaction with a magnetic dipole via e.g. dipole-dipole interaction, and thus the present idea can be extended to the detection of single atomic or nuclear dipole moments [@Kotler14]. The above setting arises naturally in the presence of non-Markovian environments as $\lambda_k(t)\sigma_x^{(k)}$ corresponds to applying a continuous drive to protect a qubit against noise [@Cai12; @Tim11]. It has be demonstrated that the lifetime of the eigenstates of the above Hamiltonian that we will exploit in our scheme can be dramatically prolonged [@Cai12].
![(Color online) Schematic diabatic quantum parameter estimation. The sensing system is adiabatically prepared into the ground state (AGP) of a local Hamiltonian which contains the interaction with a field quantified by the parameter $b$. The probe system couples with the sensing system via parameter-independent interactions that serves to amplify the field by engineering the interaction to be state dependent, and to extract the parameter information following a Ramsay interferometry procedure.[]{data-label="fig:model"}](setup_figure.pdf){width="8.5cm"}
Instead of preparing the $N$ systems into a specific state and then applying the parameter-dependent Hamiltonian followed by a final measurement, we encode the parameter information into the ground state of $N$ independent systems, then amplify and transfer the parameter information to the auxiliary probe system, see Fig.\[fig:model\]. To this end, we chose the initial value of the time-dependent parameter $\lambda_k(t)$ as $\lambda_k(0)\gg b$, and prepare the $k$-th system in the state ${\left\vert \psi_k(0) \right\rangle}\equiv {\left\vert {\downarrow}_x^{(k)} \right\rangle} = \sqrt{\frac{1}{2}}{\left( {\left\vert 0_k \right\rangle}-{\left\vert 1_k \right\rangle} \right)}$ that very closely approximates the ground state of the initial Hamiltonian $H_k(0)$. By slowly decreasing the field $\lambda_k(t)$ until $\lambda_k(T_a)=h$, we adiabatically prepare the system to the ground state of the Hamiltonian $H_k(T_a)=b\sigma_z^{(k)}+h\sigma_x^{(k)}$, namely $${\left\vert \psi_k(T_a) \right\rangle} = {\left\vert G_{k}(b,h) \right\rangle}\equiv -\sin(\frac{\theta}{2}){\left\vert 0_k \right\rangle}+\cos(\frac{\theta}{2}){\left\vert 1_k \right\rangle},$$ where $\cos\theta={b/\omega(b,h)}$ and $\sin\theta={h/\omega(b,h)}$ with $\omega(b,h)={\left( h^2+b^2 \right)}^{1/2}$. The ground state ${\left\vert G_k(b,h) \right\rangle}$ encodes the information on the value of the parameter $b$ albeit with a sensitivity that is reduced due to the additional field $h$ as compared with the standard Ramsey interferometry method. Moreover, the $N$ sensing systems are in a separable state, and thus independent measurements can only achieve standard quantum limit.
![(Color online) The dynamical decoupling scheme for the engineering of parameter-dependent effective interaction Hamiltonian. [**(a)**]{} The first order decoupling cycle consists of four decoupling gates equidistant in time. The dynamical decoupling gate $U_k=\mbox{diag}\{1,e^{i\theta(b)}\}$ with $\theta(b)\approx \pi$ is realized by the free evolution of the $k$-th sensing system for time $\tau_d={\pi/2\omega(b_0,h)}$, and $U_k^{\dagger}=\sigma_{y}^{(1)} U_k \sigma_{y}^{(1)}$ is realized by the same free evolution as sandwiched by $\hat{y}$-$\pi$ pulse $\sigma_y^{(k)}$. The effective Hamiltonian for such a cycle (as indicated in the dashed block) is given by $\tilde{H}_{ka}$. [**(b)**]{} The second order decoupling cycle is constructed by the concatenating strategy from four evolution cycles (blocks) by the first order decoupling.[]{data-label="fig:decoupling"}](decoupling_gate_scheme.pdf){width="8cm"}
{width="8cm"} {width="8cm"}
To enhance the sensitivity and achieve Heisenberg limit scaling, we introduce a probe system interacting with these $N$ adiabatically prepared systems via the Hamiltonian $$H_{ka}=-\gamma_k \sigma_z^{(k)}\otimes \sigma_z^{(a)},$$ where $\gamma_k$ is the interaction strength that is independent on the parameter $b$. By rewriting the above Hamiltonian in the basis of the $k$-th system $\{{\left\vert G_{k} \right\rangle},{\left\vert E_{k} \right\rangle}\}$, we obtain $$H_{ka}=\gamma_k \cos\theta \tilde{\sigma}_z^{(k)}\otimes \sigma_z^{(a)}
+\gamma_k \sin \theta \tilde{\sigma}_x^{(k)}\otimes \sigma_z^{(a)}, \label{eq:H12}$$ where the effective Pauli operators are defined as $\tilde{\sigma}_z^{(k)} = {\left( {{\left\vert G_{k} \right\rangle}{\left\langle G_{k} \right\vert} } - {{\left\vert E_{k} \right\rangle}{\left\langle E_{k} \right\vert} } \right)}$, and $\tilde{\sigma}_x^{(k)}={\left( {{\left\vert G_{k} \right\rangle}{\left\langle E_{k} \right\vert} }+{{\left\vert E_{k} \right\rangle}{\left\langle G_{k} \right\vert} } \right)}$. The first term $H_{k\rightarrow a}=\gamma_k \cos\theta \tilde{\sigma}_z^{(k)}\otimes \sigma_z^{(a)} $ in the above Hamiltonian represents a parameter-dependent effective field acting on the auxiliary system, which depends on the state of the $k$-th system (namely its ground state ${\left\vert G_k \right\rangle}$ or excited state ${\left\vert E_k \right\rangle}$). Thus, once the sensing systems are in the ground state of their local Hamiltonian, the effective Hamiltonian for the probe system becomes $$H_a^{(E)}(t)=\sum_k \gamma_k \cos\theta \sigma_z^{(a)} . \label{eq:H2E}$$ The second term $H_{ka}^{u}=\gamma_k \sin \theta \tilde{\sigma}_x^{(1)}\otimes \sigma_z^{(2)}$ in the Hamiltonian Eq.(\[eq:H12\]), however would lead to the transition of the $k$-th system between its ground state and excited state and would lead to deviations from the Hamiltonian in Eq.(\[eq:H2E\]) by introducing effective dephasing [@SI].
In order to eliminate such an unwanted effect, we devise a dynamical decoupling strategy and engineer an effective Hamiltonian that eliminates the effect of the second term in Eq.(\[eq:H12\]). We assume that the interaction described by Eq.(\[eq:H12\]) is switched off during the realization of the decoupling gate, which is achievable in a wide variety of physical system. The Hamiltonian of the $k$-th sensing system after the adiabatic preparation is $H_k(T_a) =\omega(b,h) \tilde{\sigma}_z^{(k)}$. As in quantum parameter estimation, the parameter is $b=b_0+\Delta b$, the value of $\Delta b$ is the quantity that we want to estimate precisely. The free evolution for time $\tau_d={\pi/2\omega(b_0,h)}$ leads to an unitary transformation $U_k=\mbox{diag}\{1,e^{i\theta(b_0)}\}$, in the basis of $\{{\left\vert G_k \right\rangle},{\left\vert E_k \right\rangle}\}$ where $\theta(b_0)= \pi+\delta_z$, with the decoupling gate error $\delta_z \approx \pi
{\left( {2b_0\Delta b} +{\Delta b^2} \right)}/2{\left( h^2+b_0^2 \right)}$ [@SI]. Such a unitary transformation $U_k(x)$ sandwiched by a $\hat{y}$ rotation $\sigma_{y}^{(k)}$ gives us $\sigma_{y}^{(k)} U_k (b) \sigma_{y}^{(k)} =U_k^{\dagger}(b)$. Up to the second order of the time interval $\tau$ between decoupling gates, according to the Baker-Campbell-Hausdorff relation, the effective interaction Hamiltonian under the dynamical decoupling sequence, see Fig.\[fig:decoupling\](a), can be written as [@SI] $$\tilde{H}_{ka}\approx \gamma_k \cos\theta \tilde{\sigma}_z^{(k)}\otimes \sigma_z^{(a)}
+ \delta_x \tilde{\sigma}_x^{(k)}\otimes \sigma_z^{(a)}, \label{eq:amp-Hamiltonian}$$ where we denote the quantity $\delta_x={\left( a_x^2+a_y^2 \right)}^{1/2}$ with $a_x=\gamma_k\sin\theta\sin^2{\left( {\delta_z}/{2} \right)}$, $a_y=-\frac{1}{2}\gamma_k^2\tau\cos\theta\sin\theta \cos^2{\left( {\delta_z}/{2} \right)} $. It can be seen that the undesirable second term in Eq.(\[eq:amp-Hamiltonian\]) arises from both the inaccuracy of dynamical decoupling protocol and the second order correction to the effective Hamiltonian. The influence of the former can be further suppressed by using a higher order dynamical decoupling sequences [@SI] which come at the price of a slightly increased time cost due to an increased number of decoupling gates, see Fig.\[fig:decoupling\](b). In our procedure, the number of decoupling gates increases by a factor of $5/4$ which represents a moderate time cost. Therefore, our dynamical decoupling procedure can engineer the effective interaction $\tilde{H}_a = \sum_k H_{k\rightarrow a} = \sum_k \gamma_k \cos\theta \tilde{\sigma}_z^{(k)}\otimes
\sigma_z^{(a)}$ as required for adiabatic parameter amplification. The total effective field acting on the probe system is $b_a=b\sum_k \gamma_k/\omega(b,h)$. Without loss of generality, hereafter we assume that $\gamma_k=\gamma$, and the field $b$ is amplified by a factor $\eta=N \gamma/\omega(b,h)$ which scales linearly with the number of sensing systems $N$.
[*Analysis of achievable sensitivity.—*]{} Once the $N$ sensing systems are prepared into their local ground state ${\left\vert G_k(b,h) \right\rangle}$, we use the probe system to measure the parameter information encoded in ${\left\vert G_k(b,h) \right\rangle}$ via the above engineered parameter-dependent effective interaction Hamiltonian. In order to estimate the parameter, we apply a Ramsey sequence, namely we initially prepare the probe system into a superposition state ${\left\vert \psi_a(0) \right\rangle} = \frac{1}{\sqrt{2}}
{\left( {\left\vert 0_a \right\rangle}+{\left\vert 1_a \right\rangle} \right)}$ by applying a $\pi/2$ pulse to the state ${\left\vert 0_a \right\rangle}$, after the interrogation time $T_s$, the state evolves to ${\left\vert \psi_a(T_s) \right\rangle} = \frac{1}{\sqrt{2}}{\left( {\left\vert 0_a \right\rangle} + e^{i b_a T_s}{\left\vert 1_a \right\rangle} \right)}$. To extract the information about the unknown parameter, we apply a $\pi/2$ pulse and measure the observable $\hat{P}={{\left\vert 0_a \right\rangle}{\left\langle 0_a \right\vert} }$ of the probe system, see Fig.\[fig:efficiency\](a).
The total time $T$ for one experiment run includes the adiabatic ground state preparation time $T_a$ for the $N$ systems, and the actually time cost $\tilde{T}_s$ (accounting for the required time for the realization of the dynamical decoupling gates) for an effective interrogation time $T_s$. The time required to maintain the adiabatic condition during the state preparation $T_a={\left( 4b \epsilon_a \right)}^{-1}
\left[A/\sqrt{1+A^2}-c/\sqrt{c^2+1} \right]$ with the ratio $A = \lambda_k(0)/b $ and $c =h /b $, is obtained from the adiabatic condition ${\vert {\left\langle e(t) \right\vert}\dot{H}_{k}(t){\left\vert g(t) \right\rangle}\vert }/{\vert E_e -
E_g\vert ^2 }\equiv \epsilon_a \ll 1$, where ${\left\vert g_k(t) \right\rangle}$ and ${\left\vert e_k(t) \right\rangle}$ are the instantaneous eigenstates of the Hamiltonian $H_k(t)$, and $E_{g,e}=\pm {\left[ \lambda_k(t)^2+b ^2 \right]}^{1/2}$ are the corresponding eigenenergies. The small quantity $ \epsilon_a$ characterizes how well the adiabatic condition is fulfilled. The adiabatic ground state fidelity is $1-\delta \approx 1-e^{-T_a\Delta/v}$ [@Landau32; @Zener32], where $v$ is the relative slope of the energy levels and $\Delta$ is the minimal energy gap between the ground and excited state. An effective interrogation time $T_s$ will require a realization time $\tilde{T}_s=T_s{\left( 1+\tau_d/\tau \right)}$, where $\tau$ is the time interval between dynamical decoupling gates. Therefore, the total time for one experiment run is $T=T_a+T_s{\left( 1+\tau_d/\tau \right)}$. The achievable shot-noise limited sensitivity for the estimation of the parameter $b$ is given by $\delta b=\sqrt{\langle \Delta^2 \hat{P} \rangle}/
{\left( \frac{\partial}{\partial b}{\langle \hat{P} \rangle} \sqrt{1/T} \right)}$. In the absence of noise and under the assumption that $h\gg b$, we obtain the achievable sensitivity as [@SI] $$\delta b \approx \frac{1}{N}{\left( \frac{h} {\gamma} \right)} \frac{\sqrt{T}}{{T_s}} =
\frac{1}{N}{\left( \frac{h} {\gamma} \right)} \frac{\sqrt{T_a+T_s{\left( 1+\tau_d/\tau \right)}}}{{T_s}}$$ It can be seen that the sensitivity reaches a Heisenberg limit scaling $ \delta b \sim 1/N$, although the parameter amplification is accompanied by the extra time cost required for the realization of dynamical decoupling gates leading to an N-independent additional factor. As compared with standard quantum limit ${\left( \delta b \right)}_{SQL}=1/\sqrt{N T}$, the enhancement factor is $\delta b / {\left( \delta b \right)}_{SQL} ={\left( h/\sqrt{N}\gamma \right)}{\left( T/T_s \right)}$, which is more pronounced as the value of $N$ increases, see Fig.\[fig:efficiency\](b). The additional advantage of the present proposal comes from the suppression of the effect of non-Markovian noise. In a standard quantum limited scheme, the interrogation time will be limited to $T_2^*$. While in the present scheme, the interrogation time is limited by the lifetime of the sensing systems (as protected by the energy gap) and the coherence time of the probe system. The probe system only needs to couple with the sensing system and may decouple from noise, thus the interrogation time (which may be further extended using decoherence free subspace [@Dorner12; @Kotler14]) would be much longer than $T_2^*$ of the sensing system.
The implementation of adiabatic quantum parameter estimation requires the capability of tuning driving field and engineering Hamiltonian, which is feasible in current state-of-art experiment setups. For example, we note that both analog adiabatic quantum simulation and digital quantum simulation, demonstrating these experimental capabilities, has achieved considerable progress in several types of physical systems. In particular, the techniques for the engineering of various spin-spin interactions and the coherent manipulation of ion spin state with a high fidelity have been very well developed for trapped ions [@Blatt12; @Boh16].
[*Effect of imperfection and entanglement.—*]{} We provide a detailed analysis of the two main sources of imperfect coherent control in the above protocol. In the adiabatic ground state preparation, due to the finite preparation time, the final state of the $k$-th system $\psi_k(T)=\sqrt{1-\delta}{\left\vert G_k \right\rangle}+\sqrt{\delta }{\left\vert E_k \right\rangle}$ is not the exact ground state and results in a ground state fidelity $1-\delta$ ($\delta\ll 1$). The excitation in the sensing systems leads to an effective dephasing in the probe system. The best achievable sensitivity under the influence of the imperfect ground state preparation,choosing the interrogation time $T_s$ such that ${(\gamma/h) bT_s}=k\pi$ with $k\in \mbox{odd}$, is found to be the same as the ideal case, namely $ \delta b \approx \frac{1}{N}{\left( \frac{h} {\gamma} \right)} \frac{\sqrt{T}}{{T_s}}$ [@SI]. The second imperfection lies in the high order corrections to the engineered Hamiltonian that we obtained via the the dynamical decoupling procedure shown in Eq.(\[eq:amp-Hamiltonian\]). The transformation of the probe system due to the interaction of the $k$-th system is described by a completely positive map $\mathcal{M}(\rho_a) = (1-\delta')U_z \rho_a U_z^{\dagger}+\delta' \sigma_z \rho_a \sigma_z$, where $U_z=\exp{(-iT_s \gamma\cos\theta {\sigma}_z)}$ is the ideal evolution, and $\delta'\lesssim \delta_x^2/
{\left( \gamma^2 \cos^2\theta \right)}$ [@SI]. Therefore, the final state of the probe system accounting for the high order corrections is $\rho_a(T_s)=\otimes_j \mathcal{M}_j(\rho_a(0))$, which leads to the best achievable sensitivity as $ \delta b \approx \frac{1}{N}{\left( \frac{h} {\gamma} \right)} \frac{\sqrt{T}}{{T_s}}$ under the same condition as ${(\gamma/h) bT_s}=k\pi$ with $k\in \mbox{odd}$. It can be seen that the two main sources of imperfect control generally leads to a reduction of the achievable sensitivity, but the reduction can be compensated for by choosing an appropriate interrogation time, and thus will not affect the achievable sensitivity.
Quantum entanglement appears unintentionally in the interrogation step due to the imperfect ground state preparation and due to the corrections to the ideal effective Hamiltonian. In standard quantum metrology, the role of entanglement in the initially prepared state has been carefully studied [@Cze15; @Aug16]. In the present scenario however, entanglement does not seem to play the role of quantum resource for parameter estimation. In general, the appearance of entanglement is accompanied with the reduction in the achievable sensitivity [@SI]. We choose the most suitable interrogation time $T_s$ that achieves the best sensitivity, entanglement instead disappears.
[*Conclusion.—*]{} In summary, we have proposed a paradigm of adiabatic quantum parameter estimation to achieve high measurement sensitivity in the presence of noise. In particular, the present proposal benefits from the energy gap protection and provides an efficient strategy for estimating local Hamiltonians against parallel phase noise. The techniques can be readily realized with the current state-of-art quantum technology, for example using trapped ions. We demonstrate that it allows to achieve Heisenberg limited measurement sensitivity without relying on quantum entanglement as a resource. Our proposal thus provides a platform to help elucidate the fundamental role of quantum entanglement in quantum metrology.
[*Acknowledgements.—*]{} We thank Prof. Ren-Bao Liu for the fruitful discussion. J.-M.C is supported by the National Natural Science Foundation of China (Grant No.11574103), the National Young 1000 Talents Plan. M.B.P is supported by an Alexander von Humboldt Professorship, and ERC Synergy grant and the EU project DIADEMS, EQUAM and QUCHIP.
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address: |
Technische Universität Berlin\
Institut für Mathematik, MA 7-4\
Stra[ß]{}e des 17. Juni 136\
10623 Berlin\
Germany\
author:
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title: Asymptotic Support Theorem for
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Acknowledgement {#acknowledgement .unnumbered}
===============
The present research was supported by the International Research Training Group [*Stochastic Models of Complex Processes*]{} funded by the German Research Council (DFG). The author gratefully thanks Michael Scheutzow and Holger van Bargen from TU Berlin for fruitful discussions.
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abstract: 'Distance sampling is a widely used method for estimating wildlife population abundance. The fact that conventional distance sampling methods are partly design-based constrains the spatial resolution at which animal density can be estimated using these methods. Estimates are usually obtained at survey stratum level. For an endangered species such as the blue whale, it is desirable to estimate density and abundance at a finer spatial scale than stratum. Temporal variation in the spatial structure is also important. We formulate the process generating distance sampling data as a thinned spatial point process and propose model-based inference using a spatial log-Gaussian Cox process. The method adopts a flexible stochastic partial differential equation (SPDE) approach to model spatial structure in density that is not accounted for by explanatory variables, and integrated nested Laplace approximation (INLA) for Bayesian inference. It allows simultaneous fitting of detection and density models and permits prediction of density at an arbitrarily fine scale. We estimate blue whale density in the Eastern Tropical Pacific Ocean from thirteen shipboard surveys conducted over 22 years. We find that higher blue whale density is associated with colder sea surface temperatures in space, and although there is some positive association between density and mean annual temperature, our estimates are consitent with no trend in density across years. Our analysis also indicates that there is substantial spatially structured variation in density that is not explained by available covariates.'
address:
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Address of Yuan Yuan\
School of Mathematics and Statistics\
University of St Andrews\
The Observatory, Buchanan Gardens\
St Andrews, UK\
KY16 9LZ\
- |
Address of Fabian E. Bachl\
School of Mathematics\
University of Edinburgh\
James Clerk Maxwell Building\
The King’s Buildings\
Peter Guthrie Tait Road\
Edinburgh, UK\
EH9 3FD\
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Address of Finn Lindgren:\
School of Mathematics\
University of Edinburgh\
James Clerk Maxwell Building\
The King’s Buildings\
Peter Guthrie Tait Road\
Edinburgh, UK\
EH9 3FD\
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Address of David L. Borchers\
School of Mathematics and Statistics\
University of St Andrews\
The Observatory, Buchanan Gardens\
St Andrews, UK\
KY16 9LZ\
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Address of Janine B. Illian\
School of Mathematics and Statistics\
University of St Andrews,\
The Observatory, Buchanan Gardens\
St Andrews, UK\
KY16 9LZ\
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Address of Stephen T. Buckland\
School of Mathematics and Statistics\
University of St Andrews\
The Observatory, Buchanan Gardens\
St Andrews, UK\
KY16 9LZ\
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Address of H[å]{}vard Rue\
CEMSE Division\
King Abdullah University of Science and Technology\
Thuval 23955-6900\
Saudi Arabia\
- |
Address of Tim Gerrodette\
NOAA National Marine Fisheries Service\
Southwest Fisheries Science Center\
8901 La Jolla Shores Drive\
La Jolla, California 92037, USA\
author:
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bibliography:
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title: 'Point process models for spatio-temporal distance sampling data from a large-scale survey of blue whales'
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Introduction {#sec:intro}
============
Distance sampling is a widely-used set of survey methods for estimating animal density or abundance [@Buckland+al:2001; @Buckland+al:2015book]. Conventional distance sampling methods (of which line transect and point transect methods are the most common) use a combination of model-based inference for estimating detection probability and design-based inference with Horvitz-Thompson-like estimators [@Borchers+al:1998] for estimating density and abundance conditional on the detection probability estimates. While the design-based nature of the second stage in this two-stage estimation process [see @Buckland+al:2016] confers robustness on density and abundance estimates when suitable designs are used, it severely restricts the spatial resolution at which such estimates can be obtained. This is because design-based inference requires adequate sampling units (strips for line transect surveys and circular plots for point transect surveys) in each area for which animal density or abundance is to be estimated. The low spatial resolution of estimates from this two-stage approach limits the utility of estimates obtained from conventional distance sampling methods as there is often interest in the distribution at high spatial resolution. As a result, there has been increasing interest in distance sampling methods that generate continuous spatial density surface estimates, and hence allow inference at an arbitrarily fine spatial scale.
In this paper, we consider a series of line transect surveys of blue whales ([*Balaenoptera musculus*]{}) in the Eastern Tropical Pacific Ocean [ETP, @Gerridette+Forcada:2005], in which the focus of inference is on how density changes continuously in space, with respect to available explanatory variables, and across years. The surveys were designed for dolphins, not blue whales, so there are relatively few blue whale sightings. A continuous spatial model has the potential to borrow strength from data outside the lightly-sampled strata to improve overall inference.
One can obtain a continuous density model by using a spatial model of density in the second stage, rather than basing inference on the design in this stage. This is usually done by transforming the data to counts: discretizing the sampled strips into smaller spatial units in the case of line transects and specifying a model for the counts within each unit, using estimated detection probability as an offset to correct the counts for detectability. @Hedley+al:2004 and @Hedley+Buckland:2004 pioneered this approach and @Niemi+Fernandez:2010 developed a similar approach (but ignoring detection uncertainty). The -package `dsm` [@Rpack:dsm] implements the approach of @Hedley+al:2004 and @Hedley+Buckland:2004 using generalized additive models [GAMs, @Wood:2006] to estimate a density surface from the count data. Either frequentist or Bayesian approaches can be used for the second stage [@Oedekoven+al:2013; @Oedekoven+al:2015], and bootstrapping is often used to propagate the uncertainty of detectability estimated from the first stage. @Williams+al:2011 use a more direct approach to incorporate uncertainty of detectability: a random effect term is added in the second stage to characterize the uncertainty in the estimation of the detection function from the first stage.
One can also estimate the parameters of the detection function and the count model simultaneously [@Royle+Dorazio:2008; @Royle+al:2004; @Johnson+al:2010; @Moore+Barlow:2011; @Schmidt+al:2012; @Conn+al:2012; @Oedekoven+al:2014; @Pardo+al:2015]. This is known as a full-likelihood approach, as it involves specifying a likelihood that incorporates both a detection function model and a spatial density model, allowing simultaneous estimation of both models.
Whether inference is in two stages or one, models that discretize searched strips or lines involve an element of subjectivity in choosing the size of the discrete units and a loss of spatial information because each discrete unit can have only one value of any spatial covariate attached to it, even though it might span an area incorporating a range of covariate values. In this paper, we develop a method that does not suffer from these problems, using a point process model.
Point process theory provides a flexible modeling framework for incorporating the underlying spatial or spatio-temporal stochastic processes and does not require discretization of spatial sampling units. Point process models have been used with ecological data to estimate smooth spatial density surfaces and are an obvious choice for the spatial model component of a full likelihood line transect model, although to date they have mainly been used in ecological applications with fully mapped point patterns [@Wiegand+Moloney:2014]: @Stoyan:1982 formulated line transect data as observations of stationary point processes; @Hedley+al:2004 considered point process models for point transect surveys, and @Hogmander:1991 [@Hogmander:1995] constructed a marked point process model for line transect data with detection probability of an animal treated as a mark, but they used a detection model (in which each animal has a detection circle with variable radius) that was shown by @Hayes+Buckland:1983 to be unrealistic and often resulting in biased inference.
Here we develop a full likelihood point process model for line transect data, in which the detection process thins the underlying point process, and in which the detection model and the point process model are estimated simultaneously. In the context of the blue whale survey, an unknown point process governs the number and locations of the whales in space, and points are thinned (whales missed) with a probability that depends in an unknown way on distance from the known locations of lines. Such an approach is not new for modeling distance sampling data. The -package `DSpat` [@Rpack:DSpat; @Johnson+al:2010] uses a thinned point process model for line transect survey data. However, their method assumes the absence of residual spatial structure on the intensity level (whale density in our case), which is usually not the case in practice, and may result in biased estimates. We relax the independence assumption by using the stochastic partial differential equation approach [SPDE, @Lindgren+al:2011] to incorporate a spatial or spatio-temporal random field for the underlying stochastic process of autocorrelated spatial or spatio-temporal random effects. For point process data in general, the SPDE approach avoids the need to aggregate observations [@Simpson+al:2015], and it provides a flexible modeling framework for spatio-temporal random fields. We build our models in a Bayesian framework, which gives us a tool for fitting complicated models, and the advantage of being able to use integrated nested Laplace approximation [INLA, @Rue+al:2009] for inference. INLA is a computationally efficient method for Bayesian inference using numerical approximations instead of a sampling-based method such as Markov chain Monte Carlo algorithms. In addition, our modeling framework accommodates the models of the sort used by @Johnson+al:2010 as a special case, as well as the second stage of the two-stage approach of @Miller+al:2013.
After describing the blue whale survey data in Section \[sec:surveydata\], we describe our model and computational methods in Sections \[sec:modelsoverall\] and Section \[sec:computation\]. We then analyze the surey data in Section \[sec:casestudy\], investigating the underlying spatial stochastic process of blue whale density in this area, and how the blue whales respond to sea surface temperature in space and time. Finally, in Section \[sec:discuss\], we discuss the results of the analysis, the utility of our modeling approach and extensions for more complicated scenarios.
The blue whale survey data\[sec:surveydata\]
============================================
Line-transect cetacean surveys were carried out in the Eastern Tropical Pacific Ocean (ETP) between 1986 and 2007. Fig \[fig:effortsight\] shows the survey region and transect lines over this whole period, together with blue whale sightings. The survey area is 21.353 million square kilometres and is large enough that the curvature of the earth needs to be taken into account in the analysis. A total of 182 blue whale groups were sighted over all years, with a mean group size of 1.8 (standard deviation 2.1). In 1986-1990, 1998-2000, 2003 and 2006, the entire ETP area of was sampled. These complete surveys required two oceanographic research vessels (3 in 1998) for 120 sea days each. Transect search effort was stratified by area [@Gerridette+Forcada:2005], and in 1992, 1993 and 2007, only part of the ETP area was sampled. These spatial differences in intensity of sampling need to be accounted for in modelling (see Section \[sec:numericalintegration\] for more detail). Data collection followed standardized line-transect protocols [@Kinzey+al:2000]. Briefly, in workable conditions, a visual search for cetaceans was conducted by a team of three observers on the flying bridge of each vessel during all daylight hours as the ship moved along the transect at a speed of 10 knots. Pedestal-mounted 25X binoculars were fitted with azimuth rings and reticles for angle and distance measurements. If a blue whale sighting was less than $10$ km from the transect, the team went off-effort and directed the ship to leave the transect to approach the sighted animal(s). The observers identified the sighting to species or subspecies (if possible) and made group-size estimates.
The inference problem we address is how to model the density of blue whale groups across this survey region in a way that takes account of (i) the variable survey effort (transect lines) in space, (ii) the unknown probability of detecting a group from a line, with detection probability decreasing with distance from line, (iii) the dependence of density on explanatory variables (sea surface temperature in particular), (iv) how density changes over years, and (v) spatial fluctuation in blue whale density that cannot be explained by any available explanatory variables.
We describe the statistical models and tools that we use to address this inference problem next, and then use these to address the blue whale inference problem.
The models {#sec:modelsoverall}
==========
Spatial point processes model the locations of objects in space [@Stoyan+Grabarnik:1991; @Lieshout:2000; @Diggle:2003; @Moller+Waagepetersen:2004; @illianBook]. Before incorporating distance sampling, we consider spatial point patterns formed by objects, represented as collections of locations, ${\mbox{\boldmath $Y$}}\equiv\{{\mbox{\boldmath $s$}}_i,\, i = 1, \dots, n\}$. The point set ${\mbox{\boldmath $Y$}}$ is considered as a realisation from a random point process on a bounded domain $\Omega$, where usually $\Omega\subset{\mathbb{R}}^2$. Since the ETP survey domain is large enough for the curvature of the Earth to matter (see Fig \[fig:effortsight\]), we treat $\Omega$ as a subdomain of a sphere, $\Omega\subset\mathbb{S}^2$.
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![Plot of the ETP data: the left panel shows the transect lines surveyed from 1986 to 2007, and the right panel displays the sightings of blue whale groups (red dots) on top of the mesh used in our analysis: the radius of each dot is proportional to the logarithm of the group size plus 1. The red line is the ETP survey region boundary. []{data-label="fig:effortsight"}](plotETPeffort86to07.pdf "fig:"){width="40.00000%"} ![Plot of the ETP data: the left panel shows the transect lines surveyed from 1986 to 2007, and the right panel displays the sightings of blue whale groups (red dots) on top of the mesh used in our analysis: the radius of each dot is proportional to the logarithm of the group size plus 1. The red line is the ETP survey region boundary. []{data-label="fig:effortsight"}](Sight86to07w6_bwhales_redfilled.pdf "fig:"){width="40.00000%"}
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Spatial hierarchical Poisson point process models {#sec:hierarchical}
-------------------------------------------------
For any subset $A\subseteq\Omega$, the number of objects in $A$ is denoted $N_{{\mbox{\boldmath $Y$}}}(A)$. For an inhomogeneous point process, we define an intensity function $\Lambda({\mbox{\boldmath $s$}})$ as $$\begin{aligned}
\Lambda({\mbox{\boldmath $s$}}) &= \lim_{\epsilon\rightarrow 0}
\frac{
{\textrm{E}}\{N_{{\mbox{\boldmath $Y$}}}[\mathcal{B}_\epsilon({\mbox{\boldmath $s$}},t)]\}
}{|\mathcal{B}_\epsilon({\mbox{\boldmath $s$}},t)|},\quad {\mbox{\boldmath $s$}}\in\Omega,\end{aligned}$$ where $\mathcal{B}_\epsilon({\mbox{\boldmath $s$}})$ is a ball of radius $\epsilon$ centered at ${\mbox{\boldmath $s$}}$. For all non-overlapping subsets $A_1,\dots,A_m\subset\Omega$, an inhomogeneous Poisson point process has the following two conditions, $$\begin{aligned}
&N_{{\mbox{\boldmath $Y$}}}(A_k) \sim {\textrm{Po}}\left[\int_{A_k} \Lambda({\mbox{\boldmath $s$}}) {{\,{{\mathrm{d}}}}}{\mbox{\boldmath $s$}} \right],\quad \text{$k=1,\dots,m$, and}
\\
&\text{$N_{{\mbox{\boldmath $Y$}}}(A_1),\dots,N_{{\mbox{\boldmath $Y$}}}(A_m)$ are mutually independent.}\end{aligned}$$ Finally, we let $\Lambda({\mbox{\boldmath $s$}})$ be a random process, and define the point pattern model conditionally on $\Lambda({\mbox{\boldmath $s$}})$. The conditional likelihood for the entire point pattern ${\mbox{\boldmath $Y$}}$, relative to a homogeneous Poisson process with intensity $1$, is given by $$\begin{aligned}
\label{eq:lgcplikelihood}
\pi(\mathbf{Y} | \Lambda) &= \exp\left( |\Omega| - \int_{\Omega} \Lambda({\mbox{\boldmath $s$}}) {{\,{{\mathrm{d}}}}}{\mbox{\boldmath $s$}}\right) {\prod_{i=1}^{N_{{\mbox{\boldmath $Y$}}}(\Omega)}\Lambda({\mbox{\boldmath $s$}}_i)},\end{aligned}$$ where ${\mbox{\boldmath $s$}}_i$ is the location of the $i$th observation. If $\log\Lambda({\mbox{\boldmath $s$}})$ is modeled by a latent Gaussian linear model, the resulting hierarchical model is a doubly-stochastic log-Gaussian Cox process [@Moller+Waagepetersen:2004].
Point process models in the context of distance sampling {#sec:PPmodelsDist}
--------------------------------------------------------
For wildlife surveys, only a proportion of the population in the domain of interest is observed, due to partial sampling of the domain, and failure to detect all animals in the sampled regions. Distance sampling provides a method to account for imperfect detection. In line transect surveys, an observer traces a path through space, searching a strip centered on the path. The probability of detecting an object typically decreases with distance from the observer. From a modeling perspective, this results in a *thinned* spatial point process with the intensity function scaled by the detection probability [@Dorazio:2012; @Johnson+al:2013; @Hefley+Hooten:2016].
When deriving the appropriate likelihood model for an observed point pattern, the problem-specific underlying generative structure influences the potential dependence between point locations both over space and in time. It is therefore important to note that the thinning in transect surveys is neither a thinning of a fixed spatial point pattern, nor a thinning of a regular spatio-temporal point process. Instead, each object is characterised by a temporally evolving curve in space, describing its movement, and the observations are thinned snapshots of time-slices of the resulting point process of curves. In addition, the intensity may vary over time, and we write $\lambda({\mbox{\boldmath $s$}};t)$ for the spatial point intensity for the full time-slice point pattern at time $t$, and $\Lambda({\mbox{\boldmath $s$}};t)$ for the intensity of the observationally thinned version. The assumptions about the movements of the observer and the objects affect what approximations are allowed in practical calculations.
Traditionally, the detection probability for an object located at a given perpendicular distance $z$ from the path of the observer is modeled by a *detection function* $g(z)$. Assuming that the observer is moving with constant speed $v$ along a straight line, standard Poisson process theory yields the probability of detecting an object located at ${\boldsymbol{s}}_0$ as a function of the perpendicular distance $z({\boldsymbol{s}}_0)$, $$\begin{aligned}
{\textrm{P}}(\text{object at ${\mbox{\boldmath $s$}}_0$ detected} \mid {\mbox{\boldmath $s$}}_0\in{\mbox{\boldmath $Y$}}) &=
1 - \exp\{- h[z({\mbox{\boldmath $s$}}_0)]/v\}
=
g(z({\mbox{\boldmath $s$}}_0),v),\end{aligned}$$ where $h(\cdot)$ is an aggregated detection *hazard* along the path, and $g(\cdot,\cdot)$ is the aggregated detection function, with explicit dependence on $v$. The standard approach is to model either the aggregated detection probability $g(z,v)$, or the aggregated hazard $h(z)$. Under simple assumptions about the observers, @Hayes+Buckland:1983 derived the commonly-used *hazard-rate model*, given by $h(z)=-(z/\sigma)^{-b}$, $b,\sigma>0$. The half-normal detection function $g(z)=\exp\left[-z^2/(2\sigma^2)\right]$, $\sigma > 0$ is another widely-used model. While the hazard-rate model is more flexible than the half-normal detection model, only the latter results in a log-linear probability model. For this reason, the hazard-rate model does not fit directly into the existing INLA estimation software [@Rue+al:2009], and instead we use a semi-parametric detection model, which we introduce in Section \[sec:positivegx\], to give us a more flexible model than the half-normal.
Line transect point process likelihood {#sec:linetransectlikelihood}
--------------------------------------
For line transects, assuming that environmental and other observational conditions that might affect detectability remain constant along suitably short and straight *transect segments*, we can formulate a tractably simple version of the likelihood. The region of space swept by the transect path is assumed to consist of a sequence of rectangular transect segment strips $\{{\mathcal{C}}_1,\dots,{\mathcal{C}}_K\}$, so that ${\mathcal{C}}_{k(t)}$ is the transect strip at time $t$. Writing $\lambda({\boldsymbol{s}};t)$ for the intensity of potentially observable objects, and introducing transect-dependent detection functions $g_{k(t)}({\mbox{\boldmath $s$}})$, the intensity for the thinned observational point process is $\Lambda({\mbox{\boldmath $s$}};t)=\lambda({\mbox{\boldmath $s$}};t) g_{k(t)}({\mbox{\boldmath $s$}})$. Under some loose assumptions (see Supplement \[appx:LTassumptions\]), the joint conditional likelihood for the observed point pattern is the product of the conditional likelihoods for each individual transect segment, $$\begin{aligned}
\label{eq:likelihood}
\pi({\mbox{\boldmath $Y$}} | \Lambda) &=
\exp\left( \sum_{k=1}^K |{\mathcal{C}}_k| -
\sum_{k=1}^K \int_{{\mathcal{C}}_k} \Lambda({\mbox{\boldmath $s$}}; t_{{\mathcal{C}}_k})
d{\mbox{\boldmath $s$}}\right) {\prod_{i=1}^{N_{{\mbox{\boldmath $Y$}}}} \Lambda({\mbox{\boldmath $s$}}_i; \,t_i)},\end{aligned}$$ where $N_{{\mbox{\boldmath $Y$}}}=\sum_{k=1}^K N_{{\mbox{\boldmath $Y$}}}({\mathcal{C}}_k)$ is the total number of observed objects, located at $({\mbox{\boldmath $s$}}_i,t_i)$, $i=1,\dots,N_{{\mbox{\boldmath $Y$}}}$. We do not specifically address the issue of *marks* here (features or quantities associated with detected groups or animals). Marks that do not affect the detection probability can be modeled alongside the object intensity $\lambda({\mbox{\boldmath $s$}};t)$, including possible common fixed effects and dependent random effects [@Illian+al:2012]. However, marks that do affect the detection probability, such as the sizes of groups of animals, require a joint likelihood expression for the extended dimension point process of object locations and their marks, which is a topic for further development.
A Bayesian hierarchical spatio-temporal point process model {#sec:model}
-----------------------------------------------------------
Following the classical approach for log-Gaussian Cox processes, we let the logarithm of the intensity $\lambda({\mbox{\boldmath $s$}};t)$ be a Gaussian process, with linear covariates ${\mbox{\boldmath $x$}}({\mbox{\boldmath $s$}}, t)$, and a zero mean additive Gaussian spatial or spatio-temporal random field $\xi({\mbox{\boldmath $s$}},t)$ [@Moller+al:1998; @Moller+Waagepetersen:2004; @Moller+Waagepetersen:2007]. For computational efficiency, we use the INLA method for numerical Bayesian inference with Gaussian Markov random fields [@Rue+al:2009; @Illian+al:2012; @Simpson+al:2012], but the general methodology is not tied to a specific inferential framework.
In the likelihood given by (\[eq:likelihood\]), the log of the thinned intensity is given by $$\begin{aligned}
\label{eq:logLambda}
\log[\Lambda({\mbox{\boldmath $s$}}; t)]=\log[\lambda({\mbox{\boldmath $s$}}; t)] + \log [g_{k(t)}({\mbox{\boldmath $s$}})]={\mbox{\boldmath $x$}}({\mbox{\boldmath $s$}}, t)^\top{\mbox{\boldmath $\beta$}} + \xi({\mbox{\boldmath $s$}},t)+ \log [g_{k(t)}({\mbox{\boldmath $s$}})],\end{aligned}$$ where we assume Gaussian priors for ${\mbox{\boldmath $\beta$}}$, and a Gaussian random field $\xi$. If the logarithm of the detection probability model is linear in its parameters, this results in a joint linear model with latent Gaussian components.
In general, any link function and spatially coherent linear predictor could be used for $\Lambda$. The point process likelihood only requires $\Lambda$ to be well-defined pointwise, and integrable. In practice, the numerical integration schemes used for practical likelihood evaluation (see Section \[sec:computation\]) require piecewise continuity and differentiability. Covariates affecting $\lambda({\mbox{\boldmath $s$}};t)$ need to be available throughout the transect region for parameter inference, and throughout the domain of interest for spatial prediction. For practical implementation reasons, spatial covariates are projected onto the same computational function space as the latent field $\xi$ (see Section \[sec:SPDEcomputation\]). Covariates affecting $g_k({\mbox{\boldmath $s$}})$ need to be available for each transect segment. Within-segment variation in detectability would require a more expensive numerical integration scheme in Section \[sec:SPDEcomputation\], equivalent to splitting segments until they were sufficiently short for our assumption of constant detectability within each segment to be fulfilled. As noted at the end of Section \[sec:linetransectlikelihood\], marks for individuals are currently only allowed if they do not affect the detection probability.
The full model is given by the following hierarchy, $$\begin{aligned}
\pi({\mbox{\boldmath $Y$}},\xi,{\mbox{\boldmath $\beta$}},{\mbox{\boldmath $\beta$}}_g,{\mbox{\boldmath $\theta$}}) &=
\pi({\mbox{\boldmath $Y$}}\mid\xi,{\mbox{\boldmath $\beta$}},{\mbox{\boldmath $\beta$}}_g,{\mbox{\boldmath $\theta$}})
\pi(\xi\mid{\mbox{\boldmath $\theta$}})
\pi({\mbox{\boldmath $\beta$}}\mid{\mbox{\boldmath $\theta$}})
\pi({\mbox{\boldmath $\beta$}}_g\mid{\mbox{\boldmath $\theta$}})
\pi({\mbox{\boldmath $\theta$}}),\end{aligned}$$ where ${\mbox{\boldmath $\beta$}}_g$ are parameters controlling the detection model, ${\mbox{\boldmath $\theta$}}$ are further model parameters, such as precision parameters for the latent Gaussian variables. Each of the prior densities $\pi(\xi\mid{\mbox{\boldmath $\theta$}})$, $\pi({\mbox{\boldmath $\beta$}}\mid{\mbox{\boldmath $\theta$}})$, $\pi({\mbox{\boldmath $\beta$}}_g\mid{\mbox{\boldmath $\theta$}})$, and $\pi({\mbox{\boldmath $\theta$}})$ are controlled by hyperparameters. Note that in some software packages, including INLA, the parameters ${\mbox{\boldmath $\theta$}}$ themselves are referred to as hyperparameters.
For given prior distributions, the goal is to compute the posterior densities for the latent variables, optionally with ${\mbox{\boldmath $\theta$}}$ integrated out: $$\begin{aligned}
\pi({\mbox{\boldmath $\theta$}}\mid{\mbox{\boldmath $Y$}})
&\propto
\left.
\frac{
\pi({\mbox{\boldmath $Y$}},\xi,{\mbox{\boldmath $\beta$}},{\mbox{\boldmath $\beta$}}_g,{\mbox{\boldmath $\theta$}})
}{
\pi(\xi,{\mbox{\boldmath $\beta$}},{\mbox{\boldmath $\beta$}}_g\mid{\mbox{\boldmath $Y$}},{\mbox{\boldmath $\theta$}})
}
\right|_{(\xi,{\mbox{\boldmath $\beta$}},{\mbox{\boldmath $\beta$}}_g)=(\xi^*,{\mbox{\boldmath $\beta$}}^*,{\mbox{\boldmath $\beta$}}_g^*)},
\\
\pi(\xi,{\mbox{\boldmath $\beta$}},{\mbox{\boldmath $\beta$}}_g\mid{\mbox{\boldmath $Y$}}) &=
\int
\pi(\xi,{\mbox{\boldmath $\beta$}},{\mbox{\boldmath $\beta$}}_g\mid{\mbox{\boldmath $Y$}},{\mbox{\boldmath $\theta$}})
\pi({\mbox{\boldmath $\theta$}}\mid{\mbox{\boldmath $Y$}})
{{\,{{\mathrm{d}}}}}{\mbox{\boldmath $\theta$}},\end{aligned}$$ where $(\xi^*,{\mbox{\boldmath $\beta$}}^*,{\mbox{\boldmath $\beta$}}_g^*)$ is an arbitrary latent variable state vector. In the INLA method [@Rue+al:2009] this is achieved approximately by replacing $\pi(\xi,{\mbox{\boldmath $\beta$}},{\mbox{\boldmath $\beta$}}_g\mid{\mbox{\boldmath $Y$}},{\mbox{\boldmath $\theta$}})$ with various Gaussian or near-Gaussian approximations, and integrating numerically over ${\mbox{\boldmath $\theta$}}$.
Stochastic PDE models {#sec:basicSPDE}
---------------------
The general model construction requires no particular assumptions on how the spatial or spatio-temporal random field $\xi({\mbox{\boldmath $s$}},t)$ is modeled or treated computationally. The only requirement is that the model can be written as a latent Gaussian random field in such a way that the model likelihood can be evaluated numerically. In the context of INLA, that means that we need to construct a Gaussian Markov random field representation of the continuous space process. The traditional approach is to discretize space into a lattice and count the number of sighted points in each lattice cell, but here we take an alternative approach that allows us to use the true sighting locations, and to let $\lambda({\mbox{\boldmath $s$}};t)$ vary continuously through space. The results from @Lindgren+al:2011 show how to take advantage of the connection between Gaussian Markov random fields of graphs and stochastic partial differential equations in continuous space. Some details of such models are given in Supplement \[appx:spde\] and the computational implications are discussed in Section \[sec:SPDEcomputation\].
Log-linear detection function models {#sec:positivegx}
------------------------------------
As noted in Section \[sec:PPmodelsDist\], the hazard-rate model is not a log-linear model, which means that estimating the parameters does not directly fall under the latent Gaussian model framework of the INLA estimation software [@Rue+al:2009]. In contrast, the half-normal detection model is $g_k({\boldsymbol{s}})=\exp\left[-z_k({\boldsymbol{s}})^2/(2\sigma^2_{g,k})\right]$, where $z_k({\mbox{\boldmath $s$}})$ is the perpendicular distance from ${\mbox{\boldmath $s$}}$ to the $k$th transect line segment, and $\sigma_{g,k}$ are scale parameters. This can be written in log-linear form as $\log\left[g_k({\boldsymbol{s}})\right]=\beta_{g,k} z^*_k({\boldsymbol{s}})$, where $z^*_k({\boldsymbol{s}})=-z_k({\boldsymbol{s}})^2/2$, and $\beta_{g,k}=1/\sigma^2_{g,k}$. To allow more flexibility within the log-linear framework we introduce a semi-parametric piecewise quadratic model for the logarithm of the detection function, based on a one-dimensional version of the SPDE in the previous section.
For ease of presentation, first assume that the detection probability is the same for all transects $k$, so that we can write $G[z({\boldsymbol{s}})]=-\log[g_k({\boldsymbol{s}})]$. The prior distribution for $G(z)$ is then defined by the spline-like stochastic differential equation $$\begin{aligned}
\gamma\frac{{{\mathrm{d}}}^2 G(z)}{{{\mathrm{d}}}z^2} &= \mathcal{W}(t),\quad t\in[0,z_{\text{max}}]\subset{\mathbb{R}}, \label{eq:detsde}\\
G(0)\;=\;0, &\;\;\;\;\;\;\;\;\;\;
\left.\frac{{{\mathrm{d}}}G(z)}{{{\mathrm{d}}}z}\right|_{z=0}\;=\;0,\label{eq:bnd}\end{aligned}$$ where $z_{\text{max}}$ is the maximal detection distance, $\gamma>0$ is a smoothness parameter, and $\mathcal{W}(t)$ is a white noise process. The boundary constraints ensure that the detection probability at distance $z=0$ is $1$, and that the probability is flat near $z=0$.
Let $(0,z_1,z_2,\dots,z_p)$ be breakpoints for piecewise quadratic B-spline basis functions $B_i(z)$ [@Farin:2002], such that $z_p=z_{\text{max}}$, with the simplest choice of breakpoints being $z_i=i z_{\text{max}} / p$, $i=0,1,\dots,p$. The non-parametric model for $G(z)$ can then be used to construct a finite dimensional model $$\begin{aligned}
G(z) &= \sum_{i=1}^p \beta_i B_i(z),
\label{eq:semiparGz}\end{aligned}$$ where the $p$ basis functions only include $B_i(\cdot)$ that fulfill the boundary conditions , as shown in Fig \[fig:detectionbasis\].
The joint multivariate Gaussian prior distribution for $(\beta_1,\dots,\beta_p)\sim{\textrm{N}}\left({\boldsymbol{0}},{\boldsymbol{Q}}_\beta^{-1}(\gamma)\right)$ is constructed with the same finite element technique that will be used for the spatial SPDE in Section \[sec:SPDEcomputation\]. For uniform breakpoint spacing, increasing $p$ will make this discrete model converge to the continuous domain model, but for finite $p$ the model is effectively a piecewise quadratic semi-parametric model.
Imposing a monotonicity constraint on $g(z)$ is possible by replacing the basis functions for $G(z)$ with increasing basis functions, and mandating positivity of the $\beta_i$ coefficients [@Ramsay1988]. However, because the latter is currently only implemented in INLA for independent $\beta_i$, this is restricted to small $p$, such as 2 or 3, since an independence prior would result in a non-smooth function for larger $p$. An alternative that is feasible when sampling from the posterior distribution is to simply reject all non-increasing samples of $G(z)$. If the data are informative, the smoothing from the prior can in practice be enough to yield monotonic estimates without including an explicit constraint.
Relaxing the assumption that the detection function is the same for all transects is most easily done by adding further linear terms to $\log g_k({\boldsymbol{s}})$ based on observed or constructed covariates that depend on $k$.
Computational methods {#sec:computation}
=====================
There are several practical considerations for evaluating the likelihoods and representing the random fields in such a way that large dense matrices can be avoided. In Section \[sec:SPDEcomputation\] we give a brief overview of the essentials for translating stochastic PDEs into manageable Gaussian Markov random fields (GMRF), and Section \[sec:numericalintegration\] presents a numerical integration scheme for the point process likelihood.
The stochastic partial differential equation approach {#sec:SPDEcomputation}
-----------------------------------------------------
The SPDE/GMRF approach works by replacing the continuous domain stochastic PDE model with a finite dimensional Gaussian Markov random field for basis function weights defined on a triangulation of the domain of interest, such that the sparse precision matrix leads to a good approximation of the continuous space SPDE solutions. Given a triangulation mesh (see the right panel of Fig \[fig:effortsight\] for the triangulation used for the ETP survey area), @Lindgren+al:2011 define a finite element representation [@Brenner+Scott:2007] of $\xi$ from , $$\begin{aligned}
\label{eq:fem}
\xi({\mbox{\boldmath $s$}}) = \sum_{j=1}^m w_j \phi_j({\mbox{\boldmath $s$}}),\end{aligned}$$ where $w_1, \dots, w_m$ are stochastic weights, and $\phi_j$, $j=1, \dots, m$, are deterministic piecewise linear basis functions defined for each node on the mesh: $\phi_j$ equals $1$ at mesh node $j$ and $0$ in all the other mesh nodes. The weight vector ${\mbox{\boldmath $w$}}\equiv(w_1, \dots, w_m)^\top$ is a GMRF with its Markovian properties defined by the mesh structure. It follows that ${\mbox{\boldmath $w$}}$ determines the stochastic properties of (\[eq:fem\]) and ${\mbox{\boldmath $w$}}$ is chosen in a way that the distribution of (\[eq:fem\]) approximates the distribution of the solution to the SPDE (\[eq:SPDE\]). As shown by @Lindgren+al:2011, for the SPDE in , the resulting weight distribution is ${\mbox{\boldmath $w$}}\sim{\textrm{N}}({\mbox{\boldmath $0$}},{\mbox{\boldmath $Q$}}(\tau,\kappa)^{-1})$, where the sparse precision matrix ${\mbox{\boldmath $Q$}}(\tau,\kappa)$ is a polynomial in the parameters $\tau$ and $\kappa$, and is obtained through finite element calculations.
The practical implication of this construction is that instead of directly using the covariances from , which results in dense covariance matrices and high computational cost, $\mathcal{O}(m^3)$, the SPDE/GMRF approach links the continuous and discrete domains in such a way that the computational cost is reduced to $\mathcal{O}(m^{1.5})$. The computational advantages of GMRFs [@Rue+Held:2005] is strengthened by using INLA for Bayesian inference [@Rue+al:2009]. For the case of fully observed log-Gaussian Cox point processes, the in-depth analysis by @Simpson+al:2015 of the combined approximation errors induced by the basis function expansion in combination with the likelihood approximation error shows that the resulting approximate posterior distribution is close to the true posterior distribution. Since the integration scheme in the following section is constructed in the same way, we do not include a detailed approximation error analysis here, but note that the SPDE/GMRF approximation is likely to be the largest source of approximation error. Point patterns are relatively uninformative about the latent intensity, which has the practical effect that the realizations of the fields in the posterior distribution are typically smoother than in directly observed process problems. Hence, the approximation error is very small as long as the triangle mesh edges are short compared with the spatial scales of the covariates and of the point pattern intensity variability.
Numerical point process likelihood evaluation {#sec:numericalintegration}
---------------------------------------------
Combining the general distance sampling point pattern likelihood with the log-linear model structure for $\Lambda({\mbox{\boldmath $s$}};t)$ from results in a log-likelihood for the observed point pattern, $$\begin{aligned}
\log \pi({\mbox{\boldmath $Y$}}\mid\lambda,g)=& \sum_{i=1}^{N_{{\mbox{\boldmath $Y$}}}} \left\{{\mbox{\boldmath $x$}}({\mbox{\boldmath $s$}}_i, t_i) ^\top {\mbox{\boldmath $\beta$}} +\xi({\mbox{\boldmath $s$}}_i,t_i) -\log[g_{k(t_i)}({\mbox{\boldmath $s$}}_i)]\right\} {\nonumber}\\
&- \sum_{k=1}^K\int_{{\mathcal{C}}_k}\exp\left\{{\mbox{\boldmath $x$}}({\mbox{\boldmath $u$}},t_k)^\top {\mbox{\boldmath $\beta$}} +\xi({\mbox{\boldmath $u$}},t_k) -\log[g_{k}({\mbox{\boldmath $u$}})]\right\} {{\,{{\mathrm{d}}}}}{\mbox{\boldmath $u$}}
+ \sum_{k=1}^K |{\mathcal{C}}_k|
,
\label{eq:filterPPP}\end{aligned}$$ where the first term evaluates the log-intensity at the observed locations, and the second term integrates the intensity over the sampled transect segments. The log-likelihood (\[eq:filterPPP\]) is in general analytically intractable as it requires integrals of the exponential of a random field. Therefore, we use numerical integration to evaluate (\[eq:filterPPP\]), and the remainder of this section describes an integration scheme to approximate the integrals efficiently. As noted in Section \[sec:model\], we assume that the covariates $x({\mbox{\boldmath $s$}},t)$ are expressed using the same piecewise linear basis functions as $\xi$. For cases where a covariate has a much finer resolution than the one needed for $\xi$, the efficient integration scheme developed here is not appropriate, and further research is needed to develop an integration method that can deal with that without incurring a high computational cost.
For distance sampling surveys, transect areas describe subsets of the earth’s surface. The most natural representation of transect areas would therefore be subsets ${\mathcal{C}}_k\subseteq\mathbb{S}^2$ of the sphere, leading to surface integration in the Poisson process likelihood. However, the small scale at which earthbound observers are capable of probing their environment lends itself to easily justifiable simplifications of the numerical integration. Apart from environmental conditions such as the weather, the curvature of the earth puts an upper bound on the distance at which an observer with a given elevation can actually detect an animal. We therefore approximate the surface integrals over ${\mathcal{C}}_k\subset\mathbb{S}^2$ by integrals over ${\widetilde{{\mathcal{C}}}}_k\subset{\mathbb{R}}^2$, $$\begin{aligned}
I_{{\mathcal{C}}_k} &= \int_{{\mathcal{C}}_k}{\lambda}({\mbox{\boldmath $u$}};t_k) g_k({\mbox{\boldmath $u$}}) {{\,{{\mathrm{d}}}}}{\mathcal{C}}_k({\mbox{\boldmath $u$}})\;=\; \int_{{\widetilde{{\mathcal{C}}}}_k}{\lambda}({\mbox{\boldmath $u$}}_k(l,z);t_k) g_k({\mbox{\boldmath $u$}}_k(l,z)) \left\| \frac{\partial {\mbox{\boldmath $u$}}_k}{\partial l} \times \frac{\partial {\mbox{\boldmath $u$}}_k}{\partial z} \right\| {{\,{{\mathrm{d}}}}}l {{\,{{\mathrm{d}}}}}z \\
&\approx \int_{{\widetilde{{\mathcal{C}}}}_k} {\lambda}({\mbox{\boldmath $u$}}_k(l, z);t_k) g_k({\mbox{\boldmath $u$}}_k(l,z)) {{\,{{\mathrm{d}}}}}l {{\,{{\mathrm{d}}}}}z,\end{aligned}$$ where we use a transect-specific parameterization ${\mbox{\boldmath $u$}}_k$ at coordinate $l$ along and distance $z$ to the transect line, respectively. If $R$ is the radius of the earth, then the Jacobian is $\cos(z/R)$, which gives an approximation error of a factor less than $5\cdot 10^{-6}$ even in the extreme and unrealistic case of an observer at $31$ metres above a calm sea looking at the horizon $20$ kilometres away.
Another fact that we can utilize is that the detection function $g$ does not depend on the position of the observer along the line but only on the distance of an observation from the line. Similarly, if the transect line is narrow compared to the spatial rate of change in the intensity function, we can substitute the evaluation of $\lambda$ by an evaluation at the center of the transect line, ${\widehat{z}}=0$ (see Fig \[fig:integration\_scheme\]). That is, $$\begin{aligned}
I_{{\mathcal{C}}_k} &\approx \int_{{\widetilde{{\mathcal{C}}}}_k} {\lambda}({\mbox{\boldmath $u$}}_k(l,{\widehat{z}});t_k) g_k({\mbox{\boldmath $u$}}_k({\widehat{l}},z)) {{\,{{\mathrm{d}}}}}l {{\,{{\mathrm{d}}}}}z,\end{aligned}$$ together with an arbitrary coordinate ${\widehat{l}}$ along the transect line.
In a standalone implementation, the integral could be written as a product of two one-dimensional integrals, and even evaluated exactly due to the log-linearity of the model. Unfortunately, the resulting structure cannot be expressed using only evaluations of products of $\lambda$ and $g$, which is a requirement imposed by the internal structure of the INLA implementation, so we do not use that approach here.
We can also make use of the fact that ${\lambda}$ lives on a mesh. If the mesh triangles are small enough, the log-linear function ${\lambda}(\cdot)$ is approximately linear within each triangle. By splitting a transect line ${\mathcal{C}}_k$ into segments ${\mathcal{C}}_{k,j},\,j \in 1,\dots,J$, each of which resides in a single triangle (see Fig \[fig:integration\_scheme\]), we obtain a Gaussian quadrature method of order one, $$\begin{aligned}
I_{{\mathcal{C}}_k} &\approx \sum_{j=1}^J\int_{{\widetilde{C}}_{k,j}} {\lambda}({\mbox{\boldmath $u$}}_{kj}(l,{\widehat{z}});t_k) g_k({\mbox{\boldmath $u$}}_{kj}({\widehat{l}},z)) {{\,{{\mathrm{d}}}}}l {{\,{{\mathrm{d}}}}}z
\;\approx\; \sum_{j=1}^J w_{k,j}\int_{z} {\lambda}({\mbox{\boldmath $u$}}_{kj}(l_{k,j},{\widehat{z}});t_k) g_k({\mbox{\boldmath $u$}}_{kj}({\widehat{l}},z)) {{\,{{\mathrm{d}}}}}z.\end{aligned}$$ Here, $l_{k,j}$ is half of segment $j$’s length $w_{k,j}$. The integration over the distance parameter can now be approximated by a quadrature rule with an equidistant scheme, so that $$\begin{aligned}
I_{{\mathcal{C}}_k} &\approx \sum_{j=1}^J \sum_{r=1}^R {\widetilde{w}}_{k,j} {\lambda}({\mbox{\boldmath $u$}}_{kj}(l_{k,j},{\widehat{z}});t_k) g_k({\mbox{\boldmath $u$}}_{kj}({\widehat{l}},z_r)),\end{aligned}$$ where ${\widetilde{w}}_{k,j} = \frac{2z_{\textrm{max}}}{R}w_{k,j}$ with maximal detection distance $z_{\textrm{max}}$, and we substitute ${\widehat{l}} = l_{k,j}$. We can then write $$\begin{aligned}
I_{{\mathcal{C}}_k} &\approx \sum_{j=1}^J \sum_{r=1}^R {\widetilde{w}}_{k,j} {\lambda}({\widetilde{{\mbox{\boldmath $u$}}}}_{k,j,r};t_k) g_k({\mbox{\boldmath $u$}}_{k,j,r}),\end{aligned}$$ where ${\mbox{\boldmath $u$}}_{k,j,r}$ are points on the perpendicular line through the midpoint of transect k’s segment $j$, and ${\widetilde{{\mbox{\boldmath $u$}}}}_{k,j,r}$ is the midpoint of each subsegment line.
As a last step we can again make use of the assumption that the function we are integrating over is approximately linear within a given triangle. It is straightforward to show (see Supplement \[appx:integration\_projection\]) that this means that each integration point can be expressed by an evaluation of the function at the triangle vertices weighted by the within-triangle Barycentric coordinates of the original point [@Farin:2002]. We can therefore summarize integration points that reside in the same triangle and share a common time coordinate $t_k$ to such evaluations at the mesh vertices, illustrated in Fig \[fig:integration\_scheme\]. This can, depending on the problem structure, lead to a significant reduction in the respective computational workload.
The approximation error from treating the log-linear function within each triangle as linear can be reduced by subdividing each triangle into four. However, evaluating the function at the mid-points of the original triangle edges as well as the original vertices leads to an increase in the computational cost of at least a factor of four, since the number of edges is approximately three times the number of vertices in the mesh.
Estimating blue whale density from the Eastern Tropical Pacific surveys {#sec:casestudy}
=======================================================================
The ETP surveys
---------------
We use the above methods to predict the blue whale group density over the ETP survey area for the each of the survey years, and to study the effect of sea surface temperature (SST) on the blue whale group density. The function of interest for density estimation is the intensity of the point process before thinning, denoted $\lambda({\mbox{\boldmath $s$}}, t)$ in , which we refer to as the group density.
These data have been analyzed before: @Forney+al:2012 used GAMs to estimate encounter rate, with a two-stage estimation approach and gridded data (counts of detections within small segments of transects). @Pardo+al:2015 also used gridded data, modelling log density in each grid cell as a polynomial function of the absolute dynamic topography, a spatially referenced variable that indicates vertical transport of nutrients and thus productivity. While they included a random component in their density model, it had no spatial structure, assuming independent residuals among grid cells. They estimated all model parameters simultaneously in a hierarchical Bayesian framework. Two key differences between their and our model structures are that we use the ungridded data (i.e. the point locations of each detection rather than counts in user-defined grid cells) in our analysis, and we use a *spatially structered* Gaussian random field to capture spatial variation in density that is not explained by the observed explanatory variable.
Because the blue whale group size is small (mean of 1.8 and standard deviation 2.1) and the size is easily established, it is realistic to treat the group size of blue whales as known without error [@Gerodette+al:2002]. We assume that the group size does not affect the detection probability. The detection of cetaceans on ship surveys also depends on wind conditions, but this is less important for blue whales because of their large body size and conspicuous blows. Therefore, we assume the detection probability depends only on the perpendicular distance for the blue whales in the ETP survey. In our analysis, we truncate the data at perpendicular distance $w=6$ km. We also assume that distances were observed without error for each detected animal group and we fit a semi-parametric model given by to estimate the detection function.
To build a spatio-temporal model using the SPDE approach described in Section \[sec:SPDEcomputation\], we start by constructing a mesh for the ETP survey as shown in Fig \[fig:effortsight\]. The ETP survey is bounded partially by the coastline and partially by the red line of Fig \[fig:effortsight\]. We use a simplified representation of the actual coastline as the mesh boundary to incorporate the boundary effect because a physical boundary such as the coastline has a strong effect – in this case, there will be no blue whales on land. We use a simplified representation of the coastline, because the actual coastline is too ‘angular’ and hence problematic for the SPDE approach [@Lindgren+Rue:2015]. Meanwhile, we extend the boundary further in the northwest and south directions in the Eastern Tropical Pacific, to exclude the boundary effect for the part of the survey area that is bounded by the red line in Fig \[fig:effortsight\]. Given the low sighting rate of the blue whales, there is little information contained in the data to fit a complicated spatio-temporal stochastic process for the random field, such as the AR(1) temporal process used by @Cameletti+al:2011; even the simpler version of the spatio-temporal process with replicated spatial field over time is not feasible.
We consider three models, all with Gaussian random fields in space alone. Model 0 has latitude and longitude fixed effects but ignores SST. Model 1 has a temporal SST fixed effect together with spatial residual SST fixed effects for each year. Model 2 has a temporal SST fixed effect and a spatial SST fixed effect together with spatio-temporal residual SST fixed effects.
Incorporating a spatio-temporal environmental covariate: sea surface temperature (SST) {#sec:centeringSST}
--------------------------------------------------------------------------------------
Based on the Simple Ocean Data Assimilation (SODA) model ([<http://apdrc.soest.hawaii.edu/datadoc/soda_2.2.4.php>]{}), the SST data are available on a fine grid over the ETP survey area on a monthly scale between 1986 and 2007. First, within each year, SST is averaged over the months July to December, during which the survey was conducted. Second, these temporally averaged SST values are spatially smooth, and can be projected onto the mesh of the survey area with only minor loss of fine-scale information. Piecewise linear interpolation is used to calculate the SST for any given location and year, denoted by ${\textrm{sst}}({\mbox{\boldmath $s$}}, t)$.
Fig \[fig:barSSTcYear\] shows the centered SST averaged over time and the centered SST averaged over space. There is both spatial and inter-year variation in SST, and we use hierarchical centering to separate the annual and spatial effects of SST.
![Sea surface temperature. The centered time-average temprature $\overline{{\textrm{sst}}}_c({\mbox{\boldmath $s$}})$ is shown in the left panel, while the centered space-averaged temperature $\overline{{\textrm{sst}}}_c(t)$ is shown in the right panel. Solid circles represent survey years and the empty circles non-survey years. The SST in 1997 is extreme relative to all survey years. []{data-label="fig:barSSTcYear"}](plotbarSSTcLoc_vs.png "fig:"){width="45.00000%"} ![Sea surface temperature. The centered time-average temprature $\overline{{\textrm{sst}}}_c({\mbox{\boldmath $s$}})$ is shown in the left panel, while the centered space-averaged temperature $\overline{{\textrm{sst}}}_c(t)$ is shown in the right panel. Solid circles represent survey years and the empty circles non-survey years. The SST in 1997 is extreme relative to all survey years. []{data-label="fig:barSSTcYear"}](plotbarSSTcYear_allyrs_vs.png "fig:"){width="35.00000%"}
Hierarchical centering is a commonly used technique in multilevel modeling [@Kreft+al:1995], and we consider two different centering schemes for SST here. Model 0 does not incorporate SST, Model 1 incorporates SST using within-year centering, and Model 2 uses space-time centering. They all have the same SPDE specification of the latent spatial Gaussian random field.
1. Model 1: within-year centering. This model has two SST components, the spatially averaged SST for each year, and the spatial SST patterns centered within each year. Let $\Omega$ denote the bounded ETP survey area in Fig \[fig:effortsight\]. We use $\overline{{\textrm{sst}}}_c(t)$ to denote the SST averaged over the ETP survey area for year $t$ after centering,
$$\begin{aligned}
\overline{{\textrm{sst}}}_{c}(t)&=\frac{1}{|\Omega|}\int_{\Omega} {\textrm{sst}}({\mbox{\boldmath $s$}}, t) {{\,{{\mathrm{d}}}}}{\mbox{\boldmath $s$}} -\overline{{\textrm{sst}}},
\label{eq:SSTcYr}\end{aligned}$$
where $\overline{{\textrm{sst}}}$ denotes the overall average of SST, $\overline{{\textrm{sst}}} =\int_{\Omega\times\mathbb{T}} {\textrm{sst}}({\mbox{\boldmath $s$}}, t) {{\,{{\mathrm{d}}}}}{\mbox{\boldmath $s$}} {{\,{{\mathrm{d}}}}}t\big/(|\Omega|\times |\mathbb{T}|)$, with $\mathbb{T}$ denoting the set of survey years. Then the SST centered within year $t$ for location ${\mbox{\boldmath $s$}}$, ${\textrm{sst}}_{cwy}({\mbox{\boldmath $s$}}, t)$, is defined as $$\begin{aligned}
{\textrm{sst}}_{cwy}({\mbox{\boldmath $s$}}, t) &= {\textrm{sst}}({\mbox{\boldmath $s$}}, t) - \overline{{\textrm{sst}}}_{c}(t).\label{eq:SSTcwy}\end{aligned}$$
2. Model 2: space-time centering. This model separates the spatial and temporal patters from a spatio-temporal interaction and has three SST components. These are the $\overline{{\textrm{sst}}}_c(t)$ given by , the SST averaged over years for each location, $\overline{{\textrm{sst}}}_c({\mbox{\boldmath $s$}})$ given by , and the SST residuals, ${\textrm{sst}}_{res}({\mbox{\boldmath $s$}}, t) $ given by ,
$$\begin{aligned}
\overline{{\textrm{sst}}}_c({\mbox{\boldmath $s$}}) &= \frac{1}{ |\mathbb{T}|}\int_{\mathbb{T}} {\textrm{sst}}({\mbox{\boldmath $s$}}, t) {{\,{{\mathrm{d}}}}}t - \overline{{\textrm{sst}}}, \label{eq:SSTcLoc}\\
{\textrm{sst}}_{res}({\mbox{\boldmath $s$}}, t) &= {\textrm{sst}}({\mbox{\boldmath $s$}}, t) - \overline{{\textrm{sst}}}_c(t) -\overline{{\textrm{sst}}}_c({\mbox{\boldmath $s$}}) - \overline{{\textrm{sst}}}.\label{eq:SSTres}\end{aligned}$$
$\overline{{\textrm{sst}}}_c(t)$ indicates whether a year is relatively warm or cold after averaging over the survey area, and similarly, $\overline{{\textrm{sst}}}_c({\mbox{\boldmath $s$}})$ indicates whether a location is relatively warm or cold after averaging over the survey years. The SST residual, ${\textrm{sst}}_{res}({\mbox{\boldmath $s$}}, t)$, contains information about the interaction between temporal pattern and the spatial average of SST.
Plots of raw and centered SST used in the analysis are given in Supplement \[appx:plotSST\].
La Niña conditions are characterized by a band of cooler waters in 1988, 1999 and 2007, and El Niño conditions by a much wider band of warm ocean water in 1997 [1987 is a moderately strong El Niño year according to the scale by @Wolter+Timlin:2011]. Centering SST strongly captures the El Niño/La Niña oscillations that occur at irregular intervals in the ETP survey area (see Supplement \[appx:plotSST\] for more detail). The temporal effect of SST after centering using (\[eq:SSTcYr\]) (see Fig \[fig:barSSTcYear\]) correctly reflects the La Niña conditions in 1988 and 2007, and strongly highlights the El Niño year 1997 as an outlier. Unfortunately, no survey was conducted in 1997. Given the time series of $\overline{{\textrm{sst}}}_{c}(t)$ for the survey years and non-survey years in the right panel of Fig \[fig:barSSTcYear\], it is obviously problematic to predict for 1997 using a model fitted on the data from the survey years, which are represented by the filled circles in Fig \[fig:barSSTcYear\]. Therefore, we make predictions for all years except 1997.
Results {#sec:ETPresults}
-------
Table \[tab:betasAllmodel\] summarizes the posterior density of the regression coefficients for each model.
### The effects of longitude and latitude
Model 0 contains only longitude and latitude as covariates. The 95% posterior credible intervals for the regression coefficients from this model both include zero with medians very close to zero. This suggests that there is no large-scale log-linear spatial effect that can be explained by longitude and latitude. This interpretation is supported by the results from models that include SST. Specifically, when we add longitude and latitude to Models 1 and 2, the 95% posterior credible intervals of the longitude and latitude regression parameters still include zero. We therefore exclude longitude and latitude, and henceforth consider only Models 1 and 2.
[lcccccc]{} && & &\
Model &Parameter & Mean & Std.dev. & 2.5% & 50%& 97.5%\
& $\displaystyle{\beta_0}$ &-12.29& 2.29 & -18.04&-11.99& -8.56\
& $\beta_{lon}$ & 0.10 & 0.07 & -0.05 &0.10 &0.26\
&$\beta_{lat}$ & 0.01& 0.09 & -0.22 & 0.02& 0.16\
&$\beta_0$ &-4.58& 3.04 & -11.00& -4.44& 1.06\
& $\beta_{\overline{{\textrm{sst}}}_c(t)}$ & 0.79 & 0.21 &0.38 &0.78 & 1.20\
& $\beta_{\overline{{\textrm{sst}}}_{cwy}({\mbox{\boldmath $s$}}, t)}$ & -0.28 & 0.10 & -0.48&-0.28 & -0.07\
&$\beta_0$ &-11.85 & 2.24 & -17.34&-11.60& -7.85\
&$\beta_{\overline{{\textrm{sst}}}_c(t)}$ & 0.73 & 0.21 & 0.32& 0.73 & 1.16\
&$\beta_{\overline{{\textrm{sst}}}_c({\mbox{\boldmath $s$}})}$ & -0.60& 0.14 & -0.88 &-0.60 & -0.34\
&$\beta_{\overline{{\textrm{sst}}}_{res}}$ & 0.22& 0.17 & -0.10 &0.22 & 0.55\
### The effects of SST
From , and , we have $\overline{{\textrm{sst}}}_{cwy}({\mbox{\boldmath $s$}}, t) =\overline{{\textrm{sst}}}_{c}({\mbox{\boldmath $s$}}) + \overline{{\textrm{sst}}}_{res}({\mbox{\boldmath $s$}}, t)$, so that $\beta_{\overline{{\textrm{sst}}}_{cwy}({\mbox{\boldmath $s$}}, t)}$ in Model 1 amounts to combining $\beta_{\overline{{\textrm{sst}}}_c({\mbox{\boldmath $s$}})}$ and $\beta_{\overline{{\textrm{sst}}}_{res}}$ in a single parameter. The negative posterior median and 95% credible interval (2.5% to 97.5% quantiles) of $\beta_{\overline{{\textrm{sst}}}_c({\mbox{\boldmath $s$}})}$ and of $\beta_{\overline{{\textrm{sst}}}_{cwy}({\mbox{\boldmath $s$}}, t)}$ indicate that locations that are colder on average over the years are expected to have more blue whale groups than locations that are warmer on average, while the opposite sign of the spatio-temporal interaction $\beta_{\overline{{\textrm{sst}}}_{res}}$ indicates that this effect is weaker at locations with higher temperature in a given year than the across-year average temperature at the location.
The posterior median estimates of $\beta_{\overline{{\textrm{sst}}}_c(t)}$ are similar for Models 1 and 2, indicating that the effect of warmer average temperature in a year, conditional on the spatial effect and random field, is to increase density. The ETP survey design is not balanced in that it does not have survey effort in every year along each transect that was surveyed in any year and as a result we need to be a bit cautious about interpretating parameters. To investigate the effect of annual mean temperature, we therefore also considered the posterior distribution of the predicted number of blue whale groups per unit area. This is shown in Figure \[fig:DvsT\]. While this plot is consistent with the estimates of $\beta_{\overline{{\textrm{sst}}}_c(t)}$ from Models 1 and 2, it is also consistent with an hypothesis of no change in average density across the years, as a horizontal line falls well within the 95% credible intervals of all estimates. The other notable feature of the plot is the unusually high estimated density for the second-warmest year, 2006. The reasons for this are unclear.
![The predicted number of blue whale groups per unit area ($\hat{D}$) from Model 2, together with 95% credible intervals against centered mean annual temperature $\overline{sst}_c(t)$. Numbers indicate the year in question.[]{data-label="fig:DvsT"}](plotLambdaHatSSTct_M21.pdf){width="75.00000%"}
### Posterior median density and its relative uncertainty
The posterior median of blue whale density, $\lambda({\mbox{\boldmath $s$}};t)$, for year 1986 is shown in the top panel of Fig \[fig:fvmedian1986\] for Models 0, 1 and 2, respectively. The top three plots of Fig \[fig:fvmedian1986\] are very similar, with areas of higher blue whale group density in the north off the coast of Baja California, in the area of the Costa Rica Dome off the coast of Central America, and in the south-east in the vicinity of the Galapagos Islands. This pattern of the blue whale group density is consistent across all the models implemented, and reflects what we observe in the sightings data in the right panel of Fig \[fig:effortsight\]. This observed spatial pattern is also in general agreement with previous analysis of blue whale sighting data in the ETP [@Forney+al:2012; @Pardo+al:2015]. Similar plots for 1986–2007 (omitting the very strong El Niño year 1997), are given in Supplement \[appx:densityests\].
![The posterior median (top) and RWPCI from (bottom) of the ETP blue whale groups density in 1986 using Models 0, 1, and 2 in Table \[tab:betasAllmodel\]. The RWPCI colour palette is cut off at 100 to exclude the extreme values at the western corner of the ETP survey area. []{data-label="fig:fvmedian1986"}](plotFV1986_medianRWPCI_3models_semidet_vs.pdf){width=".8\linewidth"}
We use the relative width of the 95% posterior credible interval (RWPCI) as a measure of the relative uncertainty for the predicted $\lambda({\mbox{\boldmath $s$}};t)$ of the ETP survey area. We define the RWPCI as the inter-quartile range divided by the median, $$\begin{aligned}
\mbox{RWPCI} &= (Q_3-Q_1)/Q_2.
\label{eq:RWPCI}\end{aligned}$$ When the posterior distribution is approximately Gaussian, the RWPCI is about $1.35$ times the ratio of the posterior standard deviation to the posterior median. The bottom panel of Fig \[fig:fvmedian1986\] shows the spatial structure of the RWPCI in 1986 for each of the three models, and this pattern persists across years (see Supplement \[appx:densityests\]). The far west of the survey region has very high relative uncertainty because it is close to the edge of the mesh boundary shown in Fig \[fig:effortsight\] and there are no sightings in that area. The spatial random field has high uncertainty in this area: regions of low $\lambda({\mbox{\boldmath $s$}};t)$ tend to have higher uncertainty associated with the latent field. The slowly varying standard deviation of the latent field in Fig \[fig:LFallmodels\] is likely due to a combination of large spatial range (see Fig \[fig:Matern\] of Supplement \[appx:MaternDetfun\]) and the fact that the observed point pattern is not very informative about the latent field.
![The posterior median and standard deviation of the latent field (\[eq:fem\]) for Models 0, 1 and 2.[]{data-label="fig:LFallmodels"}](plotlatentfield_medianSD_3models_semidet_vs.pdf){width=".65\linewidth"}
### SPDE parameters and detection function
Prior sensitivity tests of the SPDE parameters showed the posterior median of $\lambda({\mbox{\boldmath $s$}};t)$ to be less sensitive to prior specification than is its variance. Details of the SPDE prior specification are given in Supplement \[appx:spdeprior\]. Fig \[fig:SPDEhyperTwoModels\] displays the posterior densities of the SPDE parameters for Models 0, 1 and 2 using the same prior.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![The posterior densities of the SPDE parameters using Models 0, 1 and 2. The left panel is for the range parameter $\rho$ (see Section \[sec:basicSPDE\] for its definition), and the right panel for the marginal standard deviation $\sigma_{\xi}$ in (\[eq:SPDE\]). []{data-label="fig:SPDEhyperTwoModels"}](plotSPDErange_3models_spdepriorv3_semidet.pdf "fig:"){width=".25\linewidth"} ![The posterior densities of the SPDE parameters using Models 0, 1 and 2. The left panel is for the range parameter $\rho$ (see Section \[sec:basicSPDE\] for its definition), and the right panel for the marginal standard deviation $\sigma_{\xi}$ in (\[eq:SPDE\]). []{data-label="fig:SPDEhyperTwoModels"}](plotSPDEsigma_3models_spdepriorv3_semidet.pdf "fig:"){width=".25\linewidth"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The large range of the Mat[é]{}rn covariance function is consistent with the latent Gaussian random field ($\xi$ of ) shown in Fig \[fig:LFallmodels\]. There is little difference among the models for either the posterior detection function or the 95% credible band (see Fig \[fig:postgx\] of Supplement \[appx:MaternDetfun\]).
Exploratory model checking
--------------------------
Let $\eta({{{\mbox{\boldmath $s$}}}},t)$ denote the log-intensity defined by the fixed effects and random field components of , $$\begin{aligned}
\eta({{{\mbox{\boldmath $s$}}}},t)&=\log[\lambda({{{\mbox{\boldmath $s$}}}};t)]={\mbox{\boldmath $x$}}({{{\mbox{\boldmath $s$}}}},t)^\top{\mbox{\boldmath $\beta$}} +
\xi({{{\mbox{\boldmath $s$}}}},t) .\end{aligned}$$ To investigate the role of the components and the possibility of confounding, we consider the variability around the posterior mean of the overall averages of $\eta({{{\mbox{\boldmath $s$}}}},t)$; $$\begin{aligned}
M_{\eta}&=\frac{1}{|\Omega|\times |\mathbb{T}|} \int_{{{{\mbox{\boldmath $s$}}}}\in \Omega} \int_{t\in\mathbb{T}} E[\eta({{{\mbox{\boldmath $s$}}}},t) | {\mbox{\boldmath $Y$}}]{{\,{{\mathrm{d}}}}}t {{\,{{\mathrm{d}}}}}{{{\mbox{\boldmath $s$}}}},\end{aligned}$$ and similarly for the components, so that $M_\eta=M_{\beta}+M_\xi$. The posterior expected squared deviation of $\eta({{{\mbox{\boldmath $s$}}}},t)$ from $M_\eta$ can be split into contributions from the fixed effects ${\mbox{\boldmath $x$}}({{{\mbox{\boldmath $s$}}}},t)^\top{\mbox{\boldmath $\beta$}}$ and the random field $\xi({{{\mbox{\boldmath $s$}}}},t)$; $$\begin{aligned}
V_\eta({{{\mbox{\boldmath $s$}}}},t) = E\{[\eta({{{\mbox{\boldmath $s$}}}},t)-M_\eta]^2|{\mbox{\boldmath $Y$}}\} &=
E\{[{\mbox{\boldmath $x$}}({{{\mbox{\boldmath $s$}}}},t)^\top{\mbox{\boldmath $\beta$}}-M_{\beta}]^2|{\mbox{\boldmath $Y$}}\}
+ E\{[\xi({{{\mbox{\boldmath $s$}}}},t)-M_\xi]^2|{\mbox{\boldmath $Y$}}\}
\\&\phantom{=\,}
+
2 E\{[{\mbox{\boldmath $x$}}({{{\mbox{\boldmath $s$}}}},t)^\top{\mbox{\boldmath $\beta$}}-M_{\beta}] [\xi({{{\mbox{\boldmath $s$}}}},t)-M_\xi]|{\mbox{\boldmath $Y$}}\}
\\&= V_\beta({{{\mbox{\boldmath $s$}}}},t) + V_\xi({{{\mbox{\boldmath $s$}}}},t) + 2C_{\beta,\xi}({{{\mbox{\boldmath $s$}}}},t) .\end{aligned}$$ For all models, $\xi({{{\mbox{\boldmath $s$}}}},t)$ is constant over time, and we define the averages across time, $V_\eta({{{\mbox{\boldmath $s$}}}},\mathbb{T})$, $V_\beta({{{\mbox{\boldmath $s$}}}},\mathbb{T})$, and $V_\xi({{{\mbox{\boldmath $s$}}}},\mathbb{T})$, shown in Fig \[fig:Vmodel1\] for Model 1 and in Supplement \[appx:latenfixed\] for all models. It is clear that the random field component $\xi({{{\mbox{\boldmath $s$}}}})$ captures information not available in the SST components.
The full space-time averages $V_\eta(\Omega,\mathbb{T})$, $V_\beta(\Omega,\mathbb{T})$, and $V_\xi(\Omega,\mathbb{T})$ are the variances when probing the posterior distributions at a uniformly chosen random locations on $\Omega\times\mathbb{T}$. The remainder term $C_{\beta,\xi}(\Omega,\mathbb{T})$ is the posterior covariance between the fixed effect and random field contributions to the variability. We also define the corellation $\rho_{\beta,\xi}(\Omega,\mathbb{T})=C_{\beta,\xi}(\Omega,\mathbb{T})/\sqrt{V_\beta(\Omega,\mathbb{T})V_\xi(\Omega,\mathbb{T})}$. A large negative value for the covariance or correlation indicates confounding. Table \[tab:gofsummary\] shows the space-time averages, covariance and correlation for the three models. The correlations are not very small, suggesting that there is some confounding, although it is not severe. While these diagnostics do not give direct guidance for model selection, they highlight the clear contribution of the random field component of each model.
$V_\eta(\Omega,\mathbb{T})$ $V_\beta(\Omega,\mathbb{T})$ $V_\xi(\Omega,\mathbb{T})$ $C_{\beta,\xi}(\Omega,\mathbb{T})$ $\rho_{\beta,\xi}(\Omega,\mathbb{T})$
--------- ----------------------------- ------------------------------ ---------------------------- ------------------------------------ ---------------------------------------
Model 0 7.32 10.05 9.86 -6.29 -0.63
Model 1 5.04 3.19 5.97 -2.06 -0.47
Model 2 6.82 7.51 9.40 -5.05 -0.60
: The posterior space-time averages $V_\eta(\Omega,\mathbb{T})$, $V_\beta(\Omega,\mathbb{T})$, $V_\xi(\Omega,\mathbb{T})$, covariance $C_{\beta,\xi}(\Omega,\mathbb{T})$ and correlation $\rho_{\beta,\xi}(\Omega,\mathbb{T})$ for Models 0, 1 and 2.[]{data-label="tab:gofsummary"}
![Variability measures $V_\eta$, $V_\beta$, and $V_\xi$ for Model 1.[]{data-label="fig:Vmodel1"}](plotV_M11_spdepr3_sharedlegend.png){width=".7\textwidth"}
Discussion {#sec:discuss}
==========
Unlike previous methods used to analyse these and similar survey data, our spatio-temporal point process model preserves the sighting locations, models the effect of explanatory variables continuously in space, and models spatial correlation that cannot be explained by such variables. It generalizes the approach of @Johnson+al:2010, which models density as a nonhomogeneous Poisson process, using actual sighting locations, but neglecting residual spatial correlation. Unlike @Johnson+al:2010, we model residual spatial intensity. It also generalises the approach of @Pardo+al:2015, who included a model for residual spatial intensity in their analysis of ETP blue whale data, but with no spatial structure on their residual model. We found substantial evidence for residual spatial structure in our analysis
It is rarely the case that spatial data are independent, and assuming independence when data are dependent can lead to biased variance estimation, spurious significance of covariates, and overfitting [@Cressie:1993; @Hanks+al:2015]. Use of a GMRF allows us to model spatially autocorrelated random effects, and model patterns in residuals that cannot be explained by available covariates. As shown in Section \[sec:ETPresults\], the spatial pattern captured by the GMRF in Fig \[fig:LFallmodels\] plays an important role in estimating the spatial distribution of ETP blue whale groups, shown in Fig \[fig:fvmedian1986\]. Because the underlying mechanisms that dictate the distribution of blue whales in space and time are probably quite complex, it is unlikely that SST alone could adequately explain the distribution, so that drawing inferences about the effect of SST based on a model without modeling spatial correlation may result in misleading biological interpretations.
The analysis of @Pardo+al:2015 modeled blue whale density spatially as a function of absolute dynamic topography (ADT), which, like SST, predicted fewer blue whales in warmer regions. Because the model did not separate the temporal and spatial effects of ADT, large changes in ETP blue whale abundance were predicted from year to year, with few whales in warm (El Niño) years and many whales in cool (La Niña) years. Because blue whales have long life spans and reproduce slowly, and because tagging has shown that blue whales migrate to tropical waters every year, regardless of El Niño variations [see tracks in @Bailey+al:2009 for example], high interannual variation in true abundance seems unlikely. The hierarchical centering scheme in Section \[sec:centeringSST\] separates the temporal and spatial effects of SST and accommodates situations in which whales make choices about habitat use *relative* to the other choices available to them, and this leads to what is arguably a more biologically plausible model with less interannual variation. While our estimates are consistent with true blue whale group density being unchanged over the surveys, point estimates of this density do tend to be higher in warmer years. This is unexpected and warrants further investigation. @Pardo+al:2015 argue that ADT is a better predictor of blue whale density than SST because ADT contains information about subsurface as well as surface conditions. Notwithstanding this, our model is able to pick up structure in the data beyond that which can be attributed to SST. For example, @Pardo+al:2015 predict high denities on the Costa Rica Dome (approximately 10$^o$N, 90$^o$E) on the basis of ADT; we do the same by means of the GMRF (see Fig \[fig:LFallmodels\]) even though SST does not suggest high densities here. By also modeling spatial autocorrelation, our model does not run the risk of drawing biased inference about the effects of explanatory variables (SST here) due to unmodeled correlation. We found that the estimated Gaussian random field is somewhat correlated with the fixed effects assocaited with SST. As a result, the interpretation of the fixed effects is not as clearcut as it would be were the Gaussian random field and fixed effects independent.
Considering our models in the more general context of point process modelling, the data structure we consider here differs from the point patterns typically analyzed in the point process literature [but see @Waagepetersen+Schweder:2006]. These usually comprise a point pattern that has been observed completely in a finite observation window that is a subset of ${\mathbb{R}}^2$, say. Unless finite point processes are explicitly considered the standard assumption is that the point process continues in the same way outside the observation window. For interpretation, this implies that the analysis is only informative if the processes of interest are operating at a spatial scale that is captured within the (frequently single) subsample that is available. Further, there is an additional assumption that every point in the observation window has been observed, so that the detection probability is one within the observation window and zero elsewhere.
Our method extends such methods to deal with situations in which the processes of interest reflected in a spatial pattern, such as habitat preference, operates at a larger spatial scale than the sampled regions, when it may be impossible to fully sample an area that captures that scale. It also accommodates situations in which detection probability is unknown and not one, even within the sampled region. In wildlife sampling literature this has often been dealt with in two stages, first estimating detection probability and then estimating spatial distribution conditional on the estimated detection probability. Our approach integrates the two, estimating detection probability simultaneously with the point process parameters.
We expect to see advances in spatio-temporal inference when there are covariates that affect both the thinning process and the density surface [@Dorazio:2012]. We also expect further development of methods to assess goodness of fit, as such methods are somewhat lacking for spatial and spatio-temporal inference.
The point process model in Section \[sec:modelsoverall\] can also be extended to a marked point process model to incorporate group size in the model and allow detection probability to depend on group size. We also anticipate that our approach will be extended to deal with more complex observation processes and for other survey types – for spatial capture-recapture sampling [@Borchers+Efford:08; @Royle+Young:08] for example, for situations in which detection probabilities change over time, or when there is unknown spatially varying sampling effort.
Acknowledgements {#acknowledgements .unnumbered}
================
This research was funded by EPSRC grants EP/K041061/1 and EP/K041053/1. We thank the captains, crews and observers on the NOAA research vessels, and the support staff at the Southwest Fisheries Science Center, for the collection of line-transect data in the ETP over many years.
Some assumptions {#appx:LTassumptions}
================
The assumptions referred to in Section \[sec:linetransectlikelihood\] are as follows:
1. A team of observers is considered as a joint *black box* system, and the aggregated detection properties are modeled.
2. Individual objects (animals or animal groups) are not uniquely identified, only their locations are observed.
3. For each segment, the observable regions behind the starting point and ahead of the endpoint are small compared with the length of the segment as a whole, and the partial overlap of segments at changes in path direction is negligible.
4. The time between any other segment overlap is large enough that the time-slice point patterns in the overlap region can be considered independent; the object curves are considered to be in equilibrium, and at least locally mixing faster than the time between revisits by the observer.
Some details of SPDE models {#appx:spde}
===========================
As noted in the body of the paper, the results from @Lindgren+al:2011 show how to take advantage of the connection between Gaussian Markov random fields of graphs and stochastic partial differential equations in continuous space. The most basic such model is based on the following stochastic partial differential equation (SPDE) defined on a 2-dimensional spatial domain $$\begin{aligned}
\label{eq:SPDE}
(\kappa^2 - \nabla\cdot\nabla)[\tau \xi({\mbox{\boldmath $s$}})] &= \mathcal{W}({\mbox{\boldmath $s$}}),
\quad {\mbox{\boldmath $s$}} \in {\mathbb{R}}^2,\end{aligned}$$ where $\nabla\cdot\nabla$ is the Laplacian, $\mathcal{W}({\mbox{\boldmath $s$}})$ is Gaussian spatial white noise, and $\tau,\kappa>0$ are variance and range scaling parameters. @Whittle:1954 [@Whittle:1963] proved that stationary solutions to (\[eq:SPDE\]) are Gaussian random fields (GRF) with Mat[é]{}rn covariance function, $$\begin{aligned}
\label{eq:materncov}
{\textrm{cov}}\left[\xi({\mbox{\boldmath $s$}}), \xi({\mbox{\boldmath $s$}}')\right]
& =
\sigma_{\xi}^2\, \kappa \, \|{\mbox{\boldmath $s$}}'-{\mbox{\boldmath $s$}}\|
\,{K}_1\left(\kappa \, \|{\mbox{\boldmath $s$}}'-{\mbox{\boldmath $s$}}\|\right),
\quad {\mbox{\boldmath $s$}},{\mbox{\boldmath $s$}}'\in{\mathbb{R}}^2,\end{aligned}$$ where $\sigma_\xi^2={1}/({4\pi\kappa^2\tau^2})$ is the marginal variance, and $\mathcal{K}_1$ is the modified Bessel function of the second kind and order 1. The corresponding correlation function is $$\begin{aligned}
\label{eq:materncorr}
\mbox{cor} \left[\xi({\mbox{\boldmath $s$}}), \xi({\mbox{\boldmath $s$}}')\right]
& =
\kappa \, \|{\mbox{\boldmath $s$}}'-{\mbox{\boldmath $s$}}\|
\,{K}_1\left(\kappa \, \|{\mbox{\boldmath $s$}}'-{\mbox{\boldmath $s$}}\|\right),
\quad {\mbox{\boldmath $s$}},{\mbox{\boldmath $s$}}'\in{\mathbb{R}}^2.\end{aligned}$$ A measure of the spatial range can be obtained from $\rho = {\sqrt{8}}/{\kappa}$, which is the distance where the spatial correlation is approximately $0.13$. More complex models can be obtained by changing the operator order or allowing the parameters to depend on the location. Spatio-temporal models can be constructed by either using a temporally continuous differential operator such as in the heat equation, or with auto-regressive constructions in discrete time, such as $$\begin{aligned}
\label{eq:xist}
\xi({\mbox{\boldmath $s$}}, t)=a\,\xi({\mbox{\boldmath $s$}}, t-1) + {\omega({\mbox{\boldmath $s$}}, t)},\end{aligned}$$ where $|a|<1$ controls the temporal autocorrelation, and $
\omega({\mbox{\boldmath $s$}}, t)$ are solutions to , independent for each $t$. These constructions can also be directly applied to non-Euclidean domains such as the sphere, making construction of globally consistent random field models straightforward.
Integration scheme linearisation {#appx:integration_projection}
================================
Consider a spatial integration scheme for a fixed time point. Reorganise the integration points from Section \[sec:numericalintegration\] so that ${\boldsymbol{u}}_{kj}$ is integration point number $j$ falling in triangle $k$, with $k=1,\dots,K$ and $j=1,\dots,J_k$. The corresponding integration weights are $w_{kj}$. For a given triangle we then obtain a linear approximation of function evaluations, $$\begin{aligned}
f({\mbox{\boldmath $u$}}_{kj}) \approx \sum_{i=1,2,3} b_{kji}f_{ki},\end{aligned}$$ where the $b_{kji}$ are the Barycentric coordinates [@Farin:2002] of $({\mbox{\boldmath $u$}}_{kj})$ with respect to the triangle and the $f_{ki}$ denote the function $f$ evaluated at the triangle vertices. It follows that the sum approximating the integration over a fixed triangle $k$ can be carried out by three function evaluations, $$\begin{aligned}
\sum_{j=1}^{J_k} w_{kj} f({\mbox{\boldmath $u$}}_{kj}) &\approx \sum_{j=1}^{J_k} w_{kj} \sum_{i=1,2,3} b_{kji}f_{ki} \\
&= \sum_{i=1,2,3} \left[\sum_{j=1}^{J_k} w_{kj} b_{kji} \right] f_{ki} \\
&= \sum_{i=1,2,3} {\widetilde{w}}_{ki} f_{ki}\end{aligned}$$ with weights ${\widetilde{w}}_{ki} = \sum_{j=1}^{J_k} w_{kj} b_{kji}$. Furthermore, vertices are shared among triangles. That is, $f_{a\cdot}$ and $f_{b\cdot}$ of two triangles $a$ and $b$ might refer to evaluations at the same mesh vertex. Hence, we can simplify the sum over all triangles as follows. Index the mesh vertices by $v=1,\dots V$ and let $v \sim (k,i)$ denote that $v$ is the $i$-th vertex of triangle $k$. The (possibly empty) set of weights associated with vertex $v$ then becomes $W_v=\{{\widetilde{w}}_{ki};\,v\sim (k,i)\}$ and we can write the full integral approximation as $$\begin{aligned}
\sum_{k=1}^K \sum_{j=1}^{J_k} w_{kj} f({\mbox{\boldmath $u$}}_{kj}) &\approx \sum_{v=1}^V {\overline{w}}_vf_v,\end{aligned}$$ where $f_v$ denotes a function evaluation at vertex $v$ and ${\overline{w}}_v = \sum_{w \in W_v} w$.
Sea surface temperature data {#appx:plotSST}
============================
In this supplement, we give details of the sea surface temperature (SST) in the form of plots. The uncentered SST data are shown in Figure \[fig:SSTorig\].
![The SST data before centering. For each location and year (1986–2007), the SST is averaged over July to December.[]{data-label="fig:SSTorig"}](plotSSTyrloc_July2Dec_vs.png){width="96.00000%"}
SST is centered using the within-year and space-time centering schemes described in Section \[sec:centeringSST\]. The two components of SST when using the within-year centering scheme are the $\overline{{\textrm{sst}}}_{c}(t)$ in Fig \[fig:barSSTcYear\] and ${{\textrm{sst}}}_{cwy}({\mbox{\boldmath $s$}},t)$ in Fig \[fig:SSTcwy\], which have been defined in by , respectively. For the space-time centering scheme, in addition to $\overline{{\textrm{sst}}}_{c}(t)$ in Fig \[fig:barSSTcYear\], the remaining two components are displayed in Fig \[fig:barSSTcLoc\] and \[fig:SSTres\], where Fig \[fig:barSSTcLoc\] shows the spatial pattern of SST averaged over all years, and Fig \[fig:SSTres\] shows the residual SST. After subtracting the SST averaged over space and SST averaged over time from the SST displayed in Fig \[fig:SSTorig\], the residual SST in Fig \[fig:SSTres\] shows residual patterns related to El Niño/La Niña oscillations, most clearly with the strong El Niño of 1997.
![The SST centered within year for the ETP survey area when using the within-year centering scheme. ${{\textrm{sst}}}_{cwy}({\mbox{\boldmath $s$}},t)$ is given by []{data-label="fig:SSTcwy"}](plotSSTcwysYearLoc.png){width="90.00000%"}
![The time invariant spatial SST pattern $\overline{{\textrm{sst}}}_c({\mbox{\boldmath $s$}})$ given by , where the overall average SST is $\overline{{\textrm{sst}}} = \int_{\Omega\times\mathbb{T}}{\textrm{sst}}({\mbox{\boldmath $s$}}, t) {{\,{{\mathrm{d}}}}}{\mbox{\boldmath $s$}} {{\,{{\mathrm{d}}}}}t \big/(|\Omega|\times |\mathbb{T}|) \approx 25 \mbox{ \degree C}$.[]{data-label="fig:barSSTcLoc"}](plotbarSSTcLoc_vs.png){width="56.00000%"}
![The SST residuals when using the space-time centering scheme. ${\textrm{sst}}_{res}({\mbox{\boldmath $s$}}, t)$ is given by (\[eq:SSTres\]).[]{data-label="fig:SSTres"}](plotSSTcResYearLoc.png){width="90.00000%"}
Plots of estimated blue whale density {#appx:densityests}
=====================================
Plots of the posterior medial blue whale group density are shown in Figures \[fig:Model0PMandPCI\], \[fig:Model1.1PMandPCI\] and \[fig:Model2.1PMandPCI\], for Model 0, Model 1 and Model 1, respectively. In the case of Model 0 a single plot is shown for all years as this model has no temporal component. Posterior denstiy estimates for Models 1 and 2 vary by year because of their dependence on SST.
![Posterior density of the blue whale groups using Model 0 in Table \[tab:betasAllmodel\]. The left panel displays the posterior median of the number of blue whale groups per square kilometre, and the right panel displays the relative width of the posterior credible interval given by (\[eq:RWPCI\]). Model 0 does not incorporate any temporal information, these estimate apply to all years.[]{data-label="fig:Model0PMandPCI"}](plotFV_Q2CI_M0.png){width=".8\linewidth"}
![Posterior density of the blue whale groups using Model 1 in Table \[tab:betasAllmodel\], years 1986–2007 (except 1997): the left panel displays the posterior median of the number of blue whale groups per square kilometre, and the right panel displays the relative width of the posterior credible interval given by (\[eq:RWPCI\]).[]{data-label="fig:Model1.1PMandPCI"}](plotFV_Q2CI_M11_vs.png){width="1\linewidth"}
![Posterior density of the blue whale groups using Model 2 in Table \[tab:betasAllmodel\], years 1986–2007 (except 1997): the left panel displays the posterior median of the number of blue whale groups per square kilometre, and the right panel displays the relative width of the posterior credible interval given by (\[eq:RWPCI\]). []{data-label="fig:Model2.1PMandPCI"}](plotFV_Q2CI_M21_vs.png){width="1\linewidth"}
SPDE parameters {#appx:spdeprior}
===============
`R-INLA` specifies the SPDE model using two internal parameters: $\kappa$ and $\tau$, both of which have an influence on the marginal variance of the random field $\sigma^2$, $$\begin{aligned}
\sigma^2 &= \frac{\Gamma(\nu)}{\Gamma(\alpha)(4\pi)^{d/2}\kappa^{2\nu}\tau^2},\end{aligned}$$ where $\nu$ is the smooth parameter and $\alpha =\nu - d/2$ with $d$ being the dimension of the domain. In `R-INLA`, the default value is $\alpha =2$. @Whittle:1954 argues that it is more natural to use $\alpha=2$ for $d=2$ models than the fractional $\alpha=3/2$, which generates exponential covariances. @Rozanov:1982 shows that using integers for $\alpha$ gives continuous domain Markov fields, for which the discrete basis representation in Section \[sec:SPDEcomputation\] is the easy to construct. $\kappa>0$ itself is related to the range parameter, denoted by $\rho$ and $\rho=\sqrt{8\nu}/\kappa$. The range parameter $\rho$ is commonly defined as the distance at which the spatial correlation function falls close to zero for all $\nu>1/2$. Therefore, in our case of $d=2$, we use $\alpha=2$ and it follows that $$\begin{aligned}
\label{eq:sigmarho_d2}
\sigma^2& = \frac{1}{4\pi \kappa^2\tau^2}\quad \mbox{and} \quad \rho=\frac{\sqrt{8}}{\kappa}.\end{aligned}$$
Working directly with the SPDE parameters $\kappa$ and $\tau$ can be difficult, as they both affect the variance of the field. It is often more natural and interpretable to consider the standard deviation $\sigma$ and spatial range $\rho$. The aim is to construct a joint prior for the internal model parameters $\kappa$ and $\tau$ such that $\sigma$ and $\rho$ get independent log-Normal priors, $$\begin{aligned}
\begin{cases}
\sigma \sim {\textrm{logNormal}}\left(\log\sigma_0, \,\sigma_{\sigma}^2\right), &\\
\rho\sim {\textrm{logNormal}}\left( \log\rho_0,\,\sigma_{\rho}^2\right) , &
\end{cases}\end{aligned}$$ where $\sigma_0$ and $\rho_0$ are the prior medians.
First, we choose $\sigma_0$ and $\rho_0$ based on the problem domain. From (\[eq:sigmarho\_d2\]), we obtain the corresponding values for the internal parameters $\kappa$ and $\tau$ from $\kappa_0
= \sqrt{8}/{\rho_0}$ and $\tau_0 ={1}\big/\sqrt{4\pi \kappa_0^2 \sigma_0^2}$. Then, if $\tau$ and $\kappa$ are parameterised through log-linear combinations of two parameters $\theta_1$ and $\theta_2$, $$\begin{aligned}
\label{eq:internalTauKappa}
\begin{cases}
\log (\tau) = \log (\tau_0) -\theta_1 + \theta_2, & \\
\log (\kappa) = \log (\kappa_0) - \theta_2, &
\end{cases}\end{aligned}$$ the $\sigma$ and $\rho$ parameters are related to $\theta_1$ and $\theta_2$ as $$\label{eq:logtaurhoInternal_d2}
\begin{cases}
\log(\sigma)=-\log(\sqrt{4\pi}\,\tau_0\kappa_0)+\theta_1, &\\
\log(\rho)=\log(\sqrt{8}/\kappa_0) + \theta_2. &
\end{cases}$$ Thus, $\theta_1$ and $\theta_2$ separately control the standard deviation and spatial range, respectively. Assigning independent Normal distributions to $\theta_1$ and $\theta_2$ leads to the desired result, since $\log\sigma_0=-\log(\sqrt{4\pi}\,\tau_0\kappa_0)$ and $\log\rho_0=\log(\sqrt{8}/\kappa_0)$.
General guidance for how to choose the prior medians and variances is difficult to provide. Here, we use $\sigma_0=1$ and $\sigma_\sigma^2=10$ for the standard deviation, and $\rho_0=\textrm{domainsize}/5$ and $\sigma_\rho^2=1$. The domain size for the ETP study is$~62$, and a fifth is a reasonable portion of that.
Correlation function and detection function posteriors {#appx:MaternDetfun}
======================================================
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![The Mat[é]{}rn correlation function (\[eq:materncorr\]) given the posterior estimates of $\kappa$ for Models 0, 1 and 2. In all plots, the solid lines represent (\[eq:materncov\]) with $\kappa$ equal to its posterior median. The shaded area represents the range of (\[eq:materncov\]) with $\kappa$ values at its posterior 2.5% and 97.5% quantiles. For comparison, the survey region extends roughly 14,000 km from east to west.[]{data-label="fig:Matern"}](plotMaternCov_M0_spdepriorv3_semidet.pdf "fig:") ![The Mat[é]{}rn correlation function (\[eq:materncorr\]) given the posterior estimates of $\kappa$ for Models 0, 1 and 2. In all plots, the solid lines represent (\[eq:materncov\]) with $\kappa$ equal to its posterior median. The shaded area represents the range of (\[eq:materncov\]) with $\kappa$ values at its posterior 2.5% and 97.5% quantiles. For comparison, the survey region extends roughly 14,000 km from east to west.[]{data-label="fig:Matern"}](plotMaternCov_M11_spdepriorv3_semidet.pdf "fig:") ![The Mat[é]{}rn correlation function (\[eq:materncorr\]) given the posterior estimates of $\kappa$ for Models 0, 1 and 2. In all plots, the solid lines represent (\[eq:materncov\]) with $\kappa$ equal to its posterior median. The shaded area represents the range of (\[eq:materncov\]) with $\kappa$ values at its posterior 2.5% and 97.5% quantiles. For comparison, the survey region extends roughly 14,000 km from east to west.[]{data-label="fig:Matern"}](plotMaternCov_M21_spdepriorv3_semidet.pdf "fig:")
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![The posterior detection function with 95% credible band using Models 0, 1 and 2. The semi-parametric detection function is given in (\[eq:semiparGz\]) and illustrated in Fig \[fig:detectionbasis\]. The posterior credible band is calculated based on the posterior distribution of $\beta$’s in (\[eq:semiparGz\]).[]{data-label="fig:postgx"}](plotDetSemi_M0.pdf "fig:") ![The posterior detection function with 95% credible band using Models 0, 1 and 2. The semi-parametric detection function is given in (\[eq:semiparGz\]) and illustrated in Fig \[fig:detectionbasis\]. The posterior credible band is calculated based on the posterior distribution of $\beta$’s in (\[eq:semiparGz\]).[]{data-label="fig:postgx"}](plotDetSemi_M11.pdf "fig:") ![The posterior detection function with 95% credible band using Models 0, 1 and 2. The semi-parametric detection function is given in (\[eq:semiparGz\]) and illustrated in Fig \[fig:detectionbasis\]. The posterior credible band is calculated based on the posterior distribution of $\beta$’s in (\[eq:semiparGz\]).[]{data-label="fig:postgx"}](plotDetSemi_M21.pdf "fig:")
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Latent field and fixed effect plots {#appx:latenfixed}
===================================
![Variability measures $V_\eta$, $V_\beta$, and $V_\xi$ for the three models. The fixed effect component of Model 0 clearly suffers from its lack of SST information.[]{data-label="fig:Vmodel2"}](plotV_sharedlegendbottom.png){width=".99\textwidth"}
|
Imperial/TP/2017/JG/01\
DCPT-16/37
1.5cm 1.5cm Aristomenis Donos$^1$, Jerome P. Gauntlett$^2$\
Tom Griffin$^2$ and Luis Melgar$^2$\
.6cm
*$^1$Centre for Particle Theory and Department of Mathematical Sciences\
Durham University, Durham, DH1 3LE, U.K.*
\
.6cm *$^2$Blackett Laboratory, Imperial College\
London, SW7 2AZ, U.K.*
\
.6cm
[*Dedicated to the 75th birthday of John H. Schwarz*]{}
**Abstract**
> For Gauss-Bonnet gravity and in the context of holography we show how the thermal DC conductivity can be obtained by solving a generalised system of Stokes equations for an auxiliary fluid on a curved black hole horizon. For more general higher derivative theories of gravity coupled to gauge-fields, we also analyse the linearised thermal and electric currents that are produced by DC thermal and electric sources. We show how suitably defined DC transport current fluxes of the dual field theory are given by current fluxes defined at the black horizon.
Introduction
============
The thermal and electric conductivities are important observables to study within the framework of holography. For Einstein-Maxwell theory, possibly coupled to other matter fields, it has been shown that the thermoelectric DC conductivity of the dual field theory at finite temperature can be obtained by solving a system of Stokes equations for an auxiliary fluid on a black hole horizon [@Donos:2015gia; @Banks:2015wha; @Donos:2015bxe]. The purpose of this paper is to investigate how this striking result can be extended to theories of gravity whose Lagrangians contain higher derivative terms. Such higher-derivative terms naturally arise in string and M-theory, leading to finite coupling and $1/N$ effects in the context of holography, and our results provide useful tools to investigate their effect on the conductivity, analogous to similar investigations for the shear viscosity [@Brigante:2007nu].
We begin by summarising some key aspects of the analysis in [@Donos:2015gia; @Banks:2015wha; @Donos:2015bxe]. We first recall that the natural framework for studying DC conductivities in holography is provided by holographic lattices [@Hartnoll:2012rj; @Horowitz:2012ky; @Donos:2012js; @Chesler:2013qla; @Donos:2013eha; @Andrade:2013gsa]. These are stationary black hole spacetimes with Killing horizons that are dual to CFTs in thermal equilibrium which have been deformed by operators that explicitly break the translation invariance of the CFT. Although not essential, one often considers spacetimes with a single black hole horizon of planar topology and the deformations are taken to be periodic in the spatial directions. The explicit breaking of the translation symmetry is imposed by demanding suitable boundary conditions on the bulk fields at the AdS boundary; this is essential, generically, in order to obtain a finite DC response.
To calculate the thermoelectric DC conductivity one analyses a linear perturbation of the bulk fields about the background holographic lattice. The boundary conditions that are imposed on the perturbation at the AdS boundary are associated with the application of external DC thermal and electric sources to the dual field theory. An important result of [@Donos:2015bxe], building on [@PhysRevB.55.2344; @Blake:2015ina; @Hartnoll:2007ih], was to correctly identify the conserved transport currents of the dual field theory. These are obtained from the usual holographic currents by suitably subtracting off terms that arise when the equilibrium dual field theory, described by the background holographic lattice, has non-vanishing magnetisation currents. Furthermore, it was also shown in [@Donos:2015gia; @Banks:2015wha; @Donos:2015bxe] how the transport currents can be obtained from the perturbed bulk geometry, including a contribution from currents defined at the black hole horizon. Importantly, the currents at the horizon, which are conserved, only involve a subset of the bulk perturbation and they take the form of constitutive relations for an auxiliary fluid. It was also shown in [@Donos:2015gia; @Banks:2015wha; @Donos:2015bxe] that the current fluxes at the horizon, which in the case of periodic lattices are just the zero modes of the currents, are equal to the transport current fluxes of the dual field theory. This last result implies that if one knows the currents at the black hole horizon as functions of the applied DC sources on the horizon[^1] then one can obtain the transport current fluxes of the dual field theory as functions of the DC sources and hence, by definition, the DC conductivity.
In sections \[gendisc\]-\[gbonsec\] we generalise all of these results to general theories of gravity coupled to a gauge field; the extension to include additional matter fields is straightforward. We will focus on holographic lattices, but some of our results are independent of the asymptotic boundary conditions and may have applications outside of holography. In section 2, by carrying out a Kaluza-Klein reduction of the gravity theory on the time direction[^2] we can obtain natural definitions of the electric and thermal transport currents of the dual field theory with the properties mentioned in the previous paragraph. In particular, we present a recipe for obtaining the conserved currents at the horizon as well as showing that the current fluxes at the horizon are the same as the transport current fluxes of the dual field theory.
To obtain explicit expressions for the currents at the horizon in terms of the perturbation requires more information about the specific theory of gravity that is being considered. In section \[emcasesec\], for the case of Einstein-Maxwell theory in $D$ spacetime dimensions and stationary black holes, we first show that we recover exactly the same currents on the horizon that were derived in [@Donos:2015bxe]. In section \[gbonsec\] we consider the case of pure gravity in $D$ spacetime dimensions including curvature squared contributions. A special case of this class of theories includes Gauss-Bonnet gravity in $D\ge 5$, which we return to below. For the class of static black holes, for which there are no background magnetisation currents, we derive explicit expressions for the conserved heat current at the horizon.
Another key aspect of [@Donos:2015gia; @Banks:2015wha; @Donos:2015bxe], for theories of gravity without higher derivative terms, was the use of a radial decomposition of the bulk equations of motion to evaluate the momentum, Hamiltonian and Gauss-law constraints on the horizon[^3] for the DC perturbed spacetime. It was shown that this leads to a set of linearised, time-independent, forced Navier-Stokes equations for a charged, incompressible fluid on the black hole horizon. These equations, which we refer to as Stokes equations, also include the conservation laws for the electric and heat currents on the horizon, that we already discussed above. A crucial feature of the system of Stokes equations is that they form a closed set of equations for a subset of the perturbation on the horizon. By solving the Stokes equations one can obtain the conserved currents at the horizon in terms of the DC sources and hence the DC conductivity of the dual field theory. The way in which the Stokes equations appear on the horizon is related to the old membrane paradigm of [@Price:1986yy]. It is worth noting, though, that for the linearised perturbations driven by a DC source studied in [@Donos:2015gia; @Banks:2015wha; @Donos:2015bxe] one obtains linearised Stokes equations for an incompressible fluid on a curved horizon, whereas in [@Price:1986yy] there are equations for a compressible fluid with negative bulk viscosity.
The approach of [@Donos:2015gia; @Banks:2015wha; @Donos:2015bxe] for obtaining the Stokes equations at the horizon can immediately be deployed to study theories of gravity with higher derivative terms that have a well-defined Cauchy problem for radial evolution. The Lovelock theories of pure gravity, which have equations of motion only involving second order time derivatives of the metric, satisfy this condition. We will focus on the particular case of Gauss-Bonnet theory in $D\ge 5$ spacetime dimensions, whose Lagrangian consists of the usual Einstein-Hilbert term and a negative cosmological constant, supplemented by the Gauss-Bonnet term with coupling $\tilde\alpha$ as given in . In section \[gbns\] we will show that when the constraints are evaluated on the horizon we obtain a generalised set of Stokes equations which have non-trivial dependence on $\tilde\alpha$. Taking a perturbative approach[^4] to the higher derivative Gauss-Bonnet theory, it is of particular interest to obtain the leading order corrections in $\tilde\alpha$. If $h_{ij}$ is the horizon metric, with Levi-Civita connection $\nabla_i$, the leading order Stokes equations can be written[^5] $$\begin{aligned}
\label{icintro}
\nabla_i(\delta^i_j-4\tilde\alpha {G}^i_{j})v^j&=0\,,{\notag \\}-2\nabla^i\left(S_{ij}^{kl}\nabla_k v_l\right)&=(\delta^i_j-4\tilde\alpha {G}^i_{j})\left(4\pi T\zeta_i-{\nabla_i p}\right)\,,\end{aligned}$$ where $G_{ij}=R_{ij}-R h_{ij}$ is the Einstein tensor on the horizon and $$\begin{aligned}
\label{essintro}
S^{kl}_{ij}=\left[1-\tilde\alpha 2(D-4)(D-1)\right]\delta^{(k}_i\delta^{l)}_j
-\tilde\alpha\left[2h^{}_{ij}R_{}^{kl}+4\delta^{(k}_{(i}R_{}{}^{l)}_{j)}+4R_{}{}_i{}^{(k}{}_j{}^{l)}\right]\,.\end{aligned}$$ Note that the horizon metric quantities depend on $\tilde \alpha$ and need to be expanded in these expressions.
It is interesting to highlight that the coefficient appearing in the first term of is precisely the same as the the ratio of the shear viscosity to entropy ratio that was calculated for a planar black brane in Gauss-Bonnet theory in [@Brigante:2007nu], namely $4\pi\frac{\eta}{s}=1-\tilde \alpha 2(D-4)(D-1)$. The reason for this will be explained in section \[gbns\]. As in [@Donos:2015gia; @Banks:2015wha; @Donos:2015bxe], we emphasise that , again form a closed set of equations for a subset of the perturbation on the horizon and allow one to obtain the conserved currents at the horizon in terms of the DC sources and hence, in turn, the DC conductivity of the dual field theory. Thus the membrane paradigm for DC response of [@Donos:2015gia; @Banks:2015wha; @Donos:2015bxe] generalises to Gauss-Bonnet gravity[^6]. We will also show in section \[gbns\] how the new Stokes equations can be solved in the special case that there is only dependence on one of the spatial directions on the horizon.
We briefly conclude in section \[fincom\] and we have several appendices containing various technical material.
Bulk physics in the DC limit {#gendisc}
============================
General set-up
--------------
We consider a general bulk theory in $D$ spacetime dimensions that is diffeomorphism invariant and has a local internal $U(1)$ gauge symmetry. More precisely, we will assume that the Lagrangian, ${\cal L}$, is gauge-invariant and transforms as a scalar under diffeomorphims. We further assume that the theory only depends on a metric and an abelian gauge field, but we note that it is straightforward to include additional matter fields and other gauge symmetries. We thus consider $$\begin{aligned}
S=\int d^Dy\sqrt{-g}\mathcal{L}(g_{\mu\nu},A_\mu)\,,\end{aligned}$$ and we note that for notational simplicity we have suppressed the dependence of $\mathcal{L}$ on the derivatives of the fields. In such a theory, we are interested in perturbing stationary charged black hole backgrounds by DC electric fields and thermal gradients and examining the currents that are produced.
We begin by writing the $D$-dimensional coordinates as $$\begin{aligned}
y^\mu=(t,y^m)\,,\end{aligned}$$ where the coordinates $y^m$ parametrise a $(D-1)$-dimensional manifold $M_{D-1}$. We will assume that $\partial_t$ is a Killing vector that leaves the gauge field invariant and hence consider the general ansatz: $$\begin{aligned}
\label{eq:diff_metric_gfield_ansatz}
ds^{2}&=-H^{2}\,\left( dt+\alpha \right)^{2}+ds^{2}\left(M_{D-1} \right)\,,{\notag \\}A&=A_{t}\,\left( dt+\alpha \right)+\beta\,.\end{aligned}$$ Here $ds^{2}\left(M_{D-1}\right)\equiv \gamma_{mn}dy^mdy^n$, while $\alpha=\alpha_m dy^m$, $\beta=\beta_m dy^m$ are one-forms and $H$ is a function, all defined on $M_{D-1}$. This ansatz is sufficient to accommodate both the stationary black hole backgrounds that we are interested in as well as the DC perturbation we wish to consider.
We will keep the discussion general for the moment, making no further assumptions on the background. However, at some point we will consider a $D$-dimensional background black hole spacetime, with a Killing horizon, and then the coordinates $y^m=(r,x^i)$ will parametrise a holographic radial direction, $r$, as well as the spatial directions, $x^i$, of the dual field theory. Thus $M_{D-1}$ will have a holographic boundary. In addition, in order to consider $M_{D-1}$ as a manifold, with non-trivial boundary, we can envisage the radial direction to be terminated at a “stretched horizon", located at a very small distance from the event horizon. When we discuss global issues concerning regularity of the spacetime at the horizon, we will, of course, discuss them in the context of the full $D$-dimensional spacetime.
We now dimensionally reduce our theory in $D$ spacetime dimensions on the time direction. The equations of motion for the fields on $M_{D-1}$ can be obtained from a $D-1$ dimensional action of the form $$\begin{aligned}
\label{eq:lower_D_functional}
S=\int_{M_{D-1}}\,d^{D-1}y\,\sqrt{\gamma_{D-1}}\,H\,\mathcal{L}\left(H,A_{t},\alpha,\beta,\gamma_{mn} \right)\,,\end{aligned}$$ where $\cal L$ is a local function of the fields appearing in the ansatz , as well as the spatial derivatives of the fields. For notational simplicity we have again suppressed the dependence of $\mathcal{L}$ on the spatial derivatives of the fields.
There are several restrictions on $\mathcal{L}$ which follow from diffeomorphism and gauge invariance of the $D$-dimensional theory. Firstly, local coordinate transformations of the form $t\rightarrow t+\Lambda_{E}(y^m)$ imply that $\mathcal{L}$ is invariant under $$\begin{aligned}
\alpha\rightarrow \alpha+d\Lambda_{E}(y^m)\,.\end{aligned}$$ Second, the gauge transformations $A(t,y^m)\rightarrow A(t,y^m)+d\Lambda_{M}(t,y^m)$ with $\Lambda(t,y^m)=c\,t+\lambda_{M}(y^m)$, for constant $c$, imply that $\mathcal{L}$ is invariant under $$\begin{aligned}
A_{t}\rightarrow A_{t}+c\,,\qquad
\beta\rightarrow \beta+d\lambda_M-c\alpha\,.\end{aligned}$$ Third, invariance under time scalings, $t\rightarrow \lambda\,t$ imply that $\mathcal{L}$ is invariant under $$\begin{aligned}
H\rightarrow \lambda\,H,\qquad \alpha\rightarrow\alpha/\lambda,\qquad A_{t}\rightarrow \lambda\,A_{t}\,.\end{aligned}$$
These conditions imply that $\cal L$ will have the following functional dependence: $$\begin{aligned}
\label{ellform}
&\mathcal{L}\left(H,A_{t},\alpha,\beta,\gamma_{mn} \right)=\mathcal{L}\left(h,u,v,w,\gamma_{mn}\right)\,,\end{aligned}$$ where $h,u$ are one-forms and $v,w$ are two-forms, all defined on $M_{D-1}$, defined by $$\begin{aligned}
\label{eq:forms_def}
&h=d\ln H,\qquad
u=H^{-1}dA_{t}, \qquad
v=d\beta+A_{t}\,d\alpha,\qquad
w=H\,d\alpha\,.\end{aligned}$$ Another restriction is that $\cal L$ must be invariant under $(u,w)\to -(u,w)$. This follows from the fact that the dimensional reduction ansatz is invariant under $H\to -H$.
The equations of motion for $\alpha$ and $\beta$ will play an important role in the sequel. For simplicity, for now we will assume that $\cal L$ does not depend on derivatives of the one- and two-forms. This is the situation that arises for theories of gravity with two derivatives, such as Einstein-Maxwell theory. The simple generalisation that arises for higher derivative theories of gravity will be presented later in section \[gbonsec\]. With this assumption we find that the equations of motion for $\alpha$ and $\beta$ are given by $$\begin{aligned}
\label{formeqmotion}
&\nabla_{m} V^{mn} =0,\qquad \nabla_{m} W^{mn} =0\,,\end{aligned}$$ where $$\begin{aligned}
\label{eq:form_eoms}
V^{mn}=H\,\frac{\delta{\mathcal{L}}}{\delta v_{mn}},\qquad
W^{mn}=H^{2}\,\frac{\delta{\mathcal{L}}}{\delta w_{mn}}+HA_{t}\frac{\delta{\mathcal{L}}}{\delta v_{mn}}
\,.\end{aligned}$$ In other words the two-forms $W=\tfrac{1}{2}W_{mn}dx^m\wedge dx^n$ and $V=\tfrac{1}{2}V_{mn}dx^m\wedge dx^n$, with indices lowered with $\gamma_{mn}$, are both co-closed. This is a key result. As we will see, $V$ and $W$ are associated with the electric and heat currents of the dual field theory in the context of holography.
DC perturbation
---------------
We now want to consider the equations that govern a specific linear perturbation about a given stationary background solution to the equations of motion. We write the stationary background solution as $$\begin{aligned}
&H=H^{(B)}, \quad \alpha=\alpha^{(B)},\quad \gamma_{mn}=\gamma^{(B)}_{mn}\,,\quad
A_{t}=A_{t}^{(B)},\quad \beta=\beta^{(B)}\,,\end{aligned}$$ with all background quantities independent of the time coordinate $t$.
The perturbation we want to consider is seeded by two closed one-forms $\zeta,E$ that are globally defined on $M_{D-1}$. To achieve this we will introduce two locally defined functions, $\phi_{T}$ and $\phi_{E}$, on $M_{D-1}$ via[^7] $$\begin{aligned}
\zeta=d\phi_{T}\,,\qquad E=d\phi_{E}\,.\end{aligned}$$ The perturbation that we consider is then given by $$\begin{aligned}
\label{expsII}
&H=H^{(B)}\left(1-\phi_{T}\right) +\delta H, \quad \alpha=\alpha^{(B)} \left(1+\phi_{T} \right)+\delta \alpha\,,
\quad \gamma_{mn}=\gamma_{mn}^{(B)}+\delta \gamma_{mn}\,,\notag\\
&A_{t}=A_{t}^{(B)}\,\left(1-\phi_{T} \right)+\phi_{E}+\delta A_{t},\qquad
\beta=\beta^{(B)}-\phi_{E}\,\alpha^{(B)}+\delta\beta\,,\end{aligned}$$ with $\delta H,\delta\alpha,\delta A_t,\delta\beta$ and $\delta \gamma_{mn}$ all globally defined perturbations on $M_{D-1}$. Thus, at linearised order, the perturbed metric and gauge-field takes the form $$\begin{aligned}
ds^2+\delta(ds^2)&=-H^{(B)2}(1+2H^{(B)-1}\delta H)[(1-\phi_T)dt+\alpha^{(B)}+\delta\alpha]^2
+(\gamma^{(B)}_{mn}+\delta\gamma_{mn})dx^mdx^n\,,{\notag \\}A+\delta A&=(A^{(B)}_t+\delta A_t)[(1-\phi_T)dt+\alpha^{(B)}+\delta \alpha]+\beta^{(B)}+\delta\beta+\phi_E dt\,.\end{aligned}$$
For the holographic applications we consider later, on the holographic boundary the one-form $E$ parametrises an applied DC electric field source term, while $\zeta$ parametrises a DC thermal gradient via[^8]: $\zeta\leftrightarrow -T^{-1}dT$, where $T$ is the locally defined temperature. After a simple coordinate transformation and gauge transformation (see below) we can easily show that this gives rise to the perturbation containing terms linear in the time coordinate $t$ that have been used in derivations of the DC conductivity, starting with [@Donos:2014uba; @Donos:2014cya]. As we will now see, it is convenient to work with the above locally defined, time independent perturbation so that we can directly utilise the Kaluza-Klein dimensional reduction formulae. Importantly, the key equations we are ultimately interested in will only involve the globally defined one-forms $(\zeta,E)$ and not the locally defined functions $(\phi_{T},\phi_{E})$. We will make some additional comments on this in the next sub-section.
To proceed we now write out the perturbation in terms of the one- and two-forms defined in . We find: $$\begin{aligned}
\label{eq:forms_pert}
\delta h&=d(H^{(BG)-1}\delta H)-\zeta\,,{\notag \\}\delta u&=H^{(B)}{}^{-1}d\delta A_{t}-H^{(B)}{}^{-2}\,\delta H \,d A_{t}^{(B)}-H^{(B)}{}^{-1}\,A_{t}^{(B)}\,\zeta+H^{(B)}{}^{-1}\,E\,,{\notag \\}\delta v&=d\delta\beta+\delta A_{t}\,d\alpha^{(B)}+A_{t}^{(B)}\,d\delta\alpha+A_{t}^{(B)}\,\zeta\wedge\alpha^{(B)}-E\wedge\alpha^{(B)}\,,{\notag \\}\delta w&=\delta H\,d\alpha^{(B)}+H^{(B)}\,d\delta\alpha+H^{(B)}\,\zeta\wedge \alpha^{(B)}\,,\end{aligned}$$ and, in particular, we observe that they involve the globally defined one-forms $(\zeta,E)$, as just mentioned.
We next examine how the two-forms $V$ and $W$ defined in depend on the linearised perturbation. Starting with $V$ we find $$\begin{aligned}
\label{delvee}
\delta V^{mn}=&-V^{(B)mn}\,\phi_{T}+\delta L^{mn}\,,\end{aligned}$$ where we have defined $$\begin{aligned}
\label{eq:L_def}
\delta L^{mn}
\equiv \left(\frac{\delta{\mathcal{L}}}{\delta v_{mn}}\right)^{(B)}\,\delta H+H^{(B)}\,\delta\left(\frac{\delta{\mathcal{L}}}{\delta v_{mn}}\right)\,.\end{aligned}$$ The two-form $\delta L_{mn}$ is globally defined on $M_{D-1}$. This is because $\mathcal{L}$ is a functional of the one- and two-forms in which we have shown, in equation , remain well defined after perturbing under . While $\phi_{T}$ does appear in $\delta V^{mn}$, it drops out of the equations of motion . Specifically, after using the background equation of motion, $\partial_{m}[(\gamma_{D-1}^{(B)})^{1/2}\, V^{(B)mn}]=0$, we find that implies $$\begin{aligned}
\label{emcase}
\partial_{m}\left((\gamma_{D-1}^{(B)})^{1/2}\, \delta L^{mn}+\delta[(\gamma_{D-1})^{1/2}]\, V^{(B)mn}\right)=(\gamma_{D-1}^{(B)})^{1/2}\zeta_{m} V^{(B)mn}\,.\end{aligned}$$
Following a similar logic for the two-form $W$ we find $$\begin{aligned}
\label{deldub}
\delta W^{mn}=&-2\,W^{(B)mn}\,\phi_{T}+V^{(B)mn}\,\phi_{E}+\delta K^{mn}\,,\end{aligned}$$ where we have defined $$\begin{aligned}
\label{eq:K_def2}
\delta K^{mn}\equiv &\,2H^{(B)}\,\left(\frac{\delta{\mathcal{L}}}{\delta w_{mn}}\right)^{(B)}\delta H+H^{(B)}{}^{2}\,\delta \left(\frac{\delta{\mathcal{L}}}{\delta w_{mn}}\right){\notag \\}&+H^{(B)}\left(\frac{\delta{\mathcal{L}}}{\delta v_{mn}}\right)^{(B)}\,\delta A_{t}
+A^{(B)}_{t}\delta L^{mn}\,.\end{aligned}$$ The two-form $\delta K_{mn}$ is again globally defined on $M_{D-1}$. While $(\phi_{T},\phi_{E})$ do appear in $\delta W^{mn}$ they again also drop out of the equations of motion . Specifically, after using the background equation of motion, $\partial_{m}[(\gamma_{D-1}^{(B)})^{1/2}\, W^{(B)mn}]=0$, we find that implies $$\begin{aligned}
\label{eq:GR_div_eqn}
&\partial_{m}\left((\gamma_{D-1}^{(B)})^{1/2}\, \delta K^{mn}+\delta[(\gamma_{D-1})^{1/2}]\, W^{(B)mn}\right)=
(\gamma_{D-1}^{(B)})^{1/2}\left[2\,\zeta_{m}W^{(B)mn}-\,E_{m}V^{(B)mn}\right]\,.\end{aligned}$$
The two equations and are key results. As we will see below, they generalise the equations given in eq. (5.6) of [@Donos:2015bxe] from a specific theory of gravity to a more general one and, in addition, it is clear how to extend the theory to include additional matter fields.
Black hole geometry in holography {#secbhgih}
---------------------------------
We now consider the results in the previous subsection in a holographic context of DC perturbations about a background black hole geometry. While it is straightforward to be significantly more general, as in section 5 of [@Donos:2015bxe], to simplify the presentation we now assume that the stationary background solution has a single black hole Killing horizon and we will also assume that the solution has a globally defined radial coordinate, $r$, outside the black hole. Introducing the coordinates $$\begin{aligned}
y^m=(r,x^i)\,,\end{aligned}$$ we assume that the black hole is located at $r=0$ and that there is also a boundary at $r=\infty$, which we take to approach a suitably deformed $AdS_D$. More precisely, as $r\to \infty$ we have $$\begin{aligned}
\label{asmet}
ds^2&\to r^{-2}dr^2+r^2\left[g^{(\infty)}_{tt}dt^2+g^{(\infty)}_{ij}dx^idx^j+2g_{ti}^{(\infty)}dtdx^i\right]\,,{\notag \\}A&\to A^{(\infty)}_t dt+A^{(\infty)}_i dx^i\,, \end{aligned}$$ where $g^{(\infty)}_{tt}$ etc. are functions of the spatial coordinates, $x^i$, only, and parametrise the spatially dependent sources for the stress tensor and abelian current operators that have been used to deform the CFT. Such black hole solutions are known as holographic lattices. Often we are interested in CFTs on spacetimes with planar topology with deformations that are periodic in the spatial directions, so-called “periodic holographic lattices". In the set-up of this paper, by considering a fundamental domain, this corresponds, effectively, to considering CFTs living on a $(D-2)$-dimensional torus with a non-trivial metric, associated with a source for the stress tensor, as well as additional periodic sources for the abelian current, the simplest of which is a periodic chemical potential $A^{(\infty)}_t$.
The manifold $M_{D-1}$ is defined for $0<r<\infty$ and the associated metric can be written $$\begin{aligned}
\label{eq:base_metric}
ds^{2}(M_{D-1})=\gamma_{mn}^{(B)}(y^p)dy^m dy^n\,.\end{aligned}$$ This metric becomes singular at the location of the black hole at $r=0$. In order that the $D$-dimensional black hole solution is regular at the horizon, which is Killing with respect to $\partial_t$, we demand that as $r\to 0$: $$\begin{aligned}
\label{nhorexp}
H^{(B)}{}^{2}= 4\pi T\,r+\mathcal{O}(r^{2})\,,\quad
\gamma_{rr}^{(B)}=\frac{1}{4\pi T\,r}+\mathcal{O}(1)\,,\quad
A_{t}^{(B)}= \,a_{t}^{(0)}(x)\,r+\mathcal{O}(r^{2})\,,\end{aligned}$$ with the remaining quantities given by $$\begin{aligned}
\label{nhorexp2}
\alpha^{(B)}_{r}&=\alpha^{(0)}_r+\mathcal{O}(r)\,,\qquad
\alpha^{(B)}_{i}=\alpha^{(0)}_i+\mathcal{O}(r)\,,\notag\\
\gamma^{(B)}_{ij}&=h^{(0)}_{ij}+\mathcal{O}(r)\,,\qquad
\gamma^{(B)}_{ir}=\mathcal{O}(r)\,,\notag\\
\beta^{(B)}_{r}&=\beta^{(0)}_r+\mathcal{O}(r)\,,\qquad
\beta^{(B)}_{i}=\beta^{(0)}_i+\mathcal{O}(r)\,.\end{aligned}$$ It is worth noting that this is a small simplification[^9] of the coordinates used in [@Donos:2015bxe]. We emphasise that when we later consider theories of gravity with higher derivatives, we will also need to keep sub-leading terms in these expansions. Observe that we have $$\begin{aligned}
\label{textup}
\gamma^{(B)rr}&={4\pi Tr}+\mathcal{O}(r^2),\quad
\gamma^{(B)ri}=\mathcal{O}(r^2),\quad
\gamma^{(B)ij}=h^{(0)ij}+\mathcal{O}(r),\quad{\notag \\}(\gamma^{(B)}_{D-1})^{1/2}&=\frac{\sqrt{h^{(0)}}}{(4\pi T r)^{1/2}}+\mathcal{O}(r^{1/2})\,.\end{aligned}$$
To proceed, we now assume that the background two-forms $V$ and $W$ are well defined in the limit as $r\to 0$. Since the vector $\partial_r$ becomes singular as $r\to 0$ this implies that $$\begin{aligned}
\label{veedubcons}
V^{(B)ri}|_H=W^{(B)ri}|_H=0\,.\end{aligned}$$ We will explicitly check that this condition is satisfied for the examples that we consider later.
We now define the bulk current densities via $$\begin{aligned}
\label{defjayque}
J^{i}\equiv 2(\gamma_{D-1})^{1/2}V^{ri}\,,\qquad
Q^{i}\equiv-2(\gamma_{D-1})^{1/2}W^{ri}\,,\end{aligned}$$ By evaluating these quantities at the AdS boundary, located at $r\to\infty$, we obtain $J^{i}_\infty$ and $Q^{i}_\infty$, which are the bulk contributions to the electric and heat currents of the dual boundary CFT. Indeed, as we explain in more detail in appendix \[currcomment\], one can show that $J^{i}_\infty$ and $Q^{i}_\infty$ are the on-shell variations of the dimensionally reduced action with respect to $\delta\beta_i^\infty=\lim_{r\to\infty}\delta \beta_i$ and $\delta\alpha_i^\infty=\lim_{r\to\infty}\delta \alpha_i$, respectively. One then just needs to recall the parametrisation of the metric and gauge field in order to make the identification.
For the unperturbed background we write the bulk current densities as $$\begin{aligned}
\label{magcexp}
J^{(B)i}\equiv 2(\gamma^{(B)}_{D-1})^{1/2}V^{(B)ri}\,,\qquad
Q^{(B)i}\equiv-2(\gamma^{(B)}_{D-1})^{1/2}W^{(B)ri}\,,\end{aligned}$$ which we notice, from , vanish at the horizon. By integrating the equations of motion in the radial direction, it is straightforward to show that we can write the background current densities at the $r=\infty$ boundary as: $$\begin{aligned}
\label{twee}
J^{(B)i}_\infty=\partial_j M^{(B)ij}\,,\qquad
Q^{(B)i}_\infty=\partial_j M^{(B)ij}_T\,,\end{aligned}$$ where we have defined the magnetisation densities $M^{(B)ji}$, $M^{(B)ji}_T$ as the background values of $$\begin{aligned}
M^{ij}\equiv\int_0^\infty dr (\gamma^{}_{D-1})^{1/2}2V^{ij}\,,\qquad
M^{ij}_T\equiv -\int_0^\infty dr (\gamma^{}_{D-1})^{1/2}2W^{ij}\,.\end{aligned}$$ Notice that both $J^{(B)i}_\infty$ and $Q^{(B)i}_\infty$ in take the form of magnetisation current densities and hence are trivially spatially conserved: $$\begin{aligned}
\partial_iJ^{(B)i}_\infty=0\,,\qquad \partial_iQ^{(B)i}_\infty=0\,.\end{aligned}$$
For the DC perturbation we will also now assume that the sources only depend on the spatial directions of the dual field theory. We therefore now take $\phi_E,\phi_T$ to be independent of the radial coordinate and we have $$\begin{aligned}
E=d\phi_E(x)=E_i(x) dx^i\,,\qquad \zeta =d\phi_T(x)=\zeta_i(x) dx^i\,.\end{aligned}$$ It is illuminating to highlight some additional points concerning these sources and to do this it is helpful to focus on the particular class of planar periodic holographic lattices. As mentioned above, by focussing on a fundamental domain, we can view the CFT as living on a torus. In this case we can write, for example, $\phi_T=\bar \zeta_i x^i+z(x)$ and $\zeta =\bar \zeta_i dx^i+dz(x)$, where $\bar\zeta_i$ are constants and $z(x)$ is a periodic function. Note that the term $\bar\zeta_i dx^i$ is associated with a constant DC thermal gradient source $\bar\zeta_i$ in the $x^i$ direction. This can be more invariantly characterised by integrating the closed one-form $\zeta$ over a basis of one-cycles on the torus: for example, one can integrate over a period in the $x^i$ direction and then average over the length of the period. It is worth noting that on the plane $\bar \zeta_i x^i$ is a well-defined but unbounded function, while on the torus it is a bounded but not a well-defined function since it is not periodic. Deformations with $z(x)\ne 0$ correspond to deforming the background but keeping the system in thermal equilibrium. The physics of most interest is the response of the system, within a fundamental domain, to the application of a DC source with non-zero $\bar\zeta_i$.
For the perturbed current densities, in the presence of the sources $(\phi_T,\phi_E)$, from , we have $$\begin{aligned}
\label{emcasepone}
\delta J^i&=\bar \delta J^i - \phi_T J^{(B)i}\,,{\notag \\}\delta Q^i&= \bar \delta Q^i - 2\phi_T Q^{(B)i} - \phi_E J^{(B)i}\,,\end{aligned}$$ where $\bar \delta J^i$, $\bar \delta Q^i$ are given by $$\begin{aligned}
\label{emcasep}
\bar \delta J^i&\equiv (\gamma_{D-1}^{(B)})^{1/2}\, 2\delta L^{ri}+\delta[(\gamma_{D-1})^{1/2}]\, 2V^{(B)ri}\,,{\notag \\}\bar \delta Q^i&\equiv -\left((\gamma_{D-1}^{(B)})^{1/2}\, 2\delta K^{ri}+\delta[(\gamma_{D-1})^{1/2}]\, 2W^{(B)ri}\right)\,.\end{aligned}$$ Notice from , that $\delta J^i$, $\delta Q^i$ are spatially conserved $$\begin{aligned}
\label{conlasttime}
\partial_i\delta J^i=\partial_i\delta Q^i=0\,.\end{aligned}$$
An important feature of is the explicit appearance of $\phi_T$ and $\phi_E$ when the background has non-vanishing magnetisation currents, $J^{(B)i},Q^{(B)i}\ne 0$. These terms are certainly physical. For example, for periodic holographic lattices, building on the discussion above, if $\phi_T$ or $\phi_E$ are periodic functions, then gives the perturbative change in the currents as we apply a periodic deformation to the CFT while keeping it in thermal equilibrium. However, we are most interested in the case in which $\phi_T$ or $\phi_E$ are not periodic functions (i.e. are not globally defined functions on the torus) and then extracting the transport currents discussed in [@PhysRevB.55.2344; @Blake:2015ina; @Hartnoll:2007ih; @Donos:2015bxe]. The transport currents of the dual field theory are conserved and are well-defined in a fundamental domain, so they should not explicitly depend on $\phi_T, \phi_E$. They can be identified as follows.
We first notice that at $r\to \infty$, the contributions to the currents $\bar \delta J^i_\infty$, $\bar \delta Q^i_\infty$ are globally defined in a fundamental domain but are not spatially conserved. Using in we calculate, for example, $$\begin{aligned}
0&=\partial_i\bar \delta J^i_\infty-J^{(B)i}_\infty\zeta_i\,,{\notag \\}&=\partial_i\bar \delta J^i_\infty-\partial_j M^{(B)ij}\zeta_i\,,{\notag \\}&=\partial_i\left(\bar \delta J^i_\infty +M^{(B)ij}\zeta_j\right)\,,\end{aligned}$$ where in the last step we used $\partial_{[i}\zeta_{j]}=0$. There is a similar calculation for the thermal currents. We thus identify the transport currents of the dual field theory as $$\begin{aligned}
\label{deftcs1}
\delta{\mathcal J}^i_\infty&\equiv \bar\delta{J}^i_\infty+M^{(B)ij}\zeta_j\,,{\notag \\}\delta{\mathcal Q}^i_\infty&\equiv\bar \delta{Q}^i_\infty+2M^{(B)ij}_T\zeta_j+M^{(B)ij}E_j \,.\end{aligned}$$ These differ from the one-point functions $\delta J^i_\infty$, $\delta Q^i_\infty$ only when there is non-vanishing magnetisation in the background. Furthermore, $\delta{\mathcal J}^i_\infty$ and $\delta{\mathcal Q}^i_\infty$ are both globally defined on a fundamental domain and conserved: $$\begin{aligned}
\partial_i\delta {\mathcal J}_\infty^ i=0\,,\qquad \partial_i\delta {\mathcal Q}^ i_\infty=0\,.\end{aligned}$$
An additional perspective on these definitions is obtained as follows. We first note that analogous to , in the presence of the sources $(\phi_T,\phi_E)$, the perturbed magnetisations can be written $$\begin{aligned}
\label{magdefs}
\delta M^{ij}&= \bar \delta M^{ij} - \phi_T M^{(B)ij}\,,{\notag \\}\delta M^{ij}_T&= \bar \delta M^{ij}_T- 2\phi_T M^{(B)ij}_T - \phi_E M^{(B)ij}\,,\end{aligned}$$ where $ \bar \delta M^{ij}$ and $ \bar \delta M^{ij}_T$ are both globally defined on a fundamental domain and given by $$\begin{aligned}
\bar \delta M^{ij}&\equiv \int_0^\infty dr (\gamma_{D-1}^{(B)})^{1/2}\, 2\delta L^{ij}+\delta[(\gamma_{D-1})^{1/2}]\, 2V^{(B)ij}\,,{\notag \\}\bar \delta M^{ij}_T&\equiv -\int_0^\infty dr\left((\gamma_{D-1}^{(B)})^{1/2}\, 2\delta K^{ij}+\delta[(\gamma_{D-1})^{1/2}]\, 2W^{(B)ij}\right)\,.\end{aligned}$$ We thus see that these give rise to magnetisation currents in the presence of the perturbation of the form $$\begin{aligned}
\partial_j\delta M^{ij}&= \partial_j\bar \delta M^{ij}- \phi_T J^{(B)i}- \zeta_jM^{(B)ij}\,,{\notag \\}\partial_j\delta M^{ij}_T&= \partial_j\bar \delta M^{ij}_T- 2\phi_T Q^{(B)i} - \phi_E J^{(B)i}-2M^{(B)ij}_T\zeta_j-M^{(B)ij}E_j\,,\end{aligned}$$ The transport current densities at $r=\infty$ can thus also be written[^10] $$\begin{aligned}
\label{deftcs}
\delta{\mathcal J}^i_\infty&=\delta J^i_\infty -(\partial_j \delta M^{ij} -\partial_j \bar\delta M^{ij} )\,, {\notag \\}\delta{\mathcal Q}^i_\infty&=\delta Q^i_\infty -(\partial_j \delta M^{ij}_T-\partial_j \bar\delta M^{ij}_T)\,,\end{aligned}$$
With these definitions for the transport currents, after integrating , in the radial direction, we deduce that the local transport current densities at $r=\infty$ are related to the local current densities at the horizon via $$\begin{aligned}
\label{lastcexp}
\delta{\mathcal J}^i_\infty -\partial_j \bar\delta M^{ij} =\delta J^i_H\,,\qquad
\delta{\mathcal Q}^i_\infty-\partial_j \bar\delta M^{ij}_T =\delta Q^i_H
\,,\end{aligned}$$ where we used $\delta J^i_H=\bar\delta J^i_H$, $\delta Q^i_H=\bar \delta Q^i_H$. We next define current flux densities ${\mkern 2.0mu\overline{\mkern-2.0mu\delta{\mathcal J}^i_\infty\mkern-2.0mu}\mkern 2.0mu}, {\mkern 2.0mu\overline{\mkern-2.0mu\delta{\mathcal Q}^i_\infty\mkern-2.0mu}\mkern 2.0mu}$ through a basis of $(D-3)$-dimensional cycles on the spatial manifold on which the dual CFT lives, exactly as in [@Donos:2015gia; @Banks:2015wha; @Donos:2015bxe]. Rather than repeat the details of the definitions here in general, let us just note that for the special case of the periodic holographic lattices in which the spatial manifold has planar topology, these current flux densities can be equivalently and simply defined as the zero modes of $\delta{\mathcal J}^i_\infty$, $\delta{\mathcal Q}^i_\infty$, which can be obtained by taking the average integral over a fundamental domain. From we then have the key result: $$\begin{aligned}
\label{result}
{\mkern 2.0mu\overline{\mkern-2.0mu\delta{\mathcal J}^i_\infty\mkern-2.0mu}\mkern 2.0mu}={\mkern 2.0mu\overline{\mkern-2.0mu\delta J^i_H\mkern-2.0mu}\mkern 2.0mu}\,,\qquad
{\mkern 2.0mu\overline{\mkern-2.0mu\delta{\mathcal Q}^i_\infty\mkern-2.0mu}\mkern 2.0mu}={\mkern 2.0mu\overline{\mkern-2.0mu\delta Q^i_H\mkern-2.0mu}\mkern 2.0mu}\,.\end{aligned}$$
Having established , we would like to determine how the local current densities at the horizon ${\delta J^i_H}$, ${\delta Q^i_H}$ depend on the perturbation. We make some general comments in the next subsection, based on the conditions imposed by demanding that the perturbation is regular at the horizon. In subsequent sections, using specific theories of gravity we show how the currents at the horizon take the form of constitutive relations for an auxiliary fluid. For the special case of Gauss-Bonnet gravity we will show the horizon currents can be obtained by solving a generalised set of Stokes equations on the horizon.
Regularity of the perturbation at the horizon
---------------------------------------------
We now examine the conditions that we need to impose in order that the perturbation is regular at the black hole horizon. To do this, near the horizon we first perform the combined coordinate and gauge transformation given by $$\begin{aligned}
\label{coordandgauge}
t&\rightarrow t\,(1+\phi_{T})\,,\notag\\
A&\rightarrow A+d\Lambda,\quad
\Lambda=-\phi_{E}\,t\,.\end{aligned}$$ After these transformations the metric and gauge field perturbations in $D$ dimensions takes the form $$\begin{aligned}
\delta ds^{2}&=-2H^{(B)}\,\delta H\,(dt+\alpha^{(B)})^{2}-2\,H^{(B)}{}^{2}\,(dt+\alpha^{(B)})(\delta\alpha+t\,\zeta)+\delta \gamma_{mn}\,dx^{m} dx^{n}\,,\notag\\
\delta A&=\delta A_{t}\,(dt+\alpha^{(B)})+A_{t}^{(B)}\,(\delta\alpha+t\,\zeta)+\delta\beta-t\,E\,.\end{aligned}$$ Notice that the transformed perturbations still satisfy the covariant equations and .
In order to impose regular, infalling boundary conditions we define the Eddington-Finkelstein coordinate $v=t+\tfrac{\ln r}{4\pi T}$ close to the horizon. To ensure that all our fields are regular functions of $v$ and $r$ in the $r\to 0$ limit we demand that the perturbation has the following expansion $$\begin{aligned}
\label{eq:nh_pert}
\delta\alpha_{i}&=\frac{1}{4\pi T\,r }\,v_{i}+\frac{\ln r}{4\pi T}\,\zeta_{i}+\mathcal{O}(1),\qquad
\delta\alpha_{r}=-\frac{1}{4\pi T\,r }\,\delta g_{tr}^{(0)}+\mathcal{O}(1)\,,\notag\\
\delta H&=-\frac{(4\pi T r)^{1/2}}{2} \delta g_{tt}^{(0)} +\mathcal{O}(r^{3/2}),\qquad
\delta \gamma_{rr}=\frac{1}{4\pi T r} \delta g_{rr}^{(0)}+\mathcal{O}(1)\,, \notag\\
\delta \gamma_{ri}&=- \frac{1}{4\pi T r}v_{i}+\mathcal{O}(1) ,\qquad\qquad\qquad
\delta \gamma_{ij}=\delta g_{ij}^{(0)}+2\alpha^{(0)}_{(i}v_{j)}+\mathcal{O}(r)\,,\notag\\
\delta A_{t}&=w+\mathcal{O}(r),\qquad\qquad\qquad\qquad\quad
\delta \beta_{r}=\frac{1}{4\pi T r}w+\mathcal{O}(1)\,,\notag\\
\delta \beta_{i}&=\frac{\ln r}{4\pi T}(-E_{i}+A_{t}^{(B)}\,\zeta_{i})+\delta\beta_{i}^{(0)}+\mathcal{O}(r)\,,\end{aligned}$$ along with $\delta g_{tt}^{(0)}+\delta g_{rr}^{(0)}-2 \delta g_{tr}^{(0)}=0$. This is the same[^11] expansion given in (4.1)-(4.3) of [@Donos:2015bxe]. In particular, $v_i$, $w$ and $\delta g_{tr}^{(0)}$ are the same quantities that entered in the Stokes equations in [@Donos:2015bxe]. Recalling that we are using a slightly different radial coordinate for the background black holes near the horizon (see footnote \[test\]) we can identify the pressure, $p$, of [@Donos:2015bxe] via $p=-(4\pi T)\delta g_{tr}^{(0)}$. For further calculations, it is helpful to note that we have $$\begin{aligned}
\label{dgamup}
\delta \gamma^{rr}&=\mathcal{O}(r),\qquad
\delta \gamma^{ir}={v^i}+\mathcal{O}(r),\qquad
\delta \gamma^{ij}=\mathcal{O}(1),{\notag \\}\delta[(\gamma_{D-1})^{1/2}]&= \mathcal{O}(r^{-1/2})\,.\end{aligned}$$ with $v^i\equiv h^{(0)ij}v_j$.
It is now straightforward to establish $$\begin{aligned}
\delta v_{ri}=-\frac{1}{4\pi T r}\left[E_i+\nabla_i w+a^{(0)}_t v_i\right]+a^{(0)}_t\frac{\ln r}{4\pi T}\zeta_i+\mathcal{O}(1)\,,\qquad
\delta v_{ij}=\mathcal{O}(1)\,,\end{aligned}$$ and hence $$\begin{aligned}
\label{delvriup}
\delta v^{ri}=-\left[E_i+\nabla_i w+a^{(0)}_t v_i+(d\beta^{(0)})^i{}_{j} v^j\right]+\mathcal{O}(r\ln r)\,.\end{aligned}$$ Similarly, $$\begin{aligned}
\delta w^{ri}&=-(4\pi T)^{1/2}\frac{1}{r^{1/2}}v^i+\mathcal{O}(r^{1/2})\,,\qquad
\delta w^{ij}=\mathcal{O}(r^{1/2})\,.\end{aligned}$$ Interestingly, the sub-leading terms in these expansions are important for higher derivative theories, as we will see later.
To obtain explicit expressions for the currents $\delta J^i_H$, $\delta Q^i_H$, we need to evaluate the expressions at the horizon. To do this we need more information on the theory of gravity that we are considering. We will illustrate with two examples, first recovering the Einstein-Maxwell results of [@Donos:2015gia; @Banks:2015wha; @Donos:2015bxe] in section \[emcasesec\], before moving on to higher derivative pure gravity in section \[gbonsec\].
DC currents in Einstein-Maxwell theory {#emcasesec}
======================================
For the case of Einstein-Maxwell theory we can now easily recover the results for the currents at the horizon that were obtained in [@Donos:2015gia; @Banks:2015wha; @Donos:2015bxe]. The $D$ dimensional bulk action is given by $$\begin{aligned}
S=\int\,d^{D}x\,\sqrt{-g}\,\left( R-V_0-\frac{1}{4}\,F^{2} \right)\,.\end{aligned}$$ with $V_0$ constant. If we choose $V_0=-(D-1)\,(D-2)$ then a unit radius $AdS_D$ solves the equation of motion. The equations of motion for the general ansatz can be obtained from the $D-1$ dimensional action given by , with $$\begin{aligned}
\label{eq:EM_reduced_action}
&\qquad \mathcal{L}\left(h,u,v,w,\gamma_{mn}\right)=R_{D-1}+\frac{1}{4}\,w^{2}-\frac{1}{4}\,v^{2}+\frac{1}{2}\,u^{2}-V_0\,,\end{aligned}$$ where $R_{D-1}$ is the Ricci scalar for the $D-1$ dimensional metric on $M_{D-1}$. Notice that for this theory $\mathcal{L}$ happens to be independent of $h$.
We now immediately obtain $$\begin{aligned}
\frac{\delta \mathcal{L}}{\delta v_{mn}}=-\frac{1}{2}\,v^{mn},\qquad
\frac{\delta \mathcal{L}}{\delta w_{mn}}=\frac{1}{2}\,w^{mn}\,.\end{aligned}$$ We thus see that the two-forms $V$, $W$ defined in are given by $$\begin{aligned}
2V_{mn}&=-H v_{mn}\,,\qquad 2W_{mn}=H^2w_{mn}-HA_t v_{mn}\,,\end{aligned}$$ where $v,w$ are defined in . Recalling the near horizon expansions , we immediately deduce that for the background black holes the two-form $V,W$ are both well defined at the black hole horizon. In particular we find $$\begin{aligned}
\label{veedubcons2}
V^{(B)ri}|_H=\mathcal{O}(r^{3/2}),\qquad W^{(B)ri}|_H=\mathcal{O}(r^{5/2})\,.\end{aligned}$$ with both vanishing at the horizon, as assumed in .
Turning to the perturbation, the quantities defined in and are given by $$\begin{aligned}
\label{delLandK}
2\delta L^{mn}&=-H^{(B)}\,\delta v^{mn}-v^{(B)mn}\delta H\,,\notag\\
2\delta K^{mn}&=2H^{(B)}w^{(B)mn}\delta H+H^{(B)}{}^{2}\,\delta w^{mn}
- H^{(B)} v^{(B)mn} \delta A_t
+A^{(B)}_{t}\delta L^{mn}\,.\end{aligned}$$ We can now evaluate the currents at the horizon. For $\delta J^i_H$ we calculate as follows $$\begin{aligned}
\label{emcaseptwo}
\delta J^i_H&=\left[(\gamma_{D-1}^{(B)})^{1/2}\, 2\delta L^{ri}+\delta[(\gamma_{D-1})^{1/2}]\, 2 V^{(B)ri}\right]_H\,,{\notag \\}&=\frac{ \sqrt{h^{(0)}}}{(4\pi T r)^{1/2}} \left[2\delta L^{ri}\right]_H\,,{\notag \\}&=-\sqrt{h^{(0)}}[\delta v^{ri}]_H\,.\end{aligned}$$ To get to the second line we use , and . To get to the third line we use , to show that $v^{(B)ri}$ is of order $\mathcal{O}(r)$ and then use to show that the second term in does not contribute. Finally, using we obtain $$\begin{aligned}
\label{emcasepfinalj}
\delta J^i_H&=\sqrt{h^{(0)}}\left[E^i+\nabla^i w+a^{(0)}_t v^i+(d\beta^{(0)})^i{}_{k} v^k\right]\,,\end{aligned}$$ where all indices are now raised with the horizon metric $h^{(0)ij}$.
We next consider the perturbed heat current at the horizon $\delta Q^i_H$ to similarly find $$\begin{aligned}
\delta Q^i_H&= -\left[(\gamma_{D-1}^{(B)})^{1/2}\, 2\delta K^{ri}+\delta[(\gamma_{D-1})^{1/2}]\, 2W^{(B)ri}\right]_H\,,{\notag \\}&= - \frac{\sqrt{h^{(0)}}}{(4\pi T r)^{1/2}}\, \left[2\delta K^{ri}\right]_H\,,{\notag \\}&= - \sqrt{h^{(0)}}(4\pi T r)^{1/2}\, \left[\delta w^{ri}\right]_H\,,\end{aligned}$$ leading to $$\begin{aligned}
\label{emcasepfinalq}
\delta Q^i_H&=(4\pi T)\sqrt{h^{(0)}} v^i\,.\end{aligned}$$
We have now obtained expressions for the currents at the horizon that take the form of constitutive relations for an auxiliary fluid at the black hole. The expressions given in , are precisely the same as those that were derived in [@Donos:2015gia; @Banks:2015wha; @Donos:2015bxe], taking into account that we are using a slightly different radial coordinate and hence, as noted in footnote \[test\] above, the function $G^{(0)}$ in [@Donos:2015gia; @Banks:2015wha; @Donos:2015bxe] is set to unity.
DC currents in higher derivative theories {#gbonsec}
=========================================
For theories of gravity that involve higher derivatives we obtain a simple modification of the two-forms $V_{mn}$ and $W_{mn}$ that were given in . This is due to the fact that after dimensionally reducing on the time direction the $(D-1)$-dimensional action will also depend on derivatives of $v$ and $w$ defined in via .
For notational reasons it is convenient to introduce the operator $\mathcal{D}^{(n)}$ acting on a $(p,q)$ tensor $\Phi$ according to $$\begin{aligned}
\mathcal{D}^{(n)}_{m_{1}\ldots m_{n}}\Phi^{\alpha_{1}\ldots \alpha_{p}}_{\beta_{1}\ldots \beta_{q}}\equiv \nabla_{m_{1}}\cdots\nabla_{m_{n}}\Phi^{\alpha_{1}\ldots \alpha_{p}}_{\beta_{1}\ldots \beta_{q}}\,.\end{aligned}$$ Since the dimensionally reduced Lagrangian $\cal L$ given in , will be a function not only of $v$ and $w$ but also of $\mathcal{D}^{(s)}v$ and $\mathcal{D}^{(s)}w$, this will slightly modify the equations of motion for $\alpha$ and $\beta$. The important point, though, is that they will have the same form given in but now with $$\begin{aligned}
\label{genVW}
V^{mn}&=\sum_{s}(-1)^{s}\,\mathcal{D}^{(s)}_{m_{1}\ldots m_{s}}\left(H\,\frac{\delta\mathcal{L}}{\delta D^{(s)}_{m_{1}\ldots m_{s}}v_{mn}}\right)\,,{\notag \\}W^{mn}&=H\,\sum_{s}(-1)^{s}\,\mathcal{D}^{(s)}_{m_{1}\ldots m_{s}}\,\left(H\,\frac{\delta\mathcal{L}}{\delta D^{(s)}_{m_{1}\ldots m_{s}}w_{mn}}\right)+
A_{t}\,\sum_{s}(-1)^{s}\,\mathcal{D}^{(s)}_{m_{1}\ldots m_{s}}\,\left(H\,\frac{\delta\mathcal{L}}{\delta D^{(s)}_{m_{1}\ldots m_{s}}v_{mn}}\right)\,.\end{aligned}$$
As a consequence the results in section \[gendisc\] that we derived for theories with two derivatives all generalise to theories with higher derivative after performing the following replacements: $$\begin{aligned}
\label{repls}
\frac{\delta\mathcal{L}}{\delta v_{mn}}&\rightarrow (\delta_{v}\mathcal{L})^{mn}\equiv H^{-1} \sum_{s}(-1)^{s}\,\mathcal{D}^{(s)}_{m_{1}\ldots m_{s}}\left(H\,\frac{\delta\mathcal{L}}{\delta D^{(s)}_{m_{1}\ldots m_{s}}v_{mn}}\right)\,,{\notag \\}\frac{\delta\mathcal{L}}{\delta w_{mn}}&\rightarrow (\delta_{w}\mathcal{L})^{mn}\equiv H^{-1} \sum_{s}(-1)^{s}\,\mathcal{D}^{(s)}_{m_{1}\ldots m_{s}}\,\left(H\,\frac{\delta\mathcal{L}}{\delta D^{(s)}_{m_{1}\ldots m_{s}}w_{mn}}\right)\,.\end{aligned}$$
Higher derivative gravity
-------------------------
For the remainder of the paper we are going to focus on the general class of theories of pure gravity in $D$ spacetime dimensions with Lagrangian of the form $$\begin{aligned}
\label{hdact}
\mathcal{L}&=R-V_0+c_{1}\mathcal{L}_1+c_{2}\,\mathcal{L}_2+c_{3}\mathcal{L}_3\,,\end{aligned}$$ where $V_0$ and $c_i$ are constants with $$\begin{aligned}
\mathcal{L}_1=R_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}R^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}},\qquad
\mathcal{L}_2=R_{\mu_{1}\mu_{2}}R^{\mu_{1}\mu_{2}}\,,\qquad
\mathcal{L}_3=R^2\,.\end{aligned}$$ Within the context of string theory, one is interested in solutions to the leading order equations with corrections that are perturbative in the $c_a$ which will be of order $\alpha^{\prime}$. The case of Gauss-Bonnet gravity in $D\ge 5$ spacetime dimensions corresponds to $c_{1}=c_{3}=-\frac{1}{4}\,c_{2}\equiv\tilde \alpha$. We now aim to calculate the perturbed heat current at the horizon after switching on a DC source, parametrised by a closed one-form $\zeta$.
We first carry out the dimensional reduction on the time coordinate[^12] to obtain the $(D-1)$-dimensional theory. We introduce the obvious orthonormal frame $(e^0,e^{\hat m})$ associated with the $D$-dimensional spacetime metric given in , with $e^0=H( dt+\alpha)$ and $e^{\hat m}e^{\hat m}=ds^{2}\left(M_{D-1} \right)$. We calculate the various components of the Riemann tensor of the $D$-dimensional metric to get $$\begin{aligned}
\label{riemkk}
R_{\hat m\hat n\hat p\hat q}&=\bar R_{\hat m\hat n\hat p\hat q}+\frac{1}{2}w_{\hat m\hat n}w_{\hat p\hat q}-\frac{1}{2}w_{\hat m[\hat p}w_{\hat q]\hat n}\equiv\Sigma^{(4)}_{\hat m\hat n\hat p\hat q}\,,{\notag \\}R_{\hat m0\hat n\hat p}&=\frac{1}{2}\nabla_{\hat m} w_{\hat n\hat p}+\frac{1}{2}h_{\hat m}w_{\hat n\hat p}-h_{[\hat n}w_{\hat p]\hat m}\equiv\Sigma^{(3)}_{\hat m\hat n\hat p}\,,{\notag \\}R_{\hat m0\hat n0}&=\nabla_{\hat m}h_{\hat n}+h_{\hat m}h_{\hat n}-\frac{1}{4}w_{\hat m\hat p} w^{\hat p}{}_{\hat n}\equiv\Sigma^{(2)}_{\hat m\hat n}\,,\end{aligned}$$ where $\bar R_{\hat m\hat n\hat p\hat q}$ are the components of the Riemann tensor for the $(D-1)$-dimensional metric $\gamma_{D-1}$ in the orthonormal frame. It is worth noting the Bianchi identity that arises from the definition of $w$ takes the form $\nabla_{[\hat m} w_{\hat n\hat p]}=h_{[\hat m}w_{\hat n\hat p]}$. For the components of the Ricci tensor in the orthonormal frame we have $$\begin{aligned}
\label{ricexps}
R_{\hat m \hat n}&=\bar R_{\hat m \hat n}+\frac{1}{2}w^2_{\hat m\hat n}-\nabla_{\hat m} h_{\hat n}- h_{\hat m}h_{\hat n}\equiv\Lambda^{(2)}_{\hat m\hat n}\,,{\notag \\}R_{\hat m0}&=h^{\hat n}w_{\hat n\hat m}+\frac{1}{2}\nabla^{\hat n} w_{\hat n\hat m}\equiv\Lambda^{(1)}_{\hat m}\,,{\notag \\}R_{00}&=\nabla_{\hat m} h^{\hat m}+h^2+\frac{1}{4}w^2\equiv\Lambda^{(0)}\,,\end{aligned}$$ where $w^2=w_{m n}w^{mn}$ and $h^2=h_{m}h^{m}$. Finally, the Ricci scalar is given by $$\begin{aligned}
\label{rsexps}
R=\bar R+\frac{1}{4}w^2-2\nabla_m h^m-2h^2\,.\end{aligned}$$
We thus have $$\begin{aligned}
\mathcal{L}^{(1)}&=\Sigma^{(4)}_{mnpq}\Sigma^{(4)}{}^{mnpq}-4\,\Sigma^{(3)}_{mnp}\Sigma^{(3)}{}^{mnp}+4\,\Sigma^{(2)}_{mn}\Sigma^{(2)}{}^{mn}\,,\notag\\
\mathcal{L}^{(2)}&=\Lambda^{(2)}_{mn}\Lambda^{(2)}{}^{mn}-2\,\Lambda^{(1)}_{m}\Lambda^{(1)}{}^{m}+\Lambda^{(0)}{}^{2}\,,{\notag \\}\mathcal{L}^{(3)}&=R^2\,.\end{aligned}$$ Using the definitions we compute $$\begin{aligned}
\label{defgenW}
(\delta_{w}\mathcal{L}^{(1)})^{mn}&=2\left(\Sigma^{(4)}{}^{mn}{}_{pq}+\,\Sigma^{(4)}{}^{\left[m\right.}{}_{p}{}^{\left. n\right]}{}_{q} \right)\,w^{pq}\notag\\
&\quad -4\,h_{p}\,\left(\Sigma^{(3)}{}^{pmn}-2\,\Sigma^{(3)}{}^{\left[ mn \right]\,p} \right)+4H^{-1}\,\nabla_{p}\left(H\,\Sigma^{(3)}{}^{pmn}\right) -4\,\Sigma^{(2)}{}^{p\left[ m\right.}w^{\left. n\right]}{}_{p}\,,\notag\\
(\delta_{w}\mathcal{L}^{(2)})^{mn}&=-2\,\Lambda^{(2)}{}^{p\,\left[m\right.}w^{\left. n\right]}{}_{p}+4\,\Lambda^{(1)}{}^{\left[m\right.}h^{\left. n\right]}+2H^{-1}\,\nabla^{\left[m \right.}H\,\Lambda^{(1)}{}^{\left. n\right]}+\Lambda^{(0)}\,w^{mn}\,,\notag\\
(\delta_{w}\mathcal{L}^{(3)})^{mn}&=R\,w^{mn}\,.\end{aligned}$$
We also record here that for this theory instead of and we have $$\begin{aligned}
\label{wkay}
W^{mn}&=H^2(\delta_w\mathcal{L})^{mn}\,,{\notag \\}\delta K^{mn}&\equiv \,2W^{(B)mn}H^{(B)-1}\,\delta H+H^{(B)}{}^{2}\,\delta \left[\delta_w\mathcal{L}\right]^{mn}\,.\end{aligned}$$
Heat current in a static background
-----------------------------------
For simplicity we now focus on static background solutions and set $\alpha^{(B)}=0$. For this higher derivative theory, it turns out that we need to keep sub-leading terms in the expansion of the background fields about the black hole horizon. We thus take, for the background as $r\to 0$, $$\begin{aligned}
\label{nhorexpa}
&\gamma_{rr}^{(B)}=\frac{1}{4\pi T\,r}\,\left( 1+\gamma_{rr}^{(1)}\,r\right)+\mathcal{O}(r)\,,\quad
\gamma^{(B)}_{ij}=h^{(0)}_{ij}+h^{(1)}_{ij}\,r+\mathcal{O}(r^{2})\,,\quad
\gamma^{(B)}_{ir}=\mathcal{O}(r)\,,{\notag \\}&H^{(B)}{}^{2}= 4\pi T\,r\,\left(1+2\,H^{(1)}\,r \right)+\mathcal{O}(r^{3})\,.\end{aligned}$$ There is still some residual freedom in our choice of the radial coordinate. If we make the shift $r\to r+r^2f(x)$ then this leads to the same fall-offs but with $H^{(1)}\to H^{(1)}+f/2$ and $\gamma^{(1)}_{rr}\to
\gamma^{(1)}_{rr}+3f$. As a consequence, $H^{(1)}$ and $\gamma^{(1)}_{rr}$ can only appear in our final expressions in the combination $3H^{(1)}-\gamma^{(1)}_{rr}/2$, as we shall see. An important point is that while it is is necessary to keep sub-leading terms in the expansion of the background, we find that it is not necessary to include the sub-leading terms in the expansion of the perturbation at the horizon.
We are now ready to calculate $\delta Q^i$ at the horizon. From the definition we have $$\begin{aligned}
\label{defdelqueue}
\delta Q^i_H&= -\left[(\gamma_{D-1}^{(B)})^{1/2}\, 2\delta K^{ri}+\delta[(\gamma_{D-1})^{1/2}]\, 2W^{(B)ri}\right]_H\,.\end{aligned}$$ We are considering static backgrounds with $\alpha=w=0$. From , we deduce that in the background we have $\Sigma^{(3)}=\Lambda^{(1)}=0$ and hence from , we deduce that $W^{(B)mn}=0$. We thus have $$\begin{aligned}
\delta Q^i_H&=-\left[2\sqrt{\gamma}H^{(B)2}\delta[\delta_w\mathcal{L}]^{ri}\right]_H\,.\end{aligned}$$ The contribution to this expression from the two-derivative part of the action in is the same as before and so we have $$\begin{aligned}
\label{cqsa}
\delta Q^i_H&=(4\pi T)\sqrt{h^{(0)}} v^i
+c_1\delta Q_H^{(1)}{}^{i}+c_2\delta Q_H^{(2)}{}^{i}+c_3\delta Q_H^{(3)}{}^{i}\,,\end{aligned}$$ where $$\begin{aligned}
\frac{1}{4\pi T \sqrt{h_{(0)}}}\delta Q_H^{(a)}{}^{i}\equiv&
-\frac{2r^{1/2}}{(4\pi T)^{1/2}}\,\left[\delta(\delta_{w}\mathcal{L}^{(a)})^{ri}\right]_H\,.\end{aligned}$$
After some extensive calculations, which we describe in appendix \[details\], we can obtain the contributions from the higher-derivative terms in the action. For the Riemann squared part of the action we get $$\begin{aligned}
\label{qonefe}
\frac{1}{4\pi T \sqrt{h_{(0)}}}\delta Q_H^{(1)}{}^{i}=&
4\,\nabla_{j}\nabla^{[j}v^{i]}+4R_{(0)}^{ij}v_{j}+4\nabla^i\nabla_jv^j{\notag \\}&+2(4\pi T)\left[ 2\left(\zeta^{i}+\nabla^{i}\delta g_{tr}^{(0)}\right)-\left(3\,H^{(1)}-\frac{1}{2}\gamma_{rr}^{(0)} \right) v^{i}\right]\,.\end{aligned}$$ For the Ricci squared part of the action we get $$\begin{aligned}
\label{qtwofe}
&\frac{1}{4\pi T \sqrt{h_{(0)}}}\delta Q_H^{(2)}{}^{i}=
2\,\nabla_{j}\nabla^{[j}v^{i]}+2R_{(0)}^{ij}v_{j}+\nabla^i\nabla_jv^j{\notag \\}&\qquad\qquad+(4\pi T)\left[\left( \zeta^{i}+\nabla^{i} \delta g_{tr}^{(0)}\right)
-\left(3\,H^{(1)}-\frac{1}{2}\gamma_{rr}^{(1)} +\frac{1}{2}h^{(1)}{}^j{}_j\right)v^i \right] \,.\end{aligned}$$ Finally for the Ricci scalar squared part of the action we get $$\begin{aligned}
\label{qthreefe}
\frac{1}{4\pi T \sqrt{h_{(0)}}}\delta Q_H^{(3)}{}^{i}=
&2R^{(0)}v^i-2\,(4\pi T)
\left(3H^{(1)}-\frac{1}{2}\gamma^{(1)}_{rr}+h^{(1)}{}^j{}_j\right)v^{i}\,.\end{aligned}$$
A number of comments are now in order. Firstly, the expressions for the heat current above depend on the horizon metric as well as the sub-leading corrections to the background black hole solution that are parametrised by $3H^{(1)}-\frac{1}{2}\gamma^{(1)}_{rr}$ and $h^{(1)}_{ij}$. In principle these sub-leading corrections can be expressed in terms of the geometry of the horizon after using the equations of motion. We will not carry out this calculation here, but instead below we will analyse the results that one obtains by considering the higher-derivative corrections to the action to be perturbatively small.
Second, from our general analysis in section \[gendisc\] we know that we have $\partial_i \delta Q^{(a)i}=0$ when we use the equations of motion. In this regard, we note that $\nabla_i\nabla_j\nabla^{[j}v^{i]}=0$.
Third, we see that the expressions for the heat current are expressed in terms of the perturbation via $v^i$, as we saw for two-derivative theories of gravity, as well as $\zeta^{i}$ and $\nabla_i\delta g_{tr}^{(0)}$. For two-derivative theories of gravity, $\nabla_i\delta g_{tr}^{(0)}$ appeared[^13] in the Stokes equations, via the gradient of a pressure term $\nabla_ip=-(4\pi T)\nabla_i\delta g_{tr}^{(0)}$. Thus we can continue to interpret the expressions for the heat current as constitutive relations for an auxiliary fluid on the horizon.
Finally, for the special case of Gauss-Bonnet gravity in $D\ge 5$ spacetime dimensions, notice that when we combine the currents with $c_{1}=c_{3}=-\frac{1}{4}\,c_{2}\equiv \tilde\alpha$ there is a cancellation of many terms and we find the simple expression $$\begin{aligned}
\label{fchere}
\delta Q^{i}_H
={4\pi T \sqrt{h_{(0)}}}\left[v^i-4\tilde\alpha\left(\nabla_{j}\nabla^{[j}v^{i]}
+G_{(0)k}^{\,\,\,\,\,i}v^{k}\right)\right]\,,\end{aligned}$$ where we have defined the Einstein tensor for the background $G_{(0)}^{ij}=R_{(0)}^{ij}-\frac{1}{2}R_{(0)}\,h_{(0)}^{ij}$. The conservation of this current is equivalent to $$\begin{aligned}
\label{gbconn}
\nabla_iv^i=4\tilde\alpha G^{\,\,ij}_{(0)} \nabla_i v_j\,,\end{aligned}$$ where in the above $\nabla$ is the covariant derivative associated with $h_{(0)}^{ij}$. Our general analysis implies that must follow from the Gauss-Bonnet equations of motion and we will show that this is true in the next section.
Returning to theories of gravity with general $c_i$, we now derive expressions for the currents when the higher-derivative corrections are perturbatively small. At zeroth order in the corrections we can use the leading Einstein equations in $D$ spacetime dimensions, $R_{\mu\nu}=DV_0/(D-2)g_{\mu\nu}$, in order to obtain expressions for the leading order corrections to the background expansions near the horizon. Using the decompositions , combined with , and we easily obtain $$\begin{aligned}
\label{eq:alpha_pert}
3H^{(1)}-\frac{1}{2}\gamma_{r}^{(1)}&=\frac{1}{4\pi T}\,\left(-R_{(0)}+\frac{D-4}{D-2}V_0 \right)\,,{\notag \\}h^{(1)}_{ij}&=\frac{2}{4\pi T}\left( R_{(0)}{}_{ij}-\frac{1}{D-2} V_0h_{(0)}{}_{ij}\right)\,.\end{aligned}$$ After substituting these leading order expressions into $\delta Q_H^{(a)}{}^{i}$ we obtain $$\begin{aligned}
\frac{1}{4\pi T \sqrt{h_{(0)}}}\delta Q_H^{(1)}{}^{i}\approx&\,\,4\,\nabla_{j}\nabla^{(j}v^{i)}
+\left( 2 R_{(0)}-2\frac{D-4}{D-2}V_0\right)v^{i}
+4\,(4\pi T)\left(\zeta^{i}+\nabla^{i}\delta g_{tr}^{(0)}\right)\,,{\notag \\}\frac{1}{4\pi T \sqrt{h_{(0)}}}\delta Q_H^{(2)}{}^{i}\approx&\,\,\nabla^2 v^i+R_{(0)}^{ij}v_{j}+\frac{2V_0}{D-2} v^{i}
+\,(4\pi T)\left(\zeta^{i}+\nabla^{i}\delta g_{tr}^{(0)}\right)\,,{\notag \\}\frac{1}{4\pi T \sqrt{h_{(0)}}}\delta Q_H^{(3)}{}^{i}\approx&\,\,\frac{2D}{D-2}{V_0}v^{i}\,,\end{aligned}$$ where we used the identity $$\begin{aligned}
\label{ricid}
\nabla_{j}\nabla^{[j}v^{i]}=\nabla_{j}\nabla^{(j}v^{i)}-\nabla^{i}\nabla_{j}v^{j}-R_{(0)}^{ij}v_{j}\,.\end{aligned}$$ The expressions for $\delta Q_H^{(a)}{}^{i}$ now only depend on the intrinsic geometry of the black hole horizon, as well as the perturbation.
In the context of holography, following the discussion in section \[currcomment\], the current flux densities at the horizon, $\delta \bar Q^i_H$, are identified with the renormalised transport[^14] current flux densities of the dual field theory. Note that by treating the higher derivative terms perturbatively allows one to consider the total on-shell action, including Gibbons-Hawking and counter terms, to be still a functional of the boundary metric (for a related discussion see [@Cremonini:2009ih]).
In the next section, for the special case of Gauss-Bonnet gravity, we show how the local current densities at the horizon, $\delta Q^i_H$, can be obtained by solving a higher derivative generalisation of the Stokes equations.
Gauss-Bonnet and generalised Stokes equations {#gbns}
=============================================
In this section we will consider Gauss-Bonnet gravity in $D\ge 5$ spacetime dimensions. Once again, for simplicity, we will focus on static background black hole spacetimes. We will use a radial Hamiltonian formalism and evaluate the momentum and Hamiltonian constraints on the horizon. This will enable us to derive an expression for the local heat current density on the horizon. In fact we will obtain an expression that differs by a magnetisation term from that given in , for reasons we will explain later. In addition we will also obtain a higher derivative version of the Stokes equations for the auxiliary fluid living on the horizon, generalising the Stokes equations found in [@Donos:2015gia]. In particular, we will obtain a closed set of equations for a subset of the perturbation at the horizon which can be solved to obtain the local heat current density on the horizon. In turn, via , by then evaluating the current flux density on the horizon we can obtain the transport current flux density on the boundary.
Radial Hamiltonian formulation
------------------------------
To obtain the radial Hamiltonian formulation of Gauss-Bonnet gravity, we essentially follow [@Liu:2008zf], who adapted the results of [@Teitelboim:1987zz]. We start by writing the bulk action as $$\begin{aligned}
\label{hdact2}
S&=\int\! d^{D}\!x
\sqrt{-g}\left[
R-V_0+ \tilde{\alpha}\left(R_{\mu\nu\rho\sigma}
R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^2\right)\right]\,,\end{aligned}$$ where we have again set $16\pi G=1$ for convenience. If we set $V_0=-(D-1)\,(D-2)$ then a unit radius $AdS_D$ solves the equation of motion when $\tilde\alpha=0$. We write the spacetime coordinates as $y^\mu=(r,x^a)$ where $$\begin{aligned}
x^a=(t,x^i)\,,\end{aligned}$$ are the coordinates for the dual field theory. We perform a radial decomposition of the bulk metric in a standard way, writing $$\begin{aligned}
ds^2&=g_{\mu\nu}dy^\mu dy^\nu=N^2dr^2+{\sigma}_{ab}(dx^a+N^adr)(dx^b+N^bdr)\,.\end{aligned}$$ The normal vector to surfaces of constant $r$ has components $n^\mu=N^{-1}(1,-N^a)$, while $n_\mu=N(1,0)$. The induced metric on the surfaces of constant $r$ is given by $\sigma_{\mu\nu}=g_{\mu\nu}-n_\mu n_\nu$ and has non-vanishing components $\sigma_{ab}$. The extrinsic curvature is defined as $K_{\mu\nu}=\tfrac{1}{2}{\cal L}_n {\sigma}_{\mu\nu}=\sigma_\mu^\rho\nabla_\rho n_\nu$ and has non-vanishing components given by $$\begin{aligned}
K_{ab}=\frac{1}{2N}(\partial_r{\sigma}_{ab}-D_aN_b-D_bN_a)\,,\end{aligned}$$ where $N_a={\sigma}_{ab}N^b$.
The bulk action is supplemented by Gibbons-Hawking type terms given by [@Myers:1987yn; @Teitelboim:1987zz] $$\begin{aligned}
S_{GH}=-2\int _{\partial {\cal M}}\! d^{D-1}x \sqrt{-{\sigma}}\left[K -\tilde\alpha\left( 4 G_{ab}K^{ab}-\tfrac{2}{3}\left( K^3-3K K_{ab}K^{ab}+2K_a^bK_b^cK_cK^a \right)\right) \right]\,,\end{aligned}$$ where $G_{ab}$ is the Einstein tensor for $\sigma_{ab}$, and ensures that the on-shell action is a functional of the boundary metric. It is also supplemented by boundary counter terms, which we shall not need here, but are discussed in [@Brihaye:2008kh; @Astefanesei:2008wz; @Liu:2008zf; @Cremonini:2009ih] (the regularised holographic stress tensor is discussed in [@Astefanesei:2008wz]).
After dropping a total derivative, the bulk action can be rewritten as $$\begin{aligned}
\label{gbaction}
S_{GB} =& \int d^D x \sqrt{-{\sigma}} N \left(\mathcal{R}-V_0+ K^2- K_{ab}K^{ab} \right){\notag \\}+&\tilde \alpha \int d^D x \sqrt{-{\sigma}}N\Bigg(\left[\mathcal{R}+ K^2- K_{ab}K^{ab}\right]^2 - 4\left[\mathcal{R}_{ab} + KK_{ab} -K_{ac}K^c_{b} \right]^2{\notag \\}&\qquad\qquad
+ \left[\mathcal{R}_{abcd}+ K_{ac}K_{bd}- K_{ad}K_{bc}\right]^2 - \frac{4}{3}K^4 + 8 K^2K_{ab}K^{ab}{\notag \\}&\qquad\qquad
-\frac{32}{3}KK^b_a K^c_bK^a_c -4\left[K_{ab}K^{ab} \right]^2+8 K^b_a K^c_b K^d_c K^a_d\Bigg)\,,\end{aligned}$$ where $\mathcal{R}_{abcd}$, $\mathcal{R}_{ab}$ and $\mathcal{R}$, are the Riemann tensor, Ricci tensor and Ricci scalar associated with $\sigma_{ab}$.
The conjugate momenta, which are densities, are defined by $$\begin{aligned}
\pi^{ab}=\frac{\delta S_{GB}}{\delta\dot \sigma_{ab}}=\frac{1}{2N}\frac{\delta S_{GB}}{\delta K_{ab}}\,.\end{aligned}$$ Explicitly we have $$\begin{aligned}
\frac{1}{\sqrt{-{\sigma}}}{\pi}^b_a&=K\delta^b_a-K^b_a+\tilde\alpha\frac{1}{\sqrt{-{\sigma}}}{\pi_{GB}}^b_a\,,\end{aligned}$$ where: $$\begin{aligned}
\frac{1}{\sqrt{-{\sigma}}}{\pi_{GB}}^b_a=&2K^b_a(K^2-K^c_dK^d_c-\mathcal{R})\cr
&-4K\mathcal{R}^b_a+4K^c_a\mathcal{R}_c^b+4K_c^b\mathcal{R}^c_a +4K^{cd}{\mathcal{R}^b}_{cad} -4KK_c^bK^c_a+4K_c^bK^c_dK^d_a\cr
&+\delta^b_a(2K\mathcal{R}-\frac{2}{3}K^3+2KK_c^dK^c_d-4K_c^d\mathcal{R}^c_d-\frac{4}{3}K_c^dK^c_eK_d^e)\,.\end{aligned}$$
The Hamiltonian density is defined as $\mathcal{H}=\pi^{ab}\dot\sigma_{ab}-\sqrt{-g}\mathcal{L}$. After dropping a total derivative $2D_a(N_b \pi^{ab})$, we find that $\mathcal{H}$ is a sum of constraints: $$\begin{aligned}
\mathcal{H}=NH+N_aH^a\,,\end{aligned}$$ where $$\begin{aligned}
\label{hcon}
\frac{1}{\sqrt{-{\sigma}}}H=V_0-&\mathcal{R}+K^2-K^a_bK^b_a-\tilde\alpha\Big[(\mathcal{R}-K^2+K^a_bK^b_a)^2{\notag \\}&-4(\mathcal{R}_{ab}-KK_{ab}+K^c_aK_{bc})^2+(\mathcal{R}_{abcd}-K_{ac}K_{bd}+K_{ad}K_{bc})^2\Big]\,,\end{aligned}$$ and $$\begin{aligned}
\label{momcon}
\frac{1}{\sqrt{-{\sigma}}}H^a=-2D_b\left(\frac{1}{\sqrt{-{\sigma}}}{\pi}^{ba} \right)\,.\end{aligned}$$ Note that we will not need to express $H$ in terms of the canonical momentum (this is done perturbatively in $\tilde\alpha$ in [@Liu:2008zf]).
Evaluating the momenta and constraints at the horizon
-----------------------------------------------------
We want to evaluate the conjugate momenta as well as the momentum and Hamiltonian constraints on a surface of constant $r$ near the horizon, and then take the limit $r\to 0$. Several details are presented in appendix \[conGB\]. We first note that for the background (i.e. unperturbed metric), at the horizon we have $\pi^{(B)}_H{}^t_t=\pi^{(B)}_H{}^i_t=\pi^{(B)}_H{}^t_i=0$ and $$\begin{aligned}
\label{backstress}
2\pi^{(B)}_H{}^i_j=(4\pi T)\sqrt {h_{(0)}}(\delta^i_j-4\tilde\alpha {G}^i_{(0)j})\,,\end{aligned}$$ where ${G}_{(0)ij}$ is the Einstein tensor associated with the horizon metric $h_{(0)ij}$.
We next note that for the perturbed metric we have $$\begin{aligned}
\label{localcurq}
\delta\tilde Q_H^i\equiv -2\pi_H{}^i_t=2\pi^{(B)}_H{}^i_j v^j\,.\end{aligned}$$ We see that $\delta\tilde Q_H^i$ differs from the expression for $\delta Q^{i}_H$ that we derived in , by a magnetisation current piece, for reasons we explain later. Expressions for other background components can be found in appendix \[conGB\].
We next consider the momentum constraints $H^a=0$ with $H^a$ given Evaluating the time component $H_t=0$ on the horizon we find the conservation condition $\partial_i \delta\tilde Q_H^i=0$ which is equivalent to $$\begin{aligned}
\label{incompresstext}
\nabla_i(\pi^{(B)}_H{}^i_j v^j)=\pi^{(B)}_H{}^i_j {\nabla}_iv^j=0\,,\end{aligned}$$ where here $\nabla$ is the covariant derivative associated with the horizon metric $h_{(0)ij}$. In fact this is the same condition as $\partial_i \delta Q_H^i=0$ that we mentioned earlier in . It can also be shown that imposing the Hamiltonian constraint $H=0$ at the horizon, where $H$ is given in , gives rise to exactly the same condition as .
We next evaluate the space component of the momentum constraint $H_j=0$ on the horizon to obtain the Stokes equations $$\begin{aligned}
\label{nsgeneq}
&0=-{\nabla}^k{\nabla}_{(j}v_{k)}-(4\pi T\zeta_i-{\nabla}_ip)(\frac{1}{2}\delta_j^i-2\tilde\alpha{G}_{(0)}{}^i_j){\notag \\}&+\tilde\alpha{\nabla}_i\Bigg[-4{\nabla}_kv^k{R}_{(0)}{}^i_j +4{\nabla}_{(j}v_{k)}{G}_{(0)}^{ik}+4{\nabla}^{(k}v^{i)}{R}^{(0)}_{jk}+4{\nabla}^{(k}v^{l)}{\,{R}^i}_{(0)kjl}{\notag \\}&+4\pi T\left(h^{ik}_{(0)}{\nabla}_{(j}v_{k)}{h^l_l}^{(1)}+{h^i_j}^{(1)}{\nabla}_kv^k-{\nabla}^{(i}v^{k)}{h_{jk}}^{(1)}-{h^{ik}}^{(1)}{\nabla}_{(j}v_{k)}+\delta^i_j(-{h_k^k}^{(1)}{\nabla}_lv^l +{h^{kl}}^{(1)}{\nabla}_kv_l)\right)
\Bigg]\,.\end{aligned}$$ Notice that these equations depend on the intrinsic geometry of the metric, associated with the metric $h_{(0)ij}$, as well as the sub-leading piece $h_{(1)ij}$ in the expansion of the background at the horizon given in . In principle, the terms involving $h_{(1)ij}$ could be related to $h_{(0)ij}$ using the full equations of motion. We will not investigate this here as we are most interested in working perturbatively in $\tilde\alpha$. Using , that implies $\nabla_i v^i$ is order $\tilde \alpha$, as well as $V_0=-(D-1)(D-2)$ we find that the Stokes equations can be written compactly as $$\begin{aligned}
\label{nsspecial}
-2\nabla^i\left(S_{ij}^{kl}\nabla_k v_l\right)=&\frac{2\pi^{(B)}_H{}^i_j}{\sqrt {h_{(0)}}}\left(\zeta_i-\frac{\nabla_i p}{4\pi T}\right)\,,\end{aligned}$$ where $\pi^{(B)}_H{}^i_j$ is given in and $S^{kl}_{ij}=S^{(kl)}_{(ij)}$ is given by $$\begin{aligned}
\label{expforess}
S^{kl}_{ij}=\left[1-\tilde\alpha 2(D-4)(D-1)\right]\delta^{(k}_i\delta^{l)}_j
-\tilde\alpha\left[2h^{(0)}_{ij}R_{(0)}^{kl}+4\delta^{(k}_{(i}R_{(0)}{}^{l)}_{j)}+4R_{(0)}{}_i{}^{(k}{}_j{}^{l)}\right]\,.\end{aligned}$$
Equations and are the main results of this section. A number of comments are in order. Firstly, and depend only upon $v_i$, $p$, $\zeta_i$ and background quantities. As such, for a fixed source $\zeta$, they give a closed set of equations which can be solved for $v_i$, $p$. In turn this gives the local conserved heat current density on the horizon via . By then evaluating the current flux density on the horizon we can obtain the transport current flux density on the boundary, via , and hence the thermal DC conductivity. We have thus successfully generalised the main results of [@Donos:2015gia; @Banks:2015wha; @Donos:2015bxe] to Gauss-Bonnet gravity.
Second, it is interesting to point out that for the special case of the homogeneous black brane solution of [@Cai:2001dz], with flat horizon, the shear viscosity for Gauss-Bonnet theory was calculated in [@Brigante:2007nu] and the result was given by $$\begin{aligned}
\label{shear}
4\pi\frac{\eta}{s}=1-\tilde \alpha 2(D-4)(D-1)\,.\end{aligned}$$ Note that that this is precisely the same coefficient appearing in the first term in . This can be understood as follows, generalising the discussion of [@Banks:2016krz] in the context of ordinary two-derivative Einstein gravity. Imagine we calculate the DC conductivity in the hydrodynamic limit with $\epsilon=k/T<<1$, where $k$ is the largest wavelength of the background holographic lattice, but we keep $\tilde\alpha/T^2$ corrections. For simplicity, we also assume that the holographic lattice has $g_{tt}^\infty=-1$. In the limit $\epsilon<<1$ the holographic lattice black hole solution will be the Gauss-Bonnet black brane solution of [@Cai:2001dz], but with spatial sections given by $g_{ij}^\infty$. The DC conductivity is still obtained using the Stokes equations , but now with the horizon metric proportional to $g_{ij}^\infty$. On the other hand, one should also be able to obtain this result using a fluid gravity approach for studying CFTs on a curved manifold with metric $g_{ab}^\infty$. Although this has not been worked out in detail as far as we know, the analysis for Gauss-Bonnet theory should closely follow the fluid-gravity formalism of two-derivative gravity [@Bhattacharyya:2008mz]. In particular, the shear viscosity as in will appear as a first order transport coefficient in the fluid equations (see [@Grozdanov:2016fkt] for a recent discussion). Finally, as in the analysis of [@Banks:2016krz], the DC perturbation can be studied within the fluid-gravity formalism using a suitable fluid flow and this will lead to a system of Stokes equations with shear viscosity .
Third, we now explain the origin of the difference between the $-2\pi_H{}^i_t$ and the expression for $\delta Q^{i}_H$ that we derived in . In moving from the action to the action we dropped a total derivative. This will not modify the bulk equations of motion. However, it can give a contribution to the current at the horizon and, since the bulk equations of motion are not modified, the extra contribution should be a magnetisation current in order that it is trivially conserved. It is possible to explicitly check this in detail, but we shall not do so here.
Fourth, we notice that we can obtain the system of equations , by varying the following Lagrangian $$\begin{aligned}
L=\int d^{D-2}x\Big[-\sqrt{h_{(0)}}\nabla^iv^jS^{kl}_{ij}\nabla_k v_l+2\pi^{(B)}_H{}^i_j\left(
v^j \zeta_i
+\frac{p}{4\pi T}\nabla_i v^j
\right)\Big]\,,\end{aligned}$$ with respect to $v^i$ and $p$. It is also interesting to note that we can obtain the local heat current density at the horizon, $\pi_H{}^i_t$, given in , if we vary with respect to $\zeta^i$: $$\begin{aligned}
\delta\tilde Q_H^i=\frac{\delta L}{\delta \zeta^i}\,.\end{aligned}$$ By taking an additional derivative with respect to $\zeta_j$, we can easily deduce that the thermal DC conductivity matrix of the dual CFT is a symmetric matrix.
Fifth, returning to the Stokes equations, if we multiply by $v^j$ and integrate we get $$\begin{aligned}
\int d^{D-2}x \sqrt{h_{(0)}}2\nabla^iv^jS^{kl}_{ij}\nabla_k v_l=\int d^{D-2}x\delta\tilde Q_H^i\zeta_i\,.\end{aligned}$$ When $\tilde\alpha=0$ the left hand side is positive definite and this is associated with the fact that the DC thermal conductivity is a positive definite matrix. It would be interesting to examine the behaviour when $\tilde\alpha\ne 0$.
Sixth, observe that if the horizon admits a Killing vector, then we can solve the source free (i.e. $\zeta=0$) generalised Stokes equations by taking $v^i$ to be the Killing vector, and hence satisfying $\nabla_{(i}v_{j)}=0$, with $p=0$. This means that whenever the horizon admits a Killing vector then there will not be a unique solution to the Stokes equations. This is related to the fact that finite DC conductivities require translation invariance to be explicitly broken.
A final observation is that if we define $\bar v^i= (\delta^i_j-4\tilde\alpha {G}^i_{(0)j})v^j$, then we have $\nabla_i\bar v^i=0$ and the fluid is incompressible. The Stokes equation can be written in terms $\bar v^i$, but as the resulting expression is not particularly illuminating, we omit it.
One dimensional lattices
------------------------
We now consider the class of background black hole solutions of Gauss-Bonnet gravity in $D$ spacetime dimensions that break translations in just one of the spatial directions of the dual field theory. We assume that the $(D-2)$-dimensional horizon geometry depends on the spatial coordinate $x$ and is independent of the remaining $D-3$ spatial coordinates $x^I$. The horizon metric is given by $$\begin{aligned}
\label{onedform}
ds^{2}_{H}=\gamma(x) \,dx^{2}+k_{IJ}(x)dx^I dx^J\,,\end{aligned}$$ and both $k_{IJ}(x)$, $\gamma(x)$ depend periodically on $x$, with period $L$. We define $k\equiv \det k_{IJ}$. Note that it is possible to do a coordinate transformation at the horizon to set $\gamma=1$. However, if we want to use the same spatial coordinate $x$ on the holographic boundary and at the horizon, which is natural in numerically constructing holographic lattice black holes, generically we have $\gamma\ne 1$.
It is helpful to now define the matrix $M^I{}_J\equiv \gamma^{-1/2}k^{IK}\partial_x k_{KJ}$. We will also raise and lower indices via: $M_{IJ}\equiv k_{IK}M^K{}_J$ and $M^{IJ}\equiv k^{IK}M^J{}_K$. The non-vanishing Christoffel symbols for the horizon metric are given by $$\begin{aligned}
\Gamma^x_{xx}=\frac{1}{2}\partial_x \ln \gamma\,,\quad
\Gamma^x_{IJ}=-\frac{1}{2}\gamma^{-1/2}M_{IJ}\,,\quad
\Gamma^I_{xJ}=\frac{1}{2}\gamma^{1/2}M^{I}{}_{J}\,.\end{aligned}$$ The associated components of the Riemann tensor are given by $$\begin{aligned}
R_{IJKL}&=-\frac{1}{4}(M_{IK}M_{JL}-M_{IL}M_{JK})\,,{\notag \\}R_{xIxJ}&=-\frac{1}{2}\gamma^{1/2}k_{IK}\partial_x{M^K}_J-\frac{1}{4}\gamma M_{IK}{M^K}_J\,,\end{aligned}$$ and $R_{xIJK}=0$.
To proceed we write the perturbed heat current at the horizon as $$\begin{aligned}
\delta \tilde Q^i_H=(4\pi T)\sqrt{h_{(0)}}A^i_j v^j,\qquad A^i_j\equiv \delta^i_j-4\tilde\alpha {G}_{(0)}{}^i_j\,.\end{aligned}$$ We find that $A^x_I=A^I_x=0$ as well as $$\begin{aligned}
A\equiv A^x_x=1+\frac{\tilde{\alpha}}{2}\left[Tr (M^2)-(Tr M)^2\right]\,,\end{aligned}$$ where we have defined $Tr M\equiv {M^K}_K$, $Tr (M^2)\equiv {M^I}_K{M^K}_I$ and so on. The condition $\partial_i \delta \tilde Q^i_H=0$ can then be solved with $v^I=0$ and $$\begin{aligned}
v^x=(\gamma k)^{-1/2}A^{-1}v_0,\qquad \delta\tilde Q^x_H=4\pi T v_0\,,\end{aligned}$$ where $v_0$ is constant.
Next we write the Stokes equations as $$\begin{aligned}
-2\nabla_i(B^i_j)= A_j^i[(4\pi T)\zeta_i-\nabla_i p]\,,\qquad B_{ij}\equiv S_{ij}^{kl}\nabla_k v_l\,.\end{aligned}$$ We now take the $x$ component of this equation, multiply by $A^{-1}$ and then integrate over a period of $x$. After an integration by parts we then get $$\begin{aligned}
\label{zetaeq2}
4\pi T\zeta_x=\int \,\left[ 2 k^{1/2}B^x_x \partial_x\left(A^{-1}k^{-1/2}\right)
+A^{-1} B^{IJ}\partial_x k_{IJ}\right]\,.\end{aligned}$$ where $\int\equiv(L)^{-1}\int_0^{L} dx$. We next notice that we can express the components of $B$ in terms of the constant $v_0$ using the expressions: $$\begin{aligned}
\nabla_x v_x&=(\partial_x-\Gamma^x_{xx})\gamma^{1/2}(g_{d-1})^{-1/2}A^{-1}v_0\,,{\notag \\}\nabla_x v_I&=\nabla_I v_x=0\,,{\notag \\}\nabla_I v_J&=-\Gamma^x_{IJ} \gamma^{1/2}(g_{d-1})^{-1/2}A^{-1}v_0\,.\end{aligned}$$ Thus, can be used to obtain an expression relating $\zeta_x$ in terms of $v_0$, and hence in terms of $\delta\tilde Q^x_H$. Since the $\delta\tilde Q^x_H$ is a constant for these one-dimensional lattices, and moreover the bulk currents are independent of the radius, we can obtain $\delta\tilde Q^x_\infty$ in terms of $\zeta_x$ and hence the thermal conductivity.
After some lengthy but straightforward calculation we find that at leading order in $\tilde\alpha$ the conductivity can be written as $$\begin{aligned}
\kappa=\frac{(4\pi)^2 T}{X}\,,\end{aligned}$$ where $$\begin{aligned}
X=\int\frac{1}{2}\gamma^{1/2}k^{-1/2}\left[ Tr(M^2)+(Tr M)^2+\tilde\alpha C\right]\,,\end{aligned}$$ with $$\begin{aligned}
C=&-2(D-4)(D-1)[Tr(M^2)+(Tr M)^2)]
-\frac{2}{3}[Tr M][ Tr (M^3)]{\notag \\}&- Tr(M^4)+\frac{2}{3}(Tr M)^4
+(Tr M)^2 Tr(M^2)\,.\end{aligned}$$ A few comments are in order. Firstly, the final expression is invariant under reparametrisations of the coordinate $x$, as expected. Second, when $\tilde\alpha=0$ the result is consistent with that derived in section 4.2 of [@Banks:2015wha]. Third, when $D=4$, the matrix $M^I{}_J$ is just a number and it is simple to show that $\kappa$ is independent of $\tilde\alpha$ which is consistent with the fact that the Gauss-Bonnet term is topological for $D=4$ and hence does not contribute to the equations of motion.
In the special case that $k_{IJ}=\phi\delta_{IJ}$, for a periodic function $\phi(x)$, we can simplify the expressions a little. We find $$\begin{aligned}
X=\int \gamma^{1/2}\frac{(D-3)}{2}\phi^{-\frac{D-3}{2}}\left(\gamma^{-1/2}\partial_x \ln \phi\right)^2
\left(
D-2+\tilde\alpha C\right)\,,\end{aligned}$$ with $$\begin{aligned}
C=(D-4)\left( -2(D-1)(D-2)+\frac{(D-2)(2D-3)}{3}\left(\gamma^{-1/2}\partial_x \ln \phi\right)^2\right)\,.\end{aligned}$$ In order to determine the effect of the Gauss-Bonnet term on the conductivity one needs to take into account that the horizon metric functions $\gamma,\phi$ will also receive corrections of order $\tilde\alpha$. It would certainly be interesting to explore this further.
Final Comments {#fincom}
==============
In this paper we have successfully generalised many of the results of [@Donos:2015gia; @Banks:2015wha; @Donos:2015bxe] concerning DC conductivities to theories of gravity with higher derivative terms. The first main result is the identification of suitable bulk quantities that allow one to relate current fluxes at the black hole horizon to suitably defined transport current fluxes of the dual field theory. In section \[gendisc\] and \[gbonsec\] we achieved this using a Kaluza-Klein reduction on the time direction. The original approach that was used in [@Donos:2015gia; @Banks:2015wha; @Donos:2015bxe], which came from [@Donos:2014cya], worked in a gauge in which the DC sources arise as time dependent perturbations and used a two-form that arises in the derivation of Smarr formula. It would be interesting to see how that approach generalises to higher derivative gravity; for the case of Lovelock gravity the work of [@Kastor:2008xb; @Kastor:2010gq; @Liberati:2015xcp] should be helpful. We also anticipate connections with the work of [@Wald:1993nt; @Iyer:1994ys].
The second main result was to obtain a generalised set of Stokes equations on the black hole horizon for the case of static black hole backgrounds in the context of Gauss-Bonnet gravity. These equations form a closed set of equations for a subset of the perturbation at the horizon and by solving them one can obtain the thermal currents at the horizon. It should be possible to generalise these results from the case of static black holes to stationary black holes and it would be interesting to identify the additional terms that will enter the Stokes equations, including Coriolis terms.
We obtained the Stokes equations for Gauss-Bonnet theory using a radial Hamiltonian formalism and it should be straightforward to generalise this to arbitrary Lovelock theories. For general higher dimensional theories of gravity one will not have such a formalism to hand. Nevertheless, if we write the Einstein equations as $E_{\mu\nu}=0$ then by considering the projections $n^\mu n^\nu E_{\mu\nu}=0$ and $n^\mu\sigma_\nu^\rho E_{\mu\rho}=0$, where $n^\mu$ is the normal and $\sigma_\nu^\rho$ are the normal and projector for the radial slicing, and then evaluating at the horizon should give a generalised closed set of Stokes equations for more general theories.
With the results of this paper it would also be interesting to construct and study specific examples of holographic lattices, to investigate the impact of the higher derivative couplings on the DC conductivity. We note that a specific holographic model involving Gauss-Bonnet gravity coupled to a gauge-field and massless scalars was investigated in [@Cheng:2014tya]. In this work the momentum dissipation just arises from the massless scalar fields, as in the construction [@Andrade:2013gsa], and it was shown that the DC thermoelectric conductivity is independent of the Gauss-Bonnet coupling. However, this behaviour is an exception, arising from the simple mechanism for momentum dissipation. In a parallel direction, it would also be interesting if our results could be used to place bounds on thermal conductivities along the lines of [@Grozdanov:2015qia; @Grozdanov:2015djs; @Fadafan:2016gmx].
Acknowledgements {#acknowledgements .unnumbered}
================
The work is supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013), ERC Grant agreement ADG 339140. The work of JPG is also supported by STFC grant ST/L00044X/1, EPSRC grant EP/K034456/1, JPG is also supported as a KIAS Scholar and as a Visiting Fellow at the Perimeter Institute.
Currents in the dual CFT {#currcomment}
========================
In this appendix we discuss how the currents defined in section \[secbhgih\] are related to the dual CFT, including a discussion of the counter-term contributions.
We start with the bulk contributions to the currents. To simplify the presentation, we just discuss theories of bulk gravity without higher derivative terms; the generalisation to theories with higher derivatives is straightforward, using the analysis and discussion at the beginning of section \[gbonsec\]. We recall the derivation of the two co-closed two forms as given in . In varying the dimensionally reduced action we obtain a boundary term and on shell we find $$\begin{aligned}
\delta S=2\int_{M_{D-1}}\,d^{D-1}y\,\sqrt{\gamma_{D-1}}\nabla_m(V^{mn}\delta\beta_n+W^{mn}\delta\alpha_n)\,.\end{aligned}$$ If we define variations at the AdS boundary at $r\to\infty$ via: $\delta\beta_i^\infty=\lim_{r\to\infty}\delta \beta_i$ and $\delta\alpha_i^\infty=\lim_{r\to\infty}\delta \alpha_i$ we have $$\begin{aligned}
\label{blksd}
\frac{\delta S}{\delta\beta_i^\infty}=J_{\infty}^i\,,\qquad
\frac{\delta S}{\delta\alpha_i^\infty}=-Q_{\infty}^i\,,\end{aligned}$$ where $J_{\infty}^{i}$ and $Q^{i}_{\infty}$ as the boundary limits of the bulk currents defined in .
Now the full bulk action that we should be considering in holography needs to be supplemented by a boundary contribution: $S_{Tot}\equiv S+S_{Bdy}$, where $S_{Bdy}$ has two key features. The first is that it has counterterms to ensure that all divergences are cancelled. The second is that it includes a Gibbons-Hawking term to ensure that on-shell, $S_{Tot}$ is a functional of the boundary metric, $g_{ab}^\infty$, and gauge-field $A_a^\infty$, where we have introduced the field theory coordinates, $x^a$, with $$\begin{aligned}
x^a=(t,x^i)\,.\end{aligned}$$ As usual, the holographic stress tensor density of the dual CFT is defined as $[T_{Tot}]^{ab}=2\delta S_{Tot}/\delta g_{ab}^\infty$ and the holographic boundary current density as $J^{a}_{Tot}=\delta S_{Tot}/\delta A_{a}^\infty$. Using the chain rule, and recalling the definition of $\alpha_i$ and $\beta_i$ in we have $$\begin{aligned}
\frac{\delta S_{Tot}}{\delta\beta_i^\infty}=J_{Tot}^i\,,\qquad
\frac{\delta S_{Tot}}{\delta\alpha_i^\infty}=-Q_{Tot}^i\equiv ([T_{Tot}]^{i}{}_t+A_tJ^i_{Tot})\,.\end{aligned}$$
We thus see that $J_{\infty}^{i}$, $Q^{i}_{\infty}$ are the contributions from the bulk action to the finite renormalised currents of the dual CFT given by $J_{Tot}^{i}$, $Q^{i}_{Tot}$. In general $J_{\infty}^{i}$, $Q^{i}_{\infty}$ are divergent quantities. For the background geometry currents given in , the contributions from $S_{Bdy}$ will ensure that the total magnetisation densities are finite. $S_{Bdy}$ will also give regulating contributions to the currents that depend on the DC perturbation as given in . For the specific case of Einstein-Maxwell theory in $D=4,5$ spacetime dimensions, it was explicitly shown in [@Donos:2015bxe] that the contributions from $S_{Bdy}$ regulate the magnetisation terms given in as well as regulating the magnetisation currents given on the right hand side of . The net effect of the contribution of $S_{Bdy}$ is thus to ensure that in the expressions for the perturbed current densities and the perturbed transport current densities given in - we can replace all boundary quantities with the finite renormalised quantities. In particular, associated with we have that the renormalised transport current flux densities of the dual CFT are the same as the horizon current flux densities (which are, of course, finite quantities). Why this should be true in general is expanded upon below.
Before doing that we calculate the holographic charge densities using the dimensional reduced formalism of section 2. Starting with the dimensionally reduced action with , by considering the on-shell variation of the action with respect to $A_t^\infty$ we find the charge density $$\begin{aligned}
J^t_\infty&=\left[(\gamma^{(B)}_{D-1})^{1/2}\frac{\delta \mathcal{L}}{\delta u_r}\right]_\infty\,.\end{aligned}$$ Similarly, to get $T^{tt}_\infty$ we can vary with respect to $g_{tt}^\infty$. Using the fact that $g_{tt}=-H^2$ we find that on-shell $$\begin{aligned}
[T^t{}_t]_\infty=\left[(\gamma^{(B)}_{D-1})^{1/2}H\frac{\delta \mathcal{L}}{\delta h_r}\right]_\infty\,.\end{aligned}$$ With these formulae in hand, we can now work out how they depend on the DC perturbation. Using and we immediately deduce that $\delta J^t_\infty$ is independent of $\phi_T$ and $\phi_E$ and hence is globally defined on a fundamental domain. On the other hand, with $Q^t_\infty\equiv -\left[T^t_t+A_tJ^t\right]_\infty$, we find $\delta Q^t_\infty$ is not globally defined but rather that $\delta Q^t_\infty +\phi_TQ^t_{(B)}+\phi_E J^t_{(B)}$ is globally defined.
Counterterm corrections to magnetisation {#cterms}
----------------------------------------
We now show that the counterterm contribution to the boundary action only gives corrections to the magnetisation terms that were introduced in section \[gendisc\]. The argument follows from the existence of a time-like Killing vector at the boundary, defined as the $r\rightarrow \infty$ limit of the bulk Killing vector. If $x^a=\{t, x^i\}$ are the boundary coordinates we can carry out a Kaluza-Klein reduction on the boundary with respect to $\partial_t$. Using a similar argument as in section \[gendisc\], we take the boundary counterterm action to have the following form $$\begin{aligned}
S_{ct} =\int d^{D-2}x \sqrt{\tilde{\gamma}} \tilde{H} \mathcal{L}_{ct}(\tilde{h}, \tilde{u}, \tilde{v}, \tilde{w}, \tilde{\gamma}_{ij})\,,\end{aligned}$$ where $D$ is the number of bulk dimensions, the metric and gauge field on the boundary are $$\begin{aligned}
d\tilde{s}^2 &= -\tilde{H}^2(dt+ \tilde{\alpha})^2 +\tilde{\gamma}_{ij}dx^i dx^j\,,{\notag \\}\tilde{A}&=\tilde{A}_{t}\,\left( dt+\tilde{\alpha} \right)+\tilde{\beta}\,,\end{aligned}$$ and we have defined $$\begin{aligned}
\tilde{h} = d \log \tilde{H}\,, \qquad
\tilde{u}=\tilde{H}^{-1}d\tilde{A}_{t}, \qquad
\tilde{v}=d\tilde{\beta}+\tilde{A}_{t}\,d\tilde{\alpha},\qquad
\tilde{w} = \tilde{H} d\tilde{\alpha}\,,\end{aligned}$$ where $\tilde{h},\tilde{u}$ are one-forms and $\tilde{v},\tilde{w}$ are two-forms, all living in $M_{D-2}$, obtained by taking the $r\to\infty$ limit of the bulk quantities. We have used a tilde to distinguish fields defined at the boundary with the bulk fields as used in the draft.
Now $S_{ct}$ will contribute to the electric and heat current densities via $$\begin{aligned}
J^i_{ct}&
= \frac{\delta S_{ct}}{\delta\tilde{ \beta}_i} = \partial_j {M}^{ij}_{ct} \,,{\notag \\}Q^i_{ct}&
= -\frac{\delta S_{ct}}{\delta\tilde{ \alpha}_i} = \partial_j{M}_{Tct}^{ij} \,,\end{aligned}$$ where $$\begin{aligned}
{M}^{ij}_{ct}&
= 2\sqrt{\tilde{\gamma}}\tilde{H}\frac{\delta \mathcal{L}_{ct}}{\delta \tilde{v}_{ij}}\,,{\notag \\}{M}_{Tct}^{ij} &
=-2\sqrt{\tilde{\gamma}}\tilde{H}^2 \frac{\delta \mathcal{L}_{ct}}{\delta \tilde{w}_{ij}}-2\sqrt{\tilde{\gamma}}\tilde{H}\tilde{A}_t \frac{\delta \mathcal{L}_{ct}}{\delta \tilde{v}_{ij}}\,.\end{aligned}$$ Note that $\tilde\alpha_i$, $\tilde\beta_i$ were denoted $\alpha_i^\infty$, $\beta_i^\infty$, respectively, in .
We now consider the bulk DC perturbation of , where for simplicity we again assume that the locally defined functions $\phi_E$ and $\phi_T$ are independent of the radial coordinate. On the boundary the DC perturbation takes the form $$\begin{aligned}
&\tilde{H}=\tilde{H}^{(B)}\left(1-{\phi}_{T}\right) +\delta \tilde{H}, \quad \tilde{\alpha}=\tilde{\alpha}^{(B)} \left(1+{\phi}_{T} \right)+\delta \tilde{\alpha}\,,
\quad \tilde{\gamma}_{ij}=\tilde{\gamma}_{ij}^{(B)}+\delta \tilde{\gamma}_{ij}\,,\notag\\
&\tilde{A}_{t}=\tilde{A}_{t}^{(B)}\,\left(1-{\phi}_{T} \right)+{\phi}_{E}+\delta \tilde{A}_{t},\qquad
\tilde{\beta}=\tilde{\beta}^{(B)}-{\phi}_{E}\,\tilde{\alpha}^{(B)}+\delta\tilde{\beta}\,,\end{aligned}$$ with $\delta \tilde{H},\delta\tilde{\alpha},\delta \tilde{A}_t,\delta\tilde{\beta}$ and $\delta \tilde{\gamma}_{ij}$ all globally defined perturbations on $M_{D-2}$. Note that $\tilde{h}$, $\tilde{u}$, $\tilde{v}$ and $\tilde{w}$ are all globally defined. Therefore, for this perturbation we have: $$\begin{aligned}
\delta {M}^{ij}_{ct} &= \bar\delta{M}^{ij}_{ct} - {{M}^{ij}_{ct}}^{(B)}{\phi}_T\,,{\notag \\}\delta{M}_{Tct}^{ij} &= \bar\delta{M}_{Tct}^{ij}- 2 {{M}_{Tct}^{ij}}^{(B)}{\phi}_T-{{M}_{ct}^{ij}}^{(B)}{\phi}_E \,,\end{aligned}$$ where $ \bar\delta{M}^{ij}_{ct}$ and $ \bar\delta{M}_{Tct}^{ij}$ are the pieces independent of $\phi_E,\phi_T$ and hence are globally-defined densities defined and $ {{M}_{ct}^{ij}}^{(B)}$ and $ {{M}_{Tct}^{ij}}^{(B)}$ are the corresponding values in the background. We thus see that the counter terms will give corrections that renormalise the magnetisations given in . In turn this means, in effect, that in - we can replace all boundary quantities with the finite renormalised quantities.
The above analysis is valid for two derivative theories in the bulk with counterterms that also have two-derivative terms. If we relax these conditions, either for a higher derivative bulk theory, or for a two derivative theory of gravity with higher derivative boundary counterterms, then we should carry out suitable generalisations analogous to the discussion at the beginning of section \[gbonsec\].
To conclude this appendix, we briefly illustrate the above for a counterterm action that appears for theories of gravity with two derivatives in $D=4,5$: $$\begin{aligned}
S_{ct} = -\frac{1}{D-3}\int dt d^{D-2}x \sqrt{-\tilde\sigma} \left(\tilde R_{D-1} + 2 (D-3)(D-2)\right)\,,\end{aligned}$$ where $\tilde\sigma$ is the $(D-1)$-dimensional boundary metric. Using the KK decomposition, this leads to $$\begin{aligned}
\mathcal{L}_{ct}=-\frac{1}{D-3}\left(R_{D-2} +\frac{1}{4}\tilde{w}^2+ 2 (D-3)(D-2)\right)\,.\end{aligned}$$ From here we obtain $$\begin{aligned}
\mathcal{M}_{Tct}^{ij} &= \frac{\sqrt{\tilde{\gamma}}\tilde{H}^2 }{(D-3)} w^{ij}\,,\end{aligned}$$ giving a contribution to the current of the form $$\begin{aligned}
Q^i_{ct}&= \partial_j\left(\frac{\sqrt{\tilde{\gamma}}\tilde{H}^2 }{(D-3)} w^{ij}\right) \,.\end{aligned}$$ This agrees with the result in appendix B of [@Donos:2015bxe] (to make the comparison one can use the result to write $R^{\hat m}{}_{0}=\tfrac{1}{2}H^{-2}\nabla_{\hat n}(H^2 w^{\hat n \hat m})$).
Currents at the horizon for higher derivative theories {#details}
======================================================
In this appendix we outline a few more details on how we calculated the currents for the the higher derivative theory of gravity discussed in section \[gbonsec\]. For simplicity, as in the text, we consider static backgrounds and set $\alpha^{(B)}=0$ in .
For the background black hole geometries we have the expansions: $$\begin{aligned}
\label{nhorexpaapp}
\gamma_{rr}^{(B)}&=\frac{1}{4\pi T\,r}\,\left( 1+\gamma_{rr}^{(1)}\,r\right)+\mathcal{O}(r)\,,\quad
\gamma^{(B)}_{ij}=h^{(0)}_{ij}+h^{(1)}_{ij}\,r+\mathcal{O}(r^{2})\,,\quad
\gamma^{(B)}_{ir}=\mathcal{O}(r)\,,{\notag \\}H^{(B)}{}^{2}&= 4\pi T\,r\,\left(1+2\,H^{(1)}\,r \right)+\mathcal{O}(r^{3})\,.\end{aligned}$$ We thus have $$\begin{aligned}
\label{nhorexpaup}
&\gamma^{(B)rr}={4\pi Tr}\,\left( 1-\gamma_{rr}^{(1)}\,r\right)+\mathcal{O}(r^3),\quad
\gamma^{(B)ri}=\mathcal{O}(r^2),\quad
\gamma^{(B)ij}=h^{(0)ij}-h^{(1)ij}\,r+\mathcal{O}(r^{2}),\quad{\notag \\}&(\gamma^{(B)}_{D-1})^{1/2}=\frac{\sqrt{h^{(0)}}}{(4\pi T r)^{1/2}}\left(1+r\left(\frac{1}{2}{\gamma^{(1)}_{rr}}+\frac{1}{2}h^{(1)}{}^{i}{}_{i} \right) \right)+\mathcal{O}(r^{3/2})\,,{\notag \\}&h_r=\frac{1}{2r}+H^{(1)}+\mathcal{O}(r)\,,\end{aligned}$$ where indices on $h^{(1)}$ are raised with $h^{(0)}$ e.g $h^{(1)}{}^{i}{}_{i}=h^{(0)ij}h^{(1)}_{ij}$. We next expand the Christoffel connection for the $(D-1)$-dimensional metric $\gamma_{D-1}$ at the horizon to find $$\begin{aligned}
\bar\Gamma^{r}_{rr}&=-\frac{1}{2r}+\frac{1}{2}\gamma^{(1)}_{rr}+\mathcal{O}(r)\,,{\notag \\}\bar\Gamma^{r}_{ri}&=\mathcal{O}(r)\,,{\notag \\}\bar\Gamma^{r}_{jk}&=-(4\pi T)\frac{r}{2}h^{(1)}_{jk}+\mathcal{O}(r^2)\,,{\notag \\}\bar\Gamma^{i}_{rr}&=\mathcal{O}(1)\,,{\notag \\}\bar\Gamma^{i}_{rj}&=\frac{1}{2}h^{(1)}{}^i{}_j+\mathcal{O}(r)\,,{\notag \\}\bar\Gamma^{i}_{jk}&=\gamma^{i}_{jk}+\mathcal{O}(r)\,,\end{aligned}$$ where $\gamma^{i}_{jk}$ is associated with $h^{(0)}_{ij}$. Similarly, the components of the Riemann tensor for the $(D-1)$-dimensional metric $\gamma_{D-1}$ at the horizon are given by $$\begin{aligned}
\bar R^r{}_{rri}&=\mathcal{O}(1)\,,{\notag \\}\bar R^r{}_{rij}&=\mathcal{O}(r)\,,{\notag \\}\bar R^r{}_{irj}&=-\frac{1}{4}(4\pi T) h^{(1)}_{ij}+\mathcal{O}(r)\,,{\notag \\}\bar R^r{}_{kij}&=\mathcal{O}(r)\,,{\notag \\}\bar R^k{}_{rri}&=\frac{1}{4r}h^{(1)}{}^k{}_i+\mathcal{O}(1)\,,{\notag \\}\bar R^k{}_{rij}&=\mathcal{O}(1)\,,{\notag \\}\bar R^k{}_{lri}&=\mathcal{O}(1)\,,{\notag \\}\bar R^k{}_{lij}&=R^{(0)k}{}_{lij}+\mathcal{O}(r)\,.\end{aligned}$$ Hence for the Ricci tensor we have $$\begin{aligned}
\label{tink1}
\bar R_{rr}&=-\frac{1}{4r}h^{(1)}{}^i{}_i+\mathcal{O}(1)\,,{\notag \\}\bar R_{ri}&=\mathcal{O}(1)\,,{\notag \\}\bar R_{ij}&=R^{(0)}_{ij}-\frac{1}{4}(4\pi T) h^{(1)}_{ij}+\mathcal{O}(r)\,,\end{aligned}$$ and for the Ricci scalar $$\begin{aligned}
\label{tink2}
\bar R&=R^{(0)}-\frac{1}{2}(4\pi T)h^{(1)}{}^i{}_i+\mathcal{O}(r)\,.\end{aligned}$$
Since we are in the static case with $\alpha^{(B)}=0$ we have from that $$\begin{aligned}
\delta w&=H^{(B)}\,d\delta\alpha\,,\end{aligned}$$ and hence using ,, we deduce that as $r\to 0$ $$\begin{aligned}
\delta w_{ri}&=-\frac{1}{(4\pi T)^{1/2} r^{3/2}} v_i+\frac{1}{(4\pi T)^{1/2} r^{1/2}}\left( \zeta_{i}+\partial_{i}\delta g_{tr}^{(0)}-H^{(1)}v_{i}\right)+\mathcal{O}(r^{1/2})\,,{\notag \\}\delta w_{ij}&=\frac{1}{(4\pi T)^{1/2} r^{1/2}}\,dv_{ij}+\mathcal{O}(r^{1/2})\,,\end{aligned}$$ as well as $$\begin{aligned}
\label{wuppa}
\delta w^{ri}&=-(4\pi T)^{1/2}\frac{1}{r^{1/2}}\left(v^i-r\left( \zeta^{i}+\nabla^{i} \delta g_{tr}^{(0)}
-H^{(1)}v^i+\gamma^{(1)}_{rr}v^i+h^{(1)ij}v_j\right)\right) +\mathcal{O}(r^{3/2})\,,{\notag \\}\delta w^{ij}&=\frac{1}{(4\pi T)^{1/2}}\frac{1}{r^{1/2}}2\nabla^{[i}v^{j]}+\mathcal{O}(r)^{1/2}\,.\end{aligned}$$
As explained in the text, the currents at the horizon are given by $$\begin{aligned}
\label{timnow}
\delta Q^i_H&=(4\pi T)\sqrt{h^{(0)}} v^i
+c_1\delta Q^{(1)}_H{}^{i}+c_2\delta Q^{(2)}_H{}^{i}+c_3\delta Q^{(3)}_H{}^{i}\,,\end{aligned}$$ where $$\begin{aligned}
\frac{1}{4\pi T \sqrt{h_{(0)}}}\delta Q^{(a)}_H{}^{i}\equiv&
-\frac{2r^{1/2}}{(4\pi T)^{1/2}}\,\left[\delta(\delta_{w}\mathcal{L}^{(a)})^{ri}\right]_H\,.\end{aligned}$$ We next calculate each of the three variations appearing on the right hand side of . We start by analysing $\delta [\delta_{w}\mathcal{L}^{(3)})]^{ri}$ as $r\to 0$. From , since $w^{(B)}=0$ in the static background, we find $$\begin{aligned}
\delta[\delta_{w}\mathcal{L}^{(3)}]^{ri}&=R\,\delta w^{ri}\,,{\notag \\}&=(\bar R-2\nabla_m h^m-2h^2)\delta w^{ri}\,,\end{aligned}$$ where we used . If we now evaluate at the horizon, we find a cancellation of the $1/r$ singular terms in $\nabla_m h^m$ and $h^2$. Using as well as we obtain $$\begin{aligned}
\frac{1}{4\pi T \sqrt{h_{(0)}}}\delta Q^{(3)}_H{}^{i}
&=R^{(0)}v^i-\,(4\pi T)
\left(3H^{(1)}-\frac{1}{2}\gamma^{(1)}_{rr}+h^{(1)}{}^j{}_j\right)v^{i}\,,\end{aligned}$$ which is the result given in .
Next we want to consider the limit of $\delta [\delta_{w}\mathcal{L}^{(2)})]^{ri}$ as $r\to 0$. From the definition in and for a static background with $w^{(B)}=\Lambda^{(1)(B)}=0$, we have $$\begin{aligned}
\label{lambtwovar}
\delta [\delta_{w}\mathcal{L}^{(2)}]^{ri}&=-2\,\Lambda^{(2)}{}_{p}{}^{[r}\delta w^{i]p}+H\,d\left(H^{-1}\,\delta \Lambda^{(1)} \right)^{ri}+\Lambda^{(0)}\,\delta w^{ri}\,.\end{aligned}$$ Next consider each of the three terms in as $r\to 0$. We have $$\begin{aligned}
\label{pthree}
-2\,\Lambda^{(2)}{}_{p}{}^{[r}\delta w^{i]p}&=\Lambda^{(2)}{}_r{}^r\delta w^{ri}
+\Lambda^{(2)}{}_j{}^i\delta w^{rj}+\Lambda^{(2)}{}_j{}^r\delta w^{ij}\,,{\notag \\}&\to -(4\pi T)\frac{1}{2}\left(\frac{1}{2}h^{(1)}{}_i{}^i+3 H^{(1)}-\frac{1}{2}\,\gamma_{rr}^{(1)} \right)\delta w^{ri}
+(R^{(0)}_j{}^i-\frac{(4\pi T)}{2}h^{(1)}_j{}^i)\delta w^{rj}\,,\end{aligned}$$ as $r\to 0$, and we note in particular that the last term in the first line does not contribute. We next calculate $$\begin{aligned}
\delta\Lambda^{(1)}{}^{i}&=\frac{1}{2\,r^{1/2}}\,\frac{1}{(4\pi T)^{1/2}}\left(2\,\nabla_{j}\nabla^{[j}v^{i]}+4\pi T\left( \zeta^{i}+\nabla^{i} \delta g_{tr}^{(0)}\right)-M^{ij}v_{j} \right)+\mathcal{O}(r^{1/2})\,,{\notag \\}\delta\Lambda^{(1)}{}^{r}&=\frac{(4\pi T)^{1/2}}{2\,r^{1/2}}\,\nabla_{i}v^{i}+\mathcal{O}(r^{1/2})\,,\end{aligned}$$ where, as in appendix D of [@Banks:2015wha], $$\begin{aligned}
M^{ij}=(4\pi T)\left(h^{(0)}{}^{ij}\,\left(3\,H^{(1)}-\frac{1}{2}\,\gamma_{rr}^{(1)} \right)+\frac{1}{2}h^{(0)ij}h^{(1)}{}^k{}_k
-h^{(1)ij}\right)\,.\end{aligned}$$ We then find that we can write, as $r\to 0$, $$\begin{aligned}
\label{ptwo}
H\,d\left(H^{-1}\delta\Lambda^{(1)} \right)^{ri}&\to -(4\pi T)\delta\Lambda^{(1)i}-\nabla^i\delta\Lambda^{(1)r}\,.\end{aligned}$$ As $r\to 0$, we also have $$\begin{aligned}
\label{pone}
\Lambda^{(0)}\,\delta w^{ri}&\to \frac{(4\pi T)}{2}\left[3\,H^{(1)}-\frac{1}{2}\,\gamma_{rr}^{(1)} +
\frac{1}{2}h^{(1)}{}^k{}_k\right]\delta w^{ri}\,,\end{aligned}$$ which will cancel a term in . Combining all of these three terms we conclude that as $r\to 0$ we have $$\begin{aligned}
&\delta [\delta_{w}\mathcal{L}^{(2)}]^{ri}=-\frac{(4\pi T)^{1/2}}{r^{1/2}}\,R_{(0)}^{ij}v_{j}+\frac{(4\pi T)^{3/2}}{r^{1/2}}\frac{1}{2}h^{(1)}{}^{ij}v_j-\frac{(4\pi T)^{1/2}}{r^{1/2}}\,\frac{1}{2}\,\nabla^{i}\nabla_{j}v^{j}\notag\\
&-\frac{(4\pi T)^{1/2}}{r^{1/2}}\,\frac{1}{2}\left(2\,\nabla_{j}\nabla^{[j}v^{i]}+4\pi T\left( \zeta^{i}+\nabla^{i} \delta g_{tr}^{(0)}\right)-M^{ij}v_{j} \right)
+\mathcal{O}(r^{1/2})\,.\end{aligned}$$ This leads to the expression for $\delta Q_H^{(2)}{}^{i}$ given in .
Finally, we want to calculate the limit of $\delta [\delta_{w}\mathcal{L}^{(1)})]^{ri}$ as $r\to 0$. From the definition in and for a static background with $w^{(B)}=\Sigma^{(3)(B)}=0$, we have $$\begin{aligned}
\label{vlone}
\delta[\delta_{w}\mathcal{L}^{(1)}]^{ri}&=2\left(\Sigma^{(4)}{}^{ri}{}_{pq}+\,\Sigma^{(4)}{}^{\left[r\right.}{}_{p}{}^{\left. i\right]}{}_{q} \right)\,\delta w^{pq}\notag\\
&\quad -4\,h_{p}\,\left(\delta \Sigma^{(3)}{}^{pri}-2\,\delta \Sigma^{(3)}{}^{\left[ ri \right]\,p} \right)+4H^{-1}\,\nabla_{p}\left(H\, \delta \Sigma^{(3)}{}^{pri}\right){\notag \\}&\quad -4\,\Sigma^{(2)}{}^{p\left[ r\right.}w^{\left. i\right]}{}_{p}\,.\end{aligned}$$ For the first line of , as $r\to 0$ we find $$\begin{aligned}
\label{fline}
2\left(\Sigma^{(4)}{}^{ri}{}_{pq}+\,\Sigma^{(4)}{}^{\left[r\right.}{}_{p}{}^{\left. i\right]}{}_{q} \right)\delta w^{pq}
&\to 6 \bar R^{ri}{}_{rj}\delta w^{rj}\,,{\notag \\}&=(4\pi T)^{3/2}\frac{3}{2}\frac{1}{r^{1/2}}h^{(1)ij}v_j+\mathcal{O}(r^{1/2})\,.\end{aligned}$$ Next, the third line in is given by $$\begin{aligned}
-4\Sigma^{(2)}{}_p{}^{[r}\delta w^{i]p}=-2\Sigma^{(2)}{}_j{}^{r}\delta w^{ij}+2\Sigma^{(2)}{}_r{}^{r}\delta w^{ri}
+2\Sigma^{(2)}{}_j{}^{i}\delta w^{rj}\,.\end{aligned}$$ In the background we calculate that as $r\to 0$ we have $$\begin{aligned}
\Sigma^{(2)}_{rr}\to \frac{1}{2r}\left(3 H^{(1)}-\frac{1}{2}\gamma^{(1)}_{rr}\right)\,,\quad
\Sigma^{(2)}_{ij}\to(4\pi T)\frac{1}{4}h^{(1)}_{ij}\,,\quad
\Sigma^{(2)}_{ri}=\mathcal{O}(1)\,,\end{aligned}$$ and hence as $r\to 0$ $$\begin{aligned}
\label{thline}
-4\Sigma^{(2)}{}_p{}^{[r}\delta w^{i]p}&=
-\frac{(4\pi T)^{3/2}}{r^{1/2}}\left[(3H^{(1)}-\frac{1}{2}\gamma^{(1)}_{rr})v^i+\frac{1}{2}h^{(1)ij}v_j\right]
+\mathcal{O}(r^{1/2})\,.\end{aligned}$$ We next consider the second line in which can be written $$\begin{aligned}
\label{seclll}
-4\,h_{p}\,&\left(\delta \Sigma^{(3)}{}^{pri}-2\,\delta \Sigma^{(3)}{}^{\left[ ri \right]\,p} \right)+4H^{-1}\,\nabla_{p}\left(H\,\delta \Sigma^{(3)}{}^{pri}\right)
={\notag \\}&-8h_r\delta\Sigma^{(3)rri}+4H^{-1}\,\nabla_r\left(H\,\delta\Sigma^{(3)rri}\right)+4H^{-1}\,\nabla_j\left(H\,\delta\Sigma^{(3)jri}\right)\,.
\end{aligned}$$ To proceed we evaluate the limits $$\begin{aligned}
\delta\Sigma^{(3)rij}=\mathcal{O}(r^{1/2})\,,\qquad
\delta\Sigma^{(3)ijk}=\mathcal{O}(r^{-1/2})\,,\end{aligned}$$ as well as $$\begin{aligned}
&
\delta \Sigma^{(3)rri}=r^{1/2}(4\pi T)^{3/2}\frac{1}{2}\left(-(3H^{(1)}-\frac{1}{2}\gamma^{(1)}_{rr})v^i
+\frac{1}{2} h^{(1)ij}v_j+(\zeta^i+\nabla^i\delta g_{tr}^0)\right)+\mathcal{O}(r^{3/2})\,,{\notag \\}&
\delta\Sigma^{(3)jri}
=-(4\pi T)^{1/2}\frac{1}{2}\frac{1}{r^{1/2}}\nabla^{(j} v^{i)}+\mathcal{O}(r^{1/2})\,.\end{aligned}$$ A calculation then reveals as $r\to 0$ $$\begin{aligned}
-8h_r\delta\Sigma^{(3)rri}+4H^{-1}\,\nabla_r\left(H\,\delta\Sigma^{(3)rri}\right)\to -\frac{4}{r}\delta\Sigma^{(3)rri}\,,\end{aligned}$$ and $$\begin{aligned}
4H^{-1}\nabla_j(H\delta\Sigma^{(3)jri})=4\Bigg(-(4\pi T)^{1/2}\frac{1}{2}\frac{1}{r^{1/2}}\nabla_j\nabla^{(j} v^{i)}\Bigg)+\mathcal{O}(r^{1/2})\,.\end{aligned}$$ After substituting into we obtain the second line of . Finally combining the resulting expression with and , we deduce that as $r\to 0$ can be written as $$\begin{aligned}
\delta(\delta_{w}&\mathcal{L}^{(1)})^{ri}=-2\frac{(4\pi T)^{1/2}}{r^{1/2}}\,\nabla_{j}\nabla^{(j}v^{i)}{\notag \\}&-\frac{(4\pi T)^{3/2}}{r^{1/2}}\left[ 2\left(\zeta^{i}+\nabla^{i}\delta g_{tr}^{(0)}\right)
-\left(3\,H^{(1)}-\frac{1}{2}\gamma_{rr}^{(0)}\right) v^{i}\right]\,.\end{aligned}$$ This leads to the expression for $\delta Q^{(3)}_H{}^{i}$ given in .
Constraints at the horizon for Gauss-Bonnet {#conGB}
===========================================
We write the spacetime coordinates as $y^\mu=(r,x^a)$, where $$\begin{aligned}
x^a=(t,x^i)\,,\end{aligned}$$ are the coordinates for the dual field theory. We perform a radial decomposition of the bulk metric in a standard way, writing $$\begin{aligned}
ds^2&=g_{\mu\nu}dy^\mu dy^\nu=N^2dr^2+{\sigma}_{ab}(dx^a+N^adr)(dx^b+N^bdr)\,.\end{aligned}$$ The normal vector to surfaces of constant $r$ has components $n^\mu=N^{-1}(1,-N^a)$, while $n_\mu=N(1,0)$. The induced metric on the surfaces of constant $r$ is given by $\sigma_{\mu\nu}=g_{\mu\nu}-n_\mu n_\nu$ and has non-vanishing components $\sigma_{ab}$. The extrinsic curvature is defined as $K_{\mu\nu}=\tfrac{1}{2}{\cal L}_n {\sigma}_{\mu\nu}$ and has non-vanishing components given by $$\begin{aligned}
K_{ab}=\frac{1}{2N}(\partial_r{\sigma}_{ab}-D_aN_b-D_bN_a)\,,\end{aligned}$$ where $N_a={\sigma}_{ab}N^b$.
We are interested in examining the behaviour of the perturbed geometry near the horizon. We will need to consider the expansions for the background and the perturbation separately. From the background metric ${\sigma}_{ab}$ has the following expansion near the horizon: $$\begin{aligned}
{\sigma}_{tt}=-U(1+2H^{(1)}r)+\mathcal{O}(r^2)\,,\qquad
{\sigma}_{ij}=h_{ij}^{(0)}+ h_{ij}^{(1)}r+\mathcal{O}(r^2)\,,\end{aligned}$$ with ${\sigma}_{ti}=0$. In this appendix we are using $$\begin{aligned}
U=4\pi T r\,,\end{aligned}$$ for convenience. From the perturbed metric $\delta {\sigma}_{ab}$ near the horizon behaves as: $$\begin{aligned}
\delta{\sigma}_{tt}&=U\delta g_{tt}^{(0)}+o(r^2)\,,\cr
\delta{\sigma}_{ij}&=\delta g_{ij}^{(0)}+o(r)\,,\cr
\delta{\sigma}_{ti}&=-v_i-r\ln r\zeta_i-Ut\zeta_i+o(r)+\mathcal{O}(r^2)\,.\end{aligned}$$ Here we use $o(r^n)$ to denote time-independent terms of order $r^n$ or higher, while $\mathcal{O}(r^n)$ also possibly includes time-dependent terms. Near the horizon, for the background from we also have $$\begin{aligned}
N^2=&\frac{1}{U}(1+\gamma_{rr}^{(1)}r)+\mathcal{O}(r)\,,\cr
N_i=&\mathcal{O}(r)\,,\cr
N_t=&0\,,\end{aligned}$$ while for the perturbation, from we have $$\begin{aligned}
\delta N=&\frac{1}{2\sqrt{U}}\delta g_{rr}^{(0)}+\mathcal{O}(r^{1/2})\,,\cr
\delta N_i=&-\frac{v_i}{U}+\mathcal{O}(1)\,,\cr
\delta N_t=&\delta g_{tr}^{(0)}+\mathcal{O}(r)\,,\end{aligned}$$ where $\delta g_{tt}^{(0)}+\delta g_{rr}^{(0)}=2\delta g_{tr}^{(0)}$.
Near the horizon, the components of the Christoffel symbols for the perturbed metric ${\sigma}_{ab}+\delta{\sigma}_{ab}$ on the radial slices are given by $$\begin{aligned}
\Gamma_{tt}^i=&-U(\zeta^i+\frac{1}{2}\hat{\nabla}^i\delta g_{tt}^{(0)})+\mathcal{O}(r^2)\,,\cr
\Gamma_{ti}^t=&-\frac{1}{2}\hat{\nabla}_i\delta g_{tt}^{(0)}+\mathcal{O}(r)\,,\cr
\Gamma_{ij}^k=&\tilde{\Gamma}_{ij}^k+\mathcal{O}(r)\,,\cr
\Gamma_{tt}^t=&-r v^i\partial_iH^{(1)}+\mathcal{O}(r^2)\,,\cr
\Gamma_{ti}^j=&\frac{1}{2}(\hat{\nabla}^jv_i-\hat{\nabla}_iv^j)+o(r)+\mathcal{O}(r^2)\,,\cr
\Gamma_{ij}^t=&\frac{1}{2U}(\hat{\nabla}_iv_j+\hat{\nabla}_jv_i)+\frac{t}{2}(\hat{\nabla}_i\zeta_j+\hat{\nabla}_j\zeta_i)+o(1)+\mathcal{O}(r)\,.\end{aligned}$$ In the above and in what follows, all tilde quantities are defined with respect to the full horizon metric $\tilde{h}_{ij}^{(0)}\equiv h_{ij}^{(0)}+\delta g_{ij}^{(0)}$. All indices on the tilde quantities are raised and lowered using the full horizon metric $\tilde{h}_{ij}^{(0)}\equiv h_{ij}^{(0)}+\delta g_{ij}^{(0)}$ as well. On the other hand hatted quantities refer to the horizon metric $h_{ij}^{(0)}$ and we have also raised all indices without a tilde using this metric too. In the main text we have dropped the hats to simplify the presentation.
The components of the Riemann tensor for the perturbed metric ${\sigma}_{ab}+\delta{\sigma}_{ab}$ are given by[^15]: $$\begin{aligned}
{\mathcal{R}^i}_{tjt}=&-U\hat{\nabla}_j(\zeta^i+\frac{1}{2}\hat{\nabla}^i\delta g_{tt}^{(0)})+\mathcal{O}(r^2)\,,\cr
{\mathcal{R}^i}_{jkl}=&{\,\tilde{R}^i}_{jkl}+\mathcal{O}(r)\,,\cr
{\mathcal{R}^t}_{itj}=&\hat{\nabla}_j(\zeta_i+\frac{1}{2}\hat{\nabla}_i\delta g_{tt}^{(0)})+\mathcal{O}(r)\,,\cr
{\mathcal{R}^t}_{tti}=&r v^j \hat{\nabla}_i\hat{\nabla}_jH^{(1)}+\mathcal{O}(r^2)\,,\cr
{\mathcal{R}^t}_{tij}=&\mathcal{O}(r)\,,\cr
{\mathcal{R}^t}_{ijk}=&\frac{1}{2U}\hat{\nabla}_i(\hat{\nabla}_jv_k-\hat{\nabla}_kv_j)-\frac{1}{U}{\,\hat{R}^l}_{ijk}v_l+\mathcal{O}(1)\,,\cr
{\mathcal{R}^i}_{tjk}=&\frac{1}{2}\hat{\nabla}^i(\hat{\nabla}_jv_k-\hat{\nabla}_kv_j)+\mathcal{O}(r)\,,\cr
{\mathcal{R}^i}_{jtk}=&-\frac{1}{2}\hat{\nabla}_k(\hat{\nabla}^iv_j-\hat{\nabla}_jv^i)+\mathcal{O}(r)\,.\end{aligned}$$ The components of the Ricci tensor are: $$\begin{aligned}
\mathcal{R}^t_t=&\hat{\nabla}_i(\zeta^i+\frac{1}{2}\hat{\nabla}^i\delta g_{tt}^{(0)})+\mathcal{O}(r)\,,\cr
\mathcal{R}^i_j=&\tilde{R}^i_j+\hat{\nabla}_j(\zeta^i+\frac{1}{2}\hat{\nabla}^i\delta g_{tt}^{(0)})+\mathcal{O}(r)\,,\cr
\mathcal{R}^t_i=&-\frac{1}{U}{v^j\hat{R}_{ij}}-\frac{1}{2U}\hat{\nabla}_j(\hat{\nabla}^jv_i-\hat{\nabla}_iv^j)+\mathcal{O}(1)\,,\cr
\mathcal{R}^i_t=&\frac{1}{2}\hat{\nabla}_j(\hat{\nabla}^jv^i-\hat{\nabla}^iv^j)+\mathcal{O}(r)\,,\end{aligned}$$ and the Ricci scalar is $$\begin{aligned}
\mathcal{R}=\tilde{R}+2\hat{\nabla}_i(\zeta^i+\frac{1}{2}\hat{\nabla}^i\delta g_{tt}^{(0)})+\mathcal{O}(r)\,.\end{aligned}$$
We next calculate the extrinsic curvature components $K^a_b$ and find[^16] $$\begin{aligned}
K^t_t=&\frac{\sqrt{U}}{2r}(1-\frac{1}{2}\delta g_{rr}^{(0)})+\mathcal{O}(r^{1/2})\,,\cr
K^i_j=&\frac{1}{\sqrt{U}}h^{ik}_{(0)}\hat{\nabla}_{(j}v_{k)}+\mathcal{O}(r^{1/2})\,,\cr
K^i_t=&\frac{\sqrt{U}}{2r}v^i+\mathcal{O}(r^{1/2})\,,\cr
K^t_i=&\frac{\sqrt{U}}{2r}t\zeta_i+o(r^{-1/2})+\mathcal{O}(r^{1/2})\,,\cr
K=&\frac{\sqrt{U}}{2r}(1-\frac{1}{2}\delta g_{rr}^{(0)})+\frac{1}{\sqrt{U}}\hat{\nabla}_iv^i+\mathcal{O}(r^{1/2})\,.\end{aligned}$$
Calculation of conjugate momentum
----------------------------------
We now calculate the conjugate momenta on the horizon, which are given by $$\begin{aligned}
{\pi}^b_a=\sqrt{-{\sigma}}(K\delta^b_a-K^b_a)+\tilde \alpha{\pi_{GB}}^b_a\,,\end{aligned}$$ where $$\begin{aligned}
\label{gbexpkr}
\frac{1}{\sqrt{-{\sigma}}}{\pi_{GB}}^b_a=&2K^b_a(K^2-K^c_dK^d_c-\mathcal{R})\cr
&-4K\mathcal{R}^b_a+4K^c_a\mathcal{R}_c^b+4K_c^b\mathcal{R}^c_a +4K^{cd}{\mathcal{R}^b}_{cad} -4KK_c^bK^c_a+4K_c^bK^c_dK^d_a\cr
&+\delta^b_a(2K\mathcal{R}-\frac{2}{3}K^3+2KK_c^dK^c_d-4K_c^d\mathcal{R}^c_d-\frac{4}{3}K_c^dK^c_eK_d^e)\,.\end{aligned}$$ We calculate each component of ${\pi}^b_a$ in a straightforward manner. As the calculations are rather long we have recorded a few intermediate steps.
### Calculation of $\pi^i_t$
By writing out each of the components and using the previous results we find $$\begin{aligned}
2K^i_t(K^2-K_{ab}K^{ab}-\mathcal{R})
=&2K^i_t(2K_t^tK_j^j-\mathcal{R}_k^k)+\mathcal{O}(r^{1/2})\,,\end{aligned}$$ as well as $$\begin{aligned}
-4K\mathcal{R}^i_t+4K^a_t\mathcal{R}_a^i+&4K^i_a\mathcal{R}^a_t +4K^{ab}{\mathcal{R}^i}_{atb} -4KK_a^iK^a_t+4K_a^iK^a_bK^b_t\cr
&\qquad=-4K_j^j(K^i_tK_t^t)+4K_t^j\mathcal{R}^i_j +\mathcal{O}(r^{1/2})\,,\end{aligned}$$ We then find the final result $$\begin{aligned}
\label{piit}
\pi^i_t=-\sqrt{h^{(0)}}\frac{U}{r}v^j(\frac{1}{2}\delta^i_j-2\tilde\alpha \hat{G}^i_j)+\mathcal{O}(r)\,,\end{aligned}$$ where $\hat{G}_{ij}=\hat R_{ij}-\tfrac{1}{2}h^{(0)}_{ij}\hat R$ is the Einstein tensor for the horizon metric $h^{(0)}_{ij}$.
### Calculation of $\pi_i^t$
We now have $$\begin{aligned}
2K_i^t(K^2-K_{ab}K^{ab}-\mathcal{R})
=&2K_i^t(2K_t^tK_j^j-\mathcal{R}_j^j)+\mathcal{O}( r^{1/2})\,,\end{aligned}$$ and $$\begin{aligned}
-4K\mathcal{R}^t_i+4K^t_a\mathcal{R}^a_i+&4K^a_i\mathcal{R}^t_a +4K^{ab}{\mathcal{R}^t}_{aib} -4KK_i^aK^t_a+4K^a_iK_a^bK_b^t\cr
&\qquad=-4K_k^kK_i^tK^t_t+4K^t_j\mathcal{R}^j_i +o(r^{-1/2})+\mathcal{O}( r^{1/2})\,.\end{aligned}$$ Combining these we get $$\begin{aligned}
{\pi_{GB}}_i^t
=&\sqrt{h^{(0)}}(2\frac{U}{r}t\zeta_j\hat{G}^j_i)+o(1)+\mathcal{O}( r)\,,\end{aligned}$$ and hence find the following result for the time derivative $$\begin{aligned}
\partial_t\pi^t_i=&-\sqrt{h^{(0)}}\frac{U}{r}\zeta_j(\frac{1}{2}\delta_i^j-2\tilde\alpha\hat{G}^j_i)+\mathcal{O}(r)\,.\end{aligned}$$
### Calculation of $\pi_t^t$
Now we have $$\begin{aligned}
2K^t_t(K^2-K_{ab}K^{ab}-\mathcal{R})
=&2K_i^t(2K_t^tK_j^j-\mathcal{R}_j^j)+\mathcal{O}( r^{1/2})\,,\end{aligned}$$ as well as $$\begin{aligned}
-4K\mathcal{R}^t_t+4K^t_a\mathcal{R}^a_t+4K^a_t\mathcal{R}^t_a +&4K^{ab}{\mathcal{R}^t}_{atb} -4KK_t^aK^t_a+4K^a_tK_a^bK_b^t\cr
&\qquad=4K^t_t\mathcal{R}^t_t-4K_i^iK_t^tK^t_t+\mathcal{O}(r^{1/2})\,,\end{aligned}$$ and $$\begin{aligned}
&2K\mathcal{R}-\frac{2}{3}K^3+2KK_a^bK^a_b-4K_a^b\mathcal{R}^a_b-\frac{4}{3}K_a^bK^a_cK_b^c\cr
&\qquad=2(-K_t^t\mathcal{R}_t^t+K_t^t\mathcal{R}_i^i+K_j^j\mathcal{R}_i^i-2K_i^j\mathcal{R}^i_j)-2K_t^tK_i^iK_j^j+2K_t^tK_i^jK^i_j+\mathcal{O}(r^{1/2})\,.\cr\end{aligned}$$ Combining these we are led to $$\begin{aligned}
\pi_t^t
=&\sqrt{h^{(0)}}\hat{\nabla}_{i}v^j(\delta_j^i-4\tilde\alpha \hat{G}^i_j)+\mathcal{O}(r)\,.\end{aligned}$$
### Calculation of $\pi^i_j$
We have $$\begin{aligned}
2K^i_j(K^2-K_{ab}K^{ab}-\mathcal{R})=&2K^i_j(2K_k^kK_t^t+K_k^kK_l^l-K_k^lK^k_l-\mathcal{R}_t^t-\mathcal{R}_k^k){\notag \\}=&\frac{1}{\sqrt{U}}h^{ik}_{(0)}\hat{\nabla}_{(j}v_{k)}({h^l_l}^{(1)}\frac{U}{r}-2\hat{R})+{h^i_j}^{(1)}\frac{1}{\sqrt{U}}\hat{\nabla}_kv^k\frac{U}{r}+\mathcal{O}(r^{1/2})\end{aligned}$$ as well as $$\begin{aligned}
&-4K\mathcal{R}^i_j+4K^a_j\mathcal{R}_a^i+4K_a^i\mathcal{R}^a_j +4K^{ab}{\mathcal{R}^i}_{ajb} -4KK_a^iK^a_j+4K_a^iK^a_bK^b_j{\notag \\}&=-4K_t^t\mathcal{R}^i_j-4K_k^k\mathcal{R}^i_j+4K_j^k\mathcal{R}^i_k +4K^i_k\mathcal{R}_j^k +4K^{tt}{\mathcal{R}^i}_{tjt}+4K^{kl}{\mathcal{R}^i}_{kjl}-4K_t^tK^i_kK_j^k+\mathcal{O}(r^{1/2}){\notag \\}&=-2\frac{\sqrt{U}}{r}\tilde{R}^i_j+\frac{\sqrt{U}}{r}\delta g_{rr}^{(0)}\hat{R}^i_j-4\frac{1}{\sqrt{U}}\hat{\nabla}_kv^k\hat{R}^i_j+4\frac{1}{\sqrt{U}}\hat{\nabla}_{(j}v_{k)}\hat{R}^{ik} {\notag \\}&\qquad +4\frac{1}{\sqrt{U}}\hat{\nabla}^{(k}v^{i)}\hat{R}_{jk}+4\frac{1}{\sqrt{U}}\hat{\nabla}^{(k}v^{l)}{\,\hat{R}^i}_{kjl}-\frac{\sqrt{U}}{r}\hat{\nabla}^{(i}v^{k)}{h_{jk}}^{(1)}-\frac{\sqrt{U}}{r}{h^{ik}}^{(1)}\hat{\nabla}_{(j}v_{k)}+\mathcal{O}(r^{1/2})\end{aligned}$$ and $$\begin{aligned}
&\delta^i_j(2K\mathcal{R}-\frac{2}{3}K^3+2KK_a^bK^a_b-4K_a^b\mathcal{R}^a_b-\frac{4}{3}K_a^bK^a_cK_b^c){\notag \\}&\qquad=2\delta^i_j(-K_t^t\mathcal{R}_t^t+K_t^t\mathcal{R}_k^k+K_l^l\mathcal{R}_k^k-2K_k^l\mathcal{R}^k_l-K_t^tK_k^kK_l^l+K_t^tK_k^lK^k_l)+\mathcal{O}(r^{1/2}){\notag \\}&\qquad=2\delta^i_j(\frac{\sqrt{U}}{2r}\tilde{R}+\frac{\sqrt{U}}{2r}(-\frac{1}{2}\delta g_{rr}^{(0)})\hat{R}+\frac{1}{\sqrt{U}}\hat{\nabla}_lv^l \hat{R}\cr
&\qquad\qquad-2\frac{1}{\sqrt{U}}\hat{\nabla}_kv_l\hat{R}^{kl}-\frac{\sqrt{U}}{2r}{h_k^k}^{(1)}\hat{\nabla}_lv^l +\frac{\sqrt{U}}{2r}{h^{kl}}^{(1)}\hat{\nabla}_kv_l)+\mathcal{O}(r^{1/2})\,.\end{aligned}$$ Putting these together we find $$\begin{aligned}
\frac{1}{\sqrt{-{\sigma}}}{\pi}_j^i=&(\frac{\sqrt{U}}{2r}(1-\frac{1}{2}\delta g_{rr}^{(0)})+\frac{1}{\sqrt{U}}\hat{\nabla}_kv^k)\delta_j^i-\frac{1}{\sqrt{U}}h^{ik}_{(0)}\hat{\nabla}_{(j}v_{k)}+\tilde\alpha\frac{1}{\sqrt{-{\sigma}}}{\pi_{GB}}_j^i+\mathcal{O}(r^{1/2})\end{aligned}$$ where $$\begin{aligned}
&\frac{1}{\sqrt{-{\sigma}}}{\pi_{GB}}_j^i=\frac{1}{\sqrt{U}}h^{ik}_{(0)}\hat{\nabla}_{(j}v_{k)}{h^l_l}^{(1)}\frac{U}{r}+{h^i_j}^{(1)}\frac{1}{\sqrt{U}}\hat{\nabla}_kv^k\frac{U}{r}\cr
&-2\frac{\sqrt{U}}{r}\tilde{G}^i_j+\frac{\sqrt{U}}{r}\delta g_{rr}^{(0)}\hat{G}^i_j-4\frac{1}{\sqrt{U}}\hat{\nabla}_kv^k\hat{R}^i_j \cr
& +4\frac{1}{\sqrt{U}}\hat{\nabla}_{(j}v_{k)}\hat{G}^{ik}+4\frac{1}{\sqrt{U}}\hat{\nabla}^{(k}v^{i)}\hat{R}_{jk}+4\frac{1}{\sqrt{U}}\hat{\nabla}^{(k}v^{l)}{\,\hat{R}^i}_{kjl}-\frac{\sqrt{U}}{r}\hat{\nabla}^{(i}v^{k)}{h_{jk}}^{(1)}-\frac{\sqrt{U}}{r}{h^{ik}}^{(1)}\hat{\nabla}_{(j}v_{k)}\cr
&+2\delta^i_j(-2\frac{1}{\sqrt{U}}\hat{\nabla}_kv_l\hat{G}^{kl}-\frac{\sqrt{U}}{2r}{h_k^k}^{(1)}\hat{\nabla}_lv^l +\frac{\sqrt{U}}{2r}{h^{kl}}^{(1)}\hat{\nabla}_kv_l)+\mathcal{O}(r^{1/2})\,.\end{aligned}$$
Constraints on the horizon
--------------------------
The bulk momentum constraint equations $H_a=0$, where $H^a$ is given in , can be written as $$\begin{aligned}
\sqrt{-{\sigma}}\partial_a\left(\frac{\pi^a_b}{\sqrt{-{\sigma}}}\right)+\Gamma_{ac}^a\pi^c_b-\Gamma_{ab}^c\pi^a_c=&0\,.\end{aligned}$$ We now evaluate these constraints on a surface of constant $r$ near the horizon and then take the limit $r\to 0$.
### Time component of the momentum constraint
For the time component, $H_t=0$, we find $$\begin{aligned}
\hat{\nabla}_i\pi^i_t+\mathcal{O}(r)=&0\,,\end{aligned}$$ where we recall that hat refers to the metric on the horizon $h_{ij}^{(0)}$. Using we deduce that $$\begin{aligned}
\label{incompress}
(\delta^i_j-4\tilde\alpha \hat{{G}}^i_j)\hat{\nabla}_iv^j=&0\,.\end{aligned}$$ We see that when $\tilde\alpha\ne 0$, the simple incompressibility condition for the fluid is modified. Alternatively, we can define $\bar v^i= (\delta^i_j-4\tilde\alpha \hat{G}^i_{j})v^j$, then we have $\nabla_i\bar v^i=0$ and the fluid is incompressible.
### Spatial component of the momentum constraint
We next consider the spatial component of the momentum constraint, $H_j=0$, and find that near the horizon we have $$\begin{aligned}
\sqrt{-{\sigma}}\tilde{\nabla}_i\left(\frac{\pi^i_j}{\sqrt{-{\sigma}}}\right)+\partial_t\pi^t_j+\Gamma_{ti}^t\pi^i_j+\mathcal{O}(r)&=0\,.\end{aligned}$$ After some calculation we find $$\begin{aligned}
0=&\hat{\nabla}_i(\hat{\nabla}_kv^k\delta_j^i-h^{ik}_{(0)}\hat{\nabla}_{(j}v_{k)})\cr
&+\tilde\alpha\hat{\nabla}_i\Big[h^{ik}_{(0)}\hat{\nabla}_{(j}v_{k)}{h^l_l}^{(1)}\frac{U}{r}+{h^i_j}^{(1)}\hat{\nabla}_kv^k\frac{U}{r}-4\hat{\nabla}_kv^k\hat{R}^i_j \cr
& +4\hat{\nabla}_{(j}v_{k)}\hat{{G}}^{ik}+4\hat{\nabla}^{(k}v^{i)}\hat{R}_{jk}+4\hat{\nabla}^{(k}v^{l)}{\,\hat{R}^i}_{kjl}-\frac{U}{r}\hat{\nabla}^{(i}v^{k)}{h_{jk}}^{(1)}-\frac{U}{r}{h^{ik}}^{(1)}\hat{\nabla}_{(j}v_{k)}\cr
&+2\delta^i_j(-2\hat{\nabla}_kv^l\hat{{G}}^k_l-\frac{U}{2r}{h_k^k}^{(1)}\hat{\nabla}_lv^l +\frac{U}{2r}{h^{kl}}^{(1)}\hat{\nabla}_kv_l)\Big]\cr
&-\frac{U}{r}\zeta_k(\frac{1}{2}\delta_j^k-2\tilde\alpha\hat{{G}}^k_j)-\frac{1}{2}\frac{U}{r}(\frac{1}{2}\delta^i_j-2\tilde\alpha\hat{{G}}^i_j)\hat{\nabla}_i(\delta g_{tt}^{(0)}+\delta g_{rr}^{(0)})\,,\end{aligned}$$ where we have used $\hat{\nabla}_i\hat{{G}}^i_j=0$ and $\tilde{\nabla}_i\tilde{{G}}^i_j=0$, which follow from the Bianchi identity for $h_{ij}^{(0)}$ and $\tilde{h}_{ij}^{(0)}\equiv h_{ij}^{(0)}+\delta g_{ij}$ respectively.
Defining $p\equiv -\frac{U}{r}\delta g_{rt}^{(0)}=-\frac{U}{2r}(\delta g_{rr}^{(0)}+\delta g_{tt}^{(0)})$ (and also using ) we obtain the final Stokes equation: $$\begin{aligned}
&0=-\hat{\nabla}^k\hat{\nabla}_{(j}v_{k)}-(\frac{U}{r}\zeta_i-\hat{\nabla}_ip)(\frac{1}{2}\delta_j^i-2\tilde\alpha\hat{{G}}^i_j)\cr
&+\tilde\alpha\hat{\nabla}_i\Big[\frac{U}{r}\left(h^{ik}_{(0)}\hat{\nabla}_{(j}v_{k)}{h^l_l}^{(1)}+{h^i_j}^{(1)}\hat{\nabla}_kv^k-\hat{\nabla}^{(i}v^{k)}{h_{jk}}^{(1)}-{h^{ik}}^{(1)}\hat{\nabla}_{(j}v_{k)}+\delta^i_j(-{h_k^k}^{(1)}\hat{\nabla}_lv^l +{h^{kl}}^{(1)}\hat{\nabla}_kv_l)\right)\cr
&\qquad\qquad-4\hat{\nabla}_kv^k\hat{R}^i_j +4\hat{\nabla}_{(j}v_{k)}\hat{{G}}^{ik}+4\hat{\nabla}^{(k}v^{i)}\hat{R}_{jk}+4\hat{\nabla}^{(k}v^{l)}{\,\hat{R}^i}_{kjl}\Big]\,.\end{aligned}$$ Note that this depends only upon $v_i$, $p$, $\zeta_i$ and background quantities.
We can also write this as an expansion in $\tilde\alpha$. As we derived in , to leading order in $\tilde\alpha$ we have: $$\begin{aligned}
h^{(1)}_{ij}=&\frac{2r}{U}\left( \hat{R}_{ij}-\frac{1}{D-2} V_0h^{(0)}_{ij}\right)\,,\end{aligned}$$ where $V_0=-(D-1)(D-2)$. So this becomes $$\begin{aligned}
\label{stokeseqn}
0=&-\hat{\nabla}^k\hat{\nabla}_{(j}v_{k)}-(4\pi T\zeta_i-\hat{\nabla}_ip)(\frac{1}{2}\delta_j^i-2\tilde\alpha\hat{{G}}^i_j)\cr
&+\tilde\alpha\hat{\nabla}_i\Big[2(D-4) (D-1)h_{(0)}^{ik}\hat{\nabla}_{(j}v_{k)}+2\delta^i_j\hat{R}^{kl}\hat{\nabla}_kv_l+2\hat{\nabla}_{(j}v_{k)}\hat{R}^{ik}{\notag \\}&\qquad\qquad+2\hat{\nabla}^{(k}v^{i)}\hat{R}_{jk}+4\hat{\nabla}^{(k}v^{l)}{\,\hat{R}^i}_{kjl}\Big]+\mathcal{O}(\tilde\alpha^2)\,.\end{aligned}$$
### Hamiltonian Constraint
The Hamiltonian constraint, $H=0$, where $H$ is given in , can be evaluated at the horizon and we find $$\begin{aligned}
H
=&{\sqrt{h_{(0)}}}\frac{U^{1/2}}{r}(\delta_i^j-4\tilde\alpha \hat{{G}}_i^j)\hat{\nabla}_{j}v^{i}+\mathcal{O}(r^{1/2})\,.\end{aligned}$$ Thus, the Hamiltonian constraint leads to the same condition given in .
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[^1]: For simplicity, here we are assuming that the holographic lattice has a globally defined radial coordinate outside the black hole horizon and in this case we can take the applied DC sources to be independent of the radial coordinate. For a more general discussion see [@Banks:2015wha; @Donos:2015bxe].
[^2]: To do this we need to introduce the DC sources in a slightly different gauge than the “linear in time" perturbations that were used in [@Donos:2015gia; @Banks:2015wha; @Donos:2015bxe] (building on [@Donos:2014uba; @Donos:2014cya]).
[^3]: More precisely, the constraints are evaluated on a hypersurface of fixed radial coordinate just outside the horizon, a “stretched horizon", and then a limit is taken.
[^4]: The perturbative approach is directly relevant to applications to string theory. It is also worth noting that attempting to consider the Gauss-Bonnet terms non-perturbatively leads to issues with causality [@Camanho:2014apa].
[^5]: We use a slightly different notation in the main text.
[^6]: We note that the membrane paradigm of [@Price:1986yy] was adapted to Gauss-Bonnet gravity in [@Jacobson:2011dz].
[^7]: Note that $\phi_{T}, \phi_{E}$ can depend, in general, on all spatial coordinates. Later, in section \[secbhgih\], to simplify the presentation, we will take $\phi_{T}$ and $\phi_{E}$ to be independent of the radial coordinate.
[^8]: The sign can be established as follows. Consider the metric $ds^2=-(1-2\phi_T)dt^2+dx^idx^i$. For $\dot x^i<<1$ and perturbative in $\phi_T$, the geodesic equation gives $\ddot x^i=\zeta_i$. Since heat moves from hot to cold we identify $\zeta$ with $-T^{-1}dT$.
[^9]: \[test\]If we consider the expansion (2.4), (2.5) in [@Donos:2015bxe], we first make the change $r\to r/G^{(0)}+\dots$, which effectively sets $G^{(0)}=1$. Next we can shift $x^i\to x^i+rf^i_{(1)}+\dots$ and choose the $f^i$ to eliminate the leading $r$-independent terms in $g_{ri}$.
[^10]: One might consider alternative transport currents by instead subtracting $\partial_j \delta M^{ij}$ and $\partial_j \delta M^{ij}_T$ from $ \delta J^i_\infty$ and $\delta Q^i_\infty$, respectively (and in fact this was done in [@Donos:2015bxe]). Such transport currents differ from the above definitions by the trivially conserved and globally defined magnetisation currents $\partial_j \bar\delta M^{ij}$, $\partial_j \bar\delta M^{ij}_T$. The definitions we use here have the property that they agree with $\delta J^i_\infty$ and $\delta Q^i_\infty$ in the case that there are no [*background*]{} magnetisation currents. It is worth emphasising that when $M^{(B)ij}=M^{(B)ij}_T=0$, in general we still have $\partial_j \bar\delta M^{ij}\ne 0$ and $\partial_j \bar\delta M^{ij}_T\ne 0$.
[^11]: Note that the sign of the $tg_{tt}$ term in (4.1) of [@Donos:2015bxe], should actually be a plus rather than minus. We should also make the identification $\delta g_{ij}^{(0)}$ of [@Donos:2015bxe] via $\delta g_{ij}^{(0)}=\delta \gamma_{ij}^{(0)}-2\alpha^{(0)}_{(i}v_{j)}$.
[^12]: A Kaluza-Klein reduction on a spatial coordinate is analysed in [@Baskal:2010sv].
[^13]: In comparing with e.g. equation (4.7) in [@Donos:2015bxe] on should take into account that we are using a different radial variable.
[^14]: Note that since we have assumed that the background geometry is static with $\alpha^{(B)}=0$, there are no magnetisation currents and the transport currents are the same as the currents.
[^15]: Recall that $\hat{\nabla}_j\zeta_i=\hat{\nabla}_i\zeta_j$. We have also used the Bianchi identity ${\,\hat{R}^l}_{ijk}+{\,\hat{R}^l}_{jki}+{\,\hat{R}^l}_{kij}=0$.
[^16]: Note that these correct a typo in the last line of (B.4) in [@Banks:2015wha] as well making (B.5) more precise.
|
---
abstract: 'This Letter is to investigate the physics of a newly discovered phenomenon — contracting flare loops in the early phase of solar flares. In classical flare models, which were constructed based on the phenomenon of expansion of flare loops, an energy releasing site is put above flare loops. These models can predict that there is a vertical temperature gradient in the top of flare loops due to heat conduction and cooling effects. Therefore, the centroid of an X-ray looptop source at higher energy bands will be higher in altitude, for which we can define as normal temperature distribution. With observations made by [*RHESSI*]{}, we analyzed 10 M- or X-class flares (9 limb flares). For all these flares, the movement of looptop sources shows an obvious U-shaped trajectory, which we take as the signature of contraction-to-expansion of flare loops. We find that, for all these flares, normal temperature distribution does exist, but only along the path of expansion. The temperature distribution along the path of contraction is abnormal, showing no spatial order at all. The result suggests that magnetic reconnection processes in the contraction and expansion phases of these solar flares are different.'
author:
- 'Jinhua Shen, Tuanhui Zhou, Haisheng Ji, Na Wang, Wenda Cao, Haimin Wang'
title: 'Early Abnormal Temperature Structure of X-ray Looptop Source of Solar Flares'
---
Introduction
============
It is widely accepted that solar flare energy comes from sudden release of free magnetic energy via magnetic reconnection. In commonly adopted flare models, the ever-ascending Y-type reconnection point in the solar corona results in expanding flare loops and separation motion of flare foot points (FPs) (Kopp & Pneuman 1976). The expansion of flare ribbons and flare loops is an important signature of progressive magnetic reconnection in the corona. However, the contraction of flare loops in the early phase of flares may suggest a different reconnection scenario.
The signature for the contraction of flare loops has been reported by several authors (Sui et al. 2003, 2004; Li & Gan 2005; Liu et al. 2004; Veronig et al. 2006; Ji et al. 2004b, 2006, 2007, 2008). Contraction motion includes two aspects: the converging motion of FPs and the correlated downward motion of a LT source, such as the M1.1 flare of 2004 November 1 (Ji et al. 2006). The contraction picture is different from the shrinkage of flare loops with rooted FPs being fixed or still in expansion. From our experience, the similar event like the M1.1 flare is rare, since HXR FPs are usually missing during the initial phase of a flare, such as the X3.9 flare of 2003 November 3 (Veronig et al. 2006). To investigate the converging motion of FPs, the most preferable wavelength is H$_\alpha$ blue wing, at which H$_\alpha$ emission is believed to be caused by nonthermal electrons (Canfield et al. 1984).
The classical flare model puts the energy releasing site above flare loops, from which we can expect that a higher temperature source will be located above a lower temperature source due to heat conduction and cooling effects. In this Letter, we will name this distribution as normal temperature distribution (NTD). For the X10 flare of 2003 October 29, Ji et al. (2008) reported that a HXR sigmoid structure contracts during the impulsive phase of the flare. The contraction is the result of reconnection between two highly-sheared flux ropes. For magnetic reconnection between flux ropes, the existence of NTD is not required, since the energy releases inside flux ropes. Thus, temperature distribution along the altitude is an important factor for testing the reconnection scenarios.
Based on above thinking, we re-analyzed the M2.1 flare of 2002 September 9. The flare is a well-observed sample showing the early converging and subsequent separation motion of the flare kernels. We found that the motion of the HXR LT source is well correlated with that of the flare kernels. Notably, the NTD occurs only during the expansion period of flare loops. During the contraction period, higher and lower temperature structures are mixed together. To find supporting evidences for this abnormal temperature distribution during the contraction period, we surveyed nearly all M-class and X-class limb flares from 2002 to 2005, which were well-observed by [*RHESSI*]{} (Lin et al. 2002), we found at least 9 events showing the phenomenon.
In §2 we present the result of the flare of 2002 September 9, in §3 we give the other 9 supporting events. Discussions and conclusions are briefly given in §4.
The flare of 2002 September 9
=============================
The M2.1 flare occurred on 2002 September 9, starting at 17:40. The flare was well observed at Big Bear Solar Observatory at H$_\alpha$-1.3[Å]{} with very high time resolution (40 ms). By analyzing the flare with detailed spatial, spectral and temporal information, Ji et al. (2004a) identified nonthermal and thermal (multiple temperature) HXR spikes. They further reported that the distance between the two H$_\alpha$ conjugate kernels decreases during the impulsive phase of the flare. Only after the impulsive phase, the distance has a steady increase showing the usual separation motion (Ji et al. 2004b). They also mentioned that the height of the HXR looptop source decreases during the impulsive phase. The flare has two pair of conjugate kernels in H$_\alpha$ blue wing (a1-a2, and b1-b2 in Figure 1). From Figure 1a, we can clearly see two EUV loop systems connecting a1 to a2 and b1 to b2 respectively. Judged from temporal and spatial relationship between HXR emission and H$_\alpha$ emission, HXR emissions of this event come dominantly from the flare loops connecting a1 to a2. Furthermore, there is an over 1 minute delay for kernels b1 and b2 and the magnetic reconnection producing b1 and b2 was induced by the magnetic reconnection producing a1 and a2 via loop interactions (Huang & Ji 2005).
To study the moving behavior of the HXR LT source, we constructed [*RHESSI*]{} CLEAN HXR maps in the energy ranges of 8-10, 10-13, and 13-16 keV using 16-s time bin. HXR emissions in these energy ranges originate from the top of the EIT 195 Å flare loops connecting a1 and a2 (Figure 1). The CLEAN maps were made with natural weighting using grids 3-8, in which the maps have an FWHM spatial resolution of $\sim$ 10 arcsec. Note that the emission centroids can be determined with an accuracy of $<$ 1 arcsec, depending on the count statistics (Hurford et al. 2002). We measured the LT source’s centroid positions within the contour level of 50% using a tool provided by [*RHESSI*]{} software. The temporal series of centroid positions of the LT source in the energy range of 10-13 keV is shown in Figure 1b, in which red arrows show downward moving direction and blue ones show upward motion. We measured the distance from the centroid of the LT source to the base of the flare loops. Since the flare loops are clearly seen, the distance can be reasonably regarded as the projected height. The time profiles for the ‘height’ measured at three different energy ranges are plotted in Figure 2c. For comparison, the time profile of HXR emission in the energy range 25-50 keV and the distance between the two H$_\alpha$ kernels are given in Figure 2a-b. We can find that the motion of the HXR LT source is well correlated with the relative motion between the two H$_\alpha$ kernels. This confirms the picture of early contraction and subsequent expansion of flare loops, as summarized by Ji et al. (2007).
During the expansion period, the temperature structure shows an NTD (Fig. 2c). This is in agreement with what the standard flare model predicts with an energy releasing site above flare loops. However, during the contraction period, the temperature structure of the LT source is rather complex or abnormal. The LT sources in the three energy bands are almost mixed with one another. The abnormal temperature distribution in the contraction period obviously suggests a complex magnetic reconnection process.
Supporting observations
=======================
To find supporting evidences, we surveyed 70 limb flares well observed by [*RHESSI*]{} from 2002 to 2005, most of them are M-class or X-class. For each event, we constructed [*RHESSI*]{} Clean maps in the energy ranges of 8-10, 10-13, and 13-16 keV with a time bin of 12 seconds. HXR emissions at these energy ranges usually come from flares’ LT. We selected those events with a single LT source and the LT source exhibits an obvious U-shaped trajectory (only one turning point), which is assumed to be the signature of early downward motion and subsequent upward motion. An additional requirement is that the downward motion should last more than 1 minutes. According to this criteria, 12 events could be selected for the study including the three homologous flares investigated by Sui et al. (2004). In this paper, we will not include the three homologous flares (see Table 1 for the nine flares).
Even for limb events, it is hard to measure the actual height of a HXR LT source with respect to solar photospheric surface. Actually, as Sui et al. (2003, 2004) noted, the motion of LT sources of a limb event is usually not in radial direction. A reference point is needed to estimate the height of a HXR LT source. For the flare of 2002 September 9, the reference point is chosen as the center of the line connecting the two FPs of the flare loop. For each of the 9 events, a turning point around the base of the U-shaped path is taken as the reference point, from which the ‘height’ of a LT source is estimated. The trajectories of LT sources of the nine events are plotted in the top panels of Figures 3-4, in which nine “+” signs show the positions of reference points for the events. We can see that before and after the turning point the trajectories of the LT sources can be roughly fitted by two straight lines.
The time profiles for the ‘height’ of the LT sources at different energy ranges are plotted in the middle panels of Figures 3-4. For comparison, the light curves of corresponding event in different energy ranges are plotted in the lower panels. For all nine events, we can find that well-established NTD does occur but only in the expansion phases. During the contraction periods, high energy and low energy sources are spatially mixed. It is worth mentioning that the height curve in the first column of Figure 4 for the flare of 2003 November 3 has been reported by Veronig et al. (2006). Figures 3-4 conform the results that we obtained for the 2002 September 9 event. Turning points occur during the rising phases of the flares. To some events turning points coincide with peak times of the light curves.
Discussion and Conclusions
==========================
In this Letter, we report the finding of the early abnormal temperature distribution of flares’ X-ray LT sources during contraction phase of ten solar flares. In a 2D framework for flare models, an energy releasing site is assumed to be above flare loops. Following the scenario, we can expect that a LT source at higher energies will be higher in altitude due to heat conduction and cooling effects. The results from the ten flares show that this expectation can be met, but only during the expansion phase of solar flares. During the contraction period, however, high energy and low energy LT sources are mixed, showing a kind of abnormal temperature distribution.
The physics for the contraction of flare loops is still not well-understood. From above ten events, we have seen that the turning points in the trajectories of the LT sources occur near flare peak times. Therefore, the contraction is related to magnetic reconnection in the impulsive phase of solar flares. Ji et al. (2008) reported that, during the contraction period of the X10 flare of 2003 October 29, HXR emissions at all energies share the similar sigmoidal configuration. The contraction corresponds to the shrinkage of the HXR sigmoid, which is the result of magnetic reconnection between highly-sheared flux ropes. We may propose that the abnormal temperature distribution is associated with magnetic reconnection between highly-sheared flux ropes. On the other hand, the downward and upward phases of the LT motion may reflect two regimes of reconnection: bursting and no-bursting (Karlický 2008: private communication). The abnormal temperature structure can be explained by a presence of small plasmoids in the current sheet just above contracting arcade. The plasmoids are formed in the bursting regime of the reconnection and move also downwards where they interact with the arcade. This interaction represents additional reconnection that changes the normal temperature structure (Bárta et al. 2008; Kolomański & Karlický 2007). In the phase of the upward LT motion the reconnection take place without these plasmoids (no-bursting regime of the reconnection). Therefore the temperature is in agreement with the prediction made by standard flare models.
Bárta, M., Vršnak, & Karlický, M. 2008, A&A 477, 649 Canfield, R. C., Gunkler, T. A., & Ricchiazzi, P. J. 1984, , 282, 296 Huang, G. & Ji, H. 2005, Solar Phys, 227, 236 Hurford et al. 2002, Solar Phys., 210, 61 Ji, H., Wang, H., et al. 2004a, , 605, 938 Ji, H., Wang, H., Goode, P. R., Jiang, Y., & Yurchyshyn, V. 2004b, , 607, L55 Ji, H., Huang, G., Wang, H. et al. 2006, , 636, L173 Ji, H., Huang, G., & Wang, H. 2007, , 660, 893 Ji, H., Wang, H., Liu, C., & Dennis, B.R. 2008, , 680, 734 Kolomański, S. & Karlický, M., 2007, A&A, 475, 685 Kopp, R. A. & Pneuman, G. W. 1976, Solar Phys, 108, 251 Li, Y. & Gan, W. 2005, , 629, L37 Lin, R. P. et al. 2002, Solar Phys., 210, 3 Liu, W., Jiang, Y. W., Liu, S., & Petrosian, V. 2004, , 611, L53 Sui, L. & Holman, G. D. 2003, , 596, L25 Sui, L., Holman., G. D., & Dennis, B. R. 2004, , 546, 556 Veronig, A. M. et al. 2006, A&A, 446, 657
[cccccccc]{} Date & Class & Starting & Peak Time & Turning & Descending & Expanding\
& & Time & Time$^*$ & Time & Speed (km$s^{-1}$) & Speed (km$s^{-1}$)\
2002-09-09 & M2.3 & 17:40 & 17:49 & 17:48 & 18.3 & 6.8 &\
2002-02-20 & M4.3 & 09:46 & 09:58 & 09:54 & 19.2 & 6.8 &\
2002-04-04 & M6.1 & 15:24 & 15:30 & 15:30 & 33.5 & 19.2 &\
2002-07-06 & M1.8 & 03:24 & 03:33 & 03:32 & 8.2 & 8.3 &\
2002-07-20 & X3.5 & 20:52 & 21:08 & 21:08 & 32.3 & 8.1 &\
2002-07-23 & X4.8 & 00:17 & 00:28 & 00:23 & 12.3 & 15.6 &\
2003-11-03a & X3.9 & 09:43 & 09:49 & 09:49 & 14.7 & 15.7 &\
2003-11-03b & X2.9 & 01:01 & 01:17 & 01:16 & 14.8 & 6.9 &\
2004-08-18 & X2.5 & 17:31 & 17:36 & 17:36 & 12.9 & 12.3 &\
2005-01-09 & M2.4 & 08:25 & 08:39 & 08:39 & 67.0 & 22.1 &\
\* The peak time is according to [*RHESSI*]{} light curve in the energy band of 30-50 keV.
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---
abstract: 'We argue that the textbook method for solving eigenvalue equations is simpler, more elegant and efficient than the Asymptotic Iteration Method (AIM) applied in J. Phys. A [**44**]{} 155205. We show that the Kratzer potential is not a realistic model for the vibration–rotation spectrua of diatomic molecules because it predicts the position of the absorption infrared bands too far from the experimental ones (at least for the $HCl$ and $H_2$ molecules chosen as illustrative examples in that paper).'
address: 'INIFTA (UNLP, CCT La Plata–CONICET), División Química Teórica, Blvd. 113 S/N, Sucursal 4, Casilla de Correo 16, 1900 La Plata, Argentina'
author:
- Francisco M Fernández
title: 'Comment on: “Non–relativistic treatment of diatomic molecules interacting with generalized Kratzer potential in hyperspherical coordinates”'
---
In order to study the vibration–rotation motion of diatomic molecules in $N$ dimensions Durmus[@D11] chose the Kratzer potential and solved the Schrödinger equation in hyperspherical coordinates by means of the Asymptotic Iteration Method (AIM). He obtained the well known results and as an illustrative and practical application of the model he restricted himself to the only apparently relevant case $N=3$ calculating some vibration–rotation energies for $H_{2}$ and $HCl$. In what follows we contrast the AIM derivation of the main equations with the well known and widely used textbook approach, and also compare the theoretical results for those molecules with experimental ones.
The starting point is the Schrödinger equation $$\left[ -\frac{\hbar ^{2}}{2\mu }\nabla ^{2}+V(r)\right] \Psi (\mathbf{r})=E\Psi (\mathbf{r}) \label{eq:Schro}$$ where $\mu $ is the reduced mass of the molecule and $E$ is the vibration–rotation energy. As a “realistic” model for the interaction between the nuclei the author chose the Kratzer potential $$V(r)=D_{e}\left( \frac{r-r_{e}}{r}\right) ^{2}+\eta \label{eq:V_Kratzer}$$ where $r_{e}$ is the equilibrium internuclear separation and $D_{e}$ is the molecular dissociation energy (misleadingly called intermolecular separation and dissociation energy between diatomic molecules, respectively, by the author[@D11]). He made a curious distinction between the modified Kratzer potential $\eta
=0$ and the Kratzer potential $\eta =-D_{e}$ which are just two alternative expressions of the same interaction with the energy origin shifted by $\eta $. Without a plausible justification he further argued that it is useful from a physical point of view to consider the general $N$–dimensional case where $$r^{2}=\sum_{j=1}^{N}x_{j}^{2},\;\nabla ^{2}=\sum_{j=1}^{N}\frac{\partial ^{2}}{\partial x_{j}^{2}} \label{eq:N_dim}$$
Since the potential is spherically symmetric one can separate the Schrödinger equation (\[eq:Schro\]) into its radial and angular parts, and by means of well–known transformations the former is reduced to[@D11] $$yF^{\prime \prime }(y)+\left( 2\gamma +N-1-2\beta y\right) F^{\prime
}(y)+\left[ 2\kappa -\beta (2\gamma +N-1)\right] F(y)=0 \label{eq:F(y)}$$ where $$\begin{aligned}
y &=&\frac{r}{r_{e}} \nonumber \\
\kappa &=&\frac{2\mu D_{e}r_{e}^{2}}{\hbar ^{2}} \nonumber \\
\beta ^{2} &=&\frac{2\mu r_{e}^{2}(D_{e}-E+\eta )}{\hbar ^{2}} \nonumber \\
\gamma &=&\frac{N-2}{2}+\sqrt{\kappa +\left( l+\frac{N-2}{2}\right) ^{2}}
\label{eq:parameters}\end{aligned}$$ and $l=0,1,\ldots $ is the angular–momentum quantum number (called $l_{N-1}$ in Ref. [@D11]). In order to solve this equation Durmus[@D11] applied the AIM which is an iterative approach that gives results for $n=0,1,\ldots $, where $n$ labels the number of iterations. By inspection of the particular outputs $\beta _{0l},\,\beta _{1l},\ldots $ one hopefully derives the value of $\beta _{nl}$ for an arbitrary number $n$ of iteration steps and then the allowed energy $E=E_{nl} $. With some more ingenuity one realizes that $F(y)$ is proportional to the confluent hypergeometric function ${\vphantom{F}}_{1}{F}_{1}(-n,2\gamma +N-1;2\beta y)$. This procedure offers little difficulty if one already knows the exact result beforehand which is actually the case here. Once we have the solution in terms of the confluent hypergeometric function we easily rewrite it in terms of the associated Laguerre polynomials[@D11; @AS72].
Solving equation (\[eq:F(y)\]) is a textbook problem[@F99] and the widely known approach is faster, more elegant and efficient than the AIM. If we define $y=\frac{z}{2\beta }$ then $w(z)=F(\frac{z}{2\beta })$ satisfies $$zw^{\prime \prime }(z)+\left( 2\gamma +N-1-z\right) w^{\prime }(z)+\left[
\frac{\kappa }{\beta }-\left( \gamma +\frac{N-1}{2}\right) \right] w(z)=0
\label{eq:w(z)}$$ that is a particular case of Kummer’s equation[@AS72] $$zw^{\prime \prime }(z)+\left( b-z\right) w^{\prime }(z)-aw(z)=0
\label{eq:Kummer_eq}$$ The Kummer’s function can be expanded in a power–series $$M(a,b,z)=\sum_{j=0}^{\infty }\frac{a_{j}z^{j}}{j!b_{j}} \label{eq:Kummer_M}$$ where $\xi _{j}=\xi (\xi +1)(\xi +2)...(\xi +j-1)$ and $\xi _{0}=1$. It becomes a polynomial of degree $n$ when $a=-n$ and $b\neq -m$ ($m$ and $n$ positive integers). On comparing equations (\[eq:w(z)\]) and (\[eq:Kummer\_eq\]) we directly obtain $$\beta _{nl}=\frac{\kappa }{n+\gamma +\frac{N-1}{2}} \label{eq:beta_n}$$ and[@AS72] $$F_{n}(y)=M(-n,2\gamma +N-1,2\beta y)={\vphantom{F}}_{1}{F}_{1}(-n,2\gamma
+N-1;2\beta y)$$ It is clear that one can derive the solution to equation (\[eq:F(y)\]) directly from comparison of the appropriate equations in a way that makes the AIM utterly unnecessary.
Durmus[@D11] did not show any physical application of the $N$–dimensional model except for the obvious case $N=3$. Most curiously he showed results in the form of a table and figure for both $\eta =0$ and $\eta =-D_{e}$. Apparently, he did not realize that both spectra are the same but for an energy shift that is irrelevant from a physical point of view. Suffice to say that the physical observables are not affected by this shift.
Another curious fact is that in Table 2 the author only showed the energies for $n\geq $ $l=0,1,\ldots $. In the case of a diatomic molecule, to which the model is supposed to apply, $n$ is the vibrational quantum number $v$ and $l$ is the rotational quantum number $J$ so that such selection of quantum numbers is of scarce utility from a spectroscopic point of view (one would expect increasing $J$ for a given $v$).
Durmus[@D11] argued that the Kratzer potential provides a realistic description of molecular vibrations, but it is far from true as discussed by Plíva[@P99] who stated that “However, in its basic form, this function provides only a rather crude approximation for the molecular potential, and for this reason it has not been popular with spectroscopists”. In the first place, the Kratzer potential supports an infinite number of vibration–rotation levels which is not the case of actual diatomic molecules. In addition to it, it does not describe the spectrum correctly as we will show in what follows. From the vibration–rotation energies written in the usual spectroscopic way $$E_{v,J}=-\frac{\kappa D_{e}}{\left[ v+\frac{1}{2}+\sqrt{\kappa
+\left( J+\frac{1}{2}\right) ^{2}}\right] ^{2}} \label{eq:EvJ}$$ we obtain the spectral lines in wavenumber units $$\tilde{\nu}=\frac{E_{\nu ^{\prime },J^{\prime }}-E_{v,J}}{hc} \label{eq:nu}$$ according to the selection rules $\Delta v=\pm 1$ and $\Delta
J=\pm 1$.[@H50] The results in Durmus’ Table 2 do not even allow us to obtain the P and R branches of the fundamental absorption band ($v=0$, $v^{\prime }=1$, $J^{\prime }=J\pm
1$).[@H50] The first two entries for $HCl$ in that table predict the center of this band to appear at $E_{1,0}-E_{0,0}=0.1482\,eV$ or $1195\,cm^{-1}$. However, it is well known that the center of the fundamental band is located at $\tilde{\nu}=2886\,cm^{-1}$,[@H50] more than twice the value given by Durmus’ energies. It is not better for $H_2$ (theoretical$=2715.7\,cm^{-1}$, experimental$=4160.2\,cm^{-1}$) as expected from a model that no spectroscopist would take seriously.
Summarizing the main conclusions of this comment we may say that the application of the AIM for solving the Schrödinger equation with the Kratzer potential is far from being the best strategy. One obtains the same expressions in a more direct, easy and elegant way by means of the standard textbook method.[@F99] In addition to it, since Durmus did not show any plausible physical application of the model to space dimensions other than $N=3$ and the results for ordinary diatomic molecules are extremely poor we conclude that the model is of scarce physical utility.
[9]{} Durmus A 2011 *J. Phys. A* **44** 155205.
Flügge S 1999 *Practical Quantum Mechanics* (Springer-Verlag, Berlin).
Abramowitz M and Stegun I A 1972 *Handbook of Mathematical Functions* (Dover, New York).
Plíva J 1999 *J. Mol. Spectrosc.* **193** 7.
Herzberg G 1950 *Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules* (Van Nostrand Reinhold, New York).
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abstract: 'Real-world data processing problems often involve various image modalities associated with a certain scene, including RGB images, infrared images or multispectral images. The fact that different image modalities often share diverse attributes, such as certain edges, textures and other structure primitives, represents an opportunity to enhance various image processing tasks. This paper proposes a new approach to construct a high-resolution (HR) version of a low-resolution (LR) image given another HR image modality as reference, based on joint sparse representations induced by coupled dictionaries. Our approach, which captures the similarities and disparities between different image modalities in a learned sparse feature domain in *lieu* of the original image domain, consists of two phases. The coupled dictionary learning phase is used to learn a set of dictionaries that couple different image modalities in the sparse feature domain given a set of training data. In turn, the coupled super-resolution phase leverages such coupled dictionaries to construct a HR version of the LR target image given another related image modality. One of the merits of our sparsity-driven approach relates to the fact that it overcomes drawbacks such as the texture copying artifacts commonly resulting from inconsistency between the guidance and target images. Experiments on both synthetic data and real multimodal images demonstrate that incorporating appropriate guidance information via joint sparse representation induced by coupled dictionary learning brings notable benefits in the super-resolution task with respect to the state-of-the-art.'
author:
- |
Pingfan Song, *Student Member, IEEE*, Xin Deng, *Student Member, IEEE*,\
João F. C. Mota, *Member, IEEE*, Nikos Deligiannis, *Member, IEEE*,\
Pier Luigi Dragotti, *Fellow, IEEE*, and Miguel R. D. Rodrigues, *Senior Member, IEEE* [^1] [^2] [^3] [^4] [^5]
title: 'Multimodal Image Super-resolution via Joint Sparse Representations induced by Coupled Dictionaries'
---
=1
Multimodal image super-resolution, coupled dictionary learning, joint sparse representation, side information
Introduction {#sec:intro}
============
Image super-resolution (SR) is an operation that involves the enhancement of pixel-based image resolution, while minimizing visual artifacts. However, the construction of a high-resolution (HR) version of a low-resolution (LR) image requires inferring the values of missing pixels, making image SR a severely ill-conditioned problem. Various image models and approaches have been proposed to regularize this ill-posed problem via employing some prior knowledge, including natural priors [@li2001new; @dai2007soft; @sun2008image; @zhang2008image], local and non-local similarity [@yang2013fast; @dong2013nonlocally], sparse representation over fixed or learned dictionaries [@mallat2010super; @yang2010image; @yang2012coupled; @zeyde2010single; @timofte2013anchored; @timofte2014a+; @wei2016fresh], and sophisticated features from deep learning [@dong2016image; @dong2016accelerating; @kim2016accurate; @kim2016deeply]. These typical super-resolution approaches focus only on single modality images without exploiting the availability of other modalities as guidance. However, in many practical application scenarios, a certain scene is often imaged using various sensors that yield different image modalities. For example, in remote sensing it is typical to have various image modalities of earth observations, such as a panchromatic band version, a multispectral bands version, and an infrared (IR) band version [@gomez2015multimodal; @loncan2015hyperspectral]. In order to balance cost, bandwidth and complexity, these multimodal images are usually acquired with different resolutions [@gomez2015multimodal]. These scenarios call for approaches that can capitalize on the availability of multiple image modalities of the same scene – which typically share textures, edges, corners, boundaries, or other salient features – in order to super-resolve the LR images with the aid of the HR images of a different modality.
Therefore, a variety of joint super-resolution/upsampling approaches have been proposed to capitalize on the availability of additional *guidance images*, also referred to as *side information*[@renna2016classification; @mota2017compressed], to aid the super-resolution of target LR modalities[@kopf2007joint; @he2010guided; @ham2017robust; @li2016deep; @shen2015multispectral; @zhang2014rolling]. The basic idea behind these methods is that the structural details of the guidance image can be transferred to the target image. However, these methods tend to introduce notable texture-copying artifacts, i.e. erroneous structure details that are not originally present in the target image because such methods typically fail to distinguish similarities and disparities between the different image modalities. The motivation of this work is to introduce a new image SR approach, based on joint sparse representations induced by coupled dictionaries, that has the ability to take into account both similarities and disparities between target and guidance images in order to deliver superior SR performance.
[**Proposed Scheme**]{}. The proposed scheme is based on three elements: (1) a data model; (2) a coupled dictionary learning algorithm; and (3) a coupled image super-resolution algorithm.
- *Data Model*: This is a patch-based model that relies on the use of coupled dictionaries to jointly sparsely represent a pair of patches from the different image modalities. Of particular relevance is the ability to represent the similarities and disparities between the different image modalities in this sparse feature domain in *lieu* of the original image domain, which leads to a higher super-resolution accuracy.
- *Coupled Dictionary Learning*: This algorithm learns the data model – including the coupled dictionaries along with the joint sparse representations of the different image modalities – from a set of training images.
- *Coupled Image Super-Resolution*: Given the coupled dictionaries, this algorithm is then used to obtain the joint sparse representations of the target image and the guidance image, and estimates a HR version of the target image.
In comparison with state-of-the-art approaches [@kopf2007joint; @he2010guided; @ham2017robust; @li2016deep; @shen2015multispectral], our approach can better model the common and distinct features of the different data modalities. This capability makes our approach more robust to inconsistencies between the guidance and the target images, as both the target LR image and the guidance image are taken into account during the estimation of the target HR image, instead of unilaterally transferring the structure details from the HR guidance image. In addition, the proposed approach leverages explicit regularization to formulate a task-specific model, with more flexibility than implicit regularization[@kopf2007joint; @he2010guided]. Further, the proposed approach requires much less training data, computing resources and training time than the deep-learning-based approach [@li2016deep]. Overall, by running the coupled dictionary learning algorithm followed by the coupled image super-resolution algorithm, the proposed approach achieves better performance in the multimodal image super-resolution task than state-of-the-art methods including the deep-learning-based method [@li2016deep] in scenarios where the amount of training data is limited.
[[**Contributions.**]{}]{} Our contributions are as follows:
- We devise a data model for multimodal signals that captures the similarities and disparities between different modalities using joint sparse representations induced by coupled dictionaries. Compared with our previous work [@song2016coupled], the present model is more general, because it does not require the matrix that models the conversion of a HR version of the image to the LR counterpart to be known.
- We also propose a coupled dictionary learning algorithm to learn the coupled dictionaries from different data modalities. Compared with [@song2016coupled], in the learning stage, the proposed algorithm does not require the knowledge of the matrix that converts a HR image to a LR version.
- We also propose a multimodal image super-resolution algorithm that enhances the resolution of the target LR image with the aid of another guidance HR image modality.
- Finally, we conduct a series of experiments both on synthetic and real images that demonstrate that our approach provides notable benefits in the joint image super-resolution task. In particular, our approach provides significant PSNR and SSIM gains over other state-of-the-art approaches such as [@kopf2007joint; @he2010guided; @ham2017robust; @li2016deep; @shen2015multispectral].
[[**Organization.**]{}]{} The remainder of this paper is organized as follows. We review related work in Section \[sec:RelatedWork\] and propose our multimodal image super-resolution framework, including the data model, the coupled dictionary learning algorithm, and the multimodal image super-resolution algorithm in Section \[sec:SIMIS\]. Section \[sec:Experiments\] is devoted to various simulation and practical experiments which demonstrate that our approach can lead to significant gains over the state-of-the-art. We summarize the main contributions of the paper in Section V.
Related Work {#sec:RelatedWork}
============
There are various image super-resolution approaches in the literature. Single image super-resolution approaches do not leverage other guidance images, whereas joint image super-resolution approaches explicitly leverage the availability of other image modalities.
Single image SR
---------------
In general, conventional single image SR approaches can be categorized into three classes: interpolation-based, reconstruction-based and learning-based SR approaches.
[[**Interpolation-based SR approaches.**]{}]{} Advanced interpolation approaches exploit natural image priors, such as edges [@li2001new], image smoothness [@dai2007soft], gradient profile [@sun2008image] and other geometric regularity of image structures [@zhang2008image]. These methods are simple and fast, but tend to overly smooth image edges and generate ringing and jagged artifacts. [[**Reconstruction-based SR approaches.**]{}]{} Reconstruction-based SR approaches, also referred to as model-based SR methods, attempt to regularize the highly undetermined image SR inverse problem by exploiting various image priors, including self-similarity of images patches[@yang2013fast], sparsity in the wavelet domain [@mallat2010super], analysis operator [@hawe2013analysis], and other fused versions[@dong2013nonlocally]. Recent work[@wei2016fresh] proposes a piecewise smooth image model and makes use of the finite rate of innovation (FRI) theory to reconstruct HR target images. These reconstruction-based methods usually offer better performance than interpolation-based methods. [[**Learning-based SR approaches.**]{}]{} These SR approaches typically consist of two phases: (1) a learning phase where one learns certain image priors from training images and (2) a testing phase where one obtains the HR image from the LR version with the aid of the prior knowledge.
In particular, patch-wise learning-based approaches leverage learned mappings or co-occurrence priors between LR and HR training image patches to predict the fine details in the testing target HR images according to their corresponding LR versions [@bevilacqua2012low; @yang2008image; @yang2010image; @yang2012coupled; @zeyde2010single; @timofte2013anchored; @timofte2014a+; @wang2012semi; @jia2013image]. For example, motivated by Compressive Sensing [@donoho2006compressed; @candes2006robust], Yang *et al.* [@yang2008image; @yang2010image; @yang2012coupled] propose a sparse-coding based image SR strategy, which is improved further by Zeyde, *et al.*[@zeyde2010single]. The key idea is a sparse representation invariance assumption which states that HR/LR image pairs share the same sparse coefficients with respect to a pair of HR and LR dictionaries. Along similar lines, Timofte *et al.* [@timofte2013anchored; @timofte2014a+] propose a strategy, referred to as anchored neighbourhood regression, that combines the advantage of neighbor embedding and dictionary learning. In order to achieve better flexibility and stability of signal recovery, semi-coupled dictionary learning[@wang2012semi] and coupled dictionary learning [@jia2013image] are proposed to relax the sparse representation invariance assumption to the same support assumption, allowing more flexible mappings. Note that, even though the terminology related to “coupled dictionary learning” also appears in these works[@yang2012coupled; @wang2012semi; @jia2013image], their approaches focus only on coupling LR and HR images of the same modality. In addition, their assumptions, models and algorithms are also different from ours.
Inspired by sparse-coding-based SR methods, Dong *et al.*[@dong2016image] propose a single image super-resolution convolutional neural network (SRCNN) consisting of a patch extraction and representation layer, a non-linear mapping layer and a reconstruction layer. A faster and deeper version FSRCNN was proposed in [@dong2016accelerating], where the previous interpolation operation is removed and a deconvolution layer is introduced at the end of the network to perform upsampling. Kim *et al.*[@kim2016accurate] propose a very deep SR network (VDSR) which exploits residual-learning for fast converging and multi-scale training datasets for handling multiple scale factors. Different from the above CNN-based SR approaches, [@kim2016deeply] propose a deeply-recursive convolutional network (DRCN) with recursive-supervision and skip-connection to ease the training.
Joint image SR
--------------
Compared with single image SR, joint image SR attempts to leverage an additional guidance image to aid the SR process for the target image, by transferring structural information of the guidance image to the target image. It is a particular application of joint image filtering or guided image filtering[@he2010guided; @kopf2007joint].
The bilateral filter[@tomasi1998bilateral] is a widely used translation-variant edge-preserving filter that outputs a pixel as a weighted average of neighboring pixels. The weights are computed by a spatial filter kernel and a range filter kernel evaluated on the data values themselves. It smoothes the image while preserving edges. The joint bilateral upsampling[@kopf2007joint] generalizes the bilateral filter by computing the weights with respect to another guidance image rather than the input image. In particular, it applies the range filter kernel to a HR guidance image, expecting to incorporate the high frequencies of the guidance image into the LR target image. However, it has been noticed that joint bilateral image filtering may introduce gradient reversal artifacts as it does not preserve gradient information[@he2010guided]. Later, the guided image filtering[@he2010guided] was proposed to overcome this limitation. Sometimes, directly transferring guidance gradients also results in notable appearance change[@shen2015multispectral]. To address this problem, [@shen2015multispectral] proposes a framework that optimizes a novel scale map to capture the nature of structure discrepancy between images. However, as the construction of these filters considers unilaterally the static guidance image, [@shen2015multispectral] suffers from the inconsistency of the local structures in the guidance and target images, and may therefore transfer incorrect structure details to the target images. The study in [@ham2017robust] proposes robust guided image filtering, referred to as static/dynamic (SD) filtering, which jointly leverages static guidance image and dynamic target image to iteratively refine the target image. These techniques often use hand-crafted objective functions that may not reflect natural image priors well. Recent work[@li2016deep] proposes a Convolutional Neural Networks (CNN) based joint image filtering approach. This approach considers the structures of both input and guidance images, but requires numerous annotated images and intensive computing resources to train the deep model for each task.
Our joint image SR based on coupled dictionary learning falls into the learning-based category. Therefore, the priors used in our approach are learned from a training dataset rather than being hand-crafted and thus adapt to the target modality and guidance modality.
Multimodal Image SR via Joint Sparse Representations Induced by Coupled Dictionaries {#sec:SIMIS}
====================================================================================
We now introduce our SR approach. In particular, we describe the data model that couples different image modalities and also the joint image SR framework that encompasses both a coupled dictionary learning phase and a coupled super-resolution phase.
[ ]{}
Multi-modal Data Model
----------------------
[[**Basic Data Model.**]{}]{} We first introduce a basic data model that captures the relationships – including similarities and disparities – between two different image modalities. In particular, we propose to use joint sparse representations to express a pair of registered, vectorized image patches $\mathbf{x} \in \mathbb{R}^{N_x} $ and $\mathbf{y} \in \mathbb{R}^{N_y}$ associated with different modalities as follows: $$\label{Eq:MultimodalDataModel}
\begin{split}
\mathbf{x}
&= \boldsymbol{\Psi}_{c} \, \mathbf{\mathbf{z}} + \boldsymbol{\Psi} \, \mathbf{u} \,,
\\
\mathbf{y}
&= \boldsymbol{\Phi}_{c} \, \mathbf{\mathbf{z}} + \boldsymbol{\Phi} \, \mathbf{v} \,,
\end{split}$$ where $\mathbf{z} \in \mathbb{R}^{K_c}$ is a sparse representation that is common to both modalities, $\mathbf{u} \in \mathbb{R}^{K_u}$ is a sparse representation specific to modality $\mathbf{x}$, while $\mathbf{v} \in \mathbb{R}^{K_v}$ is a sparse representation specific to modality $\mathbf{y}$. In turn, $\boldsymbol{\Psi}_{c} \in \mathbb{R}^{N_x \times K_c}$ and $\boldsymbol{\Phi}_{c} \in \mathbb{R}^{N_y \times K_c}$ are a pair of dictionaries associated with the common sparse representation $\mathbf{z}$, whereas $\boldsymbol{\Psi} \in \mathbb{R}^{N_x \times K_u}$ and $\boldsymbol{\Phi} \in \mathbb{R}^{N_y \times K_v}$ are dictionaries associated with the specific sparse representations $\mathbf{u}$ and $\mathbf{v}$, respectively. (For simplicity, we take $N = N_x = N_y$, $K = K_c = K_u = K_v$ hereafter.)
This data model is inspired by the model in[@deligiannis2017multi; @deligiannis2016Xray] where we propose to learn a group of coupled dictionaries from a X-ray version and a RGB version of a painting to aid the separation of two overlapped X-ray scans from double-sided paintings. It captures the similarities and disparities between two different image modalities in the sparse representation domain rather than the original image domain (as guided approaches do[@kopf2007joint; @he2010guided; @shen2015multispectral; @ham2017robust]). It is also the basis of the data model that underlies our super-resolution approach.
[[**SR Data Model.**]{}]{} We now introduce another data model – which capitalizes on the data model in – that underlies our proposed super-resolution process. This model is based on two main assumptions:
1\. We assume – as in – that similarities and disparities between the LR and HR versions of the patches of the different image modalities can be captured using sparse representations.
2\. We also assume – as in [@yang2008image; @yang2010image; @zeyde2010single] – that the LR and HR versions of a patch of a certain image modality share the same sparse representation.
In particular, we express the LR image patch $\mathbf{x}^{l} \in \mathbb{R}^{M}$ and HR image patch $\mathbf{x}^{h} \in \mathbb{R}^{N} $ of a certain image modality, and another HR registered patch of another corresponding image modality $\mathbf{y} \in \mathbb{R}^{N}$ as follows: $$\begin{aligned}
\mathbf{x}^{h} &= \boldsymbol{\Psi}_{c}^{h} \, \mathbf{z} + \boldsymbol{\Psi}^{h} \, \mathbf{u} \,,
\label{Eq:SparseModelX}
\\
\mathbf{x}^{l}
&= \boldsymbol{\Psi}_{c}^{l} \, \mathbf{z} + \boldsymbol{\Psi}^{l} \, \mathbf{u} \,,
\label{Eq:SparseModelX_low}
\\
\mathbf{y} &= \boldsymbol{\Phi}_{c} \, \mathbf{z} + \boldsymbol{\Phi} \, \mathbf{v} \,,
\label{Eq:SparseModelY}\end{aligned}$$ where, as in the basic data model , $\mathbf{z} \in \mathbb{R}^{K}$ is the common sparse representation shared by both modalities, $\mathbf{u} \in \mathbb{R}^{K}$ is the unique sparse representation specific to modality $\mathbf{x}$ while $\mathbf{v} \in \mathbb{R}^{K}$ is the unique sparse representation specific to modality $\mathbf{y}$. In turn, $\boldsymbol{\Psi}_{c}^{h} \in \mathbb{R}^{N \times K}$, $\boldsymbol{\Psi}_{c}^{l} \in \mathbb{R}^{M \times K}$ and $\boldsymbol{\Phi}_{c} \in \mathbb{R}^{N \times K}$ are the dictionaries associated with the common sparse representation $\mathbf{z}$, whereas $\boldsymbol{\Psi}^{h} \in \mathbb{R}^{N \times K}$, $\boldsymbol{\Psi}^{l} \in \mathbb{R}^{M \times K}$ and $\boldsymbol{\Phi} \in \mathbb{R}^{N \times K}$ are dictionaries associated with the specific sparse representations $\mathbf{u}$ and $\mathbf{v}$, respectively. Please note that the sparse vectors $\mathbf{z}$ and $\mathbf{u}$ capture the relationship between the LR and HR patches of the same modality in and . Moreover, the common sparse vector $\mathbf{z}$ connects the various patches of the two different modalities in - . Note also that the disparities between modalities $\mathbf{x}$ and $\mathbf{y}$ are distinguished by the sparse vectors $\mathbf{u}$ and $\mathbf{v}$.
By capitalizing on this model, we propose in the sequel a novel joint image SR scheme that consists of two stages: (1) a training stage referred to as coupled dictionary learning (CDL) and (2) a testing stage referred to as coupled image super-resolution (CSR) (see Figure \[Fig:Diagram\]). In the training stage, we learn the dictionaries in - from a set of training image patches to couple different data modalities together. Then, in the testing stage, we use the learned dictionaries to find the representations of the LR testing patch and corresponding HR guidance patch, according to and . These sparse representations are then used to reconstruct the HR target image patch via .
Coupled Dictionary Learning (CDL) {#ssec:CDL}
---------------------------------
Training data matrices $\mathbf{X}^l$, $\mathbf{X}^h$ and $\mathbf{Y}$.
Dictionary pairs $[\boldsymbol{\Psi}_{c}^l,\boldsymbol{\Psi}^l]$, $[\boldsymbol{\Psi}_{c}^h,\boldsymbol{\Psi}^h]$ and $[\boldsymbol{\Phi}_{c},\boldsymbol{\Phi}]$.
Initialize each dictionary. Set the training iterations $OutIter$ and $InIter$.
**Step 1:** \[AlgOpt:OutIterStart\] \[AlgOpt:GlobalSC1\] \[Code:SRO:GlobalSparseCoding\] **Global Sparse Coding**. Fix all the dictionaries, then solve to obtain updated sparse representations $\mathbf{Z}$, $\mathbf{U}$ and $\mathbf{V}$. \[AlgOpt:LocalDictUpdate1\] **Local Common Dictionary Update**. Fix $\boldsymbol{\Psi}^l$, $\boldsymbol{\Phi}$, and only update $\boldsymbol{\Psi}^l_{c}$ and $\boldsymbol{\Phi}_{c}$ by solving . Specifically, for each atom pair $\begin{bmatrix}
\boldsymbol{\psi}_{ck}^l \\ \boldsymbol{\phi}_{ck}
\end{bmatrix}$ of $\begin{bmatrix}
\boldsymbol{\Psi}_{c}^l \\ \boldsymbol{\Phi}_{c}
\end{bmatrix}$, denote by $\mathbf{z}^k$ the $k$-th row vector in $\mathbf{Z}$, and $\Omega_k = \{ i | 1 \leq i \leq T, \mathbf{z}^k(i) \neq 0 \}$ the index set of those training samples that use $k$-th atom pair. Then, compute the representation residual $$\mathbf{E}_k =
\left(
\begin{bmatrix}
\mathbf{X}^l - \boldsymbol{\Psi}^l \mathbf{U}
\\
\mathbf{Y} - \boldsymbol{\Phi} \mathbf{V} \end{bmatrix}
-
\begin{bmatrix}
\boldsymbol{\Psi}_{c}^l \\ \boldsymbol{\Phi}_{c}
\end{bmatrix}
\mathbf{Z}
+
\begin{bmatrix}
\boldsymbol{\psi}_{ck}^l \\ \boldsymbol{\phi}_{ck}
\end{bmatrix} \mathbf{z}^k
\right)_{(:,\Omega_k)}$$ Apply SVD on $\mathbf{E}_k = \mathbf{P} \mathbf{\Sigma} \mathbf{Q}^\text{T}$ and choose the first column of $ \mathbf{P} $ as the updated atom pair $\begin{bmatrix}
\boldsymbol{\psi}_{ck}^l \\ \boldsymbol{\phi}_{ck}
\end{bmatrix}$. \[AlgOpt:GlobalSC2\] **Global Sparse Coding**. The same as step \[Code:SRO:GlobalSparseCoding\]. \[AlgOpt:LocalDictUpdate2\] **Local Unique Dictionary Update**. Fix $\boldsymbol{\Psi}^l_{c}$, $\boldsymbol{\Phi}_{c}$, and only update $\boldsymbol{\Psi}^l$ and $\boldsymbol{\Phi}$ by solving and . For each atom $\boldsymbol{\psi}_{k}^l$ of $\boldsymbol{\Psi}^l$, denote by $\mathbf{u}^k$ the $k$-th row vector in $\mathbf{U}$, and $\Omega_k = \{ i | 1 \leq i \leq T, \mathbf{u}^k(i) \neq 0 \}$. Then, compute the representation residual $$\mathbf{E}_k =
\left(
\begin{bmatrix}
\mathbf{X}^l - \boldsymbol{\Psi}^l_{c} \mathbf{Z}
\end{bmatrix}
-
\boldsymbol{\Psi}^l \mathbf{U}
+
\boldsymbol{\psi}_{k}^l \mathbf{u}^k
\right)_{(:,\Omega_k)}$$ Apply SVD on $\mathbf{E}_k = \mathbf{P} \mathbf{\Sigma} \mathbf{Q}^\text{T}$ and choose the first column of $ \mathbf{P} $ as the updated atom $\boldsymbol{\psi}_{k}^l$. Each atom $\boldsymbol{\phi}_{k}$ of $\boldsymbol{\Phi}$ is updated with $\Omega_k = \{ i | 1 \leq i \leq T, \mathbf{v}^k(i) \neq 0 \}$ and $
\mathbf{E}_k =
\left(
\begin{bmatrix}
\mathbf{Y} - \boldsymbol{\Phi}_{c} \mathbf{Z}
\end{bmatrix}
-
\boldsymbol{\Phi} \mathbf{V}
+
\boldsymbol{\phi}_{k} \mathbf{v}^k
\right)_{(:,\Omega_k)}
$ in a similar manner.
\[AlgOpt:OutIterEnd\]
**Step 2:** Construct $[\boldsymbol{\Psi}_{c}^h,\boldsymbol{\Psi}^h]$ as in . Return dictionaries.
We assume that we have access to $T$ registered patches of LR, HR and guidance images for learning our data model in - . In particular, let $\mathbf{x}^l_i$, $\mathbf{x}^h_i$ and $\mathbf{y}_i$ ($i=1 \ldots T$) denote the registered patches corresponding to the LR, HR, and the guidance training image patches, and let $\mathbf{z}_i$, $\mathbf{u}_i$ and $\mathbf{v}_i$ ($i=1 \ldots T$) denote their sparse representations. Our coupled dictionary learning problem can now be posed as follows: $$\label{Eq:CoupledDL}
\begin{array}{cl}
\underset{
\begin{subarray}{c}
\left\{ \boldsymbol{\Psi}_{c}^l, \boldsymbol{\Psi}^l, \boldsymbol{\Psi}_{c}^h, \right. \\
\left. \boldsymbol{\Psi}^h, \boldsymbol{\Phi}_{c}, \boldsymbol{\Phi} \right\} \\
\{ \mathbf{Z}, \mathbf{U}, \mathbf{V} \}
\end{subarray}}
{ \text{minimize}}
& \left\|
\begin{bmatrix}
\mathbf{X}^l \\ \mathbf{X}^h \\ \mathbf{Y}
\end{bmatrix}
-
\begin{bmatrix}
\boldsymbol{\Psi}_{c}^l & \boldsymbol{\Psi}^l & \mathbf{0} \\
\boldsymbol{\Psi}_{c}^h & \boldsymbol{\Psi}^h & \mathbf{0} \\
\boldsymbol{\Phi}_{c} & \mathbf{0} & \boldsymbol{\Phi} \\
\end{bmatrix}
\begin{bmatrix}
\mathbf{Z} \\
\mathbf{U} \\
\mathbf{V} \\
\end{bmatrix}
\right\|_F^2
\\
\text{subject to}
& \|\mathbf{z}_i \|_0
+ \|\mathbf{u}_i \|_0
+ \|\mathbf{v}_i \|_0 \leq s, \; \forall i.
\end{array}$$ where $\mathbf{X}^l = \left[\mathbf{x}^l_1,...,\mathbf{x}^l_T\right]$ $\in \mathbb{R}^{M \times T}$, $\mathbf{X}^h = \left[\mathbf{x}^h_1,...,\mathbf{x}^h_T\right]$ $\in \mathbb{R}^{N \times T}$ and $\mathbf{Y} = \left[\mathbf{y}_1,...,\mathbf{y}_T \right]$ $\in \mathbb{R}^{N \times T}$, $\mathbf{Z} = \left[\mathbf{z}_1, ..., \mathbf{z}_T \right]$ $\in \mathbb{R}^{K \times T}$, $\mathbf{U} = \left[\mathbf{u}_1, ..., \mathbf{u}_T \right]$ $\in \mathbb{R}^{K \times T}$ and $\mathbf{V} = \left[\mathbf{v}_1, ..., \mathbf{v}_T \right]$ $\in \mathbb{R}^{K \times T}$, and $\| \cdot \|_F $ and $\| \cdot \|_0 $ denote the Frobenius norm and $\ell_0$ norm, respectively.
Note that – akin to other dictionary learning formulations[@aharon2006img] – the objective in the optimization problem encourages the data representation to approximate the data, and the constraint in encourages the data representation to be sparse (i.e. the overall sparsity of the data representations is constrained to be less than or equal to $s$)[^6].
The coupled dictionary learning problem in is a non-convex optimization problem and differs from other standard dictionary learning problems in terms of its structure. Algorithm \[Alg:CK-SVD\] shows how we adapt K-SVD[@aharon2006img] to solve problem . Our coupled dictionary learning algorithm consists of two steps. In the first step, the algorithm uses $\mathbf{X}^l$ and $\mathbf{Y}$ to learn the two pairs of dictionaries $[\boldsymbol{\Psi}_c^l, \boldsymbol{\Psi}^l]$ and $[\boldsymbol{\Phi}_c, \boldsymbol{\Phi}]$ and the sparse codes $\mathbf{Z}$, $\mathbf{U}$, $\mathbf{V}$. In the second step, the algorithm uses $\mathbf{X}^h$ and the sparse codes $\mathbf{U}$, $\mathbf{V}$ to learn the HR dictionaries $[\boldsymbol{\Psi}_c^h, \boldsymbol{\Psi}^h]$.[^7]
### Step 1
In the first step, we learn the dictionary pairs $[\boldsymbol{\Psi}_c^l, \boldsymbol{\Psi}^l]$, $[\boldsymbol{\Phi}_c, \boldsymbol{\Phi}]$ and the sparse codes $\mathbf{Z}$, $\mathbf{U}$, $\mathbf{V}$ from $\mathbf{X}^l$ and $\mathbf{Y}$ by solving the following optimization problem: $$\label{Eq:CoupledDL_DL1}
\begin{array}{cl}
\underset{
\begin{subarray}{c}
\left\{ \boldsymbol{\Psi}_{c}^l, \boldsymbol{\Psi}^l, \boldsymbol{\Phi}_{c}, \boldsymbol{\Phi} \right\} \\
\{ \mathbf{Z}, \mathbf{U}, \mathbf{V} \}
\end{subarray}}
{ \text{minimize}}
& \left\|
\begin{bmatrix}
\mathbf{X}^l \\ \mathbf{Y}
\end{bmatrix}
-
\begin{bmatrix}
\boldsymbol{\Psi}_{c}^l & \boldsymbol{\Psi}^l & \mathbf{0} \\
\boldsymbol{\Phi}_{c} & \mathbf{0} & \boldsymbol{\Phi} \\
\end{bmatrix}
\begin{bmatrix}
\mathbf{Z} \\
\mathbf{U} \\
\mathbf{V} \\
\end{bmatrix}
\right\|_F^2
\\
\text{subject to}
& \|\mathbf{z}_i \|_0
+ \|\mathbf{u}_i \|_0
+ \|\mathbf{v}_i \|_0 \leq s, \; \forall i.
\end{array}$$
For Problem , the columns of the whole dictionary are constrained to have less than unit $\ell_2$ norm to avoid trivial solution. Otherwise, the columns may become infinitely large and accordingly the sparse codes may become infinitely small, which result in the sparsity constraint term ineffective. In order to handle this non-convex optimization problem, we adopt an alternating optimization approach that performs sparse coding and dictionary update alternatively. During the sparse coding stage, we first fix the global dictionaries and obtain the sparse representations by solving: $$\label{Eq:CDL_SparseCoding}
\begin{array}{cl}
\underset{\mathbf{Z}, \mathbf{U}, \mathbf{V}}{ \text{min}}
& \left\|
\begin{bmatrix} \mathbf{X}^l \\ \mathbf{Y} \end{bmatrix}
-
\begin{bmatrix}
\boldsymbol{\Psi}^l_{c} & \boldsymbol{\Psi}^l & \mathbf{0} \\
\boldsymbol{\Phi}_{c} & \mathbf{0} & \boldsymbol{\Phi} \\
\end{bmatrix}
\begin{bmatrix}
\mathbf{Z} \\
\mathbf{U} \\
\mathbf{V} \\
\end{bmatrix}
\right\|_F^2
\\
\text{s.t.}
& \|\mathbf{z}_i \|_0
+ \|\mathbf{u}_i \|_0
+ \|\mathbf{v}_i \|_0 \leq s, \; \forall i.
\end{array}$$ This problem – which we call global sparse coding because it updates all the sparse representations $\mathbf{Z}$, $\mathbf{U}$ and $\mathbf{V}$ – is solved using the orthogonal matching pursuit (OMP) algorithm [@tropp2007signal].
During the dictionary updating stage, we fix the sparse codes and update the global dictionaries via solving: $$\label{Eq:CDL_DictUpdate}
\begin{array}{cl}
\underset{
\begin{subarray}{c}
\boldsymbol{\Psi}_{c}^l, \boldsymbol{\Psi}^l, \boldsymbol{\Phi}_{c}, \boldsymbol{\Phi}
\end{subarray}}
{ \text{minimize}}
& \left\|
\begin{bmatrix}
\mathbf{X}^l \\ \mathbf{Y}
\end{bmatrix}
-
\begin{bmatrix}
\boldsymbol{\Psi}_{c}^l & \boldsymbol{\Psi}^l & \mathbf{0} \\
\boldsymbol{\Phi}_{c} & \mathbf{0} & \boldsymbol{\Phi} \\
\end{bmatrix}
\begin{bmatrix}
\mathbf{Z} \\
\mathbf{U} \\
\mathbf{V} \\
\end{bmatrix}
\right\|_F^2
\end{array}$$ To this end, we adapt K-SVD[@aharon2006img] algorithm for our coupled dictionary learning case. The key idea is to update common dictionaries simultaneously while updating unique dictionaries individually. Specifically, we further decompose Problem into the following convex sub-problems - , so that we can sequentially train the common dictionaries and the unique dictionaries. That is, we fix the unique dictionaries $\boldsymbol{\Psi}^l$, $\boldsymbol{\Phi}$ and only update the common dictionaries $\boldsymbol{\Psi}^l_{c}$ and $\boldsymbol{\Phi}_{c}$ by solving $$\label{Eq:PartialDictionaryUpdate_Com}
\begin{array}{cl}
\underset{
\begin{subarray}{c}
\boldsymbol{\Psi}^l_{c}, \boldsymbol{\Phi}_{c} \\
\end{subarray}}{ \text{min}}
&
\left\|
\begin{bmatrix}
\mathbf{X}^l - \boldsymbol{\Psi}^l \mathbf{U} \\
\mathbf{Y} - \boldsymbol{\Phi} \mathbf{V}
\end{bmatrix}
-
\begin{bmatrix}
\boldsymbol{\Psi}^l_{c} \\
\boldsymbol{\Phi}_{c} \\
\end{bmatrix}
\mathbf{Z}
\right\|_F^2
\end{array}$$ The algorithm alternates between global sparse coding and local common dictionary update for a few iterations until the procedure converges. Next, we fix the already learned common dictionaries and train the unique dictionaries in a similar alternating manner, including global sparse coding and following two local unique dictionary update operations:[^8] $$\label{Eq:PartialDictionaryUpdate_Psi}
\begin{array}{cl}
\underset{\boldsymbol{\Psi}^l}{ \text{min}}
&
\left\|
\left(
\mathbf{X}^l - \boldsymbol{\Psi}^l_{c} \mathbf{Z}
\right)
- \boldsymbol{\Psi}^l \mathbf{U}
\right\|_F^2
\end{array}$$ $$\label{Eq:PartialDictionaryUpdate_Phi}
\begin{array}{cl}
\underset{\boldsymbol{\Phi}}{ \text{min}}
&
\left\|
\left(
\mathbf{Y}
-
\boldsymbol{\Phi}_{c} \mathbf{Z}
\right)
-
\boldsymbol{\Phi} \mathbf{V}
\right\|_F^2
\end{array}$$
### Step 2
In the second step, once the dictionary pairs $[\boldsymbol{\Psi}_c^l, \boldsymbol{\Psi}^l]$ and $[\boldsymbol{\Phi}_c, \boldsymbol{\Phi}]$ are learned from $\mathbf{X}^l$ and $\mathbf{Y}$, we construct the HR dictionaries $[\boldsymbol{\Psi}_c^h, \boldsymbol{\Psi}^h]$ based on $\mathbf{X}^h$ and sparse codes $\mathbf{Z}$ and $\mathbf{U}$ by solving the optimization problem: $$\label{Eq:CoupledDL_DL2}
\begin{array}{cl}
\underset{ \boldsymbol{\Psi}_{c}^h, \boldsymbol{\Psi}^h }
{ \text{min}}
& \! \! \!
\left\|
\mathbf{X}^h
-
\boldsymbol{\Psi}_{c}^h \mathbf{Z} - \boldsymbol{\Psi}^h \mathbf{U}
\right\|_F^2
+
\lambda
\left\|
\begin{bmatrix}
\boldsymbol{\Psi}_{c}^h & \boldsymbol{\Psi}^h \\
\end{bmatrix}
\right\|_F^2
\end{array}$$ where the second term serves as a regularizer that makes the solution more stable[^9]. This optimization problem – which exploits the conventional sparse representation invariance assumption that HR image patches $\mathbf{X}^h$ share the same sparse codes with the corresponding LR version $\mathbf{X}^l$ – admits the closed form solution $$\label{Eq:CoupledDL_DL3}
\begin{bmatrix}
\boldsymbol{\Psi}_{c}^h & \boldsymbol{\Psi}^h \\
\end{bmatrix}
=
\mathbf{X}^h \Gamma^T (\Gamma \Gamma^T + \lambda \mathbf{I})^{-1} \,,
\text{where, }
\Gamma = \begin{bmatrix} \mathbf{Z} \\ \mathbf{U} \end{bmatrix}$$ Similar to the conventional dictionary learning, our CDL algorithm can not guarantee the convergence to a global optimum due to the non-convexity nature of Problem . However, CDL is convex with respect to the dictionaries when the sparse codes are fixed or vice versa. This property ensures that dictionary algorithms usually converge to a local optimum and perform reasonably well. Indeed, experiments on both real and synthetic data, presented in Section \[sec:Experiments\], confirm that the coupled dictionaries found in the local optimum lead to satisfactory image SR performance.
Coupled Super Resolution (CSR)
------------------------------
Given the learned coupled dictionaries associated with the model in - , we now assume that we have access to a LR testing image and a corresponding registered HR guidance image as side information. We extract (overlapping) image patch pairs from these two modalities. In particular, let $\mathbf{x}^{l}_{test} \in \mathbb{R}^M$ denote a LR testing image patch and let $\mathbf{y}^{h}_{test} \in \mathbb{R}^N$ denote the corresponding HR guidance image patch. We can now pose a coupled super-resolution problem that involves two steps.
[**Step 1**]{}. First, we solve the optimization problem
$$\label{Eq:SR}
\begin{array}{cl}
\underset{\mathbf{z},\mathbf{u},\mathbf{v}}{\text{min}}
& \left\|
\begin{bmatrix}
\mathbf{x}^{l}_{test} \\ \mathbf{y}_{test}
\end{bmatrix}
-
\begin{bmatrix}
\boldsymbol{\Psi}_{c}^l & \boldsymbol{\Psi}^l & \mathbf{0} \\
\boldsymbol{\Phi}_{c} & \mathbf{0} & \boldsymbol{\Phi} \\
\end{bmatrix}
\begin{bmatrix}
\mathbf{z} \\
\mathbf{u} \\
\mathbf{v} \\
\end{bmatrix}
\right\|_2^2
\\
\text{s.t.}
& \|\mathbf{z} \|_0
+ \|\mathbf{u} \|_0
+ \|\mathbf{v} \|_0 \leq s
\end{array}$$
where the $\ell_2$ norm promotes the fidelity of sparse representations to the signals and the $\ell_0$ norm promotes sparsity for the sparse codes.
Alternatively, we can also solve the optimization problem: $$\label{Eq:SR_L1}
\begin{array}{cl}
\underset{\mathbf{z},\mathbf{u},\mathbf{v}}{\text{min}}
& \! \! \! \! \!
\left\|
\begin{bmatrix}
\mathbf{x}^{l}_{test} \\ \mathbf{y}_{test}
\end{bmatrix}
-
\begin{bmatrix}
\boldsymbol{\Psi}_{c}^l & \boldsymbol{\Psi}^l & \mathbf{0} \\
\boldsymbol{\Phi}_{c} & \mathbf{0} & \boldsymbol{\Phi} \\
\end{bmatrix}
\begin{bmatrix}
\mathbf{z} \\
\mathbf{u} \\
\mathbf{v} \\
\end{bmatrix}
\right\|_2^2
+
\lambda
\left\|
\begin{bmatrix}
\mathbf{z} \\
\mathbf{u} \\
\mathbf{v} \\
\end{bmatrix}
\right\|_1
\end{array}$$ Note that we can use standard algorithms to approximate the solution to – such as orthogonal matching pursuit (OMP) algorithm [@tropp2007signal] and iterative hard-thresholding algorithm [@blumensath2009iterative] – and to compute the solution to , including iterative soft-thresholding algorithm [@daubechies2004iterative], least angle regression algorithm [@efron2004least]. Note also that, compared with conventional standard sparse coding problems involving only LR image patch $\mathbf{x}^{l}$, our formulations and also integrate the side information $\mathbf{y}_{test}$ into the sparse coding task. Since the increase in the amount of available information is akin to the increase of the number of measurements in a Compressive Sensing scenario [@donoho2006compressed; @candes2006robust; @mota2017compressed; @renna2016classification], one can expect to obtain a more accurate estimate of the sparse codes.
[**Step 2**]{}. Finally, we can obtain an estimate of the HR patch of the target image $\mathbf{x}^{h}_{test}$ from the HR dictionaries $[\boldsymbol{\Psi}_c^h, \boldsymbol{\Psi}^h]$ and sparse codes $\mathbf{z}$ and $\mathbf{u}$ as follows
$$\label{Eq:SR_2}
\mathbf{x}^{h}_{test} = \boldsymbol{\Psi}_{c}^{h} \mathbf{z} + \boldsymbol{\Psi}^{h} \mathbf{u} \,.$$
The coupled super-resolution algorithm is described in Algorithm \[Alg:CSR\]. Note that once all the HR patches are recovered, they are integrated into a whole image with average on the overlapping areas.
Testing signals $\mathbf{x}^{l}_{test}$ and side information $\mathbf{y}_{test}$.\
Learned dictionaries $[\boldsymbol{\Psi}_{c}^l,\boldsymbol{\Psi}^l]$, $[\boldsymbol{\Psi}_{c}^h,\boldsymbol{\Psi}^h]$ and $[\boldsymbol{\Phi}_{c},\boldsymbol{\Phi}]$.
High resolution estimation $\mathbf{x}^{h}_{test}$.
. Use off-the-shelf sparse coding algorithms to solve the problem or problem to obtain the sparse codes $\mathbf{z}$, $\mathbf{u}$ and $\mathbf{v}$. . Reconstruct $\mathbf{x}^{h}_{test}$ as in .
Experiments {#sec:Experiments}
===========
We now present a series of experiments to validate the effectiveness of the proposed joint image SR approach. These experiments involve both synthetic and real data.
Simulation with synthetic data
------------------------------
We conducted a series of experiments using synthetic data to test the performance of the proposed CDL and CSR algorithms in both noise-free and noisy situations.
### CDL phase
In this experiment, we explore the dictionary recoverability of our coupled dictionary learning algorithm. The performance is evaluated in terms of the root-mean-square error (RMSE) and the atom recovery ratio[^10]. Note that the atom recovery ratio corresponds to the ratio of successfully retrieved atoms among all the atoms in the true dictionary. Specifically, one atom in the true dictionary is regarded to be successfully retrieved when the Euclidean distance between the atom and any one of the atoms in the learned dictionary is less than a threshold[^11]. The RMSE is adopted to measure the training error convergence. [[**Generation of Synthetic Data.**]{}]{} We generate the $N \times K$ dimensional dictionaries $\boldsymbol{\Psi}_{c}^h,\boldsymbol{\Psi}^h$, $\boldsymbol{\Phi}_{c}, \boldsymbol{\Phi}$ by drawing their elements independently from a zero-mean unit-variance Gaussian distribution, followed by an operation of column-wise $\ell_2$ normalization. We also generate the corresponding two $M \times K$ dimensional LR dictionaries as follows $\left[\boldsymbol{\Psi}_{c}^l,\boldsymbol{\Psi}^l\right] = \mathbf{A} \left[\boldsymbol{\Psi}_{c}^h,\boldsymbol{\Psi}^h\right]$, where $\mathbf{A} \in \mathbb{R}^{M \times N}$ a sub-sampling matrix ($M<N$) consisting of rows uniformly selected from an identity matrix of dimension $N$. Matrix $\mathbf{A}$ serves as an observation operator that extracts LR signals from corresponding HR version.
The set of $T$ sparse representations $\mathbf{Z} \in \mathbb{R}^{K \times T}$ is generated as follows: For each column vector $\mathbf{z}$ of $\mathbf{Z}$, we choose a support of size $s_z$ uniformly at random. The elements in this support are drawn i.i.d. from a Gaussian distribution and the other elements are set to be equal to zero. The set of sparse representations $\mathbf{U}$ and $\mathbf{V}$ are also generated in a similar manner with sparsity $s_u$ and $s_v$, respectively, for each column vector.
Finally, the training datasets $\mathbf{X}^l$, $\mathbf{X}^h$ and $\mathbf{Y}$ are synthesized via our model - . The parameter setting in the coupled dictionary learning simulation is $M = 16$, $N = 64$, $T = 10000$. The number of atoms $K$ is set to be equal to 128 or 256 or 512. The ground truth individual sparsity combination $[s_z,s_u,s_v]$ varies from $[2,1,1]$ to $[4,2,2]$ and the total sparsity is set to be equal to $s=s_z+s_u+s_v$. The training iterations in Algorithm \[Alg:CK-SVD\] are set to be equal to $OutIter = 10$ and $InIter = 20$. The dictionaries are initialized with i.i.d. Gaussian random matrices. We conduct 10 trials for each group of parameters and average the results.
[[**Noise-free Case.**]{}]{} We train the coupled dictionaries on the clean synthetic training datasets using Algorithm \[Alg:CK-SVD\]. Figure \[Fig:AtomsRatioErrConv\] and Table \[Tab:CDLsimu\_Sparsity\] show the averaged training results in the noise-free situation. These illustrate that the proposed training algorithm asymptotically converges to an RMSE level less than 0.02. In addition, these also indicate that for different dictionary sizes, and sparsity levels, our algorithm achieves a reasonable atom recovery ratio around 94% on average, which implies that the learned coupled dictionaries closely approximate to the ground truth dictionaries. As we will see, the learned coupled dictionaries using our approach lead to better super-resolution results than competing ones.
$s_z,s_u,s_v$ RMSE$_\mathbf{X}$ RMSE$_\mathbf{Y}$ $\boldsymbol{\Psi}_{c}$ $\boldsymbol{\Phi}_{c}$ $\boldsymbol{\Psi}$ $\boldsymbol{\Phi}$
--------------- ------------------- ------------------- ------------------------- ------------------------- --------------------- ---------------------
2, 1, 1 0.0162 0.0178 94% 95% 92% 90%
3, 1, 1 0.0121 0.0144 97% 97% 94% 94%
4, 1, 1 0.0121 0.0108 98% 98% 94% 95%
4, 2, 2 0.0148 0.0135 96% 96% 93% 93%
3, 2, 2 0.0198 0.0189 94% 94% 90% 91%
: RMSE and Atom Recovery Ratio with different sparsity levels
\[Tab:CDLsimu\_Sparsity\]
[[**Noisy Case.**]{}]{} We also explore the effect of noise on coupled dictionary learning by adding white Gaussian noise to the synthetic training datasets. Specifically, we add noise to the training signals by varying the input SNR from -2 dB to 16 dB with a step of 2dB. Then we examine the performance of our training algorithm. The dictionary sizes are set to be equal to $64 \times 128$, $64 \times 256$ and $64 \times 512$ and the sparsities are set to be equal to $s_z,s_u,s_v = 4,2,2$.
Figure \[Fig:NoiseCDL\] shows that the final training error in terms of RMSE decreases gradually with the increase of the input SNR. That is, the cleaner the training datasets, the better the training performance is. Accordingly, the average atom recovery ratio increases with the increase of the input SNR and is well above 90% when the input SNR is larger than 8dB, which demonstrates the robustness of our CDL algorithm to the noise. In addition, we can find that a larger dictionary size gives worse training performance, which indicates that more over-complete dictionaries are more sensitive to noise during training. This is due to the fact that larger dictionaries imply smaller mutual-coherence, so it is more difficult to distinguish between atoms in the dictionary. Consequently, the sparse coding process in the training is more susceptible to noise.
### CSR phase
In this simulation, we test the signal reconstruction performance of our CSR algorithm. Considering the low complexity and effectiveness of greedy algorithms, we obtain the sparse representations by solving with the OMP algorithm and we then obtain the final estimation by using . We adopt SNR and RMSE as the measure criteria in both the noise-free case and the noisy one. We first perform coupled dictionary learning as in the previous simulation and then apply the learned dictionaries into the SR simulation. We also compare the super-resolution results in the presence of side information to those without side information, such as[@zeyde2010single] where the model involves only the single modality of interest.
The testing datasets were generated as before. The parameters were set as follows: dictionary size $N \times K = 64 \times 256$, the number of measurements $M$ varied from $2$ to $64$ in steps of 2. The other parameters were the same as the training stage. 1000 trials are conducted and the results are averaged.
Figure \[Fig:SimuCSR\] indicates that – aided by appropriate side information – our approach allows to achieve better reconstruction performance, even though the learned coupled dictionaries are not the ground truth dictionaries. Particularly, the proposed algorithm outperforms the counterpart approach that does not leverage side information when the number of measurements $M$ is less than 50, i.e. around 78% of the ambient dimension $N=64$ of the testing signal. As fewer measurements represent input testing signals with lower resolution, our approach exhibits significant advantages over the counterpart for super-resolving LR signals. The results for the noisy case also demonstrate that sparse recovery is robust to noise. For example, comparing the noise-free and noisy case with 32 measurements, we can find that the output SNR of our CSR algorithm decreases from 22.16 dB to 22.03 dB with 0.13 dB drop, while the counterpart decreases from 9.23 dB to 8.98 dB with 0.25 dB drop.
In the sequel, we show that our approach can also lead to better super-resolution results compared with various state-of-the-art approaches, such as [@kopf2007joint; @he2010guided; @ham2017robust; @li2016deep; @shen2015multispectral] in the presence of real data.
Experiments with real data
--------------------------
With real data, we evaluate our algorithm in two scenarios: (1) super-resolution of multi-spectral images aided by RGB images of the same scene; and (2) super-resolution of infrared images aided by RGB images of the same scene.
We compare our approach with state-of-the-art joint image filtering approaches, including Joint Bilateral Filtering (JBF)[@kopf2007joint], Guided image Filtering (GF)[@he2010guided], Static/Dynamic Filtering (SDF)[@ham2017robust], Deep Joint image Filtering (DJF)[@li2016deep] and Joint Filtering via optimizing a Scale Map (JFSM)[@shen2015multispectral] where the same guidance images as in our approach are leveraged and the parameters are tuned to be optimum for the modalities and scale factors. Our approach is also compared with several representative single image SR approaches, such as A+ [@timofte2014a+], ANR (Anchored neighbourhood regression) [@timofte2013anchored], the sparse coding algorithm of Zeyde *et al.*[@zeyde2010single].[^12] Furthermore, we select bicubic interpolation as the baseline method. We adopt the Peak Signal to Noise Ratio (PSNR), the RMSE and the Structure SIMilarity (SSIM) index[@wang2004image] as the image quality evaluation metrics which are commonly used in the image processing literature. The practical multispectral/RGB datasets are obtained from the Columbia multispectral database[^13]. The infrared/RGB images datasets are obtained from the EPFL RGB-NIR Scene database[^14]. All these datasets are registered for both modalities.
### Multispectral image SR
We separate both multispectral and RGB image datasets into the training group (consisting of 28 scenes) and testing group (consisting of the remaining 8 scenes). Then, similar to [@yang2008image; @yang2010image], we blur and downsample each multispectral image by a factor of 4 and 6, using the MATLAB imresize function to generate subsampled versions.
[[**Training Phase with CDL.**]{}]{} We adopt a common operation to construct the training dataset. Specifically, we first upscale the LR multispectral training images to the desired size using bicubic interpolation. The RGB images are converted to grayscale form as the guidance images. Then, the interpolated LR images, the target HR images and the corresponding guidance images are divided into a set of $\sqrt{N} \times \sqrt{N}$ patch pairs. We remove the mean from each patch, as the DC component is always preserved well during the upscaling process. Then, we vectorize the patches to form the training datasets $\mathbf{X}^l$, $\mathbf{X}^h$ and $\mathbf{Y}$ of dimension $N \times T$. Smooth patches with variance less than 0.02 have been eliminated as they are less informative. Once the training dataset is prepared, we apply our coupled dictionary learning algorithm, shown in Algorithm \[Alg:CK-SVD\], to learn the dictionary pairs $[\boldsymbol{\Psi}_{c}^l,\boldsymbol{\Psi}^l]$ and $[\boldsymbol{\Phi}_{c},\boldsymbol{\Phi}]$ from $\mathbf{X}^l$ and $\mathbf{Y}$. Then, HR dictionary pair $[\boldsymbol{\Psi}_{c}^h,\boldsymbol{\Psi}^h]$ are computed based on $\mathbf{X}^h$ and acquired sparse codes $\mathbf{Z}$ and $\mathbf{U}$. The parameter setting is as follows: patch size $\sqrt{N} \times \sqrt{N} = 8 \times 8$ for 4$\times$ upscaling and $16 \times 16$ for 6$\times$ upscaling, dictionary size $K= 1024$, total sparsity constraint $s = 8$, training size $T \approx 15,000$ or $T \approx 160,000$.
Figure \[Fig:CDL\_PracticalResuts\] shows the learned coupled dictionaries for multispectral images of wavelength 640 nm and the corresponding RGB version. We can find that any pair of LR and HR atoms from $\boldsymbol{\Psi}_{c}^l$ and $\boldsymbol{\Psi}_{c}^h$ capture associated edges, blobs, textures with the same direction and location. Similar behavior can also be observed in $\boldsymbol{\Psi}^l$ and $\boldsymbol{\Psi}^h$. This implies that LR and HR dictionaries are indeed closely related to each other. On the other hand, LR and HR atom pairs also exhibit some difference. Specifically, the edges and textures captured by LR atoms tend to be blurred and smoothed, while they tend to be clearer and sharper in the corresponding HR atoms. More importantly, the common dictionary $\boldsymbol{\Phi}_{c}^h$ from the guidance images exhibits considerable resemblance and strong correlation to $\boldsymbol{\Psi}_{c}^h$ and $\boldsymbol{\Psi}_{c}^l$ from the HR/LR modalities of interest. This inspiring outcome indicates that the three common dictionaries have indeed captured the similarities between multispectral and RGB modalities. In contrast, the learned unique dictionaries $\boldsymbol{\Psi}^h$ and $\boldsymbol{\Phi}$ represent the disparities of these modalities and therefore rarely exhibit resemblance.
[[**Testing Phase with CSR.**]{}]{} During the coupled super-resolution phase, given a new pair of LR multispectral and HR RGB images for test, we upscale the LR multispectral image to the desired size as before. Then the testing image pair are subdivided into overlapping patches of size $\sqrt{N} \times \sqrt{N}$ pixels with overlap stride equal to 1 pixel.[^15] The DC component is also removed from each patch and stored. We vectorize these patches to construct the testing datasets $\mathbf{x}^l_{test}$ and $\mathbf{y}_{test}$. Then, we perform coupled sparse coding on $\mathbf{x}^l_{test}$ and $\mathbf{y}_{test}$ with respect to learned dictionary pairs $[\boldsymbol{\Psi}_{c}^l,\boldsymbol{\Psi}^l]$ and $[\boldsymbol{\Phi}_{c},\boldsymbol{\Phi}]$ to obtain the approximated sparse codes $\mathbf{z}_{test}$, $\mathbf{u}_{test}$ and $\mathbf{v}_{test}$, which are then multiplied with the HR dictionary pair $[\boldsymbol{\Psi}_{c}^h,\boldsymbol{\Psi}^h]$ to predict the HR patches $\mathbf{x}^h_{test}$, shown in Algorithm \[Alg:CSR\]. Finally, the DC component of each patch is added back to the corresponding estimated HR patch. These HR patches are tiled together and the overlapping areas are averaged to reconstruct the HR image of interest.
\
\
\
\
-------------- -------- ------- -------- ------- -------- ------- -------- ------- ------------ ----------- ------------ ------------ ------------ ----------- -------- -------
SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR
chart toy 0.9451 29.14 0.9528 30.69 0.9514 30.70 0.9523 30.74 0.9842 33.91 0.9215 33.300 **0.9855** **34.50** 0.9952 39.07
cloth 0.7571 26.91 0.7640 27.62 0.7699 27.79 0.7315 27.18 0.9489 31.54 **0.9770** **35.330** 0.9506 32.75 0.9854 36.33
egyptian 0.9761 36.22 0.9788 37.82 0.9788 37.96 0.9677 37.16 0.9861 41.31 0.9428 39.680 **0.9935** **42.63** 0.9939 46.75
feathers 0.9530 30.46 0.9599 31.80 0.9618 32.12 0.9434 30.92 0.9848 36.01 0.9096 33.540 **0.9871** **36.25** 0.9943 40.98
glass tiles 0.9215 26.38 0.9339 27.15 0.9326 27.45 0.9188 27.01 **0.9814** **31.83** 0.9407 29.340 0.9791 31.05 0.9935 36.20
jelly beans 0.9269 27.45 0.9474 28.97 0.9488 29.54 0.9279 27.87 0.9820 32.77 0.9356 30.820 **0.9866** **34.38** 0.9952 37.99
oil painting 0.9025 32.23 0.9034 33.23 0.9033 33.30 0.9001 32.80 0.9493 34.39 0.9439 34.160 **0.9601** **36.24** 0.9902 40.73
paints 0.9569 30.47 0.9714 32.08 0.9698 32.23 0.9569 31.35 0.9897 **37.74** 0.9321 32.960 **0.9900** 36.99 0.9962 41.43
average 0.9174 29.91 0.9265 31.17 0.9270 31.39 0.9123 30.63 0.9758 34.94 0.9379 33.640 **0.9791** **35.60** 0.9930 39.94
-------------- -------- ------- -------- ------- -------- ------- -------- ------- ------------ ----------- ------------ ------------ ------------ ----------- -------- -------
\[Tab:MS4x\_Joint\]
-------------- -------- ------- -------- ------- -------- ------- -------- ------- ------------ ----------- ------------ ----------- ------------ ----------- -------- -------
SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR
chart toy 0.8774 26.83 0.8992 28.08 0.8932 27.86 0.9006 28.12 **0.9772** **32.61** 0.9144 31.22 0.9682 32.55 0.9909 37.12
cloth 0.6143 25.55 0.6424 26.06 0.6394 26.07 0.6158 25.80 0.9226 30.09 **0.9723** **33.79** 0.9256 31.73 0.9761 34.83
egyptian 0.9459 33.79 0.9560 34.95 0.9536 34.80 0.9466 34.83 0.9681 40.24 0.9444 38.43 **0.9872** **40.75** 0.9947 45.95
feathers 0.8973 27.68 0.9177 28.80 0.9138 28.76 0.9062 28.50 **0.9765** **34.09** 0.9042 31.32 0.9727 33.75 0.9904 38.32
glass tiles 0.8401 24.45 0.8652 25.05 0.8585 25.06 0.8556 25.03 **0.9705** **30.33** 0.9233 27.33 0.9646 29.87 0.9903 35.78
jelly beans 0.8424 24.93 0.8835 26.24 0.8801 26.36 0.8681 25.60 0.9721 31.22 0.9225 28.58 **0.9734** **32.73** 0.9915 36.48
oil painting 0.8511 30.90 0.8664 31.87 0.8626 31.78 0.8574 31.49 0.9392 33.77 **0.9462** 34.09 0.9427 **35.18** 0.9860 39.81
paints 0.9005 27.51 0.9328 29.04 0.9253 28.90 0.9226 28.69 **0.9842** **36.60** 0.9363 31.25 0.9792 34.93 0.9927 39.57
average 0.8461 27.70 0.8704 28.76 0.8658 28.70 0.8591 28.50 0.9638 33.62 0.9330 32.00 **0.9642** **33.93** 0.9891 38.48
-------------- -------- ------- -------- ------- -------- ------- -------- ------- ------------ ----------- ------------ ----------- ------------ ----------- -------- -------
\[Tab:MS6x\_Joint\]
Figure \[Fig:MS4x\_Joint\] shows the multispectral image SR results for the 640 nm wavelength band. As we can see, our approach is able to reliably recover more accurate image details and, at the same time, substantially suppresses ringing artifacts, thus making our reconstruction more photo-realistic than the counterparts. The overall comparison of the image SR performance is shown in Table \[Tab:MS4x\_Joint\] and Table \[Tab:MS6x\_Joint\] for 4$\times$ and 6$\times$ upscaling, respectively.[^16] The results show that our approach outperforms bicubic interpolation with significant gains of average 5.6dB, 6.2dB and also exhibits notable advantage over the state-of-the-art joint image filtering approaches. For both 4$\times$ and 6$\times$ upscaling, our approach outperforms JBF[@kopf2007joint], GF[@he2010guided], SDF[@ham2017robust], JFSM[@shen2015multispectral] with gains of at least 1.9dB in terms of average PSNR.
Our approach also leads to results better than the deep-learning-based approach DJF[@li2016deep] for a small number of training samples (e.g. $T$ = 15,000). However, [@li2016deep] eventually outperforms our approach for a large number of training samples (e.g. $T$ = 160,000). This superior performance of deep learning is at the expense of both a vast amount of training data as well as training time. For example, DJF[@li2016deep] takes almost half a day to train through about 50 epochs with a NVIDIA Titan black GPU for acceleration, but our approach takes less than an hour for training a group of coupled dictionaries without any GPU acceleration. Therefore, our approach may be preferable in scenarios where the number of training examples, the amount of training time, or the amount of computing resources is limited.[^17]
\
\
\
\
------------- -------- ------- -------- ------- -------- ------- -------- ------- ------------ ------- -------- ------- ------------ ----------- -------- -------
SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR
urban\_0004 0.9029 25.93 0.9359 28.47 0.9391 28.75 0.9066 26.82 0.9789 31.02 0.9721 30.86 **0.9811** **34.14** 0.9895 35.85
urban\_0006 0.9458 30.89 0.9311 32.10 0.9400 32.66 0.8918 30.60 **0.9894** 36.04 0.9741 32.86 0.9868 **36.79** 0.9917 37.92
urban\_0017 0.9527 30.45 0.9172 31.11 0.9205 31.32 0.9281 30.72 **0.9815** 34.18 0.9500 32.85 0.9777 **35.27** 0.9860 36.99
urban\_0018 0.9298 25.19 0.9308 27.59 0.9251 27.70 0.9196 26.09 **0.9888** 30.72 0.9774 30.80 0.9874 **33.01** 0.9932 34.44
urban\_0020 0.9577 28.03 0.9523 30.67 0.9494 30.69 0.9505 29.09 **0.9915** 33.60 0.9797 32.61 0.9893 **36.66** 0.9953 38.22
urban\_0026 0.8704 26.27 0.8627 26.82 0.8571 26.89 0.8558 26.61 0.9397 29.21 0.9332 28.97 **0.9482** **30.35** 0.9635 31.52
urban\_0030 0.8401 26.54 0.8476 27.58 0.8383 27.59 0.8415 27.21 0.9345 31.27 0.9064 30.56 **0.9443** **32.71** 0.9601 35.07
urban\_0050 0.9434 26.65 0.9099 27.32 0.9116 27.35 0.9207 27.07 0.9616 28.58 0.9251 27.58 **0.9663** **29.37** 0.9589 28.34
average 0.9179 27.49 0.9109 28.96 0.9101 29.12 0.9018 28.03 0.9707 31.83 0.9522 30.89 **0.9726** **33.54** 0.9798 34.79
------------- -------- ------- -------- ------- -------- ------- -------- ------- ------------ ------- -------- ------- ------------ ----------- -------- -------
\[Tab:NIR4x\_Joint\]
------------- -------- ------- -------- ------- -------- ------- -------- ------- ------------ ----------- -------- ------- ------------ ----------- -------- -------
SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR
urban\_0004 0.8094 23.87 0.8858 25.96 0.8817 25.94 0.8413 24.62 **0.9670** 29.68 0.9527 27.97 0.9558 **30.77** 0.9814 33.85
urban\_0006 0.8671 28.48 0.8861 30.00 0.8876 30.17 0.8377 28.76 **0.9830** **34.92** 0.9716 32.28 0.9664 34.15 0.9851 36.19
urban\_0017 0.8998 28.64 0.8864 29.63 0.8860 29.61 0.8910 29.13 **0.9599** 32.80 0.9434 32.01 0.9515 **32.98** 0.9713 35.28
urban\_0018 0.8393 23.07 0.8718 25.09 0.8591 24.98 0.8439 23.79 **0.9844** 29.92 0.9470 27.47 0.9727 **31.03** 0.9880 33.05
urban\_0020 0.9053 26.03 0.9200 28.19 0.9118 28.01 0.9089 26.93 **0.9873** 32.61 0.9673 30.33 0.9763 **33.85** 0.9908 36.88
urban\_0026 0.7850 24.71 0.8235 25.64 0.8131 25.63 0.7989 25.17 **0.9183** 28.38 0.9128 27.54 0.9172 **28.88** 0.9421 30.19
urban\_0030 0.7517 25.19 0.7994 26.32 0.7855 26.22 0.7748 25.80 0.9063 30.00 0.8902 29.38 **0.9099** **30.52** 0.9365 33.34
urban\_0050 0.8921 25.17 0.8837 26.26 0.8846 26.26 0.8837 25.90 **0.9414** 27.64 0.9068 26.67 0.9402 **28.37** 0.9318 27.12
average 0.8437 25.65 0.8696 27.13 0.8637 27.10 0.8475 26.26 **0.9559** 30.75 0.9365 29.21 0.9487 **31.32** 0.9659 33.24
------------- -------- ------- -------- ------- -------- ------- -------- ------- ------------ ----------- -------- ------- ------------ ----------- -------- -------
\[Tab:NIR6x\_Joint\]
### Near-infrared image SR
We also evaluate our approach on near-infrared (NIR) images with registered RGB images as side information. As the response of NIR band has poor correlation with the response of the visible band, it is usually difficult to infer the brightness of a NIR image given a corresponding RGB modality. Thus, it is more challenging to take good advantage of the RGB version to super-resolve the infrared version. The LR and HR training and testing dataset and the side information are prepared in a manner similar to the previous multispectral case. We set patch size to $ = 8\times8$, dictionary size to $= 64 \times 1024$. All the images in the dataset are houses and buildings, and thus contain many fine textures and sharp edges. This makes the SR task more challenging than super-resolving images with smoother textures.
Figure \[Fig:NIR4x\_Joint\] compares the visual quality of the reconstructed HR infrared images and the corresponding error maps. As before, DJF[@li2016deep] shows state-of-the-art performance with a larger training sizes ($T$ = 160,000), but for the case of limited dataset ($T$ = 15,000), our approach recovers more visually plausible images, exhibiting less error than the competing methods. Table \[Tab:NIR4x\_Joint\] and \[Tab:NIR6x\_Joint\] also confirm the significant advantage of the proposed approach over other state-of-the-art methods. In particular, this indicates that detailed structure information can be effectively captured by coupled dictionary learning, especially on images such as buildings and houses that contain a lot of sharp edges, textures and stripes.
### Impact of parameters
The previous experiments demonstrate the effectiveness of the coupled sparsity prior on regularizing ill-conditioned problems. In this section, we explore the effect of key parameters and factors, including the dictionary size, the length of each atom, the robustness to noise, and the sparsity constraints, on the performance of the proposed algorithms. The following experiments are conducted using multispectral images of wavelength 640 nm on which we perform $4 \times$ upscaling using a computer equipped with a quadro-core i7 CPU at 3.4GHz with 32GB of memory. Each evaluation metric value is averaged on all the testing images.
[[**Dictionary Size.**]{}]{} Intuitively, more atoms tend to capture more features. Thus, a larger dictionary may yield a more accurate sparse approximation to the signal of interest. On the other hand, a large dictionary size increases the complexity of the non-convex problem, thus requiring more computation. Under the multispectral image SR experimental setting, we evaluate the performance of the proposed approach for various dictionary sizes, ranging from 64, to 128, 256, 512, and 1024 atoms.
Table \[Tab:EffectDsize\] shows that the PSNR increases gradually with the increase of the dictionary size. On the other hand, the computation cost, represented by the training time and testing time, approximately increases linearly with the dictionary size. The results imply that the choice of the dictionary size depends on the balance between approximation accuracy and computational expense. We find that a dictionary size of 1024 atoms can yield decent accuracy while allowing affordable computational complexity.
\# of atoms 64 128 256 512 1024 2048
------------- -------- -------- -------- -------- -------- --------
Train Time 1.8 5.5 10.0 21.9 62.7 253.8
Test Time 92.5 95.6 117.6 138.8 142.0 216.3
PSNR (dB) 33.81 34.38 34.57 34.83 34.95 35.33
RMSE 0.0204 0.0191 0.0187 0.0181 0.0179 0.0171
: Effect of dictionary size
\[Tab:EffectDsize\]
[[**Patch Size.**]{}]{} Since each image patch is vectorized into a column vector, the length of a vectorized image patch is equivalent to the length of each atom. A larger patch size yields longer atoms and larger dictionaries, which require more resources for training and storage. In this experiment, the number of atoms is set to be equal to 1024 and we compare 4 different patch size choices: $6 \times 6$, $8 \times 8$, $10 \times 10$ and $12 \times 12$ pixels. As shown in Table \[Tab:EffectPatchSize\], a larger patch size requires more memory and training time. In addition, the SR performance, in terms of PSNR and RMSE, improves gradually with the patch size, but the improvement diminishes when the patch size goes beyond $8 \times 8$ pixels. The testing time seems rarely affected by the patch size, because a larger patch size implies a smaller number of patches extracted from a given image, although it increases the computational complexity of sparse coding. In sum, a larger patch size improves the SR performance but costs more time and memory. We chose $8 \times 8$ patch size as it strikes a good balance between performance and cost.
Patch Size $6 \times 6$ $8 \times 8$ $10 \times 10$ $12 \times 12$
---------------- -------------- -------------- ---------------- ---------------- -- --
Train Time (s) 57.1 62.7 81.8 170.1
Test Time (s) 142.5 142.0 149.6 138.2
PSNR (dB) 34.95 35.67 35.82 35.85
RMSE 0.0186 0.0172 0.0170 0.0169
Memory (KB) 1660 2949 4608 6636
: Effect of patch size
\[Tab:EffectPatchSize\]
[[**Robustness to Noise.**]{}]{} The previous experiments are conducted with clean input images free of noise. In practice, images can be contaminated by noise. To test the robustness of our algorithm to noise, white Gaussian noise of different levels is added to the LR input testing images to reduce the input PSNR. The standard deviation of different level noise ranges from 0 to 8 for uint8 precision. Then, we repeat previous multispectral image SR task. Our algorithm demonstrates reasonable stability and robustness to noise, as shown in Table \[Tab:EffectNoise\]. In particular, at smaller noise levels, our approach outperforms the competing algorithms.
Noise Levels $\sigma$ 0 2 4 6 8
------------------------------ ------------ ------------ ------------ ------------ ------------ --
Bicubic 0.0338 0.0344 0.0363 0.0392 0.0428
GF[@he2010guided] 0.0287 0.0303 0.0305 0.0308 0.0312
JFSM[@shen2015multispectral] 0.0219 0.0219 0.0220 **0.0222** **0.0224**
Proposed **0.0171** **0.0171** **0.0208** [0.0230]{} [0.0251]{}
: Robustness to noise in terms of RMSE
\[Tab:EffectNoise\]
[[**sparsity constraints.**]{}]{} A larger sparsity constraint, i.e., a larger $s$, can lead to a better approximation of the data. On the other hand, larger sparsity constraints also require more iterations to find these non-zeros via OMP. As shown in Table \[Tab:EffectSparsity\], the average PSNR of the reconstruction, as well as the computational time, increase along with the total sparsity constraint. When the total sparsity constraint goes beyond a certain level, e.g., 16, the retrieved extra non-zeros coefficients are trivial and contribute very little to the PSNR.
total sparsity $8$ $12$ $16$ $20$ $24$ $28$
---------------- ------- ------- ------- ------- ------- ------- -- --
Test Time (s) 5.7 8.8 12.6 17.9 23.2 31.9
PSNR (dB) 34.97 35.54 35.69 35.73 35.73 35.69
: Effect of sparsity constraints
\[Tab:EffectSparsity\]
Conclusion
==========
This paper proposed a new joint image SR approach based on joint sparse representations induced by coupled dictionaries. In particular, our approach explicitly captures the similarities and disparities between different image modalities in the sparse feature domain in *lieu* of the image domain.
Our approach consists of both a training phase and a testing phase. The training phase – the coupled dictionary learning algorithm – seeks to learn a number of coupled dictionaries from training data, including LR and HR images of the modality of interest and another guidance image modality. The testing phase – the coupled super-resolution algorithm – leverages the learned dictionaries in order to reconstruct a HR version of a LR image given the guidance image. This design allowed us to automatically transfer appropriate structure information to the estimated HR version and reduce texture-copying artifacts. Multispectral/RGB and NIR/RGB joint image SR experiments demonstrated that our design brings notable benefits over state-of-the-art image SR approaches.
[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[ l@\#1 =l@\#1 \#2]{}]{}
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[^1]: This work is supported by China Scholarship Council (CSC), UCL Overseas Research Scholarship (UCL-ORS), the VUB-UGent-UCL-Duke International Joint Research Group grant (VUB: DEFIS41010), and by EPSRC grant EP/K033166/1.
[^2]: Pingfan Song and Miguel R. D. Rodrigues are with the Department of Electronic & Electrical Engineering, University College London, London WC1E 6BT, UK. (e-mail: pingfan.song.14@ucl.ac.uk, m.rodrigues@ucl.ac.uk)
[^3]: Xin Deng and Pier Luigi Dragotti are with the Department of Electronic & Electrical Engineering, Imperial College London, London SW7-2AZ, UK. (e-mail: x.deng16@imperial.ac.uk, p.dragotti@imperial.ac.uk)
[^4]: João F. C. Mota is with the Department of School of Engineering & Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK. (email: j.mota@hw.ac.uk)
[^5]: N. Deligiannis is with the Department of Electronics and Informatics, Vrije Universiteit Brussel, B-1050 Brussels, Belgium, and with imec, Kapeldreef 75, B-3001 Leuven, Belgium. (e-mail: ndeligia@etrovub.be)
[^6]: Note that, we could also use alternative sparsity constraints, such as (a) $\|\mathbf{z}_i \|_0 + \|\mathbf{u}_i \|_0 \leq s_x, \|\mathbf{z}_i \|_0 + \|\mathbf{v}_i \|_0 \leq s_y$, (b) $\|\mathbf{z}_i \|_0 \leq s_z, \|\mathbf{u}_i \|_0 \leq s_u, \|\mathbf{v}_i \|_0 \leq s_v.$ We prefer to use the sparse constraint in rather than any of these constraints for various reasons. First, the proposed constraint seems to couple better the modalities than the constraint (a). Moreover, in comparison with individual upper bounds to the sparsity of the vectors $\mathbf{z}_i$, $\mathbf{u}_i$ and $\mathbf{v}_i$ in constraint (b), it appears to be more sensible to come up with an upper bound on the overall sparsity and allow the algorithm to automatically determine the individual sparsities of the various sparse representations.
[^7]: The motivation of this two-step training strategy is that the sparse codes $\mathbf{z}$ and $\mathbf{u}$ should be obtained only from $\mathbf{X}^l$ and $\mathbf{Y}$ in both training and testing stages without involving $\mathbf{X}^h$, since the HR target patches $\mathbf{X}^h$ are available only in the training stage and not in testing stage. Otherwise, the sparse coding process will be inconsistent for the two stages.
[^8]: Owing to the SVD operation in the dictionary update, atoms from the common dictionary pair \[$\boldsymbol{\Psi}^l_{c}$; $\boldsymbol{\Phi}_{c}$\] and the unique dictionaries $\boldsymbol{\Psi}^l$ and $\boldsymbol{\Phi}$ have unit $\ell_2$ norm automatically.
[^9]: Note, in order to guarantee the fidelity of the sparse approximation to the HR training datasets, the atoms in the HR dictionaries are not constrained to be unit $\ell_2$ norm, as in Zeyde, *et al.* [@zeyde2010single] and Wang, *et al.* [@wang2010resolution]. However, when there are zeros or near-zeros rows in the sparse codes $\mathbf{Z}$ or $\mathbf{U}$, the matrix inverse operation during the computation of the closed form solution will give extremely large value for corresponding atoms. Therefore, in order to make the solution more stable, a Frobenius norm is added to regularize the problem .
[^10]: The RMSE and the atom recovery ratio are given by $$\label{Eq:DefRatioRMSE}
RMSE = \sqrt{
\frac{
\| \mathbf{x} - \widehat{\mathbf{D}} \boldsymbol{\alpha} \|_F^2
}
{ N } } \,,
ratio = \frac{
\left| \left\{d = \hat{d} : d \in \mathbf{D}, \hat{d} \in \widehat{\mathbf{D}} \right\} \right|
}
{\left| \left\{d : d \in \mathbf{D} \right\} \right| } \,,
$$ where $\mathbf{x}$ and $\boldsymbol{\alpha}$ are the data and sparse representation, respectively, and $N$ denotes the number of elements in $\mathbf{x}$. $\mathbf{D}$ and $\widehat{\mathbf{D}}$ denote the ground truth and the learned dictionary, respectively. $d \in \mathbf{D}$ means that $d$ is a column, called atom, of $\mathbf{D}$. The symbol $ \left| \cdot \right|$ denotes the cardinality of a set. A pair of atoms from $\mathbf{D}$ and $\widehat{\mathbf{D}}$ are considered identical when their Euclidean distance is less than a defined threshold $\epsilon$ with $\epsilon = 0.01$ by default.
[^11]: Please refer to Ron Rubinstein’s K-SVD package for more details. <http://www.cs.technion.ac.il/~ronrubin/software.html>
[^12]: For lack of space, we refer to the supplementary materials on our website (<http://www.ee.ucl.ac.uk/~uceeong/>) for more information.
[^13]: <http://www.cs.columbia.edu/CAVE/databases/multispectral/>
[^14]: <http://ivrl.epfl.ch/supplementary_material/cvpr11/>
[^15]: The overlap stride denotes the distance between corresponding pixel locations in adjacent image patches.
[^16]: More detailed results can be found in the supplementary materials on our website(<http://www.ee.ucl.ac.uk/~uceeong/>).
[^17]: Note also that the patch size associated with DJF[@li2016deep], which is $33 \times 33$, is larger than the patch size associated with our approach, which is set to be equal to $8 \times 8$ or $16 \times 16$. It is not computationally feasible to increase much further the patch size associated with our coupled dictionary approach. These differences in patch size may also explain in part the reason why DJF performs better when the number of training samples is sufficiently large.
|
---
abstract: 'Quantum key distribution (QKD) has long been a promising area for application of quantum effects toward solving real-world problems. But two major obstacles have stood in the way of widespread applications: low secure key generation rates and short achievable operating distances. In this paper, a new physical mechanism for dealing with the first of these problems is proposed: interplay between different degrees of freedom in a hyperentangled system (parametric down conversion) is used to increase the Hilbert space dimension available for key generation while maintaining security. Polarization-based Bell tests provide security checking, while orbital angular momentum (OAM) and total angular momentum (TAM) provide higher key generation rate. Whether to measure TAM or OAM is decided randomly on each trial. The concurrent non-commutativity of TAM with OAM and polarization provides the physical basis for quantum security. TAM measurements link polarization to OAM, so that if the legitimate participants measure OAM while the eavesdropper measures TAM (or vice-versa), then polarization entanglement is lost, revealing the eavesdropper. In contrast to other OAM-based QKD methods, complex active switching between OAM bases is not required; instead, passive switching by beam splitters combined with much simpler active switching between polarization bases makes implementation at high OAM more practical.'
address:
- '1. Dept. of Physics and Astronomy, Stonehill College, 320 Washington Street, Easton, MA 02357, USA'
- '2. Department of Electrical and Computer Engineering, Boston University, 8 Saint Mary’s St., Boston, MA 02215, USA'
- '3. Photonics Center, Boston University, 8 Saint Mary’s St., Boston, MA 02215, USA'
- '4. Dept. of Physics, Boston University, 590 Commonwealth Ave., Boston, MA 02215, USA'
author:
- 'David S. Simon$^{1,2}$ and Alexander V. Sergienko$^{2,3,4}$'
title: High Capacity Quantum Key Distribution via Hyperentangled Degrees of Freedom
---
Introduction
============
Quantum key distribution
------------------------
In quantum key distribution (QKD) two experimenters (Alice and Bob) generate a shared cryptographic key, using quantum mechanics to guarantee that an eavesdropper (Eve) cannot obtain significant information about the key without being revealed. Commonly, for optical QKD schemes, key bits are derived from photon polarization. This can be done by having Alice prepare for Bob a single photon in a randomly chosen polarization state known only to her (BB84 protocol [@bb84]), or by Alice and Bob each receiving from a common source half of an entangled photon pair (Ekert protocol [@e91]). Either way, polarization measurements by Eve produce detectable disturbances. Each photon is prepared in one of two non-orthogonal, mutually unbiased bases. Eve, intercepting a photon traveling to Bob, must guess which basis to measure in; if she measures in the same basis as the two legitimate participants, she acquires full information without detection. However, half the time she guesses the wrong basis, ensuring that her outcome is uncorrelated with Alice’s; she then obtains no useful information and simultaneously exposes herself to detection by randomizing Bob’s results. This occurs because the polarization operators in the two non-orthogonal bases are not mutually commuting. Exchanging results for a subset of measurements, Alice and Bob see the decrease in correlation between their polarizations, revealing Eve’s actions. For fiber systems, phase is often used instead of polarization, but principle remains the same.
There have been two principal obstacles to widespread application of QKD outside of research labs. First, most methods have been limited in the distances over which they can operate; the simplest single-photon or weak coherent state approaches, for example, are generally limited to tens of kilometers before photon losses introduce unacceptable levels of error.
Second, most approaches to QKD with optical systems have used polarization or phase as the variables from which cryptographic key segments are generated. However, polarization can normally only encode one qubit per photon, unless substantial extra complications to the apparatus are added to allow for qutrit or ququart exploitation. Similarly, it is difficult at a practical level to increase the number of dimensions of the states encoded by phase beyond two, or at most, four. It would therefore be desirable to find a more practical means of encoding high dimensional states into a photon. This would increase the rate of key generation by allowing more than one bit of cryptographic key to be shared between the legitimate users of the system per exchanged photon.
As a means of increasing the key rate, there has been much interest (see [@grob; @gruneisen; @malik; @simonfib] and references therein) in using the photon’s orbital angular momentum (OAM) instead of polarization. The range of applications of states with OAM, such as Laguerre-Gauss states, to both classical and quantum communication has been rapidly expanding; see for example [@fickler2; @krenn]. OAM is quantized, $L_z=l_z\hbar$, with integer topological charge $l_z$. There is no fundamental upper limit to the value of $l_z$, so the alphabet size or effective Hilbert space dimension, $N$, is in principle unbounded. Using a range of $l_z$ values from $-l_0$ to $+l_0$, each photon can generate up to $\log_2N=\log_2(2l_0+1)$ bits of cryptographic key. OAM was first successfully used [@grob] to generate a quantum key by means of the three-dimensional qutrit space spanned by $l_z=0,\pm 1$. However, switching between unbiased, non-commuting bases in a higher-dimensional OAM space is now needed. This basis switching is much more difficult for OAM than for polarization and the difficulty grows with increasing basis size. Therefore the apparatus complexity and experimental difficulty increase rapidly with growing $N$.
In this paper, we wish to propose a means for increasing the number of secret key bits generated per transmitted photon, while avoiding the increasingly difficult basis modulations required by other schemes when going to higher Hilbert space dimensions. In the next section, we introduce a new physical mechanism for key generation that will allow a simpler experimental route to this goal. The approach we propose makes use of OAM for increased Hilbert space dimension, but here the OAM is employed in a fundamentally different manner than in all previous methods. In particular, we will make use of its joint entanglement with polarization, arranging the setup in such a way that the polarization is able to serve as a signal of attempts at eavesdropping on the OAM. Conceptually, rather than switching between two nonorthogonal bases in the space of *orbital* angular momentum, we switch between two nonorthogonal bases in the larger space of [*total*]{} angular momentum. This is much easier to accomplish because only the measurement basis of the photon polarization needs to be actively modulated.
Quantum Nondemolition Measurements
----------------------------------
The idea of using OAM to generate a secret key while only doing security-enforcing basis modulations in polarization seems to have an immediate problem. The two variables commute, so that one may be measured without disturbing the other. For example, the OAM eigenvalue, $l_z$, may be obtained by performing an ideal quantum nondemolition (QND) measurement [@brag1; @brag2; @thorne; @unruh]. This in principal causes no disturbance to the polarization or spin.
In practice, the situation is a little more complicated. Practical execution of QND requires nonlinear optical processes such as Kerr nonlinearity; but it has been shown that the physics of nonlinear interaction ensures that QND measurement of OAM will cause some disturbance to the signal’s polarization state [@lin]. This both reveals Eve’s presence and destroys the information she was attempting to obtain, since it prevents Alice and Bob from agreeing reliably on a shifted key. Furthermore, the low amplitudes of nonlinear processes guarantee low efficiency at the single photon level; only a small fraction of the photons will participate in the interaction, allowing Eve to determine only a small fraction of the OAM values.
But these considerations are specific to the case of QND measurements via Kerr nonlinearities. It may be possible that Eve has an advanced technology that allows her to make QND measurements of $l_z$ by some other, as yet unknown, means. There is no fundamental principal, to our knowledge, that guarantees that other such QND technologies *must* cause a similar disturbance to polarization when applied to OAM. Thus, we wish to avoid this problem by arranging a fundamental linkage between the polarization and OAM that will cause QND measurements of one variable to disturb the other, *independent of the physical mechanism used to make the measurement*. We propose a means of accomplishing this in the following sections.
Hyperentangled QKD
------------------
We propose a high-dimensional OAM-based QKD scheme that requires no random switching between OAM bases. For full security, it is necessary to treat both variables, polarization and OAM, in a fully quantum manner. The goal is to do this in such a way that active basis modulations are only needed in polarization, not in OAM. This is achieved by adding a third variable, the *total* angular momentum (TAM) $\hat{\bm J}$ about the propagation axis, and then allowing a random choice between measuring $\hat{\bm
J_z}=\hat{\bm L_z}+\hat{\bm S_z}$ or measuring $\hat{\bm L_z}$. $\hat{\bm J}$ provides a linkage between the spin $\hat{\bm S}$ (which determines the circular polarization state) and the OAM $\hat{\bm L}$, that allows the desired goal to be achieved.
The principal idea is to separate key generation and security into *different* degrees of freedom; however these variables must be closely linked in such a way that unauthorized measurements of one variable produce detectable signatures in the other. We will use OAM and TAM in tandem for key generation (due to their high dimensionality and subsequently high key-generation capacity), while employing polarization for security checks (due to the ease of alternating between polarization bases). This is possible because spontaneous parametric down conversion (SPDC) supplies photon pairs [*hyperentangled*]{} [@kwiat; @atature; @barreiro; @dossantos; @nagali; @karimi1] in polarization and OAM (among other variables). Hyperentanglement in SPDC has found applications in recent years ranging from quantum interferometry [@atature] and quantum imaging [@bonato] to quantum cryptography and dense coding [@barreiro2; @gauthier]. Polarization-OAM hyperentangled states have also been used for ultra-sensitive angular measurements [@ambrosio].
Previous uses of OAM in conjunction with polarization for QKD applications [@barreiro2; @gauthier] make use of the two variables in a sort of parallel, non-overlapping manner: the measurement of one variable has no effect on the other. The components of the two variables are simply appended to each other to form a vector with more components, thereby expanding the relevant state space to higher dimension. A complicated procedure of basis switching must still be carried out in this higher-dimensional space in order to ensure security. In contrast, in the current paper hyperentanglement is used in a more intrinsic manner; the pair of entangled variables are partially overlapping in the sense that they can be *either* independent *or* perfectly correlated with each other, depending on whether or not a third variable has also been measured. In this way, measurements by the eavesdropper on one variable become apparent through loss of entanglement in a second variable due to the pairwise noncommutativity of the first two variables with the third. The enforcement of security measures is then greatly simplified at high dimensions, since passive switching between the two measurement variables is technically much simpler than active switching between large basis sets for a single variable.
In addition to the technical simplification of QKD at high dimensions, the proposed procedure is interesting for several other reasons. Use of the noncommutativity of $\bm {\hat J}$ with $\bm {\hat L}$ and $\bm {\hat S}$ for quantum communication applications seems to be largely unexplored, as is the use in QKD of pairs of *different vector operators* in place of pairs of *different components of the same vector operator*. Further, the use of angular momentum erasure (see section (\[proceduresection\])) to maintain polarization entanglement has not previously been proposed and may be interesting in its own right as the basis for angular-momentum-based analogs of quantum eraser and delayed choice experiments.
The proposed approach relies on the fact that although OAM and polarization commute with each other, neither of them commutes with the TAM. To be explicit, the commutators of $\bm {\hat L}$ and $\bm {\hat S}$ among themselves are given by: $$\begin{aligned}
\left[ \hat L_i,\hat L_j\right] &=&
i\sum_k\epsilon_{ijk}\hbar L_k\\ \left[ \hat S_i,\hat S_j\right] &=&
i\sum_k\epsilon_{ijk}\hbar S_k \\
\left[ \hat L_i,\hat S_j\right] &=& 0
, \end{aligned}$$ so that: $$\begin{aligned}
\left[ \hat L_i,\hat J_j\right] &=& \; \left[ \hat L_i,\hat L_j+S_j\right] \; =\; \left[ \hat L_i,\hat L_j\right] +\cancel{\left[ \hat L_i,
\hat S_j\right]} \\ &=& i\sum_k\epsilon_{ijk}\hbar L_k\\
\left[ \hat S_i,\hat J_j\right] &=& \; \left[ \hat S_i,\hat L_j+S_j\right] \; =\; \cancel{\left[ \hat S_i,\hat L_j\right]} +\left[ \hat S_i,\hat S_j\right]
\\ &=& i\sum_k\epsilon_{ijk}\hbar S_k ,
.\end{aligned}$$ This lack of commutativity means that TAM measurements provide an indirect linkage between $\bm {\hat L}$ and $\bm {\hat S}$, ; we make use of this linkage in the following.
The specific procedure to be proposed in the next section allows Alice and Bob a random choice between measuring the eigenvalues of either the OAM ($\hat{\bm
L_z}$) or the TAM, ($\hat{\bm J_z}$). Only trials on which Alice and Bob both measure the same variable are kept; on these, the photon spin (polarization) state remains entangled. If Eve measures a variable different than Alice and Bob did, the spin wavefunction collapses into a definite polarization state, which will be detectable by a Bell-type test on polarization. This occurs because once the values of $\hat{\bm L_z}$ and $\hat{\bm J_z}$ have both been measured, the value of $\hat{\bm S_z}=\hat{\bm J_z}-\hat{\bm L_z}$ can be determined as well. Subsequent measurements of the photon’s linear polarization in the $x-y$ plane will then be affected by this sequence of measurements.
The higher dimension of the OAM state space increases the number of key bits generated per photon; this is done without *any* additional active modulation beyond what is needed in the usual polarization-based protocols. There is no upper limit to the number of bits possible in principle, although there are of course practical limits. $N$ can potentially be scaled up to very large size with little additional effort as long as sources with high values of entangled angular momenta and OAM sorters that work over a large enough range are both available. The range of achievable $l$ values for entangled photons has rapidly grown in recent years [@romero; @fickler]. Measures must be taken to guarantee that the range being used for the alphabet has a flat spectrum; otherwise Eve can use the differing probabilities to gain information about the key. This equalization, however, can be easily achieved, for example by using extra OAM sorters followed by filters with different transmission rates. The span of values that can be sorted by a single sorter has also grown, though more slowly [@leach2; @karimi; @guo; @berk; @sluss; @lavery; @osullivan2].
![*Each participant randomly measures either $\bm {\hat L_z}$ or $\bm {\hat J_z}$ via nondestructive sorting. After sorting is done in one of these variables, the information about the other variable is erased (see figure \[erasure\]). Regardless of which variable is sorted and which is erased, the polarization is undisturbed and available for measurement.*[]{data-label="sortfig"}](schematic.eps)
Setup and Procedure {#proceduresection}
===================
The Setup
---------
Figure \[sortfig\] shows the proposed setup in schematic form, with more detailed view of portions of it in figures \[erasure\] and \[detail\]. We will assume that the down conversion source uses a pump beam with zero OAM. The signal and idler polarizations are perfectly correlated for type I down conversion or perfectly anticorrelated for type II; either way, the OAM is perfectly anticorrelated. For specificity, we henceforth assume type II down conversion. The particular case drawn in figures \[erasure\] and \[detail\] assumes alphabet size $N=3$ ($l=0,\pm 1$; i.e. $l_0=1$), so an array of three detectors is required following each sorter. Larger alphabets require more detectors, sorters, and erasure stages, but no further changes are needed; the setup complexity therefore grows much more slowly with alphabet size than in other approaches; a change of OAM bases in the approach of [@grob], for example, requires the alignment and coordination of rapidly increasing numbers of moving stages as the dimension grows. (Note that even though the alphabet being used is $\left\{ -1,0,+1\right\}$, the sorters will need to be capable of sorting values up to $\pm 2$ to carry out the erasure procedure in figure \[erasure\].)
Alice and Bob can readily measure either OAM eigenvalue $l_z$ or spin eigenvalue $s_z$ (circular polarization), or both. In the paraxial case, $\hat {\bm S}$ and $\hat {\bm L}$ are well-defined and commute, so their components can be simultaneously measured, as verified experimentally in [@leach1]. This fully determined $j_z$. In contrast, the TAM $j_z$ about the propagation axis can be measured interferometrically in such a way [@leach1] that it leaves the separate values of *both* spin and OAM undetermined. So suppose that Alice and Bob each have a beam splitter randomly sending incoming photons either to an apparatus that measures $\hat {\bm L_z}$ or to one measuring $\hat {\bm J_z}$ (the sorters in figure \[sortfig\]). After the sorting is done on one variable, information about the other variable is erased by an arrangement of beam splitters, waveplates, and holograms (see figures \[erasure\], \[shiftfig\], and \[detail\]). The sorting by $l_z$ and $j_z$ values is nondestructive, so the spin or polarization can still be measured afterwards.
![*The procedures for erasing $\bm {\hat L_z}$ information (top) following a $\bm {\hat J_z}$ measurement, or erasing $\bm {\hat J_z}$ information (bottom) after an $\bm {\hat L_z}$ measurement. In the top case, the value of $\bm {\hat J_z}$ can still be determined from which of an array of detectors fires at the end (see figure \[detail\]), and similarly for $\bm {\hat L_z}$ in the bottom case. The figure is drawn for incoming value of $l$ or $j$ equal to $1$, but the process works the same way for other values. In each case, the undesired values are sorted, shifted to zero, then recombined so there is no way to determine what the original value was. The shifting is further illustrated in figure \[shiftfig\].* []{data-label="erasure"}](Erasure_C.eps)
Erasing unmeasured variables
----------------------------
The erasure of the unmeasured variable, as illustrated in figure \[erasure\], is necessary because otherwise there will be no interference between polarization states. To see this, imagine Alice and Bob place linear polarizers at respective angles $\theta_A$ and $\theta_B$ from the horizontal. The states passed by the polarizers will be denoted $|\theta_A\rangle $ and $|\theta_B\rangle$. After passing through the polarizers, the probability of joint detection in both labs is proportional to $|\langle \theta_A| \langle\theta_B|\psi \rangle |^2$. Provided the angles are not multiples of $\pi\over 2$, both $|H\rangle$ and $|V\rangle$ will have nonzero projections onto the $|\theta\rangle $ states, so that cross-terms between the $H$ and $V$ pieces will survive in the probability. These cross terms will be dependent on $\theta_A$ and $\theta_B$, giving rise to the desired Bell interference. However if the polarization is entangled with another variable (OAM for example), a state such as $|l_1, H\rangle_A |l_2, V\rangle_B \pm |l_2, V\rangle_A |l_1, H\rangle_B$ will produce no interference, since the $\bm {\hat L_z}$ eigenstates $|l_1\rangle $ and $|l_2\rangle$ will still be orthogonal after the polarizer, causing the cross terms to vanish; there are no intermediate states to bridge the two orthogonal OAM states in the way that the $|\theta\rangle $ states did for polarization. This can be seen in detail in the example given below (section \[example\]).
When $j_z$ has been measured, information about $l_z$ can be erased (top part of figure \[erasure\]) by sorting different $l_z$ values into different paths, inserting appropriate holograms or spiral phase plates to shift the OAM in each path to zero, then recombining the paths. In this manner, the initial OAM values are erased (shifted to zero) so that there is no way of determining which path was taken and what the initial OAM value was. The different incoming OAM states are now indistinguishable, while the polarization states are left entangled. Similarly, when $l_z$ is measured information about $j_z$ can be erased (bottom part of figure \[erasure\]) by sorting $j_z$ values, shifting the values of $s_z$ appropriately with phase plates (converting one circular polarization into the other), and recombining. Although this changes the value of $s$ for each photon, it leaves the entanglement undisturbed on the trials where both Alice and Bob measure $j_z$, since they both carry out similar shifts: an incoming entangled spin state of the form $|s_z=1\rangle_A |s_z=-1\rangle_B \pm |s_z=-1\rangle_A
|s_z=1\rangle_B$ is shifted to $|s_z=-1\rangle_A |s_z= +1\rangle_B \pm |s_z=+1\rangle_A |s_z=-1\rangle_B$, which is still entangled and in fact proportional to the original state.
Procedure
---------
Consider now the setup described in the previous section with a two-photon input state generated from type II parametric down conversion. The pump beam is assumed to have no OAM, $l_{pump}=0$. Alice and Bob each receive one photon from the state, on which to make measurements. Consider several possibilities:
$\bullet$ Suppose Alice and Bob both measure $l_z$. Using the fact that their values should be perfectly anticorrelated, they can use the resulting OAM quantum number on Alice’s side (or, equally, on Bob’s) to define a key. Since $\hat {\bm L}$ and $\hat {\bm S}$ measurements don’t affect each other, the spin eigenvalues along each axis remain undetermined and the polarization state remains entangled.
$\bullet$ Alternatively, if both measure $j_z$, the key can then be defined by the resulting TAM quantum number on Alice’s side. The spin components are again undetermined, and the polarization state remains entangled.
$\bullet$ But if one measures $j_z$ and the other measures $l_z$, this completely fixes the spin along the axis: $s_z=j_z-l_z$. The spin wavefunction collapses from an entangled state to a separable one. These trials are discarded.
In the first two cases, the spin states remain entangled after sorting, so tests on the linear polarization should yield Bell violation. In the third case, when Alice and Bob measure *different* variables, such a test should yield no violation, the spin having been reduced to a classical quantity. The measurement of one variable ($\hat {\bm J_z}$ or $\hat {\bm L_z}$) reduces the original space of states for each particle to a two-dimensional subspace, while measurement of the second variable further reduces each particle to one unique state. Consequently, the two-photon pair goes from an entangled to a separable state.
![*A more detailed view of Alice’s lab; Bob’s lab has a similar arrangement. After sorting in one variable (either $j_z$ or $l_z$), information about the other variable then needs to be erased (figure \[erasure\]) before polarization interference is measured. The outgoing arrows lead to systems for polarization measurement. For the top three outputs in the figure, which particular detector fires at the end will tell Alice the value of $j_z$ but will give no information about $l_z$; the opposite is true in the bottom three outputs. Thus Alice will know the value of only one of these two variables.*[]{data-label="detail"}](Erasure_B.eps)
If Eve attempts quantum nondemolition measurements to determine $j_z$ or $l_z$, then on half the retained trials she will measure the wrong variable (the one not measured by Alice and Bob), thus fixing $s$ and causing a detectable loss of Bell violation.
In this scheme, the beam splitter’s random choice between causing either a $\hat {\bm J_z}$ measurement or an $\hat {\bm L_z}$ measurement replaces the usual random modulation between two measurement bases for components of a single variable. All variables act in a completely quantum manner, with the “quantumness” comes from the fact that although $\hat {\bm L}$ and $\hat {\bm S}$ commute with each other, neither commutes with $\hat {\bm J}$; if $j_z$ and $l_z$ are both measured, whether by the legitimate participants or by eavesdroppers, polarization entanglement is destroyed.
Another equivalent way to view the situation, that makes the analogy to the Ekert case clearer, is that measurements can be made along a basis in angular momentum space aligned with the $\hat{\bm J_z}$ axis or one aligned with the $\hat{\bm L_z}$ axis; these are mutually unbiased on each two-dimensional subspace defined by a fixed $j_z$ value or a fixed $l_z$ value, but are also incomplete in the sense that neither measurement fully determines the state. However making *both* measurements *does* uniquely determine the state, completely fixing $j_z$, $l_z$, and $s_z$ values.
Note that the key-generating capacity grows with increasing dimension, as is the case for all OAM-based QKD methods. However, unlike in other OAM-based schemes, the security-checking remains essentially two-dimensional so that the level of security grows more slowly with increasing dimension. This is the price that is paid for reducing the complexity of generating practical high-dimensional keys. The secure key rate and mutual information between participants will be examined in section \[securitysection\] and the appendix.
Example
-------
To be more concrete, consider an entangled two-photon input state of the form $$\begin{aligned}
|\psi\rangle &=& |\psi_{oam}\rangle |\psi_{spin}\rangle \\ &=& {1\over
\sqrt{2}}\sum_{l_z=-l_0}^{l_0} |l_z\rangle_A|-l_z\rangle_B \Bigl( |H\rangle_A|V\rangle_B -|V\rangle_A |H\rangle_B\Bigr) .\label{state1}\end{aligned}$$ Such a state arises, for example, from type II down conversion after filtering to equalize the probabilities of various $l$ values. The linear and circular polarization states are related by $$\begin{aligned}
|H\rangle &=& {1\over \sqrt{2}}\left( |s_z=1\rangle + |s_z=-1\rangle \right) \; =\; {1\over \sqrt{2}}\left( |R\rangle
+|L\rangle \right) \\ |V\rangle &=& {i\over \sqrt{2}}\left( |s_z=1\rangle - |s_z=-1\rangle \right) \; =\; {i\over \sqrt{2}}\left( |R\rangle -|L\rangle \right)
,\end{aligned}$$ where $R$ and $L$ correspond to spin $s_z=+1$ and $s_z=-1$, respectively. So the two-particle state $|\psi\rangle$ can be written in terms of joint OAM-spin states $|l,s\rangle $ as $$\begin{aligned}
|\psi\rangle = -{1\over\sqrt{2}}\sum_{s_z=-1}^1 \sum_{l_z=-l_0}^{l_0} (-1)^{s_z/2} |l_z,s_z\rangle_A |-l_z,-s_z\rangle_B . \end{aligned}$$ If both experimenters measure the TAM with Alice obtaining value $j_z$, then Bob will obtain value $-j_z$, reducing the state to $$\begin{aligned}
|\psi^\prime \rangle &=& {-i\over \sqrt{2}} \left( |j_z-1,+1\rangle_A |-j_z+1,-1\rangle_B \right. \nonumber\\
& & \qquad \left.-|j_z+1,-1\rangle_A |-j_z-1,+1\rangle_B\right) .\label{psi_j_measured}\end{aligned}$$ The first entry in each ket is the $l_z$ value, the second is $s_z$. Note that $l_z$ and $s_z$ remain indeterminate after the $j_z$ measurement since multiple combinations of $l_z$ and $s_z$ can add up to the same $j_z$. If $l_z$ is not measured in either branch at any point, then information about it can be erased as in figure \[erasure\], in order to arrive at a maximally-entangled spin wavefunction, $$\begin{aligned}
|\psi_{spin}\rangle &=& -{i\over \sqrt{2}} \left( |R\rangle_A |L\rangle_B - |L\rangle_A |R\rangle_B \right)\\
&=& {1\over \sqrt{2}}\left( |H\rangle_A |V\rangle_B - |V\rangle_A |H\rangle_B \right) \label{psispin} .\end{aligned}$$ Carrying out a Bell-type test on polarization after the $j_z$-sorting then yields maximal quantum-mechanical Bell violation. However, if in addition to the $j_z$-sorting, the OAM on Bob’s side is also measured (by Bob or by Eve), the state of equation (\[psi\_j\_measured\]) collapses to either $$i|j_z-1,1 \rangle_A |-j_z+1,-1\rangle_B$$ (if Bob finds value $l_z=-j_z+1$), or else to $$-i
|j_z+1,-1\rangle_A |-j_z-1,1\rangle_B$$ (if Bob’s value is $l_z=-j_z-1$). Placing quarter-wave plates at the output, to convert from circular to linear polarization, the state becomes either$$\begin{aligned}
i |j_z-1,H\rangle_A |-j_z+1,V\rangle_B\end{aligned}$$ or $$\begin{aligned}
-i |j_z+1,V\rangle_A
|-j_z-1,H\rangle_B .\end{aligned}$$ Either way, it is now a separable state (both before and after the $l_z$ erasure) with definite polarization for each photon, so no Bell violation occurs. If Alice and Bob measure $l_z$ while Eve measures $j_z$, a similar result follows.
When Eve guesses whether to measure $\hat {\bm L_z}$ or $\hat {\bm J_z}$, half the time she guesses wrong and causes collapse of the entangled polarization state into a separable state. This lowers the interference pattern’s visibility to classical levels when Bell tests are performed, providing a clear signal of her intervention.
To examine the interference visibility, define the rotated polarization states at Alice’s location: $$\begin{aligned}
|\theta \rangle_A &=& \cos \theta |H\rangle_A +\sin \theta |V\rangle_A \\
|\theta^\perp \rangle_A &=& -\sin \theta |H\rangle_A +\cos\theta |V\rangle_A ,\end{aligned}$$ with similar states $|\phi\rangle_B$ and $|\phi^\perp\rangle_B$ defined at Bob’s lab. $\theta$ and $\phi$ are the angles of linear polarizers before Alice’s and Bob’s detectors, respectively. In the absence of eavesdropping, it is straightforward to verify that under ideal conditions (perfect detectors and no losses) the coincidence rate is proportional to $$|\langle \psi_{spin}|\theta \rangle_A |\phi \rangle_B |^2\; =\; {1\over 2}\sin^2(\theta-\phi )\; =\; {1\over
2}\left[ 1-\cos^2(\theta -\phi )\right]$$ for the two-photon entangled spin state of equation (\[psispin\]). A Clauser-Horne-Shimony-Holt (CHSH)-type interference experiment [@chsh] will then exhibit oscillations with visibility ${\cal V}$ of $100\%$. On the other hand, if $|\psi_{spin}\rangle$ is replaced by any separable state of the form $|\psi_{sep}\rangle=|\gamma\rangle_A|\gamma^\perp\rangle_B$ (where $\gamma$ is the polarization direction of the photon measured by Alice), the corresponding inner product is $|\langle \psi_{sep}|\theta \rangle_A |\phi \rangle_B |^2 =\cos^2 (\gamma-\theta )\sin^2(\gamma -\phi ); $ the dependences on $\theta $ and $\phi$ now factor so that the visibility can never be greater than the classical limit of ${1\over \sqrt{2}}\approx 71\%$. In general, if Eve is eavesdropping a fraction $\eta$ of the time, the visibility will be ${\cal V}\le 1-\left( 1-{1\over \sqrt{2}}\right) \eta $. A drop in visibility to below the value set by the Bell-CHSH inequality signals the possible presence of eavesdropping.
Information and security considerations {#securitysection}
=======================================
Instead of testing Bell inequalities on the entangled polarization states, there is a second way to check the security of the transmission, which will be more useful for arriving at quantitative estimates of information and signalling rates. On the set of trials for which Alice and Bob measure the same variable, they can choose a random subset of their measurement values ($l_z$ or $j_z$ values) for comparison. Ideally, they should both find perfectly anticorrelated values, so that the presence of discrepancies beyond the expected error rate due to the transmission method then serves as a signal of an eavesdropper’s presence. This method is more directly analogous to that of the BB84 scheme, requiring [*no*]{} active modulation of the detector settings, as opposed to the Bell-inequality-based version of the Ekert scheme, which still requires modulation of the settings for the polarization measurements. In this section, we take the BB84-like approach when we look at security considerations, since it is relatively easy to compute the probabilities of the output states and the key rate while taking Eve’s actions into account.
If the allowed OAM values are $\left\{-l_0,\dots ,0,\dots ,+l_0\right\}$, then the possible $j_z$ values are $ \left\{-l_0-1,\dots ,0,\dots ,l_0+1\right\} $. It should be noted that unlike the case of the Ekert or BB84 protocols, where there is a finite number of possible outcomes (two polarizations) and they are both used, in the current case we make use of a finite subset of a larger (in fact infinite) set of possible output values for $j_z$ or $l_z$. As a result, when Eve interferes it is possible to “run off the end” of the allowed set of values, and this possibility must be accounted for. Also, we note that there are $2l_0+3$ values of $j_z$ but only $2l_0+1$ values of $l_z$. As a result of these complications some further refinements to the method must be made. These are discussed in the appendix. The appendix also then gives the resulting probability distributions for Alice’s and Bob’s joint outcomes. In this section, we make use of those distributions to investigate the quantum security of the procedure against eavesdropping.
![*Comparison of the error rate $e$ for the scheme described in this paper (solid curves) to the BB84 protocol (dashed lines). This is done for (from top to bottom) eavesdropping ratios $\eta=1$, $\eta=.5$, and $\eta=.1$. In each case, the error rate for the current approach starts a little below the BB84 value for small $l_0$, but approaches it rapidly and asymptotically as $l_0$ increases.*[]{data-label="errorfig"}](error_rate.eps)
Error rates
-----------
If Eve intercepts fraction $\eta$ of the transmissions, she has $50\% $ chance of measuring the correct variable each time, obtaining the correct key value $k$ (either from an $\hat {\bm L_z}$ or $\hat {\bm J_z}$ measurement) without introducing errors. On the other hand, during the $50\%$ of the times in which she measures the wrong variable, she only has a further $50\% $ chance of sending the correct value to Bob. Specifically, each time she measures the wrong variable her intervention has probability ${1\over 8}$ of causing Bob to measure the value $k-2$ and overall probability ${1\over 8}$ of causing him to measure the value $k+2$, with only probability ${1\over 2}$ of obtaining the correct key value, $k$ (see figure \[splittingfig\]). So if the alphabet size were infinite, the eavesdropper-induced error rate would be $e=\eta \cdot\left( {1\over 2} \right)\cdot \left({1\over 2} \right) \; =\; {\eta\over 4}.$ However, we must take into account the fact that Eve’s actions can cause values to move out of the range being used for the key; this will alter the error rate slightly. Using the probability distribution $P_{AB}$ given in the appendix, it is then straightforward to show that the true eavesdropper-induced error rate is $$e={\eta\over 8}\left( {{4l_0+1}\over {(2l_0+1)-{\eta\over 8}}}\right).$$ Bob’s error rate is shown in figure \[errorfig\] for three different eavesdropping ratios, along with the corresponding values for polarization-based BB84. It is seen that for small $l_0$ the error rate is slightly lower than the BB84 value, but it rapidly approaches that value as $l_0$ increases.
After dropping the trials on which Alice and Bob measure different variables, the fraction $f$ of the remaining photons that are used to generate the key may also be easily computed from $P_{AB}$. It is found to be $$f={{4l_0+2-\eta}\over {4l_0+3}}.$$ This approaches $100\%$ for $l_0\to \infty$. For finite $l_0$, it ranges between a low of $f={{4l_0+1}\over {4l_0+3}}$ (for $\eta =1$) and a high of $f={{4l_0+2}\over {4l_0+3}}$ (for $\eta =0$). For the worst case ($l_0=1$ qutrits), this corresponds to a range of ${5\over 7}$ to ${6\over 7}$.
Mutual information and key rates
--------------------------------
One may also compute the mutual information between the legitimate agents, $I(A;B)$, and information gain of the eavesdropper, $I_{E}=\mbox{max}\left\{ I(A;E),I(B;E)\right\}$. From these, the secret key rate, $\kappa =max\left\{ I_{AB}-I_E,0\right\}$ may be found. Recall that it is always possible to distill a secret key using privacy amplification when $\kappa >0$. Instead of the distribution $P_{AB}$ given in the appendix (which includes *all* events, even those for which the values run off the edge of the alphabet and so generate no key) in order to compute $\kappa$, we must use the probability distribution $P_K(A,B)$ for the *key-generating events only*. This new distribution is obtained from $P_{AB}$ simply by dropping its last row and column, then dividing by the key-generating fraction $f$ in order to renormalize the total probability to unity. Straightforward calculation then gives the result that the mutual information [@cover] between Alice and Bob as a function of parameters $\eta$ and $l_0$ is: $$\begin{aligned}
I(A;B)&=& {2\over {4l_0+2-\eta}}\left\{ \left( 2l_0+1-{\eta\over 2}\right)\log_2\left( {{4l_0+2-\eta}\over 2}\right)
\right. \\
& & \quad
-8\left( 1-{\eta\over 8}\right) \log_2\left( 1-{\eta\over 8}\right) + (2l_0+1)\left( 1-{\eta\over 4}\right)\log_2\left( 1-{\eta\over 4}\right) \nonumber \\
& & \left. \quad +{\eta\over 4}(2l_0-1)\log_2 {\eta\over 8}\right\} \nonumber \end{aligned}$$ Asymptotically (for $l_0\to \infty$) this approaches $\log_2(2l_0)$, independent of $\eta$; for finite $l_0$ and no eavesdropping ($\eta =0$), it is equal to $\log_2(2l_0+1)$. Eve gains full information about the key value on half of the measurements she makes and receives none on the other half, so the information gained by Eve is simply ${\eta\over 2}$ times the information per photon. The results for the information and the secret key rate are plotted in figure \[infofig\](b-d). The case of BB84 is shown for comparison in figure \[infofig\] (a). It is seen that $\kappa$ is always greater than in the BB84 case, that it remains positive for all values of $\eta$, and that for any fixed value of $\eta$ the value of $\kappa$ increases with increasing $l_0$. Thus, the amount of key generated per transmitted photon is significantly larger than in the BB84 or Ekert schemes. Security can be further enhanced in various ways, such as replacing the four-state polarization scheme with a six-state approach [@bruss].
\
Conclusion
==========
We have proposed a method for performing QKD using high-dimensional OAM and TAM variables in a manner that does not require the complicated high-dimensional basis modulations necessary in other approaches, and which allows an increase in the rate of secure key distribution. The main ingredients are: (i) A hyperentangled system with different functions segregated into different entangled degrees of freedom. (ii) Random switching between two OAM *bases* is replaced by random switching between measurements of two distinct noncommuting but related *variables*, $\hat {\bm L_z}$ and $\hat {\bm J_z}$. (iii) Measurement of any one variable does not completely determine the state, while measurement of any two of the three relevant variables does. Together, these ingredients allow the high capacity of $\bm{\hat J_z}$ or $\bm{\hat L_z}$ eigenstates to be used while modulating only the simpler polarization states. As a result, higher-dimensional OAM spaces can be utilized and higher key-generation capacities can be achieved with only relatively minimal increases in apparatus complexity.
Previous approaches to using polarization and OAM together simply used them to generate larger keys from each photon by increasing the number of variables involved. In these past approaches, the variables still remained separate, with no interplay between them. QND measurements on one variable left no signature in the other, so that both must experience independent basis modulations to maintain security. In contrast, the approach described here makes a more fundamental use of the system’s hyperentanglement, constructing a chain of three variables such that adjacent pairs in the chain do not commute. This noncommutativity provides a linkage between the variables that enhances security, as well as increasing the number of key bits per photon.
The approach of constructing chains of pairwise noncommuting operators has not been previously used in QKD and is likely to be generalizable to other operators (aside from angular momentum), and to employment in other types of quantum protocols as well. The procedure also makes use of an angular momentum erasure process that is of interest in its own right, since it allows the possibility of conducting future quantum erasure or delayed choice experiments in angular momentum space.
At high $l_0$, the eavesdropper-induced error rate is lower than for other two-basis OAM-based QKD schemes, where $e={\eta\over 2}\left( 1-{1\over
{2l_0+1}}\right)=\eta {{l_0}\over {2l_0+1}}$; in the current scheme $e$ instead remains near the 2-dimensional BB84 error rate; this is because the eigenspaces of the two measured operators ($\bm {\hat J_z}$ and $\bm {\hat L_z}$) are effectively unbiased only on a two-dimensional subspace, due to the two-dimensional nature of the polarization. The dimension of the subspace on which the variables are unbiased does not increase with $l_0$. Because of this, the secret key rate $\kappa$ will be lower than for other OAM-based schemes at large $l_0$. However, $\kappa$ in this approach is always higher than in BB84 or E91 protocols even for the least advantageous case ($l_0=1$), and it grows logarithmically with increasing $l_0$; similarly, the BB84-level eavesdropper-induced error rate remains sufficient to detect eavesdropping regardless of dimension.
There are some technical difficulties that must be overcome for the method to become practical. Probably the chief among these is that the most common method of measuring photon OAM is to shift the input $l$-state to $l = 0$ and then to collect them in an optical fiber. This method is of low efficiency, which greatly reduces the key transmission rate of all OAM-based schemes. Further, the interferometers used to sort the OAM and TAM values [@leach1] become progressively more complex as the range of $l_z$ and $j_z$ values to be used increases.
However, when coupled to the much greater ease in this scheme of switching between $\hat J_z$ and $\hat L_z$ measurements compared to the difficult switching between measurements of different components of $\hat{\bm L}$ in other protocols, the method presented here seems to hold strong promise as a more practical way to reach higher key rates per photon while maintaining full quantum-level security.
This research was supported by the DARPA QUINESS program through US Army Research Office award W31P4Q-12-1-0015, by the DARPA InPho program through US Army Research Office award W911NF-10-1-0404, and by National Science Foundation grant ECCS-1309209. The authors would like to thank Prof. Daniel Gauthier, Dr. Thomas Brougham, and Dr. Kevin McCusker for helpful comments concerning an early version of this manuscript.
Outcome Probabilities
=====================
As mentioned in section \[securitysection\], we must make adjustments to the protocol in order to equalize the probabilities of the allowed key values, and must take into account that the values measured may lie outside the range used for key generation. We deal with those complications here, and give the joint probability distributions that result after doing so.
Undisturbed probability distributions
-------------------------------------
First, the key must have the same range of values regardless of which variable was measured. So assign no key value when $j_z=\pm (l_0+1)$ is measured; for the remaining $j_z$ values, assign key value $j_z$. Then either type measurement leads to a range of values from $-l_0$ to $l_0$, leading to alphabet size $N=2l_0+1$. Instances where $j_z$ equals $l_0-1$ or $l_0+1$, though not used for key generation, are not discarded; they are recorded for use in the security analysis. Second, all key values must have equal probability. Initially, each $l_z$ value has probability $P_L(l_z)={1\over {2(2l_0+1)}}$ (the $1\over 2$ comes from the probability that $l_z$ was measured instead of $j_z$), while each $j_z$ used for key generation has probability $P_J(j_z)={1\over {2(2l_0+2)}}$. (The two values not used for key generation each have probability ${1\over {4(2l_0+3)}}$.) To make $P_L(k)=P_J(k)$ for each key value $k$, the reflectance of the beam splitter may be adjusted away from $50\% $, so the reflection and transmission probabilities are $|r|^2={1\over 2}-\epsilon$ and $|t|^2={1\over 2}+\epsilon$, with $\epsilon= {1\over 2}\left( {1\over {4l_0+3}}\right).$ Then each $l_z$ and $j_z$ value has probability $$P_L(l_z)=P_J(j_z)={1\over {4l_0+3}},$$ and each possible key value has probability $$P(k)=P_L(k)+P_J(k)=
{2\over {4l_0+3}},$$ with probability ${1\over {4l_0+3}}$ that no key is generated.
![*The possible outcomes when Eve intervenes, assuming Alice and Bob both measure $\hat
{\bm L_z}$. (If they measure $\hat {\bm J_z}$ instead, swap the variables $l$ and $j$ everywhere in the figure.) At each splitting, each branch has equal probability. Circles containing the letters A, B, or E represent measurements by Alice, Bob, or Eve, respectively.*[]{data-label="splittingfig"}](eve_effect.eps)
In Eve’s absence, there is ideally perfect anticorrelation when Alice and Bob measure the same variable. The distributions of key values $k$ should be identical, $P_A(k)=P_B(k).$ Moreover, their joint distribution should be uniform on the diagonal and vanishing elsewhere: $$P_0(k_A,k_B)=\left( {2\over {4l_0+3}}\right)
\delta_{k_Ak_B}.$$ We therefore find entropies $H(A) = H(B) = H(A,B) =
\log_2(4l_0+3) -\left( 4l_0+2\right)/\left( 4l_0+3\right) ,$ so the mutual information $I(A;B)\equiv H(A)+H(B)-H(A,B)$ just equals the Shannon information of each participant separately, $I=\log_2(4l_0+3) -\left( 4l_0+2\right) / \left(4l_0+3\right) .$
Probability distributions with eavesdropping
--------------------------------------------
The effects of Eve’s actions are shown in figure \[splittingfig\]. (The figure assumes that Alice and Bob measure $\bm {\hat L_z}$; if they measure $\bm {\hat
J_z}$, simply interchange $l_z$ and $j_z$ everywhere in the figure.) At each splitting of branches, there is a $50\%$ chance that each branch will be taken. Suppose Alice measures $\bm {\hat L_z}$ and obtains value $-l_A$. If Eve intercepts the transmission, she may measure the same variable, in which case both she and Bob will obtain the negative of Alice’s value: $l_E=+l_A$ for Eve and $l_B=+l_A$ for Bob. But if Eve measures the other variable, $\bm {\hat J_z}$, then she has equal likelihood of measuring the value above $l_A$ or the value below it: $j_E=l_A+1$ or $j_E=l_A-1$. Similarly, Bob then has equal chances of measuring $\bm {\hat L_z}$ to have the eigenvalue one unit above or below Eve’s value: $l_B=j_E\pm 1$.
As a result, the entries of the matrix representing the undisturbed joint probability distribution, $$\begin{aligned}
P_0(k_A,k_B) = \left( {2\over {4l_0+3}}\right) \delta_{k_Ak_B} \end{aligned}$$ are now smeared out by Eve’s actions over multiple entries in Bob’s direction.
Using the diagram of figure \[splittingfig\], the new Alice-Bob joint probability distribution on trials where Eve intervenes may be determined. It is found to be $$P_1={2\over {4l_0+3}}\left( \begin{array}{cccccc|c}
{3\over 4} & 0& {1\over 8} & 0 & \dots & 0& {1\over 8} \\
0 & {3\over 4} & 0 & {1\over 8} & & &{1\over 8}\\
{1\over 8} & 0 & {3\over 4} & 0 & {1\over 8} & &0\\
& \ddots & \ddots & \ddots & & & \vdots \\
& & \ddots & \ddots & \ddots & & 0\\
& & & {1\over 8} & 0 & {3\over 4} & {1\over 8} \\ \hline
&\dots & 0& 0& {1\over {16}}& 0& {7\over {16}}
\end{array}\right) ,$$ where rows label Alice’s outcomes and columns label Bob’s. The first $2l_0+1$ rows and columns label possible key values, while the last row and column correspond to Alice or Bob, respectively, generating no key. For eavesdropping fraction $\eta$, the full joint outcome distribution for all trials becomes $$P_{AB}(k)=\left( 1-\eta\right) P_0+ \eta P_1 ,$$ with marginal probabilities for the two participants obtained by summing rows and columns.
The eavesdropper-induced error rate, the mutual information shared by Alice and Bob, and the secure key rate may all now be found using this distribution. These quantities are discussed in section \[securitysection\].
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|
---
abstract: 'The $N$-representability problem is the problem of determining whether or not there exists $N$-particle states with some prescribed property. Here we report an affirmative solution to the fermion $N$-representability problem when both the density and paramagnetic current density are prescribed. This problem arises in current-density functional theory and is a generalization of the well-studied corresponding problem (only the density prescribed) in density functional theory. Given any density and paramagnetic current density satisfying a minimal regularity condition (essentially that a von Weizäcker-like the canonical kinetic energy density is locally integrable), we prove that there exist a corresponding $N$-particle state. We prove this by constructing an explicit one-particle reduced density matrix in the form of a position-space kernel, i.e. a function of two continuous position variables. In order to make minimal assumptions, we also address mathematical subtleties regarding the diagonal of, and how to rigorously extract paramagnetic current densities from, one-particle reduced density matrices in kernel form.'
author:
- Erik Tellgren
- Simen Kvaal
- Trygve Helgaker
title: 'Fermion $N$-representability for prescribed density and paramagnetic current density'
---
Introduction
============
The question of $N$-representability has been studied extensively in quantum chemistry and related fields [@COLEMAN_RMP35_668]. In particular, it plays an important role in density-functional theory (DFT). Given prescribed values for quantities in a fermionic system, e.g. its electron density or its reduced density matrix, one may ask whether or not it can be obtained from a Slater determinant, from a pure $N$-particle state, or from a mixed $N$-particle state. Regarding the density, it is well known that any density with a finite von Weizsäcker kinetic energy may be reproduced using a Slater determinant that also has finite kinetic energy [@MACKE_PR100_992; @GILBERT_PRB12_2111; @HARRIMAN_PRA24_680; @ZUMBACH_PRA28_544; @GHOSH_JCP82_3307]. For a one-particle reduced density matrix (1-rdm), Slater-determinantal representability is equivalent to idempotency; in general, however, pure-state $N$-representability of 1-rdms is a difficult and largely unsolved problem [@RUSKAI_PR169_101; @LUDENA_JMST123_371; @RUSKAI_JPA40_F961]. On the other hand, any 1-rdm with spin-orbital occupation numbers (eigenvalues) in the range $[0,1]$ and trace $N$ may be obtained from a mixed $N$-particle state.
In this note, we report the solution to the mixed-state $N$-representability problem when both the density and paramagnetic current density are prescribed. More precisely, we answer the following question: given a density $\rho(\mathbf{r})$ and a paramagnetic current density ${{\mathbf{j}_{\mathrm{p}}}}(\mathbf{r})$, does there exist a mixed state $\Gamma$ with the prescribed density and current density, written $\Gamma\mapsto(\rho,{{\mathbf{j}_{\mathrm{p}}}})$? We answer this question affirmatively by constructing an explicit 1-rdm, from which the existence of the $N$-particle state follows.
We note that standard constructions demonstrating the Slater-determinantal $N$-representability when only the density is prescribed rely on the use of equidensity orbitals. Also, early work by Ghosh and Dhara [@GHOSH_PRA38_1149] sketched a construction of such equidensity orbitals that reproduce both densities and currents. However, such solutions have limited scope, since the vorticity vanishes when orbitals give rise to the same density.
During the preparation of the present work, one of us (S. Kvaal) met E.H. Lieb during a trimester at Institut Henri Poincaré in Paris in July 2013, and became aware that together with R. Schrader, he had shown a Slater determinant representability result for $(\rho,{{\mathbf{j}_{\mathrm{p}}}})$ and $N\geq 4$. This work has now been published [@LIEB_SCHRADER_2013].
Clearly, $N$-representability via a Slater determinant implies representabiility via a mixed state. However, Lieb and Schrader’s result requires $N\geq 4$, and they also give a counterexample for $N=2$, where no Slater determinant can exist (with continuously differentiable and single-valued orbital phase functions). Our result, while showing a weaker sense of $N$-representability, has no condition on $N$. Moreover, both the present work and Ref. [@LIEB_SCHRADER_2013] have mild regularity and decay assumptions on $(\rho,{{\mathbf{j}_{\mathrm{p}}}})$ that ensure representability, but these are different in the two approaches. The techniques of proof are also otherwise significantly different: Lieb and Schrader rely on the so-called smooth Hobby–Rice theorem, while our approach is by direct construction of a 1-rdm. The present work and the work of Lieb and Schrader are complementary, offering two different points of view and solutions to a long-standing problem.
The remainder of this paper contains five sections. In Section\[background\], we give some background information and establish notation. Following a discussion of the relationship between a reduced density matrix and its associated density and paramagnetic current density in Section\[secDIAG\], we construct in Sections\[redmat\] and \[cankin\] a reduced density matrix for a prescribed density and paramagnetic current density. Section\[discussion\] contains some concluding remarks. Finally, two appendices are also provided. Appendix A contains a brief overview of some mathematical concepts and results on Hilbert–Schmidt operators needed for the main results of Section \[secDIAG\]. Appendix B contains proofs of theorems in Section \[secDIAG\].
Background
==========
The $N$-representability problem with prescribed density $\rho$ and paramagnetic current density ${{\mathbf{j}_{\mathrm{p}}}}$ arises in current-density functional theory (CDFT) [@VIGNALE_PRL59_2360]. In CDFT, a magnetic vector potential $\mathbf{A}$, in addition the scalar potential $v$, enters the (spin-free) $N$-electron Hamiltonian. In atomic units, $$\begin{aligned}
H[v,\mathbf{A}] &= \frac{1}{2}\sum_{k=1}^N (- \mathrm i\boldsymbol \nabla_{k} + \mathbf{A}(\mathbf{r}_k))^2 \nonumber \\ & \quad\quad\quad + \sum_{k=1}^N v(\mathbf{r}_k) + \sum_{k<l} \frac{1}{r_{kl}}.\end{aligned}$$ Here $\mathbf r_k$ is the position of electron $k$, the operator $\boldsymbol \nabla_k$ differentiates with respect to $\mathbf r_k$, and $r_{kl}$ is the distance between electrons $k$ and $l$. The corresponding ground-state energy is given by the Rayleigh–Ritz variation principle, $$E[v,\mathbf{A}] = \inf_{\Gamma} \mathrm{Tr}( \Gamma H[v,\mathbf{A}])
\label{eq0:energy}$$ where the minimization is over all mixed states $\Gamma$ with a finite canonical kinetic energy $$T[\Gamma] := \frac{1}{2} {\operatorname{Tr}}\left(\boldsymbol \nabla\Gamma\boldsymbol \nabla^\dag \right).$$ Introducing the constrained-search universal functional $$F[\rho,{{\mathbf{j}_{\mathrm{p}}}}] =
\inf_{\Gamma \mapsto \rho,{{\mathbf{j}_{\mathrm{p}}}}} \mathrm{Tr} \left( \Gamma H[0,\mathbf{0}] \right) ,$$ we may rewrite the Rayleigh–Ritz variation principle in Eq.(\[eq0:energy\]) in the form of a Hohenberg–Kohn variation principle, $$E[v,\mathbf{A}]
= \inf_{\rho,{{\mathbf{j}_{\mathrm{p}}}}} \Bigl( \! F[\rho,{{\mathbf{j}_{\mathrm{p}}}}] +
\int \! \Bigl (\rho \,(v + \tfrac{1}{2} A^2) +
{{\mathbf{j}_{\mathrm{p}}}}\cdot\mathbf{A}\Bigr) \mathrm d\mathbf{r} \Bigr).
\label{eq:energy}$$ The mixed-state $N$-representability problem is directly related to how large the search domain in Eq.(\[eq:energy\]) needs to be: if no $\Gamma\mapsto(\rho,\mathbf{j}_\text{p})$ exists, then $F[\rho,{{\mathbf{j}_{\mathrm{p}}}}]=+\infty$ by definition.
In Kohn–Sham theory, the idea is to express the densities in Eq.(\[eq:energy\]) in terms of a single Slater determinant of non-interacting particles and to approximate the kinetic-energy contributions to $F[\rho,{{\mathbf{j}_{\mathrm{p}}}}]$ by the non-interacting kinetic energy, $$T_s[\rho,{{\mathbf{j}_{\mathrm{p}}}}] = \inf_{\{\phi_k\}_{k=1}^N \mapsto \rho,{{\mathbf{j}_{\mathrm{p}}}}}
\frac{1}{2}\sum_{k=1}^N {\langle {\boldsymbol \nabla\phi_k, \boldsymbol \nabla\phi_k} \rangle},
\label{sdnrep}$$ where the infimum is over an orthonormal set of orbitals $\phi_k$ or, equivalently, the corresponding Slater determinants or idempotent 1-rdms. At this point, the Slater-determinantal $N$-representability problem arises.
In general, densities $\rho$ and ${{\mathbf{j}_{\mathrm{p}}}}$ arising from a single orbital have a vanishing paramagnetic vorticity, $$\boldsymbol{\nu} = \boldsymbol \nabla\times\frac{{{\mathbf{j}_{\mathrm{p}}}}}{\rho} = 0,$$ except for possible Dirac-delta singularities at points $\mathbf r$ where $\rho(\mathbf r) =
0$. Consequently, a closed-shell two-particle Kohn–Sham system can only reproduce paramagnetic densities with vanishing vorticity. In general, therefore an extended Kohn–Sham approach with fractional occupation numbers is required (see Refs.[@CANCES_JCP114_10616; @CANCES_JCP118_5364; @KRAISLER_PRA80_032115; @NYGAARD_JCP138_094109] for work in this direction), $$\bar{T}_s[\rho,{{\mathbf{j}_{\mathrm{p}}}}] = \inf_{\{n_k\phi_k\} \mapsto \rho,{{\mathbf{j}_{\mathrm{p}}}}}
\frac{1}{2} \sum_{k=1}^{\infty} n_k {\langle {\boldsymbol \nabla\phi_k,\boldsymbol \nabla\phi_k} \rangle},$$ where orthonormality, $0 \leq n_k \leq 1$, and $\sum_k n_k = N$ are additional constraints on the infimum. Alternatively, since $n_k$ and $\phi_k$ are eigenvalues and eigenvectors of 1-rdms, the minimization may equivalently be performed over 1-rdms. Here, the mixed-state $N$-representability problem appears.
For a mixed state $\Gamma\mapsto(\rho,{{\mathbf{j}_{\mathrm{p}}}})$, it is known that $$\begin{aligned}
T_{\text{W}}[\rho] &+ T_\text{p}[\rho,\mathbf{j}_\text{p}] \leq T[\Gamma],\end{aligned}$$ where the von Weizs[ä]{}cker kinetic-energy functionals are given by $$\begin{aligned}
T_{\text{W}}[\rho] &:= \frac{1}{8}\int \!\! \rho(\mathbf{r})^{-1}|\boldsymbol \nabla\rho(\mathbf{r})|^2 \mathrm d\mathrm{r} , \\
T_\text{p}[\rho,\mathbf{j}_\text{p}] &:=
\frac{1}{2} \int \!\! \rho(\mathbf{r})^{-1}|\mathbf{j}_\text{p}(\mathbf{r})|^2 \mathrm d\mathrm{r} . \end{aligned}$$ A necessary condition for a finite-kinetic-energy representability is therefore that $T_W[\rho] +
T_\text{p}[\rho,{{\mathbf{j}_{\mathrm{p}}}}]<+\infty$. For the case ${{\mathbf{j}_{\mathrm{p}}}}=0$, this is also a sufficient condition. It is of interest to know whether this sufficiency generalizes to ${{\mathbf{j}_{\mathrm{p}}}}\neq 0$. In this paper, we shall prove sufficiency under mild additional conditions on the current density.
Diagonals of density operators {#secDIAG}
==============================
We do not explicitly consider spin and therefore take as our point of departure an $N$-electron density matrix that depends only on spatial coordinates, with the spin coordinates integrated out: $$\label{eqGAMMAINTRO}
\Gamma(\mathbf{r}_{1:N},\mathbf{s}_{1:N}) = \sum_i p_i \Psi_i(\mathbf{r}_{1:N}) \Psi^\ast_i(\mathbf{s}_{1:N}).$$ Such a density matrix is an element of a Lebesgue space, $$\Gamma \in L^2({\mathbb{R}}^{3N}\times{\mathbb{R}}^{3N}),$$ and is symmetric with respect to permutations $\pi$ of the coordinate labels, $(\mathbf{r}_k,\mathbf{s}_k)
\stackrel{\pi}{\mapsto}
(\mathbf{r}_{\pi(k)},\mathbf{s}_{\pi(k)})$. The right-hand side of Eq. is a convex combination of properly normalized pure states, $\Psi_i
\in L^2({\mathbb{R}}^{3N})$, with coefficients $p_i \geq 0$ such that $\sum_i p_i = 1$. Moreover, each (spin-free) pure state is either totally symmetric or anti-symmetric. In what follows, it does not matter whether each pure state is required to be anti-symmetric, symmetric, or either anti-symmetric or symmetric with respect to index permutations $\pi$ of the spatial coordinates. For simplicity, we shall occasionally simplify the presentation by taking $\Gamma$ to be a pure state.
Density matrices {#denmats}
----------------
The 1-rdm belonging to a pointwise defined $\Gamma$ on the form is given by the convex combination $$D_{\Gamma}({\mathbf{r}},{\mathbf{s}}) := N \sum_i p_i \!\!\int_{{\mathbb{R}}^{3N-3}}
\!\!\!\!
\!\!\!\!
\!\!\!\!
\Psi_i({\mathbf{r}},{\mathbf{r}}_{2:N}) \Psi^\ast_i({\mathbf{s}},{\mathbf{r}}_{2:N}),
\mathrm d{\mathbf{r}}_{2:N}
\label{eq:Dpsi}$$ and belongs to $L^2({\mathbb{R}}^3\times{\mathbb{R}}^3)$. For pure states $\Gamma = {| {\Psi} \rangle}
{\langle {\Psi} |}$, we may alternatively write $D_{\Psi}$. Due to permutation symmetry, the 1-rdm is independent of which $N-1$ coordinates that have been integrated out.
Given that $D_\Gamma \in L^2({\mathbb{R}}^3\times{\mathbb{R}}^3)$, $D_\Gamma$ is by definition the *kernel of a Hilbert–Schmidt integral operator*. Our discussion makes extensive use of this basic fact about the reduced density matrix. In particular, $\Gamma$ and $D_\Gamma$ are both *trace-class operators*—that is, Hilbert–Schmidt operators for which the matrix trace has a meaningful generalization. Let $\{\phi_k\}\subset L^2(X)$ be an orthonormal basis. By definition, $A$ is a trace-class operator if and only if the trace $${\operatorname{Tr}}A := \sum_k {\langle {\phi_k, A\phi_k} \rangle}$$ has a finite value, independent of the orthonormal basis.
For two Hilbert–Schmidt operators $B$ and $C$, the kernel of the operator product $A = B \ast C$ is easily seen to be $$(B*C)(x,y) := \int_X \!\!
B(x,z)C(z,y) \,\mathrm d z, \label{eq:star}$$ which is also Hilbert–Schmidt. Importantly, it can be shown that $A$ is (the kernel of) a trace-class operator if and only if $A = B\ast C$ with $B$ and $C$ Hilbert–Schmidt. (Indeed, this is often taken as an alternative definition of trace-class operators.) The trace is then given by [@BRISLAWN_1988] the integral of the diagonal, $${\operatorname{Tr}}A = \int_{X} (A\ast B)(x,x)\, \mathrm{d} x.$$ If $A$ is diagonable (e.g., symmetric positive semidefinite), then ${\operatorname{Tr}}A$ is the sum of the eigenvalues, like in the finite-dimensional case. For further information on these operator classes, see for example the standard textbook [@REED_SIMON_1980].
We denote by $\mathcal{D}_N$ the set of mixed $N$-electron states $\Gamma$ and by $\mathcal{D}_{N,1}$ the set of 1-rdms that belong to some mixed $N$-electron state. The set $\mathcal{D}_{N,1}$ has the following well-known characterization:
\[thm:DN1properties\] $\mathcal{D}_{N,1}$ consists of those $D\in L^2({\mathbb{R}}^3\times
{\mathbb{R}}^3)$ with the following properties:
1. \[item:prop1\] $D$ is the kernel of a trace-class operator on $L^2({\mathbb{R}}^3)$.
2. \[item:prop2\] $D$ is Hermitian: $D({\mathbf{r}},{\mathbf{s}}) = D^\ast({\mathbf{s}},{\mathbf{r}})$ for almost all $({\mathbf{r}},{\mathbf{s}})$.
3. \[item:prop3\] $D$ is positive semidefinite: $0 \leq \int \! \phi^\ast({\mathbf{r}})
D({\mathbf{r}},{\mathbf{s}})\phi({\mathbf{s}}) \mathrm d{\mathbf{s}}$ for all $\phi\in L^2({\mathbb{R}}^3)$
4. \[item:prop4\] $D$ has no eigenvalues greater than two: $2 \geq \int \! \phi^\ast({\mathbf{r}})
D({\mathbf{r}},{\mathbf{s}})\phi({\mathbf{s}}) \mathrm d{\mathbf{s}}$ for all $\phi\in
L^2({\mathbb{R}}^3)$
5. \[item:prop5\] $D$ has eigenvalues that add up to $N$: ${\operatorname{Tr}}D = N$.
See Ref. [@PARR_YANG_1989], Section 2.6.
The last three conditions mean that $D(\mathbf{r},\mathbf{s})$ has eigenvalues in the interval $[0,2]$—that is, eigenvalues interpretable as fermion occupation numbers—and that the sum of the eigenvalues ${\operatorname{Tr}}D$ is equal to $N$, the number of particles.
Since $D\in\mathcal{D}_{N,1}$ is Hermitian and positive, it is easy to show that there always exists a factorization of the form $D = G^\dag
\ast G$, meaning that we may write the density matrix in the form $$D({\mathbf{r}},{\mathbf{s}}) = (G^\dag * G)({\mathbf{r}},{\mathbf{s}}) =
\int_{{\mathbb{R}}^3} \! G^\ast({\mathbf{u}},{\mathbf{s}})
G({\mathbf{u}},{\mathbf{r}}) \mathrm d{\mathbf{u}},$$ which plays an important role in the following.
Density
-------
We now define the *density* $\rho_\Psi$ associated with the wave function $\Psi$ as $$\rho_\Psi({\mathbf{r}}) := D_\Psi({\mathbf{r}},{\mathbf{r}}) = N\!\!\int_{{\mathbb{R}}^{3N-3}} \!\!
|\Psi({\mathbf{r}},{\mathbf{r}}_{2:N})|^2
\mathrm d{\mathbf{r}}_{2:N}.
\label{eq:rhopsi}$$ For almost all ${\mathbf{r}}$, it holds that $\Psi({\mathbf{r}},\cdot)\in
L^2({\mathbb{R}}^{3N-3})$. Using the Cauchy–Schwarz inequality, we see from Eq. that $\rho_\Psi({\mathbf{r}}) =
D_\Psi({\mathbf{r}},{\mathbf{r}})$ is well defined for almost all ${\mathbf{r}}$. For a mixed state $\Gamma \in \mathcal D_N$, the density $\rho_\Gamma$ is defined in the same manner but from $D_\Gamma$.
The following point is subtle but important here. We write $\Gamma\mapsto D$ whenever $\|D_\Gamma-D\|_{L^2({\mathbb{R}}^3\times{\mathbb{R}}^3)} = 0$. This statement does not imply that that $D_\Gamma = D$ everywhere, only that $D_\Gamma$ and $D$ are equal as elements of $L^2({\mathbb{R}}^3\times {\mathbb{R}}^3)$. Consequently, $D_\Gamma$ and $D$ may differ at a set of measure zero, *including the totality of the diagonal*. Therefore, we need to examine carefully the validity or meaning of the statement “$\rho_\Gamma({\mathbf{r}}) = D({\mathbf{r}},{\mathbf{r}})$” for a state and density matrix related by $\Gamma\mapsto D$.
Suppose next that we are able to assign a diagonal ${\operatorname{diag}}D$ to $D$ in some unambiguous way and let $\Gamma,\Gamma'\in\mathcal{D}_N$ be two (possibly distinct) states such that $\Gamma \mapsto D$ and $\Gamma' \mapsto D$, meaning that $D = D_\Gamma =
D_{\Gamma'}$ almost everywhere in $\mathbb R^3 \times \mathbb R^3$. Is it then true that $\rho_\Gamma = \rho_{\Gamma'} = {\operatorname{diag}}D$ almost everywhere in $\mathbb R^3$? Intuitively, this should be so.
The following theorem, which is proved in AppendixB, resolves the issue:
\[thm:densities\] Let $D\in \mathcal{D}_{N,1}$, and suppose that $G\in
L^2({\mathbb{R}}^3\times{\mathbb{R}}^3)$ is such that $$D({\mathbf{r}},{\mathbf{s}}) = (G^\dag * G)({\mathbf{r}},{\mathbf{s}})$$ almost everywhere in ${\mathbb{R}}^3\times{\mathbb{R}}^3$. Then, for every $\Gamma\in \mathcal{D}_N$ such that $\Gamma\mapsto
D$, it holds that $$\begin{aligned}
\rho_\Gamma({\mathbf{r}}) &= (G^\dag*G)({\mathbf{r}},{\mathbf{r}})
\end{aligned}$$ almost everywhere in ${\mathbb{R}}^3$.
Since the factorization of $D = G^\dag\ast G$ does exist following the discussion in Section\[denmats\], it is indeed meaningful to talk about “the density $\rho$ of $D$” without reference to a specific $\Gamma\mapsto D$: $$\rho_D({\mathbf{r}}) = {\operatorname{diag}}D({\mathbf{r}}) := (G^\dag\ast G)({\mathbf{r}},{\mathbf{r}})
\quad \text{a.e.}$$ In particular, it follows that $${\operatorname{Tr}}D = \int \!\rho_D({\mathbf{r}})\,\mathrm d{\mathbf{r}}.$$ We emphasize that we only define the diagonal when a factorization is present, and that this diagonal is independent of the factorization.
Momentum density
----------------
Before considering the momentum density, we note that all derivatives that occur in the subsequent treatment are *distributional or weak derivatives*. A function $f\in L^p(X)$, with $X\subset{\mathbb{R}}^n$ open, is said to have a weak derivative $g = \partial_\alpha f
\in L^1_{{\mathrm{loc}}}(X)$ if, for all smooth, compactly supported “test functions” $u \in
\mathcal{C}^\infty_\mathrm c(X)$, $$\int_X g(x)u(x) \mathrm dx = -\int_X f(x) \partial_\alpha u(x) \mathrm dx.$$ Thus, the weak derivative acts just like the standard derivative $\partial f/\partial x_\alpha$ when we apply integration by parts, coinciding with the classical derivative whenever this exists. Higher-order weak derivatives are defined in a similar manner. A standard monograph for weak derivatives is Ref. [@EVANS_2000].
By analogy with the density in Eq., we now define the *momentum density* ${\mathbf{c}}_\Psi$ of a state $\Psi$ as $$\begin{aligned}
{\mathbf{c}}_\Psi({\mathbf{r}}) &:= N \!\! \int_{{\mathbb{R}}^{3N-3}} \!\!\!\!\!
[-{\mathrm{i}}\nabla_{{\mathbf{r}}}\Psi({\mathbf{r}},{\mathbf{r}}_{2:N})]\Psi^\ast({\mathbf{r}},{\mathbf{r}}_{2:N})
\mathrm d{\mathbf{r}}_{2:N}
\nonumber \\
&= -{\mathrm{i}}\nabla_{\mathbf{r}}D_\Psi({\mathbf{r}},{\mathbf{s}})|_{{\mathbf{r}}={\mathbf{s}}},
\label{eq:cpsi}\end{aligned}$$ whose real part is the paramagnetic current density: $${{\mathbf{j}_{\mathrm{p}}}}_\Psi({\mathbf{r}}) = {\operatorname{Re}}{\mathbf{c}}_\Psi({\mathbf{r}})$$ with an analogous definitions for a mixed state $\Gamma$. We note, however, that this definition may not make sense without additional assumptions on the wave function $\Psi$, beyond those needed for the definition of the density. We also observe that the second equality in Eq. needs to be justified further since $\nabla_{\mathbf{r}}D_\Psi({\mathbf{r}},{\mathbf{s}})|_{{\mathbf{r}}={\mathbf{s}}}$ is only defined pointwise almost everywhere and since integration may not commute with differentiation.
To assign unambiguously a momentum density ${\mathbf{c}}_D({\mathbf{r}})$ to $D\in\mathcal{D}_{N,1}$, we first introduce the notion of a *locally finite kinetic energy*:
We say that $D\in\mathcal{D}_{N,1}$ has a locally finite kinetic energy if the weak derivative $\boldsymbol \nabla_1\cdot\boldsymbol \nabla_2 D$ is the kernel of a trace class operator over $L^2(K)$ for every compact $K\subset
{\mathbb{R}}^3$. Likewise, we say that $\Psi\in
L^2({\mathbb{R}}^{3N})$ has a locally finite kinetic energy if $\boldsymbol \nabla_1\Psi
\in L^2(K\times{\mathbb{R}}^{3N-3})$ for every compact $K\subset{\mathbb{R}}^3$—that is, $\boldsymbol \nabla_1\Psi\in L^2({\mathbb{R}}^3_{{\mathrm{loc}}}\times{\mathbb{R}}^{3N-3})$. (See Appendix \[secLOCINT\].) A mixed state $\Gamma\in \mathcal{D}_{N}$ has a locally finite kinetic energy if $\sum_i p_i
\|\boldsymbol \nabla_1\Psi_i\|^2_{L^2(K\times{\mathbb{R}}^{3N-3})}$ is finite for every compact $K\subset {\mathbb{R}}^3$.
Note that the pure-state definition of locally finite kinetic energy follows from that of the mixed state. The various definitions of a locally finite kinetic energy are connected, as summarized in the following theorem, proved in Appendix B:
\[thm:lockin\] For $D\in\mathcal{D}_{N,1}$, the following statements are equivalent:
1. $D$ has a locally finite kinetic energy.
2. There exists a factorization $D = G^\dag \! \ast G$ (a.e.) with $G$ Hilbert–Schmidt and $\nabla_2 G\in L^2({\mathbb{R}}^3\times{\mathbb{R}}^3_{{\mathrm{loc}}})$.
3. Any $\Gamma\in\mathcal{D}_{N}$ with $\Gamma\mapsto D$ has a locally finite kinetic energy.
If $D$ has a locally finite kinetic energy, then the associated *kinetic-energy density* is defined as $$\begin{aligned}
\tau_D({\mathbf{r}}) &:= \frac{1}{2}\|\nabla_2
G(\cdot,{\mathbf{r}})\|^2_{L^2({\mathbb{R}}^3)} \notag \\ &=
\frac{1}{2}\int_{{\mathbb{R}}^3} [\nabla_2
G({\mathbf{u}},{\mathbf{r}})]^*\cdot [\nabla_2G({\mathbf{u}},{\mathbf{r}})] \,\mathrm d {\mathbf{u}}\\
&=\frac{1}{2}{\operatorname{diag}}(\nabla_1\cdot\nabla_2 D)({\mathbf{r}}),\notag\end{aligned}$$ which is finite almost everywhere. From the proof of the Theorem\[thm:lockin\] in Appendix B, it follows that $\tau_D$ is in fact the kinetic energy density of any $\Gamma\mapsto D$. We also see that $\tau_D\in L^1_{{\mathrm{loc}}}({\mathbb{R}}^3)$ and that the total kinetic energy is finite if and only if $\tau_D\in L^1({\mathbb{R}}^3)$.
Finally, the following theorem (proved in Appendix B) states that, if $D$ has a locally finite kinetic energy, then the momentum density of Eq. is also well defined:
\[thm:currents\] Let $D\in \mathcal{D}_{N,1}$ have a locally finite kinetic energy and let $G\in L^2({\mathbb{R}}^3\times{\mathbb{R}}^3)$ be such that $D = G^\dag*
G$ and $\nabla_2 G \in L^2({\mathbb{R}}^3\times {\mathbb{R}}^3_\mathrm{loc})$. For each $\Gamma\in \mathcal{D}_N$ with $\Gamma\mapsto D$, it then holds that ${\mathbf{c}}_\Gamma \in
L^1_{{\mathrm{loc}}}({\mathbb{R}}^3)$ and that $$\begin{aligned}
{\mathbf{c}}_\Gamma({\mathbf{r}}) &= \left([-{\mathrm{i}}\nabla_2 G]^\dag *
G\right)({\mathbf{r}},{\mathbf{r}}) \quad\mathrm{a.e.} \notag\\
&= {\operatorname{diag}}(-{\mathrm{i}}\nabla_1 D)({\mathbf{r}}) .\end{aligned}$$
This result implies that if $D$ has locally finite kinetic energy, then ${{\mathbf{j}_{\mathrm{p}}}}_\Gamma\in L^1_{{\mathrm{loc}}}({\mathbb{R}}^3)$. Moreover, $\nabla\rho_\Gamma
= -2{\operatorname{Im}}{\mathbf{c}}_\Gamma \in L^1_{{\mathrm{loc}}}({\mathbb{R}}^3)$ as well.
Summary
-------
For easy reference, we collect the main conclusions of this section in a separate theorem:
\[thm:summary\] Let $D = G^\dag\! \ast G$ with $G\in L^2({\mathbb{R}}^3\times{\mathbb{R}}^3) \in \mathcal{D}_{N,1} $, $\nabla_2 G\in
L^2({\mathbb{R}}^3\times {\mathbb{R}}^3_{{\mathrm{loc}}})$. For every $\Gamma\in \mathcal{D}_N$ such that $\Gamma\mapsto D$, it then holds that the density $\rho_\Gamma = \rho\in L^1({\mathbb{R}}^3)$, the momentum density ${\mathbf{c}}_\Gamma = {\mathbf{c}} \in L^1_{{\mathrm{loc}}}({\mathbb{R}}^3)$, and the kinetic energy density $\tau_\Gamma = \tau\in L^1_{{\mathrm{loc}}}({\mathbb{R}}^3)$ are given almost everywhere by the expressions $$\begin{aligned}
\rho({\mathbf{r}}) &= \int_{{\mathbb{R}}^3} G^\ast ({\mathbf{u}},{\mathbf{r}})
G({\mathbf{u}},{\mathbf{s}}) \,\mathrm d {\mathbf{u}}, \\
{\mathbf{c}}({\mathbf{r}}) &= \int_{{\mathbb{R}}^3} [-{\mathrm{i}}\nabla_2
G^\ast({\mathbf{u}},{\mathbf{r}})] \,G({\mathbf{u}},{\mathbf{s}}) \,\mathrm d {\mathbf{u}},\\
\tau({\mathbf{r}}) &= \frac{1}{2} \int_{{\mathbb{R}}^3} [\nabla_2
G^\ast({\mathbf{u}},{\mathbf{r}})] \cdot [\nabla_2 G({\mathbf{u}},{\mathbf{s}})]\, \mathrm d {\mathbf{u}}.
\end{aligned}$$
A reduced density matrix for a prescribed density and paramagnetic current density {#redmat}
==================================================================================
Let a density $\rho$ be given. We assume that the density is non-negative and that it belongs to the intersection of two Lebesgue spaces, $$\rho(\mathbf{r}) \geq 0, \quad \text{and} \quad \rho \in L^1(\mathbb{R}^3) \cap L^q(\mathbb{R}^3),$$ for some $q > 1$. The latter condition amounts to $$\begin{aligned}
N := \|\rho\|_1 &= \int \! |\rho(\mathbf{r})| \,\mathrm d\mathbf{r} < +\infty, \\ \|\rho\|_q^q &= \int \! |\rho(\mathbf{r})|^q \,\mathrm d\mathbf{r} < +\infty,\end{aligned}$$ where $N$ is the number of particles in the density $\rho$. (For simplicity, we restrict ourselves to states with integral $N$ but note the 1-rdm constructions given below are valid also for fractional $N$.) Furthermore, let an arbitrary measurable vector-valued function $\boldsymbol{\kappa}:{\mathbb{R}}^3\rightarrow{\mathbb{R}}^3$ be given and let it prescribe a paramagnetic current density by the relation $${{\mathbf{j}_{\mathrm{p}}}}(\mathbf{r}) = \frac{1}{2} \rho(\mathbf{r}) \boldsymbol{\kappa}(\mathbf{r}).$$ We now consider the question: does there, for every pair of $\rho$ and ${{\mathbf{j}_{\mathrm{p}}}}$ satisfying these minimal requirements, exist a $D\in\mathcal{D}_{N,1}$ that reproduces $\rho$ and ${{\mathbf{j}_{\mathrm{p}}}}$? In short, we seek a reduced density matrix $D$ such that
1. \[itema\] $\rho(\mathbf{r}) = \rho_D({\mathbf{r}}) = ({\operatorname{diag}}D)(\mathbf{r})$
2. \[itemb\] ${{\mathbf{j}_{\mathrm{p}}}}(\mathbf{r}) = {\operatorname{Re}}{\mathbf{c}}_D({\mathbf{r}}) =
-\frac{\mathrm i}{2} ({\operatorname{diag}}\nabla_1 D)({\mathbf{r}}) + \text{c.c.}$,
assuming that $D$ has a locally finite kinetic energy for (b) to be well defined. We can indeed find such a density matrix $D\in\mathcal{D}_{N,1}$ but shall see that the condition of a locally finite kinetic energy of $D$ implies mild additional conditions on $\rho$ and $\boldsymbol{\kappa}$.
Factorized elements $P_\lambda$ and $Q_\lambda$
-----------------------------------------------
Our strategy is to construct explicitly factorized elements $P_\lambda=G^\dag_\lambda\!\ast G_\lambda$ and $Q_\mu=H^\dag_\mu\ast
H_\mu$ in $\mathcal{D}_{N,1}$ with a locally finite kinetic energy. Here, $\lambda,\mu>0$ are real parameters that allow some freedom, noting that a convex combination $D_{\lambda\mu} = (P_\lambda+Q_\mu)/2$ remains in $\mathcal{D}_{N,1}$, also with a locally finite kinetic energy. The flexibility of having several independent factorized reduced density matrices $P_\lambda$ and $Q_\mu$ allows the convex combination to reproduce the desired current.
The two terms are defined by the factorized expressions $$\begin{aligned}
\label{eq:PQ1}
P_{\lambda}(\mathbf{r},\mathbf{s}) & = \sqrt{\rho(\mathbf{r}) \rho(\mathbf{s})} \int_{{\mathbb{R}}^3} \! g^\ast(\mathbf{u},\mathbf{r}) g(\mathbf{u},\mathbf{s}) \,\mathrm d\mathbf{u}, \\
\label{eq:PQ2}
Q_{\mu}(\mathbf{r},\mathbf{s}) & = \sqrt{\rho(\mathbf{r}) \rho(\mathbf{s})} \int_{{\mathbb{R}}^3} \! h^\ast(\mathbf{u},\mathbf{r}) h(\mathbf{u},\mathbf{s}) \,\mathrm d\mathbf{u},
\end{aligned}$$ where $$\begin{aligned}
\label{eq:gh1}
g(\mathbf{u},\mathbf{v}) & = \frac{\sqrt{8} \lambda^{3/4}}{\pi^{3/4}} \mathrm e^{-\mathrm i\mathbf{v}\cdot\boldsymbol{\kappa}(\mathbf{v})} \mathrm e^{-2\lambda (\mathbf{u}-\mathbf{v})^2}, \\
\label{eq:gh2}
h(\mathbf{u},\mathbf{v}) & = \frac{\sqrt{8} \mu^{3/4}}{\pi^{3/4}} \mathrm e^{\mathrm i\mathbf{u}\cdot\boldsymbol{\kappa}(\mathbf{v})} \mathrm e^{-2\mu (\mathbf{u}-\mathbf{v})^2}.
\end{aligned}$$ Clearly, these operators may be written in the form $$\begin{aligned}
{2}
\label{eq:GH1}
P_\lambda &= G_\lambda^\dag \!\ast G_\lambda, &\quad
G_\lambda({\mathbf{r}},{\mathbf{s}}) &= g({\mathbf{r}},{\mathbf{s}})\sqrt{\rho({\mathbf{s}})}, \\
\label{eq:GH2}
Q_\mu &= H_\mu^\dag \ast H_\mu &
H_\mu({\mathbf{r}},{\mathbf{s}}) &= h({\mathbf{r}},{\mathbf{s}})\sqrt{\rho({\mathbf{s}})},
\end{aligned}$$ It is straightforward to verify that $G_\lambda,H_\lambda\in L^2({\mathbb{R}}^3\times{\mathbb{R}}^3)$.
The integration over $\mathbf{u}$ may be performed analytically, yielding the alternative expressions $$\begin{aligned}
P_{\lambda}(\mathbf{r},\mathbf{s}) & = \sqrt{\rho(\mathbf{r}) \rho(\mathbf{s})} \mathrm e^{-\lambda |\mathbf{r}-\mathbf{s}|^2} \mathrm e^{\mathrm i(\mathbf{r}\cdot\boldsymbol{\kappa}(\mathbf{r})-\mathbf{s}\cdot\boldsymbol{\kappa}(\mathbf{s}))}, \label{eq:expanded1}\\
Q_{\mu}(\mathbf{r},\mathbf{s}) & = \sqrt{\rho(\mathbf{r})
\rho(\mathbf{s})} \mathrm e^{-\mu|\mathbf{r}-\mathbf{s}|^2} \notag\\ &\quad
\times \mathrm e^{-\tfrac{\mathrm i}{2} (\mathbf{r}+\mathbf{s})\cdot(\boldsymbol{\kappa}(\mathbf{r})-\boldsymbol{\kappa}(\mathbf{s}))-|\boldsymbol{\kappa}(\mathbf{r})-\boldsymbol{\kappa}(\mathbf{s})|^2/16\mu}.\label{eq:expanded2}
\end{aligned}$$ These operators were found by making the initial ansatz $\phi(\mathbf{r}) = \sqrt{\rho(\mathbf{r})}
\mathrm e^{\mathrm i\mathbf{r}\cdot\boldsymbol{\kappa}(\mathbf{r})}$ for an unnormalized natural orbital. The corresponding paramagnetic current is then almost correct but contains an extra term that is most easily canceled if the density matrix contains exponential factors of the form $\mathrm e^{\mathrm i\mathbf{r}\cdot\boldsymbol{\kappa}(\mathbf{s})}$. Since the elements of $\mathcal{D}_{N,1}$ and their properties are conveniently described if an explicit factorization is available (see Theorem \[thm:summary\]), Gaussian kernels are suitable since since they allow mixed phase factors of the type $\mathrm e^{\mathrm i\mathbf{r}\cdot\boldsymbol{\kappa}(\mathbf{s})}$ to survive the integration.
The density of $P_\lambda$ and $Q_\lambda$
------------------------------------------
We now need to verify that $P_\lambda$ and $Q_\mu$ are elements of $\mathcal{D}_{N,1}$ by checking points (1)–(5) of Theorem \[thm:DN1properties\].
Let $\rho\in L^1({\mathbb{R}}^3)\cap L^q({\mathbb{R}}^3)$ for some $q > 1$, $\rho\geq
0$ a.e., $\|\rho\|_1=N$, and let $\lambda,\mu\in{\mathbb{R}}$ be such that $$\label{eqLAMBDAMUCOND}
\lambda,\mu \geq \frac{2p}{\pi} ( \tfrac{1}{4} N \|\rho\|_q )^{2p/3}$$ where $1/p + 1/q = 1$. Then $P_\lambda$ and $Q_\mu$ in Eqs. – are elements of $\mathcal{D}_{N,1}$, with $$\rho_{P_\lambda}({\mathbf{r}}) = \rho_{Q_\mu}({\mathbf{r}}) = \rho({\mathbf{r}})$$ almost everywhere. The same is true for any convex combination $\theta P_\lambda + (1-\theta)Q_\mu \in
\mathcal{D}_{N,1}$ with $\theta\in[0,1]$.
Both operators are Hermitian and positive semidefinite. From the expressions in Eqs. and , $({\operatorname{diag}}P_\lambda)({\mathbf{r}}) = ({\operatorname{diag}}Q_\lambda)({\mathbf{r}}) = \rho({\mathbf{r}})$ almost everywhere. It follows that ${\operatorname{Tr}}P_\lambda = {\operatorname{Tr}}Q_\lambda
= \int \! \rho({\mathbf{r}}) \mathrm d{\mathbf{r}} = N$.
It remains to compute a bound on the largest eigenvalues, demonstrating point (4) of Theorem \[thm:DN1properties\] for the corresponding parameter values $\lambda$ and $\mu$. For an arbitrary normalized orbital, $$\begin{aligned}
n^2 & \leq \left| \int \!\!\phi^\ast(\mathbf{r}) P_{\lambda}(\mathbf{r},\mathbf{s}) \phi(\mathbf{s}) \,\mathrm d\mathbf{r} \mathrm d\mathbf{s} \right|^2 \notag\\
& \leq \left( \int \! |\phi(\mathbf{r})| \sqrt{\rho(\mathbf{r}) \rho(\mathbf{s})} \mathrm e^{-\lambda |\mathbf{r}-\mathbf{s}|^2} |\phi(\mathbf{s})| \, \mathrm d\mathbf{r} \, \mathrm d\mathbf{s} \right)^2 \\
& = \left( \int \! |\phi(\mathbf{r})| \sqrt{\rho(\mathbf{r})} \left( \int \! \sqrt{\rho(\mathbf{s})} \mathrm e^{-\lambda |\mathbf{r}-\mathbf{s}|^2} |\phi(\mathbf{s})| \mathrm d\mathbf{s} \right) \! \mathrm d\mathbf{r} \right)^2.\notag\end{aligned}$$ Given that $\phi, \sqrt{\rho} \in L^2(\mathbb{R}^3)$, the Cauchy–Schwarz inequality may be applied twice to give $$\begin{aligned}
n^2 & \leq \left( \int \! |\phi(\mathbf{r}')|^2 \mathrm d\mathbf{r}' \right) \times \nonumber
\\ &\quad \times \left( \int \! \rho(\mathbf{r}) \left( \int \!\! \sqrt{\rho(\mathbf{s})} \mathrm e^{-\lambda |\mathbf{r}-\mathbf{s}|^2} |\phi(\mathbf{s})| \mathrm d\mathbf{s} \right)^2 \! \mathrm d\mathbf{r} \! \right)
\nonumber \\
& \leq \int \rho(\mathbf{r}) \left( \int |\phi(\mathbf{s}')|^2 \mathrm d\mathbf{s}' \int \!\rho(\mathbf{s}) \mathrm e^{-2\lambda |\mathbf{r}-\mathbf{s}|^2} \mathrm d\mathbf{s} \right) \! \mathrm d\mathbf{r}
\nonumber \\
& \leq \int \! \rho(\mathbf{r}) \left( \sup_{\mathbf{c}} \int \! \rho(\mathbf{s}) \mathrm e^{-2\lambda |\mathbf{c}-\mathbf{s}|^2} \mathrm d\mathbf{s} \right) \mathrm d\mathbf{r}
\nonumber \\
& = N \sup_{\mathbf{c}} \int \! \rho(\mathbf{s}) \mathrm e^{-2\lambda |\mathbf{c}-\mathbf{s}|^2} \mathrm d\mathbf{s}\end{aligned}$$ Finally, exploiting the fact that $\rho \in L^q(\mathbb{R}^3)$, the integral over $\mathbf{s}$ may be bounded by invoking the Hölder inequality, $$\!\! n^2 \leq N \|\rho\|_q \sup_{\mathbf{c}} \|\mathrm e^{-2\lambda |\mathbf{c}-\mathbf{s}|^2}\|_p = N \|\rho\|_q \left( \frac{\pi}{2p\lambda} \right)^{3/2p}$$ where $1/p + 1/q = 1$. This bound is independent of the current density. Hence, $P_{\lambda}$ has no eigenvalues greater than two if $$\begin{split}
\label{eqLAMBDACONDITION}
\lambda \geq \frac{2p}{\pi} \left(\tfrac{1}{4} N \|\rho\|_q \right)^{2p/3}.
\end{split}$$ These steps hold also for $Q_{\mu}$, showing that it has no eigenvalues greater than 2 when $\mu \geq \frac{2p}{\pi}
(\tfrac{1}{4} N \|\rho\|_q )^{2p/3}$.
Finally, consider a convex combination $D_\theta = \theta P_\lambda +
(1-\theta)Q_\mu$, which belongs to $\mathcal{D}_{N,1}$ since this set convex. Moreover, ${\operatorname{diag}}A$ is linear in $A$ since ${\operatorname{diag}}(A+B)({\mathbf{r}}) =
{\operatorname{diag}}(A)({\mathbf{r}}) + {\operatorname{diag}}(B)({\mathbf{r}})$ almost everywhere. Therefore, ${\operatorname{diag}}D_\theta =
\theta {\operatorname{diag}}P_\lambda + (1-\theta){\operatorname{diag}}Q_\lambda = \rho$ almost everywhere.
The canonical kinetic energy of $P_\lambda$ and $Q_\lambda$ {#cankin}
-----------------------------------------------------------
We now turn to the question of whether the current ${{\mathbf{j}_{\mathrm{p}}}}$ can be reproduced by $D$. Indeed, a formal calculation shows that $$\begin{aligned}
-\frac{\mathrm i}{2} \frac{\partial}{\partial r_{\alpha}}
& P_{\lambda}(\mathbf{r},\mathbf{s})) \big|_{\mathbf{s}=\mathbf{r}} +
\text{c.c.} = \rho(\mathbf{r}) \!\left( \!\kappa_{\alpha}(\mathbf{r})
+ \mathbf{r}\cdot \frac{\partial
\boldsymbol{\kappa}(\mathbf{r})}{\partial r_{\alpha}} \right)
\notag \\&= 2{j_{\mathrm{p};{\alpha}}}(\mathbf{r}) + \rho(\mathbf{r})\
\mathbf{r}\cdot \frac{\partial
\boldsymbol{\kappa}(\mathbf{r})}{\partial r_{\alpha}} \end{aligned}$$ and $$\begin{aligned}
-\frac{\mathrm i}{2} \frac{\partial}{\partial r_{\alpha}}
Q_{\mu}(\mathbf{r},\mathbf{s})) \big|_{\mathbf{s}=\mathbf{r}} +
\text{c.c.} & = -\rho(\mathbf{r}) \ \mathbf{r}\cdot\frac{\partial
\boldsymbol{\kappa}(\mathbf{r})}{\partial r_{\alpha}}. \label{eq:formalcalculation}\end{aligned}$$ Thus, we expect ${{\mathbf{j}_{\mathrm{p}}}}_{D_{\lambda\mu}} =
\frac{1}{2}{{\mathbf{j}_{\mathrm{p}}}}_{P_\lambda}({\mathbf{r}}) +
\frac{1}{2}{{\mathbf{j}_{\mathrm{p}}}}_{Q_\mu}({\mathbf{r}}) = {{\mathbf{j}_{\mathrm{p}}}}({\mathbf{r}})$ to hold almost everywhere. To prove this result, it suffices to find conditions on $\rho$ and $\kappa$ such that $P_\lambda$ and $Q_\mu$ have a locally finite kinetic energy. The kinetic energy density of $P_\lambda$ is $$\begin{split}
\tau_P(\mathbf{r}) & =
\frac{1}{2}\|\nabla_2 G_\lambda(\cdot,{\mathbf{r}})\|_{L^2({\mathbb{R}}^3)}^2 = \frac{1}{2} \nabla_{\mathbf{r}}\cdot \nabla_{\mathbf{s}} P_{\lambda}(\mathbf{r},\mathbf{s}) \big|_{\mathbf{s}=\mathbf{r}} \\
& = \frac{|\nabla \rho(\mathbf{r})|^2}{8 \rho(\mathbf{r})} + \frac{1}{2} |\nabla (\mathbf{r}\cdot\boldsymbol{\kappa}(\mathbf{r}))|^2 \rho(\mathbf{r}) + \lambda \rho(\mathbf{r}).
\end{split}$$ Here, we have used the fact that the integral in $\|\nabla_2
G(\cdot,{\mathbf{r}})\|^2_{L^2({\mathbb{R}}^3)}$ can be performed analytically, so that the evaluation at ${\mathbf{s}}={\mathbf{r}}$ after the second equality is in fact well-defined. Similarly,
$$\tau_Q(\mathbf{r}) = \frac{1}{2} \nabla_{\mathbf{r}}\cdot \nabla_{\mathbf{s}} Q_{\mu}(\mathbf{r},\mathbf{s}) \big|_{\mathbf{s}=\mathbf{r}}
= \frac{|\nabla \rho(\mathbf{r})|^2}{8 \rho(\mathbf{r})} \nonumber +
\frac{1}{2} \left( \sum_{\alpha=1}^3 \left( \sum_{\beta=1}^3
r_{\beta} \frac{\partial \kappa_{\beta}(\mathbf{r})}{\partial
r_{\alpha}} \right)^2 + \tfrac{1}{8\mu} \sum_{\beta=1}^3
|\nabla \kappa_{\beta}(\mathbf{r})|^2 \right) \rho(\mathbf{r})
+ \mu \rho(\mathbf{r}).$$
The total kinetic energy density becomes $$\begin{split}
\tau_D(\mathbf{r}) & = \frac{|\nabla \rho(\mathbf{r})|^2}{8 \rho(\mathbf{r})} + \frac{1}{4} \sum_{\alpha=1}^3 \left[ \left(\frac{\partial}{\partial r_{\alpha}} \mathbf{r}\cdot\boldsymbol{\kappa}(\mathbf{r})\right)^2 + \left( \mathbf{r}\cdot \frac{\partial \boldsymbol{\kappa}(\mathbf{r})}{\partial r_{\alpha}} \right)^2 + \tfrac{1}{8\mu} \sum_{\beta=1}^3 \left( \frac{\partial \kappa_{\beta}(\mathbf{r})}{\partial r_{\alpha}} \right)^2 \right] \rho(\mathbf{r})
+ \frac{1}{2} (\lambda+\mu) \rho(\mathbf{r}),
\end{split}$$
where we have used the fact that the kinetic energy density is linear in the density matrix. Using the special form $2|ab|\leq a^2+b^2$ of Young’s inequality, with $a = \kappa_{\alpha}(\mathbf{r})$ and $b=\mathbf{r} \cdot \partial \boldsymbol{\kappa}(\mathbf{r})/\partial r_{\alpha}$, we find that $$\begin{aligned}
\left| \frac{\partial}{\partial r_{\alpha}}
\mathbf{r}\cdot\boldsymbol{\kappa}(\mathbf{r}) \right|^2 & =
\left|\kappa_{\alpha}(\mathbf{r}) + \mathbf{r} \cdot \frac{\partial
\boldsymbol{\kappa}(\mathbf{r})}{\partial r_{\alpha}} \right|^2
\notag \\ & \leq 2 |\kappa_{\alpha}(\mathbf{r})|^2 + 2 \left|\mathbf{r} \cdot \frac{\partial \boldsymbol{\kappa}(\mathbf{r})}{\partial r_{\alpha}} \right|^2.\end{aligned}$$ Hence, the canonical kinetic energy density is bounded by
$$\begin{split}
\tau_D(\mathbf{r}) & \leq \frac{|\nabla \rho(\mathbf{r})|^2}{8 \rho(\mathbf{r})} + \frac{1}{2} \left[ |\boldsymbol{\kappa}(\mathbf{r})|^2 + \tfrac{3}{2} \left( \mathbf{r}\cdot \frac{\partial \boldsymbol{\kappa}(\mathbf{r})}{\partial r_{\alpha}} \right)^2 + \tfrac{1}{16\mu} \sum_{\beta=1}^3 \left( \frac{\partial \kappa_{\beta}(\mathbf{r})}{\partial r_{\alpha}} \right)^2 \right] \rho(\mathbf{r})
+ \frac{1}{2} (\lambda+\mu) \rho(\mathbf{r}) \\
& \leq \frac{|\nabla \rho(\mathbf{r})|^2}{8 \rho(\mathbf{r})} + \frac{1}{2} \left[ |\boldsymbol{\kappa}(\mathbf{r})|^2 + (\tfrac{3}{2} r^2 + \tfrac{1}{16\mu}) \sum_{\alpha,\beta=1}^3 \left( \frac{\partial \kappa_{\beta}(\mathbf{r})}{\partial r_{\alpha}} \right)^2 \right] \rho(\mathbf{r})
+ \frac{1}{2} (\lambda+\mu) \rho(\mathbf{r}),
\end{split}$$
where the second inequality was obtained by using $|r_{\beta}| \leq
|\mathbf{r}|$. A finite canonical kinetic energy of $P_\lambda$, $Q_\mu$ and $D_{\lambda\mu}$ is ensured if $$\begin{aligned}
\label{eqFINTW}
T_W[\rho] & = \int \! \frac{|\nabla\rho|^2}{8\rho} \,\mathrm d\mathbf{r} =
\frac{1}{2}\int \! |\nabla\sqrt{\rho}|^2 \,\mathrm d{\mathbf{r}} < \infty, \\
\label{eqFINTp}
T_{\mathrm{p}}[\rho,{{\mathbf{j}_{\mathrm{p}}}}] & = \int \! \frac{|{{\mathbf{j}_{\mathrm{p}}}}|^2}{2\rho} \,\mathrm d\mathbf{r} = \frac{1}{8} \int \! \rho \kappa^2 \,\mathrm d\mathbf{r} < \infty, \\
\label{eqFINTab}
T_{\alpha\beta}[\rho,{{\mathbf{j}_{\mathrm{p}}}}] & = \int \! (1+r^2) \rho \left( \frac{\partial \kappa_{\beta}(\mathbf{r})}{\partial r_{\alpha}} \right)^2 \mathrm d\mathbf{r} < \infty.\end{aligned}$$ We remark that, by the definition of the vorticity, $\boldsymbol{\nu} = \nabla\times\rho^{-1} {{\mathbf{j}_{\mathrm{p}}}}=
\frac{1}{2} \nabla\times\boldsymbol{\kappa}$, a consequence of the last condition is that $$\int \! (1+r^2) \rho \nu^2 \mathrm d\mathbf{r} < \infty.$$ We have thus proved the following result:
\[thm:globalrep\] Let $\rho$ and $\boldsymbol{\kappa}$ be given such that $\rho\geq
0$, $\sqrt{\rho} \in H^1(\mathbb{R}^3)$, $\rho \kappa_{\alpha}^2 \in
L^1(\mathbb{R}^3)$, $(1+r^2) \rho (\partial\kappa_{\beta}/\partial
r_{\alpha})^2 \in L^1(\mathbb{R}^3)$ for all Cartesian components $\alpha,\beta\in\{1,2,3\}$. Then there exist real constants $\lambda,\mu\geq 0$ and a 1-rdm $D$ with density $\rho$ and current ${{\mathbf{j}_{\mathrm{p}}}}= \frac{1}{2} \rho \boldsymbol{\kappa}$ such that the canonical kinetic energy is bounded by $$\begin{aligned}
\frac{1}{2} &{\operatorname{Tr}}(\boldsymbol \nabla_1 \cdot\boldsymbol \nabla_2 D ) \leq T_W[\rho] + 4
T_{\mathrm{p}}\left[\rho, \tfrac{1}{2} \rho\kappa\right] + \frac{1}{2}
(\mu+\nu)N \notag\\
& \quad + \int \! \!\rho(\mathbf{r}) \left(\tfrac{3}{4} r^2 + \tfrac{1}{32\mu}\right) \!\sum_{\alpha,\beta=1}^3 \!\left( \frac{\partial \kappa_{\beta}(\mathbf{r})}{\partial r_{\alpha}} \right)^2 \!\mathrm d\mathbf{r}.
\end{aligned}$$
Note that, by a Sobolev inequality, $\sqrt{\rho}\in H^1({\mathbb{R}}^3)$ implies that $\rho\in L^q({\mathbb{R}}^3)$ for all $q\in [1,3]$.
Lifting the global integrability condition
------------------------------------------
Theorem \[thm:globalrep\] may be strengthened by replacing the global integrability conditions on the total kinetic energy by local integrability conditions, replacing integrals ${\mathbb{R}}^3$ by integrals over arbitrary compact sets $K\subset{\mathbb{R}}^3$. A larger class of $\rho$ and $\boldsymbol{\kappa}$ are then seen to be reproducible, albeit with merely a locally finite kinetic energy.
Let $\rho$ and $\boldsymbol{\kappa}$ be given such that $\rho\geq
0$, $\rho \in L^1(\mathbb{R}^3)\cap L^q({\mathbb{R}}^3)$, $q>1$, $\rho^{-1}
|\boldsymbol \nabla\rho|^2 \in L^1_{{{\mathrm{loc}}}}(\mathbb{R}^3)$, $\rho
\kappa_{\alpha}^2 \in L^1_{{{\mathrm{loc}}}}(\mathbb{R}^3)$, $(1+r^2) \rho
(\partial\kappa_{\beta}/\partial r_{\alpha})^2 \in
L^1_{{{\mathrm{loc}}}}(\mathbb{R}^3)$ for all Cartesian components $\alpha,\beta\in\{1,2,3\}$. Then there exist a 1-rdm $D$ with density $\rho$ and current ${{\mathbf{j}_{\mathrm{p}}}}= \frac{1}{2} \rho
\boldsymbol{\kappa}$.
Discussion
==========
We have provided an explicit construction of a 1-rdm that reproduces a prescribed density and paramagnetic current density. This type of $N$-representability problem arises in Kohn–Sham CDFT, as it is known that not all current densities can be represented by a single Kohn–Sham orbital. Lieb and Schrader have recently proved [@LIEB_SCHRADER_2013], under some additional assumptions, that there also exist current densities that cannot be represented by two Kohn–Sham orbitals. The question is open for three orbitals. For four or more orbitals, Lieb and Schrader provide an explicit Slater determinant that reproduces any density and paramagnetic current that satisfy mild regularity conditions. Our results are complementary in that we establish that an *extended* Kohn–Sham approach, where fractional occupation numbers are allowed even if there is an integral total number of electrons, is flexible enough to represent any density and paramagnetic current density, under minimal regularity assumptions (finite $T_{\mathrm{W}}$, $T_{\mathrm{p}}$, and $T_{\alpha\beta}$).
The generalization from finite total canonical kinetic energy to finite local canonical kinetic energy is of some value in light of gauge freedom. The kinetic energy $T_{\mathrm{p}}[\rho,{{\mathbf{j}_{\mathrm{p}}}}]$ is not gauge invariant; on the contrary, it can made to become infinite by applying a gauge transformation ${{\mathbf{j}_{\mathrm{p}}}}\mapsto {{\mathbf{j}_{\mathrm{p}}}}+ \rho \nabla \chi$ with a rapidly growing gauge function $\chi$. Our results establish that such gauge transformations do not affect $N$-representability, as long as $\chi$ exhibits some minimal regularity.
The explicit constructions of density matrices can be used to provide orbital-free upper bounds on the canonical kinetic energy $T_s[\rho,{{\mathbf{j}_{\mathrm{p}}}}]$ for an extended Kohn–Sham formalism. Combining the above results with the standard lower bound $T_W + T_{\mathrm{p}}$ on the kinetic energy, we get the following orbital-free bounds on the extended Kohn–Sham kinetic energy, $$\begin{aligned}
T_W + T_{\mathrm{p}} &\leq \bar{T}[\rho,{{\mathbf{j}_{\mathrm{p}}}}] \notag \\
& \leq T_W + 4 T_{\mathrm{p}} + (\lambda+\mu) N \notag \\
& \quad + \int \! \rho(\mathbf{r}) \left(\tfrac{3}{2} r^2 + \tfrac{1}{16\mu}\right) \left|\nabla_{\alpha} \kappa_{\beta}(\mathbf{r})\right|^2 \mathrm d\mathbf{r}.\end{aligned}$$
Noting that several authors, following Vignale and Rasolt [@VIGNALE_PRB37_10685], have discussed CDFT formulations in terms of spin-resolved densities $(\rho_{\uparrow},\rho_{\downarrow},{\mathbf{j}_{\mathrm{p};{\uparrow}}},{\mathbf{j}_{\mathrm{p};{\downarrow}}})$, we also remark that our 1-rdm construction is easily modified for spin-resolved 1-rdms $D^{\uparrow\uparrow}$ and $D^{\downarrow\downarrow}$. The eigenvectors then correspond to natural spin-orbitals with eigenvalues bounded by one rather than by two as in the case of natural spatial orbitals. The modifications to the above presentation are trivial—condition \[item:prop4\] in Theorem \[thm:DN1properties\] becomes that no occupation is larger than 1, and the factors $\tfrac{1}{4}$ consequently disappear from Eqs. and .
The authors would like to thank E.H. Lieb and R. Schrader for giving one of us (S. Kvaal) early access to their manuscript of Ref. [@LIEB_SCHRADER_2013] and for interesting discussions, which spurred the completion of the present work.
This work was supported by the Norwegian Research Council through the CoE Centre for Theoretical and Computational Chemistry (CTCC) Grant No. 179568/V30 and the Grant No. 171185/V30 and through the European Research Council under the European Union Seventh Framework Program through the Advanced Grant ABACUS, ERC Grant Agreement No. 267683.
The regular representation of trace-class operators
===================================================
The density matrix $D({\mathbf{r}},{\mathbf{s}})$ is an element of $L^2({\mathbb{R}}^3\times {\mathbb{R}}^3)$ and also the kernel of a trace-class operator over $L^2({\mathbb{R}}^3)$. As such, it is not pointwise defined everywhere. At the same time, we wish to make sense of “the diagonal $D({\mathbf{r}},{\mathbf{r}})$” in order to define the density in an unambiguous manner.
Brislawn [@BRISLAWN_1988] has presented a thorough study of trace-class operators and their kernels. The basic tools are found in this reference, but we restate some results for a self-contained treatment. We begin by clarifying some points concerning Lebesgue spaces that are often glossed over but are important here.
Lebesgue spaces
---------------
Let $X\subset{\mathbb{R}}^n$ be an open set. The $L^p(X)$ norm of a measurable function $f : X\rightarrow
{\mathbb{C}}$ is defined by $$\|f\|_p := \left( \int_X |f(x)|^p \,\mathrm dx \right)^{1/p}.$$ The vector space $\mathcal{L}^p(X)$ consists of all functions $f$ such that $\|f\|_p<+\infty$. The space $\mathcal{L}^p(X)$ is not a normed space, since $\|f\|_p = 0$ if and only if $f(x) = 0$ for almost all $x\in
X$ (rather than for all $x \in X$). On the other hand, the set $L^p(X)$ consisting of all *equivalence classes* $[f] =
\{ g\in\mathcal{L}^p(X) : \|f-g\|_p = 0 \}$ is a normed space. It is customary to speak of a function $f$ as an element of $L^p(X)$ even though, strictly speaking, it is a *representative* of $[f]\in L^p(X)$.
This distinction between $f$ and $[f]$ is not merely academic: two pointwise defined wave functions $\Psi$ and $\Phi$ describe the same physical state if and only if $\|\Psi
-\Phi\|_2 = 0$. Thus, $[\Psi]\in L^2({\mathbb{R}}^{3N})$ is the wave function. Similarly, a reduced density matrix $D\in
L^2({\mathbb{R}}^3\times{\mathbb{R}}^3)$ is not defined pointwise: its formal diagonal $D({\mathbf{r}},{\mathbf{r}})$ may therefore be redefined without changing the physics. If $[\Psi] = [\Phi]$, then $D_\Psi = D_\Phi$ almost everywhere, but if $D$ is *given* there is no *a priori* way to know how the pointwise values $D({\mathbf{r}},{\mathbf{s}})$ are affected by modifying the wave function on a set of zero measure.
Locally integrable functions {#secLOCINT}
----------------------------
A function $f\in L^p_{{\mathrm{loc}}}({\mathbb{R}}^n)$ if and only if $f\in L^p(K)$ for every compact measurable $K\subset{\mathbb{R}}^n$. We furthermore have $L^{q}_{{\mathrm{loc}}}\subset
L^p_{{\mathrm{loc}}}$ for $q\geq p$, and $$L^p({\mathbb{R}}^n)\subset L^p_{{\mathrm{loc}}}({\mathbb{R}}^n)
\subset L^1_{{\mathrm{loc}}}({\mathbb{R}}^n).\label{eq:localspaces}$$ Clearly, $L^1_{{\mathrm{loc}}}$ is a large class of functions and functions in $L^1_{{\mathrm{loc}}}$ are said to be “locally integrable”. We also need a slightly more general notion of local integrability as follows:
Let $X\subset{\mathbb{R}}^n$, $Y\subset{\mathbb{R}}^m$ be open sets. The set $L^p(X_{{\mathrm{loc}}}\times Y)$ is the set of (equivalence classes of) all measurable functions $u : X\times Y\rightarrow {\mathbb{C}}$ such that for all compact measurable $K\subset X$, $u\in L^p(K\times Y)$. A similar definition is made for arbitrary products and positions of the subscript “${{\mathrm{loc}}}$”. In particular, $L^p_{{\mathrm{loc}}}(X) =
L^p(X_{{\mathrm{loc}}})$.
The regular representation
--------------------------
The goal of this section is to establish a unique representative $\tilde{f}$ of $[f]\in L^1_{{\mathrm{loc}}}$, called the regular representative of $f$. This representative will aid in defining the diagonal of $D\in
\mathcal{D}_{N,1}$. The first step is to introduce the local averaging operator $A_\epsilon$:
Let $f\in L^1_{{\mathrm{loc}}}({\mathbb{R}}^n)$ and $\epsilon>0$. For a box $C_\epsilon = [-\epsilon,\epsilon]^n$ of Lebesgue measure $|C_\epsilon| = (2\epsilon)^n$, the (linear) local averaging operator $A_\epsilon : L^1_{{\mathrm{loc}}}\rightarrow L^1_{{\mathrm{loc}}}$ is defined by $$A_\epsilon f(x) := \frac{1}{|C_\epsilon|} \int_{C_\epsilon} \! f(x+y) \,\mathrm dy. \label{Aeps}$$
Since $C_\epsilon$ is compact, $A_\epsilon f(x)$ is everywhere finite and is independent of the particular representative $f$ of $[f]$ that appears in the integrand. It can be shown that $A_\epsilon f(x)$ is continuous both in $x$ and in $\epsilon>0$ [@STEIN_1970]. We are here interested in the limit $\epsilon\rightarrow 0$ and therefore invoke the Lebesgue differentiation theorem:
\[thm:lebesguediff\] Let $f\in L^1_{{\mathrm{loc}}}({\mathbb{R}}^n)$. Then for almost all $x\in {\mathbb{R}}^n$, $$\lim_{\epsilon\rightarrow 0} A_\epsilon f(x) = f(x). \label{eq:lebesguediff}$$
See Ref. [@STEIN_1970]
Since $A_\epsilon f(x)$ is independent of the particular $f\in[f]$, this limit determines a unique representative:
The regular representative $\tilde{f}$ of $f \in L^1_{{\mathrm{loc}}}({\mathbb{R}}^n)$ is defined by $$\tilde{f}(x) := \lim_{\epsilon\rightarrow 0} A_\epsilon f(x)$$ whenever the limit in Eq. exists.
Since $\tilde{f}(x)=f(x)$ almost everywhere, $\tilde{f}$ and $f$ represent the same element $[f] \in L^1_{{\mathrm{loc}}}$. Moreover, it is easy to see that $\tilde{f}$ is independent of the starting representative $f$ and that the set of zero measure (where $\tilde{f}$ is undefined) is uniquely given by $[f]\in
L^1_{{\mathrm{loc}}}$. Intuitively, $\tilde{f}$ is more regular than $f$, “smoothing out” unnecessary discontinuities, and so on.
Related to the regular representative is the Hardy–Littlewood maximal function and associated inequality:
For $f \in L^1_{{\mathrm{loc}}}({\mathbb{R}}^n)$ and $C_\epsilon =
[-\epsilon,\epsilon]^n$ of Lebesgue measure $|C_\epsilon| =
(2\epsilon)^n$, the Hardy–Littlewood maximal function $Mf$ is defined by $$Mf(x) := \sup_{\epsilon>0}
\frac{1}{|C_\epsilon|} \int_{C_\epsilon} \! |f(x+y)|\, \mathrm d y.$$
The following theorem is also called the Maximal Theorem:
If $f\in L^p({\mathbb{R}}^n)$, then $Mf(x)$ is finite almost everywhere. Moreover, there exists a constant $C_p$ (independent of $f$ and $n$) such that $$\|Mf\|_p \leq C_p \|f\|_p.$$
See Ref. [@STEIN_1970].
Since $|A_\epsilon f(x)| \leq Mf(x)$ for all $x$, we obtain as a corollary that $A_\epsilon$ is a bounded linear operator from $L^p$ to $L^p$. Using this fact, it is straightforward to show that $A_\epsilon$ not only smoothes $f$, but also the mode of convergence:
Suppose $f_n\rightarrow f$ in $L^p(X)$. For all $\epsilon>0$, $A_\epsilon f_n\rightarrow A_\epsilon f$ uniformly (i.e., in $L^\infty(X)$).
We show that, for every $\epsilon>0$, there exists a constant $K(\epsilon)$ such that, for all $f\in L^p$, $$\|A_\epsilon f\|_\infty \leq K(\epsilon)\|f\|_p.$$ We have $$\begin{aligned}
|A_\epsilon f(x)| \leq \frac{1}{|C_\epsilon|} \|f\|_{L^1(x + C_\epsilon)}.
\end{aligned}$$ Since $C_\epsilon$ is bounded in ${\mathbb{R}}^n$, $$\int_{x + C_\epsilon} \!\!\!\!\!\!1\times |g(x)| \,\mathrm d x \leq
|C_\epsilon|^{1/q}\|g\|_{L^p(x + C_\epsilon)}$$ where $1/q + 1/p = 1$. Thus, $$|A_\epsilon f(x)| \leq |C_\epsilon|^{1/q-1}
\|f\|_{L^p({\mathbb{R}}^n)},$$ independent of $x$.
The diagonal of a factorized kernel
-----------------------------------
Based on our intuition, we may now hypothesize that, given an arbitrary reduced density matrix $D({\mathbf{r}},{\mathbf{s}})\in\mathcal{D}_{N,1}$, the diagonal of $\tilde{D}$ is the proper definition of the density: $$\rho({\mathbf{r}}) = \tilde{D}({\mathbf{r}},{\mathbf{r}}).$$ This is indeed true, as we shall show. To this end, a slight reformulation and generalization of Theorem 3.5 in [@BRISLAWN_1988] is useful for us. The reformulation states that, if an operator kernel is factorized, then the diagonal of the regular representative is given by the diagonal of the factorization, almost everywhere. The proof carries over with only trivial modifications, but since it is important, we rephrase it here.
\[thm:factorization\] Let $(X,\mathrm dx)$ and $(Y,\mathrm dy)$ be open subsets of Euclidean spaces equipped with the standard Lebesgue measures. For $P\in L^2(X\times Y)$ and $Q\in L^2(Y\times X)$, let $C : X\times X\rightarrow {\mathbb{C}}$ be given by $$C(x,x') = (P\ast Q)(x,x') = \int_Y \!\! P(x,y)Q(y,x') \, \mathrm dy.$$ Then $C\in L^2(X\times X)$ (a pointwise representative) and $$\tilde{C}(x,x) = C(x,x)$$ for almost all $x\in X$. Moreover, the map $x \mapsto C(x,x) = (P\ast Q)(x,x)$ belongs to $L^1(X)$.
We now demonstrate that $C\in L^2(X\times X)$. For almost all $x\in X$ and for almost all $x'\in
X$, it holds that $P(x,\cdot),Q(\cdot,x') \in L^2(Y)$. From the Cauchy–Schwarz inequality, we obtain $$\begin{aligned}
\left|C(x,x')\right| &\leq \int \!
\left|P(x,y)\right|\left|Q(y,x')\right|\,\mathrm d y \nonumber \\
&\leq \|P(x,\cdot)\|_{L^2(Y)}\|Q(\cdot,x')\|_{L^2(Y)} < +\infty
\end{aligned}$$ for almost all $x$ and almost all $x'$ and hence also for almost all $(x,x^\prime)\in X\times X$. Squaring and integrating, we obtain $\|C\|_{L^2(X\times X)}^2 \leq
\|P\|^2_{L^2(X\times Y)} \|Q\|^2_{L^2(Y\times X)}<+\infty$.
Next, we demonstrate that the diagonal is in $L^1(X\times X)$. For $\epsilon>0$, let $A_{\epsilon,i} P(x,y)$ be the averaging operator acting on the $i$th argument and let $M_i P(x,y)$ be the maximal operator acting on the $i$th argument. For almost all $x,x',y$, we then obtain $$|A_{\epsilon,1} P(x,y) A_{\epsilon,2} Q(y,x')| \leq M_1 P(x,y) M_2
Q(y,x').
\label{eq:bound1}$$ By the Cauchy–Schwarz inequality, we obtain $$\begin{aligned}
\int \! & \left|M_1 P(x,y) M_2 Q(y,x')\right|^2 \mathrm d y\leq \notag \\
& \left(\int \! \left|M_1 P(x,y)\right|^2\mathrm d y\!\right)\left(\int \! \left|M_2 Q(y,x')\right|^2\mathrm d y\!\right)
\label{eq:bound2}
\end{aligned}$$ where both factors on the right-hand side are finite by the maximal theorem, for almost all $x$ and almost all $x'$. These bounds justify the use of Fubini’s theorem to write $$\begin{aligned}
A_\epsilon C(x,x') &= \frac{1}{|C_\epsilon|^2}
\int_{C_\epsilon\times C_\epsilon\times Y}
\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!
P(x+t,y)Q(y,x'+t') \,\mathrm dt \mathrm dt' \mathrm dy
\notag\\&= \int_{Y} \! A_{\epsilon,1} P(x,y) \,
A_{\epsilon,2} Q(y,x') \,\mathrm d y ,
\label{eq:fubini}
\end{aligned}$$ which holds for almost every $x$ and $x'$.
We now observe that, for each factor on the right-hand side, $$\begin{aligned}
{2}
\lim_{\epsilon\rightarrow 0} A_{\epsilon,1} P(x,y) &= P(x,y) &\quad
&\mbox{a.a.~$x\in X$} \\
\lim_{\epsilon\rightarrow 0} A_{\epsilon,2} Q(y,x') &= Q(y,x') &
&\mbox{a.a.~$x'\in X$} .
\end{aligned}$$ The dominated convergence theorem together with the bounds in Eqs. and now imply that we can take the limit in Eq. to get $$\tilde{C}(x,x) = \lim_{\epsilon\rightarrow 0} A_\epsilon C(x,x) = \int_Y \! P(x,y)Q(y,x) \,\mathrm d y$$ for almost all $x$. We have $$\int_X \! (P*Q)(x,x) \, \mathrm d x= {\langle {\hat{Q},P} \rangle}_{L^2(X\times Y)},$$ with $\hat{Q}(x,y) = Q^\ast(y,x)$. Being an inner product on $L^2$, this expression is finite, completing the proof.
Remark 1: Although the diagonal of $P*Q$ is in $L^1$, we cannot conclude that $P*Q$ is trace class—see Ref.[@BRISLAWN_1988] for a counterexample. On the other hand, if $X = Y$ in Theorem \[thm:factorization\], then $P*Q$ is by definition trace class and it is also true that ${\operatorname{Tr}}P*Q = \int_X({\operatorname{diag}}P*Q)(x)\,\mathrm d x$.
Remark 2: $C = P*Q$ is the kernel of a Hilbert–Schmidt operator over $L^2(X)$. We see that it is meaningful to define the diagonal ${\operatorname{diag}}C$ of any Hilbert–Schmidt operator on an explicitly factorized form from the expression $$[{\operatorname{diag}}P*Q](x) := (P*Q)(x,x),$$ and the theorem states that this function belongs to $L^1(X)$, independent of the factorization.
Remark 3: As a corollary, if $P\in L^2({\mathbb{R}}^n_{{\mathrm{loc}}}\times {\mathbb{R}}^m)$, $Q\in L^2({\mathbb{R}}^m\times
{\mathbb{R}}^n_{{\mathrm{loc}}})$, then $P\ast Q \in L^1_{{\mathrm{loc}}}({\mathbb{R}}^n)$.
Remark 4: If $P(x,y) = Q^\ast(y,x)$, then $P\ast Q$ is positive semidefinite. Since ${\operatorname{diag}}P\ast Q$ is integrable, it follows from a theorem in Ref. [@BRISLAWN_1988] that $P\ast Q$ is trace class over $L^2(X)$.
Some proofs from Section \[secDIAG\]
====================================
Proof of Theorem\[thm:densities\]
---------------------------------
For this proof, we use Theorem \[thm:factorization\] in Appendix A.
Let $\Gamma\in\mathcal{D}_N$ be given. Assume that that $D_\Gamma({\mathbf{r}},{\mathbf{s}}) =
D({\mathbf{r}},{\mathbf{s}})$ almost everywhere in ${\mathbb{R}}^3\times{\mathbb{R}}^3$. It follows that $\tilde{D}_{\Gamma}({\mathbf{r}},{\mathbf{r}}) =
\tilde{D}({\mathbf{r}},{\mathbf{r}}) = (G^\dag*G)({\mathbf{r}},{\mathbf{r}})$ for almost all ${\mathbf{r}}$, since the regular representative is unique, and by using Theorem \[thm:factorization\], with $P({\mathbf{r}},{\mathbf{s}}) = G({\mathbf{s}},{\mathbf{r}})^*$ and $Q({\mathbf{r}},{\mathbf{s}})=G({\mathbf{s}},{\mathbf{r}})$ ($X = Y = {\mathbb{R}}^3$).
We need to show that $\rho_\Gamma({\mathbf{r}}) =
(G^\dag*G)({\mathbf{r}},{\mathbf{r}})$ for almost all ${\mathbf{r}}$. Assume that $\Gamma = {| {\Psi} \rangle}{\langle {\Psi} |}$. Now, $\rho_\Gamma({\mathbf{r}}) =
\rho_\Psi({\mathbf{r}}) = D_\Gamma({\mathbf{r}},{\mathbf{r}})$ for almost every ${\mathbf{r}}$, by definition of $\rho_\Psi({\mathbf{r}})$. Applying Theorem \[thm:factorization\] to $P({\mathbf{r}},{\mathbf{r}}_{2:N}) =
\Psi({\mathbf{r}},{\mathbf{r}}_{2:N})$ and $Q({\mathbf{r}}_{2:N},{\mathbf{r}}) =
\Psi({\mathbf{r}},{\mathbf{r}}_{2:N})^*$, $X = {\mathbb{R}}^3$ and $Y = {\mathbb{R}}^{3N-3}$, we see that $\rho_\Gamma({\mathbf{r}}) = \tilde{D}_\Gamma({\mathbf{r}},{\mathbf{r}}) =
(G^\dag*G)({\mathbf{r}},{\mathbf{r}})$.
We invite the reader to fill in the details when $\Gamma$ is a general mixed state.
Proof for Theorem\[thm:lockin\]
-------------------------------
$2 \Rightarrow 1$: Let a $G\in L^2({\mathbb{R}}^3\times{\mathbb{R}}^3)$ be given such that $\nabla_2 G \in
L^2({\mathbb{R}}^3\times{\mathbb{R}}^3_{{\mathrm{loc}}})$. Then, for every compact $K\subset
{\mathbb{R}}^3$, $$\begin{aligned}
T_\alpha({\mathbf{r}},{\mathbf{s}}) &:= \frac{1}{2} [\partial_{2,\alpha} G]^\dag *
[\partial_{2,\alpha}G]({\mathbf{r}},{\mathbf{s}}) \notag \\ &= \frac{1}{2} \int
d{\mathbf{u}} \partial_{2,\alpha}G({\mathbf{u}},{\mathbf{r}})^*\partial_{2,\alpha} G({\mathbf{u}},{\mathbf{s}})
\end{aligned}$$ is in $L^2(K\times K)$ by Theorem \[thm:factorization\]. $T_\alpha$ is positive semidefinite, so by Remark 4 after Theorem \[thm:factorization\], $T_\alpha$ is trace class over $L^2(K)$.
By the definition of the weak derivative and Fubini’s Theorem, we easily verify that in fact $T_\alpha
= \frac{1}{2} \partial_{1,\alpha}\partial_{2,\alpha} D$ almost everywhere. Thus $\nabla_1\cdot\nabla_2 D$ is trace-class, and $D$ has locally finite kinetic energy.
$1 \Rightarrow 2$:
Since $D\in\mathcal{D}_{N,1}$ there exists a spectral decomposition $$B({\mathbf{r}},{\mathbf{s}}) = \sum_k \lambda_k \phi_k({\mathbf{r}})\phi_k({\mathbf{s}})^*,$$ where $\{\phi_k\}\subset L^2({\mathbb{R}}^3)$ is a complete, orthonormal set, and where $0\leq\lambda_k\leq 2$ such that $\sum_k \lambda_k =
N$. Of course $B({\mathbf{r}},{\mathbf{s}}) = D({\mathbf{r}},{\mathbf{s}})$ almost everywhere, but they may be pointwise different.
Let $K\subset{\mathbb{R}}^3$ be compact. Restricted to $K\times K$, $\nabla_1\cdot\nabla_2 D = \nabla_1\cdot \nabla_2 B$ (a.e.) is trace-class, and we compute $$\nabla_1\cdot\nabla_2 B({\mathbf{r}},{\mathbf{s}}) = \sum_k \lambda_k
\nabla\phi_k({\mathbf{r}})\cdot\nabla\phi_k({\mathbf{s}})^*\quad \text{a.e.}$$ Let $A_k({\mathbf{r}},{\mathbf{s}}) =
\nabla\phi_k({\mathbf{r}})\cdot\nabla\phi_k({\mathbf{s}})^*$. By assumption, $${\operatorname{Tr}}(\nabla_1\cdot\nabla_2 B) = \sum_k \lambda_k {\operatorname{Tr}}A_k = \sum_k
\lambda_k \|\nabla\phi_k\|^2_{L^2(K)} < +\infty,$$ implying that $\nabla\phi_k\in L^2(K)$ for every $K$, hence $\nabla\phi_k\in L^2_{{\mathrm{loc}}}({\mathbb{R}}^3)$.
Let $G$ be given by $$G({\mathbf{r}},{\mathbf{s}}) = \sum_k \lambda_k^{1/2} \phi_k({\mathbf{r}})\phi_k({\mathbf{s}})^*.$$ Clearly, $G \in L^2({\mathbb{R}}^3\times{\mathbb{R}}^3)$ and $D = G^\dag * G$. Moreover, $$\nabla_2 G({\mathbf{r}},{\mathbf{s}}) = \sum_k \lambda_k^{1/2} \phi_k({\mathbf{r}})\nabla\phi_k({\mathbf{s}})^*.$$ Computing the $L^2({\mathbb{R}}^3\times K)$ norm, $$\begin{aligned}
\|\nabla_2 G\|^2 &= \sum_{k\ell} \lambda_k^{1/2}\lambda_\ell^{1/2}
{\langle {\phi_\ell,\phi_k} \rangle}_{L^2({\mathbb{R}}^3)}{\langle {\nabla\phi_k,\nabla\phi_\ell} \rangle}_{L^2(K)}
\notag \\ &=
\sum_k \lambda_k \|\nabla\phi_k\|^2_{L^2(K)}.
\end{aligned}$$
$3\Leftrightarrow 1$:
Let $\Gamma$ be such that $D_\Gamma = D$ a.e. We have, $$D_\Gamma({\mathbf{r}},{\mathbf{s}}) = \sum_i p_i \int d{\mathbf{r}}_{2:N}
\Psi_i({\mathbf{r}},{\mathbf{r}}_{2:N})\Psi_i({\mathbf{s}},{\mathbf{r}}_{2:N})^*.$$ Furthermore, $$\begin{aligned}
T_\alpha({\mathbf{r}}, & {\mathbf{s}})
:= \frac{1}{2} \partial_{1,\alpha}\partial_{2,\alpha}D({\mathbf{r}},{\mathbf{s}})
\notag \\ & =
\frac{1}{2} \sum_i p_i \int d{\mathbf{r}}_{2:N}
\partial_{1,\alpha}\Psi_i({\mathbf{r}},{\mathbf{r}}_{2:N})\partial_{1,\alpha}\Psi_i({\mathbf{s}},{\mathbf{r}}_{2:N})^*,
\end{aligned}$$ using the definition of the weak derivative and Fubini’s theorem. By Theorem \[thm:factorization\], $$\tilde{T}_\alpha({\mathbf{r}},{\mathbf{r}}) = \frac{1}{2} \sum_i p_i \int
|\partial_{1,\alpha}\Psi_i({\mathbf{r}},{\mathbf{r}}_{2:N})|^2 d{\mathbf{r}}_{2:N}$$ for almost all ${\mathbf{r}}$. For any compact $K\subset{\mathbb{R}}^3$, integration yields $$\int_K d{\mathbf{r}} \tilde{T}_\alpha({\mathbf{r}},{\mathbf{r}}) = \frac{1}{2} \sum_i p_i
\|\partial_{1,\alpha}\Psi_i\|^2_{L^2(K\times{\mathbb{R}}^{3N-3})}.$$ Since $T_\alpha$ is positive semidefinite, the left hand side is the trace of $\frac{1}{2} \partial_{1,\alpha}\partial_{2,\alpha}D$. Thus, $D$ has locally finite kinetic energy if and only if any representing $\Gamma\mapsto D$ has locally finite kinetic energy.
Proof of Theorem\[thm:currents\]
--------------------------------
Most of the proof is similar that of Theorem \[thm:lockin\], so we skip some details.
Let $\Gamma$ be such that $D_\Gamma = D$ almost everywhere. The state $\Gamma$ has a locally finite kinetic energy by Theorem \[thm:lockin\]. By a reasoning similar to that of the proof of this lemma, we obtain $$c_{\Gamma,\alpha}({\mathbf{r}}) = c_\alpha({\mathbf{r}}) = [{\operatorname{diag}}(-{\mathrm{i}}\partial_{2,\alpha} G)^\dag\ast G]({\mathbf{r}},{\mathbf{r}})$$ almost everywhere, independently of $\Gamma$. Taking the absolute value, integrating over a compact $K\subset{\mathbb{R}}^3$ and applying the Cauchy–Schwarz inequality, we obtain the bound $$\begin{aligned}
\int_K |c_\alpha({\mathbf{r}})| \mathrm d{\mathbf{r}} &\leq \|\partial_{2,\alpha}
G\|_{L^2({\mathbb{R}}^3\times K)} \|G\|_{L^2({\mathbb{R}}^3\times K)} \nonumber \\ &< +\infty.
\end{aligned}$$
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---
abstract: 'Since 1993, a muon telescope located at Forschungszentrum Karlsruhe (Karlsruhe Muon Telescope) has been recording the flux of single muons mostly originating from primary cosmic-ray protons with dominant energies in the $10-20$ GeV range. The data are used to investigate the influence of solar effects on the flux of cosmic-rays measured at Earth. Non-periodic events like Forbush decreases and Ground Level Enhancements are detected in the registered muon flux. A selection of recent events will be presented and compared to data from the Jungfraujoch neutron monitor. The data of the Karlsruhe Muon Telescope help to extend the knowledge about Forbush decreases and Ground Level Enhancements to energies beyond the neutron monitor regime.'
address: |
$^a$Institut für Experimentelle Kernphysik, Universiät Karlsruhe, D-76021 Karlsruhe, Germany\
$^b$ Institut für Kernphysik, Forschungszentrum Karlsruhe, D-76021 Karlsruhe, Germany
author:
- 'I. Braun$^{a,1}$, J. Engler$^b$, J.R. Hörandel$^{a,2}$, and J. Milke$^{b,3}$'
title: |
Forbush decreases and solar events seen in the $10-20$ GeV energy range\
by the Karlsruhe Muon Telescope
---
[^1] [^2]
[^3]
cosmic rays ,muon telescope ,heliosphere ,Forbush decrease ,solar energetic event ,ground level enhancement 94.20.wq ,96.50.S-,96.50.Xy,96.60.Q-,98.70.Sa
=0.5 cm
Introduction
============
The association of solar activity with the cosmic-ray intensity has been studied for various observed effects including Forbush decreases [@For54], i.e. a rapid decrease in the observed galactic cosmic-ray intensity, and Ground Level Enhancements, which are connected to large solar flares. They can be related to magnetic disturbances in the heliosphere that create transient cosmic-ray intensity variations [@Par65; @kallenrode]. From the observation of such events with different experiments, an energy dependent description can be obtained. The heliospheric influence is mostly pronounced for primary particles with low rigidity and has been studied mainly using data of the worldwide neutron monitor network [@Sim00]. With its unique median primary energy of 40 GeV for protons, the Karlsruhe Muon Telescope fills the energy gap between neutron monitors (from $\approx 11-15$ GeV, depending on solar activity state, to $\approx33$ GeV) and other muon telescopes ($\approx 53-119$ GV rigidity). In the following, we report on the detection of Forbush decreases and the investigation of Ground Level Enhancements with the Karlsruhe Muon Telescope.
Experimental Set-Up
===================
The flux of single muons from the zenith region has been recorded continuously since 1993 with the Karlsruhe Muon Telescope located at Forschungszentrum Karlsruhe, Germany (49.094$^\circ$N, 8.431$^\circ$E, 120 m a.s.l). The set-up is sketched in , details are given by @Eng99. Two double layers of scintillation counters are arranged on top of each other, separated by a 16 cm lead absorber, forming a “tower”. Each scintillation counter comprises a scintillator (NE 102) with the dimensions $0.6~\mbox{m} \times
0.25~\mbox{m}\times 0.02$ m, read out by a photomultiplier via an adiabatic light guide. A double layer is formed by two scintillation counters arranged parallel to each other and two counters oriented perpendicular to them, forming a $2\times2$ detector matrix. The lead absorber selects muons with energies larger than 0.8 GeV. Two such towers with a separation of 1.8 m are operated with a veto trigger logic selecting vertical particles and rejecting showers in which more than one particle hits the detector, thus suppressing the hadronic background to about 0.8% of the events [@kfk5320]. The muon detector is operated in a climatized room at a stable temperature. The towers of the instrument can be used to calibrate up to 32 liquid ionization chambers for the KASCADE hadron calorimeter, their data are not included in the present analysis. This analysis includes 80017 h of data between October 1993 and November 2006.
From simulations with CORSIKA [@corsika] and GEANT 3.21 [@geant], properties of the primary particles were investigated. The simulations include the physics processes in the atmosphere and the propagation through the building surrounding the detector, as well as the trigger conditions of the muon telescope. It turns out that the muons originate mostly from cosmic-ray protons, of which 95% have zenith angles smaller than $18^\circ$. The differential energy spectrum of primary protons triggering the telescope was derived for parameterized primary spectra in different solar activity states [@Urc72], taking into account modulation parameters according to @Uso03. The differential trigger rate obtained, in other words the detector response function folded with the primary particle spectrum, is presented in . The maximum occurs at primary energies of about 15 GeV. The expected counting rate difference between the two activity states is 2.5%. The median energy $E_M$ of a detector is defined such that one half of the detected events originate from primary particles below (or above) $E_M$. The median energy of the Karlsruhe Muon Telescope for both simulated spectra is 40 GeV.
Atmospheric Corrections
=======================
Muons loose energy and decay on the way from their production site in the atmosphere to the detector, yielding a dependence of the detected rate on the height of the production layer and the amount of material traversed above the detector. Corrections were applied to the recorded muon rate using the atmospheric pressure measured at the Forschungszentrum Karlsruhe. In addition, balloon ascends at noon and midnight conducted by the German weather service (DWD) in Stuttgart provide the heights of specific pressure layers including the 150 g/cm$^2$ layer ($\approx13.6$ km) which is close to the typical production layer of muons triggering the telescope at 130 g/cm$^2$, as determined from simulations.
For each year, the muon rate was iteratively corrected for a pressure of 1013 hPa and a nominal height of the 150 g/cm$^2$ layer of 13.6 km, yielding correction parameters of $ d(Rate) / dp=(-0.12 \pm 0.04)~ \% $/hPa and $ d(Rate) / dh=(-3.8 \pm 1.2)~ \% $/km. This correction eliminates rate variations from the data-set, which are caused by changing atmospheric conditions. For a consistency check, a rough estimate of the muon lifetime can be deduced from these values, assuming that all muons are produced with the same energy at the same atmospheric depth. The obtained lifetime of $ 2\pm0.5~\mu$s is consistent with the literature value.
Forbush Decreases {#events}
=================
---- ------------ --------------------- ------------------ -------------- -----------------------
\# date significance($\mu$) amplitude($\mu$) amplitude(n) amplitude change
\[%\] \[%\] \[% / energy decade\]
1 1998/08/26 6.1$\sigma$ 10.1 10.3 -0.4
2 1998/09/25 6.2$\sigma$ 11.7 8.7 5.7
3 1999/01/23 4.7$\sigma$ 8.1 6.7 2.6
4 2000/07/15 8.3$\sigma$ 12.3 14.9 -5.0
5 2000/11/10 6.1$\sigma$ 6.8 2.1 9.0
6 2001/08/03 8.7$\sigma$ 9.4 3.4 11.7
7 2003/10/30 8.4$\sigma$ 11.3 23.2 -22.8
8 2004/11/10 5.0$\sigma$ 10.5 8.9 3.1
9 2005/01/19 10.6$\sigma$ 13.2 16.8 -6.9
10 2005/09/11 5.5$\sigma$ 8.4 13.5 -9.8
---- ------------ --------------------- ------------------ -------------- -----------------------
The muon data were searched for days where the average rate was significantly lower than that of a background region. The background level was determined from hourly count rates within two times two weeks (14 d before the test region and 14 d afterwards), separated by three days from the tested day. The significances for each day were computed according to @LiMa. Trial factors were not taken into account.
The Karlsruhe Muon Telescope has detected several significant structures. The strongest Forbush decreases in the years from 1998 to 2006 are compiled in . Shown are a sequential number, the date, the significance and the amplitude $A$ of the minimum rate ($r_{FD}$) relative to the average rate before the decrease ($r_b$) computed as $$A =(r_{b} - r_{FD}) / r_b.$$ The amplitudes $A_\mu$ and $A_n$ have been calculated according to (1) for the muon telescope (based on hourly rates) and the Jungfraujoch 18-IGY neutron monitor (46.55$^\circ$N / 7.98$^\circ$E, 3570 m asl), respectively. The latter has an effective vertical cutoff rigidity of 4.49 GV [@Bern]. It was chosen for this comparison because of its geographic proximity to Karlsruhe. The detected events compared to the neutron monitor counting rate are depicted in and \[events2\]. To display their development, the muon telescope rates are smoothed by a running mean over 24 hours. Attention should be paid to the different scales for the muon rate (left-hand scale) and for the neutron monitor rate (right-hand scale). The apparently significant excesses on 1998/09/24, 2003/11/08, 2005/09/09-10 and 2005/09/22 are artefacts of the smoothing and caused by individual high data-points at the boundaries of detector down-time. It is worth to point out that the rate development observed at 4.5 GV (Jungfraujoch) and for the muon telescope (15 GeV) are quite similar, despite of their different energy thresholds. This illustrates that Forbush decreases are clearly detectable with a muon detector with 15 GeV peak energy. Forbush decreases were detected already with the GRAND muon detector [@poirier11] at 10 GeV peak energy. With the Karlsruhe Muon Telescope we push the detection towards higher energies.
Many structures in these Forbush decreases are visible at both energies. A closer look reveals that for events 7, 8, 9, and 10 (close to the solar minimum) the rates of both detectors follow each other extremely closely. On the other hand, for events 1, 2, 4, and 6 there are systematic differences between the two energies in the behavior before or after the Forbush decrease. For the Forbush decrease in the year of the solar maximum (\# 5) the strongest differences between the two rates are observed. It appears as during solar maximum there are significant differences between the fluxes observed at 4.5 GV and 15 GeV, while the fluxes are correlated well during periods of low solar activity.
To study the energy dependence of the amplitudes of a Forbush decrease, the spectral index $\gamma$, i.e. the change of amplitude per decade in energy has been calculated according to $$\gamma = ({A_{\mu} - A_{n}})~/~{ ( log(E_{M}^{\mu}) - log(E_{M}^{n}))},$$ $E_{M}^{\mu}\sim$ 15 GeV and $E_{M}^{n}\sim$ 4.5 GeV being the most propable primary energies for the muon telescope and the neutron monitor, respectively. $\gamma$ is listed in the last column of Table\[tabevt\]. To investigate a possible dependence on the solar activity, the amplitude change per energy decade is depicted as function of the international sunspot number (taken from @SIDC) in . No clear correlation between the two quantities can be inferred from the figure. Thus, earlier claims by @ifedili cannot be confirmed. A study of the energy dependence of the recovery time of Forbush decreases including data from Karlsruhe Muon Telescope is published elsewhere [@Uso08].
Ground Level Enhancements {#gle}
=========================
GLE number event date status
------------ ------------ --------
55 1997/11/06 i
56 1998/05/02 i
57 1998/05/06 a (+)
58 1998/08/24 a
59 2000/07/14 a +
60 2001/04/15 a +
61 2001/04/18 a
62 2001/11/04 i
63 2001/12/26 i
64 2002/08/24 i
65 2003/10/28 i
66 2003/10/29 a
67 2003/11/02 i
68 2005/01/17 a
69 2005/01/20 a
: Solar Ground Level Enhancements according to the Bartol group database [@Bartol]. The last column denotes the status of the Karlsruhe Muon Telescope: a: active, i: inactive, +: GLE detected. \[gletab\]
Due to their relatively short duration, Ground Level Enhancements (GLEs) are difficult to detect with the Karlsruhe Muon Telescope. Therefore, the data were scanned for correlations with all events marked in the GLE database, as provided by the Bartol group [@Bartol] and listed in . The muon flux was recorded for events marked with an “a”. During events marked with “i” the muon telescope was not active. The hourly rates for GLEs 56 to 67 as registered by the Karlsruhe Muon Telescope and the Jungfraujoch neutron monitor are depicted in . Jungfraujoch data are scaled by a factor 100 and the muon counting rates are smoothed over a period of three hours.
For GLE 57, no significant excess was observed in the muon counting rate. However, about seven hours before the Ground Level Enhancement a small peak is visible in the registered muon flux. For GLE 58, no significant muon excess has been observed.
GLE 59, the “Bastille day event” on July 14, 2000 has been registered by many detectors, including neutron monitors and space crafts [@bieber; @vashenyukasr]. In particular, the event could be measured for primary cosmic rays with GeV energies [@wang]. It has been detected by the GRAND muon detector (10 GeV most probable energy) [@poirier14], by the L3+C detector at CERN ($\approx40$ GeV primary energy) [@l3], and also by the Karlsruhe Muon Telescope. An excess in the muon counting rate can be recognized a few hours before the event. The significance of this structure is under investigation. If real, it is a possible hint for energy dependent propagation effects or the strongly anisotropic nature of this event.
On Easter day 2001 (April 15) an event occured (GLE 60) which has been observed and discussed by several groups [@shea; @dandrea; @biebereaster; @tylka; @vashenyukasr]. A muon count excess can be recognized at the time of GLE 60, while no signal is observed from GLE 61. It should also be noted that the Jungfraujoch neutron monitor detects GLE 60 with a large signal. On the other hand, the muon flux is only slightly increased at the time of the event.
Some of the greatest bursts in the 23rd solar cycle occurred on 28/29 October and 2 November 2003 (GLE $65-67$). They are extensively discussed in the literature, [@watanabe; @liu; @hurford; @eroshenko e.g.]. Unfortunately, the muon telescope was not active during GLEs 65 and 67. At the time of GLE 66, no significant signal is seen in the muon count rate. However, about one day before GLEs 65 and 66 a peak can be recognized in the registered muon flux. It is not clear if these increases are statistically significant, since there are gaps in the observing time. Thus, it is not obvious if the detected rate variations are correlated with the Ground Level Enhancements.
Unsmoothed hourly count rates of the Karlsruhe Muon Telescope compared to the Jungfraujoch neutron monitor data during the Forbush Decrease in January 2005 are depicted in . For comparison the reader may refer to for the smoothed counting rate of the same event. In the interval shown, two Ground Level Enhancements (GLE 68 and 69) have been observed, the corresponding times are marked in the figure. GLE 69 occoured on January 20, 2005 and was the second largest GLE in fifty years [@gle69-1; @gle69-2; @gle69-3; @vashenyukasr]. Measurements of the Aragats multidirectional muon monitor indicate that protons were accelerated at the Sun up to energies of 20 GeV in this GLE [@bostanjyan]. Protons accelerated during the main phase have a softer energy spectrum than during the initial phase of the event. It is assumed that protons were accelerated in a process or processes directly related to a solar flare [@simnett]. The right-hand panel of shows the region around GLE 69. No indication for a significant increase in the muon rate associated with GLE 69 can be seen in the figure. The Aragats data indicate that the time interval of solar proton flux with very high energies was only very short, this could explain why nothing is seen in the Karlsruhe Muon Telescope data. In addition, the solar cosmic-ray flux during the initial phase was very anisotropic, another potential reason for the non-observation in the muon rate.
Conclusions
===========
The Karlsruhe Muon Telescope provides information about effects of solar activity on the cosmic-ray flux observed at Earth since 1993. The recorded muon flux corresponds to 15 GeV peak energy (40 GeV median energy) for primary protons.
Several strong Forbush decreases, i.e. a rapid decrease in the observed galactic cosmic-ray intensity, could be measured with the muon telescope, indicating that these effects can be seen at energies exceeding the typical energies of neutron monitors. Comparing the observed amplitudes to the Jungfraujoch neutron monitor data, the spectral index of the events has been estimated. No dependence of the spectral index on the sunspot number has been found. However, there are significant differences in the timely development of the rates observed at 4.5 GV and 15 GeV for different states of solar activity. For Forbush decreases during solar maximum, the rates of the muon telescope and the neutron monitor behave quite differently, while they are well correlated for periods of low solar activity.
It has been investigated whether Ground Level Enhancements, which are connected to large solar flares, observed between 1997 and 2005 can be detected in the registered muon flux. For the strong Ground Level Enhancements 59 and 60 a clear signal can be seen in the muon count rate at the times of the events. This provides direct evidence for particles being accelerated to energies as high as 15 GeV during solar flares. Indirect evidence has been obtained previously by observations of lines in the gamma ray spectrum measured during solar flares [@rieger; @fletcher]. On the other hand, no signal has been detected for the GLEs 58, 61, 66, 68, and 69. If the underlying physics processes of all Ground Level Enhancements are the same, this means that the energy spectra of GLEs 59 and 60 differ from the spectra of the other GLEs. Another possibility is that the angular distribution of the emitted particles is different for different GLEs, i.e. in cases with highly anisotropic emission no signal was detected in the muon counting rate.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to Mrs. Heike Bolz for her enthusiastic efforts in continuously operating the Karlsruhe Muon Telescope and to Jürgen Wochele for his help during the construction of the detector. We thank the team operating the Jungfraujoch neutron monitor for making their data publicly available. We acknowledge the help of the Deutscher Wetter-Dienst (DWD) and the Institut für Meteorologie und Klimaforschung of Forschungszentrum Karlsruhe providing atmospheric data. We thank Jan Kuijpers for critically reading the manuscript and the anonymous referees for useful advice.
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[^1]: corresponding author, email:Isabel.Braun@phys.ethz.ch, now at: Institute for Particle Physics, Schafmattstr. 20, 8093 Zürich, Switzerland
[^2]: now at: Radboud University Nijmegen, Department of Astrophysics, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands
[^3]: now at: Institut für Wissenschaftliches Rechnen, Forschungszentrum Karlsruhe, D-76021 Karlsruhe, Germany
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Careful studies of the deuteron are fundamental to nuclear physics. Due to its relatively simple structure, reliable calculations can be performed in both non-relativistic and relativistic models [@Arenhoevel97; @Mosconi; @Tjon; @Jeschonnek; @Gross; @Forest] for a given nucleon-nucleon (NN) potential, making the deuteron the first testing ground for any realistic nuclear model. The electromagnetic probe is of particular importance because it is well understood and weak enough to allow a simple perturbative interpretation of the observables in terms of charge and current matrix elements. For these reasons, the electrodisintegration of the deuteron provides precise information on both the ground-state wave function [@Bernheim; @Turk; @mainzdeut; @zhouprl] and the electromagnetic currents arising from meson-exchange (MEC) and isobar configurations (IC) [@Arenhoevel82; @Gilad]. As the deuteron is often used as a neutron target, such detailed understanding of both its structure and currents, as well as the dynamics of final-state interactions (FSI), is crucial for applications such as the extraction of precise information on the neutron electromagnetic form factors [@Bruins; @nikhefgen; @mainzgen]. While in the past the study of realistic NN potentials has been the main point of interest, the roles of MEC and IC, and the question of relativistic corrections (RC), have come into focus recently.
Stringent constraints on nuclear models can be provided through measurements of the individual interference response functions in electron–deuteron scattering [@Arenhoevel97; @Tjon; @Jeschonnek]. The reason for this is that small but dynamically interesting amplitudes can be considerably amplified by interference with dominant amplitudes and thus may become accessible. For example, the longitudinal-transverse response $f_{\rm LT}$ is particularly sensitive to the inclusion of relativistic effects [@Scha92], while the so-called fifth response $f_{\rm LT}^{\,\prime}$ arises purely through final-state interactions [@Dolfini]. The transverse–transverse response $f_{\rm TT}$ appears to be mostly sensitive to MEC and IC [@Pellegrino], and this sensitivity increases as the kinematics are moved away from the quasielastic (QE) ridge. By properly choosing kinematical regimes and performing systematic studies of these three response functions, the role played by various interaction effects can be disentangled [@Gilad].
However, very few data on $f_{\rm LT}^{\,\prime}$ and $f_{\rm TT}$ exist [@Dolfini; @Pellegrino; @Tamae]. This is due to the fact that they require the detection of protons out of the electron scattering plane which became possible only recently. A limited set of data on the cross-section asymmetry, $A_{\rm LT}$, or $f_{\rm LT}$ [@Scha92; @Duc94; @Jordan; @From94; @Bulten; @Kasdorp] is available mainly in QE kinematics and in the region of low missing momentum. The data were obtained with sequential measurements left and right of the momentum transfer, which may be vulnerable to systematic errors in aspects such as luminosity variations and kinematic phase-space matching when forming $A_{\rm LT}$ or extracting $f_{\rm LT}$. Obtaining precise and consistent data on all three response functions is therefore desirable, in particular in the region of high missing momenta where the sensitivity to the various currents and dynamical effects is large. This is precisely the aim of the unique out-of-plane spectrometer facility (OOPS) at the MIT-Bates Linear Accelerator. Recently, we exploited this novel technique of performing precise extractions of the interference responses by simultaneous and symmetric measurements about the direction of the momentum transfer. This method minimizes possible systematic uncertainties in the extraction. Furthermore, we made simultaneous measurements of these interference response functions over a wide kinematical region, especially where the effects to be studied are enhanced.
In the one-photon exchange approximation, the cross section for the ${^2}$H($\vec{\rm e}$,e$^{\prime}$p)n reaction with an unpolarized target can be written with five independent terms as a function of the energy and momentum transfer ($\omega$, $q$) and the polar and azimuthal angles of knocked-out protons with respect to the momentum transfer direction in the center-of-mass frame of the np pair ($\theta_{\rm pq}^{\rm cm}$, $\phi_{\rm pq}^{\rm cm}$) [@Arenhoevel79]: $$\begin{aligned}
\label{eq: crosssection}
\displaystyle\frac{d^5 \sigma}{d \omega d \Omega_{\rm e} d \Omega_{\rm p}}
=
c [ \rho_{\rm L} f_{\rm L} ~+~
\rho_{\rm T} f_{\rm T}
~ + ~
\rho_{\rm LT} f_{\rm LT} \cos ( \phi_{\rm pq}^{\rm cm} ) \nonumber\\
~ + ~
\rho_{\rm TT} f_{\rm TT} \cos ( 2 \phi_{\rm pq}^{\rm cm} )
~ + ~
h \rho_{\rm LT}^\prime f_{\rm LT}^\prime \sin (\phi_{\rm pq}^{\rm cm})].\end{aligned}$$ Here $c$ is proportional to the Mott cross section, $h$ is the helicity ($\pm 1$) of the incident electrons, $\rho$ are the virtual-photon density matrix elements which depend only on the electron kinematics, and $f$ are the response functions in the center-of-mass system as functions of $\omega$, $q$, and $\theta_{\rm pq}^{\rm cm}$. In “parallel" kinematics where $\theta_{\rm pq}^{\rm cm}$ = 0, the interference response functions ($f_{\rm LT}$, $f_{\rm LT}^{\,\prime}$, and $f_{\rm TT}$) vanish. The angle $\theta_{\rm pq}^{\rm cm}$ is directly related to the missing momentum, $\vec{p}_{\rm m}$, which is the difference between $\vec{q}$ and the ejected proton momentum. In the plane-wave impulse approximation, $\vec{p}_{\rm m}$ is equal to the initial proton momentum in the deuteron.
By measuring the differential cross sections at fixed values of $\omega$, $q$, and $\theta_{\rm pq}^{\rm cm}$, but at angles $\phi_{\rm pq}^{\rm cm} = 0^{\circ}$, 90$^{\circ}$ (and/or 270$^{\circ}$) and 180$^{\circ}$ around the $\vec{q}$ vector, one can extract the interference responses: $$\begin{aligned}
f_{\rm LT} & = &
[ d\sigma_{\phi=0^\circ} - d\sigma_{180^\circ} ]
\cdot (2c\rho_{\rm LT})^{-1} \;, \\\label{eq: flt}
f_{\rm LT}^{\prime} & = &
[ d\sigma_{90^\circ}^{(h=+1)} - d\sigma_{90^\circ}^{(h=-1)} ]
\cdot (2c\rho_{\rm LT}^\prime)^{-1} \;, \\ \label{eq: fltprime}
f_{\rm TT} & = &
[ d\sigma_{0^\circ} + d\sigma_{180^\circ}
- 2 d\sigma_{90^\circ} ]
\cdot (4c\rho_{\rm TT})^{-1} \;, \label{eq: ftt} \end{aligned}$$ and form various asymmetries to study the contributions of each individual interference term to the cross-section: $$\begin{aligned}
A_{\rm LT} & = &
\displaystyle\frac { d\sigma_{0^\circ} - d\sigma_{180^\circ} }
{ d\sigma_{0^\circ} + d\sigma_{180^\circ} }
=
\displaystyle\frac{\rho_{\rm LT} f_{\rm LT}}
{ \rho_{\rm L} f_{\rm L} + \rho_{\rm T} f_{\rm T} +
\rho_{\rm TT} f_{\rm TT} }, \\\label{eq: alt} \nonumber \\
A_{\rm LT}^{\prime} & = &
\displaystyle\frac{ d\sigma_{90^\circ}^{(+1)} - d\sigma_{90^\circ}^{(-1)} }
{ d\sigma_{90^\circ}^{(+1)} + d\sigma_{90^\circ}^{(-1)} }
=
\displaystyle\frac { \rho_{\rm LT}^{\prime} f_{\rm LT}^{\prime} }
{ \rho_{\rm L} f_{\rm L} + \rho_{\rm T} f_{\rm T} -
\rho_{\rm TT} f_{\rm TT} }, \\ \label{eq: altprime} \nonumber \\
A_{\rm TT} & = &
\displaystyle\frac { d\sigma_{0^\circ} + d\sigma_{180^\circ} -
2d\sigma_{90^\circ} }
{ d\sigma_{0^\circ} + d\sigma_{180^\circ} + 2d\sigma_{90^\circ} }
=
\displaystyle\frac{\rho_{\rm TT} f_{\rm TT}}
{ \rho_{\rm L} f_{\rm L} + \rho_{\rm T} f_{\rm T} } \; . \label{eq: att} \end{aligned}$$
The experiment was carried out with the OOPS system [@DolfiniNIM; @Mandeville] used to detect knock-out protons in coincidence with electrons detected in the reconfigured one-hundred-inch proton spectrometer (OHIPS) [@xjiangthesis]. We developed a detailed computer-aided alignment method which ensured a precise alignment of the OOPS with absolute accuracies in position and angles better than 0.3 mm and 0.3 mrad, respectively [@joecomfort]. Spectrometer optics data were obtained for all OOPS modules and OHIPS. We performed ${^1}$H(e,e$^{\prime}$p) coincidence studies between OHIPS and the OOPS, and extracted absolute cross sections which were within 2% of the expected values [@sbsoongthesis; @jchenthesis]. Calculations of the coincidence phase-space, the effect of the radiative tail and multiple scattering corrections were carried out by Monte-Carlo simulations [@vellidis].
The measurements of the ${^2}$H($\vec{\rm e}$,e$^{\prime}$p)n reaction were performed by using an 800-MeV, 1% duty-factor polarized electron beam with an average current of 4 $\mu$A and a 160-mg/cm$^2$ thick liquid deuterium target. The polarization of the electron beam was measured with the B-line [Møller]{} polarimeter and averaged to be $38.6\pm 4.0$%. The OHIPS was positioned at a scattering angle of $\theta_{\rm e}=31.0^{\circ}$ and its central momentum was set to 645.0 MeV/c, corresponding to $q = 414$ MeV/c, $\omega = 155$ MeV and $x_{\rm Bjorken} = 0.52$. Three OOPS were positioned at $\phi_{\rm pq}^{\rm cm} = 0^{\circ}$, 90$^{\circ}$ and 180$^{\circ}$ with $\theta_{\rm pq}^{\rm cm}$ fixed at 38.5$^{\circ}$, thus providing simultaneous measurements of all three interference responses $f_{\rm LT}$, $f_{\rm LT}^{\,\prime}$, and $f_{\rm TT}$ as well as their associated asymmetries at $p_{\rm m}=210$ MeV/c [@jchenthesis]. In these kinematics the signal-to-noise ratio in the most forward OOPS was about 1:1. Here, we report on the results from the measurements in the “dip" region between the QE ridge ($\omega\simeq 90$ MeV, $x_{\rm Bjorken}\simeq 1$) and the $\Delta$ resonance. Measurements of the $f_{\rm LT}$ and $f_{\rm LT}^{\,\prime}$ response functions on top of the QE ridge were also performed [@zhoublast] and will be reported on later.
In order to suppress possible systematic uncertainties in extracting the data and to reduce the effects of kinematic broadening when comparing to calculations, only events from the overlapping portion of the detector acceptances are selected after matching the $\phi_{\rm pq}^{\rm cm} = 0^\circ$ and $180^\circ$ detector phase-spaces. Fig. \[alt\] shows the asymmetry $A_{\rm LT}$ as a function of $p_{\rm m}$. The PWBA(plane wave Born approximation)+RC result of Arenhövel [*et al.*]{} [@Arenhoevel97] is compared to the full relativistic calculations (PWBA) by Tjon [*et al.*]{} [@Tjon] and the $\sigma_{\rm cc1}$ prescription of de Forest [@deForest] for the off-shell electron-proton cross section. It seems that all these plane-wave approaches do not differ much and they obviously fail to describe the data.
The results show that the asymmetry is strongly sensitive to final-state interactions. The calculations with simple plane-wave approximations are inadequate and consequently a rigorous effort in including FSI is needed. Also, relativistic current operators or relativistic corrections are necessary. The Bonn potential of the NN interactions [@Bonn] is used in the calculations of Arenhövel [*et al.*]{}, but the predictions show little sensitivity to the choice of other realistic NN potentials for these kinematics [@Arenhoevel97]. In addition, the calculations show very little sensitivity to the two-body currents for $A_{\rm LT}$.
([8]{},[6.4]{}) (+0.7,-1.5)[=6.5cm ]{}
([8]{},[5.7]{}) (+0.7,-1.7)[=6.5cm ]{}
The out-of-plane detection also makes it possible to measure the helicity asymmetries $A_{\rm LT}^{\,\prime}$, which represent the imaginary part of the interference of the longitudinal and transverse current matrix elements. Fig. \[altprime\] shows $A_{\rm LT}^{\,\prime}$ for the OOPS at $\phi_{\rm pq}^{\rm cm} = 90^\circ$. The calculations by Arenhövel [*et al.*]{} agree well with the data when the FSI are included, because $A_{\rm LT}^{\,\prime}$ arises entirely from complex amplitudes interfering in the final-state processes. In PWBA, where only real amplitudes are involved, the asymmetry vanishes. In addition, as indicated by Eq. \[eq: crosssection\], when $\phi_{\rm pq}^{\rm cm}=0^\circ$ or $180^\circ$, $A_{\rm LT}^{\,\prime}$ vanishes. As a consistency check, our data in the two in-plane OOPS together yielded an asymmetry of $0.006 \pm 0.009$.
It is interesting to observe that the asymmetries are opposite in sign to the $p$-shell proton knock-out of the $^{12}$C($\vec{\rm e}$,e$^{\prime}$p)$^{11}$B reaction [@xjiangthesis; @mandprl], as shown by both data and calculations. In the low missing momentum region of the deuteron electrodisintegration, the FSI is dominated by the spin–spin interactions of the np pair, while in the p-$^{11}$B interactions, the spin–orbit parts are more important.
The determination of the absolute cross section makes it possible to extract also individual response functions. In Asymmetries, the denominators may cancel some of the effects in the numerators, as can be seen in our $f_{\rm LT}$ data shown in Fig. \[dip\_flt\]. Here, the data are again compared to calculations by Arenhövel [*et al.*]{} Not only relativistic corrections and detailed calculations of FSI, but also the two-body currents, are needed in order to bring the predictions into agreement with the data. However, one observes substantial cancellations between the effects of two-body currents and FSI. Accordingly, the calculations by Tjon [*et al.*]{}, which include only limited contributions from FSI, but do not contain two-body currents, are also in agreement with the data.
([8]{},[6.5]{}) (+0.7,-1.5)[=6.5cm ]{}
Isolating the contributions of the two-body currents from other competing reaction effects can be done by separating the remaining interference response, $f_{\rm TT}$. As shown in Fig. \[dip\_ftt\], various models predict that $f_{\rm TT}$ (or $A_{\rm TT}$) is strongly sensitive to the two-body currents while they do not depend so much on the relativistic effects, in contrast to $A_{\rm LT}$ and $f_{\rm LT}$. Our data agree with the full calculations by Arenhövel [*et al.*]{} [@Arenhoevel97] which have recently been improved by including retardation diagrams. The calculations by Tjon [*et al.*]{} which currently do not contain two-body contributions, fail to describe the data. The $f_{\rm TT}$ data demonstrate the power of using out-of-plane detection and new observables to isolate such small, but interesting contributions to the electromagnetic currents in deuteron disintegration.
([8]{},[6.5]{}) (+0.7,-1.5)[=6.5cm ]{}
In summary, our data clearly reveal strong effects of relativity and FSI, as well as of two-body currents arising from MEC and IC. We conclude that the two-body currents and relativity are extremely important to the understanding of the deuteron. In order to describe the data better, more rigorous relativistic calculations including all ingredients discussed here are needed. We show that competing effects in the deuteron electrodisintegration can be probed selectively by studies of multiple interference response functions using the novel out-of plane technique.
Data with higher statistical precision will be taken in the near future, especially in the region of higher missing momentum ($> 250$ MeV/c) and as a function of the energy transfer up to the $\Delta$ resonance [@zhou1997]. This comprehensive set of data will clarify the role of relativity and two-body currents, and also provide a detailed understanding of the isobar configurations and possible knowledge of the $\Delta$–$N$ interactions. The figure of merit in measuring $A_{\rm LT}^{\,\prime}$ will be improved by an order of magnitude when a highly polarized continuous-wave beam is used and two OOPS are mounted simultaneously at $\phi_{\rm pq}^{\rm cm} = \pm 90 ^\circ$, which is now possible.
We would like to thank the Bates staff in making this experiment possible. This work is supported in part by the US Department of Energy and the National Science Foundation, Grant-in-Aid for International Scientific Research by the Ministry of Education, Science, and Culture in Japan, and Deutsche Forschungsgemeinschaft.
Corresponding author, email: zzhou@lns.mit.edu. Pres. addr.: Rutgers University, Piscataway, NJ 08855. Pres. addr.: ORNL, Oak Ridge, TN 37831. Pres. addr.: TJNAF, Newport News, VA 23606. Pres. addr.: St. Mary’s Univ., Halifax, Canada, B3H 3C3.
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---
abstract: 'We consider the role of the internal kinetic energy of bound systems of matter in tests of the Einstein equivalence principle. Using the gravitational sector of the standard model extension, we show that stringent limits on equivalence principle violations in antimatter can be indirectly obtained from tests using bound systems of normal matter. We estimate the bound kinetic energy of nucleons in a range of light atomic species using Green’s function Monte Carlo calculations, and for heavier species using a Woods-Saxon model. We survey the sensitivities of existing and planned experimental tests of the equivalence principle, and report new constraints at the level of between a few parts in $10^{6}$ and parts in $10^{8}$ on violations of the equivalence principle for matter and antimatter.'
author:
- 'Michael A. Hohensee'
- Holger Müller
- 'R. B. Wiringa'
title: Equivalence Principle and Bound Kinetic Energy
---
General relativity follows from the Einstein equivalence principle (EEP), which holds that in any local Lorentz frame about any point in spacetime, the laws of physics are described by the standard model of particle physics and special relativity [@MTW]. Both general relativity and the standard model are believed to be the low energy limits of some as yet unknown complete theory of physics at high energy scales. Such a theory might generate violations of EEP at experimentally accessible energy scales [@Kostelecky:1989; @Colladay:199798; @Damour:1996], although its exact form is unknown. Thus it is important to search for EEP violation in as many different places as possible. We use the Standard Model Extension (SME) [@Colladay:199798], a flexible and widely applied [@datatables] framework for describing violations of EEP. The SME is an effective field theory that phenomenologically augments the Standard Model action with terms that break local Lorentz invariance and other tenets of EEP [@Kostelecky:2010], while preserving energy conservation, gauge invariance, and general covariance. As in other models [@Damour:1996], EEP-violation in the SME can manifest in multiple ways. In particular, it may be strongly suppressed in normal matter relative to antimatter [@Kostelecky:2010; @Hohensee:2011]. Although the equivalence principle has been validated with extremely high precision for normal matter [@adelberger], the situation for antimatter is less clear.
In this Letter, we show that in the SME, EEP violation in antimatter can be constrained by tests using bound systems of normal matter. We clearly demonstrate how an anomaly that violates the weak equivalence principle for free particles generates anomalous gravitational redshifts in the energy of systems in which they are bound, in proportion to the systems’ internal kinetic energy. Using a nuclear shell model, we estimate the sensitivity of a variety of atomic nuclei to EEP violation for matter and antimatter, and illustrate points of commonality between older representations of EEP violation based on neutron excess and baryon number, and that of the SME. We show that existing experimental [@adelberger; @Mueller:2010; @matterwaves; @Vessot:1980; @PoundRebka; @Kostelecky:2009a; @Ashby; @Blatt; @Fortier; @Hohensee:2013] limits on spin-independent EEP violation in matter and antimatter [@Hohensee:2011] are up to ten times tighter than previously thought, and could be made tighter still, provided more precise estimates of the bound kinetic energy of particles in atomic systems. We focus on EEP-violation in conventional matter (made up of protons, neutrons, and electrons), and as in prior work [@Kostelecky:2010; @Hohensee:2011; @datatables], assume that anomalies affecting force-carrying virtual particles are negligible. Using general covariance, we define our coordinates such that photons follow null geodesics, ensuring that electromagnetic fields do not violate EEP.
In the SME, spin-independent violations of EEP acting on a test particle of mass $m^{w}$ are described in its action [@Kostelecky:2010] $$S\!=\!-\!\!\!\int \!\frac{m^{w}c\sqrt{-(g_{\mu\nu}+2(\bar{c}^{w})_{\mu\nu})dx^{\mu}dx^{\nu}}}{1+(5/3){(\bar{c}^{w})}_{00}}+(a_{\rm eff}^{w})_{\mu}dx^{\mu},\!\!\label{eq:smeaction}$$ where the superscript $w=p$, $n$, or $e$ (for proton, neutron, or electron) indicates the type of particle in question, $g_{\mu\nu}$ is the metric tensor, $dx^{\mu}$ is the interval between two points in spacetime, and $c$ is the speed of light. The ${(\bar{c}^{w})}_{\mu\nu}$ tensor describes a fixed background field that modifies the effective metric that the particle experiences, and thus its inertial mass relative to its gravitational mass. The four-vector $(a^{w}_{\rm eff})_{\mu}=\{(1-U\alpha){(\bar{a}_{\rm eff}^{w})}_{0},{(\bar{a}_{\rm eff}^{w})}_{j}\}$, where $U$ is the Newtonian potential, represents the particle’s coupling to a field with a non-metric interaction with gravity. As $(a^{w}_{\rm eff})_{\mu}$ is CPT-odd [@Colladay:199798], this term enters with opposite sign in the action for an antiparticle $\bar{w}$. Both ${(\bar{c}^{w})}_{\mu\nu}$ and $(a^{w}_{\rm eff})_{\mu}$ vanish if general relativity is valid. For convenience, Eq. includes an unobservable scaling of the particle mass by $(1+\frac{5}{3}{(\bar{c}^{w})}_{00})$.
We focus on the isotropic subset of this model [@Kostelecky:2010], in which ${(\bar{c}^{w})}_{\mu\nu}$ is diagonal and traceless, and the spatial terms in the vector $(a^{w}_{\rm eff})_{\mu}$ vanish. In this limit, EEP-violation is described by the comparatively poorly constrained ${(\bar{c}^{w})}_{00}$ coefficients [@datatables], and the ${(\bar{a}_{\rm eff}^{w})}_{0}$ terms, which cannot be measured in non-gravitational experiments [@Colladay:199798]. In the non-relativistic, Newtonian limit, less the rest mass energy and assuming that any violations of EEP must be small, the single particle Hamiltonian produced by the action is given by $$H=\frac{1}{2}m^{w}v^{2}-m^{w}_{g}U,\label{eq:nonrelfreeham}$$ where the effective gravitational mass $m^{w}_{g}$ is given by$$m^{w}_{g}=m^{w}\left(1-\frac{2}{3}(\bar{c}^{w})_{00}+\frac{2\alpha}{m^{w}}(\bar{a}^{w}_{\rm eff})_{0}\right).$$ Experimentally observable EEP violations are proportional to the particle’s gravitational to inertial mass ratio $$\label{eq:betadef}
\frac{m_{g}^{w}}{m^{w}}=1-\frac{2}{3}{(\bar{c}^{w})}_{00}+\frac{2\alpha}{m^{w}}{(\bar{a}_{\rm eff}^{w})}_{0}\equiv 1+\beta^{w},$$ and are described here, as elsewhere [@Mueller:2010; @Hohensee:2011; @Hohensee:2011a], by the parameter $\beta^{w}$. From Eq. , we see that both ${(\bar{c}^{w})}_{00}$ and ${(\bar{a}_{\rm eff}^{w})}_{0}$ are responsible for violations of the weak equivalence principle, an aspect of EEP [@Will:2006], since they produce particle-dependent rescalings of the effective gravitational potential. In addition, EEP violation is not apparent in the non-relativistic motion of a free particle if $\alpha{(\bar{a}_{\rm eff}^{w})}_{0}=(m^{w}/3){(\bar{c}^{w})}_{00}$, although it remains manifest in the motion of the antiparticle $\bar{w}$, for which $\beta^{\bar{w}}=-2\alpha/m^{w}{(\bar{a}_{\rm eff}^{w})}_{0}-2/3{(\bar{c}^{w})}_{00}$ [@Kostelecky:2010]. As we now demonstrate, however, the antimatter anomaly $\beta^{\bar{w}}$ does contribute to tests involving non-gravitationally bound systems of matter, thanks to the anomalous gravitational redshift produced by ${(\bar{c}^{w})}_{00}$ in the energies of bound systems.
For a bound system of particles, the total Hamiltonian is a sum of single-particle Hamiltonians, plus an interaction energy $V_{\rm int}$ that is assumed to be free of EEP-violating terms. As implicit in Eq. , we take the system’s squared center of mass velocity $\bar{v}^{2}$ to be small, and of similar order as the relevant change $\Delta U$ it explores in the gravitational potential. Since the system is non-gravitationally bound, however, we cannot assume that the same is true of its constituent particles. Thus we must include terms proportional to $v_{w,j}^{2}U/c^{2}$ in our Hamiltonian, where $v_{w,j}$ is the instantaneous velocity of the $j$th bound particle of species $w$. In the limit that $\bar{v}\ll v_{w,j} \ll c$, we may approximate $v_{w,j}^{2}=(\bar{v}+\delta v_{w,j})^{2}\approx\bar{v}^{2}+(\delta v_{w,j})^{2}$, (dropping the mixed $\bar{v}(\delta v_{w,j})$ terms which make little contribution to the bound kinetic energy) and obtain $$\begin{gathered}
H=V_{\rm int}+\sum_{w}\Bigg[\frac{1}{2}m^{w}N^{w}\bar{v}^{2}-m^{w}N^{w}U(1+\beta^{w})\\
+\frac{1}{2}\sum_{j=1}^{N^{w}}(\delta v_{w,j})^{2}\left(1+\frac{3U}{c^{2}}+\frac{2U}{3c^{2}}{(\bar{c}^{w})}_{00}\right)\Bigg].\label{eq:newrelham}\end{gathered}$$ The second line in Eq. represents the system’s internally bound kinetic energy $T_{\rm int}$, and includes a term that contributes to the system’s conventional gravitational redshift, as well as a term proportional to ${(\bar{c}^{w})}_{00}$ and the gravitational potential $U$. This last term corresponds to an anomalous gravitational redshift of the bound state energies. To evaluate this term for bound quantum states, we recast it in terms of the momenta $\delta \vec{p}_{w,j}$ conjugate to the particle displacements $\delta x_{w,j}=x_{w,j}-\bar{x}$ from the system’s center of mass $\bar{x}$. The momenta satisfy $\delta \vec{p}_{w,j}=\partial H/\partial(\delta \vec{v}_{w,j})$, and so $$(\delta \vec{p}_{w,j})=m^{w}(\delta \vec{v}_{w,j})\left(1+\frac{3U}{c^{2}}+\frac{2U}{3c^{2}}{(\bar{c}^{w})}_{00}\right).$$ The bound kinetic energy $T_{\rm int}$ in Eq. is thus $$T_{\rm int}=\sum_{w}\sum_{j=1}^{N^{w}}\frac{(\delta p_{w,j})^{2}}{2m^{w}}\left(1-\frac{3U}{c^{2}}-\frac{2U}{3c^{2}}{(\bar{c}^{w})}_{00}\right).\label{eq:momexpr}$$ Note that in general, to ensure that the system’s mass defect is subject to a conventional gravitational redshift in the absence of EEP-violation, $V_{\rm int}$ must depend upon $U$. If EEP is satisfied, the variation of the mass defect $m'_{A}=(V_{\rm int}+T_{\rm int})/c^{2}$ for a system $A$ in a gravitational potential $U$ is such that the ratio $m'_{A}(U_{1})/m'_{A}(U_{2})=1+(U_{1}-U_{2})/c^{2}$. Due to our initial scaling of the particle mass in Eq. , the factor in parenthesis in Eq. contains terms proportional to $1$, $U$, $U{(\bar{c}^{w})}_{00}$, but not ${(\bar{c}^{w})}_{00}$ alone. This, along with our assumption that $V_{\rm int}$ is independent of ${(\bar{c}^{w})}_{00}$ and ${(\bar{a}_{\rm eff}^{w})}_{0}$, implies that the ratio $m'_{A}(U_{1})/m'_{A}(U_{2})$ does not generate additional cross terms in $U{(\bar{c}^{w})}_{00}$, and we can therefore write the total Hamiltonian for a bound system $A$ as $$H=\frac{1}{2}M_{A}\bar{v}^{2}-M_{A}U\left(1+\beta^{A}+\frac{2}{3}\sum_{w}\frac{T_{\rm int}^{w}}{M_{A}c^{2}}{(\bar{c}^{w})}_{00}\right),\label{eq:kincorr}$$ where $M_{A}=(\sum_{w}N^{w}m^{w})-m'_{A}$ incorporates the conventional components of $V_{\rm int}+T_{\rm int}$, the total kinetic energy of all $w$-particles in the system is [$T_{\rm int}^{w}=\nolinebreak\sum_{j=1}^{N^{w}}\langle(\delta p_{w,j})^{2}/2m^{w}\rangle$]{}, and $$\label{eq:betat}
\beta^{A}\equiv \frac{1}{M_{A}}\sum_{w}N^{w}m^{w}\left(\frac{2\alpha}{m^{w}}{(\bar{a}_{\rm eff}^{w})}_{0}-\frac{2}{3}{(\bar{c}^{w})}_{00}\right).$$ Since ${(\bar{c}^{w})}_{00}=-(3/4)(\beta^{w}+\beta^{\bar{w}})$, this demonstrates that EEP tests using non-gravitationally bound systems of normal matter can constrain phenomena that would otherwise only be apparent for free antimatter particles.
We now apply Eq. to evaluate the phenomenological reach of existing experiments using conventional matter. Violation of EEP is described by six independent parameters. Three for matter: $\beta^{p}$, $\beta^{n}$, and $\beta^{e}$; and three for antimatter: $\beta^{\bar{p}}$, $\beta^{\bar{n}}$, and $\beta^{\bar{e}}$. For any particular EEP test comparing the effects of gravity acting on systems $A$ and $B$, the observable anomaly is given by $\beta^{A}-\beta^{B}$, where $\beta^{A}$ and $\beta^{B}$ are defined in Eqs. and . Since all high-precision tests of EEP are performed on charge-neutral systems, and since normal matter has a substantially similar ratio of proton to neutron content, the expression for $\beta^{A}-\beta^{B}$ can be usefully expressed in terms of an effective neutron excess $\widetilde{\Delta}_{j}$, effective mass defect $\widetilde{m}'_{j}$, and kinetic energy components $T^{w}_{j,{\rm int}}$ of the two systems, where $$\begin{aligned}
\widetilde{\Delta}_{j}&\equiv&\frac{m^{n}}{m^{p}}\frac{m^{e}+m^{p}}{m^{n}}N_{j}^{n}-N_{j}^{p},\\
\widetilde{m}'_{j}&\equiv&m'_{j}-\frac{(m^{n}-m^{p})(m^{e}+m^{p})}{m^{n}}N_{j}^{p},\end{aligned}$$ and $j\in\{A,B\}$. The EEP-violating observable can then be written in terms of linear combinations of the free particle ($\beta^{w}$) and anti-particle ($\beta^{\bar{w}}$) anomalies as $$\begin{gathered}
\beta^{A}-\beta^{B}=\frac{(m^{n})^{2}}{(m^{n})^{2}+(m^{e}+m^{p})^{2}}\!\!\left[\phantom{\left(\frac{\widetilde{\Delta}_{A}}{M_{A}}-\frac{\widetilde{\Delta}_{B}}{M_{B}}\right)}\right.\\
\left.\left(\frac{\widetilde{\Delta}_{A}}{M_{A}}-\frac{\widetilde{\Delta}_{B}}{M_{B}}\right)m^{p}\beta^{e+p-n}-\left(\frac{\widetilde{m}'_{A}}{M_{A}}-\frac{\widetilde{m}'_{B}}{M_{B}}\right)\beta^{e+p+n}\right]\\
-\frac{1}{2}\sum_{w}\left(\frac{T^{w}_{A, {\rm int}}}{M^{A}c^{2}}-\frac{T^{w}_{B, {\rm int}}}{M^{B}c^{2}}\right)\left(\beta^{w}+\beta^{\bar{w}}\right)\label{eq:betadiffdefect},\end{gathered}$$ where $M_{A}$ and $M_{B}$ are the masses of the two test bodies, and $$\begin{aligned}
\beta^{e+p-n}&\equiv&\beta^{e+p}-\frac{m^{e}+m^{p}}{m^{n}}\beta^{n}\label{eq:betaeppmn}\\
\beta^{e+p+n}&\equiv&\frac{m^{e}+m^{p}}{m^{n}}\beta^{e+p}+\beta^{n},\label{eq:betaepppn}\end{aligned}$$ in which $$\beta^{e+p}\equiv\frac{m^{e}}{m^{p}}\beta^{e}+\beta^{p}\label{eq:betaepp},$$ after the notation of [@Kostelecky:2010]. We can define a similar set of terms $\beta^{\bar{e}+\bar{p}}$, $\beta^{\bar{e}+\bar{p}-\bar{n}}$, and $\beta^{\bar{e}+\bar{p}+\bar{n}}$ for antimatter. Note that Eq. has a close parallel with older studies of EEP-violation [@Damour:1996], since $$\left(\frac{\widetilde{m}'_{B}}{M_{B}}-\frac{\widetilde{m}'_{A}}{M_{A}}\right)=\left(\frac{\widetilde{A}_{B}}{M_{B}}-\frac{\widetilde{A}_{A}}{M_{A}}\right)m^{n},$$ where the effective baryon number $\widetilde{A}_{j}$ is given by $$\widetilde{A}_{j}\equiv N_{j}^{n}+\frac{m^{p}}{m^{n}}\frac{m^{e}+m^{p}}{m^{n}}N_{j}^{p}.$$ Thus the quantities $m^{p}\beta^{e+p-n}$ and $m^{n}\beta^{e+p+n}$ in the SME may be understood as parameterizing an anomalous gravitational coupling to a given particle’s neutron-excess and total baryon number “charges” [@Damour:1996].
---------- ----------------------- ----------------------- ----------------------- ------------------------
Species
$T_{\rm int}^{\rm p}$ $T_{\rm int}^{\rm n}$ $T_{\rm int}^{\rm p}$ $T_{\rm int}^{\rm n} $
$^{6}$Li 77 78 64 65
$^{7}$Li 88 108 67 84
$^{9}$Be 124 135 95 112
$^{10}$B 162 164 116 122
$^{12}$C 219 219 145 153
---------- ----------------------- ----------------------- ----------------------- ------------------------
: \[tab:gfmckine\]Comparison between calculated bound kinetic energies (in MeV) of protons and neutrons in light nuclei, obtained from many-body Green’s function Monte-Carlo (GFMC) calculations [@Pieper:2001], and a single-particle calculation using a modified Woods-Saxon potential.
In our prior analysis [@Hohensee:2011], the kinetic energy of protons and neutrons bound within a given nucleus was estimated by treating the nucleons as Fermi gases confined within a square potential well. This model did not account for the nucleons’ angular momentum, treated the Coulomb potential in a heuristic way by shifting the depth of the proton potential, and did not account for the nucleons’ spin-orbit interaction. The latter is of particular significance, because it can affect the occupation number of states with a given kinetic energy. Here, we improve upon that work by modeling the nucleons as single particles bound within fixed, spherically symmetric rounded square well potentials. These Woods-Saxon potentials [@Woods:1954] are taken to be of the form developed by Schwierz *et al.* [@Schwierz:2007]. Nuclide data is taken from Audi *et al.* [@Audi:2003], and isotopic abundances (for deriving the EEP-violating signal in bulk materials) from Laeter *et al.* [@Laeter:2003]. A complete summary of our calculated kinetic energies can be found in the Supplement [@supplement]. Better estimates of the nucleons’ bound kinetic energies are available for light nuclei using Green’s function Monte-Carlo (GFMC) calculations of the many-nucleon wave functions for nuclides with $A\leq 12$ [@Pieper:2001]. The GFMC estimates of the bound kinetic energy of the constituent protons and neutrons in $^{6}$Li, $^{7}$Li, $^{9}$Be, $^{10}$B, and $^{12}$C are summarized in Tab. \[tab:gfmckine\], and are compared with the corresponding predictions of our Woods-Saxon potential. Using these estimates, we can determine the contribution of the matter-sector $\beta^{e+p\pm n}$ and antimatter-sector $\beta^{\bar{e}+\bar{p}\pm\bar{n}}$ parameters to any observed violation of EEP in the motion of two (normal matter) test masses. These contributions are summarized in Fig. \[fig:scatterplot\]. Species with particular relevance to existing or planned tests of EEP [@Dimopoulos; @DropTower; @kasevich; @STEQuest; @SRPoem; @lithium; @GG] are explicitly labeled.
![\[fig:scatterplot\] Scatterplot of the contribution of $\beta^{e+p\pm n}$ and $\beta^{\bar{e}+\bar{p}\pm\bar{n}}$ parameters to observable EEP violation in normal nuclides with lifetimes in excess of 1 Gyr, when compared to SiO$_{2}$. Tests that compare two or more widely separated species are more sensitive than tests involving neighboring isotopes. Plot (a) shows each species’ relative sensitivity to matter-sector EEP-violation, and (b) depicts their sensitivities to antimatter-sector anomalies. Gray points in (a) indicate the range of sensitivities obtained without accounting for nucleons’ kinetic energies. Sensitivities of $^{6}$Li, $^{7}$Li, $^{9}$Be, $^{10}$B, and $^{12}$C are taken from GFMC calculations, all others from a Woods-Saxon model (see Supplement [@supplement]).](./newBetaPlotHyb.pdf){width="3.4in"}
In most experiments, $\beta^{e+p-n}$ is dominant, as it scales with the neutron excess. The next most accessible are the $\beta^{e+p+n}$ term, which scales with the mass defect, and the antimatter term $\beta^{\bar{e}+\bar{p}-\bar{n}}$, which scales with the excess of the neutrons’ kinetic energy over that of the protons, followed by $\beta^{\bar{e}+\bar{p}+\bar{n}}$. In some cases, ([*e.g.* ]{}tests comparing lead and aluminium [@SRPoem]) the signal from the antimatter $\beta^{\bar{e}+\bar{p}-\bar{n}}$ may actually be stronger than that from $\beta^{e+p+n}$. These terms represent four of the six degrees of freedom describing isotropic EEP violation, primarily for protons, neutrons and their antiparticles. Electronic EEP-violation is described by $\beta^{e-p}+\beta^{\bar{e}-\bar{p}}\equiv -\frac{4}{3}[{(\bar{c}^{e})}_{00}-\frac{m^{e}}{m^{p}}{(\bar{c}^{p})}_{00}]$, and has thus far been constrained largely by gravitational redshift tests [@Vessot:1980; @PoundRebka; @Ashby; @Blatt; @Fortier; @Hohensee:2013], and tests of local Lorentz invariance [@Kostelecky:1999; @AltschulLehnert]. The sixth degree of freedom, $\beta^{e-p}-\beta^{\bar{e}-\bar{p}}\propto \alpha{(\bar{a}_{\rm eff}^{p})}_{0}-\alpha{(\bar{a}_{\rm eff}^{e})}_{0}$, is only observable in tests on charged bodies [@Kostelecky:2010; @Hohensee:2011a].
Using multivariate normal analysis of the results of an ensemble of EEP tests, including matter-wave [@Hohensee:2011; @Mueller:2010; @matterwaves], clock comparison [@PoundRebka; @Vessot:1980; @Kostelecky:2009a; @Ashby; @Blatt; @Fortier; @Hohensee:2013], and torsion pendulum experiments [@adelberger], we obtain new limits on the five isotropic EEP-violating degrees of freedom that are observable in neutral systems, summarized in Tab. \[tab:newlimits\]. These bounds improve upon prior [@Hohensee:2011] gravitational constraints on these SME coefficients by factors of two to ten, and are also stated in terms of the five matter and anti-matter $\beta^{e+p\pm n}$, $\beta^{\bar{e}+\bar{p}\pm\bar{n}}$, and $\beta^{e-p}+\beta^{\bar{e}-\bar{p}}$ coefficients. Though the limits reported in Tab. \[tab:newlimits\] are necessarily model-dependent, they are stable against small variations in the estimated value of $T^{w}/Mc^{2}$ for the relevant nuclides, and are consistent with the limits obtained using substantially different nuclear models [@WitekTBP].
----------------------------------------- ------------------ ----------------------------- ------------------
$(\beta^{e-p}+\beta^{\bar{e}-\bar{p}})$ $0.019\pm0.037$ $(\bar{c}^{e})_{TT}$ $-0.014\pm0.028$
$\beta^{e+p-n}$ $-0.013\pm0.021$ $(\bar{c}^{n})_{TT}$ $1.1\pm1.4$
$\beta^{e+p+n}$ $2.4\pm3.9$ $(\bar{c}^{p})_{TT}$ $0.24\pm0.30$
$\beta^{\bar{e}+\bar{p}-\bar{n}}$ $1.1\pm1.8$ $\alpha(\bar{a}^{n})_{T}$ $0.51\pm0.64$
$\beta^{\bar{e}+\bar{p}+\bar{n}}$ $-4.1\pm6.7$ $\alpha(\bar{a}^{e+p})_{T}$ $0.22\pm0.28$
----------------------------------------- ------------------ ----------------------------- ------------------
: \[tab:newlimits\]Global limits ($\times 10^{6}$) on isotropic EEP-violation, obtained via multivariate normal analysis on the results of an ensemble of precision tests of EEP. Limits are stated in the Sun-Centered, Celestial Equatorial Frame [@datatables], and are expressed in terms of the $\beta^{w}$ parameters as well as the individual $(\bar{c}^{w})_{TT}$ and $\alpha(\bar{a}^{w})_{T}$, with $(\bar{a}^{e+p})_{T}\equiv(\bar{a}^{e})_{T}+(\bar{a}^{p})_{T}$. Also shown is the limit on the $1\sigma$ volume $\beta^{\Pi}$ of five-dimensional parameter space consistent with experiment.
Despite the fact that torsion pendulum tests [@adelberger] set limits on specific combinations of $\beta$ parameters at the level of $10^{-12}$ (having constrained $\Delta g/g$ to the level of $10^{-14}$), the best bounds reported in Tab. \[tab:newlimits\] are at the level of $10^{-8}$. This apparent discrepancy is due to the fact that such tests do not span the full parameter space considered here. Thus the limits on the individual $\beta$’s summarized in Tab. \[tab:newlimits\] are strongly correlated with one another. Analysis of these correlations reveals that some combinations of the $\beta$’s are indeed constrained at the level of $10^{-9}$, $10^{-11}$ and $10^{-12}$, thanks to matter-wave interferometer and torsion pendulum results. Unfortunately, the specific combinations of $\beta$’s subject to these constraints are sensitive to small errors in our estimates of the nuclides’ bound kinetic energy, due to disparities between the precision of torsion pendulums and of other EEP tests. Formal limits on EEP-violation at the level of an effective field theory like the SME must therefore await the development of more reliable nuclear models [@WitekTBP] or the results of additional high precision EEP test presently in development, using matter-waves [@DropTower; @lithium; @kasevich], clocks [@STEQuest] or macroscopic masses [@SRPoem; @GG].
We have demonstrated that EEP tests on non-gravitationally bound systems of normal particles can set indirect constraints on EEP-violation in antimatter, thanks to the interaction between the EEP-violating terms and the system’s bound kinetic energy. We have explicitly derived the link between anomalous gravitational redshifts and violations of the weak equivalence principle. This occurs whenever EEP is violated by introducing a particle-specific metric. In the context of the SME, accounting for these interactions results in significantly improved constraints on EEP-violation in the standard model lagrangian, for both matter and antimatter. The precision of these bounds is limited by that of existing nuclear models, and uneven experimental coverage of EEP-violating parameter space. New EEP tests with precision comparable to that of existing torsion pendulum experiments [@DropTower; @lithium; @kasevich; @STEQuest; @SRPoem; @GG] may substantially eliminate this model-dependent limitation. Better nuclear modeling could also improve limits on EEP violation in the SME by up to eight orders of magnitude, the pursuit of which will be the subject of future work.
We thank Brian Estey, Paul Hamilton, Alan Kostelecký and Jay Tasson for stimulating discussions. We also thank W. Nazarewicz and N. Birge for providing us with independent estimates of the bound kinetic energy of nucleons in a range of atomic species.
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Supplemental Material {#supplemental-material .unnumbered}
=====================
In a previous analysis [@Hohensee:2011], we estimated the kinetic energy of the protons and neutrons bound within a given nucleus by treating the nucleons as fermi gases confined to a square potential well. Here we improve upon that work using a shell model calculation. The nucleus is modeled as a pair of rounded, spherically symmetric square well, or Woods-Saxon, potentials which separately confine its constituent protons and neutrons. Our potential is that of [@Schwierz:2007], although we do not work with relative coordinates, and so do not use the reduced particle mass in our Hamiltonian. For a nucleon of mass $m^{w}$ in a nucleus with mass number $A=Z+N$, made up of $Z$ protons and $N$ neutrons, our model Hamiltonian is [@Schwierz:2007] $$\begin{gathered}
H=\frac{p^{2}}{2m^{w}}+V_{0}\left(1-\frac{4\kappa}{A}\langle \mathbf{t}\cdot\mathbf{T'}\rangle\right)f(r,R,a)+V_{c}(r,R)\\
+\frac{1}{2(m^{w})^{2}r}\left(\frac{\partial}{\partial r}\tilde{V}f(r,R_{SO},a)\right)\mathbf{L}\cdot\mathbf{S},\label{eq:WSHam}.\end{gathered}$$ The Woods-Saxon potential is given by $$f(r,R,a) = \frac{1}{1+e^{(r-R)/a}},$$ and $V_{0}=-52.06$ MeV, $\tilde{V}=24.1V_{0}$, $R=1.26 A^{1/3}$ fm, $R_{SO}=1.16 A^{1/3}$ fm, $a=0.662$ fm. The vectors $\mathbf{t}$ and $\mathbf{T'}$ are the isospin of the nucleon and of the nucleus less that nucleon, respectively, and as in [@Schwierz:2007], are taken to be such that $$-4\langle \mathbf{t}\cdot\mathbf{T'}\rangle=\left\{\begin{aligned}
&3, && N=Z\\
&\pm(N-Z+1)+2, && N>Z\\
&\pm(N-Z-1)+2, && N<Z,
\end{aligned}\right.$$ with $\kappa=0.639$. The Coulomb potential $V_{c}(r)$ applies only to protons, and is given by $$V_{c}(r,R)=(Z-1)e^{2}\left\{\begin{aligned} &\frac{3R^{2}-r^{2}}{2R^{3}} , && r\leq R\\ &\frac{1}{r}, && r> R.\end{aligned}\right.$$
We solve for the eigenstates of this Hamiltonian numerically, and assign the protons and neutrons respectively to the $Z$ and $N$ lowest-lying energy states. We then evaluate and sum the expectation value of the kinetic energy operator $\langle p^{2}/2m\rangle$ for each occupied state, to obtain the kinetic energy correction term of Eq. (7). The total estimated nucleon kinetic energies for all stable, and many long-lived nuclides are shown in Tables \[tab:kinenergies\] and \[tab:kinenergieslonglived\]. The accuracy of these estimates is not guaranteed, as experimental measurements of the nucleons’ bound kinetic energy are unavailable, and these results are derived from models that have been optimized for the solution of other problems [@Schwierz:2007]. Nevertheless, they do yield limits on the $\beta$ coefficients (see Tab. II) that are consistent with those derived using other nuclear models [@WitekTBP], and exhibit similar trends.
----------- ------------------- ------------------- ----------- ------------------- ------------------- ------------ ------------------- ------------------- ------------ ------------------- ------------------- ------------ ------------------- -------------------
Species $T_{\rm int}^{p}$ $T_{\rm int}^{n}$ Species $T_{\rm int}^{p}$ $T_{\rm int}^{n}$ Species $T_{\rm int}^{p}$ $T_{\rm int}^{n}$ Species $T_{\rm int}^{p}$ $T_{\rm int}^{n}$ Species $T_{\rm int}^{p}$ $T_{\rm int}^{n}$
$^{6}$Li 64 65 $^{54}$Cr 559 659 $^{94}$Mo 831 1079 $^{130}$Ba 1211 1636 $^{169}$Tm 1405 2233
$^{7}$Li 67 84 $^{54}$Fe 590 662 $^{95}$Mo 886 1097 $^{131}$Xe 1173 1692 $^{170}$Er 1383 2266
$^{9}$Be 95 112 $^{55}$Mn 576 667 $^{96}$Mo 903 1114 $^{132}$Xe 1171 1712 $^{170}$Yb 1424 2157
$^{10}$B 116 122 $^{56}$Fe 574 674 $^{96}$Ru 869 1088 $^{132}$Ba 1208 1678 $^{171}$Yb 1422 2188
$^{11}$B 124 143 $^{57}$Fe 575 680 $^{97}$Mo 903 1131 $^{133}$Cs 1189 1716 $^{172}$Yb 1421 2311
$^{12}$C 145 153 $^{58}$Fe 576 686 $^{98}$Mo 903 1148 $^{134}$Xe 1168 1753 $^{173}$Yb 1419 2249
$^{13}$C 154 165 $^{58}$Ni 608 685 $^{98}$Ru 868 1124 $^{134}$Ba 1206 1719 $^{174}$Yb 1417 2279
$^{14}$N 165 173 $^{59}$Co 593 692 $^{99}$Ru 924 1141 $^{135}$Ba 1204 1739 $^{175}$Lu 1436 2295
$^{15}$N 172 185 $^{60}$Ni 610 697 $^{100}$Ru 958 1159 $^{136}$Xe 1165 1793 $^{176}$Yb 1413 2340
$^{16}$O 183 191 $^{61}$Ni 610 721 $^{101}$Ru 958 1197 $^{136}$Ba 1203 1760 $^{176}$Hf 1454 2308
$^{17}$O 188 211 $^{62}$Ni 611 745 $^{102}$Ru 958 1217 $^{136}$Ce 1239 1724 $^{177}$Hf 1452 2338
$^{18}$O 192 215 $^{63}$Cu 616 751 $^{102}$Pd 904 1224 $^{137}$Ba 1201 1780 $^{178}$Hf 1451 2368
$^{19}$O 211 235 $^{64}$Ni 610 792 $^{103}$Rh 985 1268 $^{138}$Ba 1200 1799 $^{179}$Hf 1449 2398
$^{20}$Ne 229 240 $^{64}$Zn 622 776 $^{104}$Ru 957 1254 $^{138}$Ce 1236 1765 $^{180}$Hf 1447 2428
$^{21}$Ne 233 258 $^{65}$Cu 616 799 $^{104}$Pd 1011 1279 $^{139}$La 1217 1802 $^{180}$W 1487 2390
$^{22}$Ne 236 264 $^{66}$Zn 622 825 $^{105}$Pd 1011 1306 $^{140}$Ce 1233 1805 $^{181}$Ta 1465 2439
$^{23}$Na 254 289 $^{67}$Zn 622 849 $^{106}$Pd 1011 1333 $^{141}$Pr 1241 1807 $^{182}$W 1483 2449
$^{24}$Mg 271 300 $^{68}$Zn 622 872 $^{106}$Cd 996 1295 $^{142}$Ce 1230 1840 $^{183}$W 1481 2479
$^{25}$Mg 274 321 $^{69}$Ga 628 878 $^{107}$Ag 1038 1342 $^{142}$Nd 1249 1808 $^{184}$W 1479 2508
$^{26}$Mg 276 319 $^{70}$Zn 638 882 $^{108}$Pd 1010 1386 $^{143}$Nd 1248 1826 $^{184}$Os 1520 2517
$^{27}$Al 292 352 $^{70}$Ge 634 883 $^{108}$Cd 1064 1349 $^{144}$Sm 1265 1811 $^{185}$Re 1497 2550
$^{28}$Si 308 361 $^{71}$Ga 627 888 $^{109}$Ag 1037 1395 $^{145}$Nd 1245 1862 $^{186}$W 1475 2648
$^{29}$Si 310 361 $^{72}$Ge 633 894 $^{110}$Pd 1009 1438 $^{146}$Nd 1243 1880 $^{187}$Os 1513 2581
$^{30}$Si 349 346 $^{73}$Ge 668 920 $^{110}$Cd 1063 1402 $^{148}$Nd 1258 1915 $^{188}$Os 1511 2603
$^{31}$P 312 369 $^{74}$Ge 668 947 $^{111}$Cd 1063 1429 $^{149}$Sm 1257 1885 $^{189}$Os 1509 2624
$^{32}$S 312 375 $^{74}$Se 667 902 $^{112}$Cd 1062 1455 $^{150}$Sm 1256 1903 $^{190}$Os 1507 2645
$^{33}$S 353 396 $^{75}$As 685 952 $^{112}$Sn 1115 1414 $^{151}$Eu 1264 1905 $^{191}$Ir 1515 2653
$^{34}$S 359 402 $^{76}$Se 666 957 $^{113}$In 1089 1461 $^{152}$Sm 1290 1938 $^{192}$Os 1533 2663
$^{35}$Cl 373 424 $^{77}$Se 701 984 $^{114}$Cd 1061 1450 $^{153}$Eu 1260 1941 $^{192}$Pt 1522 2661
$^{36}$S 368 441 $^{78}$Se 701 1011 $^{114}$Sn 1114 1467 $^{154}$Sm 1287 1979 $^{193}$Ir 1519 2671
$^{36}$Ar 386 430 $^{78}$Kr 699 965 $^{115}$Sn 1114 1465 $^{154}$Gd 1268 1926 $^{194}$Pt 1535 2679
$^{37}$Cl 390 451 $^{79}$Br 717 1016 $^{116}$Sn 1113 1463 $^{155}$Gd 1267 1946 $^{195}$Pt 1532 2688
$^{38}$Ar 397 460 $^{80}$Se 701 1064 $^{117}$Sn 1113 1478 $^{156}$Gd 1321 1966 $^{196}$Pt 1543 2697
$^{39}$K 410 481 $^{80}$Kr 733 1020 $^{118}$Sn 1112 1494 $^{156}$Dy 1309 1929 $^{196}$Hg 1533 2692
$^{40}$Ar 419 501 $^{81}$Br 717 1069 $^{119}$Sn 1111 1509 $^{157}$Gd 1319 1986 $^{197}$Au 1546 2704
$^{40}$Ca 423 486 $^{82}$Kr 733 1074 $^{120}$Sn 1110 1524 $^{158}$Gd 1318 2006 $^{198}$Pt 1539 2716
$^{41}$K 440 509 $^{83}$Kr 772 1101 $^{120}$Te 1147 1504 $^{158}$Dy 1306 1970 $^{198}$Hg 1529 2711
$^{42}$Ca 460 515 $^{84}$Kr 810 1127 $^{121}$Sb 1128 1530 $^{159}$Tb 1338 2008 $^{199}$Hg 1560 2720
$^{43}$Ca 466 535 $^{84}$Sr 782 1082 $^{122}$Sn 1108 1568 $^{160}$Gd 1315 2046 $^{200}$Hg 1558 2729
$^{44}$Ca 470 555 $^{85}$Rb 777 1132 $^{122}$Te 1146 1535 $^{160}$Dy 1357 2010 $^{201}$Hg 1551 2738
$^{45}$Sc 489 560 $^{86}$Kr 809 1179 $^{123}$Sb 1126 1573 $^{161}$Dy 1356 2029 $^{202}$Hg 1549 2747
$^{46}$Ca 476 593 $^{86}$Sr 782 1136 $^{124}$Sn 1106 1610 $^{162}$Dy 1354 2049 $^{203}$Tl 1556 2753
$^{46}$Ti 507 520 $^{87}$Sr 781 1163 $^{124}$Te 1144 1579 $^{162}$Er 1343 2059 $^{204}$Hg 1590 2765
$^{47}$Ti 511 548 $^{88}$Sr 838 1189 $^{124}$Xe 1181 1544 $^{163}$Dy 1353 2069 $^{204}$Pb 1551 2758
$^{48}$Ti 514 575 $^{89}$Y 786 1193 $^{125}$Te 1143 1600 $^{164}$Dy 1351 2088 $^{205}$Tl 1552 2771
$^{49}$Ti 517 602 $^{90}$Zr 792 1197 $^{126}$Te 1142 1621 $^{164}$Er 1393 2113 $^{206}$Pb 1559 2777
$^{50}$Ti 519 628 $^{91}$Zr 791 1206 $^{126}$Xe 1179 1587 $^{165}$Ho 1370 2154 $^{207}$Pb 1557 2786
$^{50}$Cr 550 593 $^{92}$Zr 848 1214 $^{127}$I 1159 1626 $^{166}$Er 1390 2167 $^{208}$Pb 1555 2795
$^{51}$V 537 638 $^{92}$Mo 831 1045 $^{128}$Xe 1177 1630 $^{167}$Er 1388 2193
$^{52}$Cr 555 648 $^{93}$Nb 867 1218 $^{129}$Xe 1175 1651 $^{168}$Er 1386 2220
$^{53}$Cr 557 653 $^{94}$Zr 847 1231 $^{130}$Xe 1174 1671 $^{168}$Yb 1428 2119
----------- ------------------- ------------------- ----------- ------------------- ------------------- ------------ ------------------- ------------------- ------------ ------------------- ------------------- ------------ ------------------- -------------------
------------ ------------------- ------------------- ------------ ------------------- ------------------- ------------ ------------------- ------------------- ------------ ------------------- ------------------- ------------ ------------------- -------------------
Species $T_{\rm int}^{p}$ $T_{\rm int}^{n}$ Species $T_{\rm int}^{p}$ $T_{\rm int}^{n}$ Species $T_{\rm int}^{p}$ $T_{\rm int}^{n}$ Species $T_{\rm int}^{p}$ $T_{\rm int}^{n}$ Species $T_{\rm int}^{p}$ $T_{\rm int}^{n}$
$^{40}$K 435 488 $^{190}$Pt 1526 2619 $^{82}$Se 780 1115 $^{209}$Bi 1573 2783 $^{186}$Os 1515 2560
$^{87}$Rb 824 1184 $^{232}$Th 1761 3048 $^{96}$Zr 846 1249 $^{50}$V 535 612 $^{115}$In 1087 1457
$^{138}$La 1218 1782 $^{238}$U 1804 3161 $^{100}$Mo 902 1186 $^{113}$Cd 1062 1453 $^{123}$Te 1145 1557
$^{147}$Sm 1260 1848 $^{128}$Te 1139 1663 $^{116}$Cd 1060 1480 $^{144}$Nd 1246 1844 $^{152}$Gd 1271 1889
$^{176}$Lu 1434 2325 $^{76}$Ge 668 999 $^{130}$Te 1136 1704 $^{148}$Sm 1259 1867
$^{187}$Re 1493 2593 $^{48}$Ca 480 629 $^{150}$Nd 1255 1950 $^{174}$Hf 1458 2247
------------ ------------------- ------------------- ------------ ------------------- ------------------- ------------ ------------------- ------------------- ------------ ------------------- ------------------- ------------ ------------------- -------------------
|
---
abstract: 'CDM models with non-scale-free step-like spectra of adiabatic perturbations produced in a realistic double inflationary model are compared with recent observational data. The model contains two additional free parameters relatively to the standard CDM model with the flat ($n=1$) initial spectrum. Results of the COBE experiment are used for the determination of a free overall spectrum normalization. Then predictions for the galaxy biasing parameter, the variance for “counts in cells”, the galaxy angular correlation function, bulk flow peculiar velocities and the Mach number test are obtained. Also considered are conditions for galaxy and quasar formation. Observational data strongly restricts allowed values for the two remaining model parameters. However, a non-empty region for them satisfying all considered tests is found.'
author:
- |
, ,\
Astrophysikalisches Institut Potsdam,\
An der Sternwarte 16,\
D-14482 Potsdam, Germany\
and\
,\
Landau Institute for Theoretical Physics,\
Kosygina St. 2,\
Moscow, 117334, Russia
title: Confrontation of a Double Inflationary Cosmological Model with Observations
---
PS. [[**]{}]{}
å[[*A&A*]{} ]{}
Introduction {#intr}
============
Inflationary cosmological models (Starobinsky 1980, Guth 1981, Linde 1982, 1983) imply a density parameter $\Omega_{tot} \approx 1$ ($\mid\Omega_{tot} - 1\mid < 10^{-4}$) within the observable part of the Universe. Combined with the value $\Omega_{bar}\approx 0.017h^{-2}$ following from the theory of primordial nucleosynthesis (see, Walker 1991) this requires the most of matter in the Universe to be nonbaryonic ($h=H_0/100$ km/s/Mpc). Second, the simplest inflationary models (with one slowly rolling effective scalar field) predict density fluctuations with a spectrum of approximately Zeldovich-Harrison type (i.e., $({\delta \rho \over \rho})_k^2 \propto k^n$ with $n\approx 1$). This result was first consistently derived by Hawking 1982, Starobinsky 1982, and Guth & Pi 1982. The cold dark matter (CDM) model with this initial spectrum of perturbations and with biasing $b_g \simeq 1.5 - 2.5$ have most successfully explained the observed hierarchy of cosmic structures up to scales of approximately 10 $h^{-1}$ Mpc (Davies 1985). However, observations of such structures as the Great Attractor (Lynden-Bell 1988, and Dressler 1991) and the Great Wall (de Lapparent 1986), the large-scale clustering in the redshift survey of IRAS galaxies (Efstathiou 1990a, Saunders 1991), and the galaxy angular correlation function and “counts in cells” for the deep APM galaxy survey (Maddox 1990, see also Loveday 1992) imply that there is more power in the perturbation spectrum at scales larger than approximately $ 10 h^{-1}$ than expected in the standard model (biased CDM plus $n = 1$ initial spectrum of adiabatic perturbations). On the other hand, it is remarkable that the standard model is so close to these observational data: only a modest increase in amplitude of perturbations at large scales is required to fit them - no more than 2 - 3 times, and the latest results have the tendency to diminish this number (see e.g. Loveday 1992 ). Finally, recent COBE measurements (Smoot 1992) also imply the ratio of amplitude at scales $(10^3 - 10^4) h^{-1} $ Mpc to that at $(1 - 10) h^{-1}$ Mpc equal to $(1.1 \pm 0.2) b_g$ ($1\sigma $ error bars). The earlier published positive RELICT - 1 result for $\Delta T/T$ at large angles (Strukov 1992a,b) is even larger but it was obtained using only one wavelength, so it may be at least partially non-primordial.
The spectrum of perturbations observed at the present time is a product of an initial (primordial) spectrum and some transfer function $T(k)$. The latter results from a transition from the radiation-dominated era to the matter dominated one at redshifts $z \sim 10^4$ and depends on the structure of dark matter. Thus, any deviation from the standard model may be explained either by changing the initial flat spectrum, or by having a different $T(k)$ due to a more complicated matter content. The latter possibility arises, for example, in models with a mixture of hot and cold dark matter (Shafi & Stecker 1984, Holtzmann 1989, van Dalen & Schaefer 1992, and others) or with a cosmological constant and dark matter (Peebles 1984, Kofman & Starobinsky 1985, Efstathiou 1990b, and Gorski 1992). For a recent reanalysis of the former model see Pogosyan & Starobinsky 1993; the same for the latter model - see Bahcall 1993.
Here, we consider CDM models with non-scale-free primordial perturbation spectra. The scale invariance of the perturbation spectrum can be broken by different physical mechanisms during an inflationary stage (see, e.g., Kofman 1985, Salopek , 1989). Scale-free but not scale-invariant spectra with $n < 1$ seem not to be able both to provide enough power at $L \sim (25-50)h^{-1} $ Mpc and to fit the COBE data (Polarski & Starobinsky 1992, Liddle 1992, Adams 1993; see, however, Cen 1992). A non-scale-free initial spectrum with an effective step ( compared to the $n=1$ spectrum) somewhere between $1 h^{-1} $ Mpc and $10 h^{-1} $ Mpc works much better.
To obtain such a spectrum, it is necessary to abandon at least one of the main assumptions of the simplest version of the inflationary scenario, i.e. to assume either that the slow-rolling condition is temporarily violated at some moment of time (and then a step of an universal form in the spectrum arises, see Starobinsky 1992), or that there is more than one effective scalar field driving inflation. The latter possibility leads to double inflation , i.e., to a cosmological model with two subsequent inflationary stages (Kofman 1985, Silk 1987). These inflationary stages may be driven by the $R^2$ term and a scalar field (1991), where $R$ is the Ricci scalar or by two noninteracting scalar fields (Polarski & Starobinsky 1992). In this case, a step in the spectrum arises as a result of a rapid decrease of the Hubble parameter $H=\dot a/a$ in a period between two inflationary stages with slowly varying $H$ and $\dot H$. Further, we consider the first of the above mentioned double inflationary models, results for the second one are expected to be similar.
Perturbation spectrum and normalization {#pert}
=======================================
The Lagrangian density of the gravitational field including the $R^2$ term and a massive scalar field reads
$$L = {1\over 16\pi G}(-R + {R^2\over 6M^2}) + {1\over
2}(\varphi_{,\mu }\varphi^{,\mu } - m^2\varphi^2).
\label{lag}$$
The $R^2$ term is coupled via $M^2 \ll G^{-1}$ to General Relativity, $\varphi$ is the scalar field and $m^2 \ll G^{-1}$ is its mass ($c = \hbar = 1$). The model contains two more parameters than the standard model. They define the location and the relative height of an effective step in the perturbation spectrum. The height of the step depends on the ratio $M/m$ and the energy density of the scalar field at the onset of the second inflation. On the other hand, this energy density is responsible for the length of the second inflationary stage, , for the physical scale at which a break in the spectrum appears. With $\zz = \zz_0 \approx 3G^{-1/2}$ at the end of the first inflationary stage, the break occurs just at the right place, and the height of the step is $\Delta\approx M/6.5 m$. Note that due to the exponential dependence of the break location on the energy density, its shifting from 10 Mpc to 100 Mpc does not practically influence the height of the step.
We consider the rms value of the Fourier transform of an gravitational perturbation $\Phi(k)$. Here the conventions $\Phi(\vec k)=(2 \pi)^{-3/2}\int \, \Phi(\vec r)e^{-i\vec
k\vec r}\, d^3r$, $\langle \Phi(\vec k) \rangle =0$, $\langle \Phi(\vec k)\Phi^{\ast}(\vec k')\rangle =
\Phi^2(k)\delta^3(\vec k -
\vec k'),\,\, k=|\vec k|$ for initially Gaussian perturbations are adopted. Provided the limit $k \to 0$ of $\ph (k)$ coincides with a flat spectrum $\tilde\ph (k)$. Then we can construct the two quantities $\ph^{1} = \lim_{k \to k_{br}-0} \tilde\ph (k)$ and $\ph^{2} = \lim_{k \to k_{br}+0} \ph (k)$. The height of the step $\Delta$ is given by $\Delta =\ph^{(1)}/\ph^{(2)}$. In this paper we consider spectra with steps $\Delta \approx M/6.5m =
2,\; 3,\; 4,\; 5$. The step is located at $L \simeq 2\pi k^{-1}_{br}$, where $k^{-1}_{br}$ denotes the wave number where the perturbation spectrum reaches the lower plateau at large $k$, the values $k^{-1}_{br} =$ (1,3,7,10,20,30) Mpc$^{-1}$ were considered. Here and further throughout the paper, unless a scaling by $h$ is given, we assume $h = 0.5$. The length of a transition period between flat parts of the spectra is of the order of one magnitude of wave numbers.
(12,7)
Results of numerical calculations of the power spectra of density perturbations at the present epoch
$$P(k)\equiv P(k,z=0)=\left({\delta \rho \over \rho}\right)_{\vec k}^2
={1\over 36}(kR_H)^4\Phi^2(k)T^2(k)$$
are plotted in Fig.\[fig1\]. Here $T(k)$ is the transfer function for the standard CDM model (a review of best fits to it see in Liddlle & Lyth 1993), for equations for perturbations see Gottlöber 1991, the overall normalization is explained below. We adopt $\Omega_{tot}=1$ in accordance with inflation, thus $a(t)\propto
t^{2/3}$ now. $R_H=2H_0^{-1}\approx 6000h^{-1}$ Mpc is the present day Hubble radius sometimes non-rigorously called the cosmological horizon. For $\Delta \le 3$, there is practically no oscillations in the spectrum. Then a very good analytical approximation for it may be obtained by assuming that the slow-roll condition is valid for both inflaton fields $R$ and $\phi$ in the region of the break (see, e.g., Gottlöber 1991).
Let us present this solution in a different form. The background equations for the model (1) in the slow-roll approximation
$$\begin{aligned}
3H\dot \phi + m^2\phi =0, \\
{6H^2\dot H\over M^2}={4\pi Gm^2\phi^2\over 3}-H^2\end{aligned}$$
have an integral
$$s\equiv -\ln {a\over a_f}= 2\pi G\phi^2+{3H^2\over M^2},$$
where $a_f$ is the value of the scale factor at the end of inflation (Kofman 1985, Starobinsky 1985a). Then, introducing $f=H^2/s$, we arrive to the equation:
$$\begin{aligned}
{df\over d\ln s}={1\over f}(f-f_1)(f_2-f), \\
f_1={2\over 3}m^2,\; f_2={1\over 3}M^2,\; f(s)<f_2. \nonumber\end{aligned}$$
The scalar field follows from the relation:
$$2\pi G\phi^2 = s\left( 1-{f(s)\over f_2}\right) .$$
Let, e.g., $M>m\sqrt 2$. Then the solution of Eq.(6) is
$${(f_2-f)^{{f_2\over f_2-f_1}}\over |f-f_1|^{{f_1\over f_2-f_1}}}
={s_1\over s}(f_2-f_1).$$
$f$ may be both larger and smaller than $f_1$. $f={f_1+f_2\over 2}$ for $s=2s_1$. For values of the quantities $M/m$ and $\phi$ used in our calculations, $f_1\ll f_2$ and $s_1\gg 1$. Using the general expression for adiabatic perturbations generated during a slow-roll multiple inflation (Starobinsky 1985a) we get the answer:
$$k^3\Phi^2(k)={9\over 100}\cdot 16\pi Gs^2f(1-{f\over 2f_2}),$$
where $s$ and $f(s)$ are taken at the moment of horizon crossing at the inflationary stage $k=a(t)H(t)$; so $s=s(k)\approx \ln {k_f\over k}$ in this expression, $k_f=a_fH_f$.
It is useful to note here that the background dynamics of an inflationary model with two massive scalar fields (see Polarsky & Starobinsky 1992) in the slow-rolling regime may be reduced to the same equation (6) with $f=H^2/s$ and $f_{1,2}={2\over 3}m_{1,2}^2$ ($m_1<m_2$). Thus, the replacement rule is $M^2\to 2m_2^2$. The integral (7) is then changes to $s=2\pi G(\phi_1^2+\phi_2^2)$. $f_1<f<f_2$ in this case, so inflationary dynamics of the model with two massive scalar fields is isomorphous to only a finite part of possible inflationary regimes of the model (1). The expression for the power spectrum of adiabatic perturbations is slightly different from (9):
$$k^3\Phi^2(k)={9\over 100}\cdot 16\pi Gs^2f.$$
Now let us turn to an overall normalization of the power spectra. It is chosen to fit the COBE data (Smoot 1992) on large-angle $\Delta T/T$ anisotropy ($2\le l <30$, where $l$ is the multipole number). If $a_l$ is a rms multipole value summed over all $m$ and averaged over the sky:
$$a_l^2={1\over 4\pi}\sum_{m=-l}^l\langle \left({\Delta T\over T}\right)_
{lm}^2\rangle
={2l+1\over 4\pi}\langle \left({\Delta T\over T}\right)_{lm}^2\rangle\, ,$$
then the expected variance of the COBE data $\sigma_T^2(\theta_{FWHM})$ may be expressed as
$$\sigma_T^2(\theta_{FWHM})=\sum_{l\ge 2}a_l^2\exp \left( -l(l+1)
\theta_s^2\right),$$
where $\theta_s=\theta_{FWHM}/2\sqrt {\ln 4}$ is the Gaussian angle characterizing smearing due to a finite antenna beam size, as well as due to an additional Gaussian smearing of raw data. We take $\sigma_T(10^{\circ})=(30\pm 7.5)\mu K/2.735 K$ (Smoot 1992), so $\theta_s=4.25^{\circ}\approx 1/13.5$.
We have numerically calculated the rms multipole values $a_l$ for the perturbation spectra plotted in Fig.\[fig1\] (& 1993). In addition, it is possible to obtain an analytical expression for $a_l$. Indeed, if $f_1\ll f\ll f_2$, then Eq.(8) simplifies to $f\approx f_2(s-s_1)/s_1$ for $0<s-s_1\ll s_1$. Thus, $s_1\approx
\ln {k_f\over k_{br}}$. Then it follows from Eq.(9) that
$$k^3\Phi^2(k)\approx {9\over 100}\cdot 16\pi Gf_2s_1(s-s_1)={9\over
100}A^2\ln {k_{br}\over k}$$
for $k_{br}e^{-s_1}\ll k\ll k_{br}$ ( here we introduce the quantity A used in Starobinsky 1983). Using the standard formula for the Sachs-Wolfe effect which is the only significant one for $2\le l<30$ and the case of adiabatic perturbations, it is straightforward to derive that
$$\begin{aligned}
a_l^2 = {A^2(2l+1)\over 400\pi}\int_0^{\infty} J_{l+1/2}^2(kR_H){\ln
(k_{br}/k)\over k^2R_H}dk= \nonumber \\
{A^2 \over 400\pi^2}{2l+1\over l(l+1)} \left(\ln{(2 l_0)} -1 - \Psi (l) -
{l + {1\over 2}\over l(l+1)}\right),\end{aligned}$$
where $l_0 = k_{br} R_{H}\gg 1$ is the multipole number corresponding to the step location ($l\ll l_0$) and $\Psi$ is the logarithmic derivative of the $\Gamma$-function, the difference between $R_H$ and the radius of the last scattering surface may be neglected here. Substituting $a_l^2$ into Eq.(12) and choosing $A$ to get the correct value for $\sigma_T(10^{\circ})$, we obtain properly normalized power spectra $P(k)$ which are plotted in Fig.\[fig1\].
It should be noted that, for the model considered, the contribution of primordial gravitational waves (GW) to large-angle $\Delta T/T$ fluctuations is small as compared to the contribution from adiabatic perturbations (AP) and may be neglected. Really, using formulas derived in Starobinsky 1985a,b, one gets
$${a_{l(GW)}\over a_{l(AP)}}\approx 2.5\sqrt{3\over 4s_1\ln {k_{br}\over
k_H}}\approx 0.1$$
for $1\ll l<30$, the value of this ratio for $l=2$ being $\approx 6\%$ more, $k_H=2\pi R_H^{-1}$. Thus, the account of the GW contribution increases the total rms value of $a_{l(tot)}=\sqrt
{a_{l(AP)}^2+a_{l(GW)}^2}$ by less than $1\%$.
Comparison with observational data {#com}
==================================
Biasing parameter
------------------
By help of the density perturbation spectrum $P(k)$ we are able to compute the variance of the mass fluctuation $\sigma_M^2(R)\equiv
\left({\delta M \over M}(R)\right)^2 $ in a sphere of a given radius $R$ (see e.g. Peebles 1980):
$$\sigma_M^2 (R) = {1\over 2 \pi^2} \int^{\infty}_0 k^2 P(k) W(kR) dk,
\label{sigma1}$$
where the window function $W(kR)$ is expressed as
$$W(kR) = {9\over (kR)^6}(\sin kR - kR \cos kR)^2
\label{window}$$
The quantity $\sigma_M^2$ gives the variance of total matter density perturbations. It becomes clear now that the distribution of galaxies is biased with respect to the distribution of total matter, so some biasing parameter $b_g$ (assumed to be scale-independent for simplicity) has to be introduced:
$$\xi_g(r)=b_g^2\xi (r),\, \sigma_{Mg}^2(R) = b_g^2 \sigma_M^2(R),
\label{sigma2}$$
where $\xi(r)$ is the matter density correlation function. Usually it is adopted that $\sigma_{Mg}^2(8 h^{-1}$ Mpc) = 1, so $b_g=\sigma_M^{-1} (8 h^{-1}$ Mpc). Comparison of N-body simulations with the observed $\xi_g(r)$ and with the mean pairwise velocity of galaxies at $r=1h^{-1}$ Mpc (Davies 1985, for more recent calculations see e.g. Gelb 1993) shows that $b_g$ probably lies between $2$ and $2.5$ ($0.4\le \sigma_M(8h^{-1})\le 0.5$). We adopt rather conservative limits $1.5<b_g<3$ ($0.33< \sigma_M(8h^{-1})<0.67$). Different bias factors obtained for the considered spectra are given in Table 1 (one should not forget about $ 25\% $ error bars at $ 1.5\sigma $ level in all these quantities due to the error bars in $\sigma_{T}(10^{\circ})$). This test excludes both the standard model and the $P(4)$ case (apart from the region $k_{br}^{-1}\approx 3$ Mpc for the latter). The $P(3)$ case is marginally admissible, and a broad range of $k_{br}$ remain allowed for the $P(1)$ and $P(2)$ cases.
0.5cm
------------------------------- ------------- ------------------ ------------------
Spectrum $P_i[k_{br}^{-1}]$ Bias factor $\sigma_v(R)$ $\sigma_v(R)$
depending on step $b_g$ km/s km/s
location at $k_{br}^{-1}$ Mpc $R=40h^{-1}$ Mpc $R=60h^{-1}$ Mpc
P1(03) 1.60 269 221
P2(03) 2.18 257 213
P3(03) 2.43 259 215
P4(03) 2.19 268 221
P1(07) 1.74 255 211
P2(07) 2.77 230 195
P3(07) 3.49 229 195
P4(07) 3.68 242 204
P1(10) 1.77 248 205
P2(10) 2.83 215 184
P3(10) 3.59 210 182
P4(10) 4.18 225 193
P1(20) 1.76 234 195
P2(20) 2.67 182 159
P3(20) 3.46 168 151
P4(20) 4.48 180 162
------------------------------- ------------- ------------------ ------------------
\
TABLE 1: The bias factors and the velocity variances for the models. P1, P2, P3 and P4 denote models with $M=13m; 20m; 26m; 32.2m$ corresponding to $\Delta = $2, 3, 4, 5. The step location is given by $k_{br}^{-1}$.
Counts-in-cells
----------------
Further we compare the variance $\sigma_c^2(l)$ for the counts-in-cells analysis of large-scale clustering in the redshift survey of IRAS galaxies (Efstathiou 1990) and in the Stromlo-APM survey (Loveday 1992) with our predictions obtained for corresponding scales. The variance $\sigma_c^2(l)$ is related to the two-point galaxy correlation function according to
$$\sigma_c^2(l) = {1\over V^2}\int \int_{V=l^3} \xi_g(r_{12})dV_1 dV_2 \, .
\label{IRAS1}$$
In terms of the density perturbation spectrum $P(k)$, it can be written as
$$\sigma_c^2(l) = {1\over 2\pi^2}\int_0^\infty k^2 P(k) W_1(kl) dk,
\label{IRAS2}$$
where $$W_1(\rho) = 8\int^1_0 dx\int^1_0 dy\int^1_0 dz (1-x)(1-y)(1-z)
{\sin(\rho \sqrt{x^2+y^2+z^2})\over \rho \sqrt{x^2+y^2+z^2}}.
\label{IRASw}$$
(12,7)
A comparison between $\sigma_c^2(l)$ for galaxies from the Stromlo-APM survey and IRAS galaxies (see Fig.\[fig2\]) shows that the biasing parameter for IRAS galaxies $b_{IRAS}<b_{APM}$ for $l=10h^{-1}$ Mpc and $b_{IRAS}
\approx b_{APM}$ for larger scales (apart from the point $l=40h^{-1}$ Mpc where $b_{IRAS}^2\approx 1.5 b_{APM}^2$ that is, however, inside $2\sigma$ error bars). Predictions for $\sigma_c^2(l)$ for the considered spectra normalized by the condition $\sigma_{Mg}(8h^{-1}$ Mpc)=1 are shown in Fig.\[fig2\]. It is seen that another normalization condition $\sigma_c(12.5h^{-1}$ Mpc)=1 proposed in Pogosyan & Starobinsky 1993 works as well. The Kaiser correction (Kaiser 1987) was not taken into account. In any case, it is clear that this correction should be rather small that presents one more argument for a sufficienly large value of $b_g$.
In order to determine values of the model parameters which give the best fit to these data, we formally apply the $\chi^2$ test considering the error bars given in Loveday 1992 as $2\sigma$ ones (in the logarithmic scale of $\sigma_c(l)$). The results are plotted in Fig.\[fig3\]. Taking into account the possibility of a change in the overall normalization, the estimated number of degrees of freedom is $N=7$, so we consider fits with $\chi^2<7$ as good and fits with $\chi^2>20$ as unacceptable. The best fit $\chi^2\approx 2.2$ occurs for $\Delta=3$ and $k_{br}^{-1}=3$ Mpc.
(12,7)
Angular correlation function
-----------------------------
Let us consider the predictions of our model for the galaxy angular correlation function $w(\theta)$. To this aim we put the Fourier transform of the 3-D correlation function $\xi_g(r)$ into the Limber equation and find
$$w(\theta)={\int_0^{\infty}\,
kP(k)dk\int_0^{\infty}y^4\phi^2(y)J_0(k\theta y)dy\over 2\pi^2
(\int_0^{\infty}y^2\phi(y)dy)^2}.$$
We insert our power spectra into Eq.(22) and scale the angular correlation function to the Lick depth ($y^{\ast}\approx 240 h^{-1}$ Mpc) with the selection function $\phi = y^{-0.5}\exp
\left(-(y/y^{\ast})^2\right)$. Results of the calculations along with the recent observational data for $w(\theta)$ for the APM survey (provided by S. Maddox, see also Maddox 1990) are shown in Fig.\[fig4\]. The use of linear approximation in the calculation of $w(\theta)$ is justified for $\theta >0.5^{\circ}$ only. This is the reason of a discrepancy between our theoretical curves and the data at small angles that should not be taken into account. At larger angles up to $\theta
\sim 10^{\circ}$, the agreement with the $\Delta=3$ and $\Delta=4$ cases is very good (and errors are too large to make any definite conclusion beyond this angle).
------------------------------- --------------------- --------------------- ----------------------
Spectrum $P_i[k_{br}^{-1}]$
depending on step $R_L = 4 h^{-1}$Mpc $R_L = 8 h^{-1}$Mpc $R_L = 18 h^{-1}$Mpc
location at $k_{br}^{-1}$ Mpc
P1(03) 2.76 1.48 0.76
P2(03) 3.16 1.68 0.86
P3(03) 3.35 1.76 0.89
P4(03) 3.21 1.69 0.85
P1(07) 2.80 1.51 0.79
P2(07) 3.45 1.88 0.98
P3(07) 4.01 2.17 1.10
P4(07) 4.18 2.19 1.07
P1(10) 2.78 1.51 0.79
P2(10) 3.37 1.87 1.01
P3(10) 3.90 2.20 1.19
P4(10) 4.51 2.44 1.22
P1(20) 2.68 1.46 0.77
P2(20) 2.85 1.61 0.93
P3(20) 3.09 1.76 1.07
P4(20) 3.87 2.25 1.35
------------------------------- --------------------- --------------------- ----------------------
\
TABLE 2: The Mach number test (notation as in Table 1)
Large-scale bulk flows
-----------------------
Next we investigate our spectra with respect to large-scale bulk flows (peculiar velocities) of matter. The rms peculiar velocity inside a sphere of a radius $R$ is given by the expression:
$$\sigma_v^2(R) = {2\over \pi^2R_H^2}\int^\infty_0 \; P(k)
W(kR) \exp {(-k^2 R^2_S)}dk,
\label{velocity}$$
where $W(kR)$ is the window function given in eq. (\[window\]) and the length scale $R_S$ characterizes the Gaussian smoothing of raw observational data. In the Table 1, the corresponding predictions for our model are shown. They may be compared with recent data from the POTENT reconstruction of the 3-D peculiar velocity field: $\sigma_v(40h^{-1}$ Mpc) $ =(405\pm 60)$ km/s and $\sigma_v(60h^{-1}$Mpc) $=(340\pm 50)$ km/s with $R_S=12h^{-1}$ Mpc, error bars are $1\sigma$ (Dekel 1992, see also Bertschinger 1990).
This test presents the most serious problem for the model because expected values of $\sigma_v$ are lower than in models having exactly flat spectrum at large scales, e.g. in the CDM+HDM $n=1$ model or in the CDM model with a step in the initial spectrum produced by one scalar field (Starobinsky 1992), due to the logarithmic increase of $P(k)$ at $k\ll k_{br}$, see Eq. (13). On the other hand, for $k_{br}^{-1}\le 7$ Mpc, they are higher than in tilted models with $n<0.75$ and without significant contribution of gravitational waves to large-angle $\Delta
T/T$ fluctuations. It should be mentioned here that bulk flow velocities obtained from observations contain large uncertainties. Second, it is fairly possible that actual values of bulk flow velocities at our location in the Universe are larger than average ones (“cosmic variance”). So, taking the data at their lower $2\sigma$ limit (this is $\approx
1.4$ times less than the average values that is permitted by cosmic variance) and pushing the overall normalization of the model to the upper $1.5\sigma$ limit of the COBE data, we find that the allowed region is $k_{br}^{-1}\le 7$ Mpc.
Mach number test
-----------------
Now we consider the Mach number test for our model (Ostriker & Suto 1990). The expression for the Mach number in terms of the quantities introduced above is given as
$$M^2(R_L) = {\int^{\infty}_0 P(k) \exp(-k^2R_S^2) \exp(-k^2 R_L^2) dk\over
\int^{\infty}_0 P(k) \exp(-k^2 R_S^2)
\left(1-(1+{k^2 R_L^2/9})\exp(-k^2 R_L^2)\right) dk},
\label{mach}$$
where $R_S = 5 h^{-1} $ Mpc characterizes the Gaussian smoothing. Ostriker & Suto (1990) found the Mach numbers $4.2\pm1.0$, $2.2 \pm 0.5$ and $1.3 \pm 0.4$ at the scales $R_L=4h^{-1}$ Mpc, $8h^{-1}$ Mpc, $18 h^{-1}$ Mpc, correspondingly. The Mach numbers predicted for our spectra are given in Table 2. Here, once more, the values of $M^2$ obtained from observations in our vicinity may be larger than average ones due to the numerator of Eq. (\[mach\]), i.e., due to local values of large-scale bulk velocities being larger than average. This test clearly shows that $\Delta >2$.
----------------------------------------------- ----------
Spectrum $P_i[k_{br}^{-1}]$ depending on step $\chi^2$
location at $k_{br}^{-1}$ Mpc
P0 18.00
P1(03) 13.41
P2(03) 9.59
P3(03) 11.08
P4(03) 10.17
P1(07) 14.44
P2(07) 11.18
P3(07) 13.69
P4(07) 28.06
P1(10) 15.80
P2(10) 14.27
P3(10) 13.69
P4(10) 20.04
P1(20) 19.56
P2(20) 49.82
P3(20) 24.17
P4(20) 17.27
----------------------------------------------- ----------
\
TABLE 3: $\chi^2$-values for the models
0.5cm
Quasar and galaxy formation
----------------------------
Finally, we consider the compatibility of our model with the existence of a sufficient number of large galaxies at the redshift $z=1$ and host galaxies for quasars at $z=4$. The standard model of quasar formation assumes that they arise as a result of formation of massive black holes ($M\approx 10^9M_{\odot}$) in nuclei of galaxies with total masses $(10^{11}-10^{12})M_{\odot}$ (incuding the dark matter component). We estimate $f(\ge M)$ - the fraction of matter in gravitationally bound objects with masses beginning from $M$ and higher - using a very simple though rather crude formula by Press & Schechter (1974):
$$f(\ge M)= 1-\hbox{erfc}\left({\delta_c\over \sqrt 2 \sigma_M(R,z)}\right),
\label{press}$$
where $M={4\over 3}\pi R^3\rho_0$, $\sigma_M(R,z)=\sigma_M(R)/(1+z)$, $\sigma_M(R)$ is the rms value of the mass fluctuation in the linear approximation at the present moment, it is defined in Eqs. (16,17). The choice of the quantity $\delta_c$ that is equal to the value of ${\delta \rho \over \rho}$ in the linear approximation at the moment when the considered region collapses in the course of a fully non-linear evolution is critical for this approach. For the spherical model, $\delta_c=3(12\pi )^{2/3}/20\approx
1.69$, but attempts to fit results of N-body simulations to the Press-Schechter formula did not produce a unique answer. It seems that the best fit for $\delta_c$ lies between 1.33 and 2.
If we accept the most recent estimates of the mass fraction in hosts of quasars at $z=4$ (see, e.g. Haehnelt 1993): $f(\ge 10^{11}M_{\odot})\approx
10^{-4}$, when it follows from Eq.(\[press\]) that the corresponding rms linear mass fluctuation at the present moment
$$\sigma_M(10^{11}M_{\odot})\approx 1.3\delta_c\approx 2.2\pm 0.5
\label{quas}$$
for $\delta_c$ lying in the range mentioned above. Clearly, this is only a lower limit on $\sigma_M$ because quasars may form not in all host galaxies and the presenly observed quasar density at $z=4$ may be less than the real one. However, due to extreme sensitivity of the expression (\[press\]) to a change in $\sigma_M$, the actual value of the latter cannot be significantly larger than that given in Eq.(\[quas\]). A similar estimate was presented in recent paper by Blanchard (1993) (it is $30\%$ higher for their value $\alpha =1.7$).
Another estimate may be obtained from the fact that large galaxies (or, at least, a significant part of them) seem to be already existent at $z=1$. Then, assuming $f(\ge 10^{12}M_{\odot})\ge 0.1$ at $z=1$, we get
$$\sigma_M(10^{12}M_{\odot})\ge 1.2\delta_c\approx 2.0\pm 0.4\, .$$
Note also that the estimate of Haehnelt (1993): $f(\ge 10^{12}M_{\odot})
\ge 10^{-5}$ at $z=4$ - leads to almost the same result: $\sigma_M(10^{12}M_{\odot})\ge 1.1\delta_c$.
(12,7)
Now the same quantities may be calculated for our spectra. Analysis of the results shows that the predicted values of $\sigma_M(10^{11}M_{\odot})$ and $\sigma_M(10^{12}M_{\odot})$ for $\Delta =3$ and 1 Mpc $\le
k_{br}^{-1}\le 20$ Mpc lie just inside the required ranges. For $\Delta =2$ they are probably too high, for $\Delta =4$ - too low (apart from the points $k_{br}^{-1}=1$ Mpc and 20 Mpc).
Conclusions {#con}
===========
We have compared the double inflationary model (1) characterized by two additional free parameters as compared to the standard model with a number of observational tests and have found that there is a reasonable agreement with all considered tests for the following region of the parameters: 1 Mpc $<k_{br}^{-1}<10$ Mpc (that corresponds to real length scales $L_{br}\approx (6-60)$ Mpc) and $2<\Delta <4$. The best fit seems to be given by $k_{br}^{-1}=$ (3-7) Mpc, $\Delta\approx 3$. Generally, there is a correlation between a step size and a break location giving a good fit to the data, i.e., for a break at larger scales also a higher step is necessary. However, due to a complicated structure of the obtained spectra (e.g., oscillations throughout the range of the break for $\Delta >2 $) this connection is not so straightforward.
The most restricting for the model are large scale bulk flows. To obtain even a marginal agreement we have to assume that their actual rms values are at least $\approx 1.4$ times less than the average values of Dekel’s (1992) data (due to the cosmic variance) and that the rms value of large-angle CMB temperature fluctuations is not less than the average result of COBE. These assumptions are certainly crucial tests for the considered model.
0.5cm The work of one of the authors (A.S.) was supported by the Russian research project “Cosmomicrophysics”. A.S. also thanks the Deutsche Forschungsgemeinschaft (DFG) for the finacial support of his visit to the Potsdam Astrophysical Institute. We would like to thank Steven Maddox for sending us the latest version of his $w(\theta )$ data.
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|
---
abstract: 'While the major contribution to the Cosmic Microwave Background (CMB) anisotropies are the sought-after primordial fluctuations produced at the surface of last scattering, other effects produce secondary fluctuations at lower redshifts. These secondary fluctuations must be carefully accounted for, in order to isolate primordial fluctuations. In addition, they are interesting in their own right, since they provide a wealth of information on the geometry and local properties of the universe. Here, I survey the different sources of secondary anisotropies and extragalactic foregrounds of the CMB. I show their relative importance on the multipole-frequency plane. I discuss in particular their impact in the future CMB missions and Planck Surveyor.'
author:
- 'A. Refregier'
title: Overview of Secondary Anisotropies of the CMB
---
Introduction
============
The Cosmic Microwave Background (CMB) provides a unique probe of the early universe (see White, Scott, & Silk 1994 for a review). If CMB fluctuations are consistent with inflationary models, future ground-based and satellite experiments will yield accurate measurements of most cosmological parameters (see Zaldarriaga, Spergel, & Seljak 1997; Bond, Efstathiou, & Tegmark 1997 and reference therein). These measurements rely on the detection of primordial anisotropies produced at the surface of last scattering. However, various secondary effects produce fluctuations at lower redshifts. The study of these secondary fluctuations (or extragalactic foregrounds) is important in order to isolate primordial fluctuations. In addition, secondary fluctuations are interesting in their own right since they provide a wealth of information on the local universe.
In this contribution, I present an overview of the different extragalactic foregrounds of the CMB. The foregrounds produced by discrete sources, the thermal Sunyaev-Zel’dovich (SZ) effect, the Ostriker-Vishniac (OV) effect, the Integrated Sachs-Wolfe (ISW) effect, gravitational lensing, and other effects, are briefly described. I show their relative importance on the multipole-frequency plane, and pay particular attention to their impact on the future CMB missions (Bennett et al. 1995) and Planck Surveyor(Bersanelli et al. 1996). A more detailed account of each extragalactic foreground can be found in the other contributions to this volume. In this article, I have focused on the latest literature, and have not aimed for bibliographical completeness. This overview is based on a more detailed study of extragalactic foregrounds in the context of the mission (Refregier et al. 1998).
Comparison of Extragalactic Foregrounds {#foregrounds}
=======================================
To assess the relative importance of the extragalactic foregrounds, I decompose the temperature fluctuations of the CMB into the usual spherical harmonic basis, $\frac{\delta{T}}{T_{0}}(\theta)=\sum_{\ell,m} a_{lm} Y_{lm}(\theta)$, and form the averaged multipole moments $C_{l}\equiv \langle
|a_{lm}|^{2} \rangle$. Following Tegmark & Efstathiou (1996), I consider the quantity $ \Delta T_{\ell} \equiv \left[ \ell(2 \ell+1)
C_{\ell}/4 \pi \right]^{\frac{1}{2}} T_{0}$, which gives the [*rms*]{} temperature fluctuations per $\ln \ell$ interval centered at $\ell$. Another useful quantity that they considered is the value of $\ell=\ell_{eq}$ for which foreground fluctuations are equal to the CMB fluctuations, i.e. for which $C_{\ell}^{\rm foreground} \simeq
C_{\ell}^{\rm CMB}$. Note that, since the foregrounds do not necessarily have a thermal spectrum, $\Delta T_{\ell}$ and $\ell_{eq}$ generally depend on frequency.
The comparison is summarized in table \[tab:foregrounds\] and in figure \[fig:lnu\]. Table \[tab:foregrounds\] shows $\Delta
T_{\ell}$ and $\ell_{eq}$ for each of the major extragalactic foregrounds at $\nu=94$ GHz and $\ell=450$, which corresponds to a FWHM angular scale of about $\theta \sim .3$ deg. These values were chosen to be relevant to the W-band ($\nu \simeq 94$ GHz and $\theta_{\rm beam} \simeq 0\fdg21$. I also indicate whether each foreground component has a thermal spectrum.
Figure \[fig:lnu\] summarizes the importance of each of the extragalactic foregrounds in the multipole-frequency plane. It should be compared to the analogous plot for galactic foregrounds (and discrete sources) shown in Tegmark & Efstathiou (1996; see also Tegmark 1997 for an updated version). These figures show regions on the $\ell$-$\nu$ plane in which the foreground fluctuations exceed the CMB fluctuations, i.e. in which $C_{\ell}^{\rm foreground} >
C_{\ell}^{\rm CMB}$. As a reference for $C_{\ell}^{\rm CMB}$, a COBE normalized CDM model with $\Omega_{b}=0.05$ and $h=0.5$ was used. Also shown in figure \[fig:lnu\] is the region in which and Planck Surveyor are sensitive, i.e. in which $\Delta C_{\ell}^{\rm
noise} < C_{\ell}^{\rm CMB}$, where $\Delta C_{\ell}^{\rm noise}$ is the [*rms*]{} uncertainty for the instrument. Note that this figure is only intended to illustrate the domains of importance of the different foregrounds qualitatively.
In the following, I briefly describe each extragalactic foreground and comment on its respective entries in table \[tab:foregrounds\] and figure \[fig:lnu\].
---------- ---------------------------- ----------------- --------- ----------------- ------
Source $\Delta T_{\ell}$ ($\mu$K) $\ell_{\rm eq}$ Thermal Note Ref.
CMB 50 yes 1
Discrete 5 1800 no $S<1.0$ Jy 2
2 3100 no $S<0.1$ Jy 2
SZ 10 1900 no C 3
7 2300 no NC 3
OV 2 2900 yes $z_{r}=50$ 4
1 3100 yes $z_{r}=10$ 4
ISW 1 5000 yes $\Omega h=0.25$ 5
0.9 5800 yes $\Omega h=0.50$ 5
Lensing 5 2400 yes 6
---------- ---------------------------- ----------------- --------- ----------------- ------
: Summary of Extragalactic Foregrounds for $\nu=94$ GHz and $\ell=450$.[]{data-label="tab:foregrounds"}
Discrete Sources
----------------
Discrete sources produce positive, point-like, non-thermal fluctuations. While not much is known about discrete source counts around $\nu \sim 100$ GHz, several models have been constructed by interpolating between radio and IR observations (Toffolatti et al. 1998; Gawiser & Smoot 1997; Gawiser et al. 1998; Sokasian et al. 1998). Here, I adopt the model of Toffolatti et al. and consider the two flux limits $S <1$ and 0.1 Jy for the source removal in table \[tab:foregrounds\]. The sparsely dotted region figure \[fig:lnu\] shows the discrete source region for $S <1$ Jy. In the context of CMB experiments, the Poisson shot noise dominates over clustering for discrete sources (see Toffolatti et al. ). As a result, the discrete source power spectrum, $C_{\ell}^{\rm discrete}$, is essentially independent of $\ell$.
Thermal Sunyaev-Zel’dovich Effect {#foregrounds_sz}
---------------------------------
The hot gas in clusters and superclusters of galaxies affect the spectrum of the CMB through inverse Compton scattering. This effect, known as the Sunyaev-Zel’dovich effect (for reviews see Sunyaev & Zel’dovich 1980; Rephaeli 1995), results from both the thermal and bulk motion of the gas. We first consider the thermal SZ effect, which typically has a larger amplitude and has a non-thermal spectrum (see the §\[OV\] below for a discussion of the kinetic SZ effect). The CMB fluctuations produced by the thermal SZ effect have been studied using the Press-Schechter formalism (see Bartlett 1997 for a review), and on large scales using numerical simulations (Cen & Ostriker 1992; Scaramella, Cen, & Ostriker 1993) and semi-analytical methods (Persi et al. 1995). Here, I consider the SZ power spectrum, $C_{\ell}^{\rm
SZ}$, calculated by Persi et al. (see their figure 5). In table \[tab:foregrounds\], I consider their calculation both with and without bright cluster removal. In figure \[fig:lnu\], only the spectrum without cluster removal is shown.
Ostriker-Vishniac Effect {#OV}
------------------------
In addition to the thermal SZ effect described above, the hot intergalactic medium can produce thermal CMB fluctuations as a result of its bulk motion. While this effect essentially vanishes to first order, the second order term in perturbation theory, the Ostriker-Vishniac effect (Ostriker & Vishniac 1986; Vishniac 1987), can be significant on small angular scales. The power spectrum of the OV effect depends on the ionization history of the universe, and has been calculated by Hu & White (1996), and Jaffe & Kamionkowski (1998; see also Persi et al. 1995). We use the results of Hu & White (see their figure 5) who assumed that the universe was fully reionized beyond a redshift $z_{r}$. In table \[tab:foregrounds\], I consider the two values $z_{r}=10$ and 50, while in figure \[fig:lnu\], I only plot the region corresponding to $z_{r}=50$. For consistency, the standard CDM power spectrum is still used as a reference, even though the primordial power spectrum would be damped in the event of early reionization. (Using the damped primordial spectrum makes, at any rate, only small corrections to both table \[tab:foregrounds\] and figure \[fig:lnu\].)
Integrated Sachs-Wolfe Effect
-----------------------------
The Integrated Sachs-Wolfe Effect (ISW) describes thermal CMB fluctuations produced by time variations of the gravitational potential along the photon path (Sachs & Wolfe 1967). Linear density perturbations produce non-zero ISW fluctuations in a $\Omega_m \neq 1$ universe only. Non-linear perturbations produce fluctuations for any geometry, an effect often called the Rees-Sciama effect (Rees & Sciama 1968). Tuluie & Laguna (1995) have shown that anisotropies due to intrinsic changes in the gravitational potentials of the inhomogeneities and anisotropies generated by the bulk motion of the structures across the sky generate CMB anisotropies in the range of $10^{-7} \la \frac{\Delta T}{T} \la 10^{-6}$ on scales of about $1^{\circ}$ (see also Tuluie et al. 1996). The power spectrum of the ISW effect in a CDM universe was computed by Seljak (1996a; see also references therein). In table \[tab:foregrounds\], I consider values of the density parameter, namely $\Omega h=0.25$ and $0.5$. In figure \[fig:lnu\], only the $\Omega h=0.25$ case is shown. As above, the standard CDM ($\Omega =1$, $h=0.5$) spectrum is still used as a reference.
Gravitational Lensing
---------------------
Gravitational lensing is produced by spatial perturbations in the gravitational potential along the line of sight (see Schneider, Ehlers, & Falco 1992; Narayan & Bartelmann 1996). This effect does not directly generate CMB fluctuations, but modifies existing background fluctuations. The effect of lensing on the CMB power spectrum was calculated by Seljak (1996b) and Metcalf & Silk (1997). Recently, Zaldarriaga & Seljak (1998a) included the lensing effect in their CMB spectrum code (CMBFAST; Seljak & Zaldarriaga 1996). This code was used to compute the absolute lensing correction $|\Delta C_{\ell}^{\rm lens}|$ to the standard CDM spectrum, including nonlinear evolution. The results are shown in table \[tab:foregrounds\] and figure \[fig:lnu\].
Other Extragalactic Foregrounds
-------------------------------
In addition to the effects discussed above, other extragalactic foregrounds can cause secondary anisotropies. For instance, patchy reionization produced by the first generation of stars or quasars can cause second order CMB fluctuations through the doppler effect (Aghanim et al. 1996a,b; Gruzinov & Hu 1998; Knox, Scoccimaro, & Dodelson 1998; Peebles & Juzkiewicz 1998). Calculations of the spectrum of this effect are highly uncertain, but show that the resulting CMB fluctuations could be of the order of 1 $\mu$K on 10 arcminute scales, for extreme patchiness. More likely patchiness parameters make the effect negligible on these scales, but potentially important on arcminute scales. Another potential extragalactic foreground is that produced by the kinetic SZ effect from Ly$_{\alpha}$ absorption systems, as was recently proposed by Loeb (1996). The resulting CMB fluctuations are of the order of a few $\mu$K on arcminute scales, and about one order of magnitude lower on 10 arcminute scales. Because of the uncertainties in the models for these two foregrounds and because they are small on 10 arcminute scales, they are not included in table \[tab:foregrounds\] and figure \[fig:lnu\].
Discussion and Conclusion
=========================
An inspection of table \[tab:foregrounds\] shows that, at 94 GHz and $\ell=450$, the power spectra of the largest extragalactic foregrounds considered are a factor of 5 below the primordial CDM spectrum. As can be seen in figure \[fig:lnu\], the dominant foregrounds for and Planck Surveyor are discrete sources, the thermal SZ effect and gravitational lensing. Note that, for Planck surveyor, these three effects produce fluctuations which are close to the sensitivity of the instrument. The spectra of the OV and ISW effects will produce fluctuations of the order of $1 \mu$K , and are thus less important for a measurement of the power spectrum. The effect of gravitational lensing is now incorporated in CMB codes such as CMBFAST, and can thus be taken into account in the estimation of cosmological parameters. The other two dominant extragalactic contributions, discrete sources and the thermal SZ effect, must also be accounted for, but are more difficult to model. Note that, on large angular scales, extragalactic foregrounds produce relatively small fluctuations, and are thus not detectable in the COBE maps (Boughn & Jahoda 1993; Bennett et al. 1993; Banday et al. 1996; Kneissl et al. 1997)
While I have concentrated above on the power spectrum, secondary anisotropies are also a source of non-gaussianity in CMB maps. Discrete sources and the SZ effect from clusters of galaxies mainly produce Poisson fluctuations and are thus clearly non-gaussian. The other extragalactic foregrounds (SZ, OV, ISW, and lensing) are also non-gaussian and trace large-scale structures in the local universe. As a consequence of the latter fact, extragalactic foregrounds can be probed by cross-correlating CMB maps with galaxy catalogs, which act as tracers of the large scale structure. Such technique can be used to detect the ISW effect (Boughn et al. 1998, and reference therein), gravitational lensing (Suginohara et al. 1998) and the SZ effect by superclusters (Refregier et al. 1998). Gravitational lensing is particularly interesting since it produces a specific non-gaussian signature (Bernardeau 1998). This signature can be used to reconstruct the gravitational potential projected along the line of sight (Zaldarriaga & Seljak 1998b). Further non-gaussian signatures result from the fact that the different extragalactic foregrounds are spatially correlated. For instance, a detection of the cross-correlation signal between gravitational lensing and the ISW and SZ effects would allow us to determine the fraction of the ionized gas and the time evolution of gravitational potential (Goldberg & spergel, 1998; Seljak & Zaldarriaga 1998). A detection of secondary anisotropies would help break the degeneracy between cosmological parameters measured from primary anisotropies alone.
I thank David Spergel and Thomas Herbig for active collaboration and discussions on this project. This work was supported by the MIDEX program.
Aghanim, N., Desert, F.-X., Puget, J. L., & Gispert, R. 1996a, , 311, 1; see also erratum to appear in , preprint astro-ph/9811054 Aghanim, N., Puget, J. L., & Gispert, R. 1996b, in Microwave Background Anisotropies, proceedings des XVIièmes Rencontres de Moriond, p. 407, eds. Bouchet, F.R., Gispert, R., Guiderdoni, B., & Trân Thanh Vân, J. Bartlett, J. G. 1997, course given at From Quantum Fluctuations to Cosmological Structures, Casablanca, Dec. 1996, preprint astro-ph/9703090 Banday, A. J., Górski, K. M., Bennett, C. L., Hinshaw, G., Kogut, A., & Smoot, G. F. 1996, , 468, L85 Bennett, C. L., Hinshaw, W. G., Banday, A., Kogut, A., Wright, E. L., Loewenstein, K., & Cheng, E. S. 1993, , 414, L77 Bennett, C.L. et al. 1995, BAAS 187.7109; see also http://map.gsfc.nasa.gov Bernardeau, F. 1998, , 338, 767 Bersanelli, M. et al. 1996, COBRAS/SAMBA, Report on Phase A Study, ESA Report D/SCI(96)3; see also http://astro.estec.esa.nl/Planck/ Bond, J.R., Efstathiou, G., & Tegmark, M. 1997, , 291, L33 Boughn, S. P., & Jahoda, K. 1993, , 412, L1 Boughn, S. P., Crittenden, R.G., & Turok, N.G. 1998, NewA, 3, 275 Cen, R., & Ostriker, J. P. 1992, , 393, 22 Gawiser, E., & Smoot, G. 1997, , 480, L1 Gawiser, E., Jaffe, A., & Silk, J., 1998, submitted to , preprint astro-ph/9811148 Goldberg, D., & Spergel, D. 1998, preprint astro-ph/9811251 Gruzinov, A. & Hu, W. 1998, submitted to , preprint astro-ph/9803188 Hu, W. & White, M. 1996, , 315, 33 Jaffe, A. H., & Kamionkowski, M. 1998, to appear in , preprint astro-ph/9801022 Kneissl, R., Egger, R., Hasinger, G., Soltan, A. M., & Trümper, J. 1997, , 320, 685 Knox, L., Scoccimaro, R., & Dodelson, S. 1998, preprint astro-ph/9805012 Loeb, A. 1996, , 471, L1 Metcalf, R. B., & Silk, J. 1997, , 489, 1 Narayan, R., & Bartelmann, M. 1996, Lectures on Gravitational Lensing, preprint astro-ph/9606001 Ostriker, J. P. & Vishniac, E. T. 1986, , 306, L51 Peebles, P. J. E. & Juszkiewicz, R. 1998, preprint astro-ph/9804260 Persi, F. M., Spergel, D. N., Cen, R., & Ostriker, J. P. 1995, ApJ, 442, 1 Rees, M. & Sciama, D. W. 1968, Nature, 517, 611 Sachs, R. K. & Wolfe, A.M. 1967, ApJ, 147, 73 Refregier, A., Spergel, D., & Herbig, T. 1998, submitted to , preprint astro-ph/9806349 Rephaeli, Y. 1995, , 33, 541 Scaramella, R., Cen, R., & Ostriker, J. P. 1993, , 416, 399 Schneider, P., Ehlers, J., & Falco, E. E. 1992, Gravitational Lenses, (New York: Springer-Verlag) Seljak, U. 1996a, , 460, 549 Seljak, U. 1996b, , 463, 1 Seljak, U. & Zaldarriaga, M. 1996, , 469, 437 Seljak, U. & Zaldarriaga, M. 1998, preprint astro-ph/9811123 Spergel, D. & Goldberg, D. 1998, preprint astro-ph/9811252 Sokasian, A., Gawiser, E., & Smoot, G. F. 1998, submitted to , preprint astro-ph/9811311 Suginohara, M., Suginohara, T., & Spergel, D 1998, , 495, 511 Sunyaev, R. A. & Zeldovich, Y. B. 1980, , 18, 537 Tegmark, M. & Efstathiou, G. 1996, , 281, 1297 Tegmark, M. 1997, to appear in ApJ, preprint astro-ph/9712038 Toffolatti, L., Argüeso Gómez, F., De Zotti, G., Mazzei, P., Franceschini, A., Danese, L., & Burigana, C. 1998, , 297, 117 Tuluie, R., Laguna, P. 1995, , 445, L73 Tuluie, R., Laguna, P., & Anninos, P. 1996, , 463, 15 Vishniac, E. T. 1987, , 322, 597 White, M., Scott, D., & Silk, J. 1994, ARAA, 32,319 Zaldarriaga, M., Spergel, D. N., & Seljak, U. 1997, , 488, 1 Zaldarriaga, M. & Seljak, U. 1998a, preprint astro-ph/9803150 Zaldarriaga, M. & Seljak, U. 1998b, preprint astro-ph/9810257
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---
abstract: 'Using Taubes’ periodic ends theorem, Auckly gave examples of toroidal and hyperbolic irreducible integer homology spheres which are not surgery on a knot in the three-sphere. We give an obstruction to a homology sphere being surgery on a knot coming from Heegaard Floer homology. This is used to construct infinitely many small Seifert fibered examples.'
address:
- 'Department of Mathematics, Columbia University, New York, NY 10027'
- 'Department of Mathematics, Bo[ğ]{}azi[ç]{}i University, Bebek, [İ]{}stanbul, TR-34342, Turkey, and, The University of Texas, Austin, TX 78712'
- 'Department of Mathematics, The University of Texas, Austin, TX 78712'
author:
- Jennifer Hom
- 'Çağr[i]{} Karakurt'
- Tye Lidman
bibliography:
- 'References.bib'
title: Surgery obstructions and Heegaard Floer homology
---
|
[****]{}\
Vincenzo Branchina[^1]\[one\]
Department of Physics, University of Catania and\
INFN, Sezione di Catania, Via Santa Sofia 64, I-95123, Catania, Italy
Marco Di Liberto[^2]\[two\]
Scuola Superiore di Catania, Via S. Nullo 5/i, Catania, Italy
Ivano Lodato[^3]\[three\]
Scuola Superiore di Catania, Via S. Nullo 5/i, Catania, Italy and\
INFN, Sezione di Catania, Via Santa Sofia 64, I-95123, Catania, Italy
[Abstract]{}\
It has been recently claimed that dark energy can be (and has been) observed in laboratory experiments by measuring the power spectrum $S_I(\omega)$ of the noise current in a resistively shunted Josephson junction and that in new dedicated experiments, which will soon test a higher frequency range, $S_I(\omega)$ should show a deviation from the linear rising observed in the lower frequency region because higher frequencies should not contribute to dark energy. Based on previous work on theoretical aspects of the fluctuation-dissipation theorem, we carefully investigate these issues and show that these claims are based on a misunderstanding of the physical origin of the spectral function $S_I(\omega)$. According to our analysis, dark energy has never been (and will never be) observed in Josephson junctions experiments. We also predict that no deviation from the linear rising behavior of $S_I(\omega)$ will be observed in forthcoming experiments. Our findings provide new (we believe definite) arguments which strongly support previous criticisms.
Introduction
============
The origin of dark energy is one of the greatest mysteries confronting theoretical and experimental physics. Different proposals for the solution of this so called cosmological constant problem are put forward and many review articles nowadays discuss and compare these alternative approaches (see for instance [@peeb],[@padam],[@cope],[@nobe]).
Among many other issues, Copeland et al.[@cope] discuss the suggestion of Beck and Mackey[@bema1; @bema2] according to which dark energy can be (and has been) observed in laboratory experiments [@koch] by measuring the power spectrum of the noise current in a resistively shunted Josephson junction. If true, this would mark a dramatic progress in our understanding of the origin of dark energy.
According to[@bema1; @bema2], this power spectrum is due to thermal and vacuum fluctuations of the electromagnetic field in the resistor and these experiments[@koch] provide a measurement of the electromagnetic zero point energies, which they consider as being at the origin of dark energy.
These ideas have generated a certain debate and some authors[@jetz1; @doran; @maha] have argued against them. Beck and Mackey have rebutted these criticisms[@bema3] but they were again criticized in [@jetz2]. Copeland et al.close the section of their review devoted to this issue by saying that “time will tell who (if either) are correct”[@cope]. Needless to say, this issue is of the greatest importance and deserves further investigation.
Scope of this work is to bring additional elements to the analysis of this problem in the hope that the question posed in[@cope] could finally find an answer. To this end, it is necessary to review in some detail the Beck and Mackey proposal[@bema1; @bema2].
These authors begin by considering the work of Koch et al.[@koch], where the spectral density $S_I(\omega)$ of the noise current in the resistor of a resistively shunted Josephson junction was measured and confronted against the theoretical prediction, \[spectr\] S\_I() = (+ ), and good agreement was found between experimental results and theory ($T$ is the temperature and $R$ the resistance of the resistive shunt).
Eq.(\[spectr\]) comes from an application of the fluctuation-dissipation theorem (FDT)[@cawe] and we immediately recognize the term in parenthesis as the mean energy of a quantum harmonic oscillator of frequency $\omega$ in a thermal bath. Where does this Bose-Einstein (BE) distribution factor come from? Does it reflect an underlying “harmonic oscillator structure” of the system[@taylor]? If yes, which harmonic oscillators are involved in Eq.(\[spectr\])? A correct answer to these questions will turn out to be crucial in understanding the status of the Beck and Mackey proposal[@bema1; @bema2] and, we believe, in settling the controversy.
Beck and Mackey interpret this factor as coming from the modes of the electromagnetic field in interaction with the charged particles [@bema2] and claim that this experiment provides a direct measurement of vacuum fluctuations of the electromagnetic field (the $\frac{\hbar\omega}{2}$ term). Moreover, they assume that dark energy originates from vacuum fluctuations of fundamental quantum fields and conjecture that only those fluctuations which can be measured in terms of a physical power spectrum are gravitationally active, i.e. contribute to dark energy. Then, by observing that for strong and electroweak interactions it is unlikely that a suitable macroscopic detector exists that can measure the corresponding vacuum spectra, they conclude that the only candidate where we know that a suitable macroscopic detector exists is the electromagnetic interaction[@bema2].
Accordingly, by noting that astrophysical measurements give $\varrho_{dark} \sim 10^{-47} GeV^4$ (in natural units), they argue that there should be a physical cut-off frequency $\nu_c =\frac{\omega_c}{2\pi}\sim 1.7\, {\rm THz}$ such that, for frequencies above this cut-off, the spectral function of the noise current in the Josephson junction should behave differently than in Eq.(\[spectr\]). According to their hypothesis, in fact, for $\omega \geq \omega_c $ the $\frac{\hbar\omega}{2}$ term should be absent.
Coming back to the BE distribution factor in Eq.(\[spectr\]), Jetzer and Straumann[@jetz1] (see also [@kubo]) observed that this term simply comes from ratios of Boltzmann factors which appear in the derivation of the FDT and stressed that the $\frac{\hbar\omega}{2}$ term has nothing to do with zero point energies, while Beck and Mackey reply that this is contrary to the view commonly expressed in the literature[@bema3].
A simple look to the derivation of the FDT (see Section 2) shows that, as for the ratios of Boltzmann factors, Jetzer and Straumann[@jetz1] are definitely right. Nevertheless, in a sense that we are going to make clear in the following, there is an element of truth in the common lore according to which this factor can be regarded as due to a sort of underlying harmonic oscillator structure of the system (the resistive shunt in the case of the Koch et al. experiment[@koch]).
In a recent paper[@noi] we have shown that whenever linear response theory applies, which is the main hypothesis under which the FDT is derived, any generic bosonic and/or fermionic system can be mapped onto a fictitious system of harmonic oscillators in such a manner that the quantities appearing in the FDT coincide with the corresponding quantities of the fictitious one (for completeness, in sections 2 and 3 we briefly review these results. For a comprehensive exposition, however, see[@noi]).
This allows us to understand [*in which sense*]{} the harmonic oscillator interpretation can be put forward so that we shall be able to say whether the Beck and Mackey’s proposal is tenable or not. We shall see that it is not.
The rest of the paper is organized as follows. In Section 2 we briefly review the derivation of the FDT and consider two convenient expressions for the power spectrum of the fluctuating observable and for the imaginary part of the corresponding generalized susceptibility respectively. In Section 3 we consider the special case of a system of harmonic oscillators in interaction with an external field and show how the above mentioned mapping is constructed. In Section 4 we apply the results of the two previous sections to our problem, namely the dark energy interpretation[@bema1; @bema2] of the measured power spectrum $S_I(\omega)$[@koch] and show that this interpretation is untenable. Section 5 is for our conclusions.
The fluctuation-dissipation theorem
===================================
In the present section we briefly review the derivation of the FDT (see[@kubo] for more details) and provide expressions for the spectral function and the imaginary part of the generalized susceptibility which will be useful for our following considerations.
Consider a macroscopic system with unperturbed hamiltonian $\hat{H}_0$ under the influence of the perturbation \[inter\] = - f(t)(t), where $\hat{A}(t)$ is an observable (a bosonic operator) of the system and $f(t)$ an external generalized force[^4]. Let $|E_n\rangle$ be the $\hat{H}_0$ eigenstates (with eigenvalues $E_n$) and $\langle E_n|\hat{A}(t)|E_n \rangle =0$. Within the framework of linear response theory, the quantum-statistical average $\langle\hat{A}(t)\rangle_f$ of the observable $\hat{A}(t)$ in the presence of $\hat{V}$ is given by \[resp2\] (t)\_f = \_[-]{}\^t d t’ \_[\_[A]{}]{}(t-t’) f(t’) where $\chi_{_{A}}(t - t')$ is the generalized susceptibility, \[chi\] \_[\_[A]{}]{}(t - t’)=(t-t’) = -G\_R(t - t’) , with $\langle ... \rangle =
\sum_{n} \varrho_n \langle E_n| ... |E_n \rangle$, $\varrho_n= e^{-\beta E_n}/Z$ , $Z=\sum_n e^{- \beta E_n}$, $G_R(t-t')$ being the retarded Green’s function and $\hat{A}(t)=e^{i\hat{H_0}t/\hbar}\hat{A}e^{-i\hat{H_0}t/\hbar}$.
If we now consider the mean square of the observable $\hat A(t)$ and write the generalized susceptibility $\chi_{_{A}}$ as $\chi_{_{A}} = \chi^{'}_{_{A}} + i\,\chi^{''}_{_{A}}$ (with $\chi^{'}_{_{A}}$ the real part and $\chi^{''}_{_{A}}$ the imaginary part of $\chi_{_{A}}$), it is not difficult to show (see [@kubo] and [@noi]) that the Fourier transform $\langle \hat{A}^2(\omega)\rangle$ of $\langle \hat{A}^2(t)\rangle$ is related to the Fourier transform $\chi_{_{A}}^{\,''}(\omega)$ through the relation \[fddt\] \^2()= \_[\_[A]{}]{}\^[”]{}() = \_[\_[A]{}]{}\^[”]{}() [coth]{}( ) =2\_[\_[A]{}]{}\^[”]{}() (1 2 + ), which is the celebrated FDT.
In Eq.(\[fddt\]) we recognize the ratio of Boltzmann factors alluded by Jetzer and Straumann[@jetz1]. Actually, it was already observed by Kubo et al.[@kubo] that the BE factor in Eq.(\[fddt\]) is simply due to a peculiar combination of Boltzmann weights and that there is no reference to physical harmonic oscillators of the system whatsoever. However, as we already said in the Introduction, there is an element of truth in the common lore which considers this term as due to a sort of harmonic oscillator structure of the system (the resistive shunt in the case of the Koch et al.experiment[@koch]).
In order to show that, we now refer to our recent work[@noi], where we have derived the following useful expressions for $\langle \hat{A}^2(\omega)\rangle$ and $\chi_{_A}^{\,''}(\omega)$ : \^2()&=& \_[j > i]{}(\_i - \_j) |A\_[ij]{}|\^2 [coth]{}( ) \[o5\]\
&=& [coth]{}()\_[j > i]{} (\_i - \_j) |A\_[ij]{}|\^2 ,\[o6\]\
”()&=&\_[j > i]{}(\_i - \_j) |A\_[i j]{}|\^2 \[chii3\]. Clearly, from Eqs.(\[o6\]) and (\[chii3\]) the FDT (Eq.(\[fddt\])) is immediately recovered. However, what matters for our scopes are the explicit expressions in Eqs.(\[o5\]) and (\[chii3\]). Starting from these equations, in fact, we can easily show that it is possible to build up a mapping between the real system and a fictitious system of harmonic oscillators[@noi] in such a manner that $\chi_{_A}^{\,''}(\omega)$ and $\langle \hat{A}^2(\omega)\rangle$ are exactly reproduced by considering the corresponding quantities of the fictitious system. In the following section we outline the main steps for this construction (see [@noi] for details).
The Mapping
===========
In order to build up this mapping, we consider first a system ${\cal S}_{osc}$ of harmonic oscillators (each of which is labeled below by the double index $\{ji\}$ for reasons that will become clear in the following) whose free hamiltonian is: \[armonico\] H\_[osc]{} = \_[j > i]{}( + q\_[ji]{}\^[2]{}), where $\omega_{ji}$ are the proper frequencies of the individual harmonic oscillators and $M_{ji}$ their masses. Let $| n_{j i}\rangle$ ($n_{ji}=0, 1,2,...$) be the occupation number states of the $\{ji\}$ oscillator out of which the Fock space of ${\cal S}_{osc}$ is built up. Let us consider also ${\cal S}_{osc}$ in interaction with an external system through the one-particle operator: \[armint\] V\_[osc]{} = - f(t) A\_[osc]{}, with \[onepart\] [A]{}\_[osc]{} = \_[j > i]{} (\_[j i]{} [q]{}\_[ji]{} ). Obviously, the FDT applied to ${\cal S}_{osc}$ gives $\langle {\hat A}_{osc}^2(\omega)\rangle =
\hbar \chi_{osc}^{\,''}(\omega) \,{\rm coth}\left( \frac{\beta\hbar\omega}{2}\right)$, but this is not what matters to us.
What is important for our purposes is that, as shown in[@noi], for ${\cal S}_{osc}$ we can exactly compute $\langle {\hat A}_{osc}^2(\omega) \rangle$ and $\chi_{osc}^{\,''}(\omega)$. The reason is that for this system, differently from any other generic system, we can explicitly compute the matrix elements of ${\hat A}_{osc}$. The result is (compare with Eqs.(\[o5\]), (\[o6\]) and (\[chii3\])): \_[osc]{}\^2()&=&\_[j>i]{} \_[ji]{}\^2 [coth]{}() \[(-\_[ji]{}) +(+\_[ji]{})\]\[osc5\]\
&=& [coth]{}()\_[j>i]{} \_[ji]{}\^2 \[(-\_[ji]{}) - (+\_[ji]{})\]\[oscc5\];\
\_[osc]{}\^[”]{} ()&=& \_[j>i]{} \_[ji]{}\^2 \[(-\_[ji]{}) - (+\_[ji]{})\]\[chi3\].
Naturally, comparing Eq.(\[oscc5\]) with Eq.(\[chi3\]) we see that for ${\cal S}_{osc}$ the FDT holds true, as it should. However, for our scopes it is important to note the following. For this system, the ${\rm coth}\left(\frac{\beta \hbar\omega}{2}\right)$ factor of the FDT originates from the [*individual contributions*]{} ${\rm coth}\left(\frac{\beta \hbar\omega_{ji}}{2}\right)$ of each of the harmonic oscillators of ${\cal S}_{osc}$.
We can now build up our mapping. Let us consider the original system ${\cal S}$, described by the unperturbed hamiltonian $\hat H_0$, in interaction with an external field $f(t)$ through the interaction term $\hat V = - f (t)\,\hat A$ (see Eq.(\[inter\])), and construct a fictitious system of harmonic oscillators ${\cal S}_{osc}$, described by the free hamiltonian ${\hat H}_{osc}$ of Eq.(\[armonico\]), in interaction with the same external field $f(t)$ through the interaction term ${\hat V}_{osc}$ of Eq.(\[armint\]), with $\hat A_{osc}$ given by Eq.(\[onepart\]), where for $\alpha_{j i}$ we choose \[alfa\] \_[j i]{} = ()\^[12]{} (\_i - \_j)\^[12]{} |A\_[ij]{}| and for the proper frequencies $\omega_{ji}$ of the oscillators \[omega\] \_[ji]{}= (E\_j-E\_i)/> 0, with $E_i$ the eigenvalues of the hamiltonian ${\hat H}_0$ of the real system.
By comparing Eq.(\[oscc5\]) with Eq.(\[o6\]) and Eq.(\[chi3\]) with Eq.(\[chii3\]), it is immediate to see that with the above choices of $\alpha_{ji}$ and $\omega_{ji}$ we have: \^2()&=& \_[osc]{}\^2()\[cen1\]\
\_[\_A]{}\^[”]{} () &=& \_[osc]{}\^[”]{} ()\[cen2\]. Eqs.(\[cen1\]) and (\[cen2\]) define the mapping we are looking for. They show that it is possible to map the real system ${\cal S}$ onto a fictitious system of harmonic oscillators ${\cal S}_{osc}$, [S]{}\_[osc]{}, in such a manner that $\chi_{_A}^{''} (\omega)$ and $\langle \hat{A}^2(\omega)\rangle$ of the real system are equivalently obtained by computing the corresponding quantities of the fictitious one. The key ingredient to construct such a mapping is the hypothesis that linear response theory is applicable (which is the central hypothesis under which the FDT is established).
Now, by considering the “equivalent” harmonic oscillators system ${\cal S}_{osc}$ rather than the real one, we can somehow regard the BE distribution factor ${\rm coth}\left( \frac{\beta\hbar\omega}{2}\right)$ of the FDT in Eq.(\[fddt\]) as originating from the individual contributions ${\rm coth}\left( \frac{\beta\hbar\omega_{ji}}{2}\right)$ of each of the oscillators of the equivalent fictitious system (see above, Eqs.(\[osc5\]), (\[oscc5\]) and (\[chi3\])). In this sense, this mapping allows for an oscillator interpretation of the BE term in the FDT.
At the same time, however, the above findings clearly teach us that the BE distribution term in the FDT [*does not describe the physical nature of the system*]{}. It rather encodes a fundamental property of any bosonic and/or fermionic system: whenever linear response theory is applicable, any generic system is equivalent (in the sense defined above) to a system of quantum harmonic oscillators.
Dark energy and laboratory experiments
======================================
We are now in the position to apply the results of the two previous sections to our problem. As we said in the Introduction, Beck and Mackey[@bema1; @bema2] interpret the Koch et al.experimental results[@koch] for the spectral density $S_I(\omega)$ of the noise current in a resistively shunted Josephson junction as a direct measurement of [*vacuum fluctuations of the electromagnetic field*]{} in the shunt resistor. Moreover, according to their ideas, these zero-point energies are nothing but the dark energy of the universe.
In view of our results, however, this interpretation seems to be untenable. Eq.(\[spectr\]) for $S_I(\omega)$ comes from an application of the FDT to the case of the noise current in the shunt resistor. Therefore, according to our findings, the BE distribution factor which appears in $S_I(\omega)$ has nothing to do with thermal and vacuum fluctuations of the electromagnetic field in the resistor. Our analysis shows that this factor rather reflects a general property of any quantum system valid whenever linear response theory applies. The resistor (as well as any other generic system) can be mapped onto a system of fictitious harmonic oscillators in such a manner that the the power spectrum of the noise current and the related susceptibility can be reproduced by considering the equivalent quantities for the fictitious oscillators.
It is in this sense, and [*only in this sense*]{}, that the BE factor can be interpreted in terms of harmonic oscillators, no other physical meaning can be superimposed on it. According to these considerations, we conclude that the claim that dark energy is observed in laboratory experiments[@bema1; @bema2] is based on an incorrect interpretation of the origin of the BE factor in the FDT.
We believe that this should help in solving the controversy, which is left open in the Copeland et al.review[@cope], between the proponents[@bema1; @bema2] of the dark energy interpretation of the Koch et al.experiments[@koch] and the opponents[@jetz1; @doran; @maha; @jetz2]. In this respect, it is worth to stress that our analysis provides new arguments which strongly support the conclusions of these latter works[@jetz1; @doran; @maha; @jetz2].
A distinctive new element of our work, which in our opinion should greatly help in settling the question, concerns the interpretation of the FDT presented in section 5 of[@bema2]. These authors note that, although the FDT is valid for arbitrary hamiltonians $H$, where $H$ need not to describe harmonic oscillators, in the FDT appears a [*universal function*]{} $H_{uni}$, $H_{uni}= \frac12\hbar\omega +\hbar\omega/(exp(\hbar\omega/kT)-1)$, which can always be interpreted as the mean energy of a harmonic oscillator. Then, they identify the $\frac12\hbar\omega$ in $H_{uni}$ as the source of dark energy.
The distinctive feature of our analysis is that, with the help of the formal mapping discussed in the previous section, which is valid for [*any generic system*]{}, the reason for the appearance of this [*universal*]{} function is immediately apparent. At the same time, however, this clearly shows that it cannot be claimed that the Koch et al.[@koch] experimental device is measuring zero point energies. As already noted in[@jetz1; @jetz2], these experiments simply measure a general quantum property of the system, the $\frac12\hbar\omega$ in $S_I(\omega)$ has nothing to do with zero point energies.
Another very important point related to these issues concerns future measurements[@barb; @warb] of the power spectrum $S_I(\omega)$ for values of the frequency higher than those measured by Koch et al.[@koch]. In fact, according to Beck and Mackey[@bema1; @bema2], in forthcoming experiments[@barb; @warb], which are purposely designed to test a higher frequency range of $S_I(\omega)$, we should observe a dramatic change in the behavior of the spectral function $S_I(\omega)$ for these higher values of the frequency due to the presence of a cut-off which separates the gravitationally active modes from those which are not gravitationally active (see the Introduction). In view of our findings, however, we do not expect to observe in these experiments[@barb; @warb] any change in the behavior of $S_I(\omega)$. We simply state that such a cut-off does not exist.
In this respect, we note that Beck and Mackey have recently proposed a new model for dark energy which should naturally incorporate such a cut-off[@bm3]. According to our analysis, this model seems to be deprived of any experimental and theoretical support.
As a consequence of our results, a deviation of $S_I(\omega)$ from the behavior given in Eq.(\[spectr\]) could be observed only if the central hypothesis on which the derivation of the FDT is based, namely the applicability of linear response theory, no longer holds true in this higher frequency region.
Summary and Conclusions
=======================
With the help of a general theorem, which shows that (under the assumption that linear response theory is applicable) any bosonic and/or fermionic fermionic system can be mapped onto a fictitious system of harmonic oscillators, we have shown that the appearance of a Bose-Einstein distribution factor in the power spectrum of the noise current of a resistively shunted Josephson junction[@koch] has nothing to do with a real (physical) harmonic oscillator structure of the shunt resistor. We then conclude that, contrary to recent claims[@bema1; @bema2], experiments where this power spectrum was measured[@koch] do not provide any direct measurement of zero point energies and, as a consequence, no dark energy has ever been measured in these laboratory experiments.
A direct consequence of our analysis is that, contrary to what is predicted in[@bema1; @bema2], we do not expect any deviation from the linear rising behavior of $S_I(\omega)$ with $\omega$. According to our analysis, in fact, the $\frac12\hbar\omega$ term in $S_I(\omega)$ has nothing to do with the dark energy in the universe, therefore we do not expect any cut-off which separates the gravitationally active zero point energies from the gravitationally non-active ones.
Finally, our analysis suggests that the theory which should naturally incorporate such a cut-off[@bm3] is deprived of any experimental and theoretical foundation.
We believe that our work provides a satisfactory answer to the intriguing and important question left open by Copeland et al. in the section of their review devoted to the possibility of measuring dark energy in laboratory experiments[@cope]: “time will tell who (if either) are correct”. According to our analysis, the opponents to the dark energy interpretation of the Koch et al.experiments[@koch] are correct.
We would like to thank Luigi Amico, Marcello Baldo, Pino Falci and Dario Zappalà for many useful discussions.
[99]{} P.J.E. Peebles, B. Ratra, Rev. Mod. Phys. 75, 559 (2003). T.Padmanahban, AIP Conf.Proc.[**[861]{}**]{}, 179 (2006). E.J.Copeland, M.Sami, S.Tsujikawa, Int.J.Mod.Phys.[**[D15]{}**]{}, 1753 (2006). S. Nobbenhuis, Found. Phys. [**36**]{}, 613 (2006). C.Beck, M.C.Mackey, Phys.Lett.[**[B605]{}**]{}, 295 (2005). C.Beck, M.C.Mackey, Physica [**[A379]{}**]{}, 101 (2007). R.H.Koch, D.J.Van Harlingen, J.Clarke, Phys.Rev.[**[B26]{}**]{}, 74 (1982). P.Jetzer, N.Straumann, Phys.Lett.[**[B606]{}**]{} 77 (2005). M. Doran, J.Jaeckel (DESY), [**[JCAP]{}**]{} 0608:010 (2006). G.Mahajan, S.Sarkar, T.Padmanahban, Phys.Lett.[**[B641]{}**]{}, 6 (2006). C.Beck, M.C.Mackey, Fluct.Noise Lett.7:C31 (2007). P.Jetzer, N.Straumann, Phys.Lett.[**[B639]{}**]{} 57 (2006). H.B.Callen, T.A.Welton, Phys.Rev.[**[83]{}**]{}, 34 (1951). J.C.Taylor, J.Phys.: Condens.Matter [**[19]{}**]{}, 106223 (2007). R.Kubo, M.Toda, N.Hashitsume [*Statistical Physics II*]{}, Springer-Verlag, Berlin (1985). V.Branchina, M.Di Liberto, I.Lodato, [*Fluctuation-dissipation theorem and harmonic oscillators*]{}, arXiv:0905.4254v1 \[cond-mat.stat-mech\]. Z.H.Barber, M.G.Blamire, [*Externally-Shunted High-Gap Josephson Junctions: Design, Fabrication and Noise Measurements*]{}, EPSRC grants EP/D029872/1. P.A.Warb, [*Externally-Shunted High-Gap Josephson Junctions: Design, Fabrication and Noise Measurements*]{}, EPSRC grants EP/D029783/1. C.Beck, M.C.Mackey, Int.J.Mod.Phys.[**[D17]{}**]{}, 71 (2008).
[^1]: vincenzo.branchina@ct.infn.it
[^2]: madiliberto@ssc.unict.it
[^3]: ivlodato@ssc.unict.it
[^4]: More generally, we could consider a local observable and a local generalized force, in which case we would have $\hat{V} = -\int d^3\,\vec r \hat{A}(\vec{r})f(\vec{r},t)$, and successively define a local susceptibility $\chi(\vec{r},t;\vec{r'},t')$ (see Eq.(\[chi\]) below). As this would add nothing to our argument, we shall restrict ourselves to $\vec r$-independent quantities. The extension to include local operators is immediate.
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abstract: 'We investigate the expected radio emission from the reverse shock of short GRBs, using the afterglow parameters derived from the observed short GRB light curves. In light of recent results suggesting that in some cases the radio afterglow is due to emission from the reverse shock, we examine the extent to which this component is detectable for short GRBs. In some GRBs, the standard synchrotron shock model predicts detectable radio emission from the reverse shock when none was seen. Because many physical parameters play a role in these estimates, our results highlight the need to more deeply explore the fundamental processes involved in GRB particle acceleration and emission. However, with more rapid follow-up, we can test our standard model of GRBs, which predicts an early, radio bright reverse shock in many cases.'
author:
- 'Nicole M. Lloyd-Ronning'
title: 'Estimates of Reverse Shock Emission from Short Gamma-ray Bursts'
---
Introduction {#sec:intro}
============
Perhaps the most robust model for short gamma-ray bursts (SGRBs) is the merger of two compact objects, such as two neutrons stars (NS-NS) or a neutron star and a black hole (NS-BH). The timescales and energetics involved in the merger have always made this a plausible model for SGRBs [@Eich89; @Nar92], but other clues including the location of these bursts in their host galaxies, the lack of associated supernovae, and of course the recent detection of gravitational waves from a neutron star merger coincident with a SGRB [@Ab17] have provided convincing evidence that these bursts are associated with the older stellar populations expected of compact objects [@Fox05; @Sod06a; @Berg09; @Koc10; @LB10; @Fong10; @Berg10; @Fong13; @Fong14].
There has been a concerted effort to follow up short GRBs with the goal of detecting the afterglow and potentially learning more about this class of gamma-ray bursts (for a review, see [@Berg14]). To date, about $93\%, 84\%,$, and $58\%$ of SGRBs have been followed up in the X-ray, optical, and radio respectively [@Fong2015]. Of these follow-up efforts, $74\%$ have an X-ray afterglow, $34 \%$ have been seen in the optical, and only $7\%$ detected in the radio.
Recently, @LR17 investigated a sample of long GRBs that were followed up in the radio, and found bright bursts (with isotropic equivalent energy $E_{iso} > 10^{52}erg$) [*without*]{} radio afterglows had a significantly shorter intrinsic prompt duration. They explored various reasons for the lack of afterglow in the context of different progenitor models; one possibility for the lack of radio afterglow is that this emission comes primarily from the reverse shock and that those with no a radio afterglow are in a parameter space with a weak reverse shock signal.
On the other hand, @Las13 [@Las16] and @Alex17 have recently reported the detection of a distinct reverse shock component in the afterglows of GRB130427A, GRB160509A, and 160625B. They suggested that the external medium density must be low ($n < 1cm^{-3}$) in order to give a long-lived radio afterglow from the reverse shock (the low density allows for the emitting electrons to be in the so-called slow-cooling regime thereby giving rise to longer-lived reverse shock emission). These results combined with those from @LR17 prompted us to investigate why more short GRBs (with their presumed low circumstellar densities) do not have a detected radio afterglow from the reverse shock.
Using the multi-band afterglow fits from @Fong2015, we explore the detectability of the reverse shock component from SGRBs. Using their fitted parameters for emission from the forward shock, we estimate the emission from the reverse shock, using the same formalism as @Las13 [@Las16]. We find that in some cases (depending on the microphysical parameters), there should be a detectable radio signal at the time of the afterglow follow-up, when none was seen.
Our paper is organized as follows. In Section 2 we describe how we calculate the radio flux from the forward and reverse shock using the standard formalism of synchrotron emission from a relativistic jet, using the fitted parameters from [@Fong2015]. In Section 3 we present our results. We find that most of the reverse shock emission occurs too early to be detected in the radio, but in some cases this emission [*should have been detected*]{}. In Section 4, we summarize and present our conclusions.
Materials and Methods
=====================
@Fong2015 carried out an extensive effort, compiling all of the available afterglow data for 103 SGRBs, and fitting these data to the standard synchrotron forward external shock model. Table 3 of @Fong2015 gives the results of these fits - in particular, the values of $p$, $\epsilon_{B}$, the average isotropic kinetic energy $E_{iso}$, and the external density $n$ (assumed a constant, as expected for NS-NS or NS-BH progenitors). Note that they assume the fraction of energy in the electrons is a fixed value of $\epsilon_{e}=0.1$. They performed two sets of fits to each burst, one in which the fraction of energy in the magnetic field $\epsilon_{B}$ is $0.1$ and one in which the value of $\epsilon_{B} = 0.01$. If neither gave an acceptable fit, they allowed $\epsilon_{B}$ to be a free parameter (hence explaining the couple of entries with $\epsilon_{B} \neq 0.1$ or $0.01$).
We point out that four individual bursts were detected by @Fong2015 in the radio band. These bursts are GRB050724A, GRB051221A, GRB130603B, and GRB140903A. Table \[tab:radiodetect\] of this paper gives the time of observation in days and the flux in $\mu Jy$ detected at these times for these SGRBs.
GRB $t_{obs}$ (days) Flux ($\mu Jy$)
--------- ------------------ -----------------
150724A 0.57, 1.68 173, 465
051221A 0.91 155
130603B 0.37,1.43 125, 65
140903A 0.4,2.4,9.2 110, 187, 81
: Radio afterglow detections of short GRBs.[]{data-label="tab:radiodetect"}
In the standard picture of a relativistic external blast wave, the onset of the afterglow occurs around the deceleration time $t_{dec}$ - i.e. when the blast wave has swept up enough external material to begin to decelerate $t_{dec} \propto (E/n)^{1/3}\Gamma^{-8/3}$ [@BM76], where $E$ is the energy in the blast wave, $n$ is the external particle number density and $\Gamma$ is the Lorentz factor of the blast wave. One can calculate the characteristic synchrotron break frequencies at this time, depending on the global and microphysical parameters of the burst. These expressions are given in Table 2 of [@GS02] for both a constant density and wind medium. Figure \[fig:freq\] shows the characteristic break frequencies (and the corresponding flux at these frequencies), using the parameters fitted from the @Fong2015 data at the deceleration time (when the afterglow begins). The light blue dots indicate the self-absorption frequency $\nu_{a}$ of the forward shock, the green dots show the frequency corresponding to the minimum or characteristic electron energy $\nu_{m}$ of the forward shock, and the pink dots show the so-called cooling frequency $\nu_{c}$ of the forward shock (see, e.g. @SPN98, for more detailed explanations of these frequencies). In general, $\nu_{a} < \nu_{m} < \nu_{c}$ for the forward shock component. The red stars indicate the minimum electron frequency for the [*reverse shock*]{}, $\nu_{m,RS} \approx \nu_{m}/\Gamma^{2}$ (note that this assumes the fraction of energy in the magnetic field is roughly the same for the forward and reverse shock, as explained below). Again, to calculate both the characteristic frequencies and the fluxes at these frequencies, we employed the expression given in Table 2 of [@GS02].
{width="6.0in"}
Jet Reverse Shock {#sec:rs}
-----------------
There have been many studies of the reverse shock from a relativistic blast wave (e.g., @MR97 [@SP99; @Kob00; @ZKM03; @KZ03; @ZWD05] and references therein), and the early-time radio flare observation of GRB 990123 has been attributed to the reverse shock [@KFS99; @NP05]. In addition, @SRR03 examined the expected strength of the reverse shock in six long GRBs, and were able to constrain the hydrodynamic evolution and bulk Lorentz factors of these bursts from this component.
As pointed out by these references and others, the evolution of the flux and break frequencies in the reverse shock depends on whether the blast wave is Newtonian or relativistic (among other factors), which in turn is related to the shell thickness $\Delta$ estimated from the observed duration $T$ by $\Delta \sim cT/(1+z)$. For a thick shell, $\Delta > l/2\Gamma^{8/3}$, where $l$ is the Sedov length in an interstellar medium $ \equiv (3E/4\pi n m_{p}c^{2})^{1/3}$, the reverse shock has time to become relativistic and the standard Blandford-McKee solution applies. For a thin shell, the reverse shock remains Newtonian and the Lorentz factor of this shock evolves as $\Gamma_{RS} \sim r^{-g}$, with $g \sim 2$ [@Kob00]. Short bursts with $T< 1s$ are likely in the thin shell - and therefore Newtonian - regime. However, we note that for a range of $g$ values, the time evolution of the flux and characteristic frequencies are fairly similar between the relativistic and Newtonian regimes.
This standard treatment overly simplifies reverse shock emission by separating it into two distinct regimes (thick shell and thin shell), when in reality the shell thickness covers a range of values and could fall in between these regimes [@Kop15]. This simplified treatment also assumes that $\epsilon_{B}$ and $\epsilon_{e}$ are constant in the shell, which is not necessarily justified. An evolving $\epsilon_{B}$ and $\epsilon_{e}$ will complicate the evolution of the flux and characteristic frequencies and allow additional degree of freedom in the treatment of the reverse shock.
However, generally speaking, because of the higher mass density in the shell, the peak flux in the reverse shock $f_{p, RS}$ will be higher by a factor of $\Gamma$ relative to the forward shock, $$f_{p, RS} \approx \Gamma f_{p, FS}$$ but the minimum electron frequency in the reverse shock $\nu_{m, RS}$ will be lower by a factor of $\Gamma^{2}$, $$\nu_{m, RS} \approx \nu_{m, FS}/\Gamma^{2}$$ assuming the forward and reverse shock have the same fraction of energy in the magnetic field (also not necessarily a well-justified assumption; see discussion below). For the purposes of comparing with others’ analyses of reverse shock emission [@Las13; @Las16], we employ this prescription in our estimates below.
### Self-absorbed Reverse Shock
Because we are examining the radio emission, we need to be concerned with synchrotron self-absorption - under certain conditions lower energy photons are self-absorbed, and the flux is suppressed. Self-absorption may be particularly relevant in the region of reverse shock, where the density is higher relative to the forward shock region. @RZ16 calculated the relevance of the self-absorption frequency and flux in the reverse shock, before and after shock crossing. For our purposes - because we are looking at later time radio emission - we consider the frequencies and fluxes after the shock crosses the thin shell (but see their Appendix A.1 for expressions in all ranges of parameter space).
Roughly, at the high radio frequencies we are considering here, the flux at the time of the peak can be obtained from equation 30 of @RZ16: $$f_{p,RS} = f_{p, RS, \nu_{m}}(\nu_{a, RS}/\nu_{m, RS})^{-\beta}$$ where $\beta=(p-1)/2$.\
The reverse shock flux is suppressed at a minimum by factors ranging from about 0.3 to 0.01. We emphasize again, therefore, that our estimates are upper limits to the emission from the reverse shock.
Results
=======
Figure \[fig:fpFSRSa\] shows our estimates of the peak flux at $\nu_{m}$ for the forward (blue circles) and reverse (red stars) shocks at the time $\nu_{m}$ reaches the radio band of 8.46GHz. The left panels are for a Lorentz factor $\Gamma=100$, while the right panels are for $\Gamma=10$. The top panels of Figure \[fig:fpFSRSa\] utilize the $\epsilon_{B} = 0.1$ fits from @Fong2015, while the bottom panels utilize the parameters from their $\epsilon_{B} = 0.01$ fits. Note that @Fong2015 report the median of the observing time response for the radio afterglow follow-up observations to be about $1$ day. This is reflected in Figure \[fig:fpFSRSa\] by the vertical shaded regions. The horizontal shaded regions show roughly the detector flux limits. The red dashed lines show the standard synchrotron flux decay as a function of time for a few representative bursts, assuming the reverse shock has become relativistic and a Blandford-McKee solution applies. This temporal decay is computed using the fitted parameters of [@Fong2015] (which determine the relative values of the characteristic synchrotron break frequencies) and the expressions for synchrotron flux given in Table 2 of [@GS02].
We point out that although many sGRBs have apparent non-thermal gamma-ray photons that constrain the Lorentz factor to be large, $\Gamma \geq 100$ (a compact region will be optically thick to pair production at gamma-ray energies, unless the region is moving relativistically, [@LS01]), some sGRBs do not impose such stringent constraints, and a $\Gamma \sim 10$ is sufficient to allow for their non-thermal spectra (the most famous example is GRB170817 [@Ab17], but see also [@Bur17] which show a sample of GRBs with a lack of high energy photons). We display both $\Gamma = 100$ and $\Gamma = 10$ not necessarily to argue for low Lorentz factor sGRBs but to show how the reverse shock flux and its peak time vary as a function of Lorentz factor.
It is clear that in the context of this model, many of the reverse shock bursts are missed because they tend to peak before the beginning of the observations.
{width="3.2in"}{width="3.2in"}\
{width="3.2in"}{width="3.2in"}
Note that a few bursts that went undetected in the radio (i.e. not one of the four listed in Table \[tab:radiodetect\]) indicate a forward shock component in the observational window in Figure \[fig:fpFSRSa\]. However, on closer examination, comparing the time of observations of these particular bursts to the predicted time of the peak (at $\nu_{m}$), we see that the radio observations occurred [*well before*]{} the predicted peak time (which occurs at $ \approx 10$’s of days in all of our models), and may be why it was not detected. However, as discussed above, [*the reverse shock emission falls above the flux limit*]{} for several bursts (in particular for GRB11112A, GRB121226A, GRB131004A, GRB150101B) during the time of their radio observations, particularly for the lower Lorentz factor cases ($\Gamma = 10$; right panels of Figure \[fig:fpFSRSa\]). The fact that this emission was not detected suggests that - at least in some cases - the reverse shock flux derived from this standard prescription of GRB afterglow emission is overly simplistic and give misleading values for the flux (we again emphasize that we are looking in the optically thin limit here and it may also be that the reverse shock emission was self-absorbed in these cases).
In any case, it is clear that rapid follow-up in the radio gives us a better chance to detect and/or constrain this important component, potentially breaking some of the degeneracies amongst the physical parameters in the models and allowing us to better understand the physics behind SGRB emission.
Conclusions
===========
We have investigated radio emission from short gamma-ray bursts, using fits from existing broadband afterglow data [@Fong2015] in the context of the standard synchrotron shock model for GRB emission. In particular, we have looked at the peak flux from the forward and reverse shock components of the relativistic jet. We find in some cases that the reverse shock component should have been detected in the context of this standard model. The lack of detection suggests any number of oversimplifications in the model, including potentially variable microphysical parameters, a mis-estimated bulk Lorentz factor, and/or not properly accounting for self-absorption.\
We can get additional important information on short gamma-ray bursts if there is rapid follow-up ($< 1 day$) in the radio - this will give the best chance of detecting the reverse shock emission component. High Lorentz factor outflows $\Gamma \sim 100$ peak very early ($t \sim 0.05$ day) and may be very challenging to detect. Lower Lorentz factor outflows $\Gamma \sim 10$ peak later and give us a better chance of temporally catching the reverse shock; however, the flux will be lower for the less relativistic outflows. The circumburst density must also be low enough to allow for a slow-cooling reverse shock (as mentioned in @Las13 [@Las16]), but such densities are expected for compact object binary progenitors of sGRBs.\
The electromagnetic signals can be very sensitive to the values of the microphysical parameters, such as the fraction of energy in the electrons and magnetic field, so a concerted effort to a concerted effort to more definitively constrain those parameters from a theoretical standpoint would be helpful in breaking the degeneracies and pinning down global burst parameters like kinetic energy, circumburst density, etc. These latter parameters can help constrain the progenitor.
Once again, more rapid follow-up (ideally within hours) with greater sensitivity could produce significant number of detections of radio emission from the reverse shock. A lack of detection would also constrain models to some extent and point us toward areas in which we are oversimplifying our treatment of GRB emission.
We point out again, however, that the radio emission is just one piece of the puzzle in understanding GRB emission and it is only through multi-wavelength follow-up that we will really be able to constrain the underlying physics of the outflow producing gamma-ray bursts. Efforts in this vein are particularly timely in light of the near era of gravitational wave detection from a double neutron star merger. A better understanding of the various components of electromagnetic emission from these objects will provide a more complete picture of these systems and ultimately help us understand their role in the context of stellar evolution in the universe.
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abstract: 'The relationship between the large cardinal notions of strong compactness and supercompactness cannot be determined under the standard ZFC axioms of set theory. Under a hypothesis called the Ultrapower Axiom, we prove that the notions are equivalent except for a class of counterexamples identified by Menas. This is evidence that strongly compact and supercompact cardinals are equiconsistent.'
author:
- Gabriel Goldberg
bibliography:
- 'Bibliography.bib'
title: The Ultrapower Axiom and the equivalence between strong compactness and supercompactness
---
Introduction
============
[*How large is the least strongly compact cardinal?*]{} Keisler-Tarski [@Tarski] asked whether it must be larger than the least measurable cardinal, and Solovay later conjectured that it is much larger: in fact, he conjectured that every strongly compact cardinal is supercompact. His conjecture was refuted by Menas [@Menas], who showed that the least strongly compact limit of strongly compact cardinals is not supercompact. But Tarski’s question was left unresolved until in a remarkable pair of independence results, Magidor showed that the size of the least strongly compact cannot be determined using only the standard axioms of set theory (ZFC). More precisely, it is consistent with ZFC that the least strongly compact cardinal is the least measurable cardinal, but it is also consistent that the least strongly compact cardinal is the least supercompact cardinal.
One of the most prominent open questions in set theory asks whether a weak variant of Solovay’s conjecture might still be true: [*is the existence of a strongly compact cardinal equiconsistent with the existence of a supercompact cardinal?*]{} Since inner model theory is essentially the only known way of proving nontrivial consistency strength lower bounds, answering this question probably requires generalizing inner model theory to the level of strongly compact and supercompact cardinals. In this paper, we show that in any canonical inner model built by anything like today’s inner model theoretic methodology, the least strongly compact cardinal is supercompact. This suggests that strong compactness and supercompactness are equiconsistent.
The precise statement of our theorem involves a combinatorial principle called the Ultrapower Axiom (UA). This principle holds in all known inner models as a direct consequence of the current methodology of inner model theory. It is therefore expected to hold in any future inner model. We prove:
The least strongly compact cardinal is supercompact.
The proof of this fact takes up the first half of this paper.
Given the supercompactness of the first strongly compact cardinal under UA, it is natural to wonder about the second one. The proof that the first strongly compact cardinal is supercompact [*does not*]{} generalize to show that the second strongly compact is supercompact, at least not in any obvious way. The second half of the paper is devoted to proving that assuming UA, every strongly compact cardinal is supercompact with the possible exception of limits:
If $\kappa$ is a strongly compact cardinal, either $\kappa$ is supercompact or $\kappa$ is a limit of supercompact cardinals.
In other words, the Ultrapower Axiom implies that the only counterexamples to Solovay’s conjecture are the ones discovered by Menas. Given the amount of structure that appears in the course of our proof, without even a hint of inconsistency, it seems likely that the case of strongly compact limits of supercompact cardinals really can occur. In other words, we believe the Ultrapower Axiom can hold simultaneously with the existence of Menas’s counterexamples. This would mean that UA is consistent with the existence of a strongly compact limit of supercompact cardinals. The only known way of proving the consistency of UA with large cardinals is by building canonical inner models[^1], so the only conceivable explanation of the consistency of UA with a strongly compact limit of supercompact cardinals would be that there is a canonical inner model with many supercompact cardinals. We therefore think the results of this paper provide compelling evidence that such an inner model exists.
This paper is in a sense the sequel to [@Frechet] and we will need to cite many of the results of that paper. This paper can be understood without reading [@Frechet], however, if one is willing to take the results of that paper on faith.
Preliminaries
=============
Uniform ultrafilters
--------------------
In this section we define two notions of “uniform ultrafilter" and describe how they are related.
If $\alpha$ is an ordinal, the [*tail filter*]{} on $\alpha$ is the filter generated by sets of the form $\alpha\setminus \beta$ where $\beta < \alpha$. An ultrafilter on $\alpha$ is [*tail uniform*]{} (or just [*uniform*]{}) if it extends the tail filter on $\alpha$.
Thus an ultrafilter $U$ on an ordinal is uniform if and only if all sets in $U$ have the same supremum.
Notice that if $\alpha$ is an ordinal, there is a [*uniform principal ultrafilter*]{} on $\alpha+1$:
The [*uniform principal ultrafilter*]{} on $\alpha+1$ is the ultrafilter $P_\alpha = \{X\subseteq \alpha+1 : \alpha\in X\}$.
Every ultrafilter on an ordinal restricts to a uniform ultrafilter on an ordinal in the following sense:
If $U$ is an ultrafilter and $X\in U$, then the [*restriction of $U$ to $X$*]{} is the ultrafilter $U\restriction X = U\cap P(X)$.
If $U$ is an ultrafilter on an ordinal, then there is a unique ordinal $\alpha$ such that $\alpha\in U$ and $U\restriction \alpha$ is uniform.
If $U$ is a uniform ultrafilter, the [*space*]{} of $U$, denoted $\textsc{sp}(U)$, is the unique ordinal $\alpha$ such that $\alpha\in U$.
A somewhat different notion of uniformity is also in use:
If $X$ is an infinite set, the [*Fréchet filter*]{} on $X$ is the filter generated by sets of the form $X\setminus S$ where $|S| < |X|$. An ultrafilter on $X$ is [*Fréchet uniform*]{} if it extends the Fréchet filter on $X$.
Thus an ultrafilter $U$ is Fréchet uniform if and only if all sets in $U$ have the same cardinality.
Arbitrary ultrafilters restrict to Fréchet uniform ultrafilters:
If $U$ is an ultrafilter, there is some $X\in U$ such that $U\restriction X$ is Fréchet uniform.
The following fact explains the relationship between tail uniform and Fréchet uniform ultrafilters:
\[RegTailFrec\] If $\delta$ is a regular cardinal, then an ultrafilter on $\delta$ is uniform if and only if it is Fréchet uniform.
Since we will mostly be concerned with ultrafilters on regular cardinals in this paper, the distinction will not be that important here.
\[CofUniform\] Suppose $\alpha$ is an ordinal and $f : \alpha\to \alpha'$ is a weakly increasing cofinal function. Then for any uniform ultrafilter $U$ on $\delta$, $f_*(U)$ is a uniform ultrafilter on $\alpha'$.
An ordinal $\alpha$ is [*tail uniform*]{} (or just [*uniform*]{}) if it carries a tail uniform countably complete ultrafilter. A cardinal $\lambda$ is [*Fréchet uniform*]{} (or just [*Fréchet*]{}) if it carries a Fréchet uniform countably complete ultrafilter.
The following is a consequence of \[RegTailFrec\]:
If $\delta$ is a regular cardinal, then $\delta$ is tail uniform if and only if $\delta$ is Fréchet.
If $\delta$ is a regular cardinal, we say $\delta$ is [*uniform*]{} to mean that $\delta$ is tail uniform or equivalently that $\delta$ is Fréchet.
The following is a consequence of \[CofUniform\]:
\[CofUniform2\] An ordinal $\alpha$ is tail uniform if and only if $\textnormal{cf}(\alpha)$ is uniform.
There are characterizations of uniform ordinals that are closely analogous to Scott’s elementary embedding characterization of measurable cardinals.
\[TailUnifChar\] An ordinal $\alpha$ is tail uniform if and only if there is an elementary embedding $j :V\to M$ that is discontinuous at $\alpha$.
\[FrechUnifChar\] A cardinal $\lambda$ is Fréchet if and only if there exist elementary embeddings $V\stackrel{j}\longrightarrow M\stackrel{k}{\longrightarrow}N$ such that $\sup j[\lambda]\leq \textsc{crt}(k) < j(\lambda)$.
Large cardinals
---------------
Suppose $\kappa\leq \lambda$ are cardinals.
We say $\kappa$ is [*$\lambda$-supercompact*]{} if there is an elementary embedding with critical point $\kappa$ from the universe of sets into an inner model that is closed under $\lambda$-sequences.
We say $\kappa$ is [*$\lambda$-strongly compact*]{} if there is an elementary embedding $j$ with critical point $\kappa$ from the universe of sets into an inner model $M$ such that every set $A\subseteq M$ with $|A|\leq \lambda$ is contained in a set $A'\in M$ with $|A'|^M < j(\kappa)$.
We state the equivalence between the embedding formulations of these large cardinal axioms and the ultrafilter versions.
Suppose $A$ is a set and $\mathcal U$ is an ultrafilter on $X\subseteq P(A)$. We say $\mathcal U$ is [*fine*]{} if any $a\in A$ belongs to $U$-almost all $\sigma\in X$. We say $\mathcal U$ is [*normal*]{} if any choice function on $X$ is constant on a $\mathcal U$-large set.
The following theorems appear in [@Kanamori] except for \[StrongcompactnessEquiv\] (4) which is due to Ketonen [@Ketonen]:
\[SupercompactnessEquiv\] Suppose $\lambda$ is a cardinal. Then the following are equivalent:
(1) $\kappa$ is $\lambda$-supercompact.
(2) There is a normal fine $\kappa$-complete ultrafilter on $P_\kappa(\lambda)$.
(3) There is a normal fine ultrafilter on $P(\lambda)$ with completeness $\kappa$.
\[StrongcompactnessEquiv\] Suppose $\lambda$ is a cardinal. Then the following are equivalent:
(1) $\kappa$ is $\lambda$-strongly compact.
(2) There is a $\kappa$-complete fine ultrafilter on $P_\kappa(\lambda)$.
(3) Every $\kappa$-complete filter generated by at most $\lambda$ sets extends to a $\kappa$-complete ultrafilter.
(4) Every regular cardinal $\delta$ such that $\kappa\leq \delta\leq\lambda$ carries a $\kappa$-complete uniform ultrafilter.
Comparisons and the seed order
------------------------------
We say $j$ is an [*ultrapower embedding*]{} of $M$ if $N = H^N(j[M]\cup \{x\})$ for some $x\in N$. We say $j$ is an [*internal ultrapower embedding*]{} of $M$ if $j$ is an ultrapower embedding of $M$ and $j\restriction x\in M$ for all $x\in M$.
If $P$ is an inner model and $U$ is a $P$-ultrafilter, we denote by $j_U^P : P\to M_U^P$ the ultrapower of $P$ by $U$ using functions in $P$. For any function $f\in P$ defined on a set in $U$, $[f]^P_U$ denotes the point in $M_U^P$ represented by $f$.
Given this terminology, an embedding $j :M \to N$ is an ultrapower embedding if and only if there is an $M$-ultrafilter $U$ such that $j = j_U^M$, and an internal ultrapower embedding if and only if $U$ can be chosen to belong to $M$. (In this case, any $M$-ultrafilter $U$ such that $j = j_U^M$ will belong to $M$.)
Suppose $M_0,M_1,$ and $N$ are transitive models. We write $$(i_0,i_1) : (M_0,M_1)\to N$$ to denote that $i_0 : M_0\to N$ and $i_1 : M_1\to N$ are elementary embeddings.
Suppose $j_0 : V\to M_0$ and $j_1 : V\to M_1$ are ultrapower embeddings. A pair of internal ultrapower embeddings $(i_0,i_1) : (M_0,M_1)\to N$ is a [*comparison*]{} of $(j_0,j_1)$ if $i_0\circ j_0 = i_1 \circ j_1$.
The notion of a comparison leads to the definitions of the Ultrapower Axiom and the seed order.
Every pair of ultrapower embeddings has a comparison.
The [*seed order*]{} is the relation defined on $U_0,U_1\in \Un$ by setting $U_0\swo U_1$ if there is a comparison $(i_0,i_1) : (M_{U_0},M_{U_1})\to N$ of $(j_{U_0},j_{U_1})$ such that $i_0([\text{id}]_{U_0}) < i_1([\text{id}]_{U_1})$.
The seed order is a wellorder of $\Un$.
We discuss some results from [@SO] that single out for any pair of ultrapowers a unique “optimal" comparison which we call a [*canonical comparison*]{}.
Suppose $j_0 : V\to M_0$ and $j_1 : V\to M_1$ are ultrapower embeddings. A comparison $(i_0,i_1) :(M_0,M_1)\to N$ of $(j_0,j_1)$ is
(1) [*minimal*]{} if $N = H^N(i_0[M_0]\cup i_1[M_1])$
(2) [*canonical*]{} if for any comparison $(i_0',i_1') : (M_0,M_1)\to N'$, there is an elementary embedding $h : N \to N'$ such that $h\circ i_0 = i_0'$ and $h\circ i_1 = i_1'$.
(3) a [*pushout*]{} if for any comparison $(i_0',i_1') : (M_0,M_1)\to N'$, there is an internal ultrapower embedding $h : N \to N'$ such that $h\circ i_0 = i_0'$ and $h\circ i_1 = i_1'$.
Under UA, these notions all coincide. The following facts come from [@RF] Section 5.
A comparison of a pair of ultrapower embeddings is canonical if and only if it is their unique minimal comparison.
\[Pushout\] Every pair of ultrapower embeddings has a pushout.
It follows that every pair of ultrapower embeddings has a unique pushout, a unique canonical comparison, and a unique minimal comparison, and these all coincide. For simplicity, in the context of UA, we will refer to this comparison as [*the canonical comparison*]{}, and largely avoid using the other terms. We will make use of the following fact:
\[CanonicalInternal\] Suppose $j_0 : V\to M_0$ and $j_1 : V\to M_1$ are ultrapower embeddings and $(i_0,i_1) : (M_0,M_1)\to N$ is a pushout of $(j_0,j_1)$. Suppose $k : N\to N'$ is an ultrapower embedding of $N$. Then the following are equivalent:
(1) $k$ is an internal ultrapower embedding of $N$
(2) $k$ is amenable to $M_0$ and $M_1$.
Limits and the Ketonen order
----------------------------
We need a slight generalization of the notion of a limit of ultrafilters defined in [@SO].
Suppose $U$ is a countably complete ultrafilter and $Z$ is an $M_U$-ultrafilter on an ordinal $\beta$. Then the [*$U$-limit of $Z$*]{} is the ultrafilter $$U^-(Z) = \{X\subseteq \alpha : j_U(X)\cap \beta\in Z\}$$ where $\alpha$ is least such that $j_U(\alpha) \geq \beta$.
The main difference here is that we do not assume $Z\in M_U$. For the most part, however, we will only be concerned with the case $Z\in M_U$.
The [*Ketonen order*]{} is defined on $U,W\in\Un$ by setting $U\sE W$ if there is some $Z\in \Un_{\leq [\text{id}]_W}^{M_W}$ such that $U = W^-(Z)$.
The following fact restates the two main theorems of [@SO].
The Ketonen order extends the seed order. The two orders coincide if and only if the Ultrapower Axiom holds.
We will make use of the notion of the [*translation function*]{} $t_U$ associated to a countably complete ultrafilter $U$ under UA:
Suppose $U$ and $W$ are uniform countably complete ultrafilters. Then $\tr U W$ is the $\sE^{M_U}$-least $Z\in \Un^{M_U}$ such that $U^-(Z) = W$.
The following trivial bound is worth noting:
\[BoundingLemma\] For any $U,W\in \Un$, $\tr U W \E^{M_U} j_U(W)$.
Note that $U^-(j_U(W)) = W$.
This bound is interesting given the following theorem, [@SO] Lemma 5.6, which relates translation functions to comparisons:
\[Reciprocity1\] Suppose $U_0,U_1\in \Un$. Let $M_0 = M_{U_0}$ and $M_1 = M_{U_1}$. Let $i_0 = j^{M_0}_{\tr {U_0} {U_1}}$ and $i_1 = j^{M_1}_{\tr {U_1} {U_0}}$. Then $(i_0,i_1) : (M_{U_0},M_{U_1})\to N$ is the canonical comparison of $(j_{U_0},j_{U_1})$.
We also use the following fact from [@SO] Proposition 5.8:
\[OrderPreserving\] For any $U\in \Un$, the function $t_U : (\Un,\sE)\to (\Un,\sE)^{M_U}$ is order preserving.
We will need the following fact:
\[kInternal\] Suppose $D,U\in \Un$ and $k : M_D\to M_U$ is an elementary embedding such that $k\circ j_D = j_U$. Then the following are equivalent:
(1) $k$ is an internal ultrapower embedding of $M_D$.
(2) $\tr U D = P_{k([\textnormal{id}]_D)}^{M_U}$.
(3) $\tr U D\in k[M_D]$.
\(1) implies (2) by \[Reciprocity1\] since assuming (1), $(k,\text{id})$ is a comparison of $(j_D,j_U)$ and $P_{k([\textnormal{id}]_D)}^{M_U}$ is the $M_U$-ultrafilter derived from $\text{id}$ using $k([\text{id}]_D)$. (2) also easily implies (1).
\(2) implies (3) trivially. We finally show that (3) implies (2). Fix $Z\in M_D$ such that $k(Z) = \tr U D$. It is easy to see that $Z = \tr D D$. We claim $Z = P^{M_D}_{[\text{id}]_D}$. Otherwise, since $D^-(P^{M_D}_{[\text{id}]_D}) = D$, we have $Z\sE P^{M_D}_{[\text{id}]_D}$ in $M_D$, or in ther words $Z\in \Un_{\leq [\text{id}]_D}^{M_D}$. But the existence of such a $Z$ implies $D\sE D$, which contradicts that $\sE$ is a strict partial order. Hence $Z = P^{M_D}_{[\text{id}]_D}$. It follows that $\tr U D = k(Z) = P_{k([\textnormal{id}]_D)}^{M_U}$.
Ketonen embeddings
==================
In order to prove the supercompactness of the least strongly compact cardinal, one must somehow define elementary embeddings witnessing its supercompactness. Of course this cannot be done in ZFC alone, and at first it is hard to see how UA will help. In this section, we define ultrafilters $U_\delta$ that will give rise to embeddings roughly witnessing $\delta$-supercompactness. It turns out that these ultrafilters [*were*]{} in a sense first isolated in the ZFC context by Ketonen [@Ketonen]. We begin by expositing some of Ketonen’s work.
Ketonen embeddings
------------------
Here we use the wellfoundedness of the Ketonen order to identify certain minimal ultrafilters which we call [*Ketonen ultrafilters*]{}. These ultrafilters will witness (approximately) the $\delta$-supercompactness of the least cardinal $\kappa$ that is strongly compact to $\delta$.
Suppose $\delta$ is a regular uncountable cardinal. An elementary embedding $j : V\to M$ is a [*Ketonen embedding*]{} at $\delta$ if the following hold where $\delta_* = \sup j[\delta]$:
(1) $M = H^{M}(j[V]\cup \{\delta_*\})$
(2) $\delta_*$ is not tail uniform in $M$.
A uniform ultrafilter $U$ is [*Ketonen*]{} at $\delta$ if it is the ultrafilter derived from a Ketonen embedding $j$ at $\delta$ using $\sup j[\delta]$.
Thus the ultrapower embedding associated to a Ketonen ultrafilter is a Ketonen embedding. It is easy to see that if $U$ is a Ketonen ultrafilter and $\delta$ is uniform, then $U$ is a uniform ultrafilter on $\delta$. Otherwise $U = P_\delta$ is a uniform ultrafilter on $P_{\delta+1}$.
There is a simple combinatorial characterization of Ketonen ultrafilters:
\[KetonenCombinatorial\] A countably complete ultrafilter $U$ is Ketonen at the regular cardinal $\delta$ if and only if the following hold:
(1) Every regressive function on $\textsc{sp}(U)$ is bounded on a $U$-large set.
(2) $\{\alpha < \textsc{sp}(U): \alpha\text{ is not tail uniform}\}\in U$.
Because it is convenient, we allow that when $\delta$ is a regular cardinal that is not tail uniform, the identity is Ketonen at $\delta$, and therefore the uniform principal ultrafilter on $\delta+ 1$ is Ketonen at $\delta$.
\[KetonenMinimal\] Suppose $\delta$ is a regular cardinal and $U$ is a uniform countably complete ultrafilter. Then the following are equivalent:
(1) $U$ is Ketonen at $\delta$.
(2) $U$ is an $\sE$-minimal element of $\Un\setminus \Un_{<\delta}$.
We first show (1) implies (2). Assume (1) and fix $W\sE U$. Let $\delta_* = [\text{id}]_U = \sup j_U[\delta]$. Then there is some $W'\in \Un^{M_U}_{\leq\delta_*}$ such that $W = U^-(W')$. Since $U$ is zero order, $\Un^{M_U}_{\delta_*} = \emptyset$, so $W'\in \Un^{M_U}_{<\delta_*}$. It follows easily that $W \in \Un_{<\delta}$. Since any $W\sE U$ belongs to $\Un_{<\delta}$, (2) holds.
We now show (2) implies (1). Assume (2). Let $\delta_* = \sup j_U[\delta]$. Assume towards a contradiction that in $M_U$, there is a uniform countable complete ultrafilter $W'$ such that $\delta_* \leq \textsc{sp}(W') \leq [\text{id}]_U$. Let $\alpha$ be least such that $j_U(\alpha) \geq \textsc{sp}(W')$, so $\alpha\geq \delta$. Let $W = U^-(W')$. Then $W \sE U$. This contradicts that $U$ is $\sE$-minimal. It follows that in $M_U$, there is no uniform countable complete ultrafilter $W'$ such that $\delta_* \leq \textsc{sp}(W') \leq [\text{id}]_U$. It follows in particular that $\Un_{\delta_*}^{M_U}= \emptyset$. Since the principal ultrafilter $P_{\delta_*}$ is a uniform countably complete ultrafilter on $\delta_*+1$, it follows that $[\text{id}]_U < \delta_* + 1$, or in other words $\delta_* = [\text{id}]_U$. That is, $U$ is Ketonen.
\[KetonenExistence\] For every regular cardinal $\delta$, there is a Ketonen ultrafilter at $\delta$.
This is immediate from the wellfoundedness of the seed order.
We end this section with Ketonen’s remarkable theorem on the covering properties of Ketonen ultrapowers.
Suppose $M$ is an inner model, $\lambda$ is a cardinal, and $\lambda'$ is an $M$-cardinal. We say $M$ has the [*$(\lambda,\lambda')$-covering property*]{} if every set $A\subseteq M$ with $|A|\leq \lambda$ is contained in a set $A'\in M$ with $|A'|^{M} \leq \lambda'$. We say $M$ has the [*$(\lambda,{<}\lambda')$-covering property*]{} if every set $A\subseteq M$ with $|A|\leq \lambda$ is contained in a set $A'\in M$ with $|A'|^{M} < \lambda'$.
\[EACov\] Suppose $j : V\to M$ is an elementary embedding. Suppose $\lambda$ is a cardinal and $\lambda'$ is an $M$-cardinal. Assume there exist sets $X$ and $X'$ such that $|X| = \lambda$, $X'\in M$, $|X'|^M = \lambda'$, and $j[X]\subseteq X'$. Then for any set $Y$ with $|Y| \leq \lambda$, $j[Y]$ is contained in a set $Y'\in M$ with $|Y'|^M \leq \lambda'$.
Fix a surjection $f : X\to Y$. Let $Y' = j(f)[X']$. Then $Y'\in M$ and since $j(f)$ is a surjection from $X'$ to $Y'$ in $M$, $|Y'|^M \leq |X'|^M = \lambda'$. Finally $j[Y] = j[f[X]] = j(f)[j[X]]\subseteq j(f)[X'] = Y'$.
The following lemma is a standard property of ultrapower embeddings (and in fact it holds in a bit more generality).
\[UltCov\] Suppose $j : V\to M$ is an ultrapower embedding. Suppose $\lambda$ is a cardinal and $\lambda'$ is an $M$-cardinal. Assume there exist sets $X$ and $X'$ such that $|X| = \lambda$, $X'\in M$, $|X'|^M = \lambda'$, and $j[X]\subseteq X'$. Then $M$ has the $(\lambda,\lambda')$-covering property.
Fix a set $A\subseteq M$ with $|A|\leq \lambda$. Fix $e\in M$ such that $M = H^M(j[V]\cup \{e\})$. Fix a set of functions $Y$ such that $|Y| = \lambda$ and $A = \{j(f)(e) : f\in Y\}$. By \[EACov\], let $Y'\in M$ be a set of functions with $|Y'|^M \leq \lambda'$ and $j[Y]\subseteq Y'$. Then $A\subseteq \{g(e) : g\in Y'\wedge e\in \text{dom}(g)\}$. Therefore let $A' = \{g(e) : g\in Y'\wedge e\in \text{dom}(g)\}$. Then $A'\in M$ contains $A$ and $|A'|^M \leq |A'|^M \leq\lambda'$. This proves the lemma.
\[KetonenCov\] Suppose $j : V\to M$ is an ultrapower embedding and $\delta$ is a regular cardinal. Let $\delta' = \textnormal{cf}^M(\sup j[\delta])$. Then $M$ has the $(\delta,\delta')$-covering property.
It suffices by \[UltCov\] to show that there exist sets $X$ and $X'$ such that $|X| = \delta$, $X'\in M$, $|X'|^M = \delta'$, and $j[X]\subseteq X'$. Let $X'\in M$ be a closed unbounded subset of $\sup j[\delta]$ such that the ordertype of $X'$ is $\delta'$. Let $X = j^{-1}[X']$. Then since $j$ is continuous at ordinals of cofinality $\omega$, $j^{-1}[X']$ is $\omega$-closed unbounded in $\delta$. Since $\delta$ is regular, $|X| = \delta$. Therefore $X$ and $X'$ are as desired.
\[KetonenRegularity\] Suppose $\delta$ is a regular cardinal and $j : V\to M$ is a Ketonen embedding at $\delta$. Suppose $\gamma \leq \delta$ is such that every regular cardinal in the interval $[\gamma,\delta]$ is uniform. Then $M$ has the $(\delta,{<}j(\gamma))$-covering property.
Let $\delta_* = \sup j[\delta]$. Let $\delta' = \text{cf}^M(\delta_*)$. By \[KetonenCov\], it suffices to show that $\delta' < j(\gamma)$. By the elementarity of $j$, every $M$-regular cardinal in the interval $[j(\gamma), j(\delta)]$ is Fréchet. On the other hand, by the definition of a Ketonen embedding, $\delta_*$ is not tail uniform in $M$, and therefore by \[CofUniform2\], $\delta'$ is not Fréchet in $M$. Since $\delta'$ is an $M$-regular cardinal, it follows that $\delta'\notin [j(\gamma),j(\delta)]$. Since $\delta' \leq j(\delta)$, it follows that $\delta' < j(\gamma)$, as desired.
The universal property of Ketonen embeddings
--------------------------------------------
Since the Ultrapower Axiom implies the MO of the Ketonen order, the following theorem is an immediate consequence of \[KetonenMinimal\]:
For any regular cardinal $\delta$, there is a unique Ketonen ultrafilter at $\delta$.
But in fact Ketonen ultrafilters are minimum ultrafilters in a second way that we now describe. We first prove the key universal property that characterizes Ketonen embeddings under UA.
Suppose $\delta$ is a regular uncountable cardinal. An ultrapower embedding $j : V\to M$ is [*zero order*]{} at $\delta$ if $\sup j[\delta]$ is not tail uniform in $N$.
Thus a Ketonen embedding is a zero order ultrapower $j$ with the additional property that $M$ is generated by $j[V]\cup \{\sup j[\delta]\}$.
\[KetonenUniversality\] Suppose $\delta$ is regular, $j : V\to M$ is Ketonen at $\delta$, and $j' : V\to M'$ is zero order at $\delta$. Then there is an internal ultrapower embedding $k : M\to M'$ such that $j' = k\circ j$.
Let $(i,i') : (M,M')\to N$ be a comparison of $(j,j')$. Since $j$ and $j'$ are zero order, $i$ and $i'$ are continuous at $\sup j[\delta]$ and $\sup j'[\delta]$ respectively, so $$i(\sup j[\delta]) = \sup i \circ j[\delta] = \sup i'\circ j'[\delta] = i'(\sup j'[\delta])$$ Since $i\circ j = i'\circ j'$, $i[j[V]] = i'[j'[V]]\subseteq i'[M']$.
Since $j$ is Ketonen, $M = H^M(j[V]\cup \{\sup j[\delta]\})$, so since $j[V]\cup \{\sup j[\delta]\}\subseteq i'[M']$, $i[M]\subseteq i'[M']$. Define $k : M\to M'$ by $$k = (i')^{-1}\circ i$$ Since $i'\circ j' = i\circ j$, $j' = (i')^{-1}\circ i\circ j = k \circ j$.
We must check that $k$ is an internal ultrapower embedding of $M$. To see that $k$ is an ultrapower embedding of $M$, we use the following trivial lemma ([@RF], Lemma 5.14):
Suppose $j : V\to M$ is an elementary embedding, $j' : V\to M'$ is an ultrapower embedding, and $k : M\to M'$ is an elementary embedding with $k\circ j = j'$. Then $k$ is an ultrapower embedding of $M$.
To conclude that $k$ is an internal ultrapower embedding of $M$, we now cite another lemma, [@RF] Lemma 5.6:
Suppose $i : M \to N$ is an internal ultrapower embedding, $k : M\to M'$ is an ultrapower embedding, and $i' : M'\to N$ is an elementary embedding such that $i = i'\circ k$. Then $k$ is an internal ultrapower embedding of $M$.
We have produced an internal ultrapower embedding $k : M \to M'$ such that $j' = k\circ j$, which proves the theorem.
The supercompactness of the least strongly compact cardinal is actually a consequence of the universal property of Ketonen embeddings alone:
If $\delta$ is a regular cardinal, $\KU$ denotes the statement that the universal property of Ketonen embeddings holds at $\delta$.
$\KU$ itself can be seen as a weak form of the Ultrapower Axiom:
\[KUA\] Suppose $\delta$ is a regular cardinal. The following are equivalent:
(1) $\KU$.
(2) Suppose $j_0 : V\to M_0$ is an ultrapower embedding and $j_1 : V\to M_1$ is a Ketonen embedding at $\delta$. Then the pair $(j_0,j_1)$ admits a comparison.
\(2) implies (1) by the proof of \[KetonenUniversality\].
We show (1) implies (2). Assume (1) and fix $j_0$ and $j_1$ as in (2). Let $\delta' = \text{cf}^{M_0}(\sup j_0[\delta])$. By \[KetonenExistence\] applied in $M_0$, in $M_0$, there is a Ketonen embedding $i_0 : M_0\to N$ at $\delta'$. In particular, $i_0$ is an internal ultrapower embedding of $M_0$.
We claim $i_0\circ j_0 : V\to N$ is a zero order ultrapower embedding at $\delta$. Note that $\text{cf}^N(\sup i_0\circ j_0[\delta]) = \text{cf}^N(\sup i_0[\delta'])$: fix an increasing cofinal function $f : \delta'\to \sup j_0[\delta]$ that belongs to $M_0$, and note that $i_0(f)\restriction \sup j_0[\delta]$ is an increasing cofinal function from $\sup i_0[\delta']$ to $\sup i_0\circ j_0[\delta]$ that belongs to $N$. Since $i_0$ is Ketonen at $\delta'$, $\sup i_0[\delta']$ is not tail uniform in $N$. Hence by \[CofUniform\], $\sup i_0\circ j_0[\delta]$ is not tail uniform in $N$. Thus $i_0\circ j_0$ is zero order at $\delta$.
Therefore by \[KetonenUniversality\], there is an internal ultrapower embedding $i_1 : M_1\to N$ such that $i_0\circ j_0 = i_1\circ j_1$. In other words, $(i_0,i_1) : (M_0,M_1)\to N$ is a comparison of $(j_0,j_1)$.
The proof of \[KUA\] yields the following partial analysis of the comparison of a Ketonen embedding with an arbitrary ultrapower embedding:
\[ZeroComparison\] Suppose $\delta$ is a uniform regular cardinal and $i: V\to M$ is an ultrapower embedding. Let $\delta' = \textnormal{cf}^{M}(\sup i[\delta])$. Then there is a comparison of $(j_\delta,i)$ of the form $(i_*,j_{\delta'}^M) : (M_\delta,M)\to N$ where $i_* : M_\delta\to N$ is an internal ultrapower embedding of $M_\delta$.
Our theorems regarding the least supercompact cardinal require only $\KU$, but we will just state them under full UA. What is interesting is that it does not seem possible to analyze larger supercompact cardinals using a principle analogous to $\KU$.
The following is a standard category theoretic argument:
\[KetonenUnique\] Suppose $\delta$ is regular. There is a unique Ketonen embedding at $\delta$.
Suppose $j_0: V\to M_0$ and $j_1: V\to M_1$ are Ketonen embeddings at $\delta$. By $\text{KU}_\delta$, there is an internal ultrapower embedding $k_0 : M_0\to M_1$ such that $j_1 = k_0\circ j_0$. By $\text{KU}_\delta$, there is an internal ultrapower embedding $k_1 : M_1\to M_0$ such that $j_0 = k_1\circ j_1$. Since $k_1 \circ k_0$ is an internal ultrapower embedding from $M_0$ to itself, $k_1 \circ k_0$ is the identity. Similarly $k_1\circ k_0$ is the identity. Therefore $M_0$ and $M_1$ are isomorphic, and so since they are transitive classes, $M_0 = M_1$ and $k_0 = k_1 = \text{id}$. It follows that $j_0 = k_0\circ j_0 = k_1\circ j_1 = j_1$, as desired.
Suppose $\delta$ is regular. We denote by $j_\delta : V \to M_\delta$ the unique Ketonen embedding at $\delta$ and by $U_\delta$ the unique Ketonen ultrafilter at $\delta$.
A very useful fact is that the internal ultrapower embeddings of $M_\delta$ are characterized by the weakest possible property:
\[KetonenInternal\] Suppose $\delta$ is regular. Suppose $i : M_\delta\to N$ is an ultrapower embedding. Then the following are equivalent:
(1) $i$ is continuous at $\sup j_\delta[\delta]$.
(2) $i$ is an internal ultrapower embedding of $M_\delta$.
That (2) implies (1) does not require $\KU$ (except to make sense of the notation $M_\delta$): since $\sup j_\delta[\delta]$ is not tail uniform in $M_\delta$, by \[TailUnifChar\], no internal ultrapower embedding of $M_\delta$ is discontinuous at $\sup j_\delta[\delta]$.
We now show that (1) implies (2). Assume (1). Let $\delta_* = \sup j_\delta[\delta]$. Then $i\circ j_\delta : V\to N$ is zero order at $\delta$: since $i$ is continuous at $\delta_*$, $\sup (i\circ j_\delta)[\delta] = i(\delta_*)$, and since $\delta_*$ is not uniform in $M_\delta$, $i(\delta_*)$ is not uniform in $N$. By $\KU$, there is an internal ultrapower embedding $k : M_\delta\to N$ such that $k\circ j_{\delta} = i\circ j_\delta$. Since $k$ is internal to $M_\delta$, $k$ is also continuous at $\delta_*$, and therefore $k(\delta_*]) = i(\delta_*)$. Thus $$k \restriction j_\delta[V]\cup \{\delta_*\} = i\restriction j_\delta[V]\cup \{\delta_*\}$$ Since $j_\delta : V\to M_\delta$ is Ketonen, $M_\delta = H^{M_{\delta}}(j_\delta[V]\cup \{\delta_*\})$. It therefore follows that $k = i$. Therefore $i$ is an internal ultrapower embedding of $M_\delta$ since $k$ is. Thus (2) holds.
Another way to write this is the following:
\[CombinatorialInternal\] Suppose $\delta$ is regular. Suppose $W$ is an $M_\delta$-ultrafilter on $X\in M_\delta$. Let $\delta' = \textnormal{cf}^{M_\delta}(\sup j_\delta[\delta])$. Then $W\in M_\delta$ if and only if for every partition $P$ of $X$ such that $P\in M_\delta$ and $|P|^{M_\delta} = \delta'$, there is some $Q\subseteq P$ such that $Q\in M_\delta$, $|Q|^{M_\delta} < \delta'$, and $\bigcup Q\in W$.
We omit the straightforward proof.
The following special case will be particularly important for us:
\[KetonenAbsorption\] Suppose $\delta$ is regular. For any $W\in \Un_{<\delta}$, $j_W\restriction M_\delta$ is an internal ultrapower embedding of $M_\delta$ and $W\cap M_\delta\in M_\delta$.
This is immediate from \[KetonenInternal\] as soon as one knows that $j_W\restriction M_\delta$ is an ultrapower embedding of $M_\delta$. This is a basic lemma proved in [@IR] Lemma 2.10.
Closure properties of $M_\delta$
================================
\[KetonenAbsorption\] shows that the Ketonen ultrapower $M_\delta$ absorbs many countably complete ultrafilters from $V$. In order to show that the embedding $j_\delta :V\to M_\delta$ is strong, and even supercompact, we use large cardinals to convert this absorption of ultrafilters into the absorption of sets. This conversion process involves
Independent families
--------------------
Suppose $X$ is a set and $\kappa$ is an infinite cardinal. A set $F\subseteq P(X)$ is called a [*$\kappa$-independent family*]{} of subsets of $X$ if for any disjoint subfamilies $\sigma$ and $\tau$ of $F$ of cardinality less than $\kappa$, there is some $x\in X$ that belongs to every element of $\sigma$ and no element of $\tau$.
Independent families come up in the theory of filters in the following way:
\[IndependentFilter\] Suppose $X$ is a set and $\kappa$ is an infinite cardinal and $F\subseteq P(X)$. Then $F$ is a $\kappa$-independent family of subsets of $X$ if and only if for any $S\subseteq F$, $S\cup \{X\setminus A : A\in F\setminus S\}$ generates a $\kappa$-complete filter on $X$.
One could take \[IndependentFilter\] to be the definition of a $\kappa$-independent family, but the advantage of our definition is that it makes the following absoluteness lemma obvious:
\[IndAbsolute\] Suppose $\kappa$ is a cardinal and $M$ is an inner model such that $M^{<\kappa}\subseteq M$. Suppose $X\in M$ and $M$ satisfies that $F$ is a $\kappa$-independent family of subsets of $X$. Then $F$ is a $\kappa$-independent family of subsets of $X$.
An important combinatorial fact, due to Hausdorff, is the existence of $\kappa$-independent families:
\[Hausdorff\] Suppose $\kappa\leq\lambda$ are cardinals such that $\lambda^{<\kappa} = \lambda$. Then there is a $\kappa$-independent family $F$ of subsets of $\lambda$ such that $|F| = 2^\lambda$.
It clearly suffices to construct a set $X$ such that $|X| = \lambda$ there is a $\kappa$-independent family $F$ of subsets of $X$ such that $|F| = 2^\lambda$. Let $$X= \{(\sigma,S) : \sigma\in P_\kappa(\lambda)\text{ and }S\subseteq \sigma\}$$ For each $A\subseteq \lambda$, let $X_A = \{(\sigma,S) \in X : \sigma\cap A\in S\}$.
The tight covering property
---------------------------
In this section we prove that $M_\delta$ has the [*tight covering property*]{} under a technical assumption proved in [@Frechet].
An inner model $M$ has the [*tight covering property*]{} at a cardinal $\lambda$ if every set $A\subseteq M$ such that $|A|\leq \lambda$ is contained in a set $A'\in M$ such that $|A'|^M\leq \lambda$.
The tight covering property at $\lambda$ is simply the $(\lambda,\lambda)$-covering property defined above. The key observation is the following fact:
\[CoverDichotomy\] Suppose $\delta$ is a uniform regular cardinal. Then exactly one of the following holds:
(1) $M_\delta$ has the tight covering property at $\delta$.
(2) For any $U\in \Un_{\leq\delta}$, $U\cap M_\delta\in M_\delta$.
We first show that (1) implies (2) fails. Assume (1). Then $\text{cf}^{M_\delta}(\sup j_\delta[\delta]) = \delta$. By \[CofUniform2\], since $\sup j_\delta[\delta]$ is not tail uniform in $M_\delta$, $\delta$ is not uniform in $M_\delta$. Since $\delta$ is uniform in $V$, it follows that (2) fails.
We now show that if (1) fails, then (2) holds. Let $\delta' = \text{cf}^{M_\delta}(\sup j_\delta[\delta])$. By \[KetonenCov\], the tight covering property at $\delta$ fails for $M_\delta$ if and only if $\delta' > \delta$. But as a consequence of \[CombinatorialInternal\], any countably complete $M_\delta$-ultrafilter on an ordinal $\alpha < \delta'$ belongs to $M_\delta$. In particular since $\delta < \delta'$, (2) holds.
By \[CoverDichotomy\], there is a bizarre consequence of the failure of tight covering for $M_\delta$: if $M_\delta$ does not have the tight covering property at $\delta$, then $U_\delta\cap M_\delta \in M_\delta$, even though $M_\delta = M_{U_\delta}$. But in fact, we do not know how to rule this out, and the following question is open:
\[TightCoverQ\] Suppose $\delta$ is a regular cardinal. Does $M_\delta$ have the tight covering property at $\delta$?
Perhaps the situation is not as bizarre as it appears on first glance, considering the following theorem from [@IR]:
If $\kappa$ is supercompact, then there is a $\kappa$-complete nonprincipal ultrafilter $U$ on a cardinal such that $U\cap M_U\in M_U$.
We do come very close to a positive answer to \[TightCoverQ\], and depending on one’s interests, one might say that the question is answered in all cases of interest (e.g., assuming GCH).
We start with the following fact:
\[TightCover\] Suppose $\delta$ is regular. Let $\kappa = \textsc{crt}(j_\delta)$ and assume $\kappa$ is $\delta$-strongly compact. Then $M_\delta$ has the tight covering property at $\delta$.
Assume towards a contradiction that $M_\delta$ does not have the tight covering propery, so by \[CoverDichotomy\], for any $U\in \Un_{\delta}$, $$\label{BadSide}U\cap M_\delta\in M_\delta$$ In order to control cardinal arithmetic, it is convenient to prove the following claim.
\[StrongCompactAbs\] In $M_\delta$, $\kappa$ is $\delta$-strongly compact.
By a theorem of Ketonen [@Ketonen] (which is closely related to \[KetonenRegularity\]), to show that $M_\delta$ satisfies that $\kappa$ is $\delta$-strongly compact, it suffices to show that every ordinal $\alpha$ such that $\kappa \leq \alpha \leq\delta$ and $\text{cf}^{M_{U_\delta}}(\alpha) \geq \kappa$ carries a uniform $\kappa$-complete ultrafilter in $M_\delta$. But note that since $M_\delta$ is closed under $\kappa$-sequences and $\text{cf}^{M_{U_\delta}}(\alpha) \geq \kappa$, in fact $\text{cf}(\alpha) \geq \kappa$. Therefore since $\kappa$ is $\delta$-strongly compact, in $V$, $\alpha$ carries a $\kappa$-complete ultrafilter $U$. Now $U\cap M_\delta\in M_\delta$ by \[BadSide\], so $U\cap M_\delta$ witnesses that $\alpha$ carries a uniform $\kappa$-complete ultrafilter in $M_\delta$.
Since $\kappa$ is $\delta$-strongly compact in $M_\delta$, by Solovay’s theorem [@Solovay], $(\delta^{<\kappa})^{M_\delta} = \delta$. Therefore we are in a position to apply \[Hausdorff\] inside $M_\delta$. This yields the existence of a $\kappa$-independent family $F$ of subsets of $\delta$ such that $|F|^{M_\delta} = \delta$. (In fact one could also have taken $|F|^{M_\delta} = (2^\delta)^{M_\delta}$, but we do not need this here.) By \[IndAbsolute\], $F$ is a $\kappa$-independent family of subsets of $\delta$ in $V$.
\[PowerClm\] $P(F)\subseteq M_\delta$.
Suppose $S\subseteq F$. Let $G$ be the filter generated by $S\cup \{\delta\setminus A : A\in F\setminus S\}$. Then by \[IndependentFilter\], $G$ is a $\kappa$-complete filter on $\delta$. Moreover $G$ is generated by $\delta$-many sets. Therefore since $\kappa$ is $\delta$-strongly compact, $G$ extends to a $\kappa$-complete ultrafilter $U$ on $\delta$. By \[BadSide\], $U\cap M_\delta\in M_\delta$. But $S = (U\cap M_\delta)\cap F$, so $S\in M_\delta$.
Since $P(F)\subseteq M_\delta$ and $|F|^{M_\delta} = \delta$, it follows that $P(\delta) \subseteq M_\delta$. But by \[BadSide\], $U_\delta = U_\delta\cap M_\delta \in M_\delta$. This is a contradiction since $M_\delta = M_{U_\delta}$ and no nonprincipal countably complete ultrafilter belongs to its own ultrapower.
A very similar argument yields the following fact:
Suppose $\delta$ is regular. Let $\kappa = \textsc{crt}(j_\delta)$ and assume $\kappa$ is $\delta$-strongly compact. Then for any $\gamma < \delta$, $P(\gamma)\subseteq M_\delta$.
Combining these facts, we obtain supercompactness:
Suppose $\delta$ is regular. Let $\kappa = \textsc{crt}(j_\delta)$ and assume $\kappa$ is $\delta$-strongly compact. Then $(M_\delta)^\gamma\subseteq M_\delta$ for all $\gamma < \delta$.
We may assume by induction that for all $\bar \gamma < \gamma$, $(M_\delta)^{\bar \gamma}\subseteq M_\delta$. Then if $\gamma$ is singular, it is immediate that $(M_\delta)^\gamma\subseteq M_\delta$. Therefore we may assume that $\gamma$ is regular.
Let $\gamma' = \text{cf}^{M_\delta}(\sup j_\delta[\gamma])$. By the tight covering property, $\gamma' \leq \delta$. But $\gamma'\neq \delta$ since $\text{cf}(\gamma') = \text{cf}(\sup j_\delta[\gamma]) = \gamma$. Therefore $\gamma' < \delta$. By \[KetonenCov\], it follows that there is a set $A'\in M_\delta$ containing $j_\delta[\gamma]$ such that $|A'|^{M_\delta} = \gamma'$. Since $\gamma' < \delta$, $P(\gamma')\subseteq M_\delta$. Since $|A'|^{M_\delta} = \gamma'$, $P(A')\subseteq M_\delta$. Since $j_\delta[\gamma]\subseteq A'$, it follows that $j_\delta[\gamma]\in M_\delta$.
The following is a glaring open question:
Let $\kappa = \textsc{crt}(j_\delta)$ and assume $\kappa$ is $\delta$-strongly compact. Is $M_\delta$ is closed under $\delta$-sequences?
We can answer this question when $\delta$ is a successor cardinal:
Suppose $\delta$ is regular but not strongly inaccessible. Let $\kappa = \textsc{crt}(j_\delta)$ and assume $\kappa$ is $\delta$-strongly compact. Then $M_\delta$ is closed under $\delta$-sequences.
We use the following fact:
Suppose $M$ is an ultrapower of $V$ and $\lambda$ is a cardinal. Then the following are equivalent:
(1) $M^\lambda\subseteq M$.
(2) $M$ has the tight covering property at $\lambda$ and $P(\lambda)\subseteq M$.
Therefore by \[TightCover\], it suffices to show that $P(\delta)\subseteq M_\delta$. There are two cases:
\[BigCofCase\] For some cardinal $\gamma < \delta$ of cofinality at least $\kappa$, $2^\gamma\geq \delta$.
Note that by the proof of \[StrongCompactAbs\] in the proof of \[TightCover\], $\kappa$ is ${<}\delta$-strongly compact in $M_\delta$. Therefore by Solovay’s theorem [@Solovay], $(\gamma^{<\kappa})^{M_\delta} =\gamma$. By \[Hausdorff\], it follows that in $M_\delta$, there is a $\kappa$-independent family $F$ of subsets of $\gamma$ such that $|F|^{M_\delta} = \delta$. As in \[PowerClm\] in the proof of \[TightCover\], $P(F)\subseteq M_\delta$, so $P(\delta)\subseteq M_\delta$.
Otherwise.
In this case, $\delta$ must be the successor of a singular cardinal of cofinality less than $\kappa$. To see this, let $\lambda < \delta$ be a cardinal such that $2^\lambda\geq \delta$. Then $2^{(\lambda^+)}\geq \delta$, and so since we are not in \[BigCofCase\], $\lambda^+ = \delta$. Similarly since we are not in \[BigCofCase\], $\text{cf}(\lambda) < \kappa$. We use the following standard fact:
\[SmallCofSuper\] Suppose $j : V\to M$ is an elementary embedding with critical point $\kappa$. Suppose $M$ is closed under $\lambda$-sequences. Then $M$ is closed under $\lambda^{<\kappa}$-sequences.
Since $|P_\kappa(\lambda)| = \lambda^{<\kappa}$, it suffices to show that $j\restriction P_\kappa(\lambda)\in M$. For $\sigma\in P_\kappa(\lambda)$, $j(\sigma) = j[\sigma]$. But $j\restriction \lambda \in M$, so the function $\sigma\mapsto j[\sigma]$ belongs to $M$.
By Konig’s theorem $\lambda^{<\kappa} \geq \lambda^+ = \delta$. Thus by \[SmallCofSuper\], $(M_\delta)^\delta\subseteq M_\delta$.
The least strongly compact cardinal
-----------------------------------
We must now say something about the hypothesis that $\textsc{crt}(j_\delta)$ is $\delta$-strongly compact. The following is Theorem 7.1 of [@Frechet]:
Suppose $\delta$ is a successor cardinal or a strongly inaccessible cardinal. Then $\textsc{crt}(j_\delta)$ is $\delta$-strongly compact.
Thus we have the following facts:
\[LeastSuper\] Suppose $\delta$ is a successor cardinal that carries a uniform countably complete ultrafilter. Then $j_\delta : V\to M_\delta$ witnesses that $\textsc{crt}(j_\delta)$ is $\delta$-supercompact.
Suppose $\delta$ is an inaccessible cardinal that carries a uniform countably complete ultrafilter. Then $j_\delta : V\to M_\delta$ witnesses that $\textsc{crt}(j_\delta)$ is ${<}\delta$-supercompact.
For any regular cardinal $\delta$, let $\kappa_\delta$ denote the least cardinal $\kappa$ such that every regular cardinal in the interval $[\kappa,\delta]$ is uniform.
We will have $\kappa_\delta = \delta^+$ when $\delta$ is not uniform.
\[crtchar\] Suppose $\delta$ is a successor cardinal or a strongly inaccessible cardinal. If $\delta$ is uniform then $\textsc{crt}(j_\delta) = \kappa_\delta$.
It suffices to show that $\textsc{crt}(j_\delta) \leq \kappa_\delta$. By \[KetonenRegularity\], every set $A\subseteq M_\delta$ with $|A|\leq \delta$ is contained in a set $A'\in M_\delta$ such that $|A'|^{M_\delta} < j_\delta(\kappa_\delta)$. Thus $j_\delta(\kappa_\delta)> \delta$, so $\textsc{crt}(j_\delta) \leq \kappa_\delta$.
Using terminology inspired by Bagaria-Magidor, $\kappa_\delta$ is the least cardinal that is $(\omega_1,\delta)$-strongly compact. (Indeed it is the least $\kappa$ such that every $\kappa$-complete filter on $\delta$ generated by $\delta$ sets extends to a countably complete ultrafilter.) In particular, we have the following fact which is a version of Ketonen’s theorem relating strongly compact cardinals to uniform ultrafilters on regular cardinals:
\[omega1Ketonen\] Assume $\kappa$ is a cardinal. Then the following are equivalent:
(1) $\kappa$ is the least $\omega_1$-strongly compact cardinal.
(2) For all regular cardinals $\delta \geq\kappa$, $\kappa_\delta = \kappa$.
A global consequence is the following fact:
The least $\omega_1$-strongly compact cardinal is supercompact.
Let $\kappa$ be the least $\omega_1$-strongly compact cardinal. Fix a regular cardinal $\delta \geq \kappa$. Then by \[omega1Ketonen\] and \[crtchar\], $\kappa = \kappa_\delta = \textsc{crt}(j_\delta)$ for all regular cardinals $\delta\geq \kappa$. Therefore if $\delta\geq \kappa$ is a successor cardinal, $j_\delta: V\to M_\delta$ witnesses that $\kappa$ is $\delta$-supercompact. Therefore $\kappa$ is $\delta$-supercompact for all successor cardinals $\delta\geq \kappa$. In other words, $\kappa$ is supercompact.
The least strongly compact cardinal is supercompact.
The next Fréchet cardinal in $M_\delta$
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In this subsection we establish a technical fact that will be of use in the proof of the Irreducibility Theorem below:
\[NextUniform\] Suppose $\delta$ is a uniform regular cardinal. Let $\kappa = \textsc{crt}(j_\delta)$, and assume $\kappa$ is $\delta$-strongly compact. Suppose that in $M_\delta$, $\lambda$ is the least Fréchet cardinal above $\delta$. Then $\lambda$ is measurable.
For the proof we need some facts about Fréchet cardinals from [@Frechet].
For any Fréchet cardinal $\lambda$, let $U_\lambda$ denote the $\sE$-least Fréchet uniform ultrafilter on $\lambda$.
Thus $U_\lambda$ generalizes $U_\delta$ to all Fréchet cardinals. It turns out that the following notion sometimes suffices as a substitute for regularity in the analysis of $U_\lambda$:
A Fréchet cardinal $\lambda$ is [*isolated*]{} if $\lambda$ is a limit cardinal but $\lambda$ is not a limit of Fréchet cardinals.
A reasonable conjecture is that any isolated cardinal is measurable, but we do not know how to prove this from UA alone. (UA + GCH suffices.)
For any cardinal $\gamma$, $\gamma^\sigma$ denotes the least Fréchet cardinal $\lambda$ such that $\lambda > \gamma$.
The following is a restatement of [@Frechet] Lemma 3.3:
\[IsolationLimit\] A cardinal $\lambda$ is isolated if and only if $\lambda = \gamma^\sigma$ for some cardinal $\gamma$ with $\gamma^+ < \lambda$.
Corollary 8.9 from [@Frechet] gives an example of how isolation can stand in for regularity in some of our arguments:
\[IsolatedInternal\] If $\lambda$ is isolated then for any countably complete $M_{U_\lambda}$-ultrafilter $W$ on an ordinal less than $\sup j_{U_\lambda}[\lambda]$, $W\in M_{U_\lambda}$.
First note that $\lambda > \delta^+$: $\delta$ is not uniform in $M_\delta$ by \[TightCover\], and a theorem of Prikry [@Prikry] shows that if a regular cardinal $\delta$ is not uniform, neither is its successor. Therefore by \[IsolationLimit\], $\lambda$ is isolated. In other words, $\lambda$ is a limit cardinal.
If $M_\delta$ satisfies that there is a measurable cardinal in the interval $[\delta,\lambda]$, then $\lambda$ is measurable. Let $U = (U_\lambda)^{M_\delta}$. To show $\lambda$ is measurable, it suffices to show that $\textsc{crt}(j^{M_\delta}_{U}) \geq \delta$. Assume towards a contradiction that $\textsc{crt}(j^{M_\delta}_{U}) < \delta$.
By [@Frechet] Theorem 9.2 applied in $M_\delta$, $M_\delta$ satisfies that $U$ is $\gamma$-supercompact for all Fréchet cardinals $\gamma < \lambda$. Since $\textsc{crt}(j_\delta)$ is $\delta$-strongly compact, $\textsc{crt}(j_\delta)$ is ${<}\delta$-strongly compact in $M_\delta$ by \[KetonenInternal\], and hence $U$ is ${<}\delta$-supercompact in $M_\delta$. By the Kunen inconsistency theorem, since $\textsc{crt}(j^{M_\delta}_{U}) < \delta$ and $U$ is ${<}\delta$-supercompact in $M_\delta$, we must have $j^{M_\delta}_{U}(\delta) > \delta$.
\[InternalClm\] Suppose $W$ is a countably complete $M_{U}^{M_\delta}$-ultrafilter on $\delta$. Then $W\in M_{U}^{M_\delta}$.
By \[IsolatedInternal\] applied in $M_\delta$, it suffices to show that $W\in M_\delta$. Let $i : M_{U}^{M_\delta}\to N$ be the ultrapower of $M_{U}^{M_\delta}$ by $W$ using functions in $M_{U}^{M_\delta}$. Since $\delta < j_{U}^{M_\delta}(\delta)$, and $j_{U}^{M_\delta}(\delta)$ is regular in $M_{U}^{M_\delta}$, $i$ is continuous at $j_{U}^{M_\delta}(\delta)$. Since $\delta$ is not uniform in $M_\delta$, $j_{U}^{M_\delta}$ is continuous at $\delta$. Therefore $i\circ j_{U}^{M_\delta}$ is continuous at $\delta$. It follows from \[KetonenInternal\] that $i\circ j_{U}^{M_\delta}$ is an internal ultrapower embedding of $M_\delta$. Let $\xi = [\text{id}]_{U}^{M_\delta}$. Then for any $f\in M_\delta$, $$i([f]_{U}^{M_\delta}) = i\circ j_{U}^{M_\delta}(f)(i(\xi))$$ Thus $i$ can be defined within $M_\delta$. Therefore $W\in M_\delta$, as desired.
By the argument of \[TightCover\], since $\textsc{crt}(j_\delta)$ is $\delta$-strongly compact, it follows from \[InternalClm\] that $P(\delta)\subseteq M_{U_\lambda}^{M_\delta}$. Applying \[InternalClm\] again, we have $U_\delta = U_\delta\cap M_{U_\lambda}^{M_\delta}\in M_{U_\lambda}^{M_\delta}\subseteq M_\delta = M_{U_\delta}$, which contradicts the fact that a nonprincipal countably complete ultrafilter cannot belong to its own ultrapower.
The Irreducibility Theorem
==========================
In this section, we prove the main result of this paper which we call the [*Irreducibility Theorem*]{}. To define the notion of irreduciblity, and to give the proof, we need some preliminary facts about factorizations of ultrapower embeddings.
The Rudin-Frolik order
----------------------
The Rudin-Frolik order is an order on ultrafilters that measures how ultrapower embeddings factor into iterated ultrapowers.
The [*Rudin-Frolik order*]{} is defined on countably complete ultrafilters $U$ and $W$ by setting $U\D W$ if there is an internal ultrapower embedding $k : M_U\to M_W$ such that $j_W = k\circ j_U$.
The Ultrapower Axiom is equivalent to the statement that the Rudin-Frolik order is directed on countably complete ultrafilters.
A countably complete ultrafilter $U$ is [*irreducible*]{} if for all $D\D U$, either $D$ is principal or $D$ is isomorphic to $U$.
An ultrapower embedding $j :V \to M$ is [*irreducible*]{} if $j = j_U$ for some irreducible ultrafilter $U$.
One version of the Irreducibility Theorem we will prove is the following:
Suppose $U$ is an irreducible Fréchet uniform ultrafilter on a successor cardinal or a strong limit singular cardinal $\lambda$. Then $M_U^\lambda\subseteq M_U$.
We continue with some more preliminary facts about the Rudin-Frolik order.
Supposer $U\in \Un$ and $D$ is a nonprincipal countably complete ultrafilter such that $D\D U$. Then $U/D$ is the uniform $M_D$-ultrafilter derived from $k$ using $[\text{id}]_U$ where $k : M_D\to M_U$ is the (unique) internal ultrapower embedding of $M_D$ such that $j_U = k\circ j_D$.
Thus $j^{M_D}_{U/D} : M_D\to M_U$ witnesses $D\D U$. The following is a key lemma in the analysis of the Rudin-Frolik order under UA, proved in [@RF] Lemma 6.6:
\[Pushdown\] Supposer $U\in \Un$ and $D$ is a nonprincipal countably complete ultrafilter such that $D\D U$. Then in $M_D$, $j_D(U)\swo U/D$.
The following fact is one of the main theorems of [@RF], Theorem 8.3:
A countably complete ultrafilter has at most finitely many predecessors in the Rudin-Frolik order up to isomorphism.
In other words, an ultrapower embedding of $V$ has at most finitely many predecessors in the Rudin-Frolik order.
The following is an easy corollary:
For any ultrapower embedding $j : V\to M$, there is a finite iterated ultrapower $$V = M_0\stackrel{j_0}{\longrightarrow} M_1 \stackrel{j_1}{\longrightarrow}\cdots \stackrel{j_{n-1}}{\longrightarrow} M_{n} = M$$ such that for each $i < n$, $j_i : M_i\to M_{i+1}$ is an irreducible ultrapower embedding of $M_i$ and $j = j_{n-1}\circ \cdots j_1\circ j_0$.
The analysis of countably complete ultrafilters under UA therefore reduces to the analysis of irreducible ultrafilters (and how they can be iterated). But even given the Irreducibility Theorem, irreducible ultrafilters remain somewhat mysterious.
Indecomposability and factorization
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The key to the proof of the irreducibility theorem is a generalization of a theorem of Silver that under favorable circumstances allows an ultrapower embedding to be “factored across a continuity point."
Suppose $\lambda$ is a cardinal. An ultrafilter $U$ on a set $X$ is [*$\lambda$-indecomposable*]{} if whenever $P$ is a partition of $X$ into $\lambda$ parts, there is some $Q\subseteq P$ such that $|Q| < \lambda$ and $\bigcup Q\in U$.
An ultrafilter is $\lambda$-decomposable if it is not $\lambda$-indecomposable.
Suppose $U$ is an ultrafilter and $\lambda$ is a cardinal. Then the following are equivalent:
(1) $U$ is $\lambda$-decomposable.
(2) There is a Fréchet uniform ultrafilter $D$ on $\lambda$ with $D\RK U$.
Assuming $U$ is countably complete, one can add to the list:
(1) There are elementary embeddings $V\stackrel j\longrightarrow M\stackrel k\longrightarrow M_U$ such that $j_U = k\circ j$ and $\sup j[\lambda] \leq \textsc{crt}(k) < j(\lambda)$.
The proof of the following theorem appears in [@Frechet] Theorem 4.8. A lucid sketch of the special case that was relevant to Silver appears in [@Silver].
\[Silver\] Suppose $U$ is an ultrafilter and $\delta$ is a regular cardinal. Assume $U$ is $\lambda$-decomposable for all cardinals $\lambda$ with $\delta \leq \lambda \leq 2^\delta$. Then there is an ultrafilter $D$ on some cardinal $\gamma <\delta$ and an elementary embedding $k : M_{D} \to M_U$ such that $j_U((2^\delta)^+)\subseteq k[M_D]$.
In the countably complete case, which is the only case in which we will be interested, $j_U((2^\delta)^+)\subseteq k[M_D]$ is equivalent to $\textsc{crt}(k) > j_U((2^\delta)^+)$. Thus we have $ j_U((2^\delta)^+) = j_D((2^\delta)^+) = (2^\delta)^+$.
Under UA, this has the following consequence:
\[SigmaSilver\] Suppose $U$ is a countably complete ultrafilter and $\delta$ is a regular cardinal. Assume $U$ is $\lambda$-decomposable for all cardinals $\lambda$ with $\delta \leq \lambda \leq 2^\delta$. Then there is a countably complete ultrafilter $D$ on some cardinal $\gamma <\delta$ and an internal ultrapower embedding $k : M_{D} \to M_U$ such that $\textsc{crt}(k) > (2^\delta)^+$.
For the proof we need the following theorem, which appears as [@Frechet] Theorem 12.1:
\[UFCounting\] For any cardinal $\lambda$, $|\Un_\lambda|\leq (2^\lambda)^+$.
By \[Silver\], fix an ultrafilter $D$ on some cardinal $\gamma <\delta$ and an elementary embedding $k : M_{D} \to M_U$ such that $j_U((2^\delta)^+)\subseteq k[M_D]$. Obviously $D$ is countably complete since $M_D$ embeds in $M_U$.
By \[kInternal\], to show $k$ is an internal ultrapower embedding, it is enough to show that $\tr U D\in k[M_D]$. By \[UFCounting\], $|\Un^{M_U}_{\leq j_U(\delta)}|^{M_U}\leq j_U((2^\delta)^+)$. Since $j_U((2^\delta)^+)\subseteq k[M_D]$ and $\Un^{M_U}_{\leq j_U(\delta)}\in k[M_D]$, it follows that $\Un^{M_U}_{\leq j_U(\delta)}\subseteq k[M_D]$. But $\tr U D\in \Un^{M_U}_{\leq j_U(\delta)}$ by \[BoundingLemma\]. So $\tr U D\in k[M_D]$ as desired.
\[DeltaFactor\] Suppose $\delta$ is a Fréchet uniform cardinal that is either a successor cardinal or a strongly inaccessible cardinal. Suppose $i : M_\delta\to N$ is an internal ultrapower embedding. Then there is some $D\in \Un_{<\delta}$ and an internal ultrapower embedding $i': M_D^{M_\delta}\to N$ such that $i = i'\circ j^{M_\delta}_D$.
Let $U\in M_\delta$ be a countably complete ultrafilter such that $j_U^{M_\delta} = i$. Then $U$ is $\lambda$-indecomposable for every cardinal $\lambda$ with $\delta\leq \lambda \leq 2^\delta$, and indeed for every cardinal $\lambda$ between $\delta$ and the next measurable cardinal, simply because none of these cardinals are Fréchet uniform by \[NextUniform\]. Applying \[SigmaSilver\] in $M_\delta$ yields the corollary.
Combinatorics of normal fine ultrafilters
-----------------------------------------
A key ingredient in the proof of the Irreducibility Theorem is a pair of combinatorial lemmas regarding normal fine ultrafilters on $P(\delta)$. The first lemma is due to Solovay, and the second is due to the author though it seems likely that it has already been discovered.
Suppose $\delta$ is a regular cardinal and $\vec S = \langle S_\alpha : \alpha < \delta\rangle$ is a stationary partition of $S^\delta_\omega$. The [*Solovay set*]{} associated to $\vec S$ is the set of all $\sigma\subseteq \delta$ such that letting $\gamma = \sup \sigma$, $\sigma = \{\alpha < \delta: S_\alpha\cap \gamma\text{ is stationary in } \gamma\}$.
\[Solovay\] Suppose $\delta$ is a regular cardinal and $\vec S$ is a stationary partition of $S^\delta_\omega$ into $\delta$ pieces. Then the Solovay set associated to $\vec S$ belongs to every normal fine ultrafilter on $P(\delta)$.
We omit the proof, which appears in [@MO]. In particular, there is a single set of cardinality $\delta$ on which all normal fine ultrafilters on $P(\delta)$ concentrate. Indeed, the function $\sup : P(\delta)\to \delta$ is one-to-one on any Solovay set. Thus any normal fine ultrafilter $\mathcal U$ on $P(\delta)$ is [*isomorphic*]{} to the ultrafilter $\sup_*(\mathcal U)$ on $\delta$.
The second lemma we need is less well-known.
\[NormalGeneration\] Suppose $\lambda$ is a cardinal and $\mathcal F$ is a normal fine filter on $P(\lambda)$. Suppose $D$ is an ultrafilter on $\lambda$. Let $S = \{\sigma\subseteq j_D(\lambda) : [\textnormal{id}]_D\in \sigma\}$. Then $j_D[\mathcal F]\cup \{S\}$ generates $j_D(\mathcal F)$ in the sense that any $X\in j_D(\mathcal F)$ contains $j_D(\bar X)\cap S$ for some $\bar X\in \mathcal F$.
Suppose $X\in j_D(\mathcal F)$. Fix $\langle X_\alpha : \alpha < \lambda\rangle$ such that letting $\langle Y_\alpha : \alpha < j_D(\lambda)\rangle$ denote $j_D(X_\alpha : \alpha < \delta\rangle)$, $X = Y_{[\text{id}]_D}$. We may assume without loss of generality that $X_\alpha\in \mathcal F$ for all $\alpha < \lambda$. Let $\bar X = \triangle_{\alpha < \delta} X_\alpha$.
We claim $j_D(\bar X)\cap S\subseteq X$. Suppose $\sigma\subseteq \lambda$ and $\sigma\in S\cap j_D(\bar X)$. The fact that $\sigma\in S$ means $[\text{id}]_D\in \sigma$. The fact that $\sigma\in j_D(\bar X)$ means $\sigma\in \triangle _{\alpha < j_D(\delta)}Y_\alpha$. By the definition of the diagonal intersection, $\sigma\in Y_\alpha$ for all $\alpha\in \sigma$. Therefore $\sigma\in Y_{[\text{id}]_D} = X$, as desired.
Suppose $M$ is an inner model, $X$ and $A\subseteq P(X)\cap M$ are sets in $M$, and $\mathcal U$ is an $M$-ultrafilter on $A$. We say $\mathcal U$ is a [*fine $M$-ultrafilter*]{} every element of $X$ belongs to $\mathcal U$-almost all elements of $A$.
Suppose $\lambda$ is a cardinal, $D$ is a countably complete ultrafilter on $\lambda$, and $\mathcal U$ is a normal fine ultrafilter on a set $A\subseteq P(\lambda)$. Then $j_D(\mathcal U)$ is the unique fine $M_D$-ultrafilter $\mathcal U'$ on $j_D(A)$ such that $j_D[\mathcal U]\subseteq \mathcal U'$.
Proof of the Irreducibility Theorem
-----------------------------------
We use a slight variant of the notion of irreducibility:
Suppose $\lambda$ is a cardinal and $U$ is a countably complete ultrafilter. Then $U$ is [*${<}\lambda$-irreducible*]{} if for all $D\in \Un_{<\lambda}$ such that $D \D U$, $D$ is principal. An ultrapower embedding $j : V\to M$ is [*${<}\lambda$-irreducible*]{} if for all $D\in \Un_{<\lambda}$ such that $j_D \D j$, $D$ is principal.
\[IrredThm\] Suppose $\delta$ is a uniform successor cardinal. Suppose $i: V\to M$ is a ${<}\delta$-irreducible ultrapower embedding. Then $M^\delta\subseteq M$.
To prove \[IrredThm\], one propagates the closure of $M_\delta$ to $M$ by comparing the two models. More specifically, the strategy of the proof of \[IrredThm\] is to take the canonical comparison $(i_*,j) : (M_\delta,M)\to N$ of $(j_\delta,i)$ and show that $\textsc{crt}(i_*) > \delta$. It follows that $N^\delta = (N^\delta)^{M_\delta}\subseteq N$ by applying in $M_\delta$ the standard fact that the ultrapower by a $\delta$-complete ultrafilter is closed under $\delta$-sequences. But $N$ is also an internal ultrapower of $M$, so $N\subseteq M$. In particular $\text{Ord}^\delta\subseteq N\subseteq M$. But this implies $i[\delta]\in M$, so $(M)^\delta\subseteq M$ by the standard criterion for closure of ultrapowers under $\delta$-sequences.
Let $\delta' = \text{cf}^M(\sup i[\delta])$ and let $i_* : M_\delta\to N$ be the internal ultrapower embedding given by \[ZeroComparison\] such that $$(i_*,j^M_{\delta'}) : (M_\delta,M)\to N$$ is a comparison of $(j_\delta,i)$. By \[DeltaFactor\], there is some $D\in \Un_{<\delta}$ and an internal ultrapower embedding $i' : (M_D)^{M_\delta}\to N$ such that $i_* = i'\circ j^{M_\delta}_D$ and $\textsc{crt}(i') > j_D^{M_\delta}(\delta)$.
To avoid superscripts, we introduce the following notation. Let $d = j_D$ and let $P = M_D$. Then $j_D^{M_\delta} = d\restriction M_\delta$ and $M_D^{M_\delta} = d(M_\delta) = (M_{d(\delta)})^P$ since $(M_\delta)^\delta\subseteq M_\delta$. Thus we have the commutative diagram \[Irred1Fig\].
![Comparing $(j_\delta,i)$, discovering $d$.[]{data-label="Irred1Fig"}](Irred1.pdf)
The bulk of the proof is contained in the following claim:
\[AgreementClm\] $d(j_\delta) \restriction N = j_{\delta'}^M\restriction N$.
We start by computing some closure properties of $N$ relative to $M$ and $P$, leading to a proof that $d(\delta) = \delta'$.
First, $N$ is closed under $d(\delta)$-sequences relative to $P$. This is because $d(M_\delta)$ is closed under $d(\delta)$-sequences relative to $P$ (by the elementarity of $d$ and the fact that $M_\delta$ is closed under $\delta$-sequences), and $N$ is closed under $d(\delta)$-sequences relative to $d(M_\delta)$ (since $i'$ is an ultrapower embedding and $\textsc{crt}(i') > d(\delta)$). This facilitates the following calculation:
$$\begin{aligned}
\text{cf}^N(\sup j_{\delta'}^M[\delta']) &= \text{cf}^N(\sup j^M_{\delta'}[\sup i[\delta]])\label{cofsup}\\
&= \text{cf}^N(\sup j^M_{\delta'}\circ i[\delta])\nonumber\\
&= \text{cf}^N(\sup i'\circ d(j_\delta)\circ d[\delta])\label{compid}\\
&= \text{cf}^N(\sup i'\circ d(j_\delta)[d(\delta)])\label{contd}\\
&= d(\delta)\label{sc}\end{aligned}$$
\[cofsup\] follows from the fact that $\delta' = \text{cf}^M(\sup i[\delta])$. \[compid\] follows from the fact that $j^M_{\delta'}\circ i = i'\circ d(j_\delta)\circ d$. \[contd\] follows from the fact that $\sup d[\delta] = d(\delta)$, since $\delta$ is regular and $d = j_D$ for some $D\in \Un_{<\delta}$. Finally \[sc\] follows from the fact that $i'\circ d(j_\delta)[d(\delta)]\in N$, since $N$ is closed under $d(\delta)$-sequences relative to $P$ and $i'\circ d(j_\delta)$ is an internal ultrapower embedding of $P$.
Second, $N$ has the tight covering property at $\delta'$ relative to $M$. To see this, we use the contrapositive of \[TightCover\]: if $N = M_{\delta'}^M$ does not have the tight covering property at $\delta'$, then $\delta'$ is weakly inaccessible in $M$ and $\sup j_{\delta'}^M[\delta']$ is regular in $N$. But then $\sup j_{\delta'}^M[\delta'] = \text{cf}^N(\sup j_{\delta'}^M[\delta']) = d(\delta)$. But $\sup j_{\delta'}^M[\delta']$ is a limit cardinal of $N$ since $\delta'$ is weakly inaccessible in $M$, while $d(\delta)$ is a successor cardinal of $P$ and hence a successor cardinal of $N$ since $\delta$ is a successor cardinal by assumption. This is a contradiction.
Now $\text{cf}^N(\sup j_{\delta'}^M[\delta']) = \delta'$ by the tight covering property. Combining this with \[sc\] above, $$d(\delta) =\delta'$$
Since $d(\delta)$ is a successor cardinal in $P$ and $N$ is closed under $d(\delta)$-sequences relative to $P$, $d(\delta)$ is a successor cardinal in $N$. Since $N\subseteq M$, $d(\delta)$ is a successor cardinal in $M$. Therefore since $N = (M_{d(\delta)})^M$, \[LeastSuper\] applied in $M$ implies that $N$ is closed under $\delta'$-sequences relative to $M$. We therefore have $$\label{MPAgreement}\text{Ord}^{d(\delta)}\cap P = \text{Ord}^{d(\delta)}\cap N = \text{Ord}^{d(\delta)}\cap M$$
We now work directly with ultrafilters instead of ultrapower embeddings. Recall that $U_\delta$ denotes the ultrafilter derived from $j_\delta$ using $\sup j_\delta[\delta]$. Let $U' = (U_{d(\delta)})^M$. We claim that $U' = d(U_\delta)$. This at least makes sense because $U'\subseteq P(d(\delta))\cap M$ and $d(U_\delta)\subseteq P(d(\delta))\cap P$ and $P(d(\delta))\cap M = P(d(\delta))\cap P$ by \[MPAgreement\].
Let $W = D^-(U') = \{X\subseteq \delta : d(X)\in U'\}$. It is easy to see that $W$ is a countably complete ultrafilter on $\delta$.
Note that $\{\alpha < \delta : \Un_\alpha = \emptyset\}\in W$ since $\{\alpha < d(\delta) :\Un^{M}_\alpha = \emptyset\}\in U'$ and $$\begin{aligned}
d(\{\alpha <\delta : \Un_\alpha = \emptyset\}) &= \{\alpha < d(\delta) :\Un^{P}_\alpha = \emptyset\}\nonumber \\
&= \{\alpha < d(\delta):\Un^{N}_\alpha = \emptyset\}\label{UniformAbs0} \\
&= \{\alpha < d(\delta) :\Un^{M}_\alpha = \emptyset\} \label{UniformAbs1}\end{aligned}$$ \[UniformAbs0\] and \[UniformAbs1\] follow from \[MPAgreement\] and \[CombinatorialInternal\], which together imply $\Un_{<d(\delta)}^P = \Un_{<d(\delta)}^N = \Un_{<d(\delta)}^P$.
Moreover, we claim $W$ is weakly normal, in the sense that any regressive function on $\delta$ is bounded below $\delta$ on a $W$-large set. This follows from the fact that $d$ is continuous at $\delta$. Suppose $f : \delta\to \delta$ is regressive. Then since $U'$ is Ketonen in $M$, by \[KetonenCombinatorial\], $d(f)$ is bounded below $d(\delta)$ on a $U'$-large set. But then there is some $\bar \xi < \delta$ such that $\xi < d(\bar \xi)$. Hence $\{\alpha < d(\delta) : d(f)(\alpha) < d(\xi)\}\in U'$. Thus $\{\alpha < \delta : f(\alpha) < \xi\}\in W$, so $f$ is bounded below $\delta$ on a $W$-large set.
By \[KetonenCombinatorial\], it follows that $W$ is Ketonen at $\delta$. Therefore by \[KetonenUnique\], $W = U_\delta$. In other words, $$\label{dimage}
d[U_\delta]\subseteq U'$$
Let $\mathcal U$ be the normal fine ultrafilter on $P(\delta)$ derived from $j_\delta[\delta]$. Working in $M$, let $\mathcal U'$ be the normal fine ultrafilter on $P(d(\delta))\cap M$ derived from $j^M_{d(\delta)}$ using $j^M_{d(\delta)}[d(\delta)]$. We want to use \[NormalGeneration\] to show essentially that $d(\mathcal U) = \mathcal U'$.
Fix a partition $\vec S = \langle S_\alpha : \alpha < \delta\rangle$ of $S^\delta_\omega$ into stationary sets. Let $A\subseteq P(\delta)$ be the Solovay set associated to $\vec S$. Thus $A$ consists of those $\sigma \subseteq \delta$ such that $\sigma = \{\xi < \gamma : S_\xi\text{ reflects to }\gamma\}$ where $\gamma = \sup \sigma$. Now $d(A)$ is the Solovay set defined from $d(\vec S)$ in $P$, but by \[MPAgreement\], $d(A)$ is also the Solovay set defined from $d(\vec S)$ in $M$. By \[Solovay\], $d(A)\in \mathcal U'$.
Since $P(A)\cap P = P(A)\cap M$ and $\mathcal U'\restriction d(A)$ is an normal fine $M$-ultrafilter on $d(A)$, $\mathcal U'\restriction d(A)$ is an normal fine $P$-ultrafilter on $d(A)$.
Let $s : \delta\to A$ be the inverse of the sup function on $A$. Then $s_*(U_\delta) = \mathcal U\restriction A$. Moreover $d(s) : d(\delta)\to d(A)$ is the inverse of the sup function on $d(A)$ so $d(s)_*(U') = \mathcal U'\restriction d(A)$. Since $d[U_\delta]\subseteq U'$, if $X\in \mathcal U\restriction A$, then $s^{-1}[X]\in U_\delta$, so $d(s^{-1}[X])\in U'$, so $X\in \mathcal U'\restriction d(A)$. Thus $$d[\mathcal U\restriction A]\subseteq \mathcal U'\restriction d(A)$$ Since $\mathcal U'$ is fine in $M$, letting $S = \{\sigma\in d(A) : [\text{id}]_D\in \sigma\}$, $S\in \mathcal U'\restriction d(A)$. But by \[NormalGeneration\], $d(\mathcal U\restriction A)$ is generated by $d[\mathcal U\restriction A]\cup \{S\}$. Thus $d(\mathcal U\restriction A) = \mathcal U'\restriction d(A)$. But then $$U' = \textstyle\sup_*(\mathcal U'\restriction d(A)) = \sup_*(d(\mathcal U\restriction A)) = d(\sup_*(\mathcal U\restriction A)) = d(U_\delta)$$ Thus $U' = d(U_\delta)$, as claimed.
Finally, applying \[MPAgreement\] again, $$d(j_\delta) \restriction N = j_{d(U_\delta)}^N = j_{U'}^N = j_{\delta'}^M \restriction N$$ This proves the claim.
The following claim will easily imply the theorem:
\[RFClm\] $d\D i$.
Let $(k,\ell) : (P,M)\to Q$ be the canonical comparison of $(d,i)$.
Since $(k,\ell)$ is the canonical comparison of $(d,i)$, by \[Pushout\], $(k,\ell)$ is the pushout of $(d,i)$. Therefore since $(i' \circ d(j_\delta),j_{\delta'}^M) : (P,M)\to N$ is also a comparison of $(d,i)$, there is an internal ultrapower embedding $h : Q\to N$ such that $j_{\delta'}^M = h\circ \ell$. Thus we have the commutative diagram \[Irred2Fig\].
![Comparing $(d,i)$, discovering $k$.[]{data-label="Irred2Fig"}](Irred2.pdf)
Since $j_{\delta'}^M$ is an irreducible ultrapower embedding of $M$ and $j_{\delta'}^M = h\circ \ell$, either $\ell$ or $h$ is the identity.
$\ell$ is the identity.
By case hypothesis, we have that $Q = M$ and $k :P\to M$ is an internal ultrapower embedding of $P$ such that $i =\ell\circ i = k\circ d$. Thus $d\D i$.
$h$ is the identity.
We will show in this case that in fact $N = M$ and $j_{\delta'}^M$ is the identity. By case hypothesis, we have the commutative diagram \[Irred3Fig\].
![The case $h = \text{id}$.[]{data-label="Irred3Fig"}](Irred3.pdf)
Let $e = d(j_\delta) \restriction N = j_{\delta'}^M\restriction N$, using \[AgreementClm\]. Then since $e = d(j_\delta)\restriction N$, $e$ is amenable to $P$, and since $e = j_{\delta'}^M\restriction N$, $e$ is amenable to $M$. Therefore by \[CanonicalInternal\], $e$ is an internal ultrapower embedding of $N$. But $N = (M_{\delta'})^M$, so if $j_{\delta'}^M\restriction N$ is amenable to $N$, then relative to $M$, $(M_{\delta'})^M$ is closed under $\alpha$-sequences for all ordinals $\alpha$. In other words, $N = M$ and so $j_{\delta'}^M$ is the identity. But then once again $k : P \to M$ is an internal ultrapower embedding of $P$ and $i =j_{\delta'}^M\circ i = k\circ d$.
Now $d \D i$, but $d = j_D$ for some $D\in \Un_{<\delta}$ and $i$ is ${<}\delta$-irreducible, so $d$ is the identity. Therefore $i_* = i'\circ d = i'$. Thus $\textsc{crt}(i_*) = \textsc{crt}(i') > d(\delta) = \delta$. Since $i_* : M_\delta\to N$ is an ultrapower embedding with critical point above $\delta$, $N$ is closed under $\delta$-sequences in $M_\delta$. By \[LeastSuper\], $M_\delta$ is itself closed under $\delta$-sequences in $V$, so it follows that $N$ is truly closed under $\delta$-sequences. But $N$ is an internal ultrapower of $M$, so $N\subseteq M$. Hence $\text{Ord}^\delta\subseteq M$, so $i[\delta]\in M$. Since $i : V\to M$ is an ultrapower embedding and $i[\delta]\in M$, $M^\delta\subseteq M$, as desired.
Supercompactness
================
In this section, we use the Irreducibility Theorem to prove the main theorem of this paper:
\[MenasGlobal\] If $\kappa$ is a cardinal, the following are equivalent:
(1) $\kappa$ is strongly compact.
(2) $\kappa$ is supercompact or a measurable limit of supercompact cardinals.
This is achieved by proving a more local result.
Level-by-level equivalence at successor cardinals
-------------------------------------------------
The title of this subsection comes from the following result:
\[MenasLocal\] If $\kappa$ is a cardinal and $\delta$ is a successor cardinal, then the following are equivalent:
(1) $\kappa$ is $\delta$-strongly compact.
(2) $\kappa$ is $\delta$-supercompact or a measurable limit of $\delta$-supercompact cardinals.
\[MenasLocal\] easily implies the main theorem of the paper:
The fact that (2) implies (1) is due to Menas [@Menas] and does not require UA.
We now prove the converse. Suppose $\kappa$ is strongly compact. If $\kappa$ is supercompact we are done, so suppose $\kappa$ is not supercompact. For each successor cardinal $\delta$, let $A_\delta\subseteq \kappa$ be the set of $\delta$-supercompact cardinals less than $\kappa$. By \[MenasLocal\], $A_\delta$ is unbounded in $\kappa$ for all successor cardinals $\delta$. Moreover, if $\delta_0 \leq \delta_1$, $A_{\delta_0}\supseteq A_{\delta_1}$. Therefore there is some set $A\subseteq \kappa$ such that for all sufficiently large $\delta$, $A_\delta = A$. It follows that $A$ is the set of supercompact cardinals below $\kappa$. But since $A = A_\delta$ for some $\delta$, $A$ is unbounded in $\kappa$. Therefore $\kappa$ is a limit of supercompact cardinals, as desired.
\[MenasLocal\] is proved by analyzing the following ultrafilters using \[IrredThm\].
If $\delta$ is a regular cardinal and $\nu \leq \delta$ is a cardinal, then $U^\nu_\delta$ denotes the $\sE$-least $\nu$-complete uniform ultrafilter on $\delta$ if it exists.
This analysis requires two simple lemmas:
\[SuccIrred\] Suppose $\nu$ is a successor cardinal and $\delta > \nu$ is a regular cardinal. Then $U^{\nu}_\delta$ is irreducible if it exists.
Suppose $D\D U$, and we will show that either $D$ is isomorphic to $U$ or $D$ is principal. By replacing $D$ with an isomorphic ultrafilter, we may assume $D\in \Un$ and $D\E U$. Given this, we will show that either $D = U$ or $D$ is principal.
$\textsc{sp}(D) = \delta$
Then $D$ is a $\nu$-complete ultrafilter $U$ on $\delta$, so $U\E D$. Therefore $U = D$.
$\textsc{sp}(D) < \delta$.
Then since $\delta$ is regular, $j_D(\delta) = \sup j_D[\delta]$. It follows that $\textsc{sp}(U/D) = j_D(\delta)$ since $\textsc{sp}(U/D)$ is derived from the internal ultrapower embedding $k : M_D\to M_U$ using $[\text{id}]_U$, and by the uniformity of $U$, $\sup j_D[\delta]$ is the least possible ordinal mapped above $[\text{id}]_U$ by $k$. Moreover $$\textsc{crt}(U/D) \geq \textsc{crt}(U) \geq \nu$$ so $U/D$ is $\nu$-complete in $M_D$. But also, and this is a key point, $$j_D(\nu) \leq j_U(\nu) = \nu$$ since $\nu$ is a successor cardinal. Therefore $M_D$ satisfies that $U/D$ is a uniform $j_D(\nu)$-complete ultrafilter on $j_D(\delta)$. Therefore $j_D(U)\E U/D$ in $M_D$. By \[Pushdown\], it follows that $D$ is principal.
Our second lemma shows that the requirement that $\nu$ is a successor cardinal is necessary above.
\[NormalTrans\] Suppose $\kappa$ is a measurable cardinal and $\delta > \kappa$ is a uniform regular cardinal. Let $U = U^\kappa_\delta$ and let $D = U^\kappa_\kappa$. Then in $M_D$, $\tr D U = U^{\kappa^+}_{j_D(\delta)}$.
Note that $D = U^\kappa_\kappa$ is the unique normal ultrafilter on $\kappa$ of Mitchell order zero.
Suppose that in $M_D$, $Z$ is a $\kappa^+$-complete uniform ultrafilter on $j_D(\delta)$. Then $D^-(Z)$ is a $\kappa$-complete uniform ultrafilter on $\delta$, so $U \E D^-(Z)$. Hence by \[OrderPreserving\], $M_D$ satisfies $$\tr D U \E \tr D {D^-(Z)} \E Z$$ Thus in $M_D$, $\tr D U$ lies $\E$-below every $\kappa^+$-complete uniform ultrafilter on $j_D(\delta)$, so to show $\tr D U = U^{\kappa^+}_{j_D(\delta)}$, it suffices to show that $\tr D U$ is $\kappa^+$-complete.
Since $D$ is a normal ultrafilter, either $D\D U$ or $D\mo U$. If $D\mo U$ then $\tr D U = j_D(U)$ so $\tr D U$ is $j_D(\kappa)$-complete; since $\kappa^+\leq j_D(\kappa)$, we are done.
If $D\D U$, then obviously $\tr D U = U/D$ is $\kappa$-complete. Since $\kappa$ is not measurable in $M_D$, $\tr D U$ is $\kappa^+$-complete, as desired.
The fact that (2) implies (1) is due to Menas [@Menas] and does not require UA.
We now prove the converse. Let $U = U^{\delta}_\kappa$ and let $D = U^\kappa_\kappa$. Working in $M_D$, let $Z = \tr D U$. By \[NormalTrans\], $Z = U^{\kappa^+}_{j_D(\delta)}$ so by \[SuccIrred\], $Z$ is an irreducible $\kappa^+$-complete uniform ultrafilter on $j_D(\delta)$. Let $\kappa' = \textsc{crt}(Z)$. Then by \[IrredThm\], $Z$ witnesses that $\kappa'$ is $j_D(\delta)$-supercompact in $M_D$. Moreover since $Z \E j_D(U) = U^{j_D(\kappa)}_{j_D(\delta)}$ by \[BoundingLemma\], $\kappa' \leq \textsc{crt}(j_D(U))$ with equality if and only if $Z = j_D(U)$. Thus in $M_D$ there is a $j_D(\delta)$-supercompact cardinal in the interval $[\kappa, j_D(\kappa)]$. By a standard reflection argument, either $\kappa$ is supercompact or $\kappa$ is a measurable limit of supercompact cardinals.
Level-by-level equivalence at singular cardinals
------------------------------------------------
In this section, we tackle the question of the local equivalence of strong compactness and supercompactness at a singular cardinal $\lambda$. This depends on the cofinality of $\lambda$ in the following way.
Suppose $\kappa < \lambda$ and $\textnormal{cf}(\lambda) < \kappa$. Then $\kappa$ is $\lambda$-strongly compact if and only if $\kappa$ is $\lambda^+$-strongly compact, and $\kappa$ is $\lambda$-supercompact if and only if $\kappa$ is $\lambda^+$-supercompact.
Thus \[MenasLocal\] implies the following fact:
Suppose $\kappa < \lambda$ and $\textnormal{cf}(\lambda) < \kappa$. Then $\kappa$ is $\lambda$-strongly compact if and only if $\kappa$ is $\lambda$-supercompact or a measurable limit of $\lambda$-supercompact cardinals.
When $\lambda$ is singular of large cofinality, equivalence provably fails. For example, we have the following fact:
Let $(\kappa_0,\lambda_0)$ be the lexicographically least pair $(\kappa, \lambda)$ such that $\kappa < \lambda$, $\kappa$ is $\lambda$-strongly compact, and $\lambda$ is a strong limit cardinal of cofinality at least $\kappa$. Then $\kappa_0$ is $\lambda_0$-strongly compact but not $\lambda_0$-supercompact.
For more on this, see [@Apter]. The question we consider here is whether there is a notion of strong compactness (i.e., a filter extension property) at singular cardinals for which level-by-level equivalence with supercompactness holds (in the sense that Menas’s theorem can be reversed).
\[LBL\] Suppose $\kappa < \lambda$ and $\lambda$ is singular. Then the following are equivalent:
(1) There is a $\kappa$-complete ultrafilter on $P_\kappa(\lambda)$ extending the club filter.
(2) $\kappa$ is $\lambda$-supercompact or a measurable limit of $\lambda$-supercompact cardinals.
To prove this we use a lemma that appears as [@GCH] Lemma 3.3:
\[ClubLemma\] Suppose $U$ is a countably complete ultrafilter, $\lambda$ is a cardinal, and $j_U[\lambda]\subseteq A\subseteq j_U(\lambda)$ has the property that $j_U(f)[A]\subseteq A$ for every $f : \lambda\to \lambda$. Suppose $D$ is a countably complete ultrafilter on an ordinal $\gamma < \lambda$. Suppose $(k,i) : (M_D,M_U)\to N$, $k$ is an internal ultrapower embedding, and $k\circ j_D = i\circ j_U$. Then $k([\textnormal{id}]_D) \in i(A)$.
\[ClubLemma\] comes into the picture through the following lemma.
\[Club2\] Suppose $\mathcal U$ extends a normal filter $\mathcal F$ on $P(\lambda)$ and $D$ is a countably complete ultrafilter on $\gamma < \lambda$ such that $D\sD \mathcal U$. Let $\mathcal U' = \mathcal U/D$. Then $\mathcal U'$ extends $j_D(\mathcal F)$.
It is easy to see that $j_D[\mathcal U]\subseteq \mathcal U'$. In particular $j_D[\mathcal F]\subseteq \mathcal U'$.
By \[ClubLemma\], $k([\text{id}]_D)\in [\text{id}]_\mathcal U$: note that $j_\mathcal U(f)[ [\text{id}]_\mathcal U]\subseteq [\text{id}]_\mathcal U$ for all $f : \lambda\to \lambda$, since this merely says that $ [\text{id}]_\mathcal U$ belongs to $j_D(C)$ where $C\subseteq P(\lambda)$ is the club of closure points of $f$. Letting $S = \{A\in j_D(P(\lambda)) : [\text{id}]_D\in A\}$, we have $[\text{id}]_\mathcal U\in k(S)$ and so $S\in \mathcal U'$. By \[NormalGeneration\], $j_D(\mathcal F)\subseteq \mathcal U$.
\[IrredExt\] For any normal fine filter $\mathcal F$ on $P(\lambda)$ and any successor cardinal $\nu < \lambda$, the $\swo$-least $\nu$-complete ultrafilter $U$ that is isomorphic to an extension of $\mathcal F$ is ${<}\lambda$-irreducible.
Suppose $D$ is an ultrafilter on $\gamma < \lambda$ and $D\sD U$. Let $U' = U/D$. To show $D$ is principal, it suffices by \[Pushdown\] to show that $j_D(U)\E U'$. Note that $U'$ is isomorphic to $\mathcal U' = \mathcal U/D$. But by \[Club2\], $\mathcal U'$ extends $j_D(\mathcal F)$. Moreover $U'$ is $j_D(\nu)$-complete since $j_D(\nu) = \nu$ and $\textsc{crt}(U') \geq \textsc{crt}(U) \geq \nu$. Therefore $U'$ is a $j_D(\nu)$-complete extension of $j_D(\mathcal F)$, and so $j_D(U)\E U'$, as desired.
The proof that (2) implies (1) follows the proof of Menas’s theorem [@Menas] and does not require UA.
We now show (1) implies (2). Assume (1).
Note that the filter $\mathcal F$ on $P(\lambda)$ generated by the club filter on $P_\kappa(\lambda)$ is normal.
Suppose first that for some cardinal $\nu < \kappa$, the least $\nu$-complete ultrafilter $U$ that is isomorphic to an extension of $\mathcal F$ is $\kappa$-complete. By replacing $\nu$ with $\nu^+$, we may assume without loss of generality that $\nu$ is a successor cardinal (notice that $\nu^+ < \kappa$). Then $U$ is ${<}\lambda$-irreducible by \[IrredExt\]. It follows from \[IrredThm\] that $U$ witnesses that $\kappa$ is ${<}\lambda$-supercompact. Since $\lambda$ is singular, $U$ witnesses that $\kappa$ is $\lambda$-supercompact, so (2) holds.
Suppose instead that for each $\nu < \kappa$, the least $\nu$-complete ultrafilter $U$ that is isomorphic to an extension of $\mathcal F$ is $\kappa$-complete. Then a similar argument shows that $\kappa$ is a limit of $\lambda$-supercompact cardinals, so (2) holds.
Ultrafilters on inaccessible cardinals {#Anom}
--------------------------------------
In this short subsection we discuss the issues with establishing a version of \[MenasLocal\] when $\delta$ is an inaccessible cardinal. The first thing we show is that the only obstruction is the fact that we do not know how to analyze $j_\delta: V \to M_\delta$ when $\delta$ is inaccessible.
\[InaccCover\] Suppose $\delta$ is a uniform inaccessible cardinal. Suppose $i :V \to M $ is a ${<}\delta$-irreducible ultrapower. Then $\textnormal{Ord}^\delta\cap M_\delta\subseteq M$.
Let $\kappa = \textsc{crt}(j_\delta)$. Then by [@Frechet] Theorem 7.8, $\kappa$ is $\delta$-strongly compact. In particular, every regular cardinal in the interval $[\kappa,\delta]$ is uniform, so by \[IrredThm\], $M_\delta$ is closed under $\lambda$-sequences for all $\lambda < \delta$. Let $(i_*,j') : (M_\delta,M)\to N$ be the canonical comparison of $(j_\delta,i)$. By \[DeltaFactor\], $i_*$ factors as $i'\circ j_D$ where $D\in \Un_{<\delta}$ and $\textsc{crt}(i') > \delta$. Since $V_\delta\subseteq M$, $D\in M$. Therefore since $M$ is closed under $\textsc{sp}(D)$-sequences, $j_D\restriction M = j_D^M$ is amenable to $M$. Therefore by \[CanonicalInternal\], $j_D\restriction N$ is an internal ultrapower embedding of $N$. Since $N\subseteq M_D^{M_\delta}$, $j_D\restriction\text{Ord}$ is amenable to $M_D^{M_{\delta}}$, and it follows that $D$ is principal. Hence $\textsc{crt}(i_*) \geq \textsc{crt}(i') > \delta$. Thus $\text{Ord}^\delta\cap M_\delta \subseteq N\subseteq M$, as desired.
Suppose $\delta$ is a uniform inaccessible cardinal. The following are equivalent:
(1) $M_\delta$ is closed under $\delta$-sequences.
(2) Every ${<}\delta$-irreducible ultrapower is closed under $\delta$-sequences.
There seems to be no clear way forward, and this suggests a number of open questions:
Suppose $\delta$ is strongly inaccessible. Can there be a countably complete ultrafilter $U$ such that $M = M_U$ has the following properties:
(1) $M$ is closed under $\lambda$-sequences for all $\lambda < \delta$.
(2) $M$ has the tight covering property at $\delta$.
(3) $M$ is not closed under $\delta$-sequences.
Even assuming only ZFC, it is not clear that it is possible for such an ultrapower to exist. Regarding supercompactness at inaccessible cardinals, we do have the following intriguing fact:
Suppose $\delta$ is a regular cardinal that carries distinct countably complete weakly normal ultrafilters. Then some $\kappa < \delta$ is $\delta$-supercompact.
We omit the proof, but this raises another question:
Suppose $\delta$ is a regular cardinal and $\nu < \delta$ is a successor cardinal. Suppose $\delta$ is a regular cardinal that carries distinct $\nu$-complete weakly normal ultrafilters. Is there a $\delta$-supercompact cardinal $\kappa$ such that $\nu < \kappa \leq \delta$?
Another interesting question is whether the ideas from the previous section suffice to characterize supercompactness at inaccessible cardinals:
Suppose $\delta$ is an inaccessible cardinal. Suppose there is a $\kappa$-complete ultrafilter extending the club filter on $P_\kappa(\delta)$. Then $\kappa$ is either $\delta$-supercompact or a measurable limit of $\delta$-supercompact cardinals.
Almost huge cardinals
---------------------
\[Huge\] Suppose there is a countably complete weakly normal ultrafilter on a regular cardinal that concentrates on a fixed cofinality. Then there is an almost huge cardinal.
Thus the same issues from \[Anom\] prevent us from showing that there is a huge cardinal under these hypotheses. We need a lemma which is useful in conjunction with \[IrredThm\].
Suppose $\nu$ is a cardinal, $\lambda\geq \nu$ is a regular cardinal, $\mathcal F$ is a normal fine filter on $P(\lambda)$. Then $U^\nu_\delta(\mathcal F)$ is the $\sE$-least countably complete weakly normal ultrafilter isomoprhic to an extension of $\mathcal F$ if it exists.
The proof of \[IrredExt\] yields the following fact:
\[FIrred\] Suppose $\nu$ is a successor cardinal, $\lambda\geq \nu$ is a cardinal, and $\mathcal F$ is a normal fine filter on $P(\lambda)$. Then $U^\nu_\delta(\mathcal F)$ is irreducible.
We just need the following corollary:
\[AIrred\] If $\delta$ is a regular cardinal and $A\subseteq \delta$, then the $\sE$-least weakly normal ultrafilter concentrating on $A$ is irreducible.
Let $\mathcal F$ be the club filter on $\delta$ restricted to $A$ viewed as a filter on $P(\delta)$. Then $\mathcal F$ is a normal fine filter on $P(\delta)$. A weakly normal ultrafilter on $\delta$ concentrates on $A$ if and only if it is isomorphic to an extension of $\mathcal F$. (For the forwards implication one needs that $\delta$ is regular: weakly normal ultrafilters on singular cardinals need not extend the club filter.) Thus the $\sE$-least weakly normal ultrafilter concentrating on $A$ is $U^{\omega_1}_\delta(\mathcal F)$, which is irreducible by \[FIrred\].
Suppose $\kappa < \delta$ are regular cardinals and $U$ is the $\sE$-least countably complete weakly normal ultrafilter such that $S^\delta_\kappa\in U$. By \[AIrred\], $U$ is an irreducible ultrafilter.
If $\delta$ is a successor cardinal then it follows immediately from \[IrredThm\] that $U$ is $\delta$-supercompact.
Suppose $\delta$ is a weakly inaccessible cardinal. By a theorem of Ketonen [@Ketonen], $U$ is $(\kappa^+,\delta)$-regular, and therefore $j_U$ is discontinuous at every regular cardinal in the interval $[\kappa^+,\delta]$. It now follows from \[IrredThm\] that $M_U$ is closed under $\lambda$-sequences for all $\lambda < \delta$. Therefore some cardinal less than $\delta$ is $\delta$-supercompact. By the main theorem of [@GCH] (Theorem 4.1), it follows that for all sufficiently large $\lambda < \delta$, $2^\lambda = \lambda^+$. In particular, $\delta$ is strongly inaccessible.
By \[TightCover\] and \[InaccCover\], it follows that $M_U$ has the tight covering property at $\delta$. Thus $\text{cf}^{M_U}(\sup j_U[\delta]) = \delta$. On the other hand, $j_U(\kappa) = \text{cf}^{M_U}(\sup j_U[\delta])$ since $U$ is weakly normal and concentrates on $S^\delta_\kappa$. Therefore $j_U(\kappa) = \delta$. Thus $j_U : V\to M$, $j_U(\kappa) = \delta$, and $M_U$ is closed under $\lambda$-sequences for all $\lambda < \delta$. It follows that $\textsc{crt}(U)$ is almost huge.
[^1]: Current forcing techniques cannot build a model of UA with even one measurable cardinal. Forcing UA in the presence of a supercompact cardinal seems like a much harder problem.
|
---
abstract: 'Using double parabola approximation for a single Bose-Einstein condensate confined between double slabs we proved that in grand canonical ensemble (GCE) the ground state with Robin boundary condition (BC) is favored, whereas in canonical ensemble (CE) our system undergoes from ground state with Robin BC to the one with Dirichlet BC in small-$L$ region and vice versa for large-$L$ region and phase transition in space of the ground state is the first order. The surface tension force and Casimir force are also considered in both CE and GCE in detail.'
author:
- Nguyen Van Thu
bibliography:
- 'mybibfile.bib'
title: 'The forces on a single interacting Bose-Einstein condensate'
---
Introduction\[sec:1\]
=====================
The original Casimir effect is discovered by H. B. G. Casimir [@Casimir], which caused by the confinement of vacumm fluctuations of the electromagnetic field between two slabs at zero temperature. In this case the author pointed out that Casimir force is attractive and varying as a power $\ell^{-4}$ with $\ell$ being inter-distance beetwen two slabs and it is proportional to area $A$ of the slab. A review for Casimir effect and its applications were mentioned in [@Bordag].
In the field of Bose-Einstein condensate (BEC), there are many papers for this subject. For two component BECs, several interesting properties were studied in Ref. [@Thu1]. This work proved that Casimir force of a BECs is not simple superposition of the one of two single component BEC because of interaction between two species and especially this force vanished in limit of strong segregation.
For a single Bose-Einstein condensate, the Casimir effect has been considered in many aspects. Using field theory in one-loop approximation, Schiefele and Henkel [@Schiefele] expressed the Casimir energy as an integral of density of state, their result shown that this energy decays as $\ell^{-3}$. At finite temperature, this effect was also investigated [@Dantchev; @Biswas2]. The Casimir force on an interacting Bose-Einstein condensate, which consists of mean field force and Casimir force, was calculated in Ref. [@Biswas3], in which system under consideration was in grand canonical ensemble (GCE). However, as our understanding, the study on Casimir and mean field forces in canonical ensemble (CE) have been still absent so far. There are two main aims in this work: (i) consider surface tension force and Casimir force of a single BEC in both CE and GCE for both Dirichlet boundary condition (BC) and Robin BC and (ii) investigate the phase transition in space of the ground state within frame work of double parabola approximation (DPA). The system under consideration is a dilute interacting BEC [@Schiefele].
This paper is organized as follow. In Section \[sec:2\] we investigate the phase transition in space of the ground state, which depends on the applied BC. The forces act on the slabs, namely, surface tension force and force are studied in Section \[sec:3\]. The conclusions and outlook are given in Section \[sec:4\] to close the paper.
Boundary condition and phase transition in space of the ground state \[sec:2\]
==============================================================================
To begin with, we consider a single BEC confined between two parallel pallates of area $A$ along the $(x,y)$-plane and they are separated along $z$ direction by a distance $\ell$. For this geometry one requests $\sqrt{A}\gg \ell$. The positions of these slabs are $z=0$ and $z=\ell$. The total Hamiltonian reads $$\begin{aligned}
{\cal H}=\int_V {\cal H}_b+\int_{W1}{\cal H}_{W1} dS+\int_{W2}{\cal H}_{W2} dS,\label{Hamilton}\end{aligned}$$ in which ${\cal H}_b$ is Hamiltonian in bulk, without an external field, which has the form $$\begin{aligned}
{\cal H}_b=\psi^*(\vec{r})\left[-\frac{\hbar^2}{2m}\nabla^2\right]\psi(\vec{r})+V_{GP},\label{Hb}\end{aligned}$$ where $$\begin{aligned}
V_{GP}=-\mu\psi(\vec{r})+\frac{g}{2}|\psi(\vec{r})|^4,\label{GPpotential}\end{aligned}$$ is Gross-Pitaevskii (GP) potential. Here we denote $\psi(\vec{r})$ is wave function of the ground state, which plays the role of order parameter, $m$ is atomic mass. The coupling constant $g$ is inter-particle interaction, which is determined via the $s$-wave scattering length $a_s$ through $g=4\pi\hbar^2a_s/m$. The chemical potential $\mu$ is read as $\mu=gn_0$ if our system contacts with “bulk reservoirs” of condensate, in other words, the system is considered in GCE. However in CE this chemical potential is determined by the relation for fixed particle number $$\begin{aligned}
N=\int \psi^2 d\vec{r}.\label{normalize}\end{aligned}$$ In mean field theory [@Schiefele; @AoChui; @Andersen], this potential is derivative of free-energy density with respect to particle density and result is $\mu= gn_0$. Here we denote $n_0=N/V$ is bulk density and $V$ is volume of system.
${\cal H}_\alpha~(\alpha=W_1,W_2)$ are Hamiltonian of hard walls, which are chosen in the phenomenological forms [@Lipowsky; @Binder], $$\begin{aligned}
H_\alpha=\frac{\hbar^2}{2m\lambda_\alpha}\psi^*_\alpha\psi_\alpha,\label{Halpha}\end{aligned}$$ with $\psi_\alpha$ being surface field at the slabs and and $\lambda_\alpha$ is extrapolation length.
Minimizing the total Hamiltonian (\[Hamilton\]) leads to the time-independent Gross-Pitaevskii (GP) equation [@Pitaevskii], $$\begin{aligned}
-\frac{\hbar^2}{2m}\nabla^2\psi(\vec{r})-\mu\psi(\vec{r})+g|\psi(\vec{r})|^3=0,\label{GP1}\end{aligned}$$ and $\psi(\vec{r})$ fulfills boundary conditions at slabs [@Thuphatsong], $$\begin{aligned}
\vec{n}\nabla\psi_\alpha=\frac{1}{\lambda_\alpha}\psi_\alpha.\label{BC}\end{aligned}$$ The unit vector $\vec{n}$ perpendiculars to the surface at slabs and points inside the system.
It is worth noting that the condensate is translation along $(0x,0y)$-directions and the motion of particles are relevant only $z$-axis so that the nabla operator is replaced by derivative with respect to $z$, then Eq. (\[BC\]) can be rewritten as follows $$\begin{aligned}
\frac{\partial \psi_{W1}}{\partial z}\bigg|_{z=0}&=&\frac{1}{\lambda_{W1}}\psi_{W1}(z=0),\nonumber\\
\frac{\partial \psi_{W2}}{\partial z}\bigg|_{z=\ell}&=&\frac{1}{\lambda_{W2}}\psi_{W2}(z=\ell).\label{RBC}\end{aligned}$$ These equations express the Robin BC at the slabs. When the surface fields are vanishing at slabs, which corresponds to Dirichlet ones. $$\begin{aligned}
\psi(0)=\psi(\ell)=0.\label{DBC}\end{aligned}$$ Eqs. (\[RBC\]) and (\[DBC\]) show that the BCs at slabs are either Robin BCs or Dirichlet BCs.
We now invoke the double parabola approximation (DPA) developed in [@Joseph] to study ground state of our system. To do this, we first introduce dimensionless coordinate $\varrho=z/\xi$ with $\xi=\hbar/\sqrt{2mgn_0}$ being healing length, the dimensionless order parameter $\phi=\psi/\sqrt{n_0}$. By this way, Eqs. (\[normalize\]) and (\[GP1\]) can be rewritten as $$\begin{aligned}
&&-\partial_\varrho^2\phi-\phi+\phi^3=0,\label{GP2}\\
&&N=n_0\xi\int_0^L \phi^2d\varrho\equiv n_0\xi I_0,\label{normalize2}\end{aligned}$$ where $L=\ell/\xi$. Next step, we note that near the slabs, because of decreasing from bulk value therefore we can expand the order parameter $$\phi\approx 1+\delta,\label{khaitrien}$$ with $\delta$ being a small real quantity. Putting (\[khaitrien\]) into (\[GPpotential\]) and keeping up to second order of $\phi$ one has DPA potential $$\begin{aligned}
V_{DPA}=2(\phi-1)^2-\frac{1}{2}.\label{DPApotential}\end{aligned}$$ At this step, instead of GP equation (\[GP1\]) we have Euler-Lagrange equation $$\begin{aligned}
-\frac{\partial^2\phi}{\partial\varrho^2}+\alpha^2(\phi-1)=0,\label{DPAGP}\end{aligned}$$ where $\alpha=\sqrt{2}$.
Coming back to our problem, the system under consideration is symmetry, this means that $\lambda=\lambda_{W1}/\xi=-\lambda_{W2}/\xi$ and the Robin BCs (\[RBC\]) are rewritten $$\begin{aligned}
\phi(0)=\lambda\partial_\varrho\phi\bigg|_{\varrho=0},~\phi(L)=-\lambda\partial_\varrho\phi\bigg|_{\varrho=L}\label{RBC2}\end{aligned}$$ in which $\lambda\geq0$. It is easily to find the solution for (\[DPAGP\]) with constraint of (\[RBC2\]), which is read as $$\begin{aligned}
\phi=1-\frac{\cosh \left[(L-2\varrho)/\alpha\right]}{\alpha \lambda \sinh \left(\frac{L}{\alpha}\right)+\cosh \left(\frac{L}{\alpha}\right)}.\label{groundstate}\end{aligned}$$
In order to determine $\lambda$ we require that when right slab goes to infinity the wave function (\[groundstate\]) becomes exactly the one for semi-infinite system [@Thu]. Therefore one gets $\lambda=0$ and $\lambda=1/\alpha$ for Dirichlet and Robin BCs, respectively. A question raises naturally is that which one of BCs is in favor? The best answer will be given after calculating surface energy of the system. Firstly, we consider in GCE, the grand potential for the condensate is defined $$\begin{aligned}
\Omega=\int_{V}{\cal H}_bdV,\end{aligned}$$ hence $$\begin{aligned}
\Omega=2P_0A\int_0^\ell \left[(\partial_\varrho\phi)^2+V_{DPA}\right]dz,\label{Omega1}\end{aligned}$$ with $P_0=gn_0^2/2$ being the bulk pressure. Combining (\[DPApotential\]) and (\[Omega1\]) one has the excess energy (or surface tension) per unit area $$\begin{aligned}
\gamma=\frac{\Omega-P_0V}{A}=2P_0\xi\int_0^L d\varrho\left[(\partial_\varrho\phi)^2+2(\phi-1)^2\right].\label{gammaGCE}\end{aligned}$$ Substituting (\[groundstate\]) into (\[gammaGCE\]) we arrive $$\begin{aligned}
\gamma=P_0\xi \frac{2 \alpha \sinh (\alpha L)}{\left[\alpha \lambda \sinh \left(\frac{L}{\alpha}\right)+\cosh \left(\frac{L}{\alpha}\right)\right]^2}.\label{gammaGCE1}\end{aligned}$$
Fig. \[f1a\] shows the $L$-dependence of surface tension for Dirichlet BC (red) and Robin BC (blue). At $L=0$ the surface tension is zero and it increases as $L$ increases and reaches constant when $L$ is large enough $$\begin{aligned}
\lim_{L\rightarrow \infty}\gamma=\frac{4 \alpha }{(\alpha \lambda +1)^2}P_0\xi.\label{gammaGCE2}\end{aligned}$$ It is clearly that the surface tension corresponding to Robin BC is always smaller than the one corresponding to Dirichlet BC therefore the state corresponds to Robin BC is in favor. Impose that at the beginning our system is set up with Dirichlet BC, sooner or later it changes into the one with Robin BC and latent heat is $$\begin{aligned}
\delta\gamma=\gamma[0,L]-\gamma[1/\alpha,L].\label{deltaEGCE}\end{aligned}$$ The $L$-dependence of the latent heat is plotted in Fig. \[f2a\]. Excepting for $L=0$, this latent heat differs from zero hence this phase transition is first order [@Thunew].
We now focus on considering the surface tension in CE. A possible definition for surface excess energy was given by Ao and Chui [@AoChui], by this means, the excess energy is total energy after a substraction of a contribution extensive in the volume $$\begin{aligned}
\Delta E=E_{CE}-\mu N=E_{CE}-N \frac{\partial E_{CE}}{\partial N}.\label{deltaE}\end{aligned}$$ Combining Eqs. (\[deltaE\]), (\[normalize2\]) and (\[GP2\]) then divided by area $A$ we obtain the surface tension $$\begin{aligned}
\sigma=\frac{\Delta E}{A}=\frac{1}{2}n_0\int_{0}^{\ell}\mathrm{d}z\phi\left(-\frac{\hbar^2}{2m}\nabla^2\right)\phi,\end{aligned}$$ or in dimensionless form $$\begin{aligned}
\sigma=-P_0\xi\int_{0}^{L}\mathrm{d}\varrho \phi\partial_\varrho^2\phi.\label{tens2}\end{aligned}$$ It is very interesting to distinguish this definition from the one which was proposed by Fetter and Walecka [@Fetter]. For the sake of simplicity, instead of (\[normalize2\]), roughly speaking we impose that $n_0=N/A\ell$. Inserting (\[groundstate\]) into (\[tens2\]) yielding $$\begin{aligned}
\sigma=-\sigma_0 \frac{I}{L^3},\label{tensCE}\end{aligned}$$ in which $\sigma_0=\frac{mg^2N^3}{\hbar^2A^3}$ and $$\begin{aligned}
I=\int_0^L \phi\partial_\varrho^2\phi d\varrho=\frac{\alpha \left[\alpha (2 \lambda +L)-2 \alpha \lambda \cosh (\alpha L)-\sinh (\alpha L)\right]}{2 \left[\alpha \lambda \sinh \left(\frac{L}{\alpha}\right)+\cosh \left(\frac{L}{\alpha}\right)\right]^2}.\end{aligned}$$ Fig. \[f1b\] shows evolution of the surface tension versus $L$ in CE. The scenario is quite different in comparing with that in GCE. At $L=0$ the surface tension for Robin BC is divergent whereas it is finite for Dirichlet BC. In region $0<L<2.0834$ the surface tension for Robin BC is larger than that for Dirichlet BC and vice versa for remaining region. This means that in region $0<L<2.0834$ the ground state with Dirichlet BC is favor (magenta region in Fig. \[f2b\]) and yellow region is supported for Robin BC. The latent heat for this phase transition is $$\begin{aligned}
\delta\sigma=\sigma[0,L]-\sigma[1/\alpha,L],\label{deltaECE}\end{aligned}$$ and it is sketched in Fig. \[f2b\].
The force act on slabs\[sec:3\]
===============================
In this section we consider the force acts on slabs, which consists of two components, namely, surface tension force caused by excess surface energy and Casimir force corresponding to the quantum fluctuation [@Biswas3].
Surface tension force
---------------------
The force corresponds to excess surface energy is defined as surface tension force. In GCE one has $$\begin{aligned}
F_\gamma=-\frac{\partial\gamma}{\partial\ell}=-\frac{1}{\xi}\frac{\partial\gamma}{\partial L}.\label{forceGCE}\end{aligned}$$ Plugging (\[gammaGCE1\]) into (\[forceGCE\]) we obtain $$\begin{aligned}
F_\gamma=P_0 \frac{4\left[\alpha \lambda \sinh \left(\frac{L}{\alpha}\right)-\cosh \left(\frac{L}{\alpha}\right)\right]}{\left[\alpha \lambda \sinh \left(\frac{L}{\alpha}\right)+\cosh \left(\frac{L}{\alpha}\right)\right]^3}.\label{forceGCE1}\end{aligned}$$
![(Color online) The surface tension force in GCE versus $L$. The red and blue lines correspond to Dirichlet and Robin BC, respectively.[]{data-label="f3"}](fig_3.eps)
It is evident that the surface tension force generated by the leading order interaction term is given by $P_0=gn_0^2/2$ as mentioned in Ref. [@Pomeau]. The distance evolution of surface tension force are shown in Fig. \[f3\] for Dirichlet BC (red line) and Ronin BC (blue line). It is obviously that these forces are always attractive. For all values of the distance $L$ surface tension force for Robin BC is smaller than that for Dirichlet BC, excepting for $L=0$ their values are the same and $F_\gamma=-4P_0$. When the distance increases there forces decrease sharply.
In CE, instead of $\gamma$, using $\sigma$ in (\[tens2\]) one obtains $$\begin{aligned}
F_\sigma=-\frac{1}{\xi}\frac{\partial\sigma}{\partial L}.\label{FCE1}\end{aligned}$$ By this way, keeping (\[normalize2\]) in mind, Eq. (\[FCE1\]) can be rewritten as $$\begin{aligned}
F_\sigma= F_0\frac{1}{L}\frac{\partial}{\partial L}\left(\frac{I}{L^3}\right),\label{FCE2}\end{aligned}$$ in which $F_0=\frac{2m^2g^3N^4}{\hbar^4A^4}$.
![(Color online) The surface tension force in CE versus $L$. The red and blue lines correspond to Dirichlet and Robin BC, respectively.[]{data-label="f4"}](fig_4.eps)
Fig. \[f4\] shows the surface tension force $F_\sigma/F_0$ as a function of distance $L$ at $\lambda=0$ (Dirichlet BC, red) and $\lambda=1/\alpha$ (Robin BC, blue). The situation basically differs from the one in GCE: surface tension force is repulsive and its strength decreases sharply as $L$ is creasing and the surface tension force for Robin BC is stronger than that for Dirichlet BC when distance $L$ is small enough. However, this force is divergent at $L=0$, the reason is the incompressibility of condensate.
Casimir force
-------------
We now consider the Casimir force caused by the quantum fluctuations on top of ground state, which corresponds to phononic excitations [@Schiefele; @Biswas2; @Biswas3; @Biswas]. To do so, one employs the field theory in one-loop approximation, which has developed for the dilute single Bose gas [@Schiefele; @Andersen] and two component Bose-Einstein condensates [@Thu1]. The Bogoliubov dispersion law for element excitation reads as $$\begin{aligned}
\varepsilon(\vec{k})=\sqrt{\frac{\hbar^2k^2}{2m}\left(\frac{\hbar^2k^2}{2m}+g\psi^2\right)},\end{aligned}$$ or, in dimensionless form $$\begin{aligned}
\varepsilon(\kappa)=gn_0\sqrt{\kappa^2(\kappa^2+\phi^2)},\label{dispersion}\end{aligned}$$ with dimensionless wave vector $\kappa=k\xi$. The density of free energy has the form $$\begin{aligned}
\Omega=\frac{gn_{0}}{2\xi^3}\int\frac{d^3\vec{\kappa}}{(2\pi)^3}\sqrt{\kappa^2(\kappa^2+\phi^2)}.\label{term1}\end{aligned}$$
Because of the confinement along $z$-axis, the wave vector is quantized as follows $$\begin{aligned}
k^2\rightarrow k_\perp^2+k_j^2,\label{k1}\end{aligned}$$ in which the wave vector component $k_\perp$ perpendicular to $0z$-axis and $k_j$ is parallel with $0z$-axis. In dimensionless form one has $$\begin{aligned}
\kappa^2\rightarrow \kappa_\perp^2+\kappa_j^2.\label{k1}\end{aligned}$$ It is well known that Gibbs and Helmholtz energies related to each other by a Legendre transform. Since we only consider here at zero temperature, i.e. only quantum fluctuation is taken into account hence the Casimir energy has the same form for both GCE and CE. Eq. (\[term1\]) leads $$\begin{aligned}
\Omega=\frac{gn_{0}}{2\xi^2}\sum_{j=-\infty}^\infty\int\frac{d^2\kappa_{\perp}}{(2\pi)^2}\sqrt{(\kappa_\perp^2+\kappa_j^2)(\kappa_\perp^2+\kappa_j^2+\phi^2)}.\label{term2}\end{aligned}$$
In order to find the parallel component $k_j$ of wave vector, we consider the ideal Bose gas confined between two slabs. The wave function and energy correspond to eigenfunction and eigenvalue of Shrodinger equation $$\begin{aligned}
-\frac{\hbar^2}{2m}\frac{\partial^2 \Phi}{\partial z^2}=E\Phi.\end{aligned}$$ The wave function $\Phi$ is required to satisfy Robin BCs in (\[RBC\]) with $\lambda_W\equiv\lambda_{W1}=-\lambda_{W2}$. It is easily to find that the wave vector has to be satisfied $$\begin{aligned}
k_j&=&\frac{\pi j}{\lambda_W+\ell},\end{aligned}$$ or $$\begin{aligned}
\kappa_j=\frac{\pi j}{\lambda+L}\equiv\frac{j}{\tilde{L}},~\widetilde{L}=\frac{\lambda+L}{\pi}.\label{k2}\end{aligned}$$ Among other calculations [@Thu1], Eq. (\[term2\]) can be read $$\begin{aligned}
\Omega=\frac{gn_{0}}{2\xi^2\widetilde{L}^2}\sum_{n=1}^\infty\int\frac{d^2\kappa_{\perp}}{(2\pi)^2}\sqrt{(\widetilde{L}^2\kappa_{\perp}^2+j^2)(M^2+j^2)},\label{term3}\end{aligned}$$ where $$\begin{aligned}
M=\widetilde{L}\sqrt{\kappa_{\perp}^2+\phi^2}.\label{massj}\end{aligned}$$ Introducing a momentum cut-off $\Lambda$ for $\kappa_\perp$ we rewrite (\[term3\]) in form $$\begin{aligned}
\Omega=\frac{gn_0}{4\pi\xi^2\widetilde{L}^2}\int_0^\Lambda \kappa_\perp d\kappa_\perp\sum_{n=0}^\infty \sqrt{(\widetilde{L}^2\kappa_{\perp}^2+j^2)(M^2+j^2)}.\label{term4}\end{aligned}$$ In order to calculate the Casimir energy (\[term4\]), one employs the Euler-Maclaurin formula [@Pomeau] and takes a limit $\Lambda\rightarrow\infty$, $$\begin{aligned}
\sum_{n=0}^\infty \theta_nF(n)-\int_0^\infty F(n)dn=-\frac{1}{12}F'(0)+\frac{1}{720}F'''(0)-\frac{1}{30240}F^{(5)}(0)+\cdots,\end{aligned}$$ leads to $$\begin{aligned}
\Omega=\frac{g n_0}{\xi ^2}\left[-\frac{\pi ^2 \phi }{1440 (\lambda +L)^3}+\frac{\pi ^4}{10080 \phi (\lambda +L)^5}\right],\label{term5}\end{aligned}$$ for both CE and GCE. Note that in (\[term5\]), instead of $\widetilde{L}$ we used (\[k2\]) to convert the result into $L$.
In GCE, because the bulk density of condensate is a constant, substituting (\[term5\]) into (\[FCE1\]) one obtains the density of Casimir force (per unit are of slab) $$\begin{aligned}
F_C^{(GCE)}=\frac{g n_0}{\xi ^2}\left[-\frac{\pi ^2 \phi }{480 (\lambda +L)^4}+\frac{\pi ^4}{2016 \phi (\lambda +L)^6}\right].\label{FC1}\end{aligned}$$ Using the dimensional quantities Eq. (\[FC1\]) gives $$\begin{aligned}
F_C^{(GCE)}\sim -\frac{\hbar v_s}{(\lambda_W+\ell)^4}+\frac{\hbar^2}{m^2v_s(\lambda_W+\ell)^6},\end{aligned}$$ in which $v_s=\sqrt{gn_0/m}$ is the speed of sound. For Dirichlet BC $\lambda_W=0$, this result coincides exactly with the one given in Ref. [@Pomeau]. Fig. \[f5a\] shows the evolution of Casimir force density in GCE, where the red and blue lines correspond to Dirichlet and Robin BC.
![(Color online) The $L$-dependence of the Casimir force density in CE. The red and blue lines correspond to Dirichlet and Robin BC, respectively.[]{data-label="f6"}](fig_6.eps)
Based on these calculations and Fig. \[f5a\] some remarkable comments are given:
- In Ref. [@Pomeau] the authors worked out the contribution due to the quantum fluctuation $$\begin{aligned}
\Omega=\frac{1}{4\pi}\int_0^\Lambda \kappa_\perp d\kappa_\perp\sum_{n=0}^\infty\left\{\left[E_0^2(k)+2gn_0E_0(k)\right]^{1/2} -E_0(k)-gn_0\right\},\label{enPo}\end{aligned}$$ in which $E_0=\hbar^2k^2/(2m)$ and $k^2=k_\perp^2+(j\pi/\ell)^2$. However, their result is the same as (\[FC1\]), this means that two last terms in right hand side of (\[enPo\]) have no contribution into the Casimir force.
- The Casimir force is attractive for both BCs and Casimir force for Robin BC is always smaller than the one for Dirichlet BC at a given value of the distance.
- When $L$ tends to zero, the density of Casimir force is divergent for Dirichlet BC whereas it is finite for Robin BC. At $L=0$ its value is $$\begin{aligned}
F_C^{(GCE)}=-\frac{\pi ^2 g n_0 \phi }{480 \xi ^2 \lambda^4},\label{FC2}\end{aligned}$$ when only the leading term in right hand side of (\[FC1\]) is taken into account.
- In comparing to that in [@Biswas3], in which the integral over was worked out with expanding the Casimir energy in power series of wave vector as shown in its Eq. (29) and keeping up to fourth order while the ultraviolet cutoff was taken to infinite limit, the first term in Eq. (\[FC1\]) is agreeable but there is a difference in second term, instead of $L^{-5}$ our result gives $L^{-6}$.
- Combining Eqs. (\[forceGCE1\]) and (\[FC1\]) we see fraction of the Casimir force over the surface tension force in GCE is approximately $$\begin{aligned}
\frac{F_\gamma}{F_C^{(GCE)}}\sim \frac{\hbar^3}{2^{5/2}m^2gv_s}.\end{aligned}$$ Experimentally, consider for sodium [@Camacho] with $m=35.2\times 10^{-27}$ kg, $a_s=2.75\times 10^{-9}$ m one has $F_\gamma/F_C^{(GCE)}\approx 7.659$. The $L$-dependence of ratio $F_\gamma/F_C^{(GCE)}$ is sketched in Fig. \[f5b\] for Robin BC, the insert shows that for Drichlet BC and magenta line corresponds to 1 in vertical axis. The parameters are chosen for sodium. It is obviously that, for Dirichlet BC the surface tension force is on top of the Casimir force, this coincides to comment in Ref. [@Biswas3]. For Robin BC one has: the surface tension force is stronger than Casimir force in region $L<3.053$ and vise versa for the other.
We now consider the Casimir force density in CE, in which only leading term in right hand side of (\[term5\]) is kept. In this case, note that $n_0$ roughly depends on the distance $L$ via $n_0=N/A\ell$, therefore combining (\[term5\]) with the leading term and (\[FCE1\]) one has $$\begin{aligned}
F_C^{(CE)}=-\frac{m^2g^2N}{\hbar^4A}F_{0}\frac{\pi ^2 \phi (4 \lambda +7 L)}{180 L^7 (\lambda +L)^4}.\label{FCE3}\end{aligned}$$ The $L$-dependence of Casimir force density in CE is plotted in Fig. \[f6\], in which scaling for vertical axis is chosen $\frac{m^2g^2N}{\hbar^4A}F_{0}$. The red and blue lines correspond to Dirichlet and Robin BCs.
![(Color online) The total force versus distance $L$ for Dirichlet BC in CE.[]{data-label="f7"}](fig_7.eps)
Although the Casimir force is attractive for both GCE and CE, there are several significant differences. Firstly, at $L=0$ this force is always divergent for both Dirichlet and Robin BCs in CE, whereas in GCE this force is finite if the Robin BC is applied. In addition, the Casimir force decays as increasing distance $L$ in law $L^{-4}$ and $L^{-9}$ for GCE and CE, respectively. Last but not least, at the same value of $L$, the Casimir force for GCE is much bigger than that for CE.
To end this section, let us compare the surface tension force with the Casimir force in CE. Combining (\[FCE2\]) with (\[FCE3\]) we first easily see that these forces are opposite, the surface tension force is repulsive and the other is attractive. The ratio of their strength approximately is $$\begin{aligned}
\frac{F_\sigma}{\left|F_C^{(CE)}\right|}\sim\frac{\hbar^4A}{m^2g^2N}.\end{aligned}$$ As already mentioned above, for sodium with total particle number $N=5.10^6$ and slab area $A=10^{-6} \text{m}^2$ one gets $F_\sigma/F_C^{(CE)}\sim 167.473$. It is worth noting that the surface tension force and Casimir one are opposite direction, furthermore both of them are divergent at $L=0$ hence one should calculate the total force $F_{total}=F_\sigma+F_C^{(CE)}$. As an example, we consider for Dirichlet BC and parameters of sodium, the graph is shown in Fig. \[f7\]. This figure points out that the surface tension force is stronger than Casimir force in large-$L$ region and vice versa.
Conclusion and outlook\[sec:4\]
===============================
In the foregoing sections, using double parabola approximation and field theory we calculated the forces on slabs immersed in a single Bose-Einstein condensate. Our main results are in order
- The surface tension of the single Bose-Einstein condensate is obtained in both GCE and CE. The corresponding excess energy causes the surface tension force. This force is either attractive or repulsive, which depends on the system under consideration in GCE or CE.
- Based on the surface tension one finds that in GCE the ground state corresponds to Robin BC is favored whereas in CE the ground state is either Dirichlet BC for region small-$L$ or Robin BC for $L>2.0834$. The phase transition in space of the ground state is first order.
- The Casimir force is divergent at $L=0$ for both statistical ensembles. However this divergence disappears for Robin BC and in GCE.
One of our interesting result is that the phase transition in space of the ground state with no-zero latent heat, especially in CE, this transition produces exothermal or endothermal heat.
Acknowledgements {#acknowledgements .unnumbered}
================
I am grateful to Prof. Tran Huu Phat and Biswas for their useful discussions. This work is supported by Ministry of Education and Training of Vietnam.
References {#references .unnumbered}
==========
H. B. G. Casimir, Proc. K. Ned. Akad. Wet. [**51**]{}, 793 (1948). M. Bordag, U. Mohideen, V. M. Mostepanenko, Phys. Rep. [**353**]{}, 1 (2001). Nguyen Van Thu and Luong Thi Theu, J. Stat. Phys [**168**]{}, 1 (2017). J. Schiefele, and C. Henkel, J. Phys. A [**42**]{}, 045401 (2009). D. Dantchev, M. Krech, S. Dietrich, Phys. Rev. E [**67**]{}, 066120 (2003). S. Biswas, Eur. Phys. J. D [**42**]{}, 109 (2007). S. Biswas [*et. al.*]{}, J. Phys. B [**43**]{}, 085305 (2010). P. Ao and S. T. Chui, Phys. Rev. A [**58**]{}, 4836 (1998). J. O. Andersen, Rev. Mod. Phys. [**76**]{}, 599 (2004). R. Lipowsky, in Random Fluctuations and Pattern Growth, ed. by H. Stanley, N. Ostrowsky, NATO ASI Series E, vol [**157**]{} ( Kluwer Akad. Publ., Dordrecht, 1988), pp. 227–245 K. Binder, [*in Phase Transitions and Critical Phenomena*]{}, ed. by C. Domb, J. Lebowitz vol 8 (Academic Press, London, 1983). L. Pitaevskii and S. Stringari, [*Bose-Einstein condensation*]{}, Oxford University Press (2003). N. V. Thu, T. H. Phat and P. T. Song, J. Low Temp. Phys. [**186**]{}, 127 (2017). J.O. Indekeu, C.-Y. Lin, N.V. Thu, B. Van Schaeybroeck, T.H. Phat, Phys. Rev. A [**91**]{}, 033615 (2015). Nguyen Van Thu, Phys. Lett. A [**380**]{}, 2920 (2016). Nguyen Van Thu, Tran Huu Phat, HoangVan Quyet,[*in preparation*]{}. A. L. Fetter, J. D. Walecka, [*Quantum theory of many-particle systems*]{}, McGraw Hill, Boston 1971. D. C. Roberts, Y. Pomeau, arxiv:cond-mat/0503757. S. Biswas, J. Phys. A [**40**]{}, 9969 (2007). A. Camacho, [*Speed of sound in a Bose-Einstein condensate*]{}, arxiv:1205.4774.
|
---
abstract: 'Let $s < t$ be two fixed positive integers. We study sufficient minimum degree conditions for a bipartite graph $G$, with both color classes of size $n=k(s+t)$, which ensure that $G$ has a $K_{s,t}$-factor. Our result extends the work of Zhao, who determined the minimum degree threshold which guarantees that a bipartite graph has a $K_{s,s}$-factor.'
author:
- 'Jan Hladký [^1]'
- 'Mathias Schacht [^2]'
bibliography:
- 'bibl.bib'
title: Note on bipartite graph tilings
---
Introduction
============
For two (finite, loopless, simple) graphs $H$ and $G$, we say that $G$ contains an [*$H$-factor*]{} if there exist $v(G)/v(H)$ vertex-disjoint copies of $H$ in $G$. As a synonym, we say that there exists an [*$H$-tiling of $G$*]{}. Obviously, if $G$ contains an $H$-factor, then $v(G)$ is a multiple of $v(H)$. For a fixed graph $H$, necessary and sufficient conditions on the minimum-degree of $G$ which guarantee that $G$ contains an $H$-factor were studied extensively. Results in this spirit include the Tutte 1-factor Theorem (see [@LP86]), the Hajnal-Szemerédi Theorem [@hajnal70:_proof_p], and series of improving results by Alon and Yuster [@alon92:_yuster_GC; @alon96:_yuster_JCTB], Komlós [@komlosTT], Zhao and Shokoufandeh [@Zhao:komlosConj], and by Kühn and Osthus [@Kuhn:Tiling]. In [@Kuhn:Tiling] the answer to the problem is settled (up to a constant) for any $H$. It was shown that the relevant parameters are the chromatic number and the critical chromatic number of $H$.
The additional information that $G$ is $r$-partite might help to decrease the minimum-degree threshold for $G$ containing an $H$-factor. The conjectured $r$-partite version of the Hajnal-Szemerédi Theorem [@FischerHS] is such an example. (Recently a proof of the approximate version of the $r$-partite Hajnal-Szemerédi Theorem was announced by Csaba.) In this paper we determine the threshold for the minimum-degree of a balanced bipartite graph $G$ which guarantees that $G$ contains a $K_{s,t}$-factor, for arbitrary integers $s<t$. If the cardinalities of both color classes of $G$ are $n$, a necessary condition for $G$ having a $K_{s,t}$-factor is that $n$ is a multiple of $s+t$. The sufficient minimum-degree condition is given in Theorem \[thm\_upper\], and a matching lower bound is provided in Theorem \[thm\_lower\]. Our work can be seen as an extension of the work of Zhao [@ZhaoBipTil], who investigated the case $s=t$.
\[thm\_upperZ\] For each $s\ge 2$ there exists a number $k_0$ such that if $G=(A,B; E)$ is a bipartite graph, $|A|=|B|=n=ks$, where $k>k_0$, and $$\delta(G)\geq\Big\{\begin{array}{ll}
\frac{n}{2}+s-1 &\mbox{if $k$ is even,} \\
\frac{n+3s}{2}-2 &\mbox{if $k$ is odd,} \\
\end{array}$$ then $G$ has a $K_{s,s}$-factor.
Moreover, Zhao showed that the bounds in Theorem \[thm\_upperZ\] are tight. We extend those results to $K_{s,t}$-factors with $s<t$.
\[thm\_upper\] Let $1\leq s<t$ be fixed integers. There exists a number $k_0\in
\mathbb{N}$ such that if $G=(A,B; E)$ is a bipartite graph, $|A|=|B|=n=k(s+t)$, with $k>k_0$, and $$\delta(G)\geq\Big\{\begin{array}{ll}
\frac{n}{2}+s-1 &\mbox{if $k$ is even,} \\
\frac{n+t+s}{2}-1 &\mbox{if $k$ is odd,} \\
\end{array}$$ then $G$ has a $K_{s,t}$-factor.
On the other hand, we show that these bounds are best possible.
\[thm\_lower\] Let $1\leq s<t$ be fixed integers. There exists a number $k_0\in
\mathbb{N}$ such that for every $k>k_0$ there exists a bipartite graph $G=(A,B; E)$, $|A|=|B|=k(s+t)=n$, such that $$\delta(G)=\Big\{\begin{array}{ll}
\frac{n}{2}+s-2 &\mbox{if $k$ is even,} \\
\frac{n+t+s}{2}-2 &\mbox{if $k$ is odd and $t\leq 2s+1$,}\\
\end{array}$$ and $G$ does not have a $K_{s,t}$-factor.
The bounds in Theorem \[thm\_upper\] and \[thm\_lower\] exhibit a somewhat surprising phenomenon: for the case when $k$ is even the bound is independent of the value $t$, while for the case $k$ is odd, the minimum-degree condition depends on $t$. Moreover, we note that our results are not tight for the case $t> 2s+1$ and $k$ odd. We are very grateful to Andrzej Czygrinow and Louis DeBiasio for drawing our attention to an oversight in Theorem \[thm\_lower\] in an earlier version of this note.
Lower bound
===========
In this section we prove Theorem \[thm\_lower\]. We treat three cases (based on the parity of $k$ and on the relation between $s$ and $t$) separately. The proof of Theorem \[thm\_lower\] is constructive, i.e., we will construct a graph $G$ with the demanded minimum-degree and then argue that $G$ does not contain a $K_{s,t}$-factor.
The building blocks of our constructions are the graphs $P(m,p)$, where $m,p\in \mathbb{N}$. The graphs $P(m,p)$ were introduced in [@ZhaoBipTil]. We just state their properties, which will be used throughout this section.
\[lemmaP\] For any $p\in \mathbb{N}$ there exists a number $m_0$ such that for any $m\in\mathbb{N}, m>m_0$ there exists a bipartite graph $P(m,p)=(P_1,P_2;E_P)$ satisfying
- $|P_1|=|P_2|=m$,
- $P(m,p)$ is $p$-regular, and
- $P(m,p)$ does not contain a copy of $K_{2,2}$.
In all constructions we assume that $n$ is large enough.
Case $k$ is even
----------------
For two integers $m$ and $q$ we write $Q(m,q)$ to denote (any of possibly many) bipartite graph $Q(m,q)=(Q_1,Q_2; E_Q)$ with the following properties:
- $|Q_1|=m, |Q_2|=m-2$,
- $Q(m,q)$ does not contain any $K_{2,2}$,
- $\deg(x)\in\{q-1,q\}$ for any vertex $x\in Q_1$, and
- $\deg(y)=q$ for any vertex $y\in Q_2$.
Such graphs $Q(m,q)$ do exist for fixed $q$ and large $m$. One way to construct them is by taking the graph $P(m,q)=(P_1, P_2; E_P)$ from Lemma \[lemmaP\], selecting two vertices $w_1, w_2\in P_2$ such that they do not share a common neighbor in $P_1$, and then take $Q(m,q)$ to be the subgraph of $P(m,q)$ induced by the vertex sets $P_1$, $P_2\setminus\{w_1,w_2\}$. In particular, the graph $Q(m,0)$ is the empty graph.
Now we describe the construction of the graph $G$. Partition $A=A_1+A_2$, $B=B_1+B_2$, $|A_1|=|B_1|=\frac{n}{2}+1$, $|A_2|=|B_2|=\frac{n}{2}-1$. The graph $G$ is described by
- $G[A_i,B_i]$ is a complete bipartite graph for $i=1,2$, and
- $G[A_1,B_2]\cong G[B_1,A_2]\cong Q(n/2+1,s-1)$.
We have $\delta(G)=\frac{n}{2}+s-2$. The fact that there exists no $K_{s,t}$-factor is implied immediately by the fact that there is no subgraph isomorphic to $K_{s,t}$ whose vertices would touch both $A_1$ and $B_2$, or $A_2$ and $B_1$.
Case $k$ is odd, $2s+1\geq t>s+1$
---------------------------------
Let $k=2l+1$, $n=k(s+t)$. Note that $\frac{n-t+s+2}{2}$ is an integer. Partition $A=A_1+A_2+A_*$, $B=B_1+B_2+B_*$, $|A_1|=|A_2|=|B_1|=|B_2|=\frac{n-t+s+2}{2}$, $|A_*|=|B_*|=t-s-2$. The graph $G$ is described by
- $G[A_i,B_i]$ is a complete bipartite graph for $i=1,2$,
- $G[A_*,B_i]$ and $G[B_*,A_i]$ are complete bipartite graphs for $i=1,2$,
- $G[A_1,B_2]\cong G[A_2,B_1]\cong P(\frac{n-t+s+2}{2},s-1)$,
- the graph $G[A_*,B_*]$ is empty.
We have $\delta(G)=\frac{n+t+s}{2}-2$. To see that $G$ does not have a $K_{s,t}$-factor, we argue as follows. Suppose for contradiction that $G$ has a $K_{s,t}$-factor. Fix a $K_{s,t}$-factor of $G$. First, observe that there cannot be a copy isomorphic to $K_{s,t}$ intersecting both $A_1\cup B_1$ and $A_2 \cup
B_2$. Let $r_1$ and $r_2$ be the number of copies of $K_{s,t}$ in the tiling whose color class of size $t$ touches $A_1$ and $B_1$, respectively. Let $A_c$ and $B_c$ be vertices covered by these $r_1+r_2$ copies. It holds $$\label{eq_subsets}
%\begin{array}{c}
A_1\subset A_c\subset A_1 \cup A_* \quad \mbox{and}\quad
B_1\subset B_c\subset B_1 \cup B_*\,.
%\end{array}$$ If $r_1\not = r_2$ then $\left||A_c|-|B_c|\right|\geq
t-s$, which contradicts (\[eq\_subsets\]). Thus, $r_1 = r_2$. We conclude that $$\frac{l(s+t)+s+1}{s+t}\leq r_1 \leq \frac{l(s+t)+t-1}{s+t} \mbox{,}$$ a contradiction to the integrality of $r_1$.
Case $k$ is odd, $t=s+1$
------------------------
By $R(m,q)$ we denote (any of possibly many) bipartite graph $R(m,q)=(R_1,R_2; E_R)$ with the following properties:
- $|R_1|=m, |R_2|=m-1$,
- $R(m,q)$ does not contain any $K_{2,2}$,
- for any vertex $x$ in $R_1$, it holds $\deg(x)\in\{q-1,q\}$, and
- for any vertex $y$ in $R_2$, it holds $\deg(y)=q$.
For fixed $q$ and large $m$ the existence of such a graph $R(m,q)$ follows by a construction analogous to the construction of the graph $Q(m,q)$.
Let $k=2l+1$. Partition $A=A_1+A_2$, $B=B_1+B_2$, $|A_1|=|B_1|=l(s+t)+s$, $|A_2|=|B_2|=l(s+t)+s+1$. The graph $G$ is described by
- $G[A_i,B_i]$ is a complete bipartite graph for $i=1,2$,
- $G[B_2,A_1]\cong G[A_2,B_1]\cong R((n+1)/2,s-1)$.
One immediately sees that $\delta(G)=\frac{n+t+s}{2}-2$ and no $K_{s,t}$-tiling of $G$ exists.
Upper bound
===========
We prove Theorem \[thm\_upper\] in this section. The proof of Theorem \[thm\_upper\] utilizes the previous work of Zhao [@ZhaoBipTil]. We will need the following lemma, which allows us to find many vertex disjoint copies of certain stars. For $h\in\mathbb{N}$, an $h$-star is a graph $K_{1,h}$, its [*center*]{} is the unique vertex in the part of size one. Moreover, for a graph $G$ and two disjoint sets $A,B\subset
V(G)$ we define $$\delta(A,B)=\min\{\deg(v,B)\::\:v\in A\}\;,\quad
\Delta(A,B)=\max\{\deg(v,B)\::\:v\in
A\}$$ and $$d(A,B)=\frac{e(A,B)}{|A||B|}\,.$$
\[lemma12\] Let $1\le h\le \delta\le M$ and $0<c<1/(6h+7)$. Suppose that $H=(U_1,U_2; E_H)$ is a bipartite graph such that $||U_i|-M|\le cM$ for $i=1,2$. If $\delta=\delta(U_1,U_2)\le cM$ and $\Delta=\Delta(V_2,V_1)\le cM$, then we can find a family of vertex-disjoint $h$-stars, $2(\delta-h+1)$ of which have centers in $U_1$ and $2(\delta-h+1)$ of which have centers in $U_2$.
As in [@ZhaoBipTil] we distinguish between an extremal and a non-extremal case. If we find a $K_{s+t,s+t}$-factor in $G$ we are done, as each copy of $K_{s+t,s+t}$ can be split into two copies of $K_{s,t}$ and hence we have a $K_{s,t}$-factor. Thus the theorem stated next is just a corollary of [@ZhaoBipTil Theorem 4.1].
For every $\alpha>0$ and positive integers $s<t$, there exist $\beta>0$ and a positive integer $k_0$ such that the following holds for all $n=k(s+t)$ with $k>k_0$. Given a bipartite graph $G=(A, B; E)$ with $|A|=|B|=n$, if $\delta(G)> (\frac12-\beta)n$, then either $G$ contains a $K_{s,t}$-factor, or there exist $$A_1\subset A, \quad B_1\subset B \quad \mbox{such that}
\quad |A_1|=|B_1|= \lfloor n/2 \rfloor, \quad d(A_1, B_1)< \alpha.$$
Therefore, we reduce the problem to the extremal case. Let $\alpha=\alpha(t)>0$ be small. As in the proof of Theorem 11 in [@ZhaoBipTil], define $$\begin{array}{r@{=}l r@{=}l}
A'_1& \left\{x\in A: \deg(x, B_1)< \alpha^{\frac13}\,\frac{n}2\right\},
& B'_1& \left\{x\in B: \deg(x, A_1)< \alpha^{\frac13} \,
\frac{n}2\right\},
\\
A'_2& \left\{x\in A: \deg(x, B_1)> (1-\alpha^{\frac13}) \frac{n}2\right\},
&B'_2& \left\{x\in B: \deg(x, A_1)> (1-\alpha^{\frac13})
\frac{n}2\right\},
\\
A_0&A-A'_1-A'_2, & B_0&B-B'_1-B'_2,
\\
G_1&G[A'_1,B'_1], & G_2&G[A'_2,B'_2].
\end{array}$$ Similarly as in the proof of Theorem 11 in [@ZhaoBipTil], we assume that removing any edge from $G$ would violate the minimum-degree condition and then change $A'_i$ and $B'_i$ a little so that $\Delta(G_1),\Delta(G_2)<\alpha^{\frac{1}{9}}n$. Vertices in $A_0\cup B_0$ are called [*special*]{}.
$k$ is even
-----------
To exhibit the existence of a tiling in this case, it is sufficient to translate carefully the proof of Case I of Theorem 11 from [@ZhaoBipTil]. We give a sketch of the proof below and refer the reader to the corresponding places in [@ZhaoBipTil] for more details.
Set $\mathcal{V}=(A'_1,B'_1,A'_2,B'_2)$. First assume, that no member of $\mathcal{V}$ contains more than $n/2$ vertices. We add vertices from $A_0$ and $B_0$ into sets of $\mathcal{V}$ in such a way, that every set has size exactly $n/2$. Then, we may apply arguments used in [@ZhaoBipTil], based on Hall’s Marriage Theorem, to find a $K_{s+t,s+t}$ tiling.
Next, assume that there is only one set in $\mathcal{V}$ which has more than $n/2$ elements. Without loss of generality, assume that it is $A'_2$, i.e., $|A'_2|=c>n/2$. Lemma \[lemma12\] applied to the graph $G[A'_2,B'_2]$ yields the existence of $c-n/2$ disjoint $s$-stars with centers in $A'_2$. We move the centers of the stars into $A'_1$ and extend each of the stars into a copy of $K_{s,t}$ (each of these copies lies entirely in $A'_1\cup B'_2$, with the color class of size $s$ being contained in $B'_2$). We distribute vertices of $B_0$ into $B'_1$ and $B'_2$ so, that $|B'_1|=|B'_2|=n/2$. Then, it is easy to finish the entire tiling. This is done in three steps. In the first step, we find in an arbitrary manner $c-n/2$ copies of $K_{s,t}$ (disjoint with the previous ones) in $G[A'_1,B'_2]$ placed in such a way, that the color-class of size $s$ lies in $A'_1$. This step ensures us, that the cardinalities of untiled (i.e., those vertices which are not covered by the partial $K_{s,t}$-factor) vertices in the both color-classes of $G[A'_1,B'_2]$ are equal and divisible by $s+t$. In the second step, all yet untiled vertices of $G[A'_1,B'_2]$ which were originally special vertices are tiled. In the third step, the tiling is in an analogous manner defined for $G[A'_2,B'_1]$.
Now, assume that two diagonal sets of $\mathcal{V}$, say $A'_2$ and $B'_1$ have sizes more than $n/2$. Then we apply separately Lemma \[lemma12\] to $G[A'_2,B'_2]$ and $G[A'_1,B'_1]$ to obtain families $\mathcal{S}_A$ and $\mathcal{S}_B$ of disjoint $s$-stars with centers in $A'_2$ and $B'_1$, such that $|A'_2|-|\mathcal{S}_A|=|B'_1|-|\mathcal{S}_B|=n/2$. We move the centers of the stars to $A'_1$ and $B'_2$ and proceed as in the previous case.
The remaining case is when two non-diagonal sets from $\mathcal{V}$ have size more than $n/2$. Assume these are $A'_2$ and $B'_1$. We apply Lemma \[lemma12\] to the graph $G[A'_2,B'_2]$ to obtain families $\mathcal{S}_A,\mathcal{S}_B$ of disjoint $s$-stars with centers in $A'_2$ and $B'_2$, such that $|A'_2|-|\mathcal{S}_A|=|B'_2|-|\mathcal{S}_B|=n/2$. We proceed as in the previous cases.
$k$ is odd
----------
Let $k=2l+1$. We say that a set of special vertices ($A_0$ and/or $B_0$) is [*small*]{} if its size is less than $t-s$. Otherwise, it is called [*big*]{}.
We distinguish four cases.
- *Both $A_0$ and $B_0$ are small.* Then there exist $i,j\in\{1,2\}$, such that $|A'_i|, |B'_j|\ge l(s+t)+s+1$. If $i=j$, then we apply Lemma \[lemma12\] to the graph $G_i$ and find families $\mathcal{S}_A$, $\mathcal{S}_B$ of pairwise disjoint $s$-stars with centers in $A'_i$ and $B'_i$ respectively, so that $|A'_i|-|\mathcal{S}_A|=|B'_i|-|\mathcal{S}_B|=l(s+t)+s$. Move the centers of the stars in $A'_{3-i}$ and $B'_{3-i}$. After the changes we shall tile two graphs: $G[A'_1,B'_2]$ and $G[A'_2,B'_1]$. Note, that both those graphs are not balanced. The tiling procedure is analogous to the previous cases (when $k$ is even); the only difference is that one copy of $K_{s,t}$ has to be found in the graphs first to make each of them balanced.
If $i\not =j$, we can assume that $|A'_j|,|B'_i|\le
l(s+t)+s$. Since if this does not hold, then we could change one index and continue as in the case when $i=j$. We will show that one can add vertices to $A'_j$ and to $B'_i$ so that $|A'_j|=l(s+t)+s$ and $|B'_i|=l(s+t)+t$. Then, the existence of the tiling will follow by standard arguments. We apply Lemma \[lemma12\] to the graph $G_j$ to obtain a family of $|B'_j|-(l(s+t)+s)$ vertex disjoint $s$-stars with centers in $B'_j$ and end-vertices in $A'_j$. If we moved all the centers to $B'_i$ and all the vertices of $B_0$, the cardinality of $B'_i$ would be $$|B'_i|+(|B'_j|-(l(s+t)+s))+|B_0|=l(s+t)+t \; .$$ The same applies for $A'_j$. Therefore, by removing some of the vertices, we may attain $|A'_j|=l(s+t)+s$ and $|B'_i|=l(s+t)+t$. Then, the existence of a tiling follows.
- *$A_0$ is small and $B_0$ is big.* Then at least one $B'_i$ (say $B'_2$) has at most $l(s+t)+s$ vertices. Lemma \[lemma12\] asserts that we can find a family $\mathcal{S}_B$ of disjoint $s$-stars with centers in $B'_1$ and end-vertices in $A'_1$, such that $|B'_1|-|\mathcal{S}_B|\le l(s+t)+s$. This implies, that we can find vertices (in $B_0$ or centers of the stars of $\mathcal{S}_B$) which can be moved to $B'_2$ so that $|B'_2|=l(s+t)+t$.
As $A_0$ is small, one of $A'_1$ and $A'_2$ must have at least $l(s+t)+s+1$ vertices. The tiling can be found by standard arguments if we achieve to have $|A'_1|=l(s+t)+s$. If $|A'_1|>l(s+t)+s$, Lemma \[lemma12\] yields existence of a family $\mathcal{S}_A$ of disjoint $s$-stars with centers in $A'_1$ and end-vertices in $B'_1$ such that $|A'_1|-|\mathcal{S}_A|=l(s+t)+s$. Moving the centers to $A'_2$, we achieve $|A'_1|=l(s+t)+s$. Assume that $|A'_1|\le l(s+t)+s$. The size of $A'_2$ is $k(s+t)-|A'_1|-|A_0|>l(s+t)+s$. Lemma \[lemma12\] yields existence of a family $\mathcal{S}_A$ of disjoint $s$-stars in $G_2$ centered in $A'_2$ with the property that $|A'_1|+|\mathcal{S}_A|=l(s+t)+s$. Moving the centers to $A'_1$ yields demanded $A'_1=l(s+t)+s$.
- *$A_0$ is big and $B_0$ is small.* The analysis of this case is analogous to the previous one.
- *Both $A_0$ and $B_0$ are big.* We shall show in the next paragraph, that we can achieve $A'_1$ to be of size $l(s+t)+s$ and of size $l(s+t)+t$. An analogous procedure can be used to show the same property for the set $B'_1$. Then, the existence of the tiling follows immediately; one prescribes the cardinalities of $A'_1$ and $B'_1$ to be equal to the same number $l(s+t)+s$.
If $|A'_i\cup A_0|<l(s+t)+t$ for some $i\in\{1,2\}$, then we have $|A'_{3-i}|>l(s+t)+s$. Appealing to Lemma \[lemma12\] we can remove centers of $s$-stars from $A'_{3-i}$ in such a way that $|A'_{3-i}|=l(s+t)+s$. Also, by moving $t-s$ vertices from the big set $A_0$ to $A'_{3-i}$ arrive at $|A'_{3-i}|=l(s+t)+t$. Then, the partial $K_{s,t}$-tiling can be extended to a $K_{s,t}$-factor.
Finally, if both $|A'_1|\le l(s+t)+s$ and $|A'_2|\le l(s+t)+s$ then we redistribute some vertices (again, appealing to Lemma \[lemma12\], and using the set $A_0$) to arrive at the situation when $|A'_1|=l(s+t)+s$, $|A'_2|=l(s+t)+t$. Then the tiling can be extended as before.
Acknowledgement {#acknowledgement .unnumbered}
---------------
We thank a careful referee for suggesting several improvements in the presentation.
[^1]: Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské náměstí 25, 118 00, Prague, Czech Republic and Zentrum Mathematik, Technische Universität München, Boltzmannstraße 3, D-85747 Garching bei München, Germany. E-mail: [ honzahladky@googlemail.com]{}. Research was supported by the Grant Agency of Charles University, and by the DFG Research Training Group 1408 “Methods for Discrete Structures” via its 2007 predoc course “Integer Points in Polyhedra”.
[^2]: Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany. E-mail: [schacht@informatik.hu-berlin.de]{}
|
---
author:
- 'Nicolas JACON [^1]'
title: 'An algorithm for the computation of the decomposition matrices for Ariki-Koike algebras'
---
Introduction
============
Ariki-Koike algebras have been independantly introduced by Ariki and Koike in [@AriKoi] and by Broué and Malle in [@BrouMAl]. According to a conjecture of Broué and Malle, this kind of algebras should play a role in the decomposition of the induced cuspidal representations of the finite groups of Lie type. Let $R$ be a commutative ring, let $d\in{\mathbb{N}_{>0}}$, $n\in{\mathbb{N}}$ and let $v$, $u_0$, $u_1$,..., $u_{d-1}$ be $d+1$ parameters in $R$. We consider the Ariki-Koike algebra $\mathcal{H}_{R,n}:=\mathcal{H}_{R,n}(v;u_0,...,u_{d-1})$ of type $G(d,1,n)$ over $R$. This is the unital associative $R$-algebra defined by:
- generators: $T_0$, $T_1$,..., $T_{n-1}$,
- relations symbolized by the following diagram:\
(240,20) ( 50,10) ( 47,18)[$T_0$]{} ( 50,8)[(1,0)[40]{}]{} ( 50,12)[(1,0)[40]{}]{} ( 90,10) ( 87,18)[$T_1$]{} ( 90,10)[(1,0)[40]{}]{} (130,10) (127,18)[$T_2$]{} (130,10)[(1,0)[20]{}]{} (160,10) (170,10) (180,10) (190,10)[(1,0)[20]{}]{} (210,10) (207,18)[$T_{n-1}$]{}
and the following ones: $$\begin{aligned}
&(T_0-u_0)(T_0-u_1)...(T_0-u_{d-1}) = 0,\\
&(T_i-v)(T_i+1) = 0\ (i\geq{1}). \end{aligned}$$
Assume that $R$ is a field of characteristic $0$. Let $\Pi^d_n$ be the set of $d$-partitions of rank $n$ that is to say the set of $d$-tuples of partitions $\underline{\lambda}=(\lambda^{(0)},...,\lambda^{(d-1)})$ such that $|\displaystyle{\lambda^{(0)}|+...+|\lambda^{(d-1)}|=n}$. For each $\underline{\lambda}\in{\Pi^d_n} $, Dipper, James and Mathas ([@DJM]) have defined a right $\mathcal{H}_{R,n}$-module $S^{\underline{\lambda}}_R$ which is called a Specht module[^2]. For each Specht module $S^{\underline{\lambda}}_R$, they have attached a natural bilinear form and a radical ${\operatorname{rad}(S^{\underline{\lambda}}_R)}$ such that the non zero $D^{\underline{\lambda}}_R:=S^{\underline{\lambda}}_R/{\operatorname{rad}(S^{\underline{\lambda}}_R)}$ form a complete set of non isomorphic irreducible modules. Let $\Phi^d_n:=\{\underline{\mu}\in{\Pi^d_n}\ |\ D_R^{\underline{\mu}}\neq{0}\}$.
Let $R_0{(\mathcal{H}_{R,n})}$ be the Grothendieck group of finitely generated $\mathcal{H}_{R,n}$-modules. This is generated by the set of simple $\mathcal{H}_{R,n}$-modules. Thus, for each $\underline{\lambda}\in{\Pi_n^d}$ and $\underline{\mu}\in{\Phi_n^d}$, there exist numbers $d_{\underline{\lambda},\underline{\mu}}$ which are called the decomposition numbers such that: $$[S^{\underline{\lambda}}_R]=\sum_{\underline{\mu}\in{\Phi_n^d}}d_{\underline{\lambda},\underline{\mu}}[D^{\underline{\mu}}_R].$$ The matrix $(d_{\underline{\lambda},\underline{\mu}})_{\underline{\lambda}\in{\Pi^d_n},\underline{\mu}\in{\Phi^d_n}}$ is called the decomposition matrix of $\mathcal{H}_{R,n}$.
One of the main problems in the representation theory of Ariki-Koike algebras is the determination of the decomposition matrix. When $\mathcal{H}_{R,n}$ is semi-simple, the decomposition matrix is just the identity. When $\mathcal{H}_{R,n}$ is not semi-simple, by using results of Dipper and Mathas, the determination of the decomposition matrix is deduced from the case where all the parameters are powers of the same number $\eta$ (see [@DipMat]). Here, we assume that $\eta$ is a primitive $e^{\textrm{th}}$-root of unity.
When $d=1$, Lascoux, Leclerc and Thibon [@LLT] have presented a fast algorithm for the computation of the canonical basis elements of a certain integrable $\mathcal{U}_q(\widehat{sl_e})$-module $\overline{\mathcal{M}}$. Moreover, they conjectured that the problem of computing the decomposition matrix of $\mathcal{H}_{R,n}$ can be translated to that of computing the canonical basis of $\overline{\mathcal{M}}$. This conjecture has been proved and generalized for all $d\in{\mathbb{N}_{>0}}$ by Ariki in [@Ari4]. Unfortunately, the generalization doesn’t give an analogue of the LLT algorithm for $d>1$. In this case, Uglov [@uglov] has given an algorithm but it computes the canonical basis for a larger space which contains $\overline{\mathcal{M}}$ as a submodule. It might be interesting to obtain a generalization of the LLT algorithm for all $d>1$.
In [@these] and [@cyclo], extending the results developed in [@geckrouq] for $d=1$ and $d=2$ by using an ordering of Specht modules by Lusztig $a$-function, we showed that there exists a “canonical basic set” $\mathcal{B}$ of Specht modules in bijection with ${\operatorname{Irr}(\mathcal{H}_{R,n})}$ and that this set is parametrized by some FLOTW $d$-partitions defined by Foda et al. in [@FLOTW]. As a consequence, this result gives a purely combinatorial triangular algorithm for the computation of the decomposition matrix for Ariki-Koike algebras which generalizes the LLT algorithm.
The aim of this paper is to present this algorithm. In the first part, we give the definitions and theorems used in the algorithm. Then, we give the different steps of the algorithm.
Ariki’s theorem and canonical basic set
=======================================
Let $e$ and $d$ be two positive integers and let $\displaystyle{\eta_e:=\textrm{exp}(\frac{2i\pi}{e})}$ and $\displaystyle{\eta_d:=\textrm{exp}(\frac{2i\pi}{d})}$. We consider the Ariki-Koike algebra $\mathcal{H}_{R,n}$ over $R:=\mathbb{Q}[\eta_d](\eta_e)$ with the following choice of parameters: $$v=\eta_e, \qquad{u_j=\eta_e^{v_j},}\qquad{\textrm{for}\ j=0,...,d-1},$$ where $0\leq{v_0}\leq ... \leq{v_{d-1}}<e$. In this section, we briefly summarize the results of Ariki which give an interpretation of the decomposition matrix in terms of the canonical basis of a certain $\mathcal{U}_q(\widehat{\textrm{sl}_e})$-module. For more details, we refer to [@Arilivre] and to [@mathas]. Next, we recall the results shown in [@these chapter 2] and in [@cyclo].\
a) We first explain the Ariki’s theorem. To do this, we need some combinatorial definitions. Let $\underline\lambda={(\lambda^{(0)} ,...,\lambda^{(d-1)})}$ be a $d$-partition of rank $n$. The diagram of $\underline{\lambda}$ is the following set: $$[\underline{\lambda}]=\left\{ (a,b,c)\ |\ 0\leq{c}\leq{d-1},\ 1\leq{b}\leq{\lambda_a^{(c)}}\right\}.$$ The elements of this diagram are called the nodes of $\underline{\lambda}$. Let $\gamma=(a,b,c)$ be a node of $\underline{\lambda}$. The residue of $\gamma$ associated to the set $\{e;{v_0},...,{v_{d-1}}\}$ is the element of $\mathbb{Z}/e\mathbb{Z}$ defined by: $$\textrm{res}{(\gamma)}=(b-a+v_{c})(\textrm{mod}\ e).$$ If $\gamma $ is a node with residue $i$, we say that $\gamma$ is an $i$-node. Let $\underline{\lambda}$ and $\underline{\mu}$ be two $d$-partitions of rank $n$ and $n+1$ such that $[\underline{\lambda}]\subset{[\underline{\mu}]}$. There exists a node $\gamma$ such that $[\underline{\mu}]=[\underline{\lambda}]\cup{\{\gamma\}}$. Then, we denote $[\underline{\mu}]/[\underline{\lambda}]=\gamma$ and if $\textrm{res}{(\gamma)}=i$, we say that $\gamma$ is an addable $i$-node for $\underline{\lambda}$ and a removable $i$-node for $\underline{\mu}$. Now, we introduce an order on the set of nodes of a $d$-partition. We say that $\gamma=(a,b,c)$ is above $\gamma'=(a',b',c')$ if: $$b-a+v_c<b'-a'+v_{c'}\ \textrm{or } \textrm{if}\ b-a+v_c=b'-a'+v_{c'}\textrm{ and }c>c'.$$ Let $\underline{\lambda}$ and $\underline{\mu}$ be two $d$-partitions of rank $n$ and $n+1$ such that there exists an $i$-node $\gamma$ such that $[\underline{\mu}]=[\underline{\lambda}]\cup{\{\gamma\}}$. We define the following numbers: $$\begin{aligned}
\overline{N}_i^{a}{(\underline{\lambda},\underline{\mu})}=& \sharp\{ \textrm{addable }\ i-\textrm{nodes of } \underline{\lambda}\ \textrm{ above } \gamma\} \\
& -\sharp\{ \textrm{removable }\ i-\textrm{nodes of } \underline{\mu}\ \textrm{ above } \gamma\},\\
\overline{N}_i^{b}{(\underline{\lambda},\underline{\mu})}= & \sharp\{ \textrm{addable } i-\textrm{nodes of } \underline{\lambda}\ \textrm{ below } \gamma\}\\
& -\sharp\{ \textrm{removable } i-\textrm{nodes of } \underline{\mu}\ \textrm{ below } \gamma\},\\
\overline{N}_{i}{(\underline{\lambda})} =& \sharp\{ \textrm{addable } i-\textrm{nodes of } \underline{\lambda}\}\\
& -\sharp\{ \textrm{removable } i-\textrm{nodes of } \underline{\lambda}\},\\
\overline{N}_{\mathfrak{d}}{(\underline{\lambda})} =& \sharp\{ 0-\textrm{nodes of } \underline{\lambda}\}.\end{aligned}$$ b) Now, let $\mathfrak{h}$ be the free $\mathbb{Z}$-module with basis $\{h_i,\mathfrak{d}\ |\ 0\leq i<e\}$ as in [@cyclo section 2.B], let $q$ be an indeterminate and let $\mathcal{U}_q:=\mathcal{U}_q(\widehat{\textrm{sl}_e})$ be the quantum group of type $A^{(1)}_{e-1}$. This is a unital associative algebra over $\mathbb{C}(q)$ which is generated by elements $\{e_i,f_i\ |\ i\in{\{0,...,e-1\}}\}$ and $\{k_h\ |\ h\in{\mathfrak{h}}\}$ (see [@Arilivre Definition 3.16] for the relations). Let $\mathcal{A}=\mathbb{Z}[q,q^{-1}]$. We consider the Kostant-Lusztig form of $\mathcal{U}_q$ which is denoted by $\mathcal{U}_{\mathcal{A}}$: this is a $\mathcal{A}$-subalgebra of $\mathcal{U}_q$ generated by the divided powers $e_i^{(r)}$, $f_j^{(r)}$ for $0\leq{i,j}<e$, $r\in{\mathbb{N}}$ and by $k_{h_i}$, $k_{\mathfrak{d}}$, $k_{h_i}^{-1}$, $k^{-1}_{\mathfrak{d}}$ for $0\leq{i}<e$. Now, if $S$ is a ring and $u$ an invertible element in $S$, we can form the specialized algebra $\mathcal{U}_{S,u}:=S\otimes_{\mathcal{A}}\mathcal{U}_{\mathcal{A}}$ by specializing the indeterminate $q$ io $u\in{S}$.
For $n\in{\mathbb{N}}$, let $\mathcal{F}_n:=\{\underline{\lambda}\ |\ \underline{\lambda}\in{\Pi_d^n}\}$ and let $\mathcal{F}:=\bigoplus_{n\in{\mathbb{N}}} \mathcal{F}_n$. $\mathcal{F}$ is called the Fock space. Then, the following theorem shows that we have a $\mathcal{U}_q$-module structure on $\mathcal{F}$.
\[jmmo\] (Jimbo, Misra, Miwa, Okado [@JMMO]) $\mathcal{F}$ is a $\mathcal{U}_q$-module with action: $$e_{i}\underline{\lambda}=\sum_{{\operatorname{res}([\underline{\lambda}]/[\underline{\mu}])}=i}{q^{-\overline{N}_i^{a}{(\underline{\mu},\underline{\lambda})}}\underline{\mu}},\qquad{f_{i}\underline{\lambda}=\sum_{{\operatorname{res}([\underline{\mu}]/[\underline{\lambda}])}=i}{q^{\overline{N}_i^{b}{(\underline{\lambda},\underline{\mu})}}\underline{\mu}}},$$ $$k_{h_i}\lambda=q^{\overline{N}_i{(\underline{\lambda})}}\underline{\lambda},\qquad{k_{\mathfrak{d}}\underline{\lambda}=q^{-\overline{N}_{\mathfrak{d}}{(\underline{\lambda})}}\underline{\lambda}},$$ where $0\leq{i}\leq{n-1}$.
Note that this action is distinct from the action used by Ariki and Mathas for example in [@AriMat1]. Let $\overline{\mathcal{M}}$ be the $\mathcal{U}_q$-submodule of $\mathcal{F}$ generated by the empty $d$-partition. This is an integrable highest weight module. Thus, we can use the canonical basis theory to obtain a basis for $\overline{\mathcal{M}}_{\mathcal{A}}$, the $\mathcal{U}_{\mathcal{A}}$-module generated by the empty $d$-partition. In particular, the canonical basis elements are known to be indexed by the vertices of some “crystal graph”. In [@FLOTW], Foda, Leclerc, Okado, Thibon and Welsh have shown that the vertices of the crystal graph of $\overline{\mathcal{M}}$ are labeled by the following $d$-partitions:
(Foda, Leclerc, Okado, Thibon, Welsh [@FLOTW]) We say that $\underline\lambda={(\lambda^{(0)} ,...,\lambda^{(d-1)})}$ is a FLOTW $d$-partition associated to the set ${\{e;{v_0},...,v_{d-1}\}}$ if and only if:
1. for all $0\leq{j}\leq{d-2}$ and $i=1,2,...$, we have: $$\begin{aligned}
&\lambda_i^{(j)}\geq{\lambda^{(j+1)}_{i+v_{j+1}-v_j}},\\
&\lambda^{(d-1)}_i\geq{\lambda^{(0)}_{i+e+v_0-v_{d-1}}};\end{aligned}$$
2. for all $k>0$, among the residues appearing at the right ends of the length $k$ rows of $\underline\lambda$, at least one element of $\{0,1,...,e-1\}$ does not occur.
We denote by $\Lambda^{1,n}_{\{e;{v_0},...,{v_{d-1}}\}}$ the set of FLOTW $d$-partitions with rank $n$ associated to ${\{e;{v_0},...,{v_{d-1}}\}}$. If there is no ambiguity concerning $\{e;{v_0},...,{v_{d-1}}\}$, we denote it by $\Lambda^{1,n}$.
Now the canonical basis of $\overline{\mathcal{M}}$ is defined by using the following theorem:
\[basedef\] (Kashiwara-Lusztig, see [@Arilivre chapter 9]) Define the bar involution to be the $\mathbb{Z}$- linear ring automorphism of $\mathcal{U}_{\mathcal{A}}$ determined for $i=0,...,e-1$ and $h\in{\mathfrak{h}}$ by: $$\overline{q}:=q^{-1},\qquad{\overline{k_h}=k_{-h}},\qquad{\overline{e_i}:=e_i,}\qquad{\overline{f_i}:=f_i}.$$ We extend it to $\overline{\mathcal{M}_{\mathcal{A}}}$ by setting $\overline{u .\underline{\emptyset}}:=\overline{u}.\underline{\emptyset}$ for all $u\in{\mathcal{U}_{\mathcal{A}}}$. Then, for each $\underline{\mu}\in{\Lambda^{1,n}}$, there exists a unique element $G({\underline{\mu}})$ in $\overline{\mathcal{M}}_{\mathcal{A}}$ such that:
- $\overline{ G({\underline{\mu}})}=G({\underline{\mu}}),$
- $G({\underline{\mu}})=\underline{\mu} \ (\textrm{mod}\ q).$
The set $\{G({\underline{\mu}})\ |\ \underline{\mu}\in{\Lambda^{1,n}}\}$ is a basis of $\overline{\mathcal{M}}_{\mathcal{A}}$ which is uniquely determined by the above conditions. It is called the canonical basis of $\overline{\mathcal{M}}$.
Now the following result of Ariki which were conjectured by Lascoux, Leclerc and Thibon in [@LLT] for $l=1$ gives an interpretation of the decomposition matrix of $\mathcal{H}_{R,n}$ in terms of the canonical basis of $\overline{\mathcal{M}}$.
(Ariki [@Ari4]) Let $\underline{\mu}\in{\Lambda^{1,n}}$, there exist polynomials $b_{\underline{\lambda},\underline{\mu}}(q)\in{q\mathbb{Z}[q]}$ such that: $$G({\underline{\mu}})=\sum_{\underline{\lambda}\in{\Pi_d^n}}b_{\underline{\lambda},\underline{\mu}}(q) \underline{\lambda}.$$ Then, there exists a unique bijection $j:\Lambda^{1,n}\to{\Phi_n^d}$ such that $b_{\underline{\lambda},\underline{\mu}}(1)=d_{\underline{\lambda},{j(\underline{\mu})}}$ for all $\underline{\lambda}\in{\Pi_n^d}$ where $(d_{\underline{\lambda},\underline{\nu}})_{\underline{\lambda}\in{\Pi^d_n},\underline{\nu}\in{\Phi^d_n}}$ is the decomposition matrix of $\mathcal{H}_{R,n}$.
Thus, the elements of the canonical basis evaluated at $q=1$ correspond to the columns of the decomposition matrix of $\mathcal{H}_{R,n}$ that is to say the indecomposable projective $\mathcal{H}_{R,n}$-modules.\
c) The aim of the work presented in [@cyclo] was to study the indecomposable projective $\mathcal{H}_{R,n}$-modules. The main result gives an interpretation of the decomposition matrix of $\mathcal{H}_{R,n}$ in terms of Lusztig $a$-function. In particular, extending results of Geck and Rouquier (see [@geckrouq]), we proved that there exists a so called “canonical basic set” of Specht modules which is in bijection with the set of simple $\mathcal{H}_{R,n}$-modules.
The first step is to define an “$a$-value” on each $d$-partition. To do this, we consider a semi-simple Ariki-Koike algebra of type $G(d,1,n)$ with a special choice of parameters and we define an $a$-value on the simple modules (which are parametrized by the $d$-partitions) using the characterization of the Schur elements which have been obtained by Geck, Iancu and Malle in [@GIM]. This leads to the following definition:
\[afonction\] Let $\underline{\lambda}:=(\lambda^{(0)},\lambda^{(1)},...,\lambda^{(d)})\in{\Lambda^{1,n}}$ where for $i=0,...,d-1$ we have $\lambda^{(i)}:=(\lambda^{(i)}_1,...,\lambda^{(i)}_{h^{(i)}})$. We assume that the rank of $\underline{\lambda}$ is $n$. For $i=0,...,d-1$ and $p=1,...,n$, we define the following rational numbers: $$\begin{aligned}
m^{(i)}&:=v_i-\frac{ie}{d}+e,\\
B'^{(i)}_p&:=\lambda^{(i)}_p-p+n+m^{(i)},\end{aligned}$$ where we use the convention that $\lambda^{(i)}_p:=0$ if $p>h^{(i)}$. For $i=0,...,d-1$, let $B'^{(i)}=(B'^{(i)}_1,...,B'^{(i)}_n)$. Then, we define: $$a_1(\underline{\lambda}):=\sum_{{0\leq{i}\leq{j}<d}\atop{{(a,b)\in{B'^{(i)}\times{B'^{(j)}}}}\atop{a>b\ \textrm{if}\ i=j}}}{\min{\{a,b\}}} -\sum_{{0\leq{i,j}<d}\atop{{a\in{B'^{(i)}}}\atop{1\leq{k}\leq{a}}}}{\min{\{k,m^{(j)}\}}}.$$
Now, the $a$-value associated to $S^{\underline{\lambda}}_R$ is the rational number $a(\underline{\lambda}):=a_1(\underline{\lambda})+f(n)$ where $f(n)$ is a rational number which only depends on the parameters $\{e;v_0,...,v_{d-1}\}$ and on $n$ (the expression of $f$ is given in [@cyclo]).
Next, we associate to each $\underline{\lambda}\in{\Lambda^{1,n}}$ a sequence of residues which will have “nice” properties with respect to the $a$-value:
\[asuite\] ([@cyclo Definition 4.4]) Let $\underline{\lambda}\in{\Lambda^{1,n}}$ and let: $$l_{\textrm{max}}:=\operatorname{max}\{\lambda^{(0)}_1,...,\lambda^{(l-1)}_1\}.$$ Then, there exists a removable node $\xi_1$ with residue $k$ on a part $\lambda_{j_1}^{(i_1)}$ with length $l_{\textrm{max}}$, such that there doesn’t exist a $k-1$-node at the right end of a part with length $l_{\textrm{max}}$ (the existence of such a node is proved in [@cyclo Lemma 4.2]).
Let $\gamma_1$, $\gamma_2$,..., $\gamma_r$ be the $k-1$-nodes at the right ends of parts $\displaystyle {\lambda_{p_1}^{(l_1)}\geq{\lambda_{p_2}^{(l_2)}\geq{...}}}$$\geq{\lambda_{p_r}^{(l_r)}}$. Let $\xi_1$, $\xi_2$,..., $\xi_{s}$ be the removable $k$-nodes of $\underline{\lambda}$ on parts $\displaystyle{\lambda_{j_1}^{(i_1)}\geq{\lambda_{j_2}^{(i_2)}\geq{...}\geq{\lambda_{j_s}^{(i_s)}}}}$ such that: $$\lambda_{j_s}^{(i_s)}>\lambda_{p_1}^{(l_1)}.$$
We remove the nodes $\xi_1$, $\xi_2$,..., $\xi_s$ from $\underline{\lambda}$. Let $\underline{\lambda}'$ be the resulting $d$-partition. Then, $\underline{\lambda}'\in{\Lambda}^{1,n-s}$ and we define recursively the $a$-sequence of residues of $\underline{\lambda}$ by: $$a\textrm{-sequence}(\underline{\lambda})=a\textrm{-sequence}(\underline{\lambda}'),\underbrace{k,...,k}_{s}.$$
**Example**:\
Let $e=4$, $d=3$, $v_0=0$, $v_1=2$ and $v_2=3$. We consider the $3$-partition $\underline{\lambda}=(1,3.1,2.1.1)$ with the following diagram: $$\left( \ \begin{tabular}{|c|}
\hline
0 \\
\hline
\end{tabular}\ ,\
\begin{tabular}{|c|c|c|}
\hline
2 & 3 & 0 \\
\hline
1 \\
\cline{1-1}
\end{tabular}\ ,\
\begin{tabular}{|c|c|}
\hline
3 & 0 \\
\hline
2 \\
\cline{1-1}
1 \\
\cline{1-1}
\end{tabular}\
\right)$$\
$\underline{\lambda}$ is a FLOTW $3$-partition.
We search the $a$-sequence of $\underline{\lambda}$: we have to find $k\in{\{0,1,2,3\}}$, $s\in{\mathbb{N}}$ and a $2$-partition $\underline{\lambda}'$ such that: $$a\textrm{-sequence}(\underline{\lambda})=a\textrm{-sequence}(\underline{\lambda}'),\underbrace{k,...,k}_{s}.$$ The part with maximal length is the part with length $3$ and the residue of the associated removable node is $0$. We remark that there are two others removable $0$-nodes on parts with length $1$ and $2$. Since there is no node with residue $0-1\equiv 3\ (\textrm{mod}\ e)$ at the right ends of the parts of $\underline{\lambda}$, we must remove these three $0$-nodes. Thus, we have to take $k=0$, $s=3$ and $\underline{\lambda}'=(\emptyset,2.1,1.1.1)$, hence: $$a\textrm{-sequence}(\underline{\lambda})=a\textrm{-sequence}(\emptyset,2.1,1.1.1),0,0,0.$$ Observe that the $3$-partition $(\emptyset,2.1,1.1.1)$ is a FLOTW $3$-partition.
Now, the residue of the removable node on the part with maximal length is $3$. Thus, we obtain: $$a\textrm{-sequence}(\underline{\lambda})=a\textrm{-sequence}(\emptyset,1.1,1.1.1),3,0,0,0.$$ Repeating the same procedure, we finally obtain: $$a\textrm{-sequence}(\underline{\lambda})=3,2,2,1,1,3,0,0,0.$$\
\[decompo\] ([@cyclo Proposition 4.14]) Let $n\in{\mathbb{N}}$, let $\underline{\lambda}\in{\Lambda}^{1,n}$ and let $a\textrm{-sequence}(\underline{\lambda})=\underbrace{i_1,...,i_1}_{a_1},\underbrace{i_{2},...,i_{2}}_{a_{2}},...,\underbrace{i_s,...,i_s}_{a_s}$ be its $a$-sequence of residues where we assume that for all $j=1,...,s-1$, we have $i_{j}\neq{i_{j+1}}$. Then, we have:
$$A(\underline{\lambda}):=f^{(a_s)}_{i_s}f^{(a_{s-1})}_{i_{s-1}} ...f^{(a_1)}_{i_1}\underline{\emptyset}=\underline{\lambda}+ \sum_{a(\underline{\mu})>a(\underline{\lambda})}{c_{\underline{\lambda},\underline{\mu}}(q)\underline{\mu}},$$ where $c_{\underline{\lambda},\underline{\mu}}(q)\in{\mathbb{Z}[q,q^{-1}]}$.
It is obvious that the set $\{A(\underline{\lambda}\ |\ \underline{\lambda}\in{\Lambda}^{1,n},\ n\in{\mathbb{N}}\}$ is a basis of $\overline{\mathcal{M}}_{\mathcal{A}}$. Using the characterization of the canonical basis, we obtain the following theorem:
\[base\] ([@cyclo Proposition 4.16]) Let $n\in{\mathbb{N}}$ and let $\underline{\lambda}\in{\Lambda}^{1,n}$, then we have: $$G({\underline{\lambda}}):=\underline{\lambda}+\sum_{a(\underline{\mu})>a(\underline{\lambda})}b_{\underline{\lambda},\underline{\mu}}(q)\underline{\mu}$$
In the following paragraph, we provide an algorithm which allows us to compute these canonical basis elements.
The algorithm
=============
We fix $n\in{\mathbb{N}}$, $d\in{\mathbb{N}_{>0}}$, $e\in{\mathbb{N}_{>0}}$ and integers $0\leq v_0\leq v_1\leq ... \leq v_{d-1} <e$. The aim of the algorithm is to compute the decomposition matrix of $\mathcal{H}_{R,n}$ following the proof of [@cyclo Proposition 4.16].\
\
**Step 1**: For each $\underline{\lambda}\in{\Lambda^{1,n}}$, we construct the $a$-sequence of residues following Proposition \[asuite\]: $$a\textrm{-sequence}(\underline{\lambda})=\underbrace{i_1,...,i_1}_{a_1},\underbrace{i_{2},...,i_{2}}_{a_{2}},...,\underbrace{i_s,...,i_s}_{a_s}.$$ Then, we compute the elements $A(\underline{\lambda})$ of Proposition \[decompo\] using the action of Theorem \[jmmo\]: $$A(\underline{\lambda}):=f^{(a_s)}_{i_s}f^{(a_{s-1})}_{i_{s-1}} ...f^{(a_1)}_{i_1}\underline{\emptyset}.$$ We obtain a basis $\{A(\underline{\lambda})\ |\ \underline{\lambda}\in{\Lambda}^1\}$ of $\overline{\mathcal{M}}_{\mathcal{A}}$ which have a “triangular decomposition”. Since $\overline{f_i}=f_i$ for all $i=0,...,e-1$, we have $\overline{A(\underline{\lambda})}=A(\underline{\lambda})$.\
\
**Step 2**: For each $\underline{\mu}\in{\Pi_n^d}$, we compute its $a$-value[^3] $a(\underline{\lambda})$ following the definition \[afonction\]. Let $\underline{\nu}$ be one of the maximal FLOTW $d$-partition with respect to the $a$-function. Then, we have: $$G({\underline{\nu}})=A(\underline{\nu}).$$ **Step 3**: Let $\underline{\lambda}\in{\Lambda^{1,n}}$. The elements $G({\underline{\mu}})$ with $a(\underline{\mu})>a(\underline{\lambda})$ are known by induction. By Theorem \[base\], there exist polynomials $\alpha_{\underline{\mu},\underline{\lambda}}(q)$ such that: $$G({\underline{\lambda}})=A(\underline{\lambda})-\sum_{a(\underline{\mu})>a(\underline{\lambda})} \alpha_{\underline{\lambda},\underline{\mu}}(q)G({\underline{\mu}}),\ \ \ \ \ \ \ \ \ \ \ (1)$$ We want to compute $\alpha_{\underline{\lambda},\underline{\mu}}(q)$ for all $\underline{\mu}\in{\Lambda^{1,n}}$. By Proposition \[decompo\], we have: $$A(\underline{\lambda})= \underline{\lambda}+ \sum_{a(\underline{\mu})>a(\underline{\lambda})}{c_{\underline{\lambda},\underline{\mu}}(q)\underline{\mu}}.$$ Now, since $\overline{G({\underline{\nu}})}=G({\underline{\nu}})$ and $\overline{A(\underline{\nu})}=A(\underline{\nu})$ for all $\underline{\nu}\in{\Lambda^{1,n}}$, we must have $\alpha_{\underline{\lambda},\underline{\mu}}(q)= \alpha_{\underline{\lambda},\underline{\mu}}(q^{-1})$ for all $\underline{\mu}$ in ${\Lambda^{1,n}}$.
Let $\underline{\nu}\in{\Lambda^{1,n}}$ be one of the minimal $d$-partition with respect to the $a$-function such that $c_{\underline{\lambda},\underline{\nu}}(q)\notin{q\mathbb{Z}[q]}$. If $\underline{\nu}$ doesn’t exist then, by unicity, we have $G({\underline{\lambda}})=A(\underline{\lambda})$. If otherwise, by existence of the canonical basis, we have $\underline{\nu}\in{\Lambda}^{1,n}$. Assume now that we have: $$c_{\underline{\lambda},\underline{\nu}}(q)=a_{i}q^{i}+a_{i-1}q^{i-1}+...+a_0+...+a_{-i}q^{-i}$$ Where $(a_i)_{j\in{[-i,i]}}$ is a sequence of elements in $\mathbb{Z}$ and $i$ is a positive integer. Then, we define: $$\alpha_{\underline{\lambda},\underline{\nu}}(q)=a_{-i}q^{i}+a_{-i+1}q^{i-1}+...+a_0+...+a_{-i}q^{-i}$$ We have $\alpha_{\underline{\lambda},\underline{\nu}}(q^{-1})=\alpha_{\underline{\lambda},\underline{\nu}}(q)$. Then, in $(1)$, we replace $A(\underline{\lambda})$ by $A(\underline{\lambda})-\alpha_{\underline{\lambda},\underline{\nu}}(q)G({\underline{\nu}})$ which is bar invariant and we repeat this step until $G({\underline{\lambda}})=A(\underline{\lambda})$.
We finally obtain elements which verify Theorem \[basedef\] that is to say the canonical basis elements.\
\
**Step 4**: We specialize the indeterminate $q$ into $1$ in the canonical basis elements to obtain the columns of the decomposition matrix of $\mathcal{H}_{R,n}$ which correpond to the indecomposable projective $\mathcal{H}_{R,n}$-modules.
We finally note that we have implemented this algorithm in .
[10]{}
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O. <span style="font-variant:small-caps;">Foda</span>, B. <span style="font-variant:small-caps;">Leclerc</span>, M. <span style="font-variant:small-caps;">Okado</span>, J-Y <span style="font-variant:small-caps;">Thibon</span>, T. <span style="font-variant:small-caps;">Welsh</span>, *Branching functions of $A\sp {(1)}\sb {n-1}$ and Jantzen-Seitz problem for Ariki-Koike algebras*. Adv. Math., 141, no. 2 : 322-365, 1999.
M. <span style="font-variant:small-caps;">Geck</span>, L. <span style="font-variant:small-caps;">Iancu</span>, G. <span style="font-variant:small-caps;">Malle</span>, *Weights of Markov traces and generic degrees*. Indag. Math., 11 : 379-397, 2000.
M. <span style="font-variant:small-caps;">Geck</span>, R.<span style="font-variant:small-caps;">Rouquier</span>, *Filtrations on projective modules for [I]{}wahori-[H]{}ecke algebras*. Modular representation theory of finite groups (Charlottesville, VA, 1998), de Gruyter, Berlin : 211-221, 2001.
N. <span style="font-variant:small-caps;">Jacon</span> *Représentations modulaires des algèbres de Hecke et des algèbres de Ariki-Koike*. PhD thesis, Université de Lyon I, 2004.
N. <span style="font-variant:small-caps;">Jacon</span> *On the parametrization of the simple modules for Ariki-Koike algebras at roots of unity*. submitted.
M. <span style="font-variant:small-caps;">Jimbo</span>, K. <span style="font-variant:small-caps;">Misra</span>, T. <span style="font-variant:small-caps;">Miwa</span>, M. <span style="font-variant:small-caps;">Okado</span>, *Combinatorics of representations of $\mathcal{U}_q(\widehat{sl}(n))$ at $q=0$*. Commun. Math. Phys., 136 : 543-566, 1991.
A. <span style="font-variant:small-caps;">Lascoux</span>, B. <span style="font-variant:small-caps;">Leclerc</span>, J-Y <span style="font-variant:small-caps;">Thibon</span>, *Hecke algebras at roots of unity and crystal bases of quantum affine algebras*. Comm. Math. Phys., 181 no. 1, 205-263 : 1996.
A. <span style="font-variant:small-caps;">Mathas</span> *The representation theory of the Ariki-Koike and cyclotomic $q$-Schur algebras* Adv. Studies Pure Math. to appear.
D. <span style="font-variant:small-caps;">Uglov</span> *Canonical bases of higher-level $q$-deformed Fock spaces and Kazhdan-Lusztig polynomials*. Physical combinatorics (Kyoto, 1999), 249-299, Progr. Math., 191 : 2000.
[^1]: Address: [Institut Girard Desargues, bat. Jean Braconnier, Université Lyon 1, 21 av Claude Bernard, F–69622 Villeurbanne cedex, France.]{} E-mail address: jacon@igd.univ-lyon1.fr
[^2]: Here, we use the definition of the classical Specht modules. Note that the results in [@DJM] are in fact given in terms of dual Specht modules. The passage from classical Specht modules to their duals is provided by the map $(\lambda^{(0)},\lambda^{(1)},...,\lambda^{(d-1)})\mapsto{(\lambda^{(d-1)'},\lambda^{(d-2)'},...,\lambda^{(0)'})}$ where, for $i=0,...,d-1$, $\lambda^{(i)'}$ is denoting the conjugate partition.
[^3]: Note that it is in fact sufficient to compute the values $a_1 (\underline{\mu})$ since we have $a (\underline{\lambda})<a(\underline{\mu})\iff a_1(\underline{\lambda})<a_1(\underline{\mu})$
|
---
address:
- 'Department of Physics and Astronomy, Glasgow University, Glasgow G12 8QQ, Scotland'
- 'Niels Bohr Institute, Blegdamsvej 17, Copenhagen $\phi$, Denmark'
- 'Institut für Physik, Humboldt Universität Berlin, Invalidenstr. 110, 10115 Berlin, Germany'
author:
- 'C. D. FROGGATT, M. GIBSON'
- 'H. B. NIELSEN'
- 'D. J. SMITH'
title: 'THE FERMION MASS PROBLEM AND THE ANTI-GRAND UNIFICATION MODEL'
---
=cmr8
1.5pt
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
Introduction
============
One of the outstanding problems in particle physics is to explain the observed pattern of quark-lepton masses and of flavour mixing. This is the problem of the hierarchy of Yukawa coupling constants in the Standard Model (SM), which range in value from of order 1 for the top quark to of order $10^{-5}$ for the electron. However there is no reason in the SM for the Higgs field to prefer to couple to one fermion rather than another; in fact one would expect them all to be of order unity. We suggest [@fn] that the natural resolution to this problem is the existence of some approximately conserved chiral charges beyond the SM. These charges, which we assume to be the gauge quantum numbers in the fundamental theory beyond the SM, provide selection rules forbidding the transitions between the various left-handed and right-handed quark-lepton states, except for the top quark. In order to generate mass terms for the other fermion states, we have to introduce new Higgs fields, which break the fundamental gauge symmetry group $G$ down to the SM group. We also need suitable intermediate fermion states to mediate the forbidden transitions, which we take to be vector-like Dirac fermions with a mass of order the fundamental scale $M_F$ of the theory. In this way effective SM Yukawa coupling constants are generated, which are suppressed by the appropriate product of Higgs field vacuum expectation values measured in units of $M_F$.
If we want to explain the observed spectrum of quarks and leptons, it is clear that we need charges which—possibly in a complicated way—separate the generations and, at least for $t-b$ and $c-s$, also quarks in the same generation. Just using the usual simple $SU(5)$ GUT charges does not help, because both ($\mu_R$ and $e_R$) and ($\mu_L$ and $e_L$) have the same $SU(5)$ quantum numbers. So we prefer to keep each SM irreducible representation in a separate irreducible representation of $G$ and introduce extra gauge quantum numbers distinguishing the generations, by adding extra cross-product factors to the SM gauge group. In this talk we consider the maximal anomaly free gauge group of this type—the anti-grand unification (AGUT) group $SMG^3\times U(1)_f$. In section 2 we discuss the structure of the AGUT model and the prediction of the values of the SM gauge coupling constants, using the so-called Multiple Point Principle (MPP). We apply this principle to the pure SM in section 3, assuming a desert up to the Planck scale, and obtain predictions for the top quark and SM Higgs particle masses. In section 4 we consider the Higgs fields responsible for breaking the AGUT gauge group and the structure of the resulting quark-lepton mass matrices, together with details of a fit to the observed spectrum. The problem of neutrino mass and mixing in models with approximately conserved chiral flavour charges are discussed in section 5. Finally we present our conclusions in section 6.
Anti-Grand Unification
======================
In the AGUT model the SM gauge group is extended in much the same way as Grand Unified $SU(5)$ is often assumed; it is just that we assume another non-simple gauge group $G = SMG^3 \times U(1)_f$, where $SMG \equiv SU(3) \times SU(2) \times U(1)$, becomes active near the Planck scale $M_{Planck} \simeq 10^{19}$ GeV. So we have a pure SM desert, without any supersymmetry, up to an order of magnitude or so below $M_{Planck}$. The existence of the $SMG^3 \times U(1)_f$ group means that, near the Planck scale, each of the three quark-lepton generations has got its own gauge group and associated gauge particles with the same structure as the SM gauge group. There is also an extra abelian $U(1)_f$ gauge boson, giving altogether $3 \times 8 = 24$ gluons, $3 \times 3 = 9$ $W$’s and $3 \times 1 + 1 =4$ abelian gauge bosons.
At first sight, this $SMG^3 \times U(1)_f$ group with its 37 generators seems to be just one among many possible SM gauge group extensions. However, it is actually not such an arbitrary choice, as it can be uniquely specified by postulating 4 reasonable requirements on the gauge group $G \supseteq SMG$:
1. $G$ should transform the presently known (left-handed, say) Weyl particles into each other, so that $G \subseteq U(45)$. Here $U(45)$ is the group of all unitary transformations of the 45 species of Weyl fields (3 generations with 15 in each) in the SM.
2. No anomalies, neither gauge nor mixed. We assume that only straightforward anomaly cancellation takes place and, as in the SM itself, do not allow for a Green-Schwarz type anomaly cancellation [@green-schwarz].
3. The various irreducible representations of Weyl fields for the SM group remain irreducible under $G$.
4. $G$ is the maximal group satisfying the other 3 postulates.
With these four postulates a somewhat complicated calculation shows that, modulo permutations of the various irreducible representations in the Standard Model fermion system, we are led to our gauge group $SMG^3\times U(1)_f$. Furthermore it shows that the SM group is embedded as the diagonal subgroup of $SMG^3$, as required in our AGUT model. The AGUT group breaks down an order of magnitude or so below the Planck scale to the SM group. The anomaly cancellation constraints are so tight that, apart from various permutations of the particle names, the $U(1)_f$ charge assignments are uniquely determined up to an overall normalisation and sign convention. In fact the $U(1)_f$ group does not couple to the left-handed particles or any first generation particles, and the $U(1)_f$ quantum numbers can be chosen as follows: $$Q_f(\tau_R) = Q_f(b_R) = Q_f(c_R) = 1$$ $$Q_f(\mu_R) = Q_f(s_R) = Q_f(t_R) = -1$$
The SM gauge coupling constants do not, of course, unify in the AGUT model, but their values have been successfully calculated using the Multiple Point Principle [@glasgowbrioni]. According to the MPP, the coupling constants should be fixed such as to ensure the existence of many vacuum states with the same energy density; in the Euclideanised version of the theory, there is a corresponding phase transition. So if several vacua are degenerate, there is a multiple point. The couplings at the multiple points have been calculated in lattice gauge theory for the groups $SU(3)$, $SU(2)$ and $U(1)$ separately. We imagine that the lattice has a truly physical significance in providing a cut-off for our model at the Planck scale. The SM fine structure constants correspond to those of the diagonal subgroup of the $SMG^3$ group and, for the non-abelian groups, this gives: $$\alpha_i(M_{Planck}) = \frac{\alpha_i^{Multiple \ Point}}{3}
\qquad i=2,\ 3$$ The situation is more complicated for the abelian groups, because it is possible to have gauge invariant cross-terms between the different $U(1)$ groups in the Lagrangian density such as: $$\frac{1}{4g^2} F_{\mu\nu}^{gen \ 1}(x) F_{gen \ 2}^{\mu \nu}(x)$$ So, in first approximation, for the SM $U(1)$ fine structure constant we get: $$\alpha_1(M_{Planck}) = \frac{\alpha_1^{Multiple \ Point}}{6}$$ The agreement of these AGUT predictions with the data is shown in figure 1.
The MPP Prediction for the Top Quark and Higgs masses in the Standard Model
===========================================================================
The application of the MPP to the pure Standard Model (SM), with a cut-off close to $M_{Planck}$, implies that the SM parameters should be adjusted, such that there exists another vacuum state degenerate in energy density with the vacuum in which we live. This means that the effective SM Higgs potential $V_{eff}(|\phi|)$ should, have a second minimum degenerate with the well-known first minimum at the electroweak scale $\langle |\phi_{vac\; 1}| \rangle = 246$ GeV. Thus we predict that our vacuum is barely stable and we just lie on the vacuum stability curve in the top quark, Higgs particle (pole) mass ($M_t$, $M_H$) plane. Furthermore we expect the second minimum to be within an order of magnitude or so of the fundamental scale, i.e. $\langle |\phi_{vac\; 2}| \rangle \simeq M_{Planck}$. In this way, we essentially select a particular point on the SM vacuum stability curve and hence the MPP condition predicts precise values for $M_t$ and $M_H$.
For the purposes of our discussion it is sufficient to consider the renormalisation group improved tree level effective potential $V_{eff}(\phi)$. We are interested in values of the Higgs field of the order $|\phi_{vac\; 2}| \simeq M_{Planck}$, which is very large compared to the electroweak scale, and for which the quartic term strongly dominates the $\phi^2$ term; so to a very good approximation we have: $$V_{eff}(\phi) \simeq
\frac{1}{8}\lambda (\mu = |\phi | ) |\phi |^4$$ The running Higgs self-coupling constant $\lambda (\mu)$ and the top quark running Yukawa coupling constant $g_t(\mu)$ are readily computed by means of the renormalisation group equations, which are in practice solved numerically, using the second order expressions for the beta functions.
The vacuum degeneracy condition is imposed by requiring: $$V_{eff}(\phi_{vac\; 1}) = V_{eff}(\phi_{vac\; 2})
\label{eqdeg}$$ Now the energy density in vacuum 1 is exceedingly small compared to $\phi_{vac\; 2}^4 \simeq M_{Planck}^4$. So we basically get the degeneracy condition, eq. (\[eqdeg\]), to mean that the coefficient $\lambda(\phi_{vac\; 2})$ of $\phi_{vac\; 2}^4$ must be zero with high accuracy. At the same $\phi$-value the derivative of the effective potential $V_{eff}(\phi)$ should be zero, because it has a minimum there. Thus at the second minimum of the effective potential the beta function $\beta_{\lambda}$ also vanishes: $$\beta_{\lambda}(\mu = \phi_{vac\; 2}) =
\lambda(\phi_{vac\; 2}) = 0$$ which gives to leading order the relationship: $$\frac{9}{4}g_2^4 + \frac{3}{2}g_2^2g_1^2 +
\frac{3}{4}g_1^4 - 12g_t^4 = 0$$ between the top quark Yukawa coupling and the electroweak gauge coupling constants $g_1(\mu)$ and $g_2(\mu)$ at the scale $\mu = \phi_{vac\; 2} \simeq M_{Planck}$. We use the renormalisation group equations to relate the couplings at the Planck scale to their values at the electroweak scale. Figures \[fig:lam19\] and \[fig:top19\] show the running coupling constants $\lambda(\phi)$ and $g_t(\phi)$ as functions of $\log(\phi)$. Their values at the electroweak scale give our predicted combination of pole masses [@fn2]: $$M_{t} = 173 \pm 5\ \mbox{GeV} \quad M_{H} = 135 \pm 9\ \mbox{GeV}$$
Fermion Mass Hierarchy in AGUT
==============================
The $SMG^3 \times U(1)_f$ gauge group is broken by a set of Higgs fields $S$, $W$, $T$ and $\xi$ down to the SM gauge group. Together with the Weinberg Salam Higgs field, $\phi_{WS}$, they are responsible for breaking the quark-lepton mass protection by the chiral AGUT quantum numbers. We have the freedom of choosing the abelian quantum numbers of the Higgs fields, which we can express as charge vectors of the form: $$\vec{Q} \equiv \left ( \frac{y_1}{2},\frac{y_2}{2},
\frac{y_3}{2},Q_f \right ),$$ where $y_i/2$ $(i= 1, 2, 3)$ are the $U(1)_i$ weak hypercharges. However we fix their non-abelian representations by imposing a natural generalisation of the SM charge quantisation rule $$y_i/2 + d_i /2 + t_i/3 = 0 \quad ( \mbox{mod} \quad 1)$$ and requiring that they be singlet or fundamental representations. The duality, $d_i$, and triality, $t_i$, here are given by $d_i = +1, 0$ for the doublet and singlet representations respectively of $SU(2)_i$, and $t_i = +1, -1, 0$ for the $\boldmath{3, \overline{3}, 1}$ representations of $SU(3)_i$.
By requiring a realistic charged fermion spectrum (with $\phi_{WS}$ giving an unsuppressed top quark mass), we are led to the following choice: $$\begin{aligned}
\vec{Q}_{\phi_{WS}} = (0, \frac{2}{3}, -\frac{1}{6}, 1)&&,\quad
\vec{Q}_W = (0, -\frac{1}{2}, \frac{1}{2}, -\frac{4}{3}), \nonumber\\
\vec{Q}_T = (0, -\frac{1}{6}, \frac{1}{6}, -\frac{2}{3})&&,\quad
\vec{Q}_{\xi} = (\frac{1}{6}, -\frac{1}{6}, 0, 0), \nonumber\\
\vec{Q}_S = (\frac{1}{6}, -\frac{1}{6}, 0 , -1)&&\end{aligned}$$ The orders of magnitude for the effective SM Yukawa coupling matrix elements are then given by: $$\begin{aligned}
Y_U & \simeq & \left ( \begin{array}{ccc}
S^{\dagger}W^{\dagger}T^2(\xi^{\dagger})^2 & W^{\dagger}T^2\xi &
(W^{\dagger})^2T\xi \\
S^{\dagger}W^{\dagger}T^2(\xi^{\dagger})^3 & W^{\dagger}T^2 &
(W^{\dagger})^2T \\
S^{\dagger}(\xi^{\dagger})^3 & 1 & W^{\dagger}T^{\dagger}
\end{array} \right ),\\
Y_D & \simeq & \left ( \begin{array}{ccc}
SW(T^{\dagger})^2\xi^2 & W(T^{\dagger})^2\xi & T^3\xi \\
SW(T^{\dagger})^2\xi & W(T^{\dagger})^2 & T^3 \\
SW^2(T^{\dagger})^4\xi & W^2(T^{\dagger})^4 & WT
\end{array} \right ),\\
Y_E & \simeq & \left (\hspace{-0.1cm}\begin{array}{ccc}
SW(T^{\dagger})^2\xi^2 & W(T^{\dagger})^2(\xi^{\dagger})^3 &
(S^{\dagger})^2WT^4\xi^{\dagger} \\
SW(T^{\dagger})^2\xi^5 & W(T^{\dagger})^2 &
(S^{\dagger})^2WT^4\xi^2 \\
S^3W(T^{\dagger})^5\xi^3 & (W^{\dagger})^2T^4 & WT
\end{array}\hspace{-0.1cm}\right)
$$ Here $W, T, \xi, S$ should be interpreted as the vacuum expectation values (VEVs) of the Higgs fields in units of $M_{Planck}$ We have used the Higgs fields $W, T, \xi, S$ and the fields $W^{\dagger}, T^{\dagger}, \xi^{\dagger}, S^{\dagger}$ (with opposite charges) equivalently here, which we can do in non-supersymmetric models. In our fit below we do not need any suppression from the Higgs field $S$ and so we set its VEV to be $S = 1$. This means that the quantum numbers of the other Higgs fields $\vec{Q}_{\phi_{WS}}$, $\vec{Q}_{W}$, $\vec{Q}_T$ and $\vec{Q}_{\xi}$ are only determined modulo $\vec{Q}_S$.
Since the diagonals of $Y_U$, $Y_D$ and $Y_E$ are equal we expect to have the approximate relations $$m_b \approx m_{\tau}, \quad m_s \approx m_{\mu}$$ at $M_{Planck}$, since these masses come from the diagonal elements. There are no such relations involving the top or charm quark masses, since they come from off-diagonal elements which dominate $Y_U$. We also note that we expect $$m_d {\raisebox{-0.5ex}{$\
\stackrel{\textstyle>}{\textstyle\sim}\ $}}m_u \approx m_e$$ at $M_{Planck}$, since there are two approximately equal contributions to the down quark mass.
The VEVs of the three Higgs fields $W$, $T$ and $\xi$ are taken to be free parameters in a fit [@fgns] to the 12 experimentally known charged fermion masses and mixing angles. The results of the best fit, which reproduces all the experimental data to within a factor of 2, are given in table \[convbestfit\] and correspond to the parameters
$$\begin{array}{|ccc|}
\hline
& {\rm Fitted} & {\rm Experimental} \\ \hline
m_u & 3.6 {\rm \; MeV} & 4 {\rm \; MeV} \\
m_d & 7.0 {\rm \; MeV} & 9 {\rm \; MeV} \\
m_e & 0.87 {\rm \; MeV} & 0.5 {\rm \; MeV} \\
m_c & 1.02 {\rm \; GeV} & 1.4 {\rm \; GeV} \\
m_s & 400 {\rm \; MeV} & 200 {\rm \; MeV} \\
m_{\mu} & 88 {\rm \; MeV} & 105 {\rm \; MeV} \\
M_t & 192 {\rm \; GeV} & 180 {\rm \; GeV} \\
m_b & 8.3 {\rm \; GeV} & 6.3 {\rm \; GeV} \\
m_{\tau} & 1.27 {\rm \; GeV} & 1.78 {\rm \; GeV} \\
V_{us} & 0.18 & 0.22 \\
V_{cb} & 0.018 & 0.041 \\
V_{ub} & 0.0039 & 0.0035 \\ \hline
\end{array}$$
\[convbestfit\]
$$\langle W\rangle = 0.179 \quad
\langle T\rangle = 0.071 \quad
\langle \xi\rangle = 0.099 \label{WTxivev},$$
in Planck units. This fit is as good as we can expect in a model making order of magnitude predictions.
Neutrino Mass and Mixing Problem
================================
There is now strong evidence that the neutrinos are not massless as they would be in the SM. Physics beyond the SM can generate an effective light neutrino mass term $${\cal L}_{\nu-mass} = \sum_{i, j} \psi_{i\alpha}
\psi_{j\beta} \epsilon^{\alpha \beta} (M_{\nu})_{ij}$$ in the Lagrangian, where $\psi_{i, j}$ are the Weyl spinors of flavour $i$ and $j$, and $\alpha, \beta = 1, 2$. Fermi-Dirac statistics mean that the mass matrix $M_{\nu}$ must be symmetric.
In models with chiral flavour symmetry we typically expect the elements of the mass matrices to have different orders of magnitude. The charged lepton matrix is then expected to give only a small contribution to the lepton mixing. As a result of the symmetry of the neutrino mass matrix and the hierarchy of the mass matrix elements it is natural to have an almost degenerate pair of neutrinos, with nearly maximal mixing[@degneut]. This occurs when an off-diagonal element dominates the mass matrix.
The recent Super-Kamiokande data on the atmospheric neutrino anomaly strongly suggests large $\nu_{\mu}-\nu_{\tau}$ mixing with $\Delta m^2_{\nu_{\mu} \nu_{\tau}} \sim 10^{-3}$ eV$^2$. Large $\nu_{\mu} - \nu_{\tau}$ mixing is given by the mass matrix $$M_{\nu} =
\left(
\begin{array}{ccc}\times & \times & \times \\
\times & \times & A \\
\times & A & \times \end{array}\right )
\label{Mnu1}$$ and we have $$\begin{aligned}
& &\Delta m^2_{23} \ll \Delta m^2_{12} \sim \Delta m^2_{13}\\
& &\sin^2 \theta_{23} \sim 1\end{aligned}$$ However, this hierarchy in $\Delta m^2$’s is inconsistent with the small angle (MSW) solution to the solar neutrino problem, which requires $\Delta m^2_{12} \sim 10^{-5} \ \mbox{eV}^2$.
Hence we need extra structure for the mass matrix such as having several elements of the same order of magnitude. [*e.g.*]{} $$M_{\nu} =
\left(
\begin{array}{ccc}a & A & B\\
A & \times & \times \\
B & \times & \times \end{array}\right )
\label{Mnu2}$$ with $A \sim B \gg a$. This gives $$\frac{\Delta m^2_{12}}{\Delta m^2_{23}} \sim \frac{a}{\sqrt{A^2 + B^2}}.$$ The mixing is between all three flavours. and is given by the mixing matrix $$U_{\nu} \sim \left( \begin{array}{ccc}
\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0\\
\frac{1}{\sqrt{2}} \cos \theta & \frac{1}{\sqrt{2}} \cos \theta &
-\sin \theta\\
\frac{1}{\sqrt{2}} \sin \theta & \frac{1}{\sqrt{2}} \sin \theta &
\cos \theta\\
\end{array} \right)$$ where $\theta = \tan^{-1} \frac{B}{A}$. So we have large $\nu_{\mu} - \nu_{\tau}$ mixing with $\Delta m^2 = \Delta m^2_{23}$, and nearly maximal electron neutrino mixing with $\Delta m^2 = \Delta m^2_{12}$. However the AGUT model naturally gives a structure like eq. (\[Mnu1\]) rather than eq. (\[Mnu2\]).
There is also some difficulty in obtaining the required mass scale for the neutrinos. In models such as the AGUT the neutrino masses are generated via super-heavy intermediate fermions in a see-saw type mechanism. This leads to too small neutrino masses: $$m_{\nu} {\raisebox{-0.5ex}{$\
\stackrel{\textstyle<}{\textstyle\sim}\ $}}\frac{\langle{\phi_{WS}} \rangle^2}{M_F}
\sim 10^{-5}\ \mbox{eV},$$ for $M_F = M_{Planck}$ (in general $m_{\nu}$ is also supressed by the chiral charges). So we need to introduce a new mass scale into the theory. Either some intermediate particles with mass $M_F {\raisebox{-0.5ex}{$\
\stackrel{\textstyle<}{\textstyle\sim}\ $}}10^{15}\ \mbox{GeV}$, or an $SU(2)$ triplet Higgs field $\Delta$ with $\langle \Delta^0 \rangle \sim 1 \mbox{eV}$ is required. Without further motivation the introduction of such particles is [*ad hoc*]{}.
Conclusions
===========
We presented two applications of the Multiple Point Principle, according to which nature should choose coupling constants such that the vacuum can exist in degenerate phases. Applied to the AGUT model, it successfully predicts the values of the three fine structure constants, as illustrated in figure 1. In the case of the pure SM, it leads to our predictions for the top quark and Higgs pole masses: $M_t = 173 \pm 5$ GeV and $M_H = 135 \pm 9$ GeV.
The maximal AGUT group $SMG^3 \times U(1)_f$ assigns a unique set of anomaly free chiral gauge charges to the quarks and leptons. With an appropriate choice of Higgs field quantum numbers, the AGUT chiral charges naturally give a realistic charged fermion mass hierarchy. An order of magnitude fit in terms of 3 Higgs VEVs is given in table \[convbestfit\], which reproduces all the masses and mixing angles within a factor of two. On the other hand, the puzzle of the neutrino masses and mixing angles presents a challenge to the model.
References {#references .unnumbered}
==========
[99]{} C.D. Froggatt and H.B. Nielsen, .
M.B. Green and J. Schwarz, .
D.L. Bennett, C.D. Froggatt and H.B. Nielsen in [*Proceedings of the 27th International Conference on High Energy Physics*]{} (Glasgow, 1994), eds. P. Bussey and I. Knowles, (IOP Publishing Ltd, 1995) p. 557; [*Perspectives in Particle Physics ’94*]{}, eds. D. Klabucar, I. Picek and D. Tadić, (World Scientific, 1995) p. 255, hep-ph/9504294.
C.D. Froggatt and H.B. Nielsen, .
C.D. Froggatt, M. Gibson, H.B. Nielsen and D.J. Smith, [*Int. J. Mod. Phys.*]{} A (to be published), hep-ph/9706212.
C.D. Froggatt and H.B. Nielsen
|
---
abstract: 'We examine in detail nonperturbative corrections for low lying energies of a symmetric triple-well potential with non-equivalent vacua, for which there have been disagreement about asymptotic formulas and controversy over the validity of the dilute gas approximation. We carry out investigations from various points of view, including not only a numerical comparison of the nonperturbative corrections with the exact values but also the prediction on the large order behavior of the perturbation series, consistency with the perturbative corrections, and comparison with the WKB approximation. We show that all the results support our formulas previously obtained from the valley method calculation beyond the dilute gas approximation.'
author:
- Masatoshi Sato
- Toshiaki Tanaka
bibliography:
- 'twprefs.bib'
title: 'On Disagreement about Nonperturbative Corrections in Triple-well Potential'
---
Introduction {#sec:intro}
============
It has been widely known that the spectral splitting of the lowest two states of a quantum mechanical symmetric double-well potential due to the quantum tunneling is successfully calculated by summing up multiple instanton contributions with the dilute gas approximation [@Co85]. On the other hand, it might not have been duly recognized that a naive application of the method to a bit more complicated system in general confronts some new difficulties and hardly yields proper results. In this respect, the problem of a symmetric triple-well potential described essentially by the following function: $$\begin{aligned}
V(q)=\frac{1}{2}q^{2}(q^{2}-1)^{2},\end{aligned}$$ have been recently attracting attention of several research groups. One of the novel features of the above potential over the symmetric double-well potential, besides the obvious difference in the number of the potential wells, comes from the fact that the harmonic frequency of the central well is different from that of the side wells. As a result, there are no degeneracies between the harmonic oscillator spectrum of the central well and the side wells while that of the left and right well completely degenerate with each other. Thus, it is difficult to expect intuitively how the quantum tunneling effect contributes to the each harmonic oscillator spectrum.
To the best of our knowledge, the multi-instanton calculation technique with the aid of the dilute gas approximation was applied to a triple-well potential problem first by Lee *et al.* [@LKYPLPY97] and later independently by Casahorrán [@Ca01]. Both of their resulting formulas of the lowest three eigenvalues are however peculiar and doubtful in the fact that they do not coincide with the harmonic oscillator spectrum of the each potential well when the instanton contribution due to the quantum tunneling effect is turned off, although both of the authors did not discuss the validity of them nor compare the obtained results with the exact eigenvalues.
A few years ago, we investigated in Ref. [@ST02] a similar problem in a different context, namely, dynamical breaking of ${\mathcal{N}}$-fold supersymmetry, by means of the valley method [@AKOSW99], which is a generalization of the semi-classical approximation and enables us to calculate nonperturbative correction beyond the dilute gas approximation. Our formulas of the spectrum are completely different from the ones previously obtained in Refs. [@LKYPLPY97; @Ca01] and we have justified our results by checking consistency with some characteristic features of ${\mathcal{N}}$-fold supersymmetry discussed earlier in Ref. [@AST01b]. We have found that the contribution from the interaction between the instantons plays a crucial role in the calculation in order to yield the correct formulas. Hence, we have asserted that the dilute gas approximation would fail in the case.
Recently, however, Alhendi and Lashin reexamined the triple-well potential problem and carried out a careful calculation of the multiple instanton contribution with the dilute gas approximation [@AL04]. Their results are different from ours but look sensible in the sense that they reduce to the harmonic oscillator spectrum when the instanton corrections tend to zero, in contrast to the ones in Refs. [@LKYPLPY97; @Ca01]. They also performed a numerical calculation of the corresponding Schrödinger equation and compared numerically their formulas with the exact values. From the comparison, they have claimed the correctness of their formulas and thus the validity of the dilute gas approximation. However, they did not make a comparison with our results nor examine the accuracy of them.
In this letter, considering the present situation described above, we would like to compare the formulas by Alhendi and Lashin in Ref. [@AL04] and ours in Ref. [@ST02] from various points of view. In the next section, we first make a numerical comparison of the nonperturbative corrections. In Section \[sec:large\], we examine the large order behavior of the perturbation series. In Section \[sec:inter\], we take into account the perturbative corrections to check the accuracy of the formulas for the ground state. In Section \[sec:wkbca\], we carry out the ordinary WKB calculation for the Schrödinger equation to provide another reference for the comparison. Finally, we summarize the results in the last section.
Purely Nonperturbative Corrections {#sec:nonpe}
==================================
In Ref. [@AL04], Alhendi and Lashin investigated the following triple-well potential: $$\begin{aligned}
\label{eq:HamAL}
H^{{\mathrm{AL}}}(x;\omega)=-\frac{1}{2}\frac{{\mathrm{d}}^2}{{\mathrm{d}}x^2}
+\frac{\omega^2}{2}x^2 (x^2-1)^2\,.\end{aligned}$$ They calculated the sum of multiple instanton contributions with the dilute gas approximation and obtained for the lowest three eigenvalues,
\[eqs:npcALs\] $$\begin{aligned}
\label{eq:AL0}
E^{{\mathrm{AL}}}_{0}(\omega)&=\omega\biggl(\frac{3}{4}-\frac{1}{4}
\sqrt{1+\frac{1024}{3\pi}\,\omega{\mathrm{e}}^{-\omega/2}}\,\biggr),\\
E^{{\mathrm{AL}}}_{1}(\omega)&=\omega\,,\\
E^{{\mathrm{AL}}}_{2}(\omega)&=\omega\biggl(\frac{3}{4}+\frac{1}{4}
\sqrt{1+\frac{1024}{3\pi}\,\omega{\mathrm{e}}^{-\omega/2}}\,\biggr).\end{aligned}$$
On the other hand, the triple-well potential we investigated in Ref. [@ST02] is the following: $$\begin{aligned}
\label{eq:HamST}
H^{{\mathrm{ST}}}(q;g,\epsilon)=-\frac{1}{2}\frac{{\mathrm{d}}^2}{{\mathrm{d}}q^2}
+\frac{1}{2}q^2 (1-g^2 q^2)^2 +\frac{\epsilon}{2}(1-3g^2 q^2).\end{aligned}$$ Utilizing the valley method, we obtained for *all* the $n_{0}$th eigenstates localized around the central well and the $n_{\pm}$th eigenstates with parity $\pm$ localized around the side wells[^1],
\[eqs:npcSTs1\] $$\begin{aligned}
E^{{\mathrm{ST}}}_{n_{0}}(g,\epsilon)&=n_{0}+\frac{1}{2}+\frac{\epsilon}{2}
+\frac{\sqrt{2}}{\pi g^2}\,{\mathrm{e}}^{-1/2g^2}E^{(2)}_{n_{0}}(g,\epsilon)
+O({\mathrm{e}}^{-1/g^2}),\\
E^{{\mathrm{ST}}}_{n_{\pm}}(g,\epsilon)&=2n_{\pm}+1-\epsilon
+\frac{\sqrt{2}}{\pi g^2}\,{\mathrm{e}}^{-1/2g^2}E^{(2)}_{n_{\pm}}(g,\epsilon)
+O({\mathrm{e}}^{-1/g^2}),\end{aligned}$$
where the coefficients $E^{(2)}_{n_{0}}$ and $E^{(2)}_{n_{\pm}}$ are given by
\[eqs:npcSTs2\] $$\begin{aligned}
E^{(2)}_{n_{0}}(g,\epsilon)&=-\frac{1}{n_{0}!}\biggl(\frac{2}{g^2}
\biggr)^{n_{0}}\biggl(-\frac{1}{g^2}\biggr)^{n_{0}/2-1/4+3\epsilon/4}
\Gamma\biggl(-\frac{n_{0}}{2}+\frac{1}{4}-\frac{3}{4}\epsilon\biggr),\\
E^{(2)}_{n_{\pm}}(g,\epsilon)&=-\frac{(-1)^{(1-3\epsilon)/2}\pm 1}{
n_{\pm}!}\biggl(\frac{2}{g^2}\biggr)^{2n_{\pm}+1/2-3\epsilon/2}\biggl(
\frac{1}{g^2}\biggr)^{n_{\pm}}\Gamma\biggl(-2n_{\pm}-\frac{1}{2}+\frac{3}{2}
\epsilon\biggr).\end{aligned}$$
In order to compare the two results, we must first establish the relation between the Hamiltonians Eqs. and . By applying a scale transformation on the coordinate $q$ in Eq. , we easily find the following relation: $$\begin{aligned}
\label{eq:scale}
H^{{\mathrm{AL}}}(x;\omega)=\omega H^{{\mathrm{ST}}}(\omega^{1/2}x;\omega^{-1/2},0)\,.\end{aligned}$$ Therefore, the quantities which we shall make comparison with Eqs. are given by[^2]
\[eqs:npcSTs3\] $$\begin{aligned}
\label{eq:ST0}
E^{{\mathrm{ST}}}_{0}(\omega)&=\omega{\operatorname{Re}}E^{{\mathrm{ST}}}_{n_{0}=0}(\omega^{-1/2},0)
=\omega\biggl(\frac{1}{2}-\frac{\Gamma(1/4)}{\pi}\,\omega^{3/4}
{\mathrm{e}}^{-\omega/2}+O({\mathrm{e}}^{-\omega})\biggr),\\
E^{{\mathrm{ST}}}_{1}(\omega)&=\omega{\operatorname{Re}}E^{{\mathrm{ST}}}_{n_{-}=0}(\omega^{-1/2},0)
=\omega\biggl(1-\frac{4}{\sqrt{\pi}}\,\omega^{3/2}{\mathrm{e}}^{-\omega/2}
+O({\mathrm{e}}^{-\omega})\biggr),\\
E^{{\mathrm{ST}}}_{2}(\omega)&=\omega{\operatorname{Re}}E^{{\mathrm{ST}}}_{n_{+}=0}(\omega^{-1/2},0)
=\omega\biggl(1+\frac{4}{\sqrt{\pi}}\,\omega^{3/2}{\mathrm{e}}^{-\omega/2}
+O({\mathrm{e}}^{-\omega})\biggr).\end{aligned}$$
Here we note that the nonperturbative corrections in Eqs. are in general complex and the real parts of them should be taken into account as the spectral shifts; the imaginary parts of them are to be canceled with the imaginary parts of the perturbative corrections arising from the Borel singularity in the framework of the valley method, see for more details Refs. [@AKOSW99; @ST02]. We will later consider the imaginary parts in order to investigate the large order behavior of the perturbation series in the next section.
For the purpose of examining the accuracy of the purely nonperturbative corrections in Eqs. and , it is important to note that in addition to the nonperturbative corrections estimated in the formulas there are perturbative corrections to the harmonic oscillator spectra[^3]. Hence it does not make sense to compare directly them to the exact eigenvalues without taking into account the perturbative contributions. Fortunately, however, the perturbative corrections to the harmonic oscillator spectrum for the first and second excited states are completely the same (cf. Section \[sec:large\]). As a result, the difference of the eigenvalues between the first and second excited states $\Delta E_{21}=E_{2}-E_{1}$ only contains the purely nonperturbative contributions. Therefore, the comparison of the quantity $\Delta E_{21}$ enables us to study the accuracy of Eqs. and adequately. In Table \[tb:dif12\], we show i) the exact results $\Delta E_{21}^{ex}$ obtained by solving numerically the Schrödinger equation for the Hamiltonian presented in Ref. [@AL04], ii) $\Delta E_{21}^{{\mathrm{AL}}}=E_{2}^{{\mathrm{AL}}}-E_{1}^{{\mathrm{AL}}}$ obtained from Eq. and the ratios $\Delta E_{21}^{{\mathrm{AL}}}/\Delta E_{21}^{ex}$, and iii) $\Delta E_{21}^{{\mathrm{ST}}}=E_{2}^{{\mathrm{ST}}}-E_{1}^{{\mathrm{ST}}}$ obtained from Eq. and the ratios $\Delta E_{21}^{{\mathrm{ST}}}/\Delta E_{21}^{ex}$.\
$$\arraycolsep = 8pt
\begin{array}{rl|lc|lc}
\hline
\omega & \Delta E_{21}^{ex}(\omega) & \Delta E_{21}^{{\mathrm{AL}}}(\omega)
& \Delta E_{21}^{{\mathrm{AL}}}/\Delta E_{21}^{ex} & \Delta E_{21}^{{\mathrm{ST}}}(\omega)
& \Delta E_{21}^{{\mathrm{ST}}}/\Delta E_{21}^{ex}\\
\hline
30 & 4.7230\cdot 10^{-3} & 3.7381\cdot 10^{-3} & 0.79147
& 6.8061\cdot 10^{-3} & 1.4411\\
50 & 9.1006\cdot 10^{-7} & 4.7154\cdot 10^{-7} & 0.51814
& 1.1081\cdot 10^{-6} & 1.2176\\
70 & 1.0186\cdot 10^{-10} & 4.1959\cdot 10^{-11} & 0.41193
& 1.1667\cdot 10^{-10} & 1.1454\\
90 & 8.9504\cdot 10^{-15} & 3.1490\cdot 10^{-15} & 0.35183
& 9.9282\cdot 10^{-15} & 1.1092\\
110 & 6.8449\cdot 10^{-19} & 2.1356\cdot 10^{-19} & 0.31201
& 7.4439\cdot 10^{-19} & 1.0875\\
\hline
\end{array}$$
From Table I, we see that our results $\Delta E^{{\mathrm{ST}}}_{21}(\omega)$ are in better agreement with the numerical exact values $\Delta E_{21}^{ex}(\omega)$ than those by Alhendi and Lashin $\Delta E^{{\mathrm{AL}}}_{21}(\omega)$. We also find that as the parameter $\omega$ becomes larger (the coupling g becomes smaller), the accuracy of our results becomes better while that of the results by Alhendi and Lashin becomes worse. This suggests that our method provides a reliable semi-classical approximation but the dilute gas approximation does not. As is easily seen from the Euclidean action of the present model $$\begin{aligned}
S^{{\mathrm{ST}}}=\frac{1}{g^{2}}\int{\mathrm{d}}\tau
\left(\frac{1}{2}\dot{x}^{2}+\frac{1}{2}x^{2}(1-x^{2})^{2}\right),
\quad x=gq,\end{aligned}$$ the coupling constant $g^{2}$ plays a similar role to the Plank constant $\hbar$. Therefore, any reliable semi-classical approximation should work better as the coupling constant $g$ becomes smaller.
Large Order Behavior of Perturbation Series {#sec:large}
===========================================
In spite of the fact that perturbation series are in general divergent and at most asymptotic, they contain much information on the property of the physical quantity under consideration. An intimate relation between nonperturbative property and large order behavior of perturbation series is a typical example [@GZJ90]. In this section, we make an examination from this point of view.
In order to evaluate the perturbative corrections to the harmonic oscillator spectrum of the each potential well, it is convenient to begin with the Hamiltonian and then make the transformation indicated by the scale relation after the perturbative calculation. The coefficients of the perturbation series are systematically calculated with the aid of the Bender–Wu method [@BW69]. First, the perturbation theory around the harmonic oscillator states of the central potential well is set up by decomposing the Hamiltonian into the harmonic oscillator part and the remaining part: $$\begin{aligned}
\label{eq:HamST0}
H^{{\mathrm{ST}}}(q;g,0)=-\frac{1}{2}\frac{{\mathrm{d}}^2}{{\mathrm{d}}q^2}+\frac{1}{2}q^{2}
-g^{2}q^{4}+\frac{1}{2}g^{4}q^{6}.\end{aligned}$$ The perturbative corrections to the eigenvalues and eigenfunctions are defined by the following formal series expansions: $$\begin{aligned}
E(g)=\sum_{m=0}^{\infty}g^{2m}c^{[2m]},\qquad
\psi(q;g)={\mathrm{e}}^{-q^{2}/2}\sum_{k=0}^{\infty}g^{2k}
\sum_{l=0}^{\infty}a_{2l+P}^{[2k]}q^{2l+P},\end{aligned}$$ where $P=0$ ($1$) for the even (odd) parity states, respectively. For the lowest state, $c^{[0]}=1/2$ and $a_{2l}^{[0]}=0$ for all $l>0$. Requiring that they satisfy the Schrödinger equation, we obtain a recursion relation for $a_{2l+P}^{[2k]}$ and $c^{[2m]}$: $$\begin{gathered}
\label{eq:recur1}
\qquad (4l+1+2P)a_{2l+P}^{[2k]}-(2l+2+P)(2l+1+P)a_{2(l+1)+P}^{[2k]}\\
-2a_{2(l-2)+P}^{[2(k-1)]}+a_{2(l-3)+P}^{[2(k-2)]}
=2\sum_{m=0}^{k}c^{[2m]}a_{2l+P}^{[2(k-m)]}.\qquad\end{gathered}$$ Second, the perturbation theory around the harmonic oscillator states of the side potential wells is defined by shifting the origin of the coordinate to one of the minimum of the side potentials $q\to q\pm 1/g$ and then decomposing the Hamiltonian: $$\begin{aligned}
H^{{\mathrm{ST}}}(q\pm 1/g;g,0)=-\frac{1}{2}\frac{{\mathrm{d}}^2}{{\mathrm{d}}q^2}+\frac{4}{2}q^{2}
\pm 6g q^{3}+\frac{13}{2}g^{2}q^{4}\pm 3g^{3}q^{5}+\frac{1}{2}g^{4}q^{6}.\end{aligned}$$ The perturbative corrections are similarly introduced by[^4] $$\begin{aligned}
E(g)=\sum_{m=0}^{\infty}g^{2m}c^{[2m]},\qquad
\psi(q;g)={\mathrm{e}}^{-q^{2}}\sum_{k=0}^{\infty}g^{k}
\sum_{l=0}^{\infty}a_{l}^{[k]}q^{l}.\end{aligned}$$ The recursion relation for $c^{[m]}$ and $a_{l}^{[k]}$ in this case is then given by $$\begin{gathered}
\label{eq:recur2}
\qquad (4l+2)a_{l}^{[k]}-(l+2)(l+1)a_{l+2}^{[k]}\pm 12 a_{l-3}^{[k-1]}\\
+13 a_{l-4}^{[k-2]}\pm 6 a_{l-5}^{[k-3]}+a_{l-6}^{[k-4]}
=2\sum_{m=0}^{[k/2]}c^{[2m]}a_{l}^{[k-2m]}.\qquad\end{gathered}$$ For the lowest state, $c^{[0]}=1$ and $a_{l}^{[0]}=0$ for all $l>0$.
On the other hand, as we have mentioned previously, the imaginary parts of the nonperturbative contributions are to be canceled with those of the perturbative ones in the framework of the valley method. This leads to the following dispersion relation [@ST02; @AKOSW99]: $$\begin{aligned}
\label{eq:dispe}
c^{[2m]}=-\frac{1}{\pi}\int_{0}^{\infty}{\mathrm{d}}g^2
\frac{{\operatorname{Im}}E_{{\mathrm{NP}}}(g)}{g^{2m+2}}\,.\end{aligned}$$ This relation enables us to predict the large order behavior of the perturbation series for the eigenvalues. For the lowest three states ($n_{0}=n_{\pm}=0$) in the present case ($\epsilon=0$), we obtain from Eqs. – and
\[eqs:lobps\] $$\begin{aligned}
\label{eq:lobps0}
c_{0}^{[2m]}&\sim -\frac{2^{5/4}}{\pi\Gamma(3/4)}\,2^{m}\Gamma\biggl(
m+\frac{3}{4}\biggr)\equiv\bar{c}_{0}^{[2m]},\\
c_{1(2)}^{[2m]}&\sim -\frac{8\sqrt{2}}{\pi^{3/2}}\,
2^{m}\Gamma\biggl(m+\frac{3}{2}\biggr)\equiv\bar{c}_{1(2)}^{[2m]}.\end{aligned}$$
Therefore, we can check the validity of the results – by comparing the predicted asymptotic forms $\bar{c}^{[2m]}$ in Eqs. with the exact perturbative coefficients $c^{[2m]}$ calculated using the recursion relations and .
$$\arraycolsep = 5pt
\begin{array}{rll}
\hline
m & c_{0}^{[2m]}/\bar{c}_{0}^{[2m]} & c_{1(2)}^{[2m]}/\bar{c}_{1(2)}^{[2m]}
\rule{0pt}{15pt}\rule[-7pt]{0pt}{0pt}\\
\hline
20 & 0.8946472445 & 0.7797002850\\
40 & 0.9493285320 & 0.8904365552\\
60 & 0.9665686152 & 0.9268736279\\
80 & 0.9750492671 & 0.9451085558\\
100 & 0.9800964967 & 0.9560611732\\
120 & 0.9834448543 & 0.9633690036\\
140 & 0.9858286759 & 0.9685922139\\
160 & 0.9876123158 & 0.9725115700\\
180 & 0.9889971178 & 0.9755611660\\
200 & 0.9901034160 & 0.9780016290\\
220 & 0.9910075563 & 0.9799989041\\
240 & 0.9917603143 & 0.9816636733\\
260 & 0.9923967744 & 0.9830725934\\
280 & 0.9929419564 & 0.9842804381\\
300 & 0.9934141831 & 0.9853273870\\
\hline
\end{array}$$
In Table \[tb:large\], we show the ratios $c^{[2m]}/\bar{c}^{[2m]}$ up to the order $m=300$. We easily see that the exact values indeed tend to the predicted asymptotic values for both the ground and excited states and thus confirm the correctness of our formulas, at least, for their imaginary parts.
Interplay between Nonperturbative and Perturbative Corrections {#sec:inter}
==============================================================
In Section \[sec:nonpe\], we have investigated the nonperturbative corrections for the excited states and confirmed that our formulas obtained from the valley method calculation are more accurate than the ones obtained from the instanton calculation with the dilute gas approximation. Although the analysis of the large order behavior in the previous section ensures the correctness of the imaginary parts of our formulas for both the ground and excited states, it does not necessarily mean that the real part of our formula for the ground state is also correct. In order to check the accuracy of the nonperturbative spectral shift for the ground state we must resort to other means.
As we have mentioned previously, there exist perturbative corrections to the harmonic oscillator spectrum in addition to nonperturbative ones. Hence, each of the spectrum is formally expressed as $$\begin{aligned}
\label{eq:decom}
E(g)=E^{(0)}+E_{{\mathrm{NP}}}(g)+E_{{\mathrm{P}}}(g),\end{aligned}$$ where $E^{(0)}$ denotes the harmonic oscillator spectrum when $g=0$, $E_{{\mathrm{NP}}}$ the purely nonperturbative part which cannot be represented by a formal power series in $g^{2}=\omega^{-1}$, and $E_{{\mathrm{P}}}$ the remaining perturbative part. The decomposition suggests that we can check the formulas for the nonperturbative corrections by examining the prediction on the *perturbative* corrections instead. That is, we can regard each of the following quantity as the prediction of the each formula on the perturbative contribution to the ground state: $$\begin{aligned}
\label{eq:prepc}
E_{{\mathrm{P}}}^{{\mathrm{AL}}}(\omega)\equiv E_{0}^{ex}(\omega)-E_{0}^{{\mathrm{AL}}}(\omega),\qquad
E_{{\mathrm{P}}}^{{\mathrm{ST}}}(\omega)\equiv E_{0}^{ex}(\omega)-E_{0}^{{\mathrm{ST}}}(\omega),\end{aligned}$$ where $E_{0}^{ex}(\omega)$ is the exact eigenvalue for the ground state, $E_{0}^{{\mathrm{AL}}}(\omega)$ and $E_{0}^{{\mathrm{ST}}}(\omega)$ are respectively given by Eqs. and , both of which consist of the harmonic oscillator eigenvalue and the predicted nonperturbative contribution.
$$\arraycolsep = 5pt
\begin{array}{rll}
\hline
\omega & E_{{\mathrm{P}}}^{{\mathrm{AL}}}(\omega) & E_{{\mathrm{P}}}^{{\mathrm{ST}}}(\omega)\\
\hline
30 & -0.8\,18251\cdots & -0.8\,21854\cdots\\
50 & -0.78839\,65537\cdots & -0.78839\,70101\cdots\\
70 & -0.7763341456\,10396\cdots & -0.7763341456\,51123\cdots\\
90 & -0.77005483676111\,02506\cdots & -0.77005483676111\,33127\cdots\\
110 & -0.76619756312233715\,58989\cdots & -0.76619756312233715\,61068\cdots\\
\hline
\end{array}$$
In Table \[tb:prepc\], we show the numerical values of the predicted perturbative contributions to the ground state energy for each value of $\omega$ calculated using Eq. . For the exact eigenvalues $E_{0}^{ex}(\omega)$, we have used again the numerical results shown in Ref. [@AL04].
To examine the accuracy of these predictions, we shall evaluate the exact perturbative contribution $E_{{\mathrm{P}}}$ by using the perturbation series. Although the perturbation series is generally divergent, as we have already observed in the previous section (cf., Eqs. ), the asymptotic property of the perturbation series nevertheless ensures that for a sufficiently small value of the expansion parameter a partial sum of the first finite terms in the perturbation series gives an asymptotic value of the perturbative correction[^5]. As a consequence, however small the value of the expansion parameter is, there exists a critical order $m_{c}$ at which the absolute value of the perturbative correction $|g^{2m}c_{0}^{[2m]}|$ becomes minimum. It is apparent that the asymptotic property is lost and replaced by the divergent one when the order of the perturbation exceeds the critical order $m_{c}$. Therefore, the asymptotic values of the exact perturbative corrections we should read from the perturbation series are given by the first finite partial sums up to at most the critical order $m_{c}$.
In Table \[tb:asymp\], we illustrate the numerical values obtained from the first finite partial sums of the perturbation series for the ground state. Here we note that from the scaling relation the perturbative quantity $E_{{\mathrm{P}}}(\omega)$ we should take for the Hamiltonian $H^{{\mathrm{AL}}}(x;\omega)$ reads $$\begin{aligned}
E_{{\mathrm{P}}}(g)=\sum_{m=1}^{M}g^{2m}c_{0}^{[2m]}\quad\longmapsto\quad
E_{{\mathrm{P}}}(\omega)=\sum_{m=1}^{M}\omega^{1-m}c_{0}^{[2m]}.\end{aligned}$$ In Table \[tb:asymp\], we show the partial sums up to the order $M$ with $m_{c}-10<M\leq m_{c}$. The critical order $m_{c}$ for all the cases $\omega=30$, $50$, $70$, $90$, and $110$ are given by $m_{c}=\omega/2$.
$$\arraycolsep = 5pt
\begin{array}{rcrcrc}
\hline
\multicolumn{2}{c}{\omega=30} &\multicolumn{2}{c}{\omega=50}
&\multicolumn{2}{c}{\omega=70}\\
M & \sum_{m=1}^{M}\omega^{1-m}c_{0}^{[2m]}
& M & \sum_{m=1}^{M}\omega^{1-m}c_{0}^{[2m]}
& M & \sum_{m=1}^{M}\omega^{1-m}c_{0}^{[2m]}
\rule[-7pt]{0pt}{0pt}\\
\hline
6 & -0.8\,21307\cdots & 16 & -0.78839\,69801\cdots
& 26 & -0.7763341456\,49363\cdots\\
7 & -0.8\,21522\cdots & 17 & -0.78839\,69881\cdots
& 27 & -0.7763341456\,49752\cdots\\
8 & -0.8\,21641\cdots & 18 & -0.78839\,69939\cdots
& 28 & -0.7763341456\,50061\cdots\\
9 & -0.8\,21714\cdots & 19 & -0.78839\,69983\cdots
& 29 & -0.7763341456\,50315\cdots\\
10 & -0.8\,21764\cdots & 20 & -0.78839\,70018\cdots
& 30 & -0.7763341456\,50532\cdots\\
11 & -0.8\,21800\cdots & 21 & -0.78839\,70047\cdots
& 31 & -0.7763341456\,50723\cdots\\
12 & -0.8\,21830\cdots & 22 & -0.78839\,70072\cdots
& 32 & -0.7763341456\,50897\cdots\\
13 & -0.8\,21856\cdots & 23 & -0.78839\,70095\cdots
& 33 & -0.7763341456\,51060\cdots\\
14 & -0.8\,21880\cdots & 24 & -0.78839\,70118\cdots
& 34 & -0.7763341456\,51217\cdots\\
m_{c} & -0.8\,21903\cdots & m_{c} & -0.78839\,70140\cdots
& m_{c} & -0.7763341456\,51373\cdots\\
\hline
\end{array}$$ $$\arraycolsep = 5pt
\begin{array}{rcrc}
\hline
\multicolumn{2}{c}{\omega=90} &\multicolumn{2}{c}{\omega=110}\\
M & \sum_{m=1}^{M}\omega^{1-m}c_{0}^{[2m]}
& M & \sum_{m=1}^{M}\omega^{1-m}c_{0}^{[2m]}
\rule[-7pt]{0pt}{0pt}\\
\hline
36 & -0.77005483676111\,32120\cdots & 46
& -0.76619756312233715\,61012\cdots\\
37 & -0.77005483676111\,32318\cdots & 47
& -0.76619756312233715\,61022\cdots\\
38 & -0.77005483676111\,32485\cdots & 48
& -0.76619756312233715\,61031\cdots\\
39 & -0.77005483676111\,32628\cdots & 49
& -0.76619756312233715\,61039\cdots\\
40 & -0.77005483676111\,32755\cdots & 50
& -0.76619756312233715\,61046\cdots\\
41 & -0.77005483676111\,32871\cdots & 51
& -0.76619756312233715\,61053\cdots\\
42 & -0.77005483676111\,32978\cdots & 52
& -0.76619756312233715\,61059\cdots\\
43 & -0.77005483676111\,33080\cdots & 53
& -0.76619756312233715\,61065\cdots\\
44 & -0.77005483676111\,33179\cdots & 54
& -0.76619756312233715\,61071\cdots\\
m_{c} & -0.77005483676111\,33277\cdots & m_{c}
& -0.76619756312233715\,61077\cdots\\
\hline
\end{array}$$
Comparing the results in Table \[tb:asymp\] with the ones in Table \[tb:prepc\], we easily see that the asymptotic values of the perturbative corrections for all the cases are in good agreement with the values $E_{{\mathrm{P}}}^{{\mathrm{ST}}}(\omega)$ predicted by our formula but apparently deviate from the values $E_{{\mathrm{P}}}^{{\mathrm{AL}}}(\omega)$ predicted by the Alhendi and Lashin’s formula .
We note that the differences between $E_{{\mathrm{P}}}^{{\mathrm{ST}}}(\omega)$ and $\sum_{m=1}^{M}\omega^{1-m}c_{0}^{[2m]}$ are minimum around $M\sim m_{c}-1$. It indicates that ${\operatorname{Re}}E_{{\mathrm{P}}}(g)$ in the case has the perturbation series as *strong asymptotic series* [@RS78], that is, there exist positive real constants $C$ and $\sigma$ so that $$\begin{aligned}
\left|{\operatorname{Re}}E_{{\mathrm{P}}}(g)-\sum_{m=1}^{M}g^{2m}c_{0}^{[2m]}\right|
\leq C\sigma^{M+1}(M+1)!\,\left|g^{2}\right|^{M+1},\end{aligned}$$ for all $M$ and all $g^{2}>0$ near the origin. Indeed, if it is the case, we have with the aid of Eq. and $\sigma=2$, $$\begin{aligned}
\left|{\operatorname{Re}}E_{{\mathrm{P}}}(g)-\sum_{m=1}^{M}g^{2m}c_{0}^{[2m]}\right|
\lesssim C'(M+7/4)^{1/4}\Bigl|g^{2(M+1)}c_{0}^{[2(M+1)]}\Bigr|,\end{aligned}$$ where $C'=C\pi\Gamma(3/4)/2^{5/4}$. The right hand side is minimum around $M\sim m_{c}-1$ by the definition of the critical order $m_{c}$, and thus the above fact would be naturally understood.
Finally, we would like to mention about the fact that in the parameter region we have examined, $30\leq\omega\leq 110$ or $0.095\lesssim g\lesssim 0.18$, the following relation is roughly satisfied: $$\begin{aligned}
\label{eq:rough}
\min_{m}\Bigl| g^{2m}c^{[2m]}\Bigr|=\Bigl| g^{2m_{c}}c^{[2m_{c}]}\Bigr|
\sim\bigl|{\operatorname{Re}}E_{{\mathrm{NP}}}(g)\bigr|\times 10^{-1}.\end{aligned}$$ Interestingly, we can show that a similar relation is observed generically as far as the system under consideration has a nonperturbative effect. Suppose the following conditions are satisfied for smaller values of the coupling constant $g^{2}$ involved in the system under consideration: $$\begin{aligned}
\label{eq:condi}
{\operatorname{Im}}E_{{\mathrm{NP}}}(g)\sim C g^{-2(\nu +1)}{\mathrm{e}}^{-1/bg^{2}},\qquad
{\operatorname{Re}}E_{{\mathrm{NP}}}(g)=A {\operatorname{Im}}E_{{\mathrm{NP}}}(g),\end{aligned}$$ where $A$, $C$, and $b>0$ are real constants. Then, we can prove the following intriguing relation for smaller $g^{2}$: $$\begin{aligned}
\label{eq:relat}
\min_{m}\Bigl| g^{2m}c^{[2m]}\Bigr|\sim\sqrt{\frac{2b\:\!{\mathrm{e}}}{A^{2}\pi}}\,
\Bigl| g{\operatorname{Re}}E_{{\mathrm{NP}}}(g)\Bigr|.\end{aligned}$$ For the proof, we first note that the first condition in Eq. implies $$\begin{aligned}
c^{[2m]}\sim -\frac{C}{\pi}\,b^{m+\nu+1}\Gamma(m+\nu+1)
\equiv\bar{c}^{[2m]},\end{aligned}$$ for larger $m$. Next, we define a function $f$ by $$\begin{aligned}
f(\mu;g)\equiv\Bigl| g^{2\mu}\bar{c}^{[2\mu]}\Bigr|.\end{aligned}$$ It is evident that for larger integer $m$ the function $f(m;g)$ well approximates the magnitude of the $m$th-order perturbative correction. The derivative with respect to $\mu$ reads, $$\begin{aligned}
\frac{\partial}{\partial\mu}f(\mu;g)
=f(\mu;g)\bigl[\,\ln(bg^{2})+\psi(\mu+\nu+1)\bigr],\end{aligned}$$ where $\psi$ denotes the digamma function. Hence $f(\mu;g)$ takes minimum value at $\mu={\bar{\mu}}$, ${\bar{\mu}}$ satisfying $$\begin{aligned}
\ln(bg^{2})+\psi({\bar{\mu}}+\nu+1)=0.\end{aligned}$$ For smaller value of $g^{2}\ll 1$, we see ${\bar{\mu}}$ becomes larger. Thus, applying the asymptotic expansion of the digamma function [@GR00]: $$\begin{aligned}
\psi(z)\sim\ln z -\frac{1}{2z}+O(z^{-2}),\end{aligned}$$ we obtain $$\begin{aligned}
\label{eq:asymp}
bg^{2}\sim\frac{1}{{\bar{\mu}}+\nu+1}\exp\biggl(\frac{1}{2({\bar{\mu}}+\nu+1)}
\bigl[1+O({\bar{\mu}}^{-1})\bigr]\biggr).\end{aligned}$$ With the aid of the Stirling formula and Eq. , we have $$\begin{aligned}
\Gamma({\bar{\mu}}+\nu+1)\sim\sqrt{2\pi}\,{\mathrm{e}}^{1/2}(bg^{2})^{-({\bar{\mu}}+\nu+1/2)}
{\mathrm{e}}^{-1/bg^{2}}\bigl[1+O({\bar{\mu}}^{-1})\bigr].\end{aligned}$$ Therefore, the minimum value of $f(\mu,g)$, which would provide a good approximation to the minimum magnitude of the perturbative correction at the critical order $m_{c}\sim{\bar{\mu}}$, is estimated as, $$\begin{aligned}
f({\bar{\mu}};g)
&=\frac{b^{\nu+1}}{\pi}|C|\,(bg^{2})^{{\bar{\mu}}}\Gamma({\bar{\mu}}+\nu+1)\notag\\
&\sim\sqrt{\frac{2b\:\!{\mathrm{e}}}{\pi}}\,|C|\, g^{-2(\nu+1/2)}{\mathrm{e}}^{-1/bg^{2}}
\bigl[1+O({\bar{\mu}}^{-1})\bigr]\notag\\
&\sim\sqrt{\frac{2b\:\!{\mathrm{e}}}{\pi}}\,\Bigl| A^{-1} g {\operatorname{Re}}E_{{\mathrm{NP}}}(g)\Bigr|
\bigl[1+O({\bar{\mu}}^{-1})\bigr],\end{aligned}$$ and thus we obtain the relation . In our case $A=1$ and $b=2$, and thus the relation for $g\sim 0.1$ follows. As a consequence, we also find that the next-order nonperturbative corrections of order $O(g^{2}{\operatorname{Re}}E_{{\mathrm{NP}}})$ becomes negligible in comparison with the perturbative correction for sufficiently small $g$; from the relation we readily obtain $$\begin{aligned}
O\left(g^{2}{\operatorname{Re}}E_{{\mathrm{NP}}}\right)\sim
O\left(g\min_{m}\left|g^{2m}c^{[2m]}\right|\right).\end{aligned}$$ Therefore, the next-order nonperturbative corrections do not affect the analysis for $g\sim 0.1$ in this section.
WKB Calculation {#sec:wkbca}
===============
So far, we have checked the accuracy of the semi-classical calculations of the path-integral by comparing with the exact values calculated numerically. In this section, we make a comparison in a different way. To this end, we employ another nonperturbative approach to derive formulas for the same physical quantities. The method we shall use here is the WKB approximation for the Schrödinger equation. In the following, we shall always consider the leading terms of the power expansion in $g$ since we are interested in the quantization condition for the nonperturbative contribution.
Let us consider the more general Hamiltonian for all $\epsilon g^{2}\ll 1$. The system has parity symmetry and thus it is sufficient to study the connection condition of the WKB wave function only on the half-line $q\in (0,\infty)$. In the vicinity of the minimum of the central potential well, the Schrödinger equation for the Hamiltonian in the leading order of $g$ is given by $$\begin{aligned}
\left(-\frac{1}{2}\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}q^{2}}+\frac{1}{2}q^{2}\right)\psi(q)
=\left(E-\frac{\epsilon}{2}\right)\psi(q).\end{aligned}$$ The local solutions possessing a definite parity $\pm$ are expressed as $$\begin{aligned}
\label{eq:cent}
\psi(q)=A_{\pm}\left(D_{\nu}(-\sqrt{2}q)\pm D_{\nu}(\sqrt{2}q)\right), \end{aligned}$$ where $A_{\pm}$ are constants and $D_{\nu}$ is the parabolic cylinder function with $\nu=E-\epsilon/2-1/2$. In a similar way, around the minimum of the right side potential well, the Schrödinger equation is approximated by $$\begin{aligned}
\left[-\frac{1}{2}\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}q^{2}}+2\Bigl(q-\frac{1}{g}\Bigr)^{2}
\right]\psi(q)=\left(E+\epsilon\right)\psi(q).\end{aligned}$$ and the local solution which vanishes at $q\rightarrow\infty$ is given by $$\begin{aligned}
\label{eq:side}
\psi(q)=B D_{\lambda}\bigl(2(q-1/g)\bigr), \end{aligned}$$ where $B$ is a constant and $\lambda=E/2+\epsilon/2-1/2$. The solutions and are to be connected with the following WKB solution in the classically forbidden region ($q_1\ll q\ll q_2$): $$\begin{aligned}
\label{eq:WKB}
\psi(q)=\frac{C_{1}}{k(q)^{1/2}}\exp\left(-\int_{q_{1}}^{q}{\mathrm{d}}x\,k(x)\right)
+\frac{C_{2}}{k(q)^{1/2}}\exp\left(\int_{q_{1}}^{q}{\mathrm{d}}x\,k(x)\right),\end{aligned}$$ where $$\begin{aligned}
k(x)=\sqrt{x^{2}(1-g^{2}x^{2})^{2}+\epsilon(1-3g^{2}x^{2})-2E}\,.\end{aligned}$$ The positive classical turning points $q_{i} (i=1,2)$ with $0<q_{1}<q_{2}$ defined by the solutions of $V(q_{i})=E$ are, $$\begin{aligned}
q_{1}=\sqrt{2E-\epsilon}+O(g^{2}),
\qquad
q_{2}=\frac{1}{g}-\sqrt{\frac{E+\epsilon}{2}}+O(g).\end{aligned}$$ In order to connect the wave functions obtained in the each region, it is important to note that the leading term in $g$ of the WKB solution varies according to the position it is viewed from. If it is viewed from the point around the central potential well, the integral in the exponent in Eq. is evaluated as $$\begin{aligned}
\int_{q_{1}}^{q}{\mathrm{d}}x\,k(x)
&=\frac{1}{g^{2}}\int_{gq_{1}}^{gq}{\mathrm{d}}\omega
\sqrt{w^{2}(1-w^{2})^{2}+\epsilon g^{2}(1-3w^{2})-2E}\notag\\
&=\frac{1}{g^{2}}\int_{gq_{1}}^{gq}{\mathrm{d}}\omega\left[w(1-w^{2})
-\frac{1}{2}\frac{(2E-\epsilon)g^{2}}{w(1-w^{2})}
-\frac{3\epsilon g^{2}}{2}\frac{w}{1-w^{2}}+\cdots\right]\notag\\
&=\left[\frac{\omega^{2}}{2g^{2}}-\frac{\omega^{4}}{4g^{2}}
+\frac{\epsilon-2E}{2}\ln|\omega|+\frac{\epsilon+E}{2}\ln|1-\omega^{2}|
+\cdots\right]_{gq_{1}}^{gq},
$$ and $k(q)\sim q+\cdots$. Thus, in the leading order of $g$ we obtain the WKB wave function as $$\begin{aligned}
\psi(q)\sim\frac{C_{1}}{q^{1/2}}\,{\mathrm{e}}^{-q^{2}/2}\left(
\frac{{\mathrm{e}}\:\! q}{\sqrt{2E-\epsilon}}\right)^{(2E-\epsilon)/2}
\!\!\!\!+\frac{C_{2}}{q^{1/2}}\,{\mathrm{e}}^{q^{2}/2}\left(
\frac{{\mathrm{e}}\:\! q}{\sqrt{2E-\epsilon}}\right)^{-(2E-\epsilon)/2}.
$$ Comparing this with the following asymptotic form for $q\gg 1$ of the wave function (cf. Ref. [@GR00]) determined in the region of the central potential well: $$\begin{aligned}
\psi(q)\sim A_{\pm}\left[\bigl((-1)^{\nu}\pm 1\bigr){\mathrm{e}}^{-q^{2}/2}
(\sqrt{2}q)^{\nu}+\frac{\sqrt{2\pi}}{\Gamma(-\nu)}
{\mathrm{e}}^{q^{2}/2}(\sqrt{2}q)^{-\nu-1}\right],\end{aligned}$$ we have the following connection condition: $$\begin{aligned}
\label{eq:conn1}
\frac{\sqrt{2\pi}}{\bigl((-1)^{\nu}\pm 1\bigr)\Gamma(-\nu)}
=\frac{C_{2}}{C_{1}}\left(\frac{2\sqrt{E-\epsilon/2}}{{\mathrm{e}}}
\right)^{2E-\epsilon}.\end{aligned}$$ On the other hand, the integral in the exponent in Eq. viewed from the point around the right side potential well is evaluated as $$\begin{aligned}
\lefteqn{
\int_{q_{1}}^{q}{\mathrm{d}}x\,k(x)
=\frac{1}{g^{2}}\int_{gq_{1}-1}^{gq-1}{\mathrm{d}}\omega
\sqrt{\omega^{2}(1+\omega)^{2}(2+\omega)^{2}-\epsilon g^{2}
(2+6\omega+3\omega^{2})-2Eg^{2}}
}\notag\\
&=-\frac{1}{g^{2}}\int_{gq_{1}-1}^{gq-1}{\mathrm{d}}\omega
\left[\omega(1+\omega)(2+\omega)-\frac{\epsilon g^{2}(2+6\omega+3\omega^{2}) +2Eg^{2}}{2\omega(1+\omega)(2+\omega)}+\cdots\right]\notag\\
&=\left[-\frac{1}{g^{2}}\left(\omega^{2}+\omega^{3}+\frac{\omega^{4}}{4}\right)
+\frac{\epsilon+E}{2}\ln|\omega(2+\omega)|+\frac{\epsilon-2E}{2}\ln|1+\omega|
+\cdots\right]_{gq_{1}-1}^{gq-1},
$$ and $k(q)\sim 1/g-q+\cdots$. Thus, in the leading order of $g$ we obtain the WKB wave function as $$\begin{aligned}
\psi(q)&\sim\frac{C_{1}\,{\mathrm{e}}^{-1/4g^{2}}g^{-3E/2}}{(1/g-q)^{1/2}}\,
{\mathrm{e}}^{(q-1/g)^{2}}\bigl(2(1/g-q)\bigr)^{-(E+\epsilon)/2}\left(
\frac{{\mathrm{e}}}{\sqrt{2E-\epsilon}}\right)^{(2E-\epsilon)/2}\notag\\
&{\phantom{=\ }}+\frac{C_{2}\,{\mathrm{e}}^{1/4g^{2}}g^{3E/2}}{(1/g-q)^{1/2}}\,{\mathrm{e}}^{-(q-1/g)^{2}}
\bigl(2(1/g-q)\bigr)^{(E+\epsilon)/2}\left(\frac{{\mathrm{e}}}{\sqrt{2E-\epsilon}}
\right)^{-(2E-\epsilon)/2}.\end{aligned}$$ Matching this with the following asymptotic form for $1/g-q\gg 1$ of the wave function determined in the region of the right side potential well: $$\begin{aligned}
\psi(q)\sim B\left[{\mathrm{e}}^{-(q-1/g)^{2}}\bigl(2(q-1/g)\bigr)^{\lambda}
-\frac{\sqrt{2\pi}(-1)^{\lambda}}{\Gamma(-\lambda)}{\mathrm{e}}^{(q-1/g)^{2}}
\bigl(2(q-1/g)\bigr)^{-\lambda-1}\right],\end{aligned}$$ we get another connection condition as follows: $$\begin{aligned}
\label{eq:conn2}
-\frac{\sqrt{2\pi}}{(-1)^{\lambda+1}\Gamma(-\lambda)}
=\frac{C_1}{C_2}\,{\mathrm{e}}^{-1/2g^{2}}g^{-3E}
\left(\frac{{\mathrm{e}}}{\sqrt{2E-\epsilon}}\right)^{2E-\epsilon}.\end{aligned}$$ Therefore, eliminating the coefficient $C_1/C_2$ in Eqs. and , we finally obtain the following quantization condition: $$\begin{gathered}
\qquad\frac{\sqrt{2}}{\pi g^{2}}\,{\mathrm{e}}^{-1/2g^{2}}
\frac{(-1)^{E-1/2-\epsilon/2}\pm 1}{2}\biggl(\frac{2}{g^2}
\biggr)^{E-1/2-\epsilon/2}\Gamma\biggl(-E+\frac{1}{2}+\frac{\epsilon}{2}
\biggr)\\
\times\biggl(-\frac{1}{g^2}\biggr)^{E/2-1/2+\epsilon/2}\Gamma\biggl(
-\frac{E}{2}+\frac{1}{2}-\frac{\epsilon}{2}\biggr)=1\,.\qquad\end{gathered}$$ Strikingly, this condition is in complete agreement with the one obtained previously by us with the valley method, Eq. (5.19) in Ref. [@ST02], from which our formulas of the nonperturbative effect – have been derived. We note that the coincidence is achieved not only for the case $\epsilon=0$ we have examined in the preceding sections but also for all $\epsilon\neq 0$ where we cannot apply the usual instanton technique since the classical configuration now becomes a bounce solution.
Summary {#sec:summa}
=======
In this letter, we have extensively examined lowest three energies of the symmetric triple-well potential with non-equivalent vacua by comparing the two different sets of the formulas, the one is calculated by means of an instanton technique with the dilute gas approximation in Ref. [@AL04] and the other is by the valley method beyond this approximation scheme in Ref. [@ST02].
First, we compared numerically both the formulas with the exact values for the spectral splitting between the first and second excited states due to the purely nonperturbative effect. We have found that in contrast to the latter formulas the prediction of the former formulas get worse as the value of the parameter tends to the region where the semi-classical methods should work better. Thus, contrary to folklore that this kind of problem can be handled by the use of the dilute gas approximation, our careful comparison with the exact results indicates that the dilute gas approximation is insufficient to produce the correct asymptotic formulas even if we restrict ourselves to examining the low lying eigenvalues.
Furthermore, we examined for the ground state both the perturbative and nonperturbative contributions. We have found that the asymptotic values of the perturbative corrections calculated from the perturbation series deviate from the predicted values obtained from the instanton calculation with the dilute gas approximation but are in good agreement with the ones obtained from the valley method calculation.
We also checked the accuracy of the imaginary parts of the latter formulas by testing the prediction on the large order behavior of the perturbation series. We evaluated the perturbative coefficients up to the order 300 and compared them with the predicted asymptotic behaviors. We have confirmed the correctness of the prediction for both the ground and excited states.
Finally, we carried out the WKB calculation in the leading order of the coupling constant. The resulting WKB quantization condition is in complete agreement with the one obtained by means of the valley method in Ref. [@ST02], from which our formulas have been derived. In other words, the dilute gas approximation in the path integral formalism does not correspond to a proper semi-classical approximation. In this respect, we would like to recall the fact that discrepancy between the dilute gas approximation and the WKB approximation has been already reported in Ref. [@RT83] for the wave functions even in the case of the symmetric double-well potential.
To conclude, all the present analyses entirely support the valley method calculation and indicate the limitation of the dilute gas approximation in the present triple-well potential problem. Therefore, we would like to repeat the assertion in Ref. [@ST02] that the applicability of the dilute gas approximation would be quite limited.
We would like to thank the organizers of the international conference “New Frontiers in Quantum Mechanics” (July 5–8, 2004, Shizuoka University, Japan) where the present work started. This work was partially supported by the Grand-in-Aid for Scientific Research No.14740158 (M. S.) and by a Spanish Ministry of Education, Culture and Sports research fellowship (T. T.).
[^1]: Except for the cases $\epsilon=\pm(2{\mathcal{N}}+1)/3$ $({\mathcal{N}}=0,1,2,\ldots)$ where a part of the harmonic oscillator spectra of the central and side wells degenerates.
[^2]: In this case, the spectral splitting takes place between the first and second excited states and hence the parity odd state is lower than the parity even state in the spectrum, which is in contrast to the case of symmetric double-well potentials.
[^3]: In this article, a contribution is called *perturbative* (*nonperturbative*) if it can (cannot) be expressed as a (formal) power series in $g=\omega^{-1/2}$, respectively. See also Section \[sec:inter\].
[^4]: Here we note that all the perturbative coefficients of odd powers in $g$ for the spectrum vanish due to the parity symmetry of the original Hamiltonian .
[^5]: We note that the asymptotic property of the perturbation series in general has nothing to do with the Borel summability. We also note that in the quantum mechanical systems there exist no renormalon singularities which originate from the IR and UV divergences in the higher-dimensional quantum field theories.
|
---
abstract: 'The Polyakov loop extended Nambu and Jona-Lasinio model (PNJL model) in a mean field framework shows astonishingly good agreement with lattice QCD calculations which needs to be better understood. The present work reports on further developments concerning both Polyakov loop and mesonic fluctuations beyond mean field approximation. Corrections beyond mean field are of special interest for the thermal expectation values of the Polyakov loop $\braket{\Phi}$ and its conjugate $\braket{\Phi^*}$, which differ once the quark chemical potential is non-zero. Mesonic fluctuations are also considered with emphasis on the role of pionic modes.'
author:
- |
S. Rö[ß]{}ner$^{a}$, T. Hell$^{a}$, C. Ratti$^{b}$ and W. Weise$^{a}$\
\
[$^a$ Physik-Department, Technische Universität München, D-85747 Garching, Germany]{}\
[$^b$ Department of Physics & Astronomy, State University of New York,]{}\
[Stony Brook, NY 11794-3800, USA ]{}
date: 'October 13, 2008'
title: |
**The chiral and deconfinement crossover transitions:\
PNJL model beyond mean field [^1]**
---
Introduction
============
Exploring the thermodynamic properties of strongly interacting matter has become a central theme of high-energy nuclear physics in recent years. On the theoretical side, great progress has been achieved thanks to lattice calculations solving discretised QCD numerically. At the present stage a major part of the numerical expense at finite quark chemical potential is caused by the fermion sign problem. The three most promising ways to address this difficulty are multi-parameter re-weighting techniques [@Fodor:2001pe; @Fodor:2002km], analytic continuation from imaginary chemical potentials [@deForcrand:2002ci; @deForcrand:2003hx] and Taylor series expansion methods [@Allton:2002zi; @Allton:2003vx; @Ejiri:2006ft; @Allton:2005gk; @Boyd:1996bx; @Kaczmarek:2002mc; @Boyd:1995cw].
It is an important task of effective field theories and models to reveal principal mechanisms and their functioning behind the otherwise hidden mechanisms of lattice QCD. Alongside with other approaches [@Schaefer:2007pw; @Barducci:1989euBarducci:1989wi; @Berges:1998rc] the Nambu and Jona-Lasinio (NJL) model [@Ratti:2005jh; @Sasaki:2006ww; @Ghosh:2006qh; @Mukherjee:2006hq; @Zhang:2006gu; @Abuki:2008nm; @Fukushima:2002ew_Fukushima:2003fm; @Fukushima:2003fw_Hatta:2003ga] is an approach that successfully describes spontaneous chiral symmetry breaking. We use the NJL model with $N_{\mathrm{f}} =2 $ quark flavours as one of our starting points. In the NJL model gluonic degrees of freedom are “integrated out”. The role of the gluons is assumed to be modelled in part by a local effective quark colour current interaction. From this effective interaction a Fierz transformation generates various quark-antiquark and diquark coupling terms. By integrating out the gluons the local $\mathrm{SU}(3)_\mathrm{c}$ gauge symmetry is lost. As a consequence the NJL model, equipped only with a global $\mathrm{SU}(3)_\mathrm{c}$ symmetry, does not feature confinement.
To bring aspects of confinement back into the model an additional homogeneous temporal background field with standard $\mathrm{SU}(3)_\mathrm{c}$ gauge invariant coupling to the quarks is introduced [@Fukushima:2002ew_Fukushima:2003fm; @Fukushima:2003fw_Hatta:2003ga], implementing the Polyakov loop. This generalised NJL model, with Polyakov loop dynamics incorporated, is called PNJL model. In the limit of static quarks (i.e. in pure gauge QCD) the Polyakov loop serves as an order parameter for confinement. In this limit the $\mathrm{Z}(3)$ centre-symmetry of the $\mathrm{SU}(3)_\mathrm{c}$ gauge group is unbroken, and the deconfinement transition is connected with the spontaneous breakdown of this symmetry. In the presence of dynamical quarks the $\mathrm{Z}(3)$ centre-symmetry is broken explicitly, such that the deconfinement transition is no longer a phase transition in the strict sense. Nevertheless, the Polyakov loop shows a rapid crossover near the deconfinement transition, still permitting to use the Polyakov loop as a measure for deconfinement. The confining gluon dynamics that was lost in the NJL model is now re-introduced via an effective potential. This potential is part of a Ginzburg-Landau model for confinement in the static quark limit. The information necessary in order to specify the effective potential is extracted from pure glue lattice QCD calculations [@Boyd:1995cw].
In Sec. \[sec:pnjlmodel\] we review the PNJL model [@Ratti:2005jh; @Sasaki:2006ww; @Ghosh:2006qh; @Mukherjee:2006hq; @Zhang:2006gu] at the level of the mean field approximation. When implementing the Polyakov loop extension to the NJL model, it is important to pay special attention to the fermion sign problem. As the Polyakov loop is coupled to the NJL model in analogy to QCD via minimal substitution the fermion sign problem in QCD and in the PNJL model appears on equal footing. Issues arising from the fermion sign problem in the PNJL model are discussed in Sec. \[sec:corr\]. In the present paper these issues are addressed more explicitly than in previous work [@Ratti:2005jh; @Roessner:2006xn; @Ratti:2006wg]. The method developed in this improved treatment introduces a systematic expansion around a leading order (mean field) approximation defined such that all physical quantities are real in this limit. It is demonstrated that the dynamics beyond mean field can be treated perturbatively. A detailed derivation of these perturbative corrections is presented in the appendix. The perturbative method is applied in Sec. \[sec:numerics\] to the PNJL model to investigate the effects of the complex phase of the action on different quantities. Here the expectation values of the Polyakov loop and its conjugate are of special interest. In the present analysis, the split of the expectation values of the Polyakov loop and its conjugate at non-zero chemical potential arises once fluctuations of the fields are taken into account. Quantities like susceptibilities, in which mean field contributions partially or completely cancel, are sensitive to the corrections beyond mean field, as we shall discuss. Finally in Sec. \[sec:pions\] further corrections beyond Hartree approximation are estimated, generated by propagating mesonic (quark-antiquark) modes. The lightest meson mode, the pseudoscalar pion with its approximate Nambu-Goldstone character, is the leading correction in this sector. Sec. \[sec:conclusio\] presents our conclusions and an outlook.
The PNJL model {#sec:pnjlmodel}
==============
The two-flavour PNJL model including diquark degrees of freedom [@Roessner:2006xn] is derived from the Euclidean action $${\cal S}_E(\psi, \psi^\dagger, \phi)= \int _0^{\beta=1/T} {\mathrm{d}}\tau\int {\mathrm{d}}^3x \left[\psi^\dagger\,\partial_\tau\,\psi + {\cal H}(\psi, \psi^\dagger, \phi)\right] + \delta{\cal S}_E(\phi,T)
\label{eqn:action}$$ with the fermionic Hamiltonian density [^2]: $${\cal H} = -{i}\psi^\dagger\,(\vec{\alpha}\cdot \vec{\nabla}+\gamma_4\,m_0 -A_4)\,\psi + {\cal V}(\psi, \psi^\dagger)~,
\label{eqn:hamiltonian}$$ where $\psi$ is the $N_\mathrm{f}=2$ doublet quark field and $m_0 = \mathrm{diag}(m_u,m_d)$ is the quark mass matrix. The quarks move in a background colour gauge field $A_4 = {i}A_0$, where $A_0 = \delta_{\mu 0}\,g{\cal A}^\mu_a\,t^a$ with the $\mathrm{SU}(3)_\mathrm{c}$ gauge fields ${\cal A}^\mu_a$ and the generators $t^a = \lambda^a/2$. The matrix valued, constant field $A_4$ relates to the (traced) Polyakov loop as follows: $$\Phi=\frac{1}{N_\mathrm{c}}{\mathrm{tr}}_\mathrm{c}\,L \qquad\text{with }L= \exp\left(i\int_{0}^{\beta}
{\mathrm{d}}\tau A_4\right)\quad\text{and }\beta = \frac{1}{T}~. \label{eqn:polyakovloop}$$ In a convenient gauge (the so-called Polyakov gauge), the matrix $L$ is given a diagonal representation $$\label{eqn:loopparametrization}
L = \exp \left[ {i}\,\left(\phi_3\,\lambda_3 + \phi_8\,\lambda_8 \right) \right]~.$$ The dimensionless effective fields $\phi_3$ and $\phi_8$ introduced here are identified with the Euclidean gauge fields in temporal direction, divided by temperature: $\phi_3={A_4^{(3)} }/{T}$ and $\phi_8={A_4^{(8)} }/{T}$. These two fields are a parametrisation of the diagonal elements of $\mathrm{SU}(3)_{\mathrm{c}}$. As such the “angles” $\phi_3$ and $\phi_8$ necessarily have to be real quantities in order to sustain the unitarity of the group. An alternative parametrisation of the diagonal elements of $\mathrm{SU}(3)_{\mathrm{c}}$ is given by the Polyakov loop, $\Phi = \frac13 {\mathrm{tr}}_\mathrm{c}\,L$, and its conjugate, $\Phi^*= \frac13 {\mathrm{tr}}_\mathrm{c}\,L^\dagger$.
The piece $\delta \mathcal{S}_\mathrm{E} = \frac{V}{T}\,\mathcal{ U}$ of the action (\[eqn:action\]) carries information about the gluon dynamics. The potential $\mathcal{ U}$ effectively models the confinement-deconfinement transition and the region up to temperatures of roughly $T \lesssim 2\,T_{\mathrm{c}}$ in quarkless, pure gauge QCD on the mean field Ginzburg-Landau level. At temperatures very far above the transition a description of the thermodynamics with just the two order parameters $\Phi$ and $\Phi^*$ is not appropriate as transverse gluons will become important. Transverse gluon degrees of freedom cannot be described by Polyakov loops.
The Polyakov loop is an order parameter for confinement in $\mathrm{SU}(3)$ gauge theory. In the confined low temperature phase the expectation value of the Polyakov loop vanishes, $\braket{\Phi} = 0$, while $\braket{\Phi} \neq 0$ implies deconfinement. Let $T_0$ be the critical temperature separating the two phases. As previously mentioned the symmetry which is restored at $T<T_0$ and broken above $T_0$ is the $\mathrm{Z}(3)$ centre-symmetry of $\mathrm{SU}(3)$[^3].
Therefore, the Landau effective potential describing the dynamics, the Polyakov loop potential $\mathcal{U}(\Phi, T)$, has to be $\mathrm{Z}(3)$-symmetric in $\Phi$. The basic building blocks for such a potential are $\Phi^*\Phi$, $\Phi^3$ and ${\Phi^*}^3$ terms. The potential used here differs from the simplest ansatz generating a first order phase transition as it is implemented in [@Ratti:2005jh]. Instead we use the ansatz given in [@Ratti:2006wg; @Roessner:2006xn] motivated by the $\mathrm{SU}(3)$ Haar measure: $$\frac{\mathcal{U}(\Phi,\,\Phi^*,\,T )}{T^4}=-\frac{1}{2}a(T)\,\Phi^*\Phi
+ b(T)\,\ln\left[1-6\,\Phi^*\Phi+4\left({\Phi^*}^3+\Phi^3\right)
-3\left(\Phi^*\Phi\right)^2\right]~,
\label{eqn:looppot}$$ where the temperature dependent prefactors are given by $$\begin{aligned}
a(T) & = a_0+a_1\left(\frac{T_0}{T}\right)
+a_2\left(\frac{T_0}{T}\right)^2 &\text{and}&& b(T) &=b_3\left(\frac{T_0}{T}
\right)^3.
\label{eqn:loopparam}\end{aligned}$$ The logarithmic divergence near $\Phi^*,\,\Phi\to 1$ properly constrains the Polyakov loop to values attainable by the normalised trace of an element of $\mathrm{SU}(3)$. The parameters of $\mathcal{U}(\Phi,\,\Phi^*,\,T )$ are chosen such that the critical temperature of the first order transition is indeed equal to $T_0$ (fixed at $270{\,\mathrm{MeV}}$ [@Karsch:2000kv]) and that $\Phi^*\,,\Phi\to 1$ as $T\to\infty$.
The numerical values using these constraints are taken as given in Refs. [@Ratti:2006wg; @Roessner:2006xn] $$\begin{aligned}
a_0 &= 3.51\;, &a_1 &= -2.47\;, &a_2 &= 15.2\;, &b_3 &= -1.75\;. \nonumber\end{aligned}$$ The resulting uncertainties are estimated to be about $6\,\%$ for $a_1$, less than $3\,\%$ for $a_2$ and $2\,\%$ for $b_3$. The value $a_0 = \frac{16\pi^2}{45}$ chosen here reproduces the Stefan-Boltzmann limit. This is not mandatory, of course, since the high-temperature limit is governed by (transverse) gluonic degrees of freedom not covered by the Polyakov loop which represents the longitudinal gauge field. Alternative parametrisations of $\mathcal{U}$ are possible, such as the two-parameter form guided by the strong-coupling approach [@Fukushima:2003fw_Hatta:2003ga], which has a different high temperature limit. In the present context these differences are not crucial as we systematically restrict ourselves to temperatures close to the transition region, $T\lesssim 2T_\mathrm{c}$, where different forms of $\mathcal{U}$ give remarkably similar results as pointed out in Ref. [@Fukushima:2008wg].
In Fig. \[fig:looppotential\] we plot the Polyakov loop potential using the parametrisation given in Refs. [@Ratti:2006wg; @Roessner:2006xn] at $T = T_0 = 0.27{\,\mathrm{GeV}}$. This illustrates the $\mathrm{Z}(3)$ symmetry. The single minimum at $T<T_0$ becomes degenerate with three minima at $T=T_0$. Above $T_0$ only these three minima survive. Of course upon spontaneous breakdown of the $\mathrm{Z}(3)$ centre-symmetry, the three minima and the $\mathrm{Z}(3)$ centre-symmetry of the potential remain intact even though the vacuum expectation value does no longer show the symmetry of the potential.
![The Polyakov loop potential ${\mathcal{U}(\Phi,\,\Phi^*,\,T )}/{T^4}$ plotted in the complex plane of $\Phi$ at $T = T_0 = 0.27{\,\mathrm{GeV}}$. \[fig:looppotential\]](./PolyakovCritUpright.eps){width=".65\textwidth"}
The NJL interaction term $\mathcal{V}$ in Eq. (\[eqn:hamiltonian\]) includes chiral $\mathrm{SU}(2)\times \mathrm{SU}(2)$ invariant four-point couplings of the quarks acting in pseudoscalar-isovector/scalar-isoscalar quark-antiquark and scalar diquark channels: $$\begin{aligned}
\mathcal{V}= -\frac
{G}{2}\left[\left(\bar{\psi}\psi\right)^2+\left(\bar{\psi}\,i\gamma_5
\vec{\tau}\,\psi
\right)^2\right]
- \frac{H}{2}\left[\left(\bar{\psi}\,{\cal C}\gamma_5\tau_2\lambda_2
\,\bar{\psi}^{T}\right)\left(\psi^{T}\gamma_5\tau_2\lambda_2 {\cal C}
\,\psi\right)\right]~,
\label{eqn:potential}\end{aligned}$$ where $\mathcal{C}$ is the charge conjugation operator. These interaction terms in Eq. (\[eqn:potential\]) are obtained from a local colour current-current interaction between quarks, $$\begin{aligned}
\mathcal{L}_{\mathrm{int}} = - G_\mathrm{c}(\bar{\psi}\gamma_\mu t^a\psi)(\bar{\psi}\gamma^\mu t^a\psi)\;,
\nonumber\end{aligned}$$ by a Fierz transformation which relates the coupling strengths $G$ and $H$ as $ G = \frac{4}{3}H$ which we choose not to alter[^4].
The NJL model with two quark flavours is usually modelled with three parameters, a current quark mass $m_{u,d}$, a local four quark coupling strength $G$ and a three-momentum cutoff $\Lambda$. The parameters used here are the ones used in [@Ratti:2005jh; @Ratti:2006wg; @Roessner:2006xn]: $$\begin{aligned}
m_{u,d} &= 5.5{\,\mathrm{MeV}}\;,&G &= \frac{4}{3}H = 10.1{\,\mathrm{GeV}}^{-2}\;,& \Lambda &= 0.65{\,\mathrm{GeV}}\;,
\nonumber\end{aligned}$$ fixed to reproduce the pion mass and decay constant in vacuum and the chiral condensate as $m_\pi =$ 139.3 MeV, $f_\pi =$ 92.3 MeV and $\langle\bar{\psi}_u\psi_u\rangle = - (251$ MeV)$^3$.
To evaluate the thermodynamic properties of the model the quark degrees of freedom are integrated out. New auxiliary fields are introduced by bosonisation, absorbing quark-antiquark and quark-quark (antiquark-antiquark) correlations. These are a scalar-pseudoscalar field $(\sigma,\,\vec{\pi}\,)$ and a diquark (antidiquark) field $\Delta$ ($\Delta^*$). The resulting thermodynamic potential then reads $$\Omega_0 = \frac{T}{V} \, \mathcal{S}_{\mathrm{bos}}= \mathcal{U}\left(\Phi,\,\Phi^*,\,T\right)-\frac{T}{2}\sum_n\int\frac{{\mathrm{d}}^3p}
{\left(2\pi\right)^3}{\mathrm{Tr}}\ln\left[\beta \tilde{S}^{-1}\left(i\omega_n,\vec{p}\,\right)
\right]+\frac{\sigma^2}{2G}+\frac{\Delta^*\Delta}{2H}~,
\label{eqn:omegageneral}$$ where the Matsubara sum runs over $\omega_n=(2n+1)\pi\,T$ reproducing antiperiodic boundary conditions in the Euclidean time direction. The inverse Nambu-Gor’kov propagator $\tilde{S}^{-1}$ in Eq. (\[eqn:omegageneral\]) is defined by $$\tilde{S}^{-1}\left(i\omega_n,\vec{p}\,\right)=\left({{\begin{array}{ccc}
i\gamma_0\,\omega_n-\vec{\gamma}\cdot\vec{p}-
m+\gamma_0\left(\mu - {i}A_4 \right)& \Delta\gamma_5\tau_2\lambda_2\\
-\Delta^*\gamma_5\tau_2\lambda_2&
i\gamma_0\, \omega_n-\vec{\gamma}\cdot\vec{p}-
m-\gamma_0\left(\mu - {i}A_4 \right)
\end{array}}}\right).
\label{eqn:ngprop}$$ The mass of the quark-quasiparticles is given as in the standard NJL model by the gap equation $$m=m_0-\langle\sigma\rangle=m_0-G\langle\bar{\psi}\psi\rangle\;.
\label{eqn:mass}$$
The Matsubara sum is evaluated analytically. The quasiparticle energies emerging in this procedure are related to the solutions of $\det\big[\tilde{S}^{-1}(p_0)\big] = 0$. The bosonised action then reads $$\begin{gathered}
\Omega_0 = \frac{T}{V} \, \mathcal{S}_{\mathrm{bos}} = \mathcal{U}\left(\Phi,\,\Phi^*,\,T\right)+\frac{\sigma^2}{2G}+
\frac{\Delta^*\Delta}{2H}\\
- 2N_f\int\frac{{\mathrm{d}}^3p}{\left(2\pi\right)^3}\sum_j \left\{
T\ln\left[1+{e}^{-E_j/T}\right]+\frac12 \Delta E_j
\right\}\;,
\label{eqn:bosaction}\end{gathered}$$ with six distinct quasiparticle energies $$\begin{aligned}
E_{1,2}&=\varepsilon(\vec{p}\,) \pm \tilde{\mu}_b~,
\nonumber\\
E_{3,4}&=\sqrt{(\varepsilon(\vec{p}\,)+\tilde{\mu}_r)^2+|\Delta|^2}\pm i\,T\,\phi_3~,
\nonumber\\
E_{5,6}&=\sqrt{(\varepsilon(\vec{p}\,)-\tilde{\mu}_r)^2+|\Delta|^2}\pm i\,T\,\phi_3~,\end{aligned}$$ where $\varepsilon(\vec{p}\,) = \sqrt{\vec{p\,}^2+m^2}$. Additionally we have introduced $$\begin{aligned}
\tilde{\mu}_b&=\mu+2i\,T\,\frac{\phi_8}{\sqrt{3}}\;, & \tilde{\mu}_r&=\mu-i\,T\,\frac{\phi_8}{\sqrt{3}}\;.\end{aligned}$$ The energy difference $\Delta E_j$ is defined as the difference of the quasiparticle energy and the energy of a free fermion, $\varepsilon_0 = \sqrt{\vec{p\,}^2+m_0^2}$: $\Delta E_j = E_j - \varepsilon_0 \pm \mu$. The form of the bosonised action, Eq. (\[eqn:bosaction\]), does not allow to factor out the Polyakov loop fields $\Phi$ and $\Phi^*$, as it was done in Ref. [@Ratti:2005jh]. Instead we keep the form of Eq. (\[eqn:bosaction\]) using $\phi_3$ and $\phi_8$ with $\phi_3,\,\phi_8\in \mathds{R}$.[^5]
The introduction of the Polyakov loop outlined above formally leads to a complex valued action as soon as $\mu\neq0$. This phenomenon is usually called fermion sign problem. Due to the connection of $\phi_3$ and $\phi_8$ to the QCD colour gauge group $\mathrm{SU}(3)_{\mathrm{c}}$, we must require $\phi_3$ and $\phi_8$ to be real fields at all times. The mean field approximation, $\Omega_{\mathrm{MF}}$, of the thermodynamic potential $\Omega$ must be introduced such that it satisfies this constraint imposed by the gauge group. Identification up to a constant of $\Omega_{\mathrm{MF}}$ with the (real) pressure $p$ in this approximation then requires that non-Hermitian structures of the inverse quasiparticle quark propagator do not contribute. One way to establish such a lowest order approximation is to use the real part of the thermodynamic potential in the mean field equations.
The necessary condition for the minimisation of the effective action in a standard situation is, in general, $$\frac{\partial\,\Omega}{\partial\theta_i}=0 \;,
\label{eqn:saddle}$$ where $\theta_i$ stands for the fields representing the relevant degrees of freedom (in our case: $\theta = \left( \sigma,\,\Delta,\,\phi_3 ,\,\phi_8 \right)$). In order to always comply with $\phi_3,\,\phi_8\in \mathds{R}$ we define the mean field thermodynamic potential, with $\Omega_0$ of Eq. (\[eqn:bosaction\]), by $$\label{eqn:omegamf}
\Omega_{\mathrm{MF}} ={\operatorname{Re}\!}\left[ \,\Omega_0\right] = {\operatorname{Re}\!}\left[\frac{T}{V}\,\mathcal{S}_{\mathrm{bos}}\right]~.$$ The mean field equations then read $$\frac{\partial\,\Omega_{\mathrm{MF}}}{\partial \left( \sigma, \Delta, \phi_3, \phi_8\right)}=\frac{\partial\,{\operatorname{Re}\!}\left[ \,\Omega_{0}\right]}{\partial \left( \sigma, \Delta, \phi_3, \phi_8\right)}=0.
\label{eqn:mfeqn}$$ The hereby neglected imaginary part of this derivative, $\frac{\partial\,{\operatorname{Im}}[\,\Omega_{0}]}{\partial \left( \sigma, \Delta, \phi_3, \phi_8\right)}$, will be taken into account by writing down a series in powers of this residual gradient. In addition it is also possible to correct for deviations of the potential from a gaussian shape which is assumed for the mean field approximation. As explained in the appendix it is most convenient to consider both types of corrections simultaneously using Feynman graphs to construct all possible terms.
A variety of PNJL model results (equations of state, phase diagrams, susceptibilities) have been obtained in previous calculations [@Ratti:2006wg; @Roessner:2006xn; @Ratti:2007jf] based on the mean field equations (\[eqn:mfeqn\]). At this point it is instructive to examine how chiral and Polyakov loop dynamics cooperate to produce crossover transitions (at zero chemical potential) which end up in a narrow overlapping range of temperatures (see Fig. \[fig:loopwithquarks\]). In isolation, the pure gauge Polyakov loop sector and the NJL sector in the chiral limit show first (second) order phase transitions with critical temperatures far separated, as demonstrated by the dashed (dash-double dotted) lines in Fig. \[fig:loopwithquarks\]. When entangled in the PNJL model, these transitions (with non-zero quark masses) move together to form a joint crossover pattern.
![Chiral condensate normalised to its value at temperature $T=0$ (dash-double-dotted line) in the NJL model with massless quarks, and Polyakov loop $\braket{\Phi}$ in the pure gauge model (dashed line). The PNJL model (with non-zero quark masses) shows dynamical entanglement of the chiral (solid line) and Polyakov loop (dash-dotted line) crossover transitions. For comparison lattice data for the Polyakov loop in pure gauge and full QCD (including quarks) are also shown [@Kaczmarek:2005ui] \[fig:loopwithquarks\].](./LoopWithQuarks.eps){width=".65\textwidth"}
Fluctuations and corrections beyond mean field {#sec:corr}
==============================================
The discussion of fluctuation corrections to the mean field approximation has two parts:
- corrections arising from the imaginary part of $\Omega_0$ and involving the Polyakov loop $\Phi$ and its complex conjugate $\Phi^*$;
- corrections from dynamical fluctuations involving propagating meson fields, with emphasis on the pion.
The first item is the primary topic of this present section. The second item will be relegated to a separate Section \[sec:pions\].
The mean field equations (\[eqn:mfeqn\]) establish a leading order approximation satisfying the reality constraints on the fields $\phi_3$ and $\phi_8$. Eq. (\[eqn:mfeqn\]) is at the same time the necessary condition for the maximisation of the modulus $\left\vert {e}^{-\mathcal{S}_{\mathrm{E}}} \right\vert$ of the thermodynamic weight in the path integral[^6]:
The direct connection $\Phi = \Phi(\phi_3,\,\phi_8)$ of the two parametrisations, ($\phi_3$, $\phi_8$) on the one hand and ($\Phi$, $\Phi^*$) on the other hand, is lost once we step away from mean field and calculate thermodynamic expectation values: $\braket{\Phi}\neq\Phi(\braket{\phi_3},\,\braket{\phi_8})$.[^7] This observation is crucial when comparing the present method of approximation to schemes in previous publications [@Ratti:2005jh; @Sasaki:2006ww; @Ghosh:2006qh; @Mukherjee:2006hq; @Zhang:2006gu]. In these publications the fields $\phi_3$, $\phi_8$ have been replaced by $\Phi$, $\Phi^*$ before doing mean field approximation. This implies that $\braket{\Phi}$ and $\braket{\Phi^*}$ (and *not* $\Phi$, $\Phi^*$) are treated as independent mean field degrees of freedom. The minimisation of $\Omega_0$ is then performed requiring that $\braket{\Phi}$ and $\braket{\Phi^*}$ are real quantities. In such approximation schemes it is therefore not possible to find a way back to the (real) quantities $\phi_3$, $\phi_8$: $\braket{\Phi}$ and $\braket{\Phi^*}$ already comprise fluctuations of $\phi_3,\, \phi_8\in\mathds{R}$. In other words, the definition of the lowest order approximation (which is usually referred to as mean field approximation) is different in Refs. [@Ratti:2005jh; @Sasaki:2006ww; @Ghosh:2006qh; @Mukherjee:2006hq; @Zhang:2006gu] and this work. The definition of the lowest order (mean field) approximation in this work allows to strictly separate contributions originating in constant and fluctuating parts of the fields.
The combination of the constraints $\Omega\in\mathds{R}$ and $\phi_3,\,\phi_8\in\mathds{R}$ allow only certain limited configurations of $\phi_3$ and $\phi_8$. In mean field approximation the condition $\phi_3,\,\phi_8\in\mathds{R}$ implies that, due to Eq. (\[eqn:loopparametrization\]), $\Phi$ and $\Phi^*$ are the complex conjugates and we find $\braket{\Phi}_{\mathrm{MF}} = \Phi_{\mathrm{MF}}$, $\braket{\Phi^*}_{\mathrm{MF}} = \Phi^*_{\mathrm{MF}}$. At $\mu=0$ the Polyakov loop $\Phi$ and its complex conjugate $\Phi^*$ are treated equally due to charge conjugation invariance. It follows that $\Phi_{\mathrm{MF}} = \Phi^*_{\mathrm{MF}} \in \mathds{R}$ in mean field approximation, fixing $\phi_8=0$. The Polyakov loop effective potential $\mathcal{U} = \mathcal{U}(T,\,\Phi,\,\Phi^*)$ in its parametrisation (\[eqn:looppot\]) is minimal for $\Phi = \Phi^*$ at fixed $\vert \Phi \vert$. We find that the Polyakov loop potential $\mathcal{U}$ is always strong enough to keep $\Phi = \Phi^*$ or, equivalently, $\phi_8=0$. Not all parametrisations of $\mathcal{U}$ will maintain this solution. If the curvature of the potential $\mathcal{U}$ is not strong enough, the solution $\phi_8=0$ becomes instable, and $\phi_8=0$ is the position of a local maximum of the potential. In Ref. [@Sasaki:2006ww] a symptom of this fact has been described: the susceptibility associated with ${\operatorname{Re}}\Phi$ may become negative. The potential used in the present work does not show such deficiencies.
After these preparatory remarks we proceed to develop a calculational scheme which systematically treats corrections to the mean field approximation, Eqs. (\[eqn:omegamf\]) and (\[eqn:mfeqn\]). The basic idea is, as usual, to expand the thermodynamic potential $\Omega$ around its mean field limit $\Omega_{\mathrm{MF}}$. Technical details of the derivation are summarized in the appendix. The result including next-to-leading order is $$\label{eqn:numOmega}
\Omega = \Omega_{\mathrm{MF}} - \left. \frac12 \left( \frac{\partial \Omega_{0}}{\partial \theta} \right)^T \cdot \left[ \frac{\partial^2 \Omega_{0}}{\partial \theta^2} \right]^{-1} \cdot \frac{\partial \Omega_{0}}{\partial \theta} \right\vert_{\theta= \theta_{\mathrm{MF}}}~,$$ starting from the (complex) $\Omega_0$ of Eq. (\[eqn:bosaction\]), with $\Omega_{\mathrm{MF}} $ defined by Eq. (\[eqn:omegamf\]), and with $\theta_{\mathrm{MF}}$ determined by Eq. (\[eqn:mfeqn\]). The gradients $\partial \Omega_0 / \partial \theta$ are understood with the set of field variables $\theta = (\theta_i) = ( \sigma, \Delta, \phi_3, \phi_8)$ arranged in vector form, and $\partial^2 \Omega_0 / \partial \theta^2$ stands for the matrix $(\partial^2 \Omega_0 / \partial \theta_i \partial \theta_j)$. The correction term in (\[eqn:numOmega\]) is taken using the mean field configuration, $ \theta = \theta_{\mathrm{MF}}$. Note that this term takes care of the contributions from ${\operatorname{Im}}\Omega_0$ in such a way that $\Omega$ remains a real quantity.
The thermal expectation value $\braket{f}$ of a physical quantity $f$ is calculated according to $$\label{eqn:numfield}
\braket{f} = f(\theta_{\mathrm{MF}}) - \left. \left( \frac{\partial \Omega_{0}}{\partial \theta} \right)^T\cdot \left[ \frac{\partial^2 \Omega_{0}}{\partial \theta^2} \right]^{-1} \cdot \frac{\partial f}{\partial \theta} \right\vert_{\theta= \theta_{\mathrm{MF}}} \;.$$ Applications will now be given, in particular, for the Polyakov loop and for susceptibilities.
Results {#sec:numerics}
=======
The numerical calculations presented in this section are performed in the thermodynamic limit, i.e. in leading order of the $\frac{T}{V}$-expansion, and up to first order in the $\delta$-expansion, as explained in detail in the appendix. The thermodynamic potential is determined by Eq. (\[eqn:numOmega\]). Thermal expectation values are computed using Eq. (\[eqn:numfield\]).
The Polyakov loop and its conjugate
------------------------------------
With the mean field definition (\[eqn:omegamf\]) the Polyakov loop expectation values $\braket{\Phi}$ and $\braket{\Phi^*}$ turn out to be equal in this limit, given the reality constraint on $\Omega_{\mathrm{MF}}$. It is the corrections from ${\operatorname{Im}}\Omega_0$ induced by the temporal gauge fields which cause the splitting of $\braket{\Phi}$ and $\braket{\Phi^*}$.
The difference $\braket{\Phi^*} - \braket{\Phi}$ vanishes at zero quark chemical potential $\mu$ and has the same sign as $\mu$, in agreement with results of Ref. [@Dumitru:2005ng]. As can be seen from Fig. \[fig:loop\] the difference $\braket{\Phi^*} - \braket{\Phi}$ is pronounced around the phase transitions. In the upper left panel of Fig. \[fig:loop\] the influence of the first order phase transition separating the chiral and the diquark phase at low temperature can be seen as a jump in both $\braket{\Phi}$ and $\braket{\Phi^*}$. The second order phase transition separating the diquark regime from the high temperature quark-gluon phase can be identified as a kink in the lower right panel of Fig. \[fig:loop\].
![\[fig:loop\] Examples of thermal expectation values of the Polyakov loop $\braket{\Phi}$ and its conjugate $\braket{\Phi^*}$. In the upper row $\braket{\Phi}$ and $\braket{\Phi^*}$ are plotted as functions of the chemical potential $\mu$ at constant temperature $T$. Below $\braket{\Phi}$ and $\braket{\Phi^*}$ are plotted as functions of temperature $T$ at constant chemical potential $\mu$.](./LoopLowTemp.eps){width="\textwidth"}
![\[fig:loop\] Examples of thermal expectation values of the Polyakov loop $\braket{\Phi}$ and its conjugate $\braket{\Phi^*}$. In the upper row $\braket{\Phi}$ and $\braket{\Phi^*}$ are plotted as functions of the chemical potential $\mu$ at constant temperature $T$. Below $\braket{\Phi}$ and $\braket{\Phi^*}$ are plotted as functions of temperature $T$ at constant chemical potential $\mu$.](./LoopHighTemp.eps){width="\textwidth"}
\
![\[fig:loop\] Examples of thermal expectation values of the Polyakov loop $\braket{\Phi}$ and its conjugate $\braket{\Phi^*}$. In the upper row $\braket{\Phi}$ and $\braket{\Phi^*}$ are plotted as functions of the chemical potential $\mu$ at constant temperature $T$. Below $\braket{\Phi}$ and $\braket{\Phi^*}$ are plotted as functions of temperature $T$ at constant chemical potential $\mu$.](./LoopLowMu.eps){width="\textwidth"}
![\[fig:loop\] Examples of thermal expectation values of the Polyakov loop $\braket{\Phi}$ and its conjugate $\braket{\Phi^*}$. In the upper row $\braket{\Phi}$ and $\braket{\Phi^*}$ are plotted as functions of the chemical potential $\mu$ at constant temperature $T$. Below $\braket{\Phi}$ and $\braket{\Phi^*}$ are plotted as functions of temperature $T$ at constant chemical potential $\mu$.](./LoopHighMu.eps){width="\textwidth"}
Susceptibilities and phase diagram
----------------------------------
A susceptibility $\chi_g$ involving a quantity $g$ is defined by $$\label{eqn:genericsusc}
\chi_g^2 = V\,\braket{(g-\braket{g})^2} = V\left(\braket{g^2}-\braket{g}^2\right)~.$$ Susceptibilities of special interest in the present context are the ones related to the dynamical quark mass, $m=m_0-\sigma$, and to the Polyakov loop. They are expressed in terms of the inverse matrix of the second derivatives of the full thermodynamic potential $\Omega$: $$\begin{aligned}
\label{eqn:chi_m}
\chi_M^2 &= V\left(\braket{m^2}-\braket{m}^2\right)= \;T\,\left[\frac{\partial^2\Omega}{\partial \theta_i\partial \theta_j}\right]^{-1}_{m,\,m} \\
\label{eqn:chi_phi}
\qquad\qquad\chi_{\Phi}^2 &= V\left(\braket{{\Phi}^2}-\braket{{\Phi}}^2\right) \;=\; T\,\left[\frac{\partial^2\Omega_{\mathrm{full}}}{\partial \theta_i\partial \theta_j}\right]^{-1}_{\Phi,\,\Phi}\qquad\qquad\\
\label{eqn:chi_rephi}
\qquad\qquad\chi_{{\operatorname{Re}}\Phi}^2 &= \; \frac{T}{4}\,\left[\frac{\partial^2\Omega}{\partial \theta_i\partial \theta_j}\right]^{-1}_{\Phi,\,\Phi} + \frac{T}{2}\,\left[\frac{\partial^2\Omega_{\mathrm{full}}}{\partial \theta_i\partial \theta_j}\right]^{-1}_{\Phi,\,\Phi^*} + \frac{T}{4}\,\left[\frac{\partial^2\Omega}{\partial \theta_i\partial \theta_j}\right]^{-1}_{\Phi^*,\,\Phi^*} ~.\end{aligned}$$ These susceptibilities are calculated using the graph rules given in Tab. \[tab:chisqr\] of the appendix which lead to the following explicit form: $$\begin{gathered}
\label{eqn:gausssus}
\chi^2_g = T \left. \left( \frac{\partial g}{\partial \theta} \right)^T \cdot \left[ \frac{\partial^2 \Omega_{0}}{\partial \theta^2} \right]^{-1} \cdot \frac{\partial g}{\partial \theta} \right\vert_{\theta= \theta_{\mathrm{MF}}} \\
-2\,T \left.\left( \frac{\partial g}{\partial \theta} \right)^T \cdot \left[ \frac{\partial^2 \Omega_{0}}{\partial \theta^2} \right]^{-1} \cdot
\frac{\partial^2 g}{\partial \theta^2} \cdot
\left[ \frac{\partial^2 \Omega_{0}}{\partial \theta^2} \right]^{-1}\cdot
\frac{\partial \Omega_{0}}{\partial \theta} \right\vert_{\theta= \theta_{\mathrm{MF}}} \qquad\qquad\rule{0pt}{0pt} \\
+ T\,\sum_{i,j,k} \frac{\partial^3 \Omega_{0}}{\partial \theta_i\partial \theta_j\partial \theta_k}
\left(\left[ \frac{\partial^2 \Omega_{0}}{\partial \theta^2} \right]^{-1}\cdot
\frac{\partial g}{\partial \theta} \right)_i
\left(\left[ \frac{\partial^2 \Omega_{0}}{\partial \theta^2} \right]^{-1}\cdot
\frac{\partial g}{\partial \theta} \right)_j \\
\left. \times \left(\left[ \frac{\partial^2 \Omega_{0}}{\partial \theta^2} \right]^{-1}\cdot
\frac{\partial \Omega_{0}}{\partial \theta} \right)_k \right\vert_{\theta= \theta_{\mathrm{MF}}} ~.$$ Here $g$ stands for $m$ or $\Phi$, respectively. The first term in Eq. (\[eqn:gausssus\]) is the susceptibility of the gaussian theory whereas the other terms are interpreted as corrections.
The susceptibilities $\chi_M$ and $\chi_\Phi$ serve as indicators for boundaries between phases when drawing a phase diagram in the plane of temperature and chemical potential. For smooth crossover transitions, such boundaries are not rigorously defined. Several criteria can be used to determine a transition line separating the region of spontaneously broken chiral symmetry from the quark-gluon phase. We use here the maxima of the chiral susceptibilities $\chi_M$ and of the Polyakov loop susceptibility $\chi_{{\operatorname{Re}}\Phi}$ in comparison with the maximal slopes ${\mathrm{d}}m/{\mathrm{d}}T$ and ${\mathrm{d}}\braket{\Phi^*+\Phi} / {\mathrm{d}}T$ of the corresponding quantities which act, respectively, as order parameters in the limiting situations of exact chiral $ \mathrm{SU}(2) \times \mathrm{SU}(2) $ symmetry or $\mathrm{Z}(3)$ symmetry.
Fig. \[fig:sustemp\] shows a comparison of crossover transition lines found with the two criteria just mentioned. As both criteria are linked via the quadratic term in the action, all curves finally converge to the same point, the critical point (here in the absence of diquark condensation). A singularity in the second derivative of the action (or equivalently in the propagator) enforces this unique intersection point where the specific heat and other quantities show singular behaviour.
Comparing our Fig. \[fig:sustemp\] with corresponding results in other publications (see Fig. 16 in Ref. [@Sasaki:2006ww] and Fig. 4 in Ref. [@Abuki:2008nm]) one finds that the detailed behaviour of the deconfinement crossover transition line depends sensitively on the parameter choice and regularisation prescription. In the present case of a strong coupling a joint course of chiral and deconfinement crossover line is observed. When the coupling becomes weaker (e.g. due to parameter choice and regularisation prescription as in Refs. [@Sasaki:2006ww; @Abuki:2008nm]) the transition lines may deviate. In any case one should note that such deviations appear at large chemical potentials approaching the typical cutoff scale of the model. Any conclusions drawn at such energy or momentum scales should be handled with care.
In Fig. \[fig:susceptiblity\] we show the chiral and the Polyakov loop susceptibilities as functions of temperature at vanishing quark chemical potential (left panel) and compare them to the temperature derivatives of the constituent quark mass and the Polyakov loop (right panel). If we consider the behaviour of $\chi_M$, $\chi_{{\operatorname{Re}}\Phi}$ and $\chi_{\Phi}$ at $T\to 0$ we find that $\chi_{{\operatorname{Re}}\Phi}$ is finite, while $\chi_M$ and $\chi_{\Phi}$ vanish. This can be explained by the fact that $({\operatorname{Re}}\Phi)^2 = \frac14(\Phi^2 + 2 \left| \Phi \right|^2 + {\Phi^*}^2)$ contains a $\mathrm{U}(1)$-symmetric term $\left| \Phi \right|$. As the $\mathrm{U}(1)$ symmetry incorporates the $Z(3)$ centre of $\mathrm{SU}(3)_\mathrm{c}$ this term does not have to vanish once the $Z(3)$ symmetry is fully restored at $T=0$.[^8] The width of the peak in the temperature derivative of the dynamical quark mass $m = m_0-\sigma$ suggests that this crossover is influenced by the crossover of the Polyakov loop. At finite current quark mass $m_0$ the PNJL model produces an approximate coincidence of the peaks in the susceptibilities of the Polyakov loop and the constituent quark mass $m$, consistent with the pattern observed in Fig. \[fig:loopwithquarks\].
![ \[fig:sustemp\] Comparison of the transition lines obtained by determination of the maximum in the chiral susceptibility (left panel solid line) and the Polyakov loop susceptibility $\chi_{{\operatorname{Re}}\Phi}$ (right panel solid line) with the transition lines fixed by the maximal change with respect to temperature of constituent quark mass (left panel dashed line) and average of the real part of the Polyakov loop $\frac12\braket{\Phi+\Phi^*}$ (right panel dashed line).](./chiralcrossover.eps){width="\textwidth"}
![ \[fig:sustemp\] Comparison of the transition lines obtained by determination of the maximum in the chiral susceptibility (left panel solid line) and the Polyakov loop susceptibility $\chi_{{\operatorname{Re}}\Phi}$ (right panel solid line) with the transition lines fixed by the maximal change with respect to temperature of constituent quark mass (left panel dashed line) and average of the real part of the Polyakov loop $\frac12\braket{\Phi+\Phi^*}$ (right panel dashed line).](./loopcrossover.eps){width="\textwidth"}
![ \[fig:susceptiblity\] The chiral susceptibility $\chi_M$ (left panel solid line) and the Polyakov loop susceptibilities $\chi_{{\operatorname{Re}}\Phi}$ (left panel dashed line) and $\chi_{\Phi}$ (left panel dotted line) plotted as functions of temperature at vanishing quark chemical potential. These susceptibilities defined by Eqs. (\[eqn:chi\_m\],\[eqn:chi\_phi\],\[eqn:chi\_rephi\]) and evaluated using Eq. (\[eqn:gausssus\]) are compared here to the derivative of the constituent quark mass (right panel solid line) and the expectation value of the real part of the Polyakov loop (right panel dashed line) with respect to temperature.](./susceptibilities.eps){width="\textwidth"}
![ \[fig:susceptiblity\] The chiral susceptibility $\chi_M$ (left panel solid line) and the Polyakov loop susceptibilities $\chi_{{\operatorname{Re}}\Phi}$ (left panel dashed line) and $\chi_{\Phi}$ (left panel dotted line) plotted as functions of temperature at vanishing quark chemical potential. These susceptibilities defined by Eqs. (\[eqn:chi\_m\],\[eqn:chi\_phi\],\[eqn:chi\_rephi\]) and evaluated using Eq. (\[eqn:gausssus\]) are compared here to the derivative of the constituent quark mass (right panel solid line) and the expectation value of the real part of the Polyakov loop (right panel dashed line) with respect to temperature.](./tempderivs.eps){width="\textwidth"}
A comparison of the phase diagram obtained in mean field approximation in Ref. [@Roessner:2006xn] and the phase diagram including corrections to the order $\beta \leq 1$ shown in Fig. \[fig:mfcomp\], explicitly approves that corrections to the phase diagram due to the fermion sign problem are indeed small [@Roessner:2006xn]: the influence of ${\operatorname{Im}}\Omega_0$ and the splitting of $\braket{\Phi^*}$ and $\braket{\Phi}$ are rather modest.
![ \[fig:mfcomp\] Phase diagram implementing corrections to the order $\beta \leq 1$. Solid lines: crossover transition of the susceptibility related to the real part of the Polyakov loop, dashed lines: first order phase transition, and dotted: second order phase transitions.](./phasediagram.eps){width=".5\textwidth"}
Moments of the pressure
-----------------------
One benchmark for the PNJL model is its surprising capability of reproducing the trends of lattice QCD calculations.[^9] One way to handle the fermion sign problem in lattice QCD is to expand the calculated pressure about $\mu = 0$ in a Taylor series. Such an expansion is given in Ref. [@Allton:2005gk]: $$\begin{aligned}
\label{eqn:pexpansion}
\frac{p(T,\mu)}{T^4}&=\sum_{n=0}^{\infty}c_n(T)
\left(\frac{\mu}{T}\right)^n & & \text{with } &c_n(T)
&=\left.\frac{1}{n!}\frac{\partial^n (p(T,\mu)/T^4)}{\partial(\mu/T)^n}\right |_{\mu=0}\end{aligned}$$ with even $n$ as the situation is charge conjugation invariant. Specifically: $$\begin{aligned}
& \;c_2 = \left.\frac{1}{2}\,\frac{\partial^2 (p/T^4)}{\partial(\mu/T)^2}\right|_{\mu=0}, &
& \;\,c_4 = \left.\frac{1}{24}\,\frac{\partial^4 (p/T^4)}{\partial(\mu/T)^4}\right|_{\mu=0}, \nonumber\\
& c_6 = \left.\frac{1}{720}\,\frac{\partial^6 (p/T^4)}{\partial(\mu/T)^6}\right|_{\mu=0}, &
& c_8 = \left.\frac{1}{40320}\,\frac{\partial^8 (p/T^4)}{\partial(\mu/T)^8}\right|_{\mu=0}.\end{aligned}$$ The pressure in the PNJL model is evaluated by subtracting the divergent vacuum contributions of the thermodynamic potential: $$\label{eqn:pressure}
p = -\left(\Omega - \Omega(T=0)\right)$$
Results for $c_2$, $c_4$ and $c_6$ are shown in Fig. \[fig:moments\]. In comparison with the plots for $c_n$ presented in a previous paper [@Roessner:2006xn] at the mean field level the moments $c_n$ show slightly more structure. The rise in $c_2$ is somewhat sharper, the peak in $c_4$ is about $5\,\%$ higher. In summary, however, the corrections induced so far by corrections involving $\braket{\Phi^*-\Phi}$ around the mean fields are small. Pionic fluctuations, to be discussed in Sec. \[sec:pions\], tend to be more important. In presently available lattice results [@Allton:2005gk], these latter effects are however suppressed by the relatively large pion masses.
![\[fig:moments\] The moments of the pressure with respect to $\frac{\mu}{T}$ as defined in Eq. (\[eqn:pexpansion\]). $c_2$ is shown in the left panel, $c_4$ is displayed to the right where $c_6$ is shown in the inset. The data deduced from lattice computations are taken from [@Allton:2005gk]. ](./c2.eps){width="\textwidth"}
![\[fig:moments\] The moments of the pressure with respect to $\frac{\mu}{T}$ as defined in Eq. (\[eqn:pexpansion\]). $c_2$ is shown in the left panel, $c_4$ is displayed to the right where $c_6$ is shown in the inset. The data deduced from lattice computations are taken from [@Allton:2005gk]. ](./c4inset.eps){width="\textwidth"}
The ratio of the moments $c_4$ and $c_2$ has been discussed [@Ejiri:2005wq] as a suitable indicator of fluctuations close to the phase transition. The quantity of interest here is the cumulant ratio $R^{\mathrm{q}}_{4,2}$ defined in [@Ejiri:2005wq] and given as $R^{\mathrm{q}}_{4,2} = 12\,c_4/c_2$. The PNJL model calculation for this ratio is shown in Fig. \[fig:rq42\]. The dashed curve is found in the mean field limit with $\braket{\Phi^*}=\braket{\Phi}$ which suppresses one of the two Polyakov loop degrees of freedom. The solid curve is computed with inclusion of corrections beyond mean field and demonstrates the role of the non-zero $\braket{\Phi^*-\Phi}$. At temperatures below $T_{\mathrm{c}}$ one reaches $R^{\mathrm{q}}_{4,2} = 9$, the value characteristic of a hadronic resonance gas [@Ejiri:2005wq].
Interaction measure
-------------------
The PNJL results for the interaction or conformal measure $\varepsilon -3p$ are illustrated in Fig. \[fig:iameasure\]. The total interaction measure normalised to $T^4$ is split into quark and Polyakov loop parts. Note the sensitive balance between quark quasiparticle and Polyakov loop contributions to $\varepsilon -3p$ close to $T_{\mathrm{c}}$. In pure gauge QCD (or with infinitely heavy quarks) the Polyakov loop interaction measure is positive throughout. The presence of light quarks and their dynamical coupling to the Polyakov loop changes their pattern significantly. The Polyakov loop parts of the presure itself that determines the dashed curve in Fig. \[fig:iameasure\], is found to be consistent with calculations reported in Ref. [@Fukushima:2008wg]. For orientation, the total PNJL interaction measure (with $N_{\mathrm{f}}=2$) is shown in Fig. \[fig:iameasure\] along with recent $N_{\mathrm{f}}=2+1$ lattice QCD results [@Karsch:2008fe].
![\[fig:iameasure\] Contributions to the conformal measure from quarks and Polyakov loop, as well as the total PNJL interaction measure for $N_{\mathrm{f}} = 2$. The $N_{\mathrm{f}} = 2+1$ lattice QCD results [@Karsch:2008fe] (with $N_\tau=8$ for p4-improved and asqtad action) are shown for orientation. ](./rq42comp.eps){width="\textwidth"}
![\[fig:iameasure\] Contributions to the conformal measure from quarks and Polyakov loop, as well as the total PNJL interaction measure for $N_{\mathrm{f}} = 2$. The $N_{\mathrm{f}} = 2+1$ lattice QCD results [@Karsch:2008fe] (with $N_\tau=8$ for p4-improved and asqtad action) are shown for orientation. ](./InteractionMeasure.eps){width="\textwidth"}
Speed of sound
--------------
In Fig. \[fig:speed\] the squared speed of sound in units of the speed of light is plotted as a solid line. The speed of sound $v_\mathrm{s}$ is defined by $$v_\mathrm{s}^2 = \left. \frac{\partial p}{\partial \varepsilon} \right\vert_{S} = \left. \frac{\partial \Omega}{\partial T}\right\vert_{V} \bigg/ \left. T \frac{\partial^2 \Omega}{\partial T^2}\right\vert_{V} ~,$$ where the denominator is the specific heat capacity $c_V$. The dashed line in Fig. \[fig:speed\] gives the size of the ratio of pressure and energy density, $\frac{p}{\varepsilon}$. In the panel to the left where the quantities are plotted at vanishing chemical potential $\mu=0$ both graphs show a pronounced dip near the crossover transition temperature. In the panel to the right the same situation is plotted at a quark chemical potential close to the chemical potential of the critical point $\mu\lesssim \mu_{\mathrm{crit.}}$.
![ \[fig:speed\] The speed of sound (solid) and the ratio of pressure over energy density (dashed) at vanishing chemical potential as function of temperature (left panel). The right panel shows the same quantities at a quark chemical potential slightly less than the one at the critical point ($\mu = 0.3{\,\mathrm{GeV}}\lesssim \mu_\mathrm{crit} \simeq 0.31{\,\mathrm{GeV}}$). ](./SpeedZeroMu.eps){width="\textwidth"}
![ \[fig:speed\] The speed of sound (solid) and the ratio of pressure over energy density (dashed) at vanishing chemical potential as function of temperature (left panel). The right panel shows the same quantities at a quark chemical potential slightly less than the one at the critical point ($\mu = 0.3{\,\mathrm{GeV}}\lesssim \mu_\mathrm{crit} \simeq 0.31{\,\mathrm{GeV}}$). ](./SpeedSubCritMu.eps){width="\textwidth"}
Dynamical fluctuations {#sec:pions}
======================
So far the formalism presented has been focused on the treatment of fluctuations around the mean fields, averaged over space and (Euclidean) time. The homogeneous, constant Polyakov loop field and its corrections beyond mean field fall in this category.
In this section we consider mesonic excitations and their propagation (i.e. $\frac{1}{N_{\mathrm{c}}}$-corrections). The NJL framework is well suited to incorporate such effects. The NJL model features a dynamical mechanism which produces spontaneous chiral symmetry breaking and, at the same time, generates the pion as a Goldstone boson in the pseudoscalar quark-antiquark channel, together with a massive scalar (sigma) boson. The thermodynamics of these modes and their changing spectral properties have been subject of several NJL model calculations in the past [@Klevansky:1992qe; @Hatsuda:1994pi; @Lutz:1992dv].
With increasing temperature, the mass of the pion is still protected by its Goldstone boson nature, whereas the sigma mass drops until at $T \sim T_\mathrm{c}$ it becomes degenerate with the pion, signalling restoration of chiral symmetry in its Wigner-Weyl realisation. For $T>T_\mathrm{c}$, the $\pi$ and $\sigma$ masses jointly increase quite rapidly while at the same time their widths for decay into $q\bar{q}$ grow continuously. This implies that at temperatures exceeding $T_\mathrm{c}$ both $\pi$ and $\sigma$ modes become thermodynamically irrelevant while correlated quark-antiquark pairs carrying the quantum numbers of $\pi$ and $\sigma$ can still be active above $T_\mathrm{c}$. One therefore expects that the corrections to the pressure from propagating pions and sigmas should be concentrated around $T_\mathrm{c}$. These mesonic modes are colour singlets[^10]. Thus their statistical weight is much smaller than the weight of the deconfined quark quasiparticles.
Meson propagators in the PNJL model
-----------------------------------
We start from the derivation of mesonic propagators in the PNJL model as performed, for example, in [@Hansen:2006ee]. We calculate the momentum dependent propagator $$\label{eqn:mesonpropagator}
\parbox{40pt}{\begin{picture}(40,40)
\put(0,20){\includegraphics{./graph2.eps}}
\put(2,10){$j$}\put(32,10){$k$}\put(17,30){$q^\mu$}
\end{picture}} = \left[ \frac{\partial^2 \mathcal{S}_{\mathrm{bos}}}{\partial \xi_j(q^\mu) \partial \xi_k(-q^\mu)}\right]^{-1}~,$$ where $\xi = \theta -\theta_{\mathrm{MF}}$ now stands for the pion field or for the deviation of the sigma field from its expectation value. Note that the functional trace in the formula for $\mathcal{S}_{\mathrm{bos}}$ ensures momentum conservation, such that the sum of the momentum arguments in the denominator always vanishes. The calculation can be done numerically as it was done in the previous section. Alternatively, we use an analytic approach as follows. Recall some useful formulae also exploited in Refs. [@Allton:2003vx; @Allton:2005gk]: $$\begin{aligned}
\frac{\partial \ln \det M}{\partial x} &= {\mathrm{tr}}\left[ M^{-1}\frac{\partial M}{\partial x} \right] &\text{and}&& \frac{\partial M^{-1}}{\partial x} &= -M^{-1}\frac{\partial M}{\partial x}M^{-1}\;,\end{aligned}$$ with $M$ an invertible matrix and $\frac{\partial M}{\partial x}$ is the component-wise derivative of this matrix. Applying this to the PNJL action $\mathcal{S}_{\mathrm{bos}}$ in (\[eqn:omegageneral\]) and neglecting the potential terms for the moment we find $$\label{eqn:firstderiv}
\frac{\partial \mathcal{S}_{\mathrm{bos}}}{\partial \theta} = -\frac{V}{2}\sum_n\int\frac{{\mathrm{d}}^3p}{\left(2\pi\right)^3}{\mathrm{Tr}}\left[ \tilde{S}\left(i\omega_n,\vec{p}\,;\theta\right) \frac{\partial \tilde{S}^{-1}\left(i\omega_n,\vec{p}\,;\theta\right)}{\partial \theta}\right]~,$$ where $\tilde{S}^{-1}\left(i\omega_n,\vec{p}\,;\theta\right)$ denotes the inverse quark propagator with emphasis on the fact that the quark propagates in the mesonic background field $\theta$. This formalism makes it possible to calculate derivatives with respect to bosonic fields (say $\theta_k$) that have not been explicitly included in the action, as long as it is ensured that the model does not produce finite vacuum expectation values for these particular fields. Not having a vacuum expectation value is equivalent to the fact, that the mean field equations corresponding to these fields are satisfied for a vanishing field, i.e. that $$\left.\frac{\partial \mathcal{S}_{\mathrm{bos}}}{\partial \theta}\right\vert_{\theta_k = 0} = 0\;.
\label{eqn:mfthetak}$$ All we need to know is the constant matrix $\frac{\partial \tilde{S}^{-1}}{\partial \theta_k}$. This matrix involves the Dirac, colour and flavour structure of a quark-antiquark pair (or a quark-quark pair) that couples to the bosonic field $\theta_k$, i.e. it is determined by the quantum numbers of $\theta_k$. The mean field equation is fulfilled if the trace in Eq. (\[eqn:firstderiv\]) vanishes for the given Dirac, colour and flavour structure. For the pion field this is true as long as there is no pion condensate.[^11] The condensate corresponding to the sigma, namely the chiral condensate, figures explicitly in the action and is therefore included in the quark propagator.
In the case of the pion and sigma propagators the functional derivative in (\[eqn:mesonpropagator\]) produces exactly the trace over Dirac, colour and flavour structures known from RPA calculations [@Klevansky:1992qe; @Hansen:2006ee; @Hatsuda:1994pi; @Lutz:1992dv]. We adopt the definition of the quark distribution functions $f^+_\Phi $ and $f^-_\Phi $ and the separation of the emerging integral into the contributions $I_1$ and $I_2$ as given in Ref. [@Hansen:2006ee]. In the treatment of the thermodynamics we have modified the cutoff prescription of the standard NJL model, such that non-divergent integrals are integrated over the whole quark-momentum range, while only divergent integrals are regularised by the usual NJL three-momentum cutoff. The separation of finite and divergent contributions is defined such that the model reproduces the classical limit at high temperatures, i.e. the Stefan-Boltzmann limit. As a downside, for consistency all newly appearing integrals have to be treated in the same manner, which leads to slightly different results from those given in Ref. [@Hansen:2006ee].
Mesonic corrections to the pressure
-----------------------------------
Once the meson propagators are given, it is possible to evaluate the contribution to the pressure from mesons propagating in the heat bath using RPA methods. Applying Bethe-Salpeter (RPA) equations generates spectral functions $$\rho_{\mathrm{M}}(\omega,\,\vec{q};\,T) = \frac{G {\operatorname{Im}}\Pi_\mathrm{M}(\omega,\,\vec{q};\,T)}{(1-G {\operatorname{Re}}\Pi_\mathrm{M})^2+ (G {\operatorname{Im}}\Pi_\mathrm{M} )^2}$$ with the thermal quark-antiquark polarisation function $$\Pi_{\mathrm{M}}(\omega,\vec{q};\,T) = T \sum_{\omega_n} \int\frac{{\mathrm{d}}^3 p}{(2\pi)^3} {\mathrm{Tr}}\left[ \Gamma_\mathrm{M} \tilde{S}({i}\omega_n+\mu,\vec{p}\,) \Gamma_\mathrm{M} \tilde{S}({i}(\omega_n-\omega)+\mu,\vec{p}-\vec{q}\,) \right]~,$$ where the sum is taken over the Matsubara frequencies $\omega_n = (2n+1)\pi\, T$. Here $\Gamma_\mathrm{M}$ is a Dirac, flavour and colour representation of a meson current labelled $\mathrm{M}$. In this work we only focus on the pseudoscalar isovector channel (i.e. pionic excitations) and the scalar isoscalar channel. $\tilde{S}({i}\omega_n,\vec{p}\,) = -\frac{m+\slashed{p}}{\omega_n^2+p^2+m^2}$ denotes the quark quasiparticle propagator with $\slashed{p}={i}\omega_n\gamma_0-\vec{\gamma}\cdot\vec{p}$.
The pressure below $T_{\mathrm{c}}$ is essentially generated by the pion pole with its almost temperature independent position. Therefore the calculated pressure below $T_{\mathrm{c}}$ basically represents the one of a pion gas with fixed (temperature independent) mass. Fig. \[fig:spectral\] shows, as examples, the spectral functions for the pion and sigma modes at threshold temperature $T_{\mathrm{thr}}$ where the breakup into a quark-antiquark pair occurs. This threshold temperature is at about $ 1.1 \,T_{\mathrm{c}}$. At this point the $\pi$ and $\sigma$ spectral functions are still distinguishable (left panel of Fig. \[fig:spectral\]), whereas they coincide (right panel) at temperatures well above threshold where $\pi$-$\sigma$ degeneracy indicates restoration of chiral symmetry in its Wigner-Weyl realisation. Their width is a measure of the decay of the (increasingly massive) pionic and sigma modes into (light) deconfined quark-antiquark pairs at temperatures above $T_{\mathrm{c}}$.
![The spectral functions $\rho_{\mathrm{M}}= \frac{G {\operatorname{Im}}\Pi_\mathrm{M}}{(1-G {\operatorname{Re}}\Pi_\mathrm{M})^2+ (G {\operatorname{Im}}\Pi_\mathrm{M} )^2} $ taken at $\vec{q}=0$ for pion and sigma at $T\approx T_{\mathrm{thr}} $ (left) and at $T> T_{\mathrm{thr}} $ (right). \[fig:spectral\]](./SpectralPlotMed.eps){width="\textwidth"}
![The spectral functions $\rho_{\mathrm{M}}= \frac{G {\operatorname{Im}}\Pi_\mathrm{M}}{(1-G {\operatorname{Re}}\Pi_\mathrm{M})^2+ (G {\operatorname{Im}}\Pi_\mathrm{M} )^2} $ taken at $\vec{q}=0$ for pion and sigma at $T\approx T_{\mathrm{thr}} $ (left) and at $T> T_{\mathrm{thr}} $ (right). \[fig:spectral\]](./SpectralPlotHigh.eps){width="\textwidth"}
The resonant interaction of instable mesons with the quark sea above $T_{\mathrm{c}}$ produces an additional pressure contribution. This contribution is not part of the quark pressure previously calculated in Hartree-Fock approximation. The meson decay products form rings of RPA chains. Such kind of pressure contributions are investigated in Ref. [@Hufner:1994ma] and calculated performing the ring sum. However, below $T_{\mathrm{c}}$ the NJL model does not handle the mesonic degrees of freedom properly. In the hadronic phase the coupling of mesonic modes to the quark-antiquark continuum is suppressed by confinement, whereas $\rho_{\mathrm{M}}$ receives contributions from decays into $q\bar{q}$ even below $T_{\mathrm{c}}$. This unphysical feature persists [@Hansen:2006ee] in the PNJL generalisation of the NJL approach. Moreover, the non-renormalisability of the NJL model requires to introduce further subtractions when following the lines of Ref. [@Hufner:1994ma]. To avoid such arbitrariness and unphysical features we ignore the decay of meson modes into $q\bar{q}$-pairs altogether when calculating an estimate for the meson contributions to the pressure: $$\label{eqn:omegacorr}
\delta\Omega =\nu \int\frac{{\mathrm{d}}^3 q}{(2\pi)^3} \,T \ln(1-{e}^{-{E_q}/{T}}) + B(T) ~,$$ where $\nu$ is the statistical weight of the corresponding meson species, $E_q = \sqrt{{\vec{q}\,}^2+m^2_{\mathrm{pole}}(T)}$ with $m_{\mathrm{pole}}(T)$ the temperature dependent pion and sigma pole mass determined by $1-G{\operatorname{Re}}\Pi = 0$. Furthermore $B(T)$ is an appropriately chosen vacuum energy constant ensuring thermodynamic consistency. $B(T)$ is fixed such that the temperature dependence of the pole mass $m_{\mathrm{pole}}(T)$ is compensated on differentiating $\Omega$ with respect to the temperature $T$. This implies that the inclusion of $B(T)$ ensures that $\left.\partial \Omega/\partial m_{\mathrm{pole}}\right\vert_T =0$.
In Fig. \[fig:totalpressure\] the calculated pressure of $\pi^{0,\pm}$ and sigma modes are compared with the quark Hartree-Fock pressure and the result for the overall pressure of Hartree-Fock plus RPA is plotted. For comparison the pressure of a Bose gas with three internal degrees of freedom is indicated by the thin solid line. Below the crossover temperature $T_\mathrm{c}$ one can clearly identify the pion gas contribution resulting from the RPA calculations. Once the meson masses reach the scale of the NJL cutoff $\Lambda$ the used approximation breaks down. The inversion of the scale hierarchy appears at temperatures of about $1.3\,T_{\mathrm{c}}$.
For larger current quark masses the meson gas contributions and correlations are reduced. This effect is illustrated by Fig. \[fig:totalpressure50\] where the pressure of the PNJL model is plotted using an increased current quark mass leading to an unphysically heavy pion. Thus for heavy pions the agreement with lattice data observed in a previous publication [@Roessner:2006xn] remains. This agreement is also confirmed by calculations in a non-local PNJL framework [@Blaschke:2007np] which does not suffer from cutoff artefacts.
![Same as Fig. \[fig:totalpressure\], but with higher current quark mass $m_0=50{\,\mathrm{MeV}}\Rightarrow m_\pi=421{\,\mathrm{MeV}}$ (compared to $m_0=5.5{\,\mathrm{MeV}}\Rightarrow m_\pi=139{\,\mathrm{MeV}}$ in Fig. \[fig:totalpressure\]). The pressure of the boson gas (thin solid line) was now plotted using the heavier pion mass. \[fig:totalpressure50\]](./totalpressurePole.eps){height=".76\textwidth"}
![Same as Fig. \[fig:totalpressure\], but with higher current quark mass $m_0=50{\,\mathrm{MeV}}\Rightarrow m_\pi=421{\,\mathrm{MeV}}$ (compared to $m_0=5.5{\,\mathrm{MeV}}\Rightarrow m_\pi=139{\,\mathrm{MeV}}$ in Fig. \[fig:totalpressure\]). The pressure of the boson gas (thin solid line) was now plotted using the heavier pion mass. \[fig:totalpressure50\]](./totalpressure50Pole.eps){height=".76\textwidth"}
The mesonic contribution to the interaction measure is rather modest. The interaction measure already shown in Fig. \[fig:iameasure\] is replotted in Fig. \[fig:mesoniameasure\] including mesonic contributions.
![ The normalised interaction measure $(\varepsilon-3p)/T^4$ from the PNJL model with and without mesonic corrections. \[fig:mesoniameasure\]](./InteractionMeasureMeson.eps){height=".46\textwidth"}
Conclusions and outlook {#sec:conclusio}
=======================
The PNJL model as an approximation to QCD thermodynamics picks up on two basic properties of low-energy QCD: spontaneous chiral symmetry breaking and confinement. In this work some of the existing calculations [@Ratti:2005jh; @Ratti:2006wg; @Roessner:2006xn] have been extended in several directions. We have reviewed the expectation values of the Polyakov loop and its complex conjugate, the phase diagram, the moments of the pressure and the speed of sound in a framework beyond mean field theory. While the phase diagram does not show significant changes when improving the mean field approximation, the moments of the pressure and the speed of sound show quantitative differences on the order of $5\,\%$. In general the structures observed become more articulate. In the case of the Polyakov loop and its complex conjugate the corrections cause qualitative differences. While the Polyakov loop and its complex conjugate are equal at mean field level in the present approach, the corrections beyond mean field generate the split of the two expectation values $\braket{\Phi}$ and $\braket{\Phi^*}$ at non-zero quark chemical potential. The numerical results show that the corrections are largest in the vicinity of phase transitions or rapid crossovers. This comes as no surprise as it is the transitional region between two phases where we expect large fluctuations.
The degrees of freedom that govern the low temperature regime, primarily the pions, produce significant corrections to the pressure only in the regime below the critical temperature $T_{\mathrm{c}}$ where constituent quarks are frozen and confined. As soon as the pressure of quark degrees of freedom starts to rise at the chiral and deconfinement crossover, mesonic pressure contributions become comparatively small.
The good agreement of the PNJL model at mean field level with lattice calculations remains in the presence of the mesonic corrections calculated in this work. The most prominent feature of the pressure, namely the steep rise near the critical temperature, is only slightly modified by the corrections due to dynamic fluctuations of pions and sigma mesons. Below the quark-antiquark threshold the pressure generated by pion fluctuations is basically the pressure of a free pion gas. As low temperatures are difficult to access by lattice QCD, the pressure in typical lattice calculations is usually normalised to zero at some finite temperature below $T_{\mathrm{c}}$. This might explain why the pressure of the PNJL model including mesonic corrections is slightly higher than the pressure resulting from lattice calculations [@Allton:2005gk]. Due to these normalisation issues the comparison suffers from this uncertainty, $\Delta (p/T^4) = \left. p/T^4\right\vert_{T=T_{\mathrm{norm}}}-\left.p/T^4\right\vert_{T\to 0}$, which in turn depends on the normalisation temperature $T_{\mathrm{norm}}$ and additionally on the realized pion mass. For large pion masses (see Fig. \[fig:totalpressure50\]) this correction is small maintaining the good agreement between PNJL and lattice results. For small pion masses the pressure contribution from pion modes is almost flat in the temperature region $T\approx m_\pi$, such that the pressures from lattice and PNJL calculations mainly differ by a shift in $p/T^4$. Shifting the lattice data to higher values of $p/T^4$ indeed reduces the difference between Stefan-Boltzmann limit and lattice data for the pressure at high temperatures around $2$–$3\, T_\mathrm{c}$ and above, improving the agreement between lattice results and PNJL. Even when taking into account these issues in the comparison of PNJL and lattice results, we conclude that there exists a good qualitative and quantitative agreement of these two approaches.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We thank Marco Cristoforetti, Kenji Fukushima and Volker Koch for stimulating discussions.
\[app:corrections\]
Detailed derivation of corrections to mean fields
=================================================
Expansion of the effective action
---------------------------------
This appendix displays some technical details concerning the treatment of fluctuation corrections beyond mean field approximation in the PNJL model (cf. Sec. \[sec:corr\]).
In the following we denote by $\theta = (\theta_i)$ the set of fields $(\sigma,\,\Delta,\,\phi_3,\,\phi_8)$ which operate as bosonic degrees of freedom in the effective action $\mathcal{S}_{\mathrm{bos}}$ of Eqs. (\[eqn:omegageneral\]) and (\[eqn:bosaction\]). Furthermore, let $\theta_0 = (\braket{\sigma}_0,\,\braket{\Delta}_0,\,\braket{\phi_3}_0,\,\braket{\phi_8}_0)$ be the set of mean field (expectation) values of these quantities, and introduce deviations from the mean fields by $\xi = (\xi_i) = \theta-\theta_0$.
A frequently used procedure that we follow here, is to expand the effective action in powers of $\xi$ around a properly chosen mean field configuration. The Gaussian part of such an expansion of the path integral can be handled analytically. In Sec. \[sec:corr\] the mean field approximation has been defined such that the (formally) complex action $\mathcal{S}_{\mathrm{bos}}$ produces, to this leading order, a real-valued thermodynamical potential (or pressure), $\Omega_{\mathrm{MF}} = {\operatorname{Re}\!}\left[\Omega_0\right]$, subject to the mean field equations (\[eqn:mfeqn\]). The expansion of $\mathcal{S}_{\mathrm{bos}}$ is then of the generic form $$\label{eqn:expansion}
\mathcal{S}_{\mathrm{bos}} = \frac{V}{T}\left( \Omega_{\mathrm{MF}} + \omega^{(1)}\cdot\xi +\frac12 \xi\cdot\omega^{(2)}\cdot\xi \cdots \right)~,$$ where we have introduced the notations $a\cdot b = \sum_i a_i\,b_i$ and $a\cdot A \cdot b = \sum_{ij} a_i\,A_{ij}\,b_j$, with summations extending over all bosonic degrees of freedom. The expansion (\[eqn:expansion\]) is performed such that the path integral is optimally approximated. This is achieved when the perturbative terms in the expansion of the action are maximally suppressed. With the thermodynamic weight ${e}^{\mathcal{-S}}\in\mathds{C}$ this approximation is optimal near the maximum of $\left\vert {e}^{-\mathcal{S}} \right\vert$. The equations to determine $\theta_0$ are the mean field equations (\[eqn:mfeqn\]) (also used in [@Roessner:2006xn]).
Given the expansion (\[eqn:expansion\]) in terms of the $\xi$ fields, thermal expectation values incorporate fluctuations around the mean field configuration $\theta_{\mathrm{MF}}\equiv\theta_0$. We refer to these corrections as “fluctuations” even if the fields themselves (such as the Polyakov loop field variables $\phi_3$ and $\phi_8$) are constant in space and time.
A perturbative approach is now used to calculate corrections to the mean field solutions. The action $\mathcal{S}_{\mathrm{bos}}$ is split into “large” and “small” parts, $\mathcal{S}_{\mathrm{bos}} = \mathcal{S}_{0} + \mathcal{S}_{\mathrm{I}}$, as follows: the “large” part $\mathcal{S}_{0}$ incorporates the leading mean field terms plus the additional Gaussian part of $O(\xi^2)$ in Eq. (\[eqn:expansion\]): $$\label{eqn:esnull}
\mathcal{S}_{0} = \frac{V}{T}\left( {\operatorname{Re}\!}\left[\Omega_0\right]+\frac12\xi\cdot\omega^{(2)}\cdot\xi \right)~,$$ while $\mathcal{S}_{\mathrm{I}}$ deals with the remaining pieces, in particular with the non-vanishing ${\operatorname{Im}\!}\left[\Omega_0\right]$. The leading correction of this sort is the term $\delta \mathcal{S}_{\mathrm{I}} = \frac{V}{T}\omega^{(1)}\cdot\xi$. In the present context we truncate Eq. (\[eqn:expansion\]) as it stands and keep only this term in $\mathcal{S}_{\mathrm{I}}$, for the moment.
The thermal expectation values of a given quantity $f(\xi)$ is proportional to $$\label{eqn:expvalue}
\int \mathcal{D}\xi\;f(\xi)\,{e}^{-\mathcal{S}_{\mathrm{bos}}} = \int \mathcal{D}\xi\;f(\xi)\,{e}^{-\mathcal{S}_{0}} \,{e}^{-\mathcal{S}_{\mathrm{I}}} ~,$$ where, for fields constant in space-time, the path integral reduces to $$\label{eqn:expvaluexi}
\int {\mathrm{d}}\xi\;f(\xi)\,{e}^{-\mathcal{S}_{0}(\xi)} \,{e}^{-\mathcal{S}_{\mathrm{I}}(\xi)} = \int {\mathrm{d}}\xi\;f(\xi)\,{e}^{-\mathcal{S}_{0}(\xi)} \,{e}^{-{i}k\cdot \xi} ~,$$ with $$\label{eqn:kdef}
k= \frac{V}{{i}T} \omega^{(1)} = \frac{V}{T} {\operatorname{Im}}\omega^{(1)}~.$$ A perturbative expansion of $f(\xi)$ about $\xi=0$ (i.e. about $\theta = \theta_{\mathrm{MF}}$) in powers of $\xi$ involves integrals of the form $$\label{eqn:fourrier}
\int {\mathrm{d}}\xi\;\xi^n\,{e}^{-\mathcal{S}_{0}(\xi)} \,{e}^{-{i}k\cdot \xi} = \left.({i}\partial_k)^n\,\mathcal{Z}_0(k)\right\vert_{k=\frac{V}{T} {\operatorname{Im}}\omega^{(1)}}~,$$ where we have introduced the generating function $ \mathcal{Z}_0(k) = \int {\mathrm{d}}\xi\; {e}^{-\mathcal{S}_{0}(\xi)} \,{e}^{-{i}k\cdot \xi} $. Each power of ${i}\partial_k$ evidently produces a factor $\frac{T}{V}$. At the same time, performing this derivative explicitly on $\mathcal{Z}_0(k)$, with $\mathcal{S}_{0}(\xi)$ specified in Eq. (\[eqn:esnull\]), produces a factor $$\delta = {i}\frac{T}{V}\left[ \omega^{(2)} \right]^{-1}\cdot k = \left[ \omega^{(2)} \right]^{-1}\cdot \omega^{(1)}~,$$ which is independent of $\frac{T}{V}$.
Hence there are two small quantities at hand to establish a perturbative expansion: $\frac{T}{V}$ and $\delta$. The smallness of $\frac{T}{V}$ is given here as we are interested in the thermodynamic limit. The size of $\delta$, however, is controlled by the action itself. Whether the expansion in $\delta$ is justified or not depends on the model and must be examined accordingly. The explicit calculations presented in the main body of this work shows that in the present version of the PNJL model the expansion in $\delta$ is indeed a good approximation.
We are now in a position to write down the thermal expectation value of a generic function $f$ as an expansion in powers of $\frac{T}{V}$ and $\delta$. We proceed here with establishing Feynman diagrams for this perturbative approach. We write generically $$\label{eqn:genpartfun}
Z \;=\; \frac1{\mathcal{N}}\int \mathcal{D}\xi\;{e}^{-\mathcal{S}_{\mathrm{bos}}} \;=\; \frac1{\mathcal{N}}\int \mathcal{D}\xi\;\;\sum_{l=0}^{\infty}\frac1{l!}\,\left(-\mathcal{S}_{\mathrm{I}}\right)^{l}\;\,{e}^{-\mathcal{S}_0}\,.$$ If corrections to the partition function of the PNJL model are to be calculated, the $\mathcal{S}_0$ part of the action only comprises zeroth and second order terms, while the “small” part $\mathcal{S}_\mathrm{I}$ is identified with all other orders. The first order term acts as a source term. We establish the following Feynman rules: $$\begin{aligned}
\label{eqn:feynrules}
\parbox{40pt}{\begin{picture}(40,40)
\put(0,20){\includegraphics{./graph1.eps}}
\put(10,10){$j$}
\end{picture}} &= - \frac{\partial \mathcal{S}_{\mathrm{bos}}}{\partial \xi_j} &
\parbox{40pt}{\begin{picture}(40,40)
\put(0,20){\includegraphics{./graph2.eps}}
\put(2,10){$j$}\put(32,10){$k$}
\end{picture}} &= +\left[ \frac{\partial^2 \mathcal{S}_{\mathrm{bos}}}{\partial \xi_j \partial \xi_k}\right]^{-1} \nonumber \\
\parbox{40pt}{\begin{picture}(40,40)
\put(0,2){\includegraphics{./graph3.eps}}
\put(6,2){$j$}\put(6,34){$k$}\put(27,10){$l$}
\end{picture}} &= - \frac{\partial^3 \mathcal{S}_{\mathrm{bos}}}{\partial \xi_j \partial \xi_k \partial \xi_l} &
\parbox{40pt}{\begin{picture}(40,40)
\put(0,0){\includegraphics{./graph4.eps}}
\put(2,10){$j$}\put(22,0){$k$}\put(14,32){$l$}\put(30,23){$m$}
\end{picture}} &= - \frac{\partial^4 \mathcal{S}_{\mathrm{bos}}}{\partial \xi_j \partial \xi_k \partial \xi_l \partial \xi_m} \\
\vdots & &\vdots& \nonumber\end{aligned}$$
In perturbation theory it can be shown that only connected diagrams contribute to the partition function, i.e. $$\label{eqn:pertpartition}
Z_\mathrm{I} = \braket{{e}^{-\mathcal{S}_\mathrm{I}}}_0 = \sum_{l=0}^{\infty}\frac1{l!}\braket{(-\mathcal{S}_\mathrm{I})^l}_0 = \exp\left\lbrace \sum_{n=1}^{\infty} \frac1{n!} \braket{(-\mathcal{S}_\mathrm{I})^n}_{0c}\right\rbrace,$$ where $\braket{\cdots}_0$ denotes the expectation value with respect to the unperturbed action, and $\braket{\cdots}_{0c}$ is the expectation value of the connected diagrams with respect to this unperturbed action. Note that here the corrections depicted by the Feynman diagrams are corrections to the negative action, $-\mathcal{S}$, as the partition function was defined by $Z={e}^{-\mathcal{S}_{\mathrm{eff.}}}$. The corrections therefore need to be subtracted from the mean field result of the action $\mathcal{S}_{\mathrm{MF}}$.
For the thermal expectation values of $f$ we write $$\label{eqn:fplusinteraction}
\braket{f} = \braket{f\,{e}^{-\mathcal{S}_{\mathrm{I}}} }_0 = \sum_{l=0}^{\infty} \frac1{l!}\,\braket{f\,(-\mathcal{S}_{\mathrm{I}})^l}_0 .$$ Here each term under the sum can be written in terms of connected expectation values $$\begin{gathered}
\braket{f\,(-\mathcal{S}_{\mathrm{I}})^l}_0 = \sum_{a_1, a_2\cdots,\,a_n,\,m = 0}^{\infty} \frac{l!}{a_1!a_2!(2!)^{a_2}\cdots (a_n!)(n!)^{a_n}m!} \braket{(-\mathcal{S}_{\mathrm{I}})}^{a_1}_{0c} \braket{(-\mathcal{S}_{\mathrm{I}})^2}^{a_2}_{0c} \cdots\\
\cdots \braket{f\,(-\mathcal{S}_{\mathrm{I}})^{m}}_{0c} \delta_{\nu,\,l}~,\end{gathered}$$ where $\nu = a_1+2a_2+\cdots+na_n+m$. Substituting back in Eq. (\[eqn:fplusinteraction\]) gives $$\braket{f\,e^{-\mathcal{S}_{\mathrm{I}}} }_0 = \exp\left\lbrace \sum_{n=1}^{\infty} \frac1{n!} \braket{(-\mathcal{S}_\mathrm{I})^n}_{0c}\right\rbrace\;\times\; \sum_{m=0}^{\infty} \frac1{m!} \braket{f\,(-\mathcal{S}_\mathrm{I})^m}_{0c}.$$ Using Eq. (\[eqn:pertpartition\]) we find the final result $$\label{eqn:expvaluef}
\braket{f} = \braket{f\,e^{-\mathcal{S}_{\mathrm{I}}} }_0 = \sum_{n=0}^{\infty} \frac1{n!} \braket{f\,(-\mathcal{S}_\mathrm{I})^n}_{0c}.$$ In terms of Feynman diagrams Eq. (\[eqn:expvaluef\]) can be translated into all those connected diagrams that contain exactly one insertion coming from the function $f$. The Feynman rules for the insertions of $f$ are $$\begin{aligned}
\label{eqn:feynrulesforf}
\parbox{40pt}{\begin{picture}(40,40)
\put(0,20){\includegraphics{./graph5.eps}}
\put(10,10){$j$}
\end{picture}} &= \frac{\partial f}{\partial \xi_j} &
\parbox{40pt}{\begin{picture}(40,40)
\put(0,20){\includegraphics{./graph6.eps}}
\put(2,10){$j$}\put(32,10){$k$}
\end{picture}} &= \frac{\partial^2 f}{\partial \xi_j \partial \xi_k} \nonumber \\
\parbox{40pt}{\begin{picture}(40,40)
\put(0,2){\includegraphics{./graph7.eps}}
\put(6,2){$j$}\put(6,34){$k$}\put(27,10){$l$}
\end{picture}} &= \frac{\partial^3 f}{\partial \xi_j \partial \xi_k \partial \xi_l} &
\parbox{40pt}{\begin{picture}(40,40)
\put(0,0){\includegraphics{./graph8.eps}}
\put(2,10){$j$}\put(22,0){$k$}\put(14,32){$l$}\put(30,23){$m$}
\end{picture}} &= \frac{\partial^4 f}{\partial \xi_j \partial \xi_k \partial \xi_l \partial \xi_m} \\
\vdots & &\vdots& \nonumber\end{aligned}$$ What is needed to use these rules systematically is a scheme that orders all possible diagrams according to their importance in powers of the small parameters $\frac{T}{V}$ and $\delta$. The lowest order corrections in $\frac{T}{V}$ and $\delta$ are shown in Table \[tab:feynmangraphs\].
[c||c|c|c||]{}
------------------------------------------------------------------------
& $\beta = 0$ & $\beta = 1$ & $\beta = 2$\
$\alpha = 0$
------------------------------------------------------------------------
& $f(\theta_{\mathrm{MF}})$ & &\
$\alpha = 1$
------------------------------------------------------------------------
& & &\
$\alpha = 2$
------------------------------------------------------------------------
& & &\
[c||c|c||]{} & $\beta = 0$ & $\beta = 1$\
$\alpha=0$
------------------------------------------------------------------------
& — & —\
$\alpha=1$
------------------------------------------------------------------------
& & $+ \quad\frac12 \times 2$\
A useful consistency check is to verify that the thermal expectation values are now closer to the properties of an order parameter than the mean field result. In other words: we examine whether the thermodynamic potential $\Omega$ is a Landau effective action minimised with respect to $\braket{\sigma},\, \braket{\Delta},\, \braket{\Phi},\,\braket{\Phi^*}$ using Eq. (\[eqn:expvaluef\]) for the expectation values. The analysis below is done for the lowest order terms, $\alpha = 0$ and $\beta = 0,\,1$. We start from the form also used for the numerical calculations, presented below Eq. (\[eqn:numOmega\]), and differentiate with respect to the expectation values $\braket{\theta} = (\braket{\sigma},\, \braket{\Delta},\, \braket{\Phi},\,\braket{\Phi^*})^T$. To orders $\alpha = 0$ and $\beta = 0,\,1$ we find that $\braket{\theta} = \theta_0+\delta \theta$, where $\delta \theta$ is given by $$\delta \theta_i \;=\; \frac12\Bigg( \left. \left[ \frac{\partial^2 \Omega_{0}}{\partial \theta^2} \right]^{-1} \cdot \frac{\partial \Omega_{0}}{\partial \theta} \right\vert_{\theta= \theta_{\mathrm{MF}}} \Bigg)_i
\label{eqn:fieldshift}$$ (which is Eq. (\[eqn:numfield\]) with $f(\theta) = \theta_i$ ). After some calculation we arrive at the lowest order term in $\beta$ $$\begin{gathered}
\left.\frac{\partial \Omega}{\partial \theta_i} \right\vert_{\theta = \braket{\theta}} \;=\;
\frac{9}{8} \sum_{jk}\left[\frac{\partial^3 \Omega_{0}}{\partial \theta^3} \right]_{ijk} \;\;\Bigg(
\left[ \frac{\partial^2 \Omega_{0}}{\partial \theta^2} \right]^{-1} \cdot \frac{\partial \Omega_{0}}{\partial \theta} \Bigg)_j \\
\left.\Bigg(\left[ \frac{\partial^2 \Omega_{0}}{\partial \theta^2} \right]^{-1} \cdot \frac{\partial \Omega_{0}}{\partial \theta} \Bigg)_k \;\;
\right\vert_{\theta= \theta_{\mathrm{MF}}}\;\cdots \;+\; \text{higher orders},\end{gathered}$$ which is of order $\beta=2$, i.e. the self consistency equations are satisfied to the order we have been working in. As a consequence the corrections necessary to account for the fermion sign problem do not modify the mean field equations. A backward reaction on the mean field equations does not occur at this level of the approximation, which focuses on the evaluation of corrections to the effective (Polyakov loop) potential. Additional effects of pionic and scalar quark-antiquark modes, as considered in Sec. \[sec:pions\], do in principle have backward effects on the mean field equation.
The formalism allows to determine susceptibilities involving a quantity $g$, $
\chi_g = [ V (\braket{g^2}-\braket{g}^2 ) ]^{1/2}$. All that needs to be done is to apply the previously developed formalism to the function $g^2$. In Table \[tab:feynmangraphs\] the Feynman rules and multiplicity factors are written down for the evaluation of $\braket{f}$. In a second step $f$ is replaced by $g^2$. In this step the product rule of differentiation has to be applied producing additional prefactors. In this procedure it will happen that vertices of $f$ with $m=2,\, 3,\dots$ or more legs will split into two vertices with $m_1+m_2 =m$ legs. The lowest orders of the expression, Eq. (\[eqn:genericsusc\]), are shown in Table \[tab:chisqr\]. The contributions of order $\left(\frac{T}{V}\right)^0$ cancel. In this framework susceptibilities scale with $V^{\frac12}$ as expected. Additionally, it becomes obvious from Table \[tab:chisqr\] that there are no mean field contributions to susceptibilities in the sense that $\braket{(g-\braket{g}_{\mathrm{MF}})^2}_{\mathrm{MF}} = \braket{g^2}_{\mathrm{MF}}-\braket{g}^2_{\mathrm{MF}} = g^2_{\mathrm{MF}}-g^2_{\mathrm{MF}} = 0$. In the framework of mean field calculations, susceptibilities are usually evaluated by inverting the second derivative of the mean field action with respect to the fields. This is seen in the present framework as well: the entry for $\alpha=1$ and $\beta=0$ in Table \[tab:chisqr\] produces exactly this expression.
[99]{}
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[^1]: Work supported in part by BMBF, GSI, INFN, the DFG excellence cluster “Origin and Structure of the Universe” and by the Elitenetzwerk Bayern.
[^2]: $\vec{\alpha} = \gamma_0\,\vec{\gamma}$ and $\gamma_4 = i\gamma_0$ in terms of the standard Dirac $\gamma$ matrices.
[^3]: The centre of $\mathrm{SU}(3)$ contains all those $\mathrm{SU}(3)$ elements that commute with all other $\mathrm{SU}(3)$ elements, i.e. the elements $ {e}^{{i}\frac{2\pi}{3}k}\,\Eins$, with $k\in\mathds{Z}$ constituting a $\mathrm{Z}(3)$ subgroup of $\mathrm{SU}(3)$.
[^4]: Additional terms generated by the Fierz transformation are of no importance in the present context and will be omitted.
[^5]: As the parameter space of $\phi_3$ and $\phi_8$ is periodic there are different parameter sets representing the same physics. We use the (triangular shaped) domain $\lbrace ( \phi_8 \geqq -\frac{\pi}{\sqrt{3}} ) \wedge (\phi_8 \leqq \sqrt{3}(\phi_3+\frac{2\pi}{3}) ) \wedge (\phi_8 \leqq \sqrt{3}(-\phi_3+\frac{2\pi}{3}) ) \rbrace $. Note that the periodic domain of $L$ and $L^\dagger$ is $3!$-times larger than the domains for $\Phi$ and $\Phi^*$ (or equivalently $\phi_3$ and $\phi_8$) due to the trace’s invariance under unitary transformations of $L$.
[^6]: In analogy to the procedure in Minkowskian space-time one might argue that the complex phase needs to become stationary. Taking the thermodynamic limit one observes that the stationary phase field configuration is favoured over any other configuration by the factor $\frac{V}{T}$, while the absolute value is favoured over other configurations by a factor ${e}^{ \frac{V}{T}}$.
[^7]: Recall that the thermal expectation value $\braket{\cdots}$ is a weighted sum (integral) over various thermal field configurations. The equality $\braket{\Phi} = \Phi(\braket{\phi_3},\,\braket{\phi_8})$ holds only if $\Phi$ and $\Phi^*$ are linear functions of $\phi_3$ and $\phi_8$. This is not the case.
[^8]: The authors thank Chihiro Sasaki for pointing this out to them.
[^9]: Note however the discussion concerning the dependence on quark masses in Ref. [@Ratti:2007jf].
[^10]: Colour octet quark-antiquark modes turn out to be heavy and far removed from the spectrum of active degrees of freedom.
[^11]: The mean field equation is satisfied as the flavour-trace ${\mathrm{tr}}_{\mathrm{f}}[\,\Eins \,\tau_i\,]=0$ with $i=1,2,3$ vanishes.
|
---
abstract: 'Heat-invariants are a class of spectral invariants of Laplace-type operators on compact Riemannian manifolds that contain information about the geometry of the manifold, e.g., the metric and connection. Since Brownian motion solves the heat equation, these invariants can be obtained studying Brownian motion on manifolds. In this article, we consider Brownian motion on the Toeplitz algebra, discrete Heisenberg group algebras, and non-commutative tori to define Laplace-type operators and heat-semigroups on these C\*-bialgebras. We show that their traces can be $\zeta$-regularized and compute “heat-traces” on these algebras, giving us a notion of dimension and volume. Furthermore, we consider $SU_q(2)$ which does not have a Brownian motion but a class of driftless Gaussians which still recover the dimension of $SU_q(2)$.'
address:
- 'Department of Mathematics and Statistics, Lancaster University, LA1 4YF, Lancaster, United Kingdom'
- 'Department of Mathematics, King’s College London, Strand, WC2R 2LS, London, United Kingdom'
author:
- Jason Hancox
- Tobias Hartung
title: 'Zeta-regularization and the heat-trace on some compact quantum semigroups'
---
Introduction {#sec:intro}
============
In this article, we want to consider $\zeta$-regularization and the heat-trace in the non-commutative settings of the Toeplitz algebra, discrete Heisenberg group, non-commutative tori, and $SU_q(2)$. $\zeta$-regularization is a means to extend tracial functionals (not necessarily bounded) that are defined on a subalgebra to a larger domain. More precisely, consider an algebra $A$, a subalgebra $A_0$, and a linear functional $\tau:\ A_0\to{\ensuremath{\mathbb{C}^{{}}}}$ such that ${\forall}x,y\in A_0:\ \tau(xy)=\tau(yx)$. Given a holomorphic family ${\varphi}:\ {\ensuremath{\mathbb{C}^{{}}}}\to A$ and $\Omega{\ensuremath{\subseteq}}{\ensuremath{\mathbb{C}^{{}}}}$ open and connected such that the restriction ${\varphi}|_\Omega$ of ${\varphi}$ to $\Omega$ takes values in $A_0$, we want to consider maximal holomorphic extensions $\zeta({\varphi})$ of $\tau\circ{\varphi}|_{\Omega}$. In a way, this is a generalized version of the Riemann $\zeta$-function $\zeta_R$ and its applications like “$\sum_{n\in{\ensuremath{\mathbb{N}^{{}}}}}n=\zeta_R(-1)$”. These ideas were pioneered by Ray and Singer [@ray; @ray-singer] whose initial works had already been successfully applied by Hawking [@hawking] to compute the energy momentum tensor on the black hole horizon. Since traces are important for studying invariants, $\zeta$-regularization has become an integral part of the pseudo-differential toolkit, especially in geometric analysis.
As such, the “classical case” to consider is where $A=\Psi$ is the algebra of classical \[classical is important because we are not looking at the entire algebra of $\psi$dos\] pseudo-differential operators on a compact Riemannian $C^\infty$-manifold $M$ without boundary, $A_0$ the dense subalgebra of classical pseudo-differential operators that are trace class on $L_2(M)$, and $\tau$ the canonical trace ${\ensuremath{{\operatorname}{tr}}}$ on the Schatten class $S^1(L_2(M))$. It is then possible to construct this holomorphic family ${\varphi}$ of pseudo-differential operators in such a way that each ${\varphi}(z)$ has affine order $qz+a$ where $q>0$. Then, ${\varphi}(z)$ is of trace class whenever $\Re(z)<\frac{-\dim M-\Re(a)}{q}$ and $\tau\circ{\varphi}$ has a meromorphic extension to ${\ensuremath{\mathbb{C}^{{}}}}$. Furthermore, all poles are simple and contained in the set ${\ensuremath{\left}}\{\frac{j-a-\dim M}{q};\ j\in{\ensuremath{\mathbb{N}^{{}}}}_0{\ensuremath{\right}}\}$. This construction, using the notion of gauged symbols, was introduced by Guillemin [@guillemin], the residues at the poles give rise to Wodzicki’s non-commutative residue [@wodzicki] which (up to a constant factor) is the unique continuous trace on $\Psi$ (if $\dim M>1$), and the constant Laurent coefficients give rise to the Kontsevich-Vishik trace [@kontsevich-vishik; @kontsevich-vishik-geometry]. It was later shown [@maniccia-schrohe-seiler] that the Kontsevich-Vishik trace (which is unbounded in general) is the unique extension of the canonical trace on $S^1(L_2(M))$ to the subspace of pseudo-differential operators of non-integer order (a dense subspace of $\Psi$ which is not an algebra). The Kontsevich-Vishik trace has also been extended to Fourier integral operators (or, more precisely, “gauged poly-$\log$-homogeneous distributions” which contain the gauged Lagrangian distributions studied by Guillemin [@guillemin] which in turn contain Fourier integral operator traces) in Hartung’s Ph.D. thesis [@hartung-phd; @hartung-scott].
Families ${\varphi}$ of the form ${\varphi}(z)=TQ^z$, which are constructed using a classical pseudo-differential operator $T$ and complex powers $Q^z$ of an appropriate invertible elliptic operator $Q$ [@seeley], are particularly important example of such $\zeta$-functions. Here, the meromorphic extension of ${\ensuremath{{\operatorname}{tr}}}TQ^z$ is denoted by $\zeta(T,Q)$ and called the $\zeta$-regularized trace of $T$ with weight $Q$. It was shown [@paycha-scott] that the constant term of the Laurent expansion of $\zeta(T,Q)$ centered at zero is of the form ${\ensuremath{{\ensuremath{{\operatorname}{tr}}}_{\mathrm{KV}}}}T-\frac{1}{q}{\ensuremath{{\operatorname}{res}}}(T\ln Q)-{\ensuremath{{\operatorname}{tr}}}(T{\ensuremath{{\operatorname}{pr}}}_{\ker Q})$ where ${\ensuremath{{\ensuremath{{\operatorname}{tr}}}_{\mathrm{KV}}}}$ denotes the Kontsevich-Vishik trace, ${\ensuremath{{\operatorname}{res}}}(T\ln Q)$ is the so called “trace anomaly”, and ${\ensuremath{{\operatorname}{res}}}$ denotes the extended Wodzicki residue (note that $T\ln Q$ is typically not a pseudo-differential operator and the formula holds only locally as neither ${\ensuremath{{\ensuremath{{\operatorname}{tr}}}_{\mathrm{KV}}}}$ nor ${\ensuremath{{\operatorname}{res}}}$ are globally defined in general). For $T=1$, ${\ensuremath{{\operatorname}{res}}}(\ln Q)$ this is called the logarithmic residue [@okikiolu; @scott]. This trace anomaly only appears in the so called “critical case” which is if there exists a degree of homogeneity $-\dim M$ in the asymptotic expansion of $T$, i.e., $\zeta(T,Q)$ has a pole in zero.[^1]
The Kontsevich-Vishik trace of $T$ can be stated in the following form. Let $T$ have symbol $\sigma$ with asymptotic expansion $\sigma(x,\xi)\sim\sum_{j\in{\ensuremath{\mathbb{N}^{{}}}}_0}\alpha_{m-j}(x,\xi)$ where each $\alpha_{m-j}$ is homogeneous of degree $m-j$ in $\xi$. In other words, $$\begin{aligned}
k(x,y):=&(2\pi)^{-\dim M}\int_{{\ensuremath{\mathbb{R}^{\dim M}}}}e^{i\langle x-y,\xi\rangle_{\ell_2(\dim M)}}\sigma(x,\xi)d\xi\\
\sim&\sum_{j\in{\ensuremath{\mathbb{N}^{{}}}}_0}{\underbrace}{(2\pi)^{-\dim M}\int_{{\ensuremath{\mathbb{R}^{\dim M}}}}e^{i\langle x-y,\xi\rangle_{\ell_2(\dim M)}}\alpha_{m-j}(x,\xi)d\xi}_{=:k_{m-j}(x,y)}\end{aligned}$$ coincides locally with the kernel of $T$ modulo smoothing operators. Then, there exists $N\in{\ensuremath{\mathbb{N}^{{}}}}$ (any $N>\dim M+\Re(m)$ will do) such that the operator $T^{\mathrm{reg}}$ with kernel $k^{\mathrm{reg}}:=k-\sum_{j=0}^Nk_{m-j}$ is of trace class and $$\begin{aligned}
{\ensuremath{{\ensuremath{{\operatorname}{tr}}}_{\mathrm{KV}}}}T={\ensuremath{{\operatorname}{tr}}}T^{\mathrm{reg}}=\int_Mk^{\mathrm{reg}}(x,x)d{\ensuremath{\mathrm{vol}}}_M(x).\end{aligned}$$ This formula has a very important consequence, namely that the Kontsevich-Vishik trace of differential operators ($m\in{\ensuremath{\mathbb{N}^{{}}}}_0$ and ${\forall}j\in{\ensuremath{\mathbb{N}^{{}}}}_{>m}:\ \alpha_{m-j}=0$) vanishes.
If $Q=\Delta+{\ensuremath{{\operatorname}{pr}}}_{\ker\Delta}$ where $\Delta$ is an elliptic differential operator and ${\ensuremath{{\operatorname}{pr}}}_{\ker\Delta}$ the projection onto its kernel, then $\Gamma(-z)TQ^{-z}$ is the Mellin transform of $Te^{-tQ}$ and, provided $\Delta$ is non-negative, $z\mapsto\Gamma(-z)\zeta(T,Q)(-z)$ is the inverse Mellin transform of $t\mapsto{\ensuremath{{\operatorname}{tr}}}Te^{-tQ}$. For $T=1$ the function $t\mapsto {\ensuremath{{\operatorname}{tr}}}e^{-tQ}$ is called the (generalized) heat-trace generated by $-Q$. An important application of these heat-traces is given in the heat-trace proof [@atiyah-bott-patodi] of the Atiyah-Singer index theorem; namely, if $D$ is a differential operator, then its Fredholm index is given by ${\ensuremath{{\operatorname}{ind}}}D={\ensuremath{{\operatorname}{tr}}}(e^{-t D^*D}- e^{-t DD^*})$.
Using the inverse Mellin mapping theorem [@azzali-levy-neira-paycha], it follows that ${\ensuremath{{\operatorname}{tr}}}Te^{-tQ}$ has an asymptotic expansion $\frac{1}{q}\sum_{j\in{\ensuremath{\mathbb{N}^{{}}}}_0}a_jt^{-d_j}+O(t^{-\gamma})$ where $d_j=\frac{j-a-n}{q}$, $a_j=-\frac{1}{q}{\ensuremath{{\operatorname}{res}}}(TQ^{-d_j})$ for $d_j>0$, and some appropriate $\gamma>\frac{a+\dim M}{q}$. If $T$ is a differential operator plus a trace class operator $T_0$, then we also know that $a_j={\ensuremath{{\operatorname}{tr}}}T_0-\frac{1}{q}{\ensuremath{{\operatorname}{res}}}(T\ln Q)$ if $d_j=0$.
For instance, let $Q$ be the positive Laplace-Beltrami operator on a compact Riemannian $C^\infty$-manifold $M$ of even dimension and without boundary. Then, $$\begin{aligned}
{\ensuremath{{\operatorname}{tr}}}e^{-tQ}=\frac{{\ensuremath{\mathrm{vol}}}(M)}{(4\pi t)^{\frac{\dim M}{2}}}+\frac{\mathrm{total\ curvature}(M)}{3(4\pi)^{\frac{\dim M}{2}}t^{\frac{\dim M}{2}-1}}+\mathrm{higher\ order\ terms}\end{aligned}$$ and, more generally, for $\dim M\in{\ensuremath{\mathbb{N}^{{}}}}$, the heat-trace has an expansion $$\begin{aligned}
{\ensuremath{{\operatorname}{tr}}}e^{-tQ}=(4\pi t)^{-\frac{\dim M}{2}}\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}_0}A_kt^{\frac{k}{2}}\end{aligned}$$ for $t\searrow0$. The $A_k$ are called heat-invariants and are spectral invariants of Laplace type operators $\nabla^*\nabla+V$ generating the corresponding “heat-semigroup” where $\nabla$ is a connection on a vector bundle over $M$ and $V$ is a multiplication operator called the potential. More precisely, the heat-invariants are functorial algebraic expressions in the jets of homogeneous components of $Q$, i.e., if $Q$ is geometric, then the heat-invariants carry information about the underlying metric and connection on $M$. We can see this quite nicely in the Laplace-Beltrami case, in which the volume and total curvature appear as lowest order heat-invariants and the dimension of the manifold in the pole order.
These properties of $\zeta$-functions and heat-traces are fundamental in geometric analysis which begs the questions whether or not they extend to non-commutative settings. Such questions have also been studied on the non-commutative torus, the Moyal plane, the Groenewold-Moyal star product, the non-commutative ${\varphi}^4$ theory on the $4$-torus, and $SU_q(2)$ [@azzali-levy-neira-paycha; @carey-gayral-rennie-sukochev; @carey-rennie-sadaev-sukochev; @connes-fathizadeh; @connes-moscovici; @connes-tretkoff; @dabrowski-sitarz-curved; @dabrowski-sitarz-asymmetric; @fathi; @fathi-ghorbanpour-khalkhali; @fathi-khalkhali; @fathizadeh; @fathizadeh-khalkhali-scalar-4; @fathizadeh-khalkhali-scalar-2; @fathizadeh-khalkhali-gauss-bonnet; @gayral-iochum-vassilevich; @iochum-masson; @levy-neira-paycha; @liu; @matassa; @sadeghi; @sitarz; @vassilevich-I; @vassilevich-II].
The non-commutative torus ${\ensuremath{\mathbb{T}^{n}}}_{\vartheta}$ is a deformation of the torus ${\ensuremath{\mathbb{R}^{n}}}/{{\ensuremath{\mathbb{Z}^{n}}}}$ using a real anti-symmetric $n\times n$ matrix ${\vartheta}$ as a twist. The corresponding C\*-algebra $A_{\vartheta}$ which has a dense subalgebra consisting of elements $a=\sum_{k\in{\ensuremath{\mathbb{Z}^{n}}}}a_kU_k$ where $(a_k)_{k\in{\ensuremath{\mathbb{Z}^{n}}}}$ is in the Schwartz space ${\ensuremath{\mathcal{S}^{{}}}}({\ensuremath{\mathbb{Z}^{n}}})$, $U_0=1$, each $U_k$ is unitary, and $U_kU_l=e^{-\pi i\langle k,{\vartheta}l\rangle_{\ell_2(n)}}U_{k+l}=e^{-2\pi i\langle k,{\vartheta}l\rangle_{\ell_2(n)}}U_lU_k$. The construction of $A_{\vartheta}$ and its algebra of pseudo-differential operators $\Psi({\ensuremath{\mathbb{T}^{n}}}_{\vartheta})$ is chosen in such a way that ${\vartheta}\to0$ recovers $A_0=C^\infty{\ensuremath{\left}}({\ensuremath{\mathbb{R}^{n}}}/{{\ensuremath{\mathbb{Z}^{n}}}}{\ensuremath{\right}})$ and $\Psi({\ensuremath{\mathbb{T}^{n}}}_0)$ is the algebra of classical pseudo-differential operators on ${\ensuremath{\mathbb{R}^{n}}}/{{\ensuremath{\mathbb{Z}^{n}}}}$. Then, it is possible to define ${\ensuremath{\mathbb{T}^{n}}}_{\vartheta}$ versions of the Wodzicki residue and Kontsevich-Vishik trace and show many of the properties described above. In particular, in [@levy-neira-paycha] it is shown that $\zeta$-functions are meromorphic on ${\ensuremath{\mathbb{C}^{{}}}}$ with isolated simple poles at $\frac{j-a-n}{q}$ for $j\in{\ensuremath{\mathbb{N}^{{}}}}_0$ and that the heat-trace pole order is $\frac{n}{2}$.
Similarly, it is possible to introduce a Dirac operator $D_q$ on $SU_q(2)$ [@kaad-senior] taking symmetries into account while constructing a twisted modular spectral triple and insuring that the classical limit $q\to 1$ recovers the Dirac operator on $SU(2)$. Using $D_q$, $\zeta$-functions and heat kernels can be constructed on $SU_q(2)$. Heat kernel expansions, heat-traces, $\zeta$-functions and their asymptotics, and their relation to the Dixmier trace (which for pseudo-differential operators coincides with the Wodzicki residue of the $\zeta$-function [@connes-action-functional]) have been studied in this context [@carey-gayral-rennie-sukochev; @carey-rennie-sadaev-sukochev; @matassa].
In this article, we want to add another layer of abstraction and consider a number of quantum semigroups. While it is perfectly possible to define a “twisted” Laplace-Beltrami operator and, more generally pseudo-differential operators, on the non-commutative torus or $SU_q(2)$ by introducing a non-commutative twist on the classical algebra of pseudo-differential operators on the torus or $SU(2)$, such a construction is not straight forward, if at all possible, for many interesting quantum semigroups. Instead, we want to make use of the fact that Brownian motion solves the heat equation. In other words, the Laplace operator and the heat-semigroup can be recovered using Brownian motion. Hence, our approach in this article is to consider driftless Gaussian processes on quantum semigroups that allow us to define an appropriate notion of Brownian motion and use these Markov semigroups to define Laplace-type operators and “heat-semigroups”.
The study of Lévy processes on \*-bialgebras (cf. [@schurmann]) gives a very satisfying theory of independent increment processes in the non-commutative framework. This was initiated in the late eighties by Accardi, Schürmann, and von Waldenfels [@accardi-schurmann-waldenfels]. The theory generalizes the notion of Lévy processes on semigroups and allows for various types of familiar Lévy processes. The most important of these types in this article, and arguably in general, is the notion of a Gaussian Lévy process. The construction of these Lévy processes is purely algebraic. Attempts at extending these methods to the C\*-algebraic framework have made great progress. Lindsay and Skalski have completed this work relying on the assumption that the generator of the Lévy process is bounded [@lindsay-skalski1; @lindsay-skalski2; @lindsay-skalski3]. More recently, Cipriani, Franz and Kula [@cipriani-franz-kula] have developed a characterization in terms of translation invariant quantum Markov semigroups on compact quantum groups that does not assume the generator to be bounded. At the time of writing an unpublished approach by Das and Lindsay will give a full characterisation for reduced compact quantum groups again which allows unbounded generators. In this article, we will introduce a C\*-algebraic Lévy process methodology that does not rely on the boundedness of the generator but will require the C\*-algebra to be universal and “nicely-generated” in some sense. This will allow us to develop Gaussian processes on C\*-bialgebras (whose generators in general are not bounded) and then by a canonical choice of Gaussian process which we will take to be Brownian motion we will have definitions for a heat-semigroup on our examples of C\*-bialgebras.
The Toeplitz algebra is an interesting choice of algebra to consider in this context since it does not have a twist structure of the form allowing us to directly model pseudo-differential operators, yet defining Brownian motion is very natural. Hence, we will start by formally introducing the Toeplitz algebra ${\ensuremath{\mathcal{T}^{{}}}}$ and give an overview of convolution semigroups (which contain the notion of Lévy processes) in section \[sec:semigroups\]. Since Brownian motion is classically generated by the Laplace-Beltrami operator, we define a class of operators (polyhomogeneous operators) on the Toeplitz algebra which play a similar role to classical pseudo-differential operators in section \[sec:zeta\], as well as their $\zeta$-functions. Then, we will study the heat-semigroup and $\zeta$-regularized heat-trace in section \[sec:heat\].
While the dynamics of the Toeplitz algebra are generated by the circle ${\partial}B_{{\ensuremath{\mathbb{C}^{{}}}}}\cong{\ensuremath{\mathbb{R}^{{}}}}/{2\pi{\ensuremath{\mathbb{Z}^{{}}}}}$, it is not obtained from “twisting” the product on $C({\partial}B_{{\ensuremath{\mathbb{C}^{{}}}}})$. In particular, it is not merely some ${\ensuremath{\mathbb{T}^{{}}}}_{\vartheta}$. In order to relate these results to more “classical” scenarios, we will consider the discrete Heisenberg group algebra in sections \[sec:heisenberg\] and \[sec:heisenberg-Z-complex\], and non-commutative tori in section \[sec:non-com-torus\]. In particular, we can relate the heat-traces of discrete Heisenberg group algebras and non-commutative tori to the “classical” heat-traces on tori.
Finally, we will consider $SU_q(2)$ which, although being a “twisted” manifold, does not have a Brownian motion. Instead all driftless Gaussian semigroups are generated by constant multiples of a unique operator (which is not the Laplacian on $SU(2)$). Still, this family of driftless Gaussians can be regularized and is formally very similar to the Brownian motion on the Toeplitz algebra.
Our main observations are the following.
1. On the Toeplitz algebra, $\zeta$-functions of polyhomogeneous operators have at most simple poles in the set ${\ensuremath{\left}}\{\frac{-2-d_\iota}{\delta};\ \iota\in I{\ensuremath{\right}}\}$ where the $d_\iota$ are the degrees of homogeneity and $\delta$ plays the same role $q$ did above. Furthermore, the “heat-trace” can be $\zeta$-regularized, has a first order pole in zero, and the sequence of heat coefficients $(A_k)_{k\in{\ensuremath{\mathbb{N}^{{}}}}_0}$ satisfies $A_0=-2\pi$ and ${\forall}k\in{\ensuremath{\mathbb{N}^{{}}}}:\ A_k=0$. This is exactly what we would expect to see if the Toeplitz algebra were a $2$-dimensional manifold of “volume” $-2\pi$ (all other heat coefficients vanishing).
2. In the case of the discrete Heisenberg group algebra ${\ensuremath{\mathbb{H}^{{}}}}_N$ we consider two cases; namely, the twist being an abstract unitary or having a complex twist.
1. If the twist is an abstract unitary, then $\zeta$-functions of polyhomogeneous operators have isolated first order poles in the set ${\ensuremath{\left}}\{\frac{-2N-1-d_{\iota}}{\delta};\ \iota\in I{\ensuremath{\right}}\}$. This corresponds to the classical case of a $2N+1$-dimensional manifold. The heat-trace however is given by $-{\ensuremath{{\operatorname}{tr}}}\circ S$ where $S$ is the heat-semigroup on the ${\ensuremath{\mathbb{R}^{2N}}}/{2\pi{\ensuremath{\mathbb{Z}^{2N}}}}$, i.e., the heat-trace appears as if ${\ensuremath{\mathbb{H}^{{}}}}_N$ were a $2N$-torus with all heat-coefficients multiplied by $-1$.
2. If we consider a complex twist, then $\zeta$-functions of polyhomogeneous operators have isolated first order poles in the set ${\ensuremath{\left}}\{\frac{-2N-d_{\iota}}{\delta};\ \iota\in I{\ensuremath{\right}}\}$ and the heat-trace coincides with the heat-trace on ${\ensuremath{\mathbb{R}^{2N}}}/{2\pi{\ensuremath{\mathbb{Z}^{2N}}}}$. In other words, ${\ensuremath{\mathbb{H}^{{}}}}_N$ looks exactly like a $2N$-dimensional torus.
3. The non-commutative torus $A_{\vartheta}^N$ is closely related to the discrete Heisenberg group algebra case. As such we will consider two cases again; (a) ${{\ensuremath{\mathfrak{T}^{{}}}}}$ twists that are abstract unitaries and (b) ${{\ensuremath{\mathfrak{T}^{{}}}}}$ complex twists. It is also possible to add another $T'$ complex twists to the case (a) without changing the results.
1. If the twists are abstract unitaries, then $\zeta$-functions of polyhomogeneous operators have isolated first order poles in ${\ensuremath{\left}}\{\frac{-N-{{\ensuremath{\mathfrak{T}^{{}}}}}-d_{\iota}}{\delta};\ \iota\in I{\ensuremath{\right}}\}$. This corresponds to the classical case of an $N+{{\ensuremath{\mathfrak{T}^{{}}}}}$-dimensional manifold. The heat-trace however is given by $(-1)^{{\ensuremath{\mathfrak{T}^{{}}}}}{\ensuremath{{\operatorname}{tr}}}\circ S$ where $S$ is the heat-semigroup on the ${\ensuremath{\mathbb{R}^{N}}}/{2\pi{\ensuremath{\mathbb{Z}^{N}}}}$, i.e., the heat-trace appears as if $A_{\vartheta}^N$ were an $N$-torus with all heat-coefficients multiplied by $(-1)^{{\ensuremath{\mathfrak{T}^{{}}}}}$.
2. If we consider a complex twists, then $\zeta$-functions of polyhomogeneous operators have isolated first order poles in the set ${\ensuremath{\left}}\{\frac{-N-d_{\iota}}{\delta};\ \iota\in I{\ensuremath{\right}}\}$ and the heat-trace coincides with the heat-trace on ${\ensuremath{\mathbb{R}^{N}}}/{2\pi{\ensuremath{\mathbb{Z}^{N}}}}$. In other words, $A_{\vartheta}^N$ looks exactly like an $N$-dimensional torus.
4. $SU_q(2)$ is a somewhat special case in the list of quantum semigroups we consider here since it does not have a Brownian motion. Hence, there is no heat-semigroup. However, there is still a class of driftless Gaussians that we can consider in lieu of alternatives. Their traces can be $\zeta$-regularized and have a pole in zero which is of order $\frac32$. This corresponds to a $3$-dimensional manifold ($SU_q(2)$ is a twisted $3$-dimensional Calabi-Yau algebra and $SU(2)$ is isomorphic to the $3$-sphere). The sequence of corresponding “heat-coefficients” $(A_k)_{k\in{\ensuremath{\mathbb{N}^{{}}}}_0}$ is given by $A_0=-2\pi^2r^{-\frac{3}{2}}$ and ${\forall}k\in{\ensuremath{\mathbb{N}^{{}}}}:\ A_k=0$ where $r\in{\ensuremath{\mathbb{R}^{{}}}}_{>0}$ is a parameter describing the family of driftless Gaussians on $SU_q(2)$. Hence, we still obtain consistent results regarding dimensionality of $SU_q(2)$ but interpreting the “heat-coefficient” $A_0$ as volume would be a bit of a stretch as it is also negative in the $SU(2)$ case. Furthermore, $\zeta$-functions of polyhomogeneous operators have isolated first order poles in the set ${\ensuremath{\left}}\{\frac{-3-d_{\iota}}{\delta};\ \iota\in I{\ensuremath{\right}}\}$.
#### **Acknowledgements**
The authors would like to express his gratitude to Prof. Martin Lindsay and Prof. Simon Scott for inspiring comments and conversations which helped us to develop the work presented in this article. The first author was funded by the Faculty of Science and Technology at Lancaster University.
Convolution semigroups and the Toeplitz algebra {#sec:semigroups}
===============================================
In this section we will introduce the Toeplitz algebra as a C\*-bialgebra. This has been introduced previously in [@aukhadiev-grigoryan-lipacheva]. We will characterize the Schürmann triples on this C\*-bialgebra which generalizes the notion of Lévy processes on a compact topological semigroup.
This will lead to a natural choice for Brownian motion and in later sections we will calculate important quantities associated to this semigroup that in the classical setting give information about the structure of the manifold involved.
The universal C\*-algebra generated by the right shift operator $R:\ \ell_2({\ensuremath{\mathbb{N}^{{}}}}_0)\to \ell_2({\ensuremath{\mathbb{N}^{{}}}}_0)$ such that $$\begin{aligned}
R(\lambda_0,\lambda_1,\dots)= (0,\lambda_0,\lambda_1,\dots)\end{aligned}$$ is called the Toeplitz algebra and denoted ${\ensuremath{\mathcal{T}^{{}}}}$.
The Toeplitz algebra has a dense \*-subalgebra with basis given by $R_{n,m}=R^nR^{*m}$. We will denote this sub \*-algebra ${\ensuremath{\mathcal{T}^{{}}}}_0$. For a more detailed account of the Toeplitz algebra see [@murphy].
We will proceed to define C\*-bialgebras, these are the non-commutative analogue to topological semigroups with identity in the same sense that C\*-algebras are a non-commutative analogue to locally compact Hausdorff topological spaces and compact quantum groups are non-commutative analogues to compact groups.
A \*-bialgebra is a unital \*-algebra $A$ with unital \*-homomorphisms $\Delta:\ A\to A\otimes A$ and ${\ensuremath{\varepsilon}}:\ A\to {\ensuremath{\mathbb{C}^{{}}}}$ that satisfy $$\begin{aligned}
(\Delta\otimes {{\operatorname}{id}})\circ\Delta=({{\operatorname}{id}}\otimes \Delta)\circ \Delta \quad\text{ and }\quad ({\ensuremath{\varepsilon}}\otimes {{\operatorname}{id}})\circ\Delta={{\operatorname}{id}}=({{\operatorname}{id}}\otimes {\ensuremath{\varepsilon}})\circ \Delta\end{aligned}$$ where $\otimes$ is the algebraic tensor product.
A C\*-bialgebra is a unital C\*-algebra $A$ with unital C\*-homo–morphisms $\Delta:\ A\to A\otimes A$ and ${\ensuremath{\varepsilon}}:\ A\to {\ensuremath{\mathbb{C}^{{}}}}$ that satisfy $$\begin{aligned}
(\Delta\otimes {{\operatorname}{id}})\circ\Delta=({{\operatorname}{id}}\otimes \Delta)\circ \Delta \quad\text{ and }\quad ({\ensuremath{\varepsilon}}\otimes {{\operatorname}{id}})\circ\Delta={{\operatorname}{id}}=({{\operatorname}{id}}\otimes {\ensuremath{\varepsilon}})\circ \Delta\end{aligned}$$ where $\otimes$ is the spatial tensor product.
If we also required that the sets $\Delta(A)(1\otimes A)$ and $\Delta(A)(1\otimes A)$ were dense in $A\otimes A$ in the definition of C\*-bialgebra we would have the definition of a compact quantum group. These conditions are the quantum cancellation properties but will not be required for this.
The map $\Delta$ is called to co-multiplication and the first identity involving only $\Delta$ is called co-associativity. This is to mirror the multiplication of a semigroup. The map ${\ensuremath{\varepsilon}}$ is called the co-unit and the second identity is called the co-unital property. This is analogous to the identity element of a semigroup.
The Toeplitz algebra can be given the structure of a C\*-bialgebra with co-multiplication $\Delta(R_{n,m})=R_{n,m}\otimes R_{n,m}$ and co-unit ${\ensuremath{\varepsilon}}(R_{n,m})=1$ for all $n,m\in{\ensuremath{\mathbb{N}^{{}}}}_0$. Furthermore, the restriction of these maps to ${\ensuremath{\mathcal{T}^{{}}}}_0$ makes ${\ensuremath{\mathcal{T}^{{}}}}_0$ a \*-bialgebra.
As the Toeplitz algebra is a universal C\*-algebra generated by the isometry $R$, we only need to show that $\Delta(R)^*\Delta(R)=I_{{\ensuremath{\mathcal{T}^{{}}}}\otimes {\ensuremath{\mathcal{T}^{{}}}}}$ and ${\ensuremath{\varepsilon}}(R)^*{\ensuremath{\varepsilon}}(R)=1$. This is straightforward: $$\begin{aligned}
\Delta(R)^*\Delta(R)&=(R^*\otimes R^*)(R\otimes R)=R^*R\otimes R^*R=I_{{\ensuremath{\mathcal{T}^{{}}}}}\otimes I_{{\ensuremath{\mathcal{T}^{{}}}}}=I_{{\ensuremath{\mathcal{T}^{{}}}}\otimes{\ensuremath{\mathcal{T}^{{}}}}}\end{aligned}$$ and $$\begin{aligned}
{\ensuremath{\varepsilon}}(R)^*{\ensuremath{\varepsilon}}(R)&=(1^*)(1)=1.\end{aligned}$$ The fact that the maps restricted to ${\ensuremath{\mathcal{T}^{{}}}}_0$ gives it the structure of a \*-bialgebra is easily seen by the identity $\Delta(R_{n,m})=R_{n,m}\otimes R_{n,m}$.
Now that we have a \*-bialgebra we can appeal to the theory of Lévy processes on \*-bialgebras [@schurmann; @lindsay-skalski].
For a pre-Hilbert space $D$, let $L^*(D)$ denote the set of adjointable operators, that is, linear maps $T:\ D\to D$ such that there exists $T^*:\ D\to D$ such that ${\forall}x,y\in D:\ \langle x,Ty \rangle=\langle T^*x,y \rangle$. This is clearly a unital \*-algebra.
Let $A$ be a \*-bialgebra . A Sch" urmann triple $({\varrho},\eta,L)$ consists of a unital \*-homomorphism ${\varrho}:\ A\to L^*(D)$ for some pre-Hilbert space $D$, a ${\varrho}-{\ensuremath{\varepsilon}}$ cocycle $\eta:\ A\to D$, i.e., $$\begin{aligned}
\eta(ab)=\eta(a){\ensuremath{\varepsilon}}(b)+{\varrho}(a)\eta(b)\end{aligned}$$ and a \*-linear functional $L:\ A\to{\ensuremath{\mathbb{C}^{{}}}}$ such that $$\begin{aligned}
L(ab)=L(a){\ensuremath{\varepsilon}}(b)+{\ensuremath{\varepsilon}}(a)L(b)+{\ensuremath{\left}}\langle \eta(a^*),\eta(b){\ensuremath{\right}}\rangle.\end{aligned}$$ A Schürmann triple will be called surjective if the cocycle $\eta$ has dense image.
If we let $\overline{D}$ be the Hilbert space completion of $D$ and we consider unital \*-homomorphisms ${\varrho}:\ {\ensuremath{\mathcal{T}^{{}}}}_0\to L^*(D)$, we can see that ${\varrho}(R)\in L^*(D)$ is an isometry and can therefore be extended to $B{\ensuremath{\left}}(\overline{D}{\ensuremath{\right}})$. As ${\ensuremath{\mathcal{T}^{{}}}}_0$ is generated by $R$ we can now use induction on word length to see that ${\varrho}(R_{n,m})$ can be extended to $B{\ensuremath{\left}}(\overline{D}{\ensuremath{\right}})$ for all $n,m\in {\ensuremath{\mathbb{N}^{{}}}}_0$.
Therefore we can replace the pre-Hilbert space in the Schürmann triple definition by a Hilbert space and the adjointable operators by the bounded operators.
We will now proceed to characterize the Schürmann triples on ${\ensuremath{\mathcal{T}^{{}}}}_0$.
\[thm:isolp\] Given an isometry $V\in B(H)$ on some Hilbert space $H$, $h\in H$, and $\lambda\in {\ensuremath{\mathbb{R}^{{}}}}$, there exists a unique Schürmann triple $({\varrho},\eta,L)$ on ${\ensuremath{\mathcal{T}^{{}}}}$ such that $$\begin{aligned}
{\varrho}(R)=V,\quad \eta(R)=h,\quad\text{ and }\quad L(R-R^*)=i\lambda.\end{aligned}$$ Furthermore, every Schürmann triple arises this way.
Clearly given any Schürmann triple we can see that ${\varrho}(R)$ is an isometry on some Hilbert space $H$. By definition $\eta(R)$ is an element of $H$ and by \*-linearity $L(R-R^*)$ is a purely imaginary number.
Starting with $V\in B(H), h\in H$, and $\lambda \in {\ensuremath{\mathbb{R}^{{}}}}$, we easily construct ${\varrho}:{\ensuremath{\mathcal{T}^{{}}}}\to B(H)$ by universality where ${\varrho}(R)=V$. If we let $\eta(R)=h$, $\eta(R^*)=-Vh$ and $\eta(ab)=\eta(a){\ensuremath{\varepsilon}}(b)+{\varrho}(a)\eta(b)$ for all $a,b\in {\ensuremath{\mathcal{T}^{{}}}}_0$, we will see that $\eta(R^*R)=0$ and $\eta:\ {\ensuremath{\mathcal{T}^{{}}}}_0\to H$ is well-defined.
Finally, let $L(R-R^*)=i\lambda$, $L(R+R^*)=-{\ensuremath{\left}}\langle h,h{\ensuremath{\right}}\rangle$, and $L(ab)=L(a){\ensuremath{\varepsilon}}(b)+{\ensuremath{\varepsilon}}(a)L(b)+{\ensuremath{\left}}\langle \eta(a^*),\eta(b){\ensuremath{\right}}\rangle$ for all $a,b\in {\ensuremath{\mathcal{T}^{{}}}}_0$. Again, we see that $L(R^*R)=0$ and $L:{\ensuremath{\mathcal{T}^{{}}}}_0\to {\ensuremath{\mathbb{C}^{{}}}}$ is well-defined.
A convolution semigroup of states is a family of linear functionals ${\varphi}_t:\ A\to {\ensuremath{\mathbb{C}^{{}}}}$ such that ${\varphi}_t(a^*a)\geq 0$ and ${\varphi}_t(1)=1$ for all $t\geq 0$, i.e., ${\varphi}_t$ is a \*-algebra state for all $t\in{\ensuremath{\mathbb{R}^{{}}}}_{\ge 0}$ and $$\begin{aligned}
{\varphi}_t*{\varphi}_s:=({\varphi}_s\otimes {\varphi}_t)\circ\Delta={\varphi}_{t+s},\quad {\varphi}_0={\ensuremath{\varepsilon}},\quad \text{ and }\quad \lim_{r\to 0}{\varphi}_r(a)={\ensuremath{\varepsilon}}(a)\end{aligned}$$ for all $t,s\in {\ensuremath{\mathbb{R}^{{}}}}_{\ge0}$ and $a\in A$.
A generating functional is a linear functional $L:A\to {\ensuremath{\mathbb{C}^{{}}}}$ such that $$L(1)=0, \quad L(a^*)=\overline{L(a)},\quad \text{ and } L((a-{\ensuremath{\varepsilon}}(a))^*(a-{\ensuremath{\varepsilon}}(a)))\geq 0$$ for all $a\in A$.
Schürmann proved that following are in one-to-one correspondence
- Schürmann triples on $A$;
- Convolution semigroups of states on $A$;
- Generating functionals on $A$.
In the classical setting given a Lévy process $X_t$ on a compact semigroup the associated probabiltity distributions $\mu_t$ form a convolution semigroup of probability measures. This motivates the definition of Lévy processes on \*-bialgebras as states act as a noncommutative analogue to probability measures by results such as the Markov-Riesz-Kakutani theorem.
In the definition of the Schürmann triple the functional $L$ is the generating functional. These will assist us in constructing contraction semigroups of operators. To extend these results to the C\*-algebraic level we will introduce the symmetric Fock space.
Let $H$ be a Hilbert space. The symmetric Fock space is given by $$\begin{aligned}
{\ensuremath{\mathbb{C}^{{}}}}\Omega\oplus\bigoplus_{n\geq 1} H^{\vee n}\end{aligned}$$ where $\Omega$ is called the vacuum vector and $H^{\vee n}{\ensuremath{\subseteq}}H^{\otimes n}$ such that elements are unchanged by the action of permutation of tensor factors. The symmetric Fock space of $H$ is denoted by $\Gamma(H)$.
If $H=L_2({\ensuremath{\mathbb{R}^{{}}}}_{\ge0}; K)$ for some Hilbert space $K$, we will call $\Gamma(H)={\ensuremath{\mathcal{F}_{{}}}}$ and for $I\subseteq {\ensuremath{\mathbb{R}^{{}}}}_{\ge0}$ call $\Gamma(L_2(I; K))={\ensuremath{\mathcal{F}_{I}}}$.
Note that the so called exponential property of Fock spaces with $$\begin{aligned}
L_2([0,b_1);K)\oplus L_2([b_1,b_2);K)\oplus \dots \oplus L_2([b_n,\infty);K)\cong L_2({\ensuremath{\mathbb{R}^{{}}}}_{\ge0};K)\end{aligned}$$ gives the decomposition $$\begin{aligned}
{\ensuremath{\mathcal{F}_{[0,b_1)}}}\otimes{\ensuremath{\mathcal{F}_{[b_1,b_2)}}}\otimes\dots\otimes {\ensuremath{\mathcal{F}_{[b_n,\infty)}}}\cong {\ensuremath{\mathcal{F}_{{}}}}\end{aligned}$$ for all $n\in {\ensuremath{\mathbb{N}^{{}}}}$ and $0< b_1<b_2<\dots<b_n$.
A very important subspace of the Fock space is the the space of exponential vectors given by the linear span of the vectors $$\begin{aligned}
e(u)={\ensuremath{\left}}(1,u,\frac{u^{\otimes 2}}{\sqrt{2}},\dots ,\frac{u^{\otimes n}}{\sqrt{n!}},\dots{\ensuremath{\right}})\in {\ensuremath{\mathcal{F}_{{}}}}\end{aligned}$$ for all $u\in L_2({\ensuremath{\mathbb{R}^{{}}}}_{\ge0};K)$. This is a dense subspace of ${\ensuremath{\mathcal{F}_{{}}}}$ and we will denote it by ${\ensuremath{\mathcal{E}_{{}}}}$.
Using this characterization of Schürmann processes we can now appeal to the Representation Theorem (Theorem 1.15 [@franz]) to realize our Lévy process on the Fock space. This gives us a family of adapted unital weak\*-homomorphisms $j_{s,t}:\ {\ensuremath{\mathcal{T}^{{}}}}_0\to L^\dagger({\ensuremath{\mathcal{E}_{{}}}})$ where $L^\dagger({\ensuremath{\mathcal{E}_{{}}}})$, in the context of ${\ensuremath{\mathcal{E}_{{}}}}$ being a subspace of a Hilbert space, is the family of linear operators $T:\ {\ensuremath{\mathcal{E}_{{}}}}\to {\ensuremath{\mathcal{F}_{{}}}}$ such that the adjoint $T^*$ has domain which contains ${\ensuremath{\mathcal{E}_{{}}}}$.
More specifically, $j_{s,t}:A\ \to L^\dagger({\ensuremath{\mathcal{E}_{{}}}})$ is a family such that $j_{s,t}(a)$ acts non-identically only on ${\ensuremath{\mathcal{F}_{[s,t)}}}$ (adapted), $j_{s,t}(1)={{\operatorname}{id}}_{{\ensuremath{\mathcal{F}_{{}}}}}$ (unital), satisfies the weak multiplicative property $$\begin{aligned}
\langle x,j_{s,t}(a^*b)y\rangle=\langle j_{s,t}(a)x,j_{s,t}(b)y\rangle,
\end{aligned}$$ and is a Fock space Lévy process, i.e., a family $(j_{s,t})_{0\leq s\leq t}$ of maps $A\to L^\dagger({\ensuremath{\mathcal{E}_{{}}}})$ such that
- $ j_{t,t} ={{\operatorname}{id}}_{{\ensuremath{\mathcal{F}_{{}}}}}$,
- $ (j_{r,s}\otimes j_{s,t})\circ \Delta =j_{r,t}$,
- $\lim_{t\to s}\langle e(0), j_{s,t}(a)e(0))\rangle =1$ and
- $\langle e(0), j_{s,t}(a)e(0))\rangle =\langle e(0), j_{s+r,t+r}(a)e(0))\rangle$.
The following result of Belton and Wills [@belton-wills] tells us that, because the Toeplitz algebra is nicely generated, algebraic unital weak\*-homomorphisms are enough to extend to C\*-algebraic unital homomorphisms.
There is a one-to-one correspondence between unital weak\*-homomorphisms $j:\ {\ensuremath{\mathcal{T}^{{}}}}_0\to L^\dagger({\ensuremath{\mathcal{E}_{{}}}})$ and unital C\*-homomorphisms $\hat{j}:\ {\ensuremath{\mathcal{T}^{{}}}}\to B({\ensuremath{\mathcal{F}_{{}}}})$.
Clearly given $\hat{j}:\ {\ensuremath{\mathcal{T}^{{}}}}\to B({\ensuremath{\mathcal{F}_{{}}}})$ then $j:=\hat{j}|_{{\ensuremath{\mathcal{T}^{{}}}}_0}$ defines a unital \*-homomorphism $j:\ {\ensuremath{\mathcal{T}^{{}}}}_0\to B({\ensuremath{\mathcal{F}_{{}}}})\subseteq L^\dagger({\ensuremath{\mathcal{E}_{{}}}})$.
Now let $j:\ {\ensuremath{\mathcal{T}^{{}}}}_0\to L^\dagger({\ensuremath{\mathcal{E}_{{}}}})$ this implies that $$\begin{aligned}
{{\left\lVert}{x}{\right\lVert}}^2=\langle x,j(R^*R)x\rangle=\langle j(R)x,j(R)x\rangle={{\left\lVert}{j(R)x}{\right\lVert}}^2\end{aligned}$$ for all $x\in {\ensuremath{\mathcal{E}_{{}}}}$. Since ${\ensuremath{\mathcal{E}_{{}}}}$ is dense in ${\ensuremath{\mathcal{F}_{{}}}}$, $j(R)$ can be extended to an isometry in $B({\ensuremath{\mathcal{F}_{{}}}})$. Adjointability implies that $j(R^*)$ is also bounded. We can now use induction on word length and linearity to show that $j(R_{n,m})\in B({\ensuremath{\mathcal{F}_{{}}}})$ for all $n,m\in {\ensuremath{\mathbb{N}^{{}}}}_0$. Now using universality there exists a unital C\*-homomorphism $\hat{j}:\ {\ensuremath{\mathcal{T}^{{}}}}\to B({\ensuremath{\mathcal{F}_{{}}}})$ such that $\hat{j}(R)=j(R)$.
There is a one-to-one correspondence between Schürmann triples on ${\ensuremath{\mathcal{T}^{{}}}}_0$ and Lévy processes $j_{s,t}:\ {\ensuremath{\mathcal{T}^{{}}}}\to B({\ensuremath{\mathcal{F}_{{}}}})$.
Given such a Lévy process we get a $C_0$-convolution semigroup of states [@lindsay-skalski1] on the C\*-algebra ${\ensuremath{\mathcal{T}^{{}}}}$ this is given by ${\varphi}_t(a)=\langle \Omega,j_{0,t}(a)\Omega\rangle$. Furthermore we get an associated $C_0$-semigroup $T(t):\ {\ensuremath{\mathcal{T}^{{}}}}\to {\ensuremath{\mathcal{T}^{{}}}}$ given by $T(t):=({{\operatorname}{id}}\otimes {\varphi}_t)\circ \Delta$.
Let $H={\ensuremath{\mathbb{C}^{{}}}}$, $V={{\operatorname}{id}}_{{\ensuremath{\mathbb{C}^{{}}}}}$, $h=1$, and $\lambda=0$ from Theorem \[thm:isolp\]. This is the natural choice of Brownian motion on the Toeplitz algebra. Firstly Schürmann triples are said to be Gaussian if and only if the associated unital \*-homomomorphism is of the form ${\varrho}=\epsilon$.
Furthermore, if the $V\in B(H)$ in Theorem \[thm:isolp\] is chosen to be unitary, then the associated Lévy process can be restricted to the quotient C\*-algebra ${\ensuremath{\mathcal{T}^{{}}}}/K(\ell_2({\ensuremath{\mathbb{N}^{{}}}}_0))\cong C({\partial}B_{{\ensuremath{\mathbb{C}^{{}}}}})$ the continuous functions on the circle group.
In the case above the associated Lévy process corresponds to the standard Brownian motion on the real line “wrapped” around the circle.
More explicitly the Schürmann triple on the dense \*-bialgebra ${\ensuremath{\mathcal{T}^{{}}}}_0$ is given by $$\begin{aligned}
{\varrho}(R_{n,m})={\ensuremath{\varepsilon}}(R_{n,m})=1,\quad \eta(R_{n,m})=n-m,\quad \text{ and }\quad L(R_{n,m})=-\frac{(n-m)^2}{2}.\end{aligned}$$
This has an associated $C_0$-convolution semigroup of states that acts on the dense \*-bialgebra by $$\begin{aligned}
{\varphi}_t(R_{n,m})=e^{-\frac{(n-m)^2}{2}t}.\end{aligned}$$
$\zeta$-regularized traces of polyhomogeneous operators {#sec:zeta}
=======================================================
In this section, we want to consider a class of operators that generate convolution semigroups on the Toeplitz algebra ${\ensuremath{\mathcal{T}^{{}}}}$ which resemble pseudo-differential operators. In particular, the generator of Brownian motion, i.e., our version of the Laplacian, is an operator of this type. We will then show, that these operators have $\zeta$-regularized traces, i.e., analogues of the Kontsevich-Vishik trace and residue trace. Recall the following properties of the Toeplitz algebra ${\ensuremath{\mathcal{T}^{{}}}}$ and generators of convolution semigroups on ${\ensuremath{\mathcal{T}^{{}}}}$.
1. The space ${\ensuremath{\mathcal{T}^{{}}}}_0:={\operatorname}{lin}{\ensuremath{\left}}\{R_{n,m};\ n,m\in{\ensuremath{\mathbb{N}^{{}}}}_0{\ensuremath{\right}}\}$ is a dense $^*$-subalgebra of ${\ensuremath{\mathcal{T}^{{}}}}$.
2. The co-unit ${\ensuremath{\varepsilon}}$ satisfies ${\forall}m,n\in{\ensuremath{\mathbb{N}^{{}}}}_0:\ {\ensuremath{\varepsilon}}(R_{n,m})=1$.
3. The co-multiplication $\Delta$ satisfies $$\begin{aligned}
{\forall}m,n\in{\ensuremath{\mathbb{N}^{{}}}}_0\ {\forall}\alpha\in{\ensuremath{\mathbb{C}^{{}}}}:\ \Delta(\alpha R_{n.m})=\alpha R_{n,m}\otimes R_{n,m}.
\end{aligned}$$
4. Let $L$ be the generating functional of a convolution semigroup $\omega$. Then, the corresponding operator semigroup is given by $t\mapsto ({{\operatorname}{id}}\otimes\omega_t)\circ\Delta$ and has generator $({{\operatorname}{id}}\otimes L)\circ\Delta$.
An operator $H$ on ${\ensuremath{\mathcal{T}^{{}}}}$ is called polyhomogeneous if and only if there exists a functional $L:\ {\ensuremath{\mathcal{T}^{{}}}}_0\to{\ensuremath{\mathbb{C}^{{}}}}$ such that $H|_{{\ensuremath{\mathcal{T}^{{}}}}_0}=({{\operatorname}{id}}\otimes L)\circ\Delta$ and $$\begin{aligned}
{\exists}r\in{\ensuremath{\mathbb{R}^{{}}}}\ {\exists}I{\ensuremath{\subseteq}}{\ensuremath{\mathbb{N}^{{}}}}\ {\exists}\alpha\in \ell_1(I)\ {\exists}d\in({\ensuremath{\mathbb{C}^{2}}}_{\Re(\cdot)<r})^I\ {\forall}m,n\in{\ensuremath{\mathbb{N}^{{}}}}_0:\ L(R_{n,m})=\sum_{\iota\in I}\alpha_\iota \sigma_{d_\iota}(m,n)
\end{aligned}$$ where ${\ensuremath{\mathbb{C}^{{}}}}_{\Re(\cdot)<r}:=\{z\in{\ensuremath{\mathbb{C}^{{}}}};\ \Re(z)<r\}$, ${\ensuremath{\mathbb{C}^{2}}}_{\Re(\cdot)<r}:={\ensuremath{\mathbb{C}^{{}}}}_{\Re(\cdot)<r}\times{\ensuremath{\mathbb{C}^{{}}}}_{\Re(\cdot)<r}$, $\sigma_{d_\iota}:\ {\ensuremath{\mathbb{R}^{2}}}\to{\ensuremath{\mathbb{C}^{{}}}}$ is homogeneous of degree $d_\iota$, i.e., $$\begin{aligned}
{\forall}\lambda\in{\ensuremath{\mathbb{R}^{{}}}}_{>0}\ {\forall}\xi\in{\ensuremath{\mathbb{R}^{2}}}:\ \sigma_{d_\iota}(\lambda\xi)=\lambda^{d_\iota}\sigma_{d_\iota}(\xi),
\end{aligned}$$ and $\sum_{\iota\in I}\alpha_\iota \sigma_{d_\iota}(m,n)$ is absolutely convergent.
The generating functional of Brownian motion is given by $$\begin{aligned}
{\forall}m,n\in{\ensuremath{\mathbb{N}^{{}}}}_0:\ L_{BM}(R_{n,m})=-\frac{(n-m)^2}{2}={\ensuremath{\left}}(\frac{-1}{2}n^2m^0{\ensuremath{\right}})+{\ensuremath{\left}}(n^1m^1{\ensuremath{\right}})+{\ensuremath{\left}}(\frac{-1}{2}n^0m^2{\ensuremath{\right}}).
\end{aligned}$$ Thus, the generator $H_{BM}$ of the Brownian motion semigroup $B$ is polyhomogeneous with finite $I$ and each $d_\iota=2$.
Since $2H_{BM}$ is the Laplace-Beltrami on compact Riemannian manifolds without boundary, we obtain the following definition of the Laplacian on ${\ensuremath{\mathcal{T}^{{}}}}$.
Let $H_{BM}=({{\operatorname}{id}}\otimes L_{BM})\circ\Delta$ be the generator of Brownian motion. Then, we call the (polyhomogeneous) operator $\Delta_{{\ensuremath{\mathcal{T}^{{}}}}}:=2H_{BM}$ the Laplacian on ${\ensuremath{\mathcal{T}^{{}}}}$.
We are now interested in $\zeta$-regularized traces of polyhomogeneous operators. Thus, in order to compute traces, the following results shine a light on their spectral properties.
Let $L$ be the generating functional of a convolution semigroup of states on ${\ensuremath{\mathcal{T}^{{}}}}$ and $H:=({{\operatorname}{id}}\otimes L)\circ\Delta$. Then, $H$ is a closed, densely defined operator and $$\begin{aligned}
{\forall}n\in{\ensuremath{\mathbb{N}^{{}}}}\ {\forall}\lambda\in{\ensuremath{\mathbb{R}^{{}}}}_{>\omega}:\ {{\left\lVert}{(\lambda-H)^{-1}}{\right\lVert}}\le\frac{1}{\lambda}.
\end{aligned}$$
Let ${\varphi}$ be the convolution semigroup of states generated by $L$. Then, $t\mapsto({{\operatorname}{id}}\otimes{\varphi}_t)\circ\Delta$ satisfies $$\begin{aligned}
{\forall}t\in{\ensuremath{\mathbb{R}^{{}}}}_{>0}:\ {{\left\lVert}{T(t)}{\right\lVert}}={{\left\lVert}{({{\operatorname}{id}}\otimes{\varphi}_t)\circ\Delta}{\right\lVert}}\le{{\left\lVert}{{{\operatorname}{id}}\otimes{\varphi}_t}{\right\lVert}}{{\left\lVert}{\Delta}{\right\lVert}}=1
\end{aligned}$$ and is a contraction semigroup on ${\ensuremath{\mathcal{T}^{{}}}}$. Since ${\ensuremath{\mathcal{T}^{{}}}}$ is a Banach space, the Theorem of Hille-Yosida-Phillips yields the result.
Let $H=({{\operatorname}{id}}\otimes L)\circ\Delta$ with $L:\ {\ensuremath{\mathcal{T}^{{}}}}_0\to{\ensuremath{\mathbb{C}^{{}}}}$ linear. Then, the point spectrum $\sigma_p(H)$ of $H$ is given by $$\begin{aligned}
{\ensuremath{\left}}\{L(R_{n,m});\ m,n\in{\ensuremath{\mathbb{N}^{{}}}}_0{\ensuremath{\right}}\}{\ensuremath{\subseteq}}\sigma_p(H)
\end{aligned}$$ including multiplicities and the spectrum $\sigma(H)$ of $H$ is given by $$\begin{aligned}
\sigma(H)={\overline}{{\ensuremath{\left}}\{L(R_{n,m});\ m,n\in{\ensuremath{\mathbb{N}^{{}}}}_0{\ensuremath{\right}}\}}.
\end{aligned}$$ In particular, if ${\ensuremath{\left}}\{L(R_{n,m});\ m,n\in{\ensuremath{\mathbb{N}^{{}}}}_0{\ensuremath{\right}}\}$ is closed in ${\ensuremath{\mathbb{C}^{{}}}}$, then $$\begin{aligned}
\sigma(H)=\sigma_p(H)={\ensuremath{\left}}\{L(R_{n,m});\ m,n\in{\ensuremath{\mathbb{N}^{{}}}}_0{\ensuremath{\right}}\}
\end{aligned}$$ including multiplicities.
Furthermore, if $\sigma(H){\ensuremath{\subsetneq}}{\ensuremath{\mathbb{C}^{{}}}}$, then $H$ is closable.
Let $S:={\ensuremath{\left}}\{L(R_{n,m});\ m,n\in{\ensuremath{\mathbb{N}^{{}}}}_0{\ensuremath{\right}}\}$. Then, we observe $$\begin{aligned}
{\forall}m,n\in{\ensuremath{\mathbb{N}^{{}}}}_0:\ HR_{n,m}=L(R_{n,m})R_{n,m},
\end{aligned}$$ i.e., $S{\ensuremath{\subseteq}}\sigma_p$ and ${\overline}{S}{\ensuremath{\subseteq}}\sigma(H)$ since the spectrum is always closed.
Let $\lambda\in{\ensuremath{\mathbb{C}^{{}}}}\setminus{\overline}{S}$. Then, $\lambda-H$ is boundedly invertible[^2] on ${\ensuremath{\mathcal{T}^{{}}}}_0$ and, since ${\ensuremath{\mathcal{T}^{{}}}}_0$ is dense in ${\ensuremath{\mathcal{T}^{{}}}}$, we obtain $\sigma(H){\ensuremath{\subseteq}}{\overline}{S}$.
Finally, assume $\sigma(H){\ensuremath{\subsetneq}}{\ensuremath{\mathbb{C}^{{}}}}$ and let $\lambda\in{\varrho}(H)$. Then, $\lambda-H$ is boundedly invertible, i.e., closable. Since $H$ is closable if and only if $\lambda-H$ is closable, we obtain the assertion.
The Laplacian $\Delta_{{\ensuremath{\mathcal{T}^{{}}}}}$ and the heat-semigroup $T$ (generated by $\Delta_{{\ensuremath{\mathcal{T}^{{}}}}}$) have pure point spectrum $$\begin{aligned}
\sigma(\Delta_{{\ensuremath{\mathcal{T}^{{}}}}})=&\sigma_p(\Delta_{{\ensuremath{\mathcal{T}^{{}}}}})={\ensuremath{\left}}\{-(n-m)^2;\ m,n\in{\ensuremath{\mathbb{N}^{{}}}}_0{\ensuremath{\right}}\}\\
{\forall}t\in{\ensuremath{\mathbb{R}^{{}}}}_{\ge0}:\ \sigma(T(t))=&\sigma_p(T(t))={\ensuremath{\left}}\{\exp{\ensuremath{\left}}(-(n-m)^2t{\ensuremath{\right}});\ m,n\in{\ensuremath{\mathbb{N}^{{}}}}_0{\ensuremath{\right}}\}
\end{aligned}$$ including multiplicities.
Here we can see two very important differences to the classical theory. Namely, $\Delta_{{\ensuremath{\mathcal{T}^{{}}}}}$ does not have compact resolvent and the heat-semigroup is not a semigroup of trace class operators (in fact, they are not even compact). In particular, this means that the heat-trace will need to be regularized.
Hence, we know which polyhomogeneous operators have pure point spectrum. However, since “${\ensuremath{{\operatorname}{tr}}}H=\sum_{\lambda\in\sigma(H)\setminus\{0\}}\mu_\lambda \lambda$” (where $\mu_\lambda$ denotes the multiplicity of $\lambda$) will not converge in general, the idea is to use a spectral $\zeta$-regularization similar to “$\sum_{n\in{\ensuremath{\mathbb{N}^{{}}}}}n=\zeta_R(-1)$” where $\zeta_R$ is the Riemann $\zeta$-function.
Let ${\ensuremath{\mathfrak{G}}}$ be a holomorphic family of operators satisfying $$\begin{aligned}
{\forall}z\in{\ensuremath{\mathbb{C}^{{}}}}:\ {\ensuremath{\mathfrak{G}}}(z)=({{\operatorname}{id}}\otimes L_{{\ensuremath{\mathfrak{G}}}(z)})\circ\Delta={\ensuremath{\mathfrak{G}}}_0(z)+{\ensuremath{\mathfrak{G}}}_p(z)
\end{aligned}$$ such that each ${\ensuremath{\left}}\{L_{{\ensuremath{\mathfrak{G}}}(z)}(R_{n,m});\ m,n\in{\ensuremath{\mathbb{N}^{{}}}}{\ensuremath{\right}}\}$ is closed, ${\ensuremath{\mathfrak{G}}}_0(z)$ is of trace class for all $z\in{\ensuremath{\mathbb{C}^{{}}}}_{\Re(\cdot)<R}$ with $R\in{\ensuremath{\mathbb{R}^{{}}}}_{>0}$, and ${\ensuremath{\mathfrak{G}}}_p$ is polyhomogeneous with $$\begin{aligned}
{\forall}m,n\in{\ensuremath{\mathbb{N}^{{}}}}_0:\ L_{p,{\ensuremath{\mathfrak{G}}}(z)}(R_{n,m})=\sum_{\iota\in I}\alpha_\iota(z)\sigma_{d_\iota+\delta z}(m,n)
\end{aligned}$$ where each $\alpha_\iota$ is holomorphic and $\delta\in{\ensuremath{\mathbb{R}^{{}}}}_{>0}$. Then, we call ${\ensuremath{\mathfrak{G}}}$ a gauged polyhomogeneous operator with index set $I$.
Furthermore, we call ${\ensuremath{\mathfrak{G}}}$ normally gauged if and only if $\delta=1$. $\delta$ is called the gauge scaling.
\[thm:existence-zeta-function\] Let ${\ensuremath{\mathfrak{G}}}={\ensuremath{\mathfrak{G}}}_0+{\ensuremath{\mathfrak{G}}}_p$ be a gauged polyhomogeneous operator on ${\ensuremath{\mathcal{T}^{{}}}}$ with $$\begin{aligned}
{\forall}m,n\in{\ensuremath{\mathbb{N}^{{}}}}_0:\ L_{p,{\ensuremath{\mathfrak{G}}}(z)}(R_{n,m})=\sum_{\iota\in I}\alpha_\iota(z)\sigma_{d_\iota+\delta z}(m,n)
\end{aligned}$$ such that $\sigma({\ensuremath{\mathfrak{G}}}_p(z))=\sigma_p({\ensuremath{\mathfrak{G}}}(z))=\{L_{p,{\ensuremath{\mathfrak{G}}}(z)}(R_{n,m});\ m,n\in{\ensuremath{\mathbb{N}^{{}}}}_0\}$. Then, ${\ensuremath{\mathfrak{G}}}$ is of trace class if $\Re(d_\iota+\delta z)<-2$.
Furthermore, the meromorphic extension $\zeta({\ensuremath{\mathfrak{G}}})$ of $$\begin{aligned}
{\ensuremath{\mathbb{C}^{{}}}}_{\Re(\cdot)<\frac{-2-\sup\{\Re(d_\iota);\ \iota\in I\}}{\delta}}\ni z\mapsto{\ensuremath{{\operatorname}{tr}}}{\ensuremath{\mathfrak{G}}}(z)\in{\ensuremath{\mathbb{C}^{{}}}}\end{aligned}$$ exists on a half space ${\ensuremath{\mathbb{C}^{{}}}}_{\Re(\cdot)<R}$ with $R\in{\ensuremath{\mathbb{R}^{{}}}}_{>0}$ and has at most simple poles at points in ${\ensuremath{\left}}\{\frac{-2-d_\iota}{\delta};\ \iota\in I{\ensuremath{\right}}\}$.
Since ${\ensuremath{\mathfrak{G}}}_0$ is of trace class on a half space ${\ensuremath{\mathbb{C}^{{}}}}_{\Re(\cdot)<R}$ with $R\in{\ensuremath{\mathbb{R}^{{}}}}_{>0}$, it suffices to consider ${\ensuremath{\mathfrak{G}}}_p$. Since ${\ensuremath{\mathfrak{G}}}_p(z)$ has the same spectrum as $D(z):=\sum_{\iota\in I}\alpha_\iota(z)\sigma_{d_\iota+\delta z}{\ensuremath{\left}}(\frac{{{\left\lvert}{\nabla}{\right\lvert}}}{2}{\ensuremath{\right}})$ on ${\ensuremath{\mathbb{R}^{2}}}/{2\pi{\ensuremath{\mathbb{Z}^{2}}}}$. Thus, the result follows from the known pseudo-differential theory.
Let ${\ensuremath{\mathfrak{G}}}$ and ${\ensuremath{\mathfrak{H}}}$ be gauged polyhomogeneous operators with ${\ensuremath{\mathfrak{G}}}(0)={\ensuremath{\mathfrak{H}}}(0)$.
1. The residue $c_{-1}(\zeta({\ensuremath{\mathfrak{G}}}),0)$ of $\zeta({\ensuremath{\mathfrak{G}}})$ in zero is gauge-invariant up to the gauge scalings $\delta_{{\ensuremath{\mathfrak{G}}}}$ and $\delta_{{\ensuremath{\mathfrak{H}}}}$. More precisely, $$\begin{aligned}
\frac{c_{-1}(\zeta({\ensuremath{\mathfrak{G}}}),0)}{\delta_{{\ensuremath{\mathfrak{G}}}}}=\frac{c_{-1}(\zeta({\ensuremath{\mathfrak{H}}}),0)}{\delta_{{\ensuremath{\mathfrak{H}}}}}.
\end{aligned}$$
2. Let ${\forall}\iota\in I:\ d_\iota\ne-2$. Then, the constant Laurent coefficient is gauge-invariant, i.e., $$\begin{aligned}
c_{0}(\zeta({\ensuremath{\mathfrak{G}}}),0)=c_{0}(\zeta({\ensuremath{\mathfrak{H}}}),0)
\end{aligned}$$
Let ${\ensuremath{\mathfrak{G}}}$ be a gauged polyhomogeneous operator.
1. ${\ensuremath{\mathfrak{G}}}(0)$ is called non-critical if and only if ${\forall}\iota\in I:\ d_\iota\ne-2$.
2. Let ${\ensuremath{\mathfrak{G}}}(0)$ be non-critical. Then, we define the $\zeta$-regularized trace of ${\ensuremath{\mathfrak{G}}}(0)$ as $$\begin{aligned}
{\ensuremath{{\operatorname}{tr}}}_\zeta({\ensuremath{\mathfrak{G}}}(0)):=\zeta({\ensuremath{\mathfrak{G}}})(0).
\end{aligned}$$
Since criticality, i.e., whether or not there is a $d_\iota=-2$, determines the possible existence of a pole in zero, we will use the following terminology. The lowest order Laurent coefficient of $\zeta({\ensuremath{\mathfrak{G}}})$ is independent of chosen gauge, i.e., it depends only on ${\ensuremath{\mathfrak{G}}}(0)$, justifying the definition $$\begin{aligned}
{\ensuremath{\mathrm{l.o.L.c.}}}({\ensuremath{\mathfrak{G}}}(0)):=
\begin{cases}
c_{-1}{\ensuremath{\left}}(\zeta({\ensuremath{\mathfrak{G}}}),0{\ensuremath{\right}})&,\ {\ensuremath{\mathfrak{G}}}(0)\text{ critical}\\
c_{0}{\ensuremath{\left}}(\zeta({\ensuremath{\mathfrak{G}}}),0{\ensuremath{\right}})&,\ {\ensuremath{\mathfrak{G}}}(0)\text{ non-critical}
\end{cases}\end{aligned}$$ independent on whether or not these values are zero.
The lowest order Laurent coefficient is tracial given any normal gauge.
More precisely, let $A=({{\operatorname}{id}}\otimes L_A)\circ\Delta$ and $B=({{\operatorname}{id}}\otimes L_B)\circ\Delta$ be polyhomogeneous operators. Then, $AB=BA$ and, if $AB$ is non-critical, ${\ensuremath{{\operatorname}{tr}}}_\zeta(AB)={\ensuremath{{\operatorname}{tr}}}_\zeta(BA)$.
Note that $AB=BA$ since $$\begin{aligned}
{\forall}m,n\in{\ensuremath{\mathbb{N}^{{}}}}_0:\ ABR_{n,m}=&A{\ensuremath{\left}}(L_B(R_{n,m})R_{n,m}{\ensuremath{\right}})
=L_A(R_{n,m})L_B(R_{n,m})R_{n,m}\\
=&BAR_{n,m}.
\end{aligned}$$ Let ${\ensuremath{\mathfrak{H}}}$ be a gauged polyhomogeneous operator with ${\ensuremath{\mathfrak{H}}}(0)=B$, ${\ensuremath{\mathfrak{G}}}_1:=A{\ensuremath{\mathfrak{H}}}$, and ${\ensuremath{\mathfrak{G}}}_2:={\ensuremath{\mathfrak{H}}}A$. Then, ${\ensuremath{\mathfrak{G}}}_1={\ensuremath{\mathfrak{G}}}_2$ and, hence, $\zeta({\ensuremath{\mathfrak{G}}}_1)=\zeta({\ensuremath{\mathfrak{G}}}_2)$. Since ${\ensuremath{\mathrm{l.o.L.c.}}}$ is gauge independent, we obtain ${\ensuremath{\mathrm{l.o.L.c.}}}(AB)={\ensuremath{\mathrm{l.o.L.c.}}}(BA)$.
Consider the Laplacian $\Delta_{{\ensuremath{\mathcal{T}^{{}}}}}=({{\operatorname}{id}}\otimes L_{\Delta_{{\ensuremath{\mathcal{T}^{{}}}}}})\circ\Delta$ with $$\begin{aligned}
{\forall}m,n\in{\ensuremath{\mathbb{N}^{{}}}}_0:\ L_{\Delta_{{\ensuremath{\mathcal{T}^{{}}}}}}{\ensuremath{\left}}(R_{n,m}{\ensuremath{\right}})=-(n-m)^2=-n^2+2mn-m^2.
\end{aligned}$$ Thus, $\Delta_{{\ensuremath{\mathcal{T}^{{}}}}}$ is non-critical and ${\ensuremath{{\operatorname}{tr}}}_\zeta(\Delta_{{\ensuremath{\mathcal{T}^{{}}}}})$ can be written as a Kontsevich-Vishik trace of a classical pseudo-differential operator which yields $$\begin{aligned}
{\ensuremath{{\operatorname}{tr}}}_\zeta(\Delta_{{\ensuremath{\mathcal{T}^{{}}}}})={\ensuremath{{\ensuremath{{\operatorname}{tr}}}_{\mathrm{KV}}}}{\ensuremath{\left}}(-\frac{({{\left\lvert}{{\partial}_1}{\right\lvert}}-{{\left\lvert}{{\partial}_2}{\right\lvert}})^2}{4}{\ensuremath{\right}})=0
\end{aligned}$$ where $({\partial}_1,{\partial}_2)$ is the gradient on ${\ensuremath{\mathbb{R}^{2}}}/{2\pi{\ensuremath{\mathbb{Z}^{2}}}}$.
More generally, let $p:\ {\ensuremath{\mathbb{R}^{2}}}\to{\ensuremath{\mathbb{C}^{{}}}}$ be a polynomial. An operator $D=({{\operatorname}{id}}\otimes L)\circ\Delta$ with $L(R_{n,m})=p(m,n)$ is called a differential operator. Then, $D$ is non-critical and ${\ensuremath{{\operatorname}{tr}}}_\zeta(D)=0$.
The $\zeta$-regularized heat-trace {#sec:heat}
==================================
For a compact Riemannian manifold $M$ without boundary and of even dimension $\dim M\in 2{\ensuremath{\mathbb{N}^{{}}}}$, the heat-trace, that is, the trace of the heat-semigroup $T$, has a polyhomogeneous expansion in the time parameter near zero which is of the form $$\begin{aligned}
{\ensuremath{{\operatorname}{tr}}}T(t)=\frac{{\ensuremath{\mathrm{vol}}}(M)}{(4\pi t)^{\frac{\dim M}{2}}}+\frac{\mathrm{total\ curvature}(M)}{3(4\pi)^{\frac{\dim M}{2}}t^{\frac{\dim M}{2}-1}}+\mathrm{higher\ order\ terms}.\end{aligned}$$ More generally, for $\dim M\in{\ensuremath{\mathbb{N}^{{}}}}$, the heat-trace has an expansion $$\begin{aligned}
{\ensuremath{{\operatorname}{tr}}}T(t)=(4\pi t)^{-\frac{\dim M}{2}}\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}_0}t^{\frac{k}{2}}A_k\end{aligned}$$ for $t\searrow0$ where the $A_k$ are called heat-invariants. These heat-invariants are spectral invariants of Laplace type operators $\nabla^*\nabla+V$ generating the corresponding “heat-semigroup” where $\nabla$ is a connection on a vector bundle over $M$ and $V$ is called the potential.
In this section, we want to consider the heat-semigroup $T$ generated by $\Delta_{{\ensuremath{\mathcal{T}^{{}}}}}$ on ${\ensuremath{\mathcal{T}^{{}}}}$ and compute the heat-coefficient. However, while the heat-semigroup is a semigroup of trace class operators, this is no longer true for the Toeplitz algebra since $$\begin{aligned}
\sigma(T(t))=\sigma_p(T(t))={\ensuremath{\left}}\{\exp{\ensuremath{\left}}(-t(n-m)^2{\ensuremath{\right}});\ m,n\in{\ensuremath{\mathbb{N}^{{}}}}_0{\ensuremath{\right}}\}\end{aligned}$$ including multiplicities. In other words, each eigenvalue has multiplicity $\aleph_0$, i.e., $T(t)$ is bounded but not compact. Thus, we need to regularize ${\ensuremath{{\operatorname}{tr}}}T(t)$. However, if we naïvely expand $\exp{\ensuremath{\left}}(-t(n-m)^2{\ensuremath{\right}})=\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}_0}\frac{(-t)^k(n-m)^{2k}}{k!}$, we obtain a polyhomogeneous representation which fails to satisfy $\sup_{\iota\in I}\Re(d_{\iota})<\infty$. In other words, we cannot simply apply the theory developed in section \[sec:zeta\]. Thus, we define a slightly more general $\zeta$-function.
Let ${\ensuremath{\mathfrak{G}}}={\ensuremath{\left}}({\ensuremath{\mathfrak{G}}}(z){\ensuremath{\right}})_{z\in{\ensuremath{\mathbb{C}^{2}}}}$ be a holomorphic family of operators on ${\ensuremath{\mathcal{T}^{{}}}}$. Then, we call ${\ensuremath{\mathfrak{G}}}$ a generalized gauged polyhomogeneous operator (or generalized gauge) if and only if ${\ensuremath{\mathfrak{G}}}(0)=1$ and ${\forall}z\in{\ensuremath{\mathbb{C}^{2}}}:\ {\ensuremath{\mathfrak{G}}}(z)={\ensuremath{\mathfrak{G}}}_0(z)+({{\operatorname}{id}}\otimes L_{p,{\ensuremath{\mathfrak{G}}}(z)})\circ\Delta$ where ${\ensuremath{\mathfrak{G}}}_0$ is of trace class for all $z\in{\ensuremath{\mathbb{C}^{2}}}_{\Re(\cdot)<R}$ for some $R\in{\ensuremath{\mathbb{R}^{{}}}}_{>0}$ and $$\begin{aligned}
{\forall}z\in{\ensuremath{\mathbb{C}^{2}}}\ {\forall}m,n\in{\ensuremath{\mathbb{N}^{{}}}}_0:\ L_{p,{\ensuremath{\mathfrak{G}}}(z)}(R_{n,m})=\sum_{\iota\in I}\alpha_\iota(z)\sigma_{d_{1,\iota}+\delta_1z_1}(m)\sigma_{d_{2,\iota}+\delta_2z_2}(n).
\end{aligned}$$ Furthermore, let $H$ be a polyhomogeneous operator on ${\ensuremath{\mathcal{T}^{{}}}}$. Then, we define $\zeta(H,{\ensuremath{\mathfrak{G}}})$ to be the maximal holomorphic extension of $z\mapsto{\ensuremath{{\operatorname}{tr}}}(H{\ensuremath{\mathfrak{G}}}(z))$ with open, connected domain containing ${\ensuremath{\mathbb{C}^{2}}}_{\Re(\cdot)<r}$ for some $r\in{\ensuremath{\mathbb{R}^{{}}}}$ sufficiently small.
Let $H=({{\operatorname}{id}}\otimes L_H)\circ\Delta$ and ${\ensuremath{\mathfrak{G}}}=({{\operatorname}{id}}\otimes L_{{\ensuremath{\mathfrak{G}}}})\circ\Delta$ a generalized gauge. Then, $$\begin{aligned}
{\forall}m,n\in{\ensuremath{\mathbb{N}^{{}}}}_0\ {\forall}z\in{\ensuremath{\mathbb{C}^{2}}}:\ H{\ensuremath{\mathfrak{G}}}(z)R_{n,m}=L_H{\ensuremath{\left}}(R_{n,m}{\ensuremath{\right}})L_{{\ensuremath{\mathfrak{G}}}}(z){\ensuremath{\left}}(R_{n,m}{\ensuremath{\right}}).
\end{aligned}$$
Eventually, we are interested in $\zeta(H,{\ensuremath{\mathfrak{G}}})(z,z)$ in a neighborhood of $z=0$. However, with the introduction of a second complex parameter, we can compute the limit $\lim_{z\to0}\zeta(H,{\ensuremath{\mathfrak{G}}})(z,z)$ by computing either $\lim_{z_1\to0}\lim_{z_2\to0}\zeta(H,{\ensuremath{\mathfrak{G}}})(z_1,z_2)$ or $\lim_{z_2\to0}\lim_{z_1\to0}\zeta(H,{\ensuremath{\mathfrak{G}}})(z_1,z_2)$ which may be significantly easier. This is possible since the identity theorem holds for holomorphic functions on $\Omega{\ensuremath{\subseteq}}{\ensuremath{\mathbb{C}^{n}}}$ in the usual sense, that is, if $\Omega$ is open and connected and a holomorphic function $f$ vanishes in an open subset of $\Omega$, then $f=0$. Thus, the $\zeta$-function in multiple variables is unique. Furthermore, since restricting a generalized gauge to the diagonal yields a gauge again, it suffices to check gauge independence of the lowest order Laurent coefficient at zero using gauges parametrized on ${\ensuremath{\mathbb{C}^{{}}}}$.
\[lemma-gauge-independence\] Let $A$ be an operator on ${\ensuremath{\mathcal{T}^{{}}}}$ and ${\ensuremath{\mathfrak{G}}}$ and ${\ensuremath{\mathfrak{H}}}$ gauged polyhomogeneous operators with ${\ensuremath{\mathfrak{G}}}(0)={\ensuremath{\mathfrak{H}}}(0)$, $L_{{\ensuremath{\mathfrak{G}}}(z)}{\ensuremath{\left}}(R_{n,m}{\ensuremath{\right}})=\sigma_{\delta z}(m,n)$, and $L_{{\ensuremath{\mathfrak{H}}}(z)}{\ensuremath{\left}}(R_{n,m}{\ensuremath{\right}})=\tilde\sigma_{\delta z}(m,n)$. Then, the lowest order Laurent coefficient of $\zeta(A,{\ensuremath{\mathfrak{G}}})$ and $\zeta(A,{\ensuremath{\mathfrak{H}}})$ in zero coincide, i.e., depend only on $A$.
Let $l$ be the order of the lowest order Laurent coefficient of $\zeta(A,{\ensuremath{\mathfrak{G}}})$ and $\zeta(A,{\ensuremath{\mathfrak{H}}})$. Then, ${\ensuremath{\mathfrak{I}}}(z):=\frac{{\ensuremath{\mathfrak{G}}}(z)-{\ensuremath{\mathfrak{H}}}(z)}{z}$ is also a gauged polyhomogeneous operator and the order of the lowest order Laurent coefficient of $\zeta(A,{\ensuremath{\mathfrak{I}}})$ is $l$. In particular, $z\mapsto z^l\zeta(A,{\ensuremath{\mathfrak{I}}})(z)=z^{l-1}{\ensuremath{\left}}(\zeta(A,{\ensuremath{\mathfrak{G}}})(z)-\zeta(A,{\ensuremath{\mathfrak{H}}})(z){\ensuremath{\right}})$ is holomorphic in zero, i.e., the lowest order Laurent coefficients of $\zeta(A,{\ensuremath{\mathfrak{G}}})$ and $\zeta(A,{\ensuremath{\mathfrak{H}}})$ must coincide.
Thus, we can compute the leading order coefficient of the $\zeta$-regularized heat-trace on ${\ensuremath{\mathcal{T}^{{}}}}$.
\[thm:heat-trace-toeplitz\] Let ${\forall}m,n\in{\ensuremath{\mathbb{N}^{{}}}}_0\ {\forall}z\in{\ensuremath{\mathbb{C}^{2}}}:\ L_{{\ensuremath{\mathfrak{G}}}}(z){\ensuremath{\left}}(R_{n,m}{\ensuremath{\right}})=\sum_{\iota\in I}\alpha_\iota(z)n^{\delta_{1,\iota}z_1}m^{\delta_{2,\iota}z_2}$ with finite $I$, ${\ensuremath{\mathfrak{G}}}=({{\operatorname}{id}}\otimes L_{{\ensuremath{\mathfrak{G}}}})\circ\Delta$ such that ${\ensuremath{\mathfrak{G}}}(0)=1$, and $T$ the heat-semigroup on ${\ensuremath{\mathcal{T}^{{}}}}$ (generated by $\Delta_{{\ensuremath{\mathcal{T}^{{}}}}}$). Then, the following assertions are true.
1. $\max{\ensuremath{\left}}\{\frac{\Re(z_1)}{\delta_{1,\iota}},\frac{\Re(z_2)}{\delta_{2,\iota}};\ \iota\in I{\ensuremath{\right}}\}<-1\ {\ensuremath{\Rightarrow}}\ {\forall}t\in{\ensuremath{\mathbb{R}^{{}}}}_{\ge0}:\ T(t){\ensuremath{\mathfrak{G}}}(z_1,z_2)$ is of trace class.
2. For all $t\in{\ensuremath{\mathbb{R}^{{}}}}_{>0}$, we obtain $$\begin{aligned}
\lim_{z_2\to0}\lim_{z_1\to0}\zeta(T(t),{\ensuremath{\mathfrak{G}}})(z_1,z_2)=-\frac{1}{2}-\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}(k+1)e^{-tk^2}
\end{aligned}$$ and $$\begin{aligned}
{\ensuremath{\mathrm{Htr}}}_{{\ensuremath{\mathcal{T}^{{}}}},\zeta,{\ensuremath{\mathfrak{G}}}}(t)=\frac{1}{2}-\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}(k-1)e^{-tk^2}.
\end{aligned}$$
3. $\lim_{t\searrow0}\ 4\pi t\ {\ensuremath{\mathrm{Htr}}}_{{\ensuremath{\mathcal{T}^{{}}}},\zeta,{\ensuremath{\mathfrak{G}}}}(t)=-2\pi$.
“(i)” The assertion follows directly from the fact that $I$ is finite and each ${{\left\lvert}{L_{T(t)}{\ensuremath{\left}}(R_{n,m}{\ensuremath{\right}})}{\right\lvert}}\le1$.
####
“(ii)” Setting $k:=n-m$ we observe for $\Re(z_1)$ and $\Re(z_2)$ sufficiently small $$\begin{aligned}
\zeta(T(t),{\ensuremath{\mathfrak{G}}})(z)=&\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}}\sum_{n\in{\ensuremath{\mathbb{N}^{{}}}}}\sum_{\iota\in I}\alpha_\iota(z)e^{-t(n-m)^2}n^{\delta_{1,\iota}z_1}m^{\delta_{2,\iota}z_2}\\
=&\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}}\sum_{k\in{\ensuremath{\mathbb{Z}^{{}}}}_{>-m}}\sum_{\iota\in I}\alpha_\iota(z)e^{-tk^2}(m+k)^{\delta_{1,\iota}z_1}m^{\delta_{2,\iota}z_2}\\
=&\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}}{\ensuremath{\left}}(\sum_{k=1-m}^0+\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}{\ensuremath{\right}})\sum_{\iota\in I}\alpha_\iota(z)e^{-tk^2}(m+k)^{\delta_{1,\iota}z_1}m^{\delta_{2,\iota}z_2}.
\end{aligned}$$ Let us consider the $\sum_{k=1-m}^0$ case first. To assist readability, we will notationally suppress $\sum_{\iota\in I}\alpha_\iota(z)$ since $I$ is finite. Then, $$\begin{aligned}
\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}}\sum_{k=1-m}^0e^{-tk^2}(m+k)^{\delta_{1,\iota}z_1}m^{\delta_{2,\iota}z_2}=&\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}}\sum_{k=0}^{m-1}e^{-tk^2}(m-k)^{\delta_{1,\iota}z_1}m^{\delta_{2,\iota}z_2}\\
=&\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}_0}e^{-tk^2}\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}_{>k}}(m-k)^{\delta_{1,\iota}z_1}m^{\delta_{2,\iota}z_2}
\end{aligned}$$ is holomorphic in $z_1$ in a neighborhood of $0$ given $\Re(z_2)<-\frac{1}{\delta_{2,\iota}}$ and the limit $z_1\to0$ yields $$\begin{aligned}
\lim_{z_1\to0}\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}}\sum_{k=1-m}^0e^{-tk^2}(m+k)^{\delta_{1,\iota}z_1}m^{\delta_{2,\iota}z_2}=&\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}_0}e^{-tk^2}\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}_{>k}}m^{\delta_{2,\iota}z_2}\\
=&\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}_0}e^{-tk^2}{\ensuremath{\left}}(\zeta_R(-\delta_{2,\iota}z_2)-\sum_{m=1}^km^{\delta_{2,\iota}z_2}{\ensuremath{\right}})
\end{aligned}$$ which itself has a holomorphic extension in $z_2$ to a neighborhood of $0$, i.e., $$\begin{aligned}
\begin{aligned}
\lim_{z_2\to0}\lim_{z_1\to0}\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}}\sum_{k=1-m}^0e^{-tk^2}(m+k)^{\delta_{1,\iota}z_1}m^{\delta_{2,\iota}z_2}=&\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}_0}e^{-tk^2}{\ensuremath{\left}}(\zeta_R(0)-k{\ensuremath{\right}})\\
=&-\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}_0}e^{-tk^2}\frac{2k+1}{2}.
\end{aligned}
\end{aligned}$$ Considering the $\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}$ term, we still have holomorphy in $z_1$ in a neighborhood of $0$ given $\Re(z_2)<-\frac{1}{\delta_{2,\iota}}$ and, thus, $$\begin{aligned}
\lim_{z_2\to0}\lim_{z_1\to0}\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-tk^2}(m+k)^{\delta_{1,\iota}z_1}m^{\delta_{2,\iota}z_2}=&\lim_{z_2\to0}\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-tk^2}m^{\delta_{2,\iota}z_2}\\
=&\lim_{z_2\to0}\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-tk^2}\zeta_R(-\delta_{2,\iota}z_2)\\
=&-\frac{1}{2}\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-tk^2}.
\end{aligned}$$ Hence, $$\begin{aligned}
\lim_{z_2\to0}\lim_{z_1\to0}\zeta(T(t),{\ensuremath{\mathfrak{G}}})(z_1,z_2)=-\frac{1}{2}-\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}(k+1)e^{-tk^2}
\end{aligned}$$ since $\sum_{\iota\in I}\alpha_\iota(0)=1$.
Since gauging terms with $m=0$ or $n=0$ yields the constant function $0$, we need to add these terms again for the heat-trace, i.e., $$\begin{aligned}
{\ensuremath{\mathrm{Htr}}}_{{\ensuremath{\mathcal{T}^{{}}}},\zeta,{\ensuremath{\mathfrak{G}}}}(t)=-\frac{1}{2}-\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}(k+1)e^{-tk^2}+1+2\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-tk^2}=\frac{1}{2}-\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}(k-1)e^{-tk^2}.
\end{aligned}$$
####
“(iii)” It suffices to consider the $-\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}(k+1)e^{-tk^2}$ term since the remainder is a “classical” heat-trace on a $1$-dimensional compact manifold without boundary and, thus, has asymptotics proportional to $t^{-\frac12}$ for $t\searrow0$. We can estimate the series using the integral comparison test since ${\ensuremath{\mathbb{R}^{{}}}}_{>0}\ni x\mapsto(x+1)e^{-tx^2}\in{\ensuremath{\mathbb{R}^{{}}}}$ is increasing on $[0,K_t]$ and decreasing on ${\ensuremath{\mathbb{R}^{{}}}}_{\ge K_t}$ where $K_t:=\frac12\sqrt{\frac{2}{t}+1}-\frac12$.
On $[0,K_t]$, we obtain $$\begin{aligned}
\sum_{k=1}^{\lfloor K_t\rfloor-1}(k+1)e^{-tk^2}\le\int_1^{\lfloor K_t\rfloor}(x+1)e^{-tx^2}dx\le\sum_{k=2}^{\lfloor K_t\rfloor}(k+1)e^{-tk^2}
\end{aligned}$$ and $$\begin{aligned}
\kappa_t:=\int_1^{\lfloor K_t\rfloor}(x+1)e^{-tx^2}dx=&\frac{\sqrt{\pi t}{\ensuremath{{\operatorname}{erf}}}(\sqrt t\lfloor K_t\rfloor)-e^{-t\lfloor K_t\rfloor^2}}{2t}-\frac{\sqrt{\pi t}{\ensuremath{{\operatorname}{erf}}}(\sqrt t)-e^{-t}}{2t}
\end{aligned}$$ where ${\ensuremath{{\operatorname}{erf}}}$ denotes the error function (with range $[-1,1]$). Then, we obtain $$\begin{aligned}
\sum_{k=1}^{\lfloor K_t\rfloor}(k+1)e^{-tk^2}\in{\ensuremath{\left}}[\kappa_t+2e^{-t},\kappa_t+(\lfloor K_t\rfloor+1)e^{-t\lfloor K_t\rfloor^2}{\ensuremath{\right}}].
\end{aligned}$$ Similarly, on ${\ensuremath{\mathbb{R}^{{}}}}_{\ge K_t}$, we obtain $$\begin{aligned}
\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}_{\ge \lfloor K_t\rfloor+2}}(k+1)e^{-tk^2}\le\int_{{\ensuremath{\mathbb{R}^{{}}}}_{\ge \lfloor K_t\rfloor+1}}(x+1)e^{-tx^2}dx\le\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}_{\ge \lfloor K_t\rfloor+1}}(k+1)e^{-tk^2}
\end{aligned}$$ and $$\begin{aligned}
\lambda_t:=\int_{{\ensuremath{\mathbb{R}^{{}}}}_{\ge \lfloor K_t\rfloor+1}}(x+1)e^{-tx^2}dx=&\frac{\sqrt{\pi}}{2\sqrt t}-\frac{\sqrt{\pi t}{\ensuremath{{\operatorname}{erf}}}(\sqrt t(\lfloor K_t\rfloor+1))-e^{-t(\lfloor K_t\rfloor+1)^2}}{2t}
\end{aligned}$$ which yields $$\begin{aligned}
\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}_{\ge \lfloor K_t\rfloor+1}}(k+1)e^{-tk^2}\in{\ensuremath{\left}}[\lambda_t,\lambda_t+(\lfloor K_t\rfloor+2)e^{-t(\lfloor K_t\rfloor+1)^2}{\ensuremath{\right}}].
\end{aligned}$$ Hence, $$\begin{aligned}
\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}(k+1)e^{-tk^2}-\kappa_t-\lambda_t\in{\ensuremath{\left}}[2e^{-t},(\lfloor K_t\rfloor+1)e^{-t\lfloor K_t\rfloor^2}+(\lfloor K_t\rfloor+2)e^{-t(\lfloor K_t\rfloor+1)^2}{\ensuremath{\right}}].
\end{aligned}$$
In order to compute $\lim_{t\searrow0}t\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}(k+1)e^{-tk^2}$, we need to study $\lim_{t\searrow0}\sqrt t\lfloor K_t\rfloor$ first. Let $\lfloor K_t\rfloor=n$. Then, $$\begin{aligned}
n\le K_t=\frac12\sqrt{\frac{2}{t}+1}-\frac12<n+1
\end{aligned}$$ implies $$\begin{aligned}
\frac{2}{(2n+3)^2-1}<t\le\frac{2}{(2n+1)^2-1}.
\end{aligned}$$ In other words, $$\begin{aligned}
\lfloor K_t\rfloor=n\quad {\ensuremath{\Rightarrow}}\quad\sqrt t\lfloor K_t\rfloor\in{\ensuremath{\left}}(\frac{n\sqrt2}{\sqrt{(2n+3)^2-1}},\frac{(n+1)\sqrt2}{\sqrt{(2n+1)^2-1}}{\ensuremath{\right}}]
\end{aligned}$$ which yields $$\begin{aligned}
\lim_{t\searrow0}\sqrt t\lfloor K_t\rfloor\in{\ensuremath{\left}}[\lim_{n\to\infty}\frac{\sqrt2}{\sqrt{\frac{(2n+3)^2-1}{n^2}}},\lim_{n\to\infty}\frac{\sqrt2}{\sqrt{\frac{(2n+1)^2-1}{(n+1)^2}}}{\ensuremath{\right}}]=\{1\}.
\end{aligned}$$ Finally, this implies $$\begin{aligned}
\lim_{t\searrow0}t\kappa_t=&\lim_{t\searrow0}\frac{\sqrt{\pi t}{\ensuremath{{\operatorname}{erf}}}(\sqrt t\lfloor K_t\rfloor)-e^{-t\lfloor K_t\rfloor^2}}{2}-\frac{\sqrt{\pi t}{\ensuremath{{\operatorname}{erf}}}(\sqrt t)-e^{-t}}{2}=\frac{1-e^{-1}}{2}\\
\lim_{t\searrow0}t\lambda_t=&\lim_{t\searrow0}\frac{\sqrt{\pi t}}{2}-\frac{\sqrt{\pi t}{\ensuremath{{\operatorname}{erf}}}(\sqrt t(\lfloor K_t\rfloor+1))-e^{-t(\lfloor K_t\rfloor+1)^2}}{2}=\frac{e^{-1}}{2},
\end{aligned}$$ i.e., $$\begin{aligned}
\lim_{t\searrow0}4\pi t\ {\ensuremath{\mathrm{Htr}}}_{{\ensuremath{\mathcal{T}^{{}}}},\zeta,{\ensuremath{\mathfrak{G}}}}(t)=\lim_{t\searrow0}-2\pi t-4\pi t\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}(k+1)e^{-tk^2}=-4\pi\lim_{t\searrow0}t\kappa_t+t\lambda_t=-2\pi.
\end{aligned}$$
This result is very interesting as well, since it says the “heat-invariants” and “criticality” in section \[sec:zeta\] indicate that ${\ensuremath{\mathcal{T}^{{}}}}$ is a “quantum manifold” of dimension $2$ with “volume” $-2\pi$. This stands in stark contrast to the classical background. The Brownian motion on ${\ensuremath{\mathcal{T}^{{}}}}$ is induced by Brownian motion on the circle ${\ensuremath{\mathbb{R}^{{}}}}/{2\pi{\ensuremath{\mathbb{Z}^{{}}}}}$. Yet, we do not observe “criticality” $-1$ and $\lim_{t\searrow0}\sqrt{4\pi t}{\ensuremath{\mathrm{Htr}}}_{{\ensuremath{\mathcal{T}^{{}}}},\zeta,{\ensuremath{\mathfrak{G}}}}(t)\in{\ensuremath{\mathbb{R}^{{}}}}\setminus\{0\}$ which would be expected if Brownian motion in ${\ensuremath{\mathcal{T}^{{}}}}$ inherited the properties of Brownian motion in ${\ensuremath{\mathbb{R}^{{}}}}/{2\pi{\ensuremath{\mathbb{Z}^{{}}}}}$.
However, there is a way of making sense of these results. ${\ensuremath{\mathcal{T}^{{}}}}_0$ can be seen as the semigroup algebra of the bicyclic semigroup and ${\ensuremath{\mathcal{T}^{{}}}}$ as the universal inverse semigroup C\*-algebra of the bicyclic semigroup. In a sense we can view ${\ensuremath{\mathcal{T}^{{}}}}$ as the “generalized Pontryagin type dual” of the bicyclic semigroup which would give rise to the dimension $2$.
The discrete Heisenberg group algebras {#sec:heisenberg}
======================================
In this section, we want to consider the $\zeta$-regularized trace and $\zeta$-heat-trace on the group algebra generated by the discrete Heisenberg groups.
\[def:heisenberg\] Let $N\in{\ensuremath{\mathbb{N}^{{}}}}$ and $P_j$, $Q_j$ ($j\in{\ensuremath{\mathbb{N}^{{}}}}_{\le N}$), and $Z$ be unitaries satisfying $P_iZ=ZP_i$, $Q_jZ=ZQ_j$, $P_iP_j=P_jP_i$, $Q_iQ_j=Q_jQ_i$, $P_iQ_j=Q_jP_i$ for $i\ne j$, and $P_iQ_i=ZQ_iP_i$.
The discrete Heisenberg group algebra ${\ensuremath{\mathbb{H}^{{}}}}_N$ of dimension $2N+1$ is the universal C\*-algebra generated by ${\ensuremath{\left}}\{R_{m,n,p}:=P^mQ^nZ^p;\ m,n\in{\ensuremath{\mathbb{Z}^{N}}}, p\in{\ensuremath{\mathbb{Z}^{{}}}}{\ensuremath{\right}}\}$. ${\ensuremath{\mathbb{H}^{{}}}}_N$ carries the structure of a C\*-bialgebra by extending the maps $\Delta(R_{m,n,p})=R_{m,n,p}\otimes R_{m,n,p}$ and ${\ensuremath{\varepsilon}}(R_{m,n,p})=1$.
Let $H$ be a Hilbert space, $\eta_{P_j}, \eta_{Q_j}$ be elements of $H$ such that $$\eta_P:=(\langle\eta_{P_i},\eta_{P_j} \rangle)_{1\leq i,j\leq N}\in M_N({\ensuremath{\mathbb{R}^{{}}}}),\quad \eta_Q:=(\langle\eta_{Q_i},\eta_{Q_j} \rangle)_{1\leq i,j\leq N}\in M_N({\ensuremath{\mathbb{R}^{{}}}})$$ and $\Im(\langle\eta_{P_i},\eta_{Q_j} \rangle)$ is constant over $i,j\in \{1,\dots N\}$ and let $\lambda_{P_j}$ and $\lambda_{Q_j}$ be real numbers then there exists a unique Gaussian Sch" urmann triple on the discrete Heisenberg group algebra such that $$\begin{aligned}
\eta(P_j)&=\eta_{P_j},&\eta(Q_j)&=\eta_{Q_j},\\
L(P_j-P_j)&=2i \lambda_{P_j} & L(Q_j-Q_j)&=2i \lambda_{Q_j}.\end{aligned}$$ Furthermore every Gaussian Schürmann triple on the discrete Heisenberg group algebra arises this way.
The method of proof is identical to that of Theorem \[thm:isolp\].
Given a Gaussian Schürmann triple on ${\ensuremath{\mathbb{H}^{{}}}}_N$ the commutativity conditions and the product rule on $L$ give the real constraint on the matrices $\eta_P$ and $\eta_Q$. Note that for Gaussian cocycles the relation $PQ=ZQP$ implies that $\eta(Z)=0$.
The constant imaginary constraint, $\Im\langle \eta_{P_i},\eta_{Q_j}\rangle$, is a result of the product rule on $L$ and the identity $L(P_{i}Q_{j})=L(ZQ_{j}P_{i})$ which implies that $L(Z)=\langle \eta_{P_i},\eta_{Q_j}\rangle-\langle \eta_{Q_j},\eta_{P_i}\rangle$ for all $i,j$.
Every Gaussian generating functional on the discrete Heisenberg group algebra is of the form $$L(P^mQ^nZ^p)=i(m\cdot\lambda_P+n\cdot \lambda_Q+p \lambda_Z)-\frac{1}{2}\begin{pmatrix}m&n\end{pmatrix}\begin{pmatrix}\eta_P &\eta_{PQ}\\ \eta_{PQ}^t& \eta_Q\end{pmatrix}\begin{pmatrix}m\\n\end{pmatrix}$$ for all $m,n\in {\ensuremath{\mathbb{Z}^{N}}}$ and $p\in {\ensuremath{\mathbb{Z}^{{}}}}$ where $\eta_{PQ}:=(\langle\eta_{P_i},\eta_{Q_j}\rangle)_{1\leq i,j\leq N}$.
Note that the generating functional in the previous Corollary strongly resembles the exponent of the characteristic function of a multivariate normal distribution. Furthermore, if $\lambda_Z=0$ then $\eta_{PQ}\in M_N({\ensuremath{\mathbb{R}^{{}}}})$ and the corresponding Lévy process can be restricted to the classical $2N$-torus via the quotient by the ideal generated by $Z-I$.
In this case the vectors $\lambda_P$ and $\lambda_Q$ dictate the drift in various directions and the matrices $\eta_P,\eta_Q$ and $\eta_{PQ}$ give us information of the covariance. The canonical choice of Brownian motion on a multidimensional object is without drift and should consist of independent one-dimensional Brownian motions in each direction.
In this scenario, this can be achieved by the choice $H={\ensuremath{\mathbb{C}^{2N}}}$, $\eta_{P_i}=e_i$, $\eta_{Q_j}=e_{N+j}$, $\lambda_{P_i}=\lambda_{Q_j}=0$. Thus, $$\begin{aligned}
{\forall}\mu=(m,n)\in{\ensuremath{\mathbb{Z}^{2N}}}\ {\forall}p\in{\ensuremath{\mathbb{Z}^{{}}}}:\ L(R_{\mu,p})=-\frac{1}{2}{{\left\lVert}{\mu}{\right\lVert}}_{\ell_2(2N)}^2.
\end{aligned}$$
This warrants the following definition of polyhomogeneous operators on ${\ensuremath{\mathbb{H}^{{}}}}_N$.
An operator $H:=({{\operatorname}{id}}\otimes L)\circ \Delta$ on ${\ensuremath{\mathbb{H}^{{}}}}_N$ is called polyhomogeneous if and only if ${\exists}r\in{\ensuremath{\mathbb{R}^{{}}}}\ {\exists}I{\ensuremath{\subseteq}}{\ensuremath{\mathbb{N}^{{}}}}\ {\exists}\alpha\in \ell_1(I)\ {\exists}d\in({\ensuremath{\mathbb{C}^{2N+1}}}_{\Re(\cdot)<r})^I\ {\forall}\mu\in{\ensuremath{\mathbb{Z}^{2N+1}}}:$ $$\begin{aligned}
L(R_\mu)=\sum_{\iota\in I}\alpha_\iota \mu^{d_\iota}
\end{aligned}$$ where we assume that each $\sum_{\iota\in I}\alpha_\iota \mu^{d_\iota}$ is absolutely convergent.
Similar to the Toeplitz case, we obtain that the spectrum of a polyhomogeneous operator $H$ is given by the closure of the point spectrum $$\begin{aligned}
\sigma_p(H)={\ensuremath{\left}}\{L{\ensuremath{\left}}(R_\mu{\ensuremath{\right}});\ \mu\in{\ensuremath{\mathbb{Z}^{2N+1}}}{\ensuremath{\right}}\}\end{aligned}$$ and we can define gauged polyhomogeneous operators in a similar fashion.
Let ${\ensuremath{\mathfrak{G}}}$ be a poly-holomorphic family of operators satisfying $$\begin{aligned}
{\forall}z\in{\ensuremath{\mathbb{C}^{{}}}}:\ {\ensuremath{\mathfrak{G}}}(z)=({{\operatorname}{id}}\otimes L(z))\circ\Delta
\end{aligned}$$ such that each ${\ensuremath{\left}}\{L(z)(R_\mu);\ \mu\in{\ensuremath{\mathbb{Z}^{2N+1}}}{\ensuremath{\right}}\}$ is closed and each ${\ensuremath{\mathfrak{G}}}(z)$ is polyhomogeneous with $$\begin{aligned}
{\forall}\mu\in{\ensuremath{\mathbb{Z}^{2N+1}}}:\ L(z)(R_\mu)=\sum_{\iota\in I}\alpha_\iota(z)\sigma_{d_\iota+\delta z}(\mu)
\end{aligned}$$ where each $\alpha_\iota$ is holomorphic, $\sigma_{d}:\ {\ensuremath{\mathbb{R}^{2N+1}}}\to{\ensuremath{\mathbb{C}^{{}}}}$ is homogeneous of degree $d$, i.e., ${\forall}\lambda\in{\ensuremath{\mathbb{R}^{{}}}}_{>0}:\ \sigma_d(\lambda\cdot)=\lambda^d\sigma_d$, and $\delta\in{\ensuremath{\mathbb{R}^{{}}}}_{>0}$. Then, we call ${\ensuremath{\mathfrak{G}}}$ a gauged polyhomogeneous operator with index set $I$.
Furthermore, we call ${\ensuremath{\mathfrak{G}}}$ normally gauged if and only if $\delta=1$.
We can then show that $\zeta$-functions exist.
\[thm:existence-zeta-function-heisenberg\] Let ${\ensuremath{\mathfrak{G}}}$ be a gauged polyhomogeneous operator with $$\begin{aligned}
{\forall}\mu\in{\ensuremath{\mathbb{Z}^{2N+1}}}:\ L(z)(R_\mu)=\sum_{\iota\in I}\alpha_\iota(z)\sigma_{d_\iota+\delta z}(\mu).
\end{aligned}$$ Then, ${\ensuremath{\mathfrak{G}}}(z)$ is of trace class if ${\forall}\iota\in I:\ \Re(d_{\iota}+\delta z)<-2N-1$ and the $\zeta$-function $\zeta({\ensuremath{\mathfrak{G}}})$ defined by meromorphic extension of $z\mapsto{\ensuremath{{\operatorname}{tr}}}{\ensuremath{\mathfrak{G}}}(z)$ has isolated first order poles in the set ${\ensuremath{\left}}\{\frac{-2N-1-d_{\iota}}{\delta};\ \iota\in I{\ensuremath{\right}}\}$. Furthermore, the lowest order Laurent coefficient is tracial.
Consider the torus $T:={\ensuremath{\mathbb{R}^{2N+1}}}/{2\pi{\ensuremath{\mathbb{Z}^{2N+1}}}}$. Then, ${\ensuremath{\mathfrak{G}}}(z)$ has the same spectrum as $D(z):=\sum_{\iota\in I}\alpha_\iota(z)(i{\partial})^{d_\iota+\delta_\iota z}$ where ${\partial}_k$ is the derivative in the $k$^th^ coordinate direction. In particular, we obtain that $D(z)$ is of trace class if all $\Re(d_\iota+\delta z)<-2N-1$ and since $D(z)$ is a pseudo-differential operator on $T$, we obtain the assertion from the established pseudo-differential theory.
Let ${\ensuremath{\mathfrak{G}}}$ be a gauged differential operator on ${\ensuremath{\mathbb{H}^{N}}}$. Then, $I$ is finite and all $d_\iota\in{\ensuremath{\mathbb{N}^{{}}}}_0$. Then, $\zeta({\ensuremath{\mathfrak{G}}})=0$.
Similarly, the heat-trace in ${\ensuremath{\mathbb{H}^{{}}}}_N$ can be reduced to the heat-trace in ${\ensuremath{\mathbb{R}^{2N}}}/{2\pi{\ensuremath{\mathbb{Z}^{2N}}}}$.
Let $T$ be the heat-semigroup on ${\ensuremath{\mathbb{H}^{{}}}}_N$, $S$ the heat-semigroup on ${\ensuremath{\mathbb{R}^{2N}}}/{2\pi{\ensuremath{\mathbb{Z}^{2N}}}}$, and ${\ensuremath{\mathfrak{G}}}$ a gauged polyhomogeneous operator on ${\ensuremath{\mathbb{H}^{{}}}}_N$ with ${\ensuremath{\mathfrak{G}}}(0)=1$. Then, the $\zeta$-regularized heat-trace ${\ensuremath{\mathrm{Htr}}}_{{\ensuremath{\mathbb{H}^{{}}}}_N,\zeta,{\ensuremath{\mathfrak{G}}}}$ on ${\ensuremath{\mathbb{H}^{{}}}}_N$ satisfies $$\begin{aligned}
{\ensuremath{\mathrm{Htr}}}_{{\ensuremath{\mathbb{H}^{{}}}}_N,\zeta,{\ensuremath{\mathfrak{G}}}}(t)=\zeta(T(t),{\ensuremath{\mathfrak{G}}})(0)=-{\ensuremath{{\operatorname}{tr}}}S(t).
\end{aligned}$$ In particular, the $k$^th^ heat coefficient $A_k({\ensuremath{\mathbb{H}^{{}}}}_N)$ of ${\ensuremath{\mathbb{H}^{{}}}}_N$ is $-A_k({\ensuremath{\mathbb{R}^{2N}}}/{2\pi{\ensuremath{\mathbb{Z}^{2N}}}})$ where $A_k({\ensuremath{\mathbb{R}^{2N}}}/{2\pi{\ensuremath{\mathbb{Z}^{2N}}}})$ is the $k$^th^ heat coefficient of ${\ensuremath{\mathbb{R}^{2N}}}/{2\pi{\ensuremath{\mathbb{Z}^{2N}}}}$.
Using the same argument as in Lemma \[lemma-gauge-independence\], we obtain gauge independence of $\zeta(T(t),{\ensuremath{\mathfrak{G}}})(0)$ and can choose $\delta',\delta''\in{\ensuremath{\mathbb{R}^{{}}}}_{>0}$ such that ${\forall}\mu\in{\ensuremath{\mathbb{Z}^{2N}}}\ {\forall}p\in{\ensuremath{\mathbb{Z}^{{}}}}:\ L_{{\ensuremath{\mathfrak{G}}}(z)}(R_{\mu,p})={{\left\lVert}{\mu}{\right\lVert}}_{\ell_2(2N)}^{\delta'z}{{\left\lvert}{p}{\right\lvert}}^{\delta'' z}$. Hence, there exists a gauged polyhomogeneous operator ${\ensuremath{\mathfrak{H}}}$ on ${\ensuremath{\mathbb{R}^{2N}}}/{2\pi{\ensuremath{\mathbb{Z}^{2N}}}}$ with ${\ensuremath{\mathfrak{H}}}(0)=1$ such that $$\begin{aligned}
{\ensuremath{{\operatorname}{tr}}}T(t){\ensuremath{\mathfrak{G}}}(z)=&\sum_{p\in{\ensuremath{\mathbb{Z}^{{}}}}}{\underbrace}{\sum_{\mu\in{\ensuremath{\mathbb{Z}^{2N}}}}e^{-t{{\left\lVert}{\mu}{\right\lVert}}_{\ell_2(2N)}^2}{{\left\lVert}{\mu}{\right\lVert}}_{\ell_2(2N)}^{\delta'z}}_{\text{gauged trace of }S(t)}{{\left\lvert}{p}{\right\lvert}}^{\delta'' z}\\
=&2{\ensuremath{{\operatorname}{tr}}}(S(t){\ensuremath{\mathfrak{H}}}(z))\sum_{p\in{\ensuremath{\mathbb{N}^{{}}}}}{{\left\lvert}{p}{\right\lvert}}^{\delta'' z}\\
=&2\zeta_R(-\delta''z){\ensuremath{{\operatorname}{tr}}}S(t){\ensuremath{\mathfrak{H}}}(z)
\end{aligned}$$ where $\zeta_R$ is the Riemann $\zeta$-function, i.e., $\zeta_R(0)=-\frac12$.
The discrete Heisenberg group algebras with $Z\in{\ensuremath{\mathbb{C}^{{}}}}$ {#sec:heisenberg-Z-complex}
================================================================================
In section \[sec:heisenberg\] we considered ${\ensuremath{\mathbb{H}^{{}}}}_N$ as generated by the $P_i$, $Q_j$, and $Z$. However, if $Z\in{\ensuremath{\mathbb{C}^{{}}}}$, then $Z$ is not a generator and we obtain the “reduced” algebra ${\ensuremath{\mathbb{H}^{r}}}_N$. Thus, ${\ensuremath{\mathbb{H}^{r}}}_N$ has the generators $\{P^mQ^n;\ m,n\in{\ensuremath{\mathbb{Z}^{{}}}}\}$ and we obtain the following two theorems.
\[thm:existence-zeta-function-heisenberg-reduced\] Let ${\ensuremath{\mathfrak{G}}}$ be a gauged polyhomogeneous operator with $$\begin{aligned}
{\forall}\mu\in{\ensuremath{\mathbb{Z}^{2N}}}:\ L(z)((P,Q)^\mu)=\sum_{\iota\in I}\alpha_\iota(z)\sigma_{d_\iota+\delta z}(\mu).
\end{aligned}$$ Then, ${\ensuremath{\mathfrak{G}}}(z)$ is of trace class if ${\forall}\iota\in I:\ \Re(d_{\iota}+\delta z)<-2N$ and the $\zeta$-function $\zeta({\ensuremath{\mathfrak{G}}})$ defined by meromorphic extension of $z\mapsto{\ensuremath{{\operatorname}{tr}}}{\ensuremath{\mathfrak{G}}}(z)$ has isolated first order poles in the set ${\ensuremath{\left}}\{\frac{-2N-d_{\iota}}{\delta};\ \iota\in I{\ensuremath{\right}}\}$. Furthermore, the lowest order Laurent coefficient is tracial.
\[thm:reduced-heisenberg-heat-trace\] Let $T$ be the heat-semigroup on ${\ensuremath{\mathbb{H}^{r}}}_N$ and $S$ the heat-semigroup on ${\ensuremath{\mathbb{R}^{2N}}}/{2\pi{\ensuremath{\mathbb{Z}^{2N}}}}$. Then, ${\forall}t\in{\ensuremath{\mathbb{R}^{{}}}}_{>0}:\ T(t)$ is of trace class and $$\begin{aligned}
{\ensuremath{\mathrm{Htr}}}_{{\ensuremath{\mathbb{H}^{r}}}_N}(t)={\ensuremath{{\operatorname}{tr}}}T(t)={\ensuremath{{\operatorname}{tr}}}S(t).
\end{aligned}$$ In particular, the $k$^th^ heat coefficient $A_k({\ensuremath{\mathbb{H}^{r}}}_N)$ of ${\ensuremath{\mathbb{H}^{r}}}_N$ coincides with the $k$^th^ heat coefficient $A_k({\ensuremath{\mathbb{R}^{2N}}}/{2\pi{\ensuremath{\mathbb{Z}^{2N}}}})$ of ${\ensuremath{\mathbb{R}^{2N}}}/{2\pi{\ensuremath{\mathbb{Z}^{2N}}}}$.
These results do not come as a surprise, since ${\ensuremath{\mathbb{H}^{r}}}_1$ is the non-commutative $2$-torus $A_{\vartheta}$ generated by two unitaries $U$ and $V$ satisfying $UV=e^{-2\pi i{\vartheta}}VU$ where ${\vartheta}\in{\ensuremath{\mathbb{R}^{{}}}}$. As such the family $(A_{\vartheta})_{{\vartheta}\in{\ensuremath{\mathbb{R}^{{}}}}}$ is a fundamental class of examples of non-commutative spaces generalizing the algebra of continuous functions on the $2$-torus; a property recovered by the Brownian motion approach to the heat-trace.
The non-commutative torus {#sec:non-com-torus}
=========================
Having observed the non-commutative $2$-torus as a special case of the discrete Heisenberg group, we want to continue studying more general non-commutative tori. This will also give us a direct means of comparison with the classical heat-trace approach on the non-commutative torus ${\ensuremath{\mathbb{T}^{n}}}_{\vartheta}$ as studied in [@azzali-levy-neira-paycha; @levy-neira-paycha]. There, too, the heat-trace recovered the dimension of the underlying torus, i.e., the commutative case ${\ensuremath{\mathbb{T}^{n}}}_0={\ensuremath{\mathbb{R}^{n}}}/{2\pi{\ensuremath{\mathbb{Z}^{n}}}}$.
Let us first recall the usual construction of the non-commutative torus. Given a real symmetric $N\times N$ matrix ${\vartheta}$ and unitaries $U_k$ with $k\in{\ensuremath{\mathbb{Z}^{N}}}$, $U_0=1$, and $$\begin{aligned}
{\forall}m,n\in{\ensuremath{\mathbb{Z}^{N}}}:\ U_mU_n=e^{-\pi i\langle m,{\vartheta}n\rangle_{\ell_2(N)}}U_{m+n},\end{aligned}$$ we consider the algebra $A_{\vartheta}:={\ensuremath{\left}}\{\sum_{k\in{\ensuremath{\mathbb{Z}^{N}}}}a_k U_k;\ a\in{\ensuremath{\mathcal{S}^{{}}}}({\ensuremath{\mathbb{Z}^{N}}}){\ensuremath{\right}}\}$ where ${\ensuremath{\mathcal{S}^{{}}}}$ denotes the Schwartz space. Then, it is possible to define a corresponding algebra of pseudo-differential operators which has been extensively studied in [@levy-neira-paycha]. The for us interesting operator is the Laplace $\Delta_{\vartheta}:=\sum_{j=1}^N{\partial}_j^2$ where ${\partial}_j\sum_{k\in{\ensuremath{\mathbb{Z}^{N}}}}a_k U_k:=\sum_{k\in{\ensuremath{\mathbb{Z}^{N}}}}k_ja_k U_k$.
Since we need our algebra to be a C\*-bialgebra, we consider the C\*-algebra ${\ensuremath{\mathcal{A}^{N}}}_{\vartheta}$ generated by the $U_k$ ($k\in{\ensuremath{\mathbb{Z}^{N}}}$) as well as a finite set of additional (unitary) generators ${\ensuremath{\left}}\{Z_\tau;\ \tau\in {{\ensuremath{\mathfrak{T}^{{}}}}}{\ensuremath{\right}}\}$ (${{\ensuremath{\mathfrak{T}^{{}}}}}\in{\ensuremath{\mathbb{N}^{{}}}}$), and set ${\ensuremath{\varepsilon}}(U_k)={\ensuremath{\varepsilon}}(Z_\tau)=1$, $\Delta(U_k)=U_k\otimes U_k$, and $\Delta(Z_\tau)=Z_\tau\otimes Z_\tau$. The $Z_\tau$ are a generalized version of the $e^{-\pi i{\vartheta}_{k,l}}$ which we consider to be generators as well (for now). Thus, the Laplacian $\Delta_{{\vartheta}}$ has the generating functional $L$ which satisfies $$\begin{aligned}
{\forall}p\in {\ensuremath{\mathbb{Z}^{{{\ensuremath{\mathfrak{T}^{{}}}}}}}}\ {\forall}k\in {\ensuremath{\mathbb{Z}^{N}}}:\ L_{\Delta_{{\vartheta}}}(Z^pU_k)=-{{\left\lVert}{k}{\right\lVert}}_{\ell_2(N)}^2.\end{aligned}$$ This follows from $\Delta_{\vartheta}\sum_{k\in{\ensuremath{\mathbb{Z}^{N}}}}a_k U_k=\sum_{k\in{\ensuremath{\mathbb{Z}^{N}}}}\sum_{j=1}^N k_j^2 a_k U_k$ and is consistent with the Brownian motion approach (cf. example below Definition \[def:heisenberg\]). In particular, we obtain similar results to the Heisenberg group algebra case.
\[thm:existence-zeta-function-nc-torus\] Let ${\ensuremath{\mathfrak{G}}}$ be a gauged polyhomogeneous operator on ${\ensuremath{\mathcal{A}^{N}}}_{\vartheta}$ with $$\begin{aligned}
{\forall}p\in {\ensuremath{\mathbb{Z}^{{{\ensuremath{\mathfrak{T}^{{}}}}}}}}\ {\forall}k\in {\ensuremath{\mathbb{Z}^{N}}}:\ L(z)(Z^pU_k)=\sum_{\iota\in I}\alpha_\iota(z)\sigma_{d_\iota+\delta z}((p,k)).
\end{aligned}$$ Then, ${\ensuremath{\mathfrak{G}}}(z)$ is of trace class if ${\forall}\iota\in I:\ \Re(d_{\iota}+\delta z)<-N-{{\ensuremath{\mathfrak{T}^{{}}}}}$ and the $\zeta$-function $\zeta({\ensuremath{\mathfrak{G}}})$ defined by meromorphic extension of $z\mapsto{\ensuremath{{\operatorname}{tr}}}{\ensuremath{\mathfrak{G}}}(z)$ has isolated first order poles in the set ${\ensuremath{\left}}\{\frac{-N-{{\ensuremath{\mathfrak{T}^{{}}}}}-d_{\iota}}{\delta};\ \iota\in I{\ensuremath{\right}}\}$. Furthermore, the lowest order Laurent coefficient is tracial.
Let ${\ensuremath{\mathfrak{G}}}$ be a gauged differential operator on ${\ensuremath{\mathcal{A}^{N}}}_{\vartheta}$. Then, $I$ is finite and all $d_\iota\in{\ensuremath{\mathbb{N}^{{}}}}_0$. Then, $\zeta({\ensuremath{\mathfrak{G}}})=0$.
Let $T$ be the heat-semigroup on ${\ensuremath{\mathcal{A}^{N}}}_{\vartheta}$, $S$ the heat-semigroup on ${\ensuremath{\mathbb{R}^{N}}}/{2\pi{\ensuremath{\mathbb{Z}^{N}}}}$, and ${\ensuremath{\mathfrak{G}}}$ a gauged polyhomogeneous operator on ${\ensuremath{\mathcal{A}^{N}}}_{\vartheta}$ with ${\ensuremath{\mathfrak{G}}}(0)=1$. Then, the $\zeta$-regularized heat-trace ${\ensuremath{\mathrm{Htr}}}_{{\ensuremath{\mathcal{A}^{N}}}_{\vartheta},\zeta,{\ensuremath{\mathfrak{G}}}}$ on ${\ensuremath{\mathcal{A}^{N}}}_{\vartheta}$ satisfies $$\begin{aligned}
{\ensuremath{\mathrm{Htr}}}_{{\ensuremath{\mathcal{A}^{N}}}_{\vartheta},\zeta,{\ensuremath{\mathfrak{G}}}}(t)=\zeta(T(t),{\ensuremath{\mathfrak{G}}})(0)=(-1)^{{\ensuremath{\mathfrak{T}^{{}}}}}{\ensuremath{{\operatorname}{tr}}}S(t).
\end{aligned}$$ In particular, the $k$^th^ heat coefficient $A_k({\ensuremath{\mathcal{A}^{N}}}_{\vartheta})$ of ${\ensuremath{\mathcal{A}^{N}}}_{\vartheta}$ is $(-1)^{{\ensuremath{\mathfrak{T}^{{}}}}}A_k({\ensuremath{\mathbb{R}^{N}}}/{2\pi{\ensuremath{\mathbb{Z}^{N}}}})$ where $A_k({\ensuremath{\mathbb{R}^{N}}}/{2\pi{\ensuremath{\mathbb{Z}^{N}}}})$ is the $k$^th^ heat coefficient of ${\ensuremath{\mathbb{R}^{N}}}/{2\pi{\ensuremath{\mathbb{Z}^{N}}}}$.
Similarly, for $A^N_{\vartheta}$, i.e., the case $Z\in{\ensuremath{\mathbb{C}^{{{\ensuremath{\mathfrak{T}^{{}}}}}}}}$, we obtain the following analogous theorems.
Let ${\ensuremath{\mathfrak{G}}}$ be a gauged polyhomogeneous operator on $A^N_{\vartheta}$ with $$\begin{aligned}
{\forall}k\in {\ensuremath{\mathbb{Z}^{N}}}:\ L(z)(U_k)=\sum_{\iota\in I}\alpha_\iota(z)\sigma_{d_\iota+\delta z}(k).
\end{aligned}$$ Then, ${\ensuremath{\mathfrak{G}}}(z)$ is of trace class if ${\forall}\iota\in I:\ \Re(d_{\iota}+\delta z)<-N$ and the $\zeta$-function $\zeta({\ensuremath{\mathfrak{G}}})$ defined by meromorphic extension of $z\mapsto{\ensuremath{{\operatorname}{tr}}}{\ensuremath{\mathfrak{G}}}(z)$ has isolated first order poles in the set ${\ensuremath{\left}}\{\frac{-N-d_{\iota}}{\delta};\ \iota\in I{\ensuremath{\right}}\}$. Furthermore, the lowest order Laurent coefficient is tracial.
Let ${\ensuremath{\mathfrak{G}}}$ be a gauged differential operator on $A^N_{\vartheta}$. Then, $I$ is finite and all $d_\iota\in{\ensuremath{\mathbb{N}^{{}}}}_0$. Then, $\zeta({\ensuremath{\mathfrak{G}}})=0$.
Let $T$ be the heat-semigroup on $A^N_{\vartheta}$ and $S$ the heat-semigroup on ${\ensuremath{\mathbb{R}^{N}}}/{2\pi{\ensuremath{\mathbb{Z}^{N}}}}$. Then, ${\forall}t\in{\ensuremath{\mathbb{R}^{{}}}}_{>0}:\ T(t)$ is of trace class and $$\begin{aligned}
{\ensuremath{\mathrm{Htr}}}_{A^N_{\vartheta}}(t)={\ensuremath{{\operatorname}{tr}}}T(t)={\ensuremath{{\operatorname}{tr}}}S(t).
\end{aligned}$$ In particular, the $k$^th^ heat coefficient $A_k(A^N_{\vartheta})$ of $A^N_{\vartheta}$ coincides with the $k$^th^ heat coefficient $A_k({\ensuremath{\mathbb{R}^{N}}}/{2\pi{\ensuremath{\mathbb{Z}^{N}}}})$ of ${\ensuremath{\mathbb{R}^{N}}}/{2\pi{\ensuremath{\mathbb{Z}^{N}}}}$.
Gaussian invariants of $SU_q(2)$ {#sec:SUq2}
================================
Finally, we want to have a look at the quantum group $SU_q(2)$. This case is particularly interesting since there is a canonical choice of Brownian motion for the classical case $SU_1(2)=SU(2)$ but not necessarily for $SU_q(2)$ for $q\ne 1$. There is a unique driftless Gaussian (up to time scaling) on each $SU_q(2)$ for $q\ne 1$ which we will treat as the heat-semigroup even though it can not be the heat-semigroup on $SU(2)$. This will allow us to compute a $\zeta$-regularized trace and recover that $SU_q(2)$ is $3$-dimensional.
We will begin with a quick summary of Section 6.2 of [@timmermann]. In order to construct $SU_q(2)$, let us start with the compact Lie group $$\begin{aligned}
SU(2):={\ensuremath{\left}}\{g_{\alpha,\gamma}:=
\begin{pmatrix}
\alpha&-\gamma^*\\
\gamma&\alpha^*
\end{pmatrix}\in B{\ensuremath{\left}}({\ensuremath{\mathbb{C}^{2}}}{\ensuremath{\right}});\ \alpha,\gamma\in{\ensuremath{\mathbb{C}^{{}}}},\ \det g_{\alpha,\gamma}=1{\ensuremath{\right}}\}\end{aligned}$$ and define $a,c\in C(SU(2))$ by $a(g_{\alpha,\gamma}):=\alpha$ and $c(g_{\alpha,\gamma})=\gamma$. Then, the C\*-algebra generated by $a$ and $c$ subject to $a^*a+c^*c=1$ is a C\*-algebraic compact quantum group with co-multiplication $\Delta$ given by $$\begin{aligned}
\Delta(a)=a\otimes a+c\otimes c\qquad\text{and}\qquad\Delta(c)=c\otimes a+a^*\otimes c,\end{aligned}$$ co-unit ${\ensuremath{\varepsilon}}(a)=1$, ${\ensuremath{\varepsilon}}(c)=0$, and antipode $S(a)=a^*$, $S(a^*)=a$, $S(c)=-c$, and $S(c^*)=-c^*$.
Let $q\in[-1,1]\setminus\{0\}$. Then, we define $SU_q(2)$ to be the universal unital C\*-algebra generated by elements $a$ and $c$ subject to the condition that $$\begin{aligned}
u:=
\begin{pmatrix}
a&-qc^*\\c&a^*
\end{pmatrix}
\end{aligned}$$ is unitary, i.e.,
1. $a^*a+c^*c=1$
2. $aa^*+q^2c^*c=1$
3. $c^*c=cc^*$
4. $ac=qca$
5. $ac^*=qc^*a$
Furthermore, $SU_q(2)$ is endowed with the co-unit ${\ensuremath{\varepsilon}}$ given by ${\ensuremath{\varepsilon}}(a)=1$ and ${\ensuremath{\varepsilon}}(c)=0$, and co-multiplication given by $$\begin{aligned}
\Delta(a)=a\otimes a-q c^*\otimes c\qquad\text{and}\qquad\Delta(c)=c\otimes a+a^*\otimes c.
\end{aligned}$$
If we let $SU^0_q(2){\ensuremath{\subseteq}}SU_q(2)$ denote the \*-subalgebra generated by $a$ and $c$ and $\Delta_0$ and ${\ensuremath{\varepsilon}}_0$ the co-multiplication $\Delta$ and co-unit ${\ensuremath{\varepsilon}}$ restricted to this \*-subalgebra then $(SU^0_q(2), \Delta_0,{\ensuremath{\varepsilon}}_0 ) = (SU_q(2), \Delta,{\ensuremath{\varepsilon}})_0$ is the associated algebraic compact quantum group to $(SU_q(2), \Delta,{\ensuremath{\varepsilon}})$.
Furthermore, the family $(a_{kmn})_{(k,m,n)\in{\ensuremath{\mathbb{Z}^{{}}}}\times{\ensuremath{\mathbb{N}^{{}}}}_0\times{\ensuremath{\mathbb{N}^{{}}}}_0}$ defined as $$\begin{aligned}
a_{k,m,n}:=
\begin{cases}
a^k(c^*)^m c^n&,\ k\in{\ensuremath{\mathbb{N}^{{}}}}_0\\
(a^*)^{-k}(c^*)^m c^n&,\ k\in-{\ensuremath{\mathbb{N}^{{}}}}\end{cases}\end{aligned}$$ is a basis of $SU_q^0(2)$.
The Gaussian generating functionals on $SU_q(2)$ for $|q|\in (0,1)$ are classified in [@schurmann-skeide]. For the quantum group we have a family of characters ${\ensuremath{\varepsilon}}_{\varphi}:\ SU_q(2)\to {\ensuremath{\mathbb{C}^{{}}}}$ for such that $$\begin{aligned}
{\ensuremath{\varepsilon}}_{\varphi}(a)=e^{i{\varphi}}\quad \text{ and }\quad {\ensuremath{\varepsilon}}_{{\varphi}}(c)=0.\end{aligned}$$ This family of characters is pointwise continuous with respect to ${\varphi}$ and satisfies ${\ensuremath{\varepsilon}}_0={\ensuremath{\varepsilon}}$. On the basis $\{a_{kmn};\ (k,m,n)\in{\ensuremath{\mathbb{Z}^{{}}}}\times{\ensuremath{\mathbb{N}^{{}}}}_0\times{\ensuremath{\mathbb{N}^{{}}}}_0\}$ of the dense subalgebra we have that ${\ensuremath{\varepsilon}}_{\varphi}(a_{kmn})=e^{ik{\varphi}}\delta_{m+n,0}$ for ${\varphi}\in {\ensuremath{\mathbb{R}^{{}}}}$.
We will define linear functionals ${\ensuremath{\varepsilon}}'(a_{kmn})={\partial}_{\varphi}{\ensuremath{\varepsilon}}_{\varphi}(a_{kmn})|_{{\varphi}=0}=ik\delta_{m+n,0}$ and ${\ensuremath{\varepsilon}}''(a_{kmn})={\partial}_{\varphi}^2{\ensuremath{\varepsilon}}_{\varphi}(a_{kmn})|_{{\varphi}=0}=-k^2\delta_{m+n,0}$.
All Gaussian generating functionals are of the form $$\begin{aligned}
L=r_D{\ensuremath{\varepsilon}}'+r{\ensuremath{\varepsilon}}''
\end{aligned}$$ where $r_D\in {\ensuremath{\mathbb{R}^{{}}}}$ and $r\in {\ensuremath{\mathbb{R}^{{}}}}_{>0}$.
By the definition of drift, the parameter $r_D$ contributes only to drift so we will only consider $r_D=0$. This leaves only positive multiples of ${\ensuremath{\varepsilon}}''$.
The operator $T_L:=({{\operatorname}{id}}\otimes {\ensuremath{\varepsilon}}'')\circ\Delta:\ SU_q(2)\to SU_q(2)$ takes the following values on $\{a_{kmn};\ (k,m,n)\in{\ensuremath{\mathbb{Z}^{{}}}}\times{\ensuremath{\mathbb{N}^{{}}}}_0\times{\ensuremath{\mathbb{N}^{{}}}}_0\}$ $$\begin{aligned}
T_L(a_{kmn})=-(k-m+n)^2a_{kmn}.
\end{aligned}$$
First note that $$\begin{aligned}
\Delta(a_{kmn})=(a\otimes a-qc^*\otimes c)^k(c^*\otimes a^*+a\otimes c^*)^m(c\otimes a+a^*\otimes c)^n
\end{aligned}$$ for all $(k,m,n)\in {\ensuremath{\mathbb{Z}^{{}}}}\times {\ensuremath{\mathbb{N}^{{}}}}_0\times {\ensuremath{\mathbb{N}^{{}}}}_0$ and, using the commutation rules, we can simplify so that $$\begin{aligned}
a^ka^{*m}a^n=a^{k-m+n}+\text{ terms with } c.
\end{aligned}$$ Then, by applying ${\ensuremath{\varepsilon}}''$ to the right leg of the tensor product, we observe that the only non-zero term is given by $a^{k-m+n}$, i.e., $$\begin{aligned}
\ T_L(a_{kmn})=a^kc^{*m}c^n\epsilon''(a^ka^{*m}a^n)=-(k-m+n)^2a_{kmn}.
\end{aligned}$$ for all $(k,m,n)\in{\ensuremath{\mathbb{Z}^{{}}}}\times {\ensuremath{\mathbb{N}^{{}}}}_0\times {\ensuremath{\mathbb{N}^{{}}}}_0$.
Let ${\ensuremath{\mathfrak{G}}}$ be a gauged polyhomogeneous operator with $$\begin{aligned}
{\forall}(k,m,n)\in{\ensuremath{\mathbb{Z}^{{}}}}\times{\ensuremath{\mathbb{N}^{{}}}}_0\times{\ensuremath{\mathbb{N}^{{}}}}_0:\ {\ensuremath{\mathfrak{G}}}(z)(a_{kmn})=\sum_{\iota\in I}\alpha_\iota(z)\sigma_{d_\iota+\delta z}((k,m,n))a_{kmn}.
\end{aligned}$$ Then, ${\ensuremath{\mathfrak{G}}}(z)$ is of trace class if ${\forall}\iota\in I:\ \Re(d_\iota+\delta z)<-3$ and the $\zeta$-function $\zeta({\ensuremath{\mathfrak{G}}})$ defined by meromorphic extension of $z\mapsto{\ensuremath{{\operatorname}{tr}}}{\ensuremath{\mathfrak{G}}}(z)$ has at most isolated first order poles in the set ${\ensuremath{\left}}\{\frac{-3-d_\iota}{\delta};\ \iota\in I{\ensuremath{\right}}\}$. Furthermore, the lowest order Laurent coefficient is tracial.
This, again, follows directly from the fact that the spectrum of ${\ensuremath{\mathfrak{G}}}(z)$ and $\sum_{\iota\in I}\alpha_\iota(z)\sigma_{d_\iota+\delta z}{\ensuremath{\left}}({\ensuremath{\left}}(i{\partial}_1,\frac{{{\left\lvert}{{\partial}_2}{\right\lvert}}}{2},\frac{{{\left\lvert}{{\partial}_3}{\right\lvert}}}{2}{\ensuremath{\right}}){\ensuremath{\right}})$ on ${\ensuremath{\mathbb{R}^{3}}}/{2\pi{\ensuremath{\mathbb{Z}^{3}}}}$ coincide.
Now taking the operator exponentiation we see that the operator semigroup $T(t):\ SU_q(2)\to SU_q(2)$ associated with $r{\ensuremath{\varepsilon}}''$ is given by $$\begin{aligned}
T(t)(a_{kmn})=e^{-rt(k-m+n)^2}a_{kmn}\end{aligned}$$ on the basis $\{a_{kmn};\ (k,m,n)\in{\ensuremath{\mathbb{Z}^{{}}}}\times{\ensuremath{\mathbb{N}^{{}}}}_0\times{\ensuremath{\mathbb{N}^{{}}}}_0\}$. It is important to note that this is not the heat-semigroup on $SU(2)$. Since $SU(2)$ is a compact Riemannian $C^\infty$-manifold without boundary, its heat-semigroup is a semigroup of trace class operators but each $T(t)$ above has multiplicity $\aleph_0$ for each of its eigenvalues. In other words, none of the $T(t)$ is compact.
\[thm:gaussian-trace-SUq(2)\] Let $T$ be a driftless Gaussian semigroup on $SU_q(2)$, i.e., $$\begin{aligned}
{\forall}(k,m,n)\in{\ensuremath{\mathbb{Z}^{{}}}}\times{\ensuremath{\mathbb{N}^{2}}}_0\ {\forall}t\in{\ensuremath{\mathbb{R}^{{}}}}_{>0}:\ T(t)(a_{kmn})=e^{-rt(k-m+n)^2}a_{kmn},
\end{aligned}$$ and ${\ensuremath{\left}}({\ensuremath{\mathfrak{G}}}_t(z){\ensuremath{\right}})_{z\in{\ensuremath{\mathbb{C}^{3}}}}$ a holomorphic family of operators on $SU_q(2)$ satisfying $$\begin{aligned}
{\ensuremath{\mathfrak{G}}}_t(z)a_{kmn}=e^{-rt(k+m-n)^2}{{\left\lvert}{k}{\right\lvert}}^{\delta_1z_1}m^{\delta_2z_2}n^{\delta_3z_3}
\end{aligned}$$ for all $z\in{\ensuremath{\mathbb{C}^{3}}}$, $(k,m,n)\in{\ensuremath{\mathbb{Z}^{{}}}}\times{\ensuremath{\mathbb{N}^{{}}}}_0\times{\ensuremath{\mathbb{N}^{{}}}}_0$, and $t\in{\ensuremath{\mathbb{R}^{{}}}}_{>0}$, where $\delta_1,\delta_2,\delta_3\in{\ensuremath{\mathbb{R}^{{}}}}_{>0}$. Then, $$\begin{aligned}
{\ensuremath{{\operatorname}{tr}}}_\zeta(T(t))=&\zeta({\ensuremath{\mathfrak{G}}}_t)(0)=\frac{1}{12}+\frac{13}{12}\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rtk^2}+\frac{13}{12}\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}ke^{-rtk^2}-\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}k^2e^{-rtk^2}
\end{aligned}$$ and $$\begin{aligned}
\lim_{t\searrow0}(4\pi t)^{\frac{3}{2}}{\ensuremath{{\operatorname}{tr}}}_\zeta(T(t))=-2\pi^2r^{-\frac{3}{2}}.
\end{aligned}$$
In order to show $$\begin{aligned}
{\ensuremath{{\operatorname}{tr}}}_\zeta(T(t))=\zeta({\ensuremath{\mathfrak{G}}}_t)(0)=\frac{1}{12}+\frac{13}{12}\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rtk^2}+\frac{13}{12}\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}ke^{-rtk^2}-\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}k^2e^{-rtk^2},
\end{aligned}$$ we need to compute the limit $z\to0$ of $$\begin{aligned}
\sum_{k\in{\ensuremath{\mathbb{Z}^{{}}}}}\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}}\sum_{n\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt(k-m+n)^2}{{\left\lvert}{k}{\right\lvert}}^{\delta_1z_1}m^{\delta_2z_2}n^{\delta_3z_3}
\end{aligned}$$ which we can alternatively write as $$\begin{aligned}
\sum_{k\in{\ensuremath{\mathbb{Z}^{{}}}}}e^{-rtk^2}{{\left\lvert}{k}{\right\lvert}}^{\delta_1z_1}\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}}\sum_{n\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt(n-m)^2}e^{-2rtk(n-m)}m^{\delta_2z_2}n^{\delta_3z_3}.
\end{aligned}$$ The inner two series are very similar to the heat-trace on the Toeplitz algebra - there is simply an additional factor $e^{-2rtk(n-m)}$ now. Hence, we will treat these series in a similar fashion. $$\begin{aligned}
\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}}\sum_{n\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt(n-m)^2}e^{-2rtk(n-m)}m^{\delta_2z_2}n^{\delta_3z_3}=\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}}\sum_{\ell\in{\ensuremath{\mathbb{Z}^{{}}}}_{>-m}}e^{-rt\ell^2}e^{-2rtk\ell}m^{\delta_2z_2}(m+\ell)^{\delta_3z_3}
\end{aligned}$$ which allows us to change gauge with respect to $z_3$ to obtain for $\Re(z_2)\ll0$ and we obtain $$\begin{aligned}
\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}}\sum_{\ell\in{\ensuremath{\mathbb{Z}^{{}}}}_{>-m}}e^{-rt\ell^2}e^{-2rtk\ell}m^{\delta_2z_2}{{\left\lvert}{\ell}{\right\lvert}}^{\delta_3z_3}.
\end{aligned}$$ Let $$\begin{aligned}
A:=\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}}\sum_{\ell=1-m}^0e^{-rt\ell^2}e^{-2rtk\ell}m^{\delta_2z_2}{{\left\lvert}{\ell}{\right\lvert}}^{\delta_3z_3}
\end{aligned}$$ and $$\begin{aligned}
B:=\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}}\sum_{\ell\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt\ell^2}e^{-2rtk\ell}m^{\delta_2z_2}{{\left\lvert}{\ell}{\right\lvert}}^{\delta_3z_3}.
\end{aligned}$$ Then, we obtain $$\begin{aligned}
\lim_{z_2\to0}A=&\lim_{z_2\to0}\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}}\sum_{\ell=0}^{m-1}e^{-rt\ell^2}e^{2rtk\ell}m^{\delta_2z_2}\ell^{\delta_3z_3}\\
=&\lim_{z_2\to0}\sum_{\ell\in{\ensuremath{\mathbb{N}^{{}}}}_0}\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}_{>\ell}}e^{-rt\ell^2}e^{2rtk\ell}m^{\delta_2z_2}\ell^{\delta_3z_3}\\
=&\lim_{z_2\to0}\sum_{\ell\in{\ensuremath{\mathbb{N}^{{}}}}_0}e^{-rt\ell^2}e^{2rtk\ell}{\ensuremath{\left}}(\zeta_R(-\delta_2z_2)-\sum_{m=1}^\ell m^{\delta_2z_2}{\ensuremath{\right}})\ell^{\delta_3z_3}\\
=&-\frac{1}{2}\sum_{\ell\in{\ensuremath{\mathbb{N}^{{}}}}_0}e^{-rt\ell^2}e^{2rtk\ell}{\ensuremath{\left}}(2\ell+1{\ensuremath{\right}})\ell^{\delta_3z_3}
\end{aligned}$$ and $$\begin{aligned}
\lim_{z_2\to0}B=&\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}}\sum_{\ell\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt\ell^2}e^{-2rtk\ell}m^{\delta_2z_2}\ell^{\delta_3z_3}=-\frac{1}{2}\sum_{\ell\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt\ell^2}e^{-2rtk\ell}\ell^{\delta_3z_3}.
\end{aligned}$$ Hence, we are looking to compute $$\begin{aligned}
\lim_{z_3\to0}\lim_{z_1\to0}\sum_{k\in{\ensuremath{\mathbb{Z}^{{}}}}}e^{-rtk^2}{{\left\lvert}{k}{\right\lvert}}^{\delta_1z_1}{\ensuremath{\left}}(-\frac{1}{2}-\frac{1}{2}\sum_{\ell\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt\ell^2}\ell^{\delta_3z_3}{\ensuremath{\left}}((2\ell+1)e^{2rtk\ell}+e^{-2rtk\ell}{\ensuremath{\right}}){\ensuremath{\right}}).
\end{aligned}$$ Using $$\begin{aligned}
\sum_{k\in{\ensuremath{\mathbb{Z}^{{}}}}}e^{-rtk^2\pm2rtk\ell}=1+\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rtk^2+2rtk\ell}+\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rtk^2-2rtk\ell},
\end{aligned}$$ we are looking for the limit $z_1,z_3\to0$ of $$\begin{aligned}
&-\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rtk^2}k^{\delta_1z_1}\\
&-\frac{1}{2}\sum_{k,\ell\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt(k-\ell)^2}(2\ell+1)k^{\delta_1z_1}\ell^{\delta_3z_3}-\frac{1}{2}\sum_{k,\ell\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt(k+\ell)^2}(2\ell+1)k^{\delta_1z_1}\ell^{\delta_3z_3}\\
&-\frac{1}{2}\sum_{k,\ell\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt(k-\ell)^2}k^{\delta_1z_1}\ell^{\delta_3z_3}-\frac{1}{2}\sum_{k,\ell\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt(k+\ell)^2}k^{\delta_1z_1}\ell^{\delta_3z_3}
\end{aligned}$$ which, in parts, we already know in terms of ${\ensuremath{\mathrm{Htr}}}_{{\ensuremath{\mathcal{T}^{{}}}},\zeta}(t)=-\frac{1}{2}-\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}(k+1)e^{-tk^2}$, the heat-trace on the Toeplitz algebra. Thus, $$\begin{aligned}
\zeta({\ensuremath{\mathfrak{G}}}_t)(0)=&-\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rtk^2}-\frac{1}{2}\sum_{k,\ell\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt(k+\ell)^2}(2\ell+1)-\frac{1}{2}{\ensuremath{\mathrm{Htr}}}_{{\ensuremath{\mathcal{T}^{{}}}},\zeta}(rt)-\frac{1}{2}\sum_{k,\ell\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt(k+\ell)^2}\\
&-\frac{1}{2}\lim_{z_1,z_3\to0}\sum_{k,\ell\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt(k-\ell)^2}(2\ell+1)k^{\delta_1z_1}\ell^{\delta_3z_3}\\
=&-\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rtk^2}-\frac{1}{2}\sum_{k,\ell\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt(k+\ell)^2}(2\ell+1)-{\ensuremath{\mathrm{Htr}}}_{{\ensuremath{\mathcal{T}^{{}}}},\zeta}(rt)-\frac{1}{2}\sum_{k,\ell\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt(k+\ell)^2}\\
&-\lim_{z_1,z_3\to0}\sum_{k,\ell\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt(k-\ell)^2}k^{\delta_1z_1}\ell^{1+\delta_3z_3} \\
=&-\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rtk^2}-\frac{1}{2}\sum_{k,\ell\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt(k+\ell)^2}(2\ell+1)-{\ensuremath{\mathrm{Htr}}}_{{\ensuremath{\mathcal{T}^{{}}}},\zeta}(rt)-\frac{1}{2}\sum_{k,\ell\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt(k+\ell)^2}\\
&-\zeta_R(-1)\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rtk^2}-\zeta_R(-1)+\zeta_R(0)+{\ensuremath{\left}}(\frac{1}{2}-\zeta_R(-1){\ensuremath{\right}})\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}ke^{-rtk^2}\\
&-\frac{1}{2}\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}k^2e^{-rtk^2}\\
=&\frac{1}{12}+\frac{1}{12}\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rtk^2}+\frac{19}{12}\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}ke^{-rtk^2}-\frac{1}{2}\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}k^2e^{-rtk^2}\\
&-\sum_{k,\ell\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt(k+\ell)^2}\ell-\sum_{k,\ell\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt(k+\ell)^2}
\end{aligned}$$ where we computed the final limit in the same way the limit in the proof of Theorem \[thm:heat-trace-toeplitz\]. Finally, the latter two series can be reduced to the former three; namely, $$\begin{aligned}
\sum_{k,\ell\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt(k+\ell)^2}=&\sum_{\ell\in{\ensuremath{\mathbb{N}^{{}}}}}\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}_{>\ell}}e^{-rtm^2}=\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}_{\ge2}}\sum_{\ell=1}^{m-1}e^{-rtm^2}=\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}}me^{-rtm^2}-\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rtm^2}
\end{aligned}$$ and $$\begin{aligned}
\sum_{k,\ell\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rt(k+\ell)^2}\ell=\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}_{\ge2}}\sum_{\ell=1}^{m-1}e^{-rtm^2}\ell=\frac{1}{2}\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}}m^2e^{-rtm^2}-\frac{1}{2}\sum_{m\in{\ensuremath{\mathbb{N}^{{}}}}}me^{-rtm^2}.
\end{aligned}$$ Hence, we obtain $$\begin{aligned}
\zeta({\ensuremath{\mathfrak{G}}}_t)(0)=&\frac{1}{12}+\frac{13}{12}\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rtk^2}+\frac{13}{12}\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}ke^{-rtk^2}-\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}k^2e^{-rtk^2}.
\end{aligned}$$
To obtain the asymptotics with respect to $t\searrow0$, we shall approximate each series using the integral comparison test again which yields $$\begin{aligned}
\lim_{t\searrow0}\sqrt{4\pi t}\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}e^{-rtk^2}=\frac{\pi}{\sqrt{r}},
\end{aligned}$$ $$\begin{aligned}
\lim_{t\searrow0}4\pi t\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}ke^{-rtk^2}=\frac{2\pi}{r},
\end{aligned}$$ and $$\begin{aligned}
\lim_{t\searrow0}(4\pi t)^{\frac{3}{2}}\sum_{k\in{\ensuremath{\mathbb{N}^{{}}}}}k^2e^{-rtk^2}=\frac{2\pi^2}{r^{\frac{3}{2}}}.
\end{aligned}$$ In other words, $$\begin{aligned}
\lim_{t\searrow0}(4\pi t)^{\frac{3}{2}}{\ensuremath{{\operatorname}{tr}}}_\zeta(T(t))=-\frac{2\pi^2}{r^{\frac{3}{2}}}.
\end{aligned}$$
Recall that we have not been computing “heat-invariants” in Theorem \[thm:gaussian-trace-SUq(2)\] since there is no Brownian motion on $SU_q(2)$. Thus, we cannot interpret $SU_q(2)$ as a “quantum manifold” of volume $-2\pi^2r^{-\frac{3}{2}}$. However, the driftless Gaussians that we still have at our disposal recovered the pole order $\frac32$ for $t\searrow0$ which is the expected result since $SU(2)$ is isomorphic to the (real) $3$-sphere. In other words, we can interpret $SU_q(2)$ as a three-dimensional “quantum manifold” which gives the correct limit at $q=1$. This stands in contrast to Connes’ observation [@connes-hochschild-dimension] that the Hochschild dimension of $SU_q(2)$ drops from $3$ ($q=1$) to $1$ ($q\ne1$). However, it is consistent with Hadfield’s and Krähmer’s results [@hadfield-kraehmer-I; @hadfield-kraehmer-II] that $SU_q(2)$ is a twisted $3$-dimensional Calabi-Yau algebra.
Conclusion {#sec:conclusion}
==========
We have considered driftless Gaussians, in particular Brownian motion, on a number of C\*-bialgebras to define Laplace-type operators and heat-semigroups. We spectrally regularized their traces using operator $\zeta$-functions and computed quantities like criticality and heat-coefficients. We noticed that the notion of dimension obtained from the critical degree of homogeneity need not coincide with the dimension obtained from the heat-trace. In particular, having an abstract twist structure is seen in the dimension obtained from criticality but not in the heat-trace. Thus, the “criticality dimension” seems (somewhat unsurprisingly) to be related to the algebraic properties of the algebra whereas the “heat-trace dimension” (being induced by the dynamics of Brownian motion) seems to be related to geometric/analytic properties of the algebra. This is particularly obvious in the case of twisted classical structures where the “heat-trace dimension” coincides with the classical dimension which is not the case for the “criticality dimension” which also counts the number of abstract twists (as these are generators of the algebra as well). In the $SU_q(2)$ case, we observed the additional obstruction that there is no Brownian motion and we had to make do with the projectively unique generator of a driftless Gaussian as our version of a “Laplacian” (whose $SU(2)$ version does not have compact resolvent). Still, we were able to recover $3$-dimensionality using the “Gauss-trace” and criticality.
In terms of the “heat-coefficients”, the leading order coefficient can hardly be interpreted as a volume since many of them are negative. On the other hand, we observed that the “heat-coefficients” can be used to differentiate between different algebras (e.g., the Toeplitz algebra and the discrete Heisenberg group algebra ${\ensuremath{\mathbb{H}^{{}}}}_1$ have different “heat-coefficients”). However, it is not possible to “hear the shape of a quantum drum” using these “heat-coefficients” alone as we have observed that the “heat-coefficients” of ${\ensuremath{\mathbb{H}^{{}}}}_N$ and $A_{\vartheta}^N$ with complex twists coincide with the heat-coefficients of classical tori.
[^1]: Here, we are ignoring the fact that the residue might be zero in which case $\zeta(T,Q)$ is holomorphic in a neighborhood of zero. However, even if this is the case $\zeta(T,Q)(0)$ behaves exactly like you expect the constant Laurent coefficient to behave in the presence of a pole, so for all intents an purposes $\zeta(T,Q)$ has a pole.
[^2]: By “boundedly invertible on ${\ensuremath{\mathcal{T}^{{}}}}_0$” we mean that $\lambda-H:\ {\ensuremath{\mathcal{T}^{{}}}}_0\to{\ensuremath{\mathcal{T}^{{}}}}_0$ is bijective, i.e., the inverse relation $(\lambda-H)^{-1}:=\{(x,y)\in{\ensuremath{\mathcal{T}^{2}}}_0;\ (y,x)\in\lambda-H\}$ is an operator, and $(\lambda-H)^{-1}:\ {\ensuremath{\mathcal{T}^{{}}}}_0\to{\ensuremath{\mathcal{T}^{{}}}}_0$ is bounded with respect to the topology induced by ${\ensuremath{\mathcal{T}^{{}}}}$.
|
---
abstract: |
The third and fourth cumulants of voltage in a current-biased diffusive metal contact of resistance $R$ are calculated for arbitrary temperatures and voltages using the semiclassical cascade approach. The third cumulant equals $e^2R^3I/3$ at high temperatures and $4e^2R^3I/15$ at low temperatures, whereas the fourth cumulant equals $2e^2R^3T/3$ at high temperatures and $(34/105)e^3R^4I$ at low temperatures.
[ PACS numbers: 73.50.Td, 05.40.Ca, 72.70.+m, 74.40+k ]{}
address: ' Institute of Radioengineering and Electronics, Russian Academy of Sciences, Mokhovaya ulica 11, 125009 Moscow, Russia\'
author:
- 'K. E. Nagaev'
title: ' Higher cumulants of voltage fluctuations in current-biased diffusive contacts '
---
[2]{}
Recently, higher cumulants of current in mesoscopic conductors received a significant attention of theorists.[@Blanter-00a] This work was pioneered by Levitov and Lesovik,[@Levitov-93] who found that the charge transmitted through a single-channel quantum contact at zero temperature is distributed according to a binomial law. Subsequently, these calculations were extended to conductors with a large number of quantum channels such as diffusive wires[@Lee-95] and chaotic cavities.[@Jalabert-94; @Baranger-94; @Brouwer-96] More recently, the third cumulant of current was calculated for a tunnel contact with interacting quasiparticles.[@Levitov-01] The third and fourth cumulants of current were also calculated for diffusive-metal contacts for arbitrary temperatures and voltages.[@Gutman-02; @Nagaev-02a]
Common to al these papers was that they considered the fluctuations of current or charge transmitted through a contact at a constant voltage drop across it. This allowed the authors to treat independently the charge transmitted through different quantum channels and at different energies. The assumption of constant voltage is justified if the resistance of the external circuit is much smaller than that of the conductor. In actual experiments, the opposite relation is quite possible and in this case one can speak of fluctuations of the voltage drop across the conductor at a constant current. For a system with an Ohmic conduction the second cumulant of voltage is just the second cumulant of current in the voltage-biased regime times $R^2$, where $R$ is the resistance of the conductor. One might think that higher cumulants of voltage and current are related in a similar way, but this is not the case. Very recently Kindermann, Nazarov, and Beenakker[@Kindermann-02] showed that higher cumulants of voltage and current in the current- and voltage-biased conductors are not related in such a simple way. In particular, they calculated the low-temperature third and fourth cumulants of charge transmitted through a multichannel conductor connected in series with a macroscopic resistor to a voltage source and found that these cumulants present nonlinear functions of current cumulants of the same conductor in the voltage-biased mode.
The purpose of this paper is to calculate the third and fourth cumulants of voltage in a current-biased diffusive-metal contact for arbitrary temperatures and to show how the recently proposed semiclassical cascade approach
should be modified in that case. The key point for this approach is that the system is described by at least two distinct variables whose fluctuations are characterized by essentially different time scales. Fluctuations of the “slow” variable modulate the intensity of noise sources for the “fast” variable and result in additional higher-order correlations. Therefore the higher cumulants of the fast variable may be recursively expressed in terms of its lower-order cumulants. Originally, this method was proposed for diffusive metals.[@Nagaev-02a] Later it was proved to be equivalent to the rigorous quantum-mechanical approach for chaotic cavities.[@Nagaev-02b] More recently similar recursive relations were obtained as a saddle-point expansion of a stochastic path integral.[@Pilgram-02] The papers[@Nagaev-02a; @Nagaev-02b] addressed the case of purely elastic scattering in the low-frequency limit at constant voltage bias where the fluctuations of the electric potential are inessential and the cascade expansions can be made with respect to only one parameter, the distribution function $f({\varepsilon})$. For a current-biased conductor this is not the case any more and the electric potential explicitly enters into the expressions. Here we show how the cascade expansion should be generalized to the case where the system is described by two different slow variables.
Consider a quasi-one-dimensional diffusive contact of length $L$ and resistance $R$ connected in series with a resistor of larger cross section that has yet much larger resistance $R_S \gg R$ (see Fig. 1). Because of a strong energy relaxation in the resistor the local distribution of electrons is Fermian with a temperature equal to that of the bath, yet the local electric potential may fluctuate. Assume that the resistor is connected to the left end of the contact and the right end of the contact is grounded. A large constant voltage is applied to the left end of the series resistor. Therefore the noise of the resistor may be considered as Gaussian. The fluctuations of current in the circuit are determined by the larger resistance, hence the spectral density of the current noise $S_I = 4T/R_S$ is extremely small because of large $R_S$. Therefore the current through the contact may be considered as constant even at a nonzero temperature. We will be interested in fluctuations of the electric potential at the left end of the contact, which actually presents the voltage drop across it.
In what follows we consider only fluctuations in the zero-frequency limit and will neglect the pile-up of charge. It will be implied that all the subsequent equations contain only low-frequency Fourier transforms of the corresponding quantities. A fluctuation of current inside the contact is given by $$\delta{{\bf j}}=
-\sigma\nabla\delta\phi
+
\delta{{\bf j}}^{ext},
\label{dj}$$ where $\sigma$ is the conductivity of the metal, $\delta\phi$ is a fluctuation of the electric potential, and $$\delta{{\bf j}}^{ext}({{\bf r}})
=
e N_F
\int d{\varepsilon}{\bf \delta F}^{ext}.
\label{dj^ext}$$ The Fourier transform of the correlator of extraneous sources $\delta{\bf F}^{ext}$ is expressed in terms of the average distribution function $f$ via a formula[@Nagaev-92] $${\langle}\delta F_{\alpha}^{ext}({\varepsilon}, {{\bf r}})
\delta F_{\beta }^{ext}({\varepsilon}', {{\bf r}}')
{\rangle}=
2\frac{D}{N_F}
\delta({{\bf r}}- {{\bf r}}')$$ $$\times
\delta({\varepsilon}- {\varepsilon}')
\delta_{\alpha\beta}
f({\varepsilon})[1 - f({\varepsilon})].
\label{<dF^2>}$$ Integrating Eq. (\[dj\]) over the contact volume, one obtains $$\delta I
=
\frac{ eN_F}{L}
\int d^3r \int d{\varepsilon}\,
\delta F_x^{ext}
+
\frac{1}{R}
\delta\phi_0.
\label{dI}$$ where $\delta\phi_0$ is a fluctuation of electric potential at the left end of the contact and $R$ is the contact resistance. As the current fluctuations are negligibly small, one may set $\delta I =
0$, so that at low frequencies $$\delta\phi_0
=
-R
\frac{eN_F}{L}
\int d^3r \int d{\varepsilon}\,
\delta F_x.
\label{dphi}$$ Hence the second cumulant of voltage $\phi_0$ is related to the second cumulant of current in the voltage-biased contact in a trivial way $${\langle}{\langle}\phi_0^2
{\rangle}{\rangle}=
R^2
\left.
{\langle}{\langle}I^2
{\rangle}{\rangle}\right|_{\phi_0=const},$$ $${\langle}{\langle}I^2
{\rangle}{\rangle}=
\frac{2}{RL}
\int_0^L dx
\int d{\varepsilon}\,
f({\varepsilon}, x)[1 - f({\varepsilon}, x)].
\label{<phi^2>}$$ The characteristic time scale for $\delta{{\bf j}}^{ext}$ and $\delta{\bf F}^{ext}$ is the elastic scattering time. As this time is much shorter than the $RC$ time and the time of diffusion across the contact that describe the evolution of fluctuations of voltage $\delta
\phi_0$ and the distribution function $\delta f({\varepsilon},{{\bf r}})$. Hence one may perform a cascade expansion of higher cumulants with respect to these slow variables.
Consider first the third cumulant of voltage ${\langle}{\langle}\phi_0^3{\rangle}{\rangle}$. Since the third cumulant of extraneous currents is vanishingly small in the diffusive limit,[@Nagaev-02a] the bare third cumulant of voltage fluctuations obtained by a direct multiplication of three equations (\[dphi\]) is also small. Hence the third cumulant of voltage should be given by a cascade correction $${\langle}{\langle}\phi_0^3
{\rangle}{\rangle}=
3
\int_0^L dx \int d{\varepsilon}\frac{
\delta
{\langle}{\langle}\phi_0^2
{\rangle}{\rangle}}{
\delta
f({\varepsilon}, x)
}
{\langle}\delta f({\varepsilon}, x)
\delta\phi_0
{\rangle},
\label{<phi^3>-1}$$ where the functional derivative $$\frac{
\delta
{\langle}{\langle}\phi_0^2
{\rangle}{\rangle}}{
\delta
f({\varepsilon}, x)
}
=
2
\frac{R}{L}
[
1 - 2f({\varepsilon}, x)
].
\label{d<phi^2>/df}$$ is easily calculated from Eq. (\[<phi\^2>\]). The key difference from the case of constant voltage is that the fluctuation $\delta
f$ should be calculated now taking into account the feedback from the environment. The fluctuations of voltage $\phi_0$ caused by random scattering in the contact result in fluctuations of the distribution function at the right end of the resistor, which presents the boundary condition for the distribution function in the contact. Hence the fluctuation of a distribution function is a sum $$\delta f({\varepsilon}, x)
=
\delta\tilde f({\varepsilon}, x)
+
\frac{
\partial f({\varepsilon}, x)
}{
\partial\phi_0
}
\delta\phi_0,
\label{df}$$ where $\delta\phi_0$ is given by Eq. (\[dphi\]) and $$\delta\tilde f({\varepsilon}, x)
=
(
D\nabla^2
)^{-1}
\nabla\delta{\bf F}^{ext}$$ is the “intrinsic” part of fluctuation directly caused by random scattering. The last term in Eq. (\[df\]) mixes together fluctuations at different energies so that they are not independent any more. In the case of purely elastic scattering in the contact, the average distribution function $f({\varepsilon}, x)$ in the contact is given by $$f({\varepsilon}, x)
=
\psi(x)
f_0({\varepsilon}- e\phi_0)
+
\bar\psi(x)
f_0({\varepsilon})
\label{f}$$ where $\phi_0 = IR$ is the voltage drop across the contact and $\psi(x) = 1-x/L$ and $\bar\psi(x) = x/L$ are the characteristic potentials[@Buttiker-93] of the left and right electrodes. Hence the derivative in Eq. (\[df\]) is just $$\frac{
\partial f({\varepsilon}, x)
}{
\partial\phi_0
}
=
-e\psi(x)
\frac{
\partial f_0
(
{\varepsilon}- e\phi_0
)
}{
\partial{\varepsilon}}.
\label{df/dphi}$$ Multiplying Eqs. (\[df\]) and (\[dphi\]) and averaging the product with use of Eq. (\[<dF\^2>\]), one easily obtains that $${\langle}\delta f({\varepsilon}, x)
\delta\phi_0
{\rangle}=
-2eR
U({\varepsilon}, x)
+
R^2
\frac{
\partial f({\varepsilon}, x)
}{
\partial\phi_0
}
{\langle}{\langle}I^2
{\rangle}{\rangle},
\label{<dfdphi>}$$ where $$U({\varepsilon}, x)
=
\frac{1}{L}
(
\nabla^2
)^{-1}
\frac{\partial}{\partial x}
[
f(1-f)
].$$ A substitution of Eqs. (\[d<phi\^2>/df\]) and (\[<dfdphi>\]) into Eq. (\[<phi\^3>-1\]) gives $${\langle}{\langle}\phi_0^3
{\rangle}{\rangle}=
\frac{1}{30}
eR^2
\Biggl[
8T^2
\sinh
\left(
\frac{eIR}{T}
\right)
-
2eIR T$$ $$+
4eIR T
\cosh
\left(
\frac{eIR}{T}
\right)
-
5e^2I^2R^2
\coth
\left(
\frac{eIR}{2T}
\right)
\Biggr]$$ $$\Biggl/
\left[
T
\sinh^2
\left(
\frac{eIR}{2T}
\right)
\right].
\label{<dphi^3>-3}$$ This expression reduces to $${\langle}{\langle}\phi_0^3
{\rangle}{\rangle}=
\frac{1}{3}
e^2 R^3 I$$ at low current or high temperature $eIR \ll T$ and to $${\langle}{\langle}\phi_0^3
{\rangle}{\rangle}=
\frac{4}{15}
e^2 R^3 I$$ at high current or low temperature $eIR \gg T$. In the former case ${\langle}{\langle}\phi_0^3{\rangle}{\rangle}$ coincides with $-R^3{\langle}{\langle}I^3{\rangle}{\rangle}$, but in the latter case it is four times larger. On the whole, the temperature dependence of ${\langle}{\langle}\phi_0^3{\rangle}{\rangle}$ at a given current appears to be more flat than that of ${\langle}{\langle}I^3{\rangle}{\rangle}$ at a given voltage. Equation (\[<dphi\^3>-3\]) is in an agreement with the temperature-dependent third cumulant of voltage obtained by Beenakker et al.[@Beenakker-03] One may also obtain its low-temperature limit from the formula for the third cumulant of current of Kindermann et al.[@Kindermann-02] using voltage-biased cumulants for a diffusive contact.[@Gutman-02]
Unlike the third cumulant, the fourth cumulant cannot be expressed in terms of only functional derivatives with respect to $\delta f$. The point is that the correlator (\[<dfdphi>\]) explicitly depends on the voltage drop $\phi_0$ through the derivative $\partial f_0({\varepsilon}- e\phi_0)/\partial{\varepsilon}$. Therefore one has to perform the cascade expansion with respect to $\delta\tilde f$ and $\delta\phi_0$ considering them as different stochastic variables. The rules for constructing the diagrams remain basically the same as for the case of a voltage-biased contact,[@Nagaev-02a] but now a variation of any fluctuating quantity should be taken twice, i.e. with respect to $\tilde\delta f({\varepsilon}, x)$ and $\delta\phi_0$. Hence the number of terms significantly increases: $${\langle}{\langle}\phi_0^4
{\rangle}{\rangle}=
6S_1
+
12S_2
+
6S_3
+
12
(
S_4
+
S_5
+
S_6
+
S_7
)$$ $$+
3S_8
+
6S_9
+
3S_{10},
\label{<dphi^4>-1}$$ where $$S_1
=
\int d{\varepsilon}_1
\int d{\varepsilon}_2
\int dx_1
\int dx_2\,
\frac{
\delta^2
{\langle}{\langle}\phi_0^2
{\rangle}{\rangle}}{
\delta f({\varepsilon}_1, x_1)
\delta f({\varepsilon}_2, x_2)
}$$ $$\times
{\langle}\delta\tilde f({\varepsilon}_1, x_1)
\delta\phi_0
{\rangle}{\langle}\delta\tilde f({\varepsilon}_2, x_2)
\delta\phi_0
{\rangle},
\label{S1-1}$$ $$S_2
=
\int d{\varepsilon}\int dx
\frac{
\delta^2
{\langle}{\langle}\phi_0^2
{\rangle}{\rangle}}{
\delta\tilde f({\varepsilon}, x)
\delta\phi_0
}
{\langle}\delta\tilde f({\varepsilon}_1, x_1)
\delta\phi_0
{\rangle}{\langle}{\langle}\phi_0^2
{\rangle}{\rangle},
\label{S2-1}$$ $$S_3
=
\frac{
\partial^2
{\langle}{\langle}\phi_0^2
{\rangle}{\rangle}}{
\partial
\phi_0^2
}
{\langle}{\langle}\phi_0^2
{\rangle}{\rangle},
\label{S3-1}$$ $$S_4
=
\int d{\varepsilon}_1
\int d{\varepsilon}_2
\int dx_1
\int dx_2\,
\frac{
\delta
{\langle}{\langle}\phi_0^2
{\rangle}{\rangle}}{
\delta f({\varepsilon}_1, x_1)
}$$ $$\times
\frac{
\delta
{\langle}\delta\tilde f({\varepsilon}_1, x_1)
\delta\phi_0
{\rangle}}{
\delta f({\varepsilon}_2, x_2)
}
{\langle}\delta\tilde f({\varepsilon}_2, x_2)
\delta\phi_0
{\rangle},
\label{S4-1}$$ $$S_5
=
\int d{\varepsilon}\int dx\,
\frac{
\delta
{\langle}{\langle}\phi_0^2
{\rangle}{\rangle}}{
\delta f({\varepsilon}, x)
}
\frac{
\delta
{\langle}\delta\tilde f({\varepsilon}, x)
\delta\phi_0
{\rangle}}{
\delta\phi_0
} {\langle}{\langle}\phi_0^2
{\rangle}{\rangle},
\label{S5-1}$$ $$S_6
=
\frac{
\partial
{\langle}{\langle}\phi_0^2
{\rangle}{\rangle}}{
\partial
\phi_0
} \int d{\varepsilon}\int dx\,
\frac{
\delta
{\langle}{\langle}\phi_0^2
{\rangle}{\rangle}}{
\delta f({\varepsilon}, x)
} {\langle}\delta\tilde f({\varepsilon}, x)
\delta\phi_0
{\rangle},
\label{S6-1}$$ $$S_7
=
\left(
\frac{
\partial
{\langle}{\langle}\phi_0^2
{\rangle}{\rangle}}{
\partial
\phi_0
} \right)^2
{\langle}{\langle}\phi_0^2
{\rangle}{\rangle},
\label{S7-1}$$ $$S_8
=
\int d{\varepsilon}_1
\int d{\varepsilon}_2
\int dx_1
\int dx_2\,
\frac{
\delta
{\langle}{\langle}\phi_0^2
{\rangle}{\rangle}}{
\delta f({\varepsilon}_1, x_1)
}$$ $$\times
{\langle}\delta\tilde f({\varepsilon}_1, x_1)
\delta\tilde f({\varepsilon}_2, x_2)
{\rangle}\frac{
\delta
{\langle}{\langle}\phi_0^2
{\rangle}{\rangle}}{
\delta f({\varepsilon}_2, x2)
},
\label{S8-1}$$ $$S_9
=
\int d{\varepsilon}\int dx\,
\frac{ \delta
{\langle}{\langle}\phi_0^2
{\rangle}{\rangle}}{
\delta f({\varepsilon}, x)
} {\langle}\delta\tilde f({\varepsilon}, x)
\delta\phi_0
{\rangle}\frac{
\partial
{\langle}{\langle}\phi_0^2
{\rangle}{\rangle}}{
\partial\phi_0
},
\label{S9-1}$$ and $$S_{10}
=
\left(
\frac{
\partial
{\langle}{\langle}\phi_0^2
{\rangle}{\rangle}}{
\partial\phi_0
}
\right)^2
{\langle}{\langle}\phi_0^2
{\rangle}{\rangle}.
\label{S10-1}$$ The numerical prefactors 6, 12, and 3 in Eq. (\[<dphi\^4>-1\]) present the numbers of inequivalent permutations of $\phi_0$ in the corresponding expressions. The functional derivative with respect to $\phi_0$ is defined as $$\frac{
\delta
{\langle}\ldots
{\rangle}}{
\delta\phi_0
} =
\frac{
\partial
{\langle}\ldots
{\rangle}}{
\partial\phi_0
}
+
\int d{\varepsilon}\int dx\,
\frac{
\delta
{\langle}\ldots
{\rangle}}{
\delta f({\varepsilon},x)
}
\frac{
\partial f({\varepsilon}, x)
}{
\partial\phi_0
}.$$ The sum $6S_1 + 12S_4 + 3S_8$ gives just $R^4{\langle}{\langle}\tilde I^4 {\rangle}{\rangle}$, where ${\langle}{\langle}\tilde I^4{\rangle}{\rangle}$ is the fourth cumulant of current for a current-biased contact with a voltage drop ${\langle}\phi_0{\rangle}$. The sum $12S_2 + 12S_5$ is easily brought to a form $$12
\int d{\varepsilon}\int dx\,
\frac{
\partial
}{
\partial\phi_0
} \left(
\frac{
\delta
{\langle}{\langle}\phi_0^2
{\rangle}{\rangle}}{
\delta f({\varepsilon}, x)
} {\langle}\delta\tilde f({\varepsilon}, x)
\delta\phi_0
{\rangle}\right)
{\langle}{\langle}\phi_0^2
{\rangle}{\rangle}$$ $$=
4R^4
\frac{
\partial
{\langle}{\langle}\tilde I^3
{\rangle}{\rangle}}{
\partial\phi_0
}
{\langle}{\langle}\tilde I^2
{\rangle}{\rangle}.$$ The integrals in $S_6$ and $S_9$ also present cumulants ${\langle}{\langle}\tilde
I^3{\rangle}{\rangle}$ for a voltage-biased contact. The whole expression (\[<dphi\^4>-1\]) assumes the form $${\langle}{\langle}\phi_0^4
{\rangle}{\rangle}=
R^4
{\langle}{\langle}\tilde I^4
{\rangle}{\rangle}+
6R^6
\frac{
\partial^2
{\langle}{\langle}\tilde I^2
{\rangle}{\rangle}}{
\partial\phi_0^2
} {\langle}{\langle}\tilde I^2
{\rangle}{\rangle}^2$$ $$+
15R^6
\left(
\frac{
\partial
{\langle}{\langle}\tilde I^2
{\rangle}{\rangle}}{
\partial\phi_0
} \right)^2
{\langle}{\langle}\tilde I^2
{\rangle}{\rangle}-
6R^5
\frac{
\partial
{\langle}{\langle}\tilde I^2
{\rangle}{\rangle}}{
\partial\phi_0
} {\langle}{\langle}\tilde I^3
{\rangle}{\rangle}$$ $$-
4R^5
\frac{
\partial
{\langle}{\langle}\tilde I^3
{\rangle}{\rangle}}{
\partial\phi_0
} {\langle}{\langle}\tilde I^2
{\rangle}{\rangle}.
\label{<dphi^4>-2}$$ Substituting the cumulants of current for the voltage-biased contact,[@Nagaev-02a] one easily obtains $${\langle}{\langle}\phi_0^4
{\rangle}{\rangle}=
\frac{1}{2520}
e^2 R^3
\Biggl\{
51
eIR T^2
\cosh
\left(
\frac{5eIR}{2T}
\right)$$ $$+
72 T^3
\sinh
\left(
\frac{5eIR}{2T}
\right)$$ $$-
( 456 T^3 + 224 e^2 I^2R^2 T )
\sinh
\left(
\frac{3eIR}{2T}
\right)$$ $$+
( 70 e^3I^3R^3 - 399 e IR T^2 )
\cosh
\left(
\frac{3eIR}{2T}
\right)$$ $$+
1008 T^3
\sinh
\left(
\frac{eIR}{2T}
\right)$$ $$+
( 560 e^3 I^3R^3 + 348 e IR T^2 )
\cosh
\left(
\frac{eIR}{2T}
\right)
\Biggr\}$$ $$\Biggr/
\left[
T^2
\sinh
\left(
\frac{eIR}{2T}
\right)
\right]^5.
\label{<dphi^4>-3}$$ This expression reduces to $${\langle}{\langle}\phi_0^4
{\rangle}{\rangle}=
\frac{2}{3}
e^2 R^3 T$$ in the high-temperature limit, which differs from the corresponding cumulant of current in a voltage-biased contact just by a factor $R^4$. In the high-current limit, $${\langle}{\langle}\phi_0^4
{\rangle}{\rangle}=
\frac{34}{105}
e^3 R^4 I.$$ This value is in an agreement with the formula for the low-temperature fourth cumulant of transmitted charge of Kindermann et al.[@Kindermann-02] Unlike the fourth cumulant of current, the fourth cumulant of voltage is positive for all currents and temperatures and its numerical prefactor in the high-current limit is larger by more than an order of magnitude than that of the fourth cumulant of current at high voltages.
I am grateful to C. W. J. Beenakker for a discussion. I am also grateful to S. Pilgram and E. V. Sukhorukov for helping me to find a term missed in the previous version of the paper.
This work was supported by Russian Foundation for Basic Research, grant 01-02-17220, and by the INTAS Open grant 2001-1B-14.
[99]{} Y. Blanter and M. Büttiker, Phys. Rep. [**336**]{}, 1 (2000). L. S. Levitov and G. B. Lesovik, Pis’ma Zh. Eksp. Teor. Fiz. [**58**]{}, 225 (1993) \[JETP Lett. [**58**]{}, 230 (1993)\]. H. Lee, L. S. Levitov. and A. Yu. Yakovets, Phys. Rev. B [**51**]{}, 4079 (1995). R. A. Jalabert, J.-L. Pichard, and C. W. J. Beenakker, Europhys. Lett. [**27**]{}, 255 (1994). H. U. Baranger and P. A. Mello, Phys. Rev. Lett. [**73**]{}, 142 (1994). P.W. Brouwer and C.W.J. Beenakker, J. Math. Phys. [**37**]{}, 4904 (1996). L. S. Levitov and M. Reznikov, cond-mat/0111057 D. B. Gutman and Y. Gefen, cond-mat/0201007. K. E. Nagaev, Phys. Rev. B [**66**]{}, 075334 (2002). M. Kindermann, Yu. V. Nazarov, and C. W. J. Beenakker, cond-mat/0210617 C. W. J. Beenakker, M. Kindermann, and Yu. V. Nazarov, cond-mat/0301476 K. E. Nagaev, P. Samuelsson, and S. Pilgram, Phys. Rev. B [**66**]{}, 195318 (2002). S. Pilgram, A. N. Jordan, E. V. Sukhorukov, and M. Büttiker, cond-mat/0212446. K. E. Nagaev, Phys. Lett. A [**169**]{}, 103 (1992). M. Büttiker, J. Phys.: Condens. Matter [**5**]{}, 9361 (1993).
|
---
author:
- '<span style="font-variant:small-caps;">Andrea Gambioli</span>'
title: |
<span style="font-variant:small-caps;">**Eight-dimensional SU(3)-manifolds\
of cohomogeneity one**</span>
---
[ 57S25; 22E46, 57S15, 53C30, 53C26, 58D05. ]{}
Introduction
============
Let $M$ be a differentiable manifold, and $G$ a compact semisimple group acting smoothly on $M$. Then $M$ is said to be a *cohomogeneity-one* $G$-space if the principal orbits are codimension-one submanifolds. A result due to Mostert [@mostert] asserts that the quotient space $M/G$ is isomorphic to $[0,1]$ or to $S^{1}$ if $M$ is compact, to $[0,1)$ or ${\mathbb{R}}$ if $M$ is non-compact. In the case of the interval $[0,1]$, there are precisely two singular orbits corresponding to the endpoints.
Manifolds with a cohomogeneity-one group action have been increasingly studied in recent years. This is mainly due to the fact that many problems concerning the existence of $G$-invariant structures on them can be reduced to ODE’s, which are sometimes straightforward to handle. As typical examples, we cite [@beber], [@brysal], [@dancwang1], in which such techniques were used to construct Einstein metrics and examples of metrics with exceptional holonomy.
More recently, cohomogeneity-one quaternion-Kähler and hyperkähler manifolds were classified in [@dancswan1], [@dancswan2]. General criteria for the classification of cohomogeneity-one manifolds were also developed in [@alek-alek], [@alek-alek2], and used to partially classify such manifolds with $\chi(M)>0$ (and a corresponding family of quaternion-Kähler manifolds) in [@alek-pod]. Cohomogeity-one $SU(3)$ manifolds of dimension 7 are the subject of [@podestaverdiani].
In this paper, we shall focus on $8$-dimensional simply-connected smooth manifolds admitting an action of $SU(3)$ of cohomogeneity one. The interest in these manifolds arises from the following considerations. Firstly, the 8-dimensional quaternion-Kähler (QK) spaces $$\label{wolfspaces}
{\mathbb{H}}{\mathbb{P}}^{2},\qquad {\mathbb{G}\mathrm{r}_{2}({\mathbb{C}}^{\,4})}\cong{\widetilde{\mathbb{G}\mathrm{r}}_{4}({\mathbb{R}}^{\,6})},\qquad G_{2}/SO(4),$$ remarkably all admit an $SU(3)$-action of cohomogeneity one. (See [@wolf65], [@alek3], [@poonsal1] for the theory of such Wolf spaces.) In [@gambioli1], the author studied the moment mapping $\mu$ of a QK space into the Grassmannian ${\widetilde{\mathbb{G}\mathrm{r}}_{3}}({\mathfrak{g}})$ of oriented 3-planes in the Lie algebra ${\mathfrak{g}}$. Whilst $\mu$ is a branched covering of $G_2/SO(4)$ onto its image in ${\widetilde{\mathbb{G}\mathrm{r}}_{3}}({\mathfrak{su}(3)})$ [@kobswn1], we shall point out that the first two spaces in (\[wolfspaces\]) give rise to 7-dimensional images.
Another observation is that $8$ is precisely the dimension of the Lie group $SU(3)$ itself, and it is natural to ask whether there is a cohomogeneity-one action of $SU(3)$ on itself. Whilst the Adjoint action has cohomogeneity-two, a positive answer to the question comes from a modification called the *$A$-twisted action* or *$\sigma$-action* (see [@conlon], [@hptt1] and [@kollross]). For the case of $SU(3)$, this coincides with the more elementary *consimilarity action*, studied independently in the theory of matrices [@hornjohnson]. In any case, the tangent space at a generic point of an $8$-dimensional Riemannian manifold with an isometric $SU(3)$-action of cohomogeneity-one can be naturally identified with the Lie algebra ${\mathfrak{su}(3)}$.
Such considerations suggest the importance of setting these four examples in a wider context, in order to understand more deeply their common structure. Although $SU(3)$ does not admit a global QK structure, we show that it has features in common with (\[wolfspaces\]) that allow it to be regarded as an “honorary Wolf space”. For example, $SU(3)$ minus a 5-sphere is $SU(3)$-diffeomorphic to $G_2/SO(4)$ minus a complex projective plane ${\mathbb{C}}{\mathbb{P}}^2$, and we explain that open dense sets of both $SU(3)$ and ${\mathbb{H}}{\mathbb{P}}^{2}$ are the total spaces of $S^1$ bundles over the vector bundle $\Lambda_-^2{\mathbb{C}}{\mathbb{P}}^2$. The manifold $SU(3)$ admits an invariant hypercomplex structure [@joyc1], and a $PSU(3)$ structure in the sense of [@hitc1]. The theory also has links with $Spin(7)$ structures [@gukospartong].
In the present article, we classify compact $8$-dimensional differentiable manifolds $M$ admitting a cohomogeneity-one $SU(3)$ action such that the quotient space $M/SU(3)$ is $[0,1]$. In this case, the generic orbit has type $SU(3)/H$ where the connected component at the identity is $S^{1}$, and there are precisely two singular orbits $M_{1},\,M_{2}$ of type $SU(3)/K_{i}$, $i=1,2$, satisfying the relations $SU(3)\supset K_{i}\supset H$ (we refer the reader to [@bredon] for this basic theory). We also give a partial classification of the case $M/G\cong S^{1}$, where in almost all cases, $M$ turns out to be a product of $S^1$ with an Aloff-Wallach space, which is the principal orbit.
The paper is organized as follows. In Section \[preliminaryresults\], we describe our approach to the classification, along with some results concerning connected principal stabilizers and the sphere-transitive representations of $U(2)$ and ${\mathrm{T}^{2}}$; in the latter case are also discussed non-connected pricipal stabilizers, which can appear only in presence of this type of singular stabilizer. In Section \[theclassification\], we carry out the classification distinguishing two possible situations: the case in which both singular stabilizers are connected (Theorem \[classu3\]), and that in which at least one is not connected (Proposition \[propstab2\]). Moreover, we discuss the case in which $M/G\cong S^{1}$ and the principal orbits are simply connected (Theorem \[mgs1\]).
In Section \[examplesandapplications\], we shall identify some of the manifolds obtained during the classification, and discuss more extensively the consimilarity action of $SU(3)$ on itself. Afterwards, in Section \[quotientsbycirclesubgroups\], we apply ideas behind the classification results to discuss the QK moment mappings induced on ${\mathbb{H}}{\mathbb{P}}^{2}$ and ${\mathbb{G}\mathrm{r}_{2}({\mathbb{C}}^{\,4})}$ under the action of $SU(3)$, and relate these $8$-dimensional manifolds with examples of $7$-dimensional $SU(3)$-manifolds via circle actions.
Preliminary results {#preliminaryresults}
===================
In general, for arbitrary $G$-manifolds $M$ with orbit space isomorphic to $[0,1]$ there are two singular orbits $M_{1}$, $M_{2}$ and a normal (or slice) representation for each of them; let ${V}$ denote such representation at a point $x$ of a singular orbit $M_{i}$; then the bundle obtained as the twisted product $$G\times_{\scriptscriptstyle K_{i}} {V}$$ is $G$-equivariantly isomorphic to a tube around $M_{i}$. If we consider the corresponding disk bundle $D_{i}$, we can describe $M$ as $$M=M_{\phi}=D_{1}\cup_{\phi} D_{2},$$ where $$\label{gluing}
\phi:\partial D_{1}\xymatrix{\ar[r]&}\partial D_{2}$$ is a $G$-equivariant diffeomorphism identifying the points of the two boundaries. The latter are precisely the principal orbits: $\partial D_{i}\cong G/H$, where $H$ is the principal stabilizer.
In [@uchida], Uchida used this approach in order to classify cohomology complex projective spaces with a cohomogeneity-one action. We cite his useful sufficient conditions to decide if the manifolds obtained using different maps $\phi$ are isomorphic as $G$-spaces (see [@uchida Lemma 5.3.1]): let $\phi,\psi:\partial D_{1}\to
\partial D_{2}$ be $G$-equivariant maps as in (\[gluing\]); then $M_{\phi}$ and $M_{\psi}$ are $G$-equivariantly diffeomorphic if one of the following conditions are satisfied:
1. $\phi$ and $\psi$ are $G$-diffeotopic, or
2. $\psi\circ \phi^{-1}$ can be extended to a $G$-equivariant diffeomorphism of $D_{1}$ on itself, or
3. $\phi\circ \psi^{-1}$ can be extended to a $G$-equivariant diffeomorphism of $D_{2}$ on itself.
Our problem can therefore be reduced to classifying automorphisms of the generic orbit $SU(3)/U(1)$ up to these conditions.
One can obtain $G$-equivariant automorphisms of homogeneous spaces $G/H$ as follows: let $a\in N(H)$; then the map $\phi^{a}$ given by $$\label{equivhom}
\phi^{a}(gH)=ga^{-1}H$$ is well defined and commutes with the left multiplication for elements $g\in G$. It can be shown that *all* $G$-equivariant automorphisms of $G/H$ have this form (see [@bredon Chap I, Th. 4.2]); we have therefore the identification $$\mathrm{Aut}_G(G/H)\cong \frac{N(H)}{H}\,.$$
Let us discuss in some detail the case that two $SU(3)$-spaces obtained by distinct gluing maps are isomorphic (as $SU(3)$-spaces). In general if $M_{\phi}$ and $M_{\psi}$ are two such manifolds, then an equivariant morphism $\Phi:M_{\phi}\to M_{\psi}$ would restrict on the two tubular neighborhoods to a couple of equivariant morphisms $\phi^{a}$ and $\phi^{b}$, as described in (\[equivhom\]), which make the following diagram commutative: $$\label{gluingdiag}
\xymatrix{G/K_{1} & G/H\ar[l]_{p_{1}} & G/H\ar[r]^{p_{2}}\ar[l]_{\psi}&G/K_{2}\\
G/K_{1}\ar[u]^{\phi^{a}}&G/H\ar[u]^{\phi^{a}}\ar[l]^{p_{1}}& G/H\ar[u]_{\phi^{b}}\ar[r]_{p_{2}}\ar[l]^{\phi}& G/K_{2}\ar[u]_{\phi^{b}}}$$ In this case, $a\in N(H)\cap N(K_{1})$ and $b\in N(H)\cap N(K_{2})$.
In general we cannot expect to have the same map $\phi^{a}$ in the first two columns of the diagram (similarly for $\phi^{b}$); instead, for instance, we will have $\phi^{a}$ and $\phi^{a'}$ repectively at $G/K_{1}$ and at $G/H$. Nevertheless, the map $\Phi$ is always diffeotopic (through $SU(3)$-invariant maps) to a map $\Phi'$ for wihich $a$ and $b$ are constant in the respective tubular neighborhoods. The homotopy between $\Phi$ and $\Phi'$ can be described as follows: the map $\Phi$ is identified on each tubular neighborhood by a continuous function $\epsilon:[0,\frac12]\to N(H)$, so that $\Phi=
\phi^{\epsilon(t)}$; we can define $$\eta(t,s):=\epsilon((1-s)t)\,,$$ and $\phi^{\eta(t,s)}$ is the required homotopy. We also observe that, for instance, $a\in N(H)\cap N(K_{1})$ in general, because the map $\epsilon$ is continuous and $N(H)$ is a closed subgroup.
In the sequel, we shall use the following notation to identify the most commonly used homogeneous spaces:
$$\hspace{30pt}\begin{array}{ll}
{\displaystyle}{\mathbb{S}}:=\frac{SU(3)}{SU(2)}, &\hbox{the 5-sphere},\\[10pt]
{\displaystyle}{\mathbb{P}}:=\frac{SU(3)}{S(U(2)\times U(1))}, &\hbox{the complex projective
plane }{\mathbb{C}}{\mathbb{P}}^2,\\[10pt]
{\displaystyle}{\mathbb{L}}:=\frac{SU(3)}{SO(3)}, &\hbox{the set of special Lagrangian
subspaces in ${\mathbb{C}}^{3}$},\\[10pt]
{\displaystyle}{\mathbb{A}}:= \frac{SU(3)}{U(1)}, &\hbox{any Aloff-Wallach type space},\\[10pt]
{\displaystyle}{\mathbb{F}}:=\frac{SU(3)}{T^2}, &\hbox{the complex flag manifold}.
\end{array}$$
We shall actually use ${\mathbb{A}}$ to stand for any homogeneous space of the form $SU(3)/U(1)$, even though the terminology “Aloff-Wallach” usually excludes one case (we shall be more precise in the next section). The Lagrangian interpretation of ${\mathbb{L}}$ can be found in [@harveylawson], and is important for making more explicit some of our constructions, such as finding geodesics from one singular orbit to another.
Connected principal stabilizers
-------------------------------
Principal stabilizers $H$ in our case are $1$-dimensional subgroups of $SU(3)$, such that $H^{0}=U(1)$. The case in which $H^{0}=H$ will be particularly significant, so we will dedicate the first part of this section to it. We begin by defining circle subgroups of $SU(3)$. Let $k,l$ be integers, and let $U_{k,l}$ denote the subgroup (isomorphic to $U(1)$) of $SU(3)$ consisting of matrices $$\begin{pmatrix}
e^{k\imath t} & 0 & 0\\
0 & e^{l \imath t} & 0 \\
0 & 0 & e^{-(k+l)\imath t}\\
\end{pmatrix}\,.$$ We shall denote the coset space $SU(3)/U_{k,l}$ by ${\mathbb{A}}_{k,l}$. Since $U_{k,l}$ is unchanged when any common factor of $k,l$ is removed, we may assume that they are coprime. The space ${\mathbb{A}}_{k,l}$ is called an *Aloff-Wallach space* provided $kl(k+l)\ne0$, since the pairs equivalent to $(1,-1)$ are excluded for geometrical reasons (they do not satisfy the conditions that guarantee the existence of homogeneous positively-curved metrics, see [@aw]). In our analysis, the subgroups $U_{1,-1}$ will however play important roles.
Denote the 1-dimensional subalgebra of ${\mathfrak{su}(3)}$ corresponding to $U_{k,l}$ by ${\mathfrak{u}\lower1.5pt\hbox{$_{k,l}$}}$. We consider the pair of orthogonal subalgebras ${\mathfrak{u}\lower1.5pt\hbox{$_{1,-1}$}},{\mathfrak{u}\lower1.5pt\hbox{$_{1,1}$}}$ generated by the respective elements $$\label{princu1}
{\mathbf{u}}=
\begin{pmatrix}
\imath & 0 & 0\\
0 & -\imath & 0 \\
0 & 0 & 0\\
\end{pmatrix}\quad\text{and}\quad
{\mathbf{v}}=
\begin{pmatrix}
\imath & 0 & 0\\
0 & \imath & 0 \\
0 & 0 & -2\imath\\
\end{pmatrix}$$ that together span a Cartan subalgebra ${\mathfrak{t}}$. It can be shown that ${\mathbf{u}}$ is a regular element, so it belongs to a unique Cartan subalgebra ${\mathfrak{t}}\subset {\mathfrak{su}(3)}$, namely that consisting of diagonal elements; if $\alpha,\,\beta,\alpha+\beta$ denote the roots in ${\mathfrak{t}}_{{\mathbb{C}}}$, we have that ${\mathbf{u}}$ corresponds to $\alpha$, and we can identify $${\mathrm{span}}\{\alpha\}=-\imath\, {\mathfrak{u}\lower1.5pt\hbox{$_{1,-1}$}}\quad {\mathrm{span}}\{\beta\}= -\imath\, {\mathfrak{u}\lower1.5pt\hbox{$_{1,0}$}}\quad {\mathrm{span}}\{\alpha+\beta\}= -\imath\,{\mathfrak{u}\lower1.5pt\hbox{$_{0,1}$}}\,.$$ On the other hand, ${\mathbf{v}}$ is a singular element and is contained in three independent Cartan subalgebras ${\mathfrak{t}},\,{\mathfrak{t}}_{1},\,{\mathfrak{t}}_{2}$; the 1-dimensional orthogonal complements ${\mathbf{v}}^{\perp},\,{\mathbf{v}}^{\perp}_{1},\,{\mathbf{v}}^{\perp}_{2}$ then span the subalgebra ${\mathfrak{su}(2)}$ corresponding to the root $\alpha$. Each root has an orthogonal singular hyperplane, which in our notation correspond to ${\mathfrak{u}\lower1.5pt\hbox{$_{1,1}$}}, {\mathfrak{u}\lower1.5pt\hbox{$_{-2,1}$}}$ and ${\mathfrak{u}\lower1.5pt\hbox{$_{1,-2}$}}$.
The first step to obtain our classification is that of determining the possible gluing maps between two principal orbits. In the case that the principal stabilizer is connected, this corresponds to identify the group $N(U(1))$: this depends from the way $U(1)$ is immersed in $SU(3)$, up to conjugacy. The subgroups $U_{1,-1}$ and $U_{1,1}$ represent distinguished cases, in this sense.
\[normu1\] The normalizer of $U_{k,l}$ in $SU(3)$ is given by $$N(U_{k,l})=
\begin{cases}
{\mathrm{T}^{2}}\cup \tau{\mathrm{T}^{2}}& \text{if $(k,l)=(1,-1)$}\\
S(U(2)\times U(1)) & \text{if $(k,l)=(1,1)$}\\
{\mathrm{T}^{2}}& \text{if $(k\pm l)\ne0$.}\\
\end{cases}$$ Here, $\tau$ denotes an element of $SU(3)$ such that $Ad_{\tau}$ is an element in the Weyl group ${W}$.
For the first case, let $g\in N(U_{1,-1})$; then we also have $g\in N({\mathrm{T}^{2}})$, as otherwise $$U_{1,-1}\subset g{\mathrm{T}^{2}}g^{-1}\neq {\mathrm{T}^{2}}$$ which is impossible as ${\mathbf{u}}$ is a regular element. Hence $N(U_{1,-1})\subset N({\mathrm{T}^{2}})$. It is well known that $${W}:=\frac{N({\mathrm{T}^{2}})}{{\mathrm{T}^{2}}}\cong \mathfrak{S}_{3}$$ is the group of permutations on $3$ elements; it acts on the Cartan subalgebra ${\mathfrak{t}}$ by permuting the three roots $\alpha,\,\beta$ and $\alpha+\beta$. The only elements fixing the subspace $t\cdot\alpha$ corresponding to ${\mathbf{u}}$ are reflections about the hyperplane ${\mathbf{u}}^{\perp}$, sending ${\mathbf{u}}$ to $-{\mathbf{u}}$ and swapping $\beta $ and $\alpha+\beta$, which can be represented by the the action $Ad_{\tau}$ with $\tau$ an appropriate element in $SU(3)$.
In the second case, an element $g\in N(U_{1,1})$ that preserves ${\mathfrak{u}\lower1.5pt\hbox{$_{1,1}$}}$ must also preserve the centralizer $C(U_{1,1})=S(U(2)\times U(1))\cong U(2)$, so that $N(U_{1,1})\subset N(U(2))=U(2)$; the reverse inclusion is obvious.
In the final case, we just have to note that roots and their orthogonal complements are the only eigenspaces for the elements of ${W}$. Hence the other regular elements in ${\mathfrak{t}}$ are normalized only by ${\mathrm{T}^{2}}\cong N({\mathrm{T}^{2}})^{0}$. $\blacksquare$
As a consequence, we obtain the required isomorphisms for the coset spaces parametrizing $SU(3)$-equivariant automorphisms of principal orbits. Firstly, $$\frac{N(U_{1,-1})}{U_{1,-1}}\cong U(1)\cup \tau U(1);$$ more explicitly, this group is generated by the matrices $$\label{normstabroot}
\begin{pmatrix}
e^{\imath t} & 0 & 0\\
0 & e^{\imath t} & 0 \\
0 & 0 & e^{-2 \imath t} \\
\end{pmatrix}
\quad\text{and}\quad
\begin{pmatrix}
0 & e^{\imath t} & 0\\
-e^{\imath t} & 0 & 0 \\
0 & 0 & e^{-2 \imath t} \\
\end{pmatrix}\,.$$ For the second case, $$\frac{N(U_{1,1})}{U_{1,1}}\cong SU(2),$$ and finally $$\frac{N(U_{k,l})}{U_{k,l}}\cong U(1),\qquad (k\pm l)\ne0.$$
*Remark.* We can already estimate the number of $SU(3)$-equivariant diffeomorphism classes in some cases. For, if the principal stabilizer is conjugate to $U_{1,-1}$ then, thanks to Lemma \[normu1\] and Uchida’s condition 1, there are at most two such classes (the number of connected components of $N(U_{1,-1})/U_{1,-1}$). In the case of singular and all other regular elements there is just one $SU(3)$-diffeomorphism class.
The next information that we need is knowledge of which tubular neighborhoods can be built around a given singular orbit. To this aim, we have to determine which representations of a singular stabilizer are sphere-transitive, and associate to it the integers $k,l$ characterizing the corresponding principal stabilizer.
$U(2)$ representations
----------------------
Let us concentrate now on the subgroup $S(U(2)\times U(1))\cong U(2)$ of $SU(3)$, classifying its sphere-transitive real $4$-dimensional representations.
First we introduce some notation. Let ${\Sigma}^n$ denote the complex irreducible representation of $SU(2)$ on ${\mathbb{C}}^2$ of dimension $n+1$, and $A^{m}$ the $U(1)$-representation of weight $m$ with $m\in{\mathbb{Z}}$.
\[su2u1\] The real $4$-dimensional sphere-transitive representations ${V}$ of $U(2)$ are given by $${V}_{{\mathbb{C}}}\cong{\mathbb{C}}^{4}\cong{\Sigma}^1\otimes (A^{m}\oplus A^{-m}),\quad m=2r+1,\,r\in{\mathbb{Z}}.$$ If $\{{\mathbf{u}},{\mathbf{v}}\}$ is a basis for ${\mathfrak{t}}\subset {\mathfrak{su}(2)}\oplus{\mathfrak{u}\lower1.5pt\hbox{$_{1,1}$}}$ (see (\[princu1\])), then the Lie algebra of the stabilizer of a point $x\in S^{3}\subset {V}\cong{\mathbb{R}}^{4}$ has the form $(3{\mathbf{u}}+
m{\mathbf{v}})^{\perp}$.
A consequence of the Peter-Weyl theorem is that a representation of $SU(2)\times U(1)$ necessarily has the form $${V}_{{\mathbb{C}}}\cong
\sum_{n,m}\Sigma^{n}\otimes A^{m}.$$ It is straightforward to see that the only possible case in which one can obtain a sphere-transitive $4$-dimensional representation is given by ${\Sigma}^1\otimes (A^{m}+A^{-m})$. Now, the sums $1+m$ and $1-m$ must be even in order to obtain an $S(U(2)\times U(1))\cong U(2)$ representation, as $SU(2)\times U(1)$ covers $U(2)$ in a two-to-one manner. Hence $m$ must be odd.
Let us restrict the representation to the maximal torus ${\mathrm{T}^{2}}$ contained in $U(2)$, whose Lie algebra is ${\mathfrak{t}}={\mathrm{span}}\{{\mathbf{u}},\,{\mathbf{v}}\}$; then we obtain $${V}_{{\mathbb{C}}}\cong (A^{1}+A^{-1})\otimes (A^{m}+A^{-m})\cong A^{m+1}+A^{-m-1}+A^{-m+1}+A^{m-1}.$$ The necessary real structure is effectively the tensor product $\jmath\otimes\jmath$ of the respective quaternionic structures on ${\Sigma}^1$ and $A^{m}+A^{-m}$. The latter act as the antilinear extensions of the maps $\jmath(x,y)=(y,-x)$ and $\jmath(e,f)=(f,-e)$ for $x,y$ a basis of ${\Sigma}$ and $e,f$ a basis of $A^{m}+A^{-m}$; the fixed point set is given by $$\begin{aligned}
{V}=\>&{\mathrm{span}}\{x\otimes e+y\otimes f,\,x\otimes f-y\otimes e,\>\imath(x\otimes e-y\otimes f),\>\imath(x\otimes f+y\otimes e)\}\notag\\
=\>&{\mathrm{span}}\{w_{1},\,w_{2},\,w_{3},\,w_{4}\}.\notag\end{aligned}$$ Let us consider now the corresponding Lie algebra representation. We choose the point $w_{1}\in S^{3}$: then the Lie algebra ${\mathfrak{su}(2)}\oplus {\mathfrak{u}\lower1.5pt\hbox{$_{1,1}$}}$ acts on $w_{1}$ spanning the $3$-dimensional tangent space of $S^{3}$. More explicitly, if $${\mathfrak{su}(2)}={\mathrm{span}}\{v_{1},\,v_{2},\,v_{3}\}={\mathrm{span}}\left\{\begin{pmatrix}\imath&0\\0&-\imath\end{pmatrix},\,\begin{pmatrix}0&\imath\\\imath&0\end{pmatrix}
,\,\begin{pmatrix}0&1\\-1&0\end{pmatrix}\right\}$$ then we obtain $$v_{1}(w_{1})=w_{3},\quad v_{2}(w_{1})=w_{4},\quad v_{3}(w_{1})=w_{2}\,;$$ moreover the generator ${\mathbf{v}}\in{\mathfrak{u}\lower1.5pt\hbox{$_{1,1}$}}$ acts by $${\mathbf{v}}(w_{1})=m w_{3}\,.$$ Returning to the inclusion ${\mathfrak{su}(2)}\oplus{\mathfrak{u}\lower1.5pt\hbox{$_{1,1}$}}\subset{\mathfrak{su}(3)}$ and identifying $v_{1}={\mathbf{u}}$, we can restrict to the Cartan subalgebra ${\mathfrak{t}}$; then the subspace spanned by $w_{1}, w_{3}$ in ${V}$ is an irreducible ${\mathfrak{t}}$-submodule, and the corresponding weight can be represented via the Killing metric by the vector $${\textstyle}\mathbf{h}=\frac{1}{2}{\mathbf{u}}+ \frac{m}{6}{\mathbf{v}};$$ its kernel, which kills the vector $w_{1}$, is given by the hyperplane $\mathbf{h}^{\perp}$: this is the Lie algebra of the stabilizer $U(1)$, hence the conclusion.$\blacksquare$
Let us analyse in more detail some examples for small $m$: for $m=1$ we get the hyperplane ${\mathfrak{u}\lower1.5pt\hbox{$_{0,1}$}}$ as the stabilizer’s Lie algebra; for $m=3$ we obtain a singular stabilizer ${\mathfrak{u}\lower1.5pt\hbox{$_{2,-1}$}}$. For $m\geq5$ we get other generic regular stabilizers, all belonging to the Weyl chambers delimited by ${\mathfrak{u}\lower1.5pt\hbox{$_{-2,1}$}}$ and ${\mathfrak{u}\lower1.5pt\hbox{$_{1,-2}$}}$; the limit stabilizing subalgebra for $m \to \infty$ corresponds to ${\mathfrak{u}\lower1.5pt\hbox{$_{1,-1}$}}$. The same results are obtained for $m\leq0$, as the representation $A^{m}$ and $A^{-m}$ are isomorphic as real reprsentations.
In the sequel, let us use ${\mathbb{P}}(m)$ to denote the bundle on ${\mathbb{P}}={\mathbb{C}}{\mathbb{P}}^{2}$ obtained as the twisted product by the representation ${V}=[{\Sigma}^2\otimes (A^{m}+A^{-m})]$. Clearly ${\mathbb{P}}(m)\cong {\mathbb{P}}(-m)$, so we can restrict to $m\in {\mathbb{N}}$.
${\mathrm{T}^{2}}$ representations
----------------------------------
Let us discuss now the case of ${\mathrm{T}^{2}}$ as a singular stabilizer: we need to determine its sphere transitive $2$-dimensional representations in order to classify the possible tubular neighborhoods around a singular orbit of type ${\mathbb{F}}$. Let us choose for the standard Cartan subalgebra ${\mathfrak{t}}$ the basis formed by $$\label{basiscartan}
{\mathbf{u}}=
\begin{pmatrix}
\imath & 0 & 0\\
0 & -\imath & 0 \\
0 & 0 & 0\\
\end{pmatrix}\quad\text{and}\quad
{\mathbf{u}}'=
\begin{pmatrix}
0 & 0 & 0\\
0 & \imath & 0 \\
0 & 0 & -\imath\\
\end{pmatrix}\,.$$
Comparing this basis with that in (\[princu1\]), we note that the relation ${\mathbf{v}}={\mathbf{u}}+2{\mathbf{u}}'$ holds, and that ${\mathbf{u}},{\mathbf{u}}'$ correspond to the two roots $\alpha,\beta$; the parallelogram $P$ determined by $2\pi{\mathbf{u}}$ and $2\pi{\mathbf{u}}'$ is a fundamental domain for the maximal torus ${\mathrm{T}^{2}}$, which can therefore be described as $${\mathrm{T}^{2}}\cong \{\exp{s{\mathbf{u}}}\times\exp{t{\mathbf{u}}'}:s,t\in{\mathbb{R}}\}\,.$$
The $2$-dimensional spere-transitive real ${\mathrm{T}^{2}}$-represenations ${V}$ are given by $${V}\cong A^{p}\otimes A^{q}$$ for $p,q\in {\mathbb{Z}}$, with $(p,q)\neq(0,0)$ and $A^{p}\otimes A^{q}\cong A^{-p}\otimes A^{-q}$. Each of them is determined by a weight ${\mathbf{z}}$ contained in ${\mathfrak{t}}$ such that $$\langle{\mathbf{z}},{\mathbf{u}}\rangle=p,\qquad
\langle{\mathbf{z}},{\mathbf{u}}'\rangle=q$$ A basis for the integer lattice of such weights is given by $${\textstyle}{\mathbf{z}}_{1}:=\frac{1}{3}\,{\mathbf{v}}\,,\quad
{\mathbf{z}}_{2}:=\frac{1}{3}\,(2{\mathbf{u}}+{\mathbf{u}}')\;\quad$$ so that a generic weight has the form ${\mathbf{z}}=p\,{\mathbf{z}}_{1}+q\,{\mathbf{z}}_{2}$ for $p,q\in{\mathbb{Z}}$, and the stabilizer for the corresponding representation is given by ${\mathbf{z}}^{\perp}$.
*Observation.* The weights described in Proposition \[su2u1\] are of this type: in fact $${\textstyle}\frac{1}{2}{\mathbf{u}}+ \frac{m}{6} {\mathbf{v}}=\frac{1}{2}{\mathbf{u}}+ \frac{m}{6} ({\mathbf{u}}+2{\mathbf{u}}')=\frac{m-1}{2}{\mathbf{z}}_{1}+{\mathbf{z}}_{2}$$ for the choice $p=(m-1)/2$ and $q=1$ (recall that $l$ is odd): this is just the result of the reduction from $U(2)$ its maximal torus ${\mathrm{T}^{2}}$. In fact the representation ring $R[U(2)]$ is isomorphic to the polynomial ring ${\mathbb{Z}}[\lambda_{1},\lambda_{2},\lambda_{2}^{-1}]$, whereas $R[{\mathrm{T}^{2}}]$ is isomorphic to ${\mathbb{Z}}[\lambda_{1},\lambda_{1}^{-1},\lambda_{2},\lambda_{2}^{-1}]$; it is well known that the inclusion ${\mathrm{T}^{2}}\subset U(2)$ induces an injective map $$\xymatrix{R[U(2)]\ar[r]&}R[{\mathrm{T}^{2}}]$$ (see [@brocktd]). The image of this inclusion coincides with the subring $$R[{\mathrm{T}^{2}}]^{{W}(U(2))}$$ of ${\mathrm{T}^{2}}$ representations which are invariant under the Weyl group $W(U(2))$; the latter is isomorphic to ${\mathbb{Z}}_{2}\subset {W}(SU(3))$, corresponding to a reflection around one of the singular hyperplanes.
In this case, generic stabilizers might not be connected:
\[lemmastabtorus\] For a ${\mathrm{T}^{2}}$-representation of type $A^{p}\otimes A^{q}$ the generic stabilizer is $$U(1)\times {\mathbb{Z}}_{h}$$ where $h=\gcd(p,q)$.
We can describe the representation by $(x,y)\mapsto
e^{2\pi\imath(px+qy)}$, where $(x,y)$ are coordinates with respect to (\[basiscartan\]), after a suitable normalization, to be considered modulo ${\mathbb{Z}}^{2}$. The stabilizer is the solution of the equation $$\label{stabtorus}
px+qy=h,\qquad h\in {\mathbb{Z}}\,;$$ so we have a $1$-dimensional solution for each $h$. On the other hand we can choose any $h_{1}$ and $h_{2}$ in ${\mathbb{Z}}$ so that $(x+h_{1},y+h_{2})$ is the same solution as $(x,y)$ on ${\mathrm{T}^{2}}$, but for a different $h$. Therefore equation (\[stabtorus\]) becomes $$\label{gcd}
px+qy=h-ph_{1}-qh_{2}\,.$$ Let us suppose that $h=\gcd(p,q)>0$: then equation (\[gcd\]) is equivalent to $px+qy=0$, as the $\gcd$ is precisely the smallest positive integer which can be obtained in the form $ph_{1}+qh_{2}$. Hence the solution $(x,y)$ for $h$ is also the solution for $0$; moreover this implies that if $0<h'<h$, then the solution $(x,y)$ for $h'$ is not a solution for $0$. This shows that the solutions are repeated modulo $h$, so that there are precisely $h$ distinct ones, each one isomorphic to a circle $U(1)$: altogether they form an abelian subgroup, isomorphic to $U(1)\times {\mathbb{Z}}_{h}\subset {\mathrm{T}^{2}}$. $\blacksquare$
We introduce some more notation at this point: we shall denote by ${\mathbb{F}}(p,q)$ a tubular neighborhood of a flag manifold obtained by a slice representation $A^{p}\otimes A^{q}$ as explained above. We observe that ${\mathbb{F}}(p,q)\cong {\mathbb{F}}(-p,-q)$.
Regarding the singular stabilizers $SO(3)$ and $SU(2)$, we have a unique sphere-transitive $3$-dimensional representation, namely the standard irreducible space ${\mathbb{R}}^3\cong
[{\Sigma}^2]$. The complete list of possible slice representations for each connected singular stabilizer is given in Table 1.
The classification {#theclassification}
==================
We are now in a position to present the main results of the paper, classifying the possible ways of gluing together tubular neighborhoods obtained from the singular orbits discussed in Section \[introduction\] and from the normal representations described in Section \[preliminaryresults\].
Connected singular stabilizers
------------------------------
We focus first on the case that both the singular stabilizers $K_1,K_2$ are connected. Connected subgroups of $SU(3)$ are in one-to-one correspondence with Lie subalgebras of ${\mathfrak{su}(3)}$. Note that the two Lie subalgebras ${\mathfrak{so}(3)}$ and ${\mathfrak{su}(2)}\oplus{\mathbb{R}}$ are maximal subalgebras. It is also well known that ${\mathfrak{so}(3)}$ and ${\mathfrak{su}(2)}$ are the only $3$-dimensional subalgebras of ${\mathfrak{su}(3)}$, up to conjugation.
Passing to subalgebras of ${\mathfrak{so}(3)}$ and ${\mathfrak{su}(2)}\oplus {\mathbb{R}}$, observe that ${\mathfrak{su}(2)}\cong{\mathfrak{so}(3)}$ does not contain any subalgebra of dimension greater than $1$; therefore we obtain only other two subalgebras, both contained in ${\mathfrak{su}(2)}\oplus{\mathbb{R}}$: namely ${\mathfrak{su}(2)}$ and the Cartan subalgebra ${\mathfrak{t}}$.
Let us also list here the normalizers of each corresponding connected subgroup. Let ${\mathbb{Z}}_{3}$ denote the center of $SU(3)$, and (again) ${W}\cong\mathfrak{S}_3$ its Weyl group. Then $$\begin{array}{c}
N(SU(2))=N(U(2))=U(2)\\[5pt]
N(SO(3))=SO(3)\times {\mathbb{Z}}_{3}\\[5pt]
N({\mathrm{T}^{2}})= \bigcup\limits_{\tau\in {W}} \tau {\mathrm{T}^{2}}\end{array}$$
*Remark.* We shall not treat immediately the case of a singular stabilizer ${\mathrm{T}^{2}}$ with slice representation $A^{p}\otimes
A^{q}$ and $\gcd(p,q)\neq1$. In fact this can imply that the second singular stabilizer is not connected even if ${\mathrm{T}^{2}}$ is (because the principal stabilizer turns out to be not connected, see Lemma \[lemmastabtorus\]), and this situation fits better in Subsection \[nonconnectedstabilizers\] (see Proposition \[propstab2\]).
We can now state the main result of this section:
\[classu3\] Tables 2 and 3 list respectively all the $SU(3)$-diffeomorphism classes of $8$-dimensional compact cohomogeneity-one $SU(3)$-manifolds with orbit space $[0,1]$ such that:\
— both stabilizers belong to the set $\{SU(2),\,U(2),\,SO(3)\}$,\
— one singular stabilizer is isomorphic to ${\mathrm{T}^{2}}$ and the normal representation is $A^{p}\otimes A^{q}$ with $\gcd(p,q)=1$.
Let us consider these connected singular stabilizers: correspondingly we have a slice representation ${V}$ of dimension $3,\,4,\,3,\,2$ (see Table 1); the representations involved must again be of cohomogeneity one, or in other words the singular stabilizer $K_{i}$ must act transitively on the unit sphere $S^{n-1}\subset
{V}$. Let us analyse the possibilities case by case.
The cases of $SU(2)$ and $SO(3)$ are rather simple, as the only $3$-dimensional representation of cohomogeneity one is the standard 3-dimensional irreducible representation ${\mathbb{R}}^{3}\cong[{\Sigma}^2]$, as already observed at the end of Section \[preliminaryresults\]; in this case the principal stabilizer turns out to be one corresponding to $U_{1,-1}$. Therefore, thanks to Lemma \[normu1\], the normalizer is ${\mathrm{T}^{2}}\cup\tau {\mathrm{T}^{2}}$ in both cases; on the other hand, it can be shown that for both the singular orbits ${\mathbb{S}}$ and ${\mathbb{L}}$, the component $\tau{\mathrm{T}^{2}}$ of the normalizer $N(U(1))$ intesects $SU(2)\subset N(SU(2))$ and $SO(3)\subset N(SO(3))$ respectively (for instance in a point $x$ obtained by putting $t=\pi/2$ in an appropriate conjugate of the second element in (\[normstabroot\])). Hence any $SU(3)$ equivariant automorphism of the principal orbit is diffeotopic to one which can be extended to an automorphism of the whole tubular neighborhood (see Uchida’s criteria in Section \[preliminaryresults\]), so that we have a unique $SU(3)$-equivariant diffeomorphism class of $M$ containing one of ${\mathbb{S}}$ or ${\mathbb{L}}$ and another singular orbit $M_{2}$.
Let us discuss the case of ${\mathbb{P}}(m)$: Proposition \[su2u1\] says that we have a singular stabilizer for $m=3$ and a root stabilizer for $m=1$, while for all other values of $m$ the stabilizer is generic regular; in all cases, except for $m=1$, we have that $N(U(1))$ is connected, hence we have $1$ possible way of gluing each of these tubular neighborood to others; for $m=1$ we have that the component $\tau{\mathrm{T}^{2}}$ does not intersect $ S(U(2)\times U(1))$, so we have $2$ distinct classes in this case. The fact that the two classe obtained form the two gluing maps $\phi^{e}$ and $\phi^{\tau}$ cannot be isomorphic follows by inspecting diagram (\[gluingdiag\]). In fact all the vertical maps must be of the form $\phi^{e}$ in order to be defined on the whole tubular neighborhoods, and this implies that the central part of the diagram can not be commutative if we put, for instance, $\psi=\phi^{\tau}$. In general we can combine two tubular neigborhoods if and only if the principal stanilizers are conjugate; therefore ${\mathbb{P}}(n)$ and ${\mathbb{P}}(m)$ can be glued together if and only if $n=m$, and the gluing map is unique for $m\neq1$, and there are two distinct for $m=1$.
Let us pass now to tubular neighborhoods of type ${\mathbb{F}}(p,q)$ assuming that $\gcd(p,q)=1$: there is precisely one gluing map for $(p,q)\neq(0,1)$, $(p,q)\neq(1,0)$ or $(p,q)\neq(1,-1)$, as in fact in this case the normalizers $N(U(1))$ are all connected. By contrast, for the remaining representations the principal stabilizer is of type $U_{1,-1}$, hence we have at first sight two possible gluing maps. These can be used to join this tubular neighorhood to others with the same type of stabilizer; nevertheless $\tau {\mathrm{T}^{2}}\subset N({\mathrm{T}^{2}})$, so as usual $\phi^{\tau}$ can be extended to the whole tubular neighborhood, and it is equivalent to the identity gluing map.
The list of all possible combinations is given in the Tables. $\blacksquare$
We end this section by examining the case in which $M/SU(3)$ is $S^{1}$ and the principal orbits are simply connected. Let us point out that the homogeneous manifold ${\mathbb{A}}_{k,l}$ is simply connected, as shown by the long exact homotopy sequence for a fibration: $$\label{lesaw}
\xymatrix{\cdots\,\pi_{1}(SU(3))\ar[r]&\pi_{1}({\mathbb{A}}_{k,l})\ar[r]&\pi_{0}(U(1))\,\cdots}\,.$$ In this case there are no singular orbits and the manifold $M$ is a bundle $$\xymatrix{G/H\>\ar@{^{(}->}[r]&M\ar[d]\\ & S^{1}}$$ where $H=U_{k,l}$ is the principal (and unique) stabilizer; the structure group for this bundle is contained in $N(H)/H$ (see [@bredon Th. 8.2, Ch. IV]). Hence we have
\[mgs1\] Let $M$ be a cohomogeneity-one $SU(3)$-manifold with $M/G\cong
S^{1}$ and such that the principal orbit is simply connected. Then the principal orbit has the form ${\mathbb{A}}_{k,l}$. Either $$M\cong {\mathbb{A}}_{k,l}\times S^{1},$$ which is possible for any $k,l$, or ${\mathbb{A}}_{k,l}={\mathbb{A}}_{1,-1}$ and $M$ is a nontrivial bundle over $S^{1}$.
We can divide the proof in three cases, corresponding to the stabilizers described in Lemma \[normu1\]. First we note that the bundle structure is given by the $N(H)/H$-valued transition functions $g_{1}$ and $g_{2}$ defined on the two points $p_{1}$ and $p_{2}$, which constitute the “equator” of the base manifold $S^{1}$.
In the first case, $N(U_{1,-1})/U_{1,-1}$ has $2$ connected components, therefore there are two possible nonequivalent choices for the maps $g_{i}$, giving rise to the trivial bundle and another nontrivial, respectively.
In the remaining two cases, $N(U_{k,l})/U_{k,l}$ is connected: we have a unique (trivial) bundle for $U_{1,1}$, and there are infinite nonconjugate generic $U_{k,l}$’s, giving rise to nonisomorphic generic fibres ${\mathbb{A}}_{k,l}$.
The $SU(3)$-manifolds obtained in this way are all trivial bundles, except for the first case. $\blacksquare$
Non-connected singular stabilizers {#nonconnectedstabilizers}
----------------------------------
We now conclude the classification, describing the more general situation in which the singular stabilizers are not connected. This implies that the singular orbits are not simply connected, and their respective universal covers are those described in Theorem \[classu3\]. Some of our arguments are inspired by those used in [@alek-pod].
\[propstab1\] If the connected components $K_{i}^{0}$ of the two singular stabilizers belong to the set $\{SO(3), SU(2), U(2)\}$, then both are connected: $K_{i}^{0}=K_{i}$ for $k=1,\,2$.
Suppose that $K_{1}^{0}$ is one of the three subgroups in the list: then the codimension of the singular orbit is at least $3$; a general position argument shows that $M\setminus (SU(3)/K_{1})$ is simply connected, as is $M$. This complement has the same homotopy type of $SU(3)/K_{2}$, so $\pi_{1}(SU(3)/K_{2})=0$ too: this implies that the stabilizer $K_{2}$ is connected. By the long exact homotopy sequence for a fibration $$\label{lesk1}
\xymatrix{\cdots\, \pi_{1}(S^{r})\ar[r]&\pi_{0}(H)\ar[r] &\pi_{0}(K_{2})\cdots}$$ the principal stabilizer $H$ must also be connected, for $r>1$, which is the case for all the representations involved with the three stabilizers under consideration. This implies that also $K_{1}$ is connected, hence the result. $\blacksquare$
This means that we cannot obtain new simply-connected manifolds by gluing together tubular neighborhoods unless they involve ${\mathrm{T}^{2}}$ as $K_{i}^{0}$ for at least one $i$. We discuss now this remaining case; the new manifolds we obtain in this way are given in Table $4$. We point out that the principal stabilizers turn out to be non-connected in these cases. Before that, we prove a result which corresponds to Lemma \[normu1\] for non-connected $H$:
\[normu1nc\] Consider the subgroup $U_{k,l}\times {\mathbb{Z}}_{h}$ of ${\mathrm{T}^{2}}\subset SU(3)$; then $$N(U_{k,l}\times {\mathbb{Z}}_{h})=N(U_{k,l})$$ if $U_{k,l}$ is regular; if $U_{k,l}$ is singular (for instance $U_{1,1}$) then $$N( U_{1,1}\times {\mathbb{Z}}_{h})= {\mathrm{T}^{2}}\cup \tau {\mathrm{T}^{2}}\,.$$
The proof in the regular case is completely analogous to that of Lemma \[normu1\]; for the singular case we just have to observe that if $h\neq1$ the group contains regular elements, and the whole ${\mathrm{T}^{2}}$ must be preserved by the normalizer of $ U_{1,1} \times
{\mathbb{Z}}_{h}$. The element $\tau\in{W}$ that normalizes $U_{1,-1}$, reflecting the root $\alpha$, is the only one which also preserves $U_{1,1}\times {\mathbb{Z}}_{h}$, hence the conclusion. $\blacksquare$
*Observation.* In this situation, we have two connected components for the normalizers of $U_{1,-1}\times {\mathbb{Z}}_{h}$ and of $U_{1,-1}\times{\mathbb{Z}}_{h}$; if these stabilizers appear in a tubular neighborhood of type ${\mathbb{F}}(p,q)$, we observe that in both cases we obtain only one $SU(3)$ diffeomorphism class, because both normalizers are contained in $N({\mathrm{T}^{2}})$ (for Uchida’s criteria, see Theorem \[classu3\]).
We pass now to the main result of this section, but before of that we recall that any subgroup $K\subset G$ is alaways contained in $N(K^{0})$, because for any $x\in K$ the adjoint action $Ad_{x}$ is continuous, preserves $K$ and fixes $e$.
\[propstab2\] Suppose that $K_{1}^{0}\in\{SO(3), SU(2), U(2)\}$ and that $K_{2}^{0}={\mathrm{T}^{2}}$: then $K_{2}=K_{2}^{0}={\mathrm{T}^{2}}$. Moreover if $K_{2}={\mathrm{T}^{2}}$ and if the slice representation at ${\mathbb{F}}$ is $A^{p}\otimes A^{q}$ there are the following possibilities:
1. if $(p,q)=(0,h)$ then $K_{1}\in\{SO(3),\,SU(2),\,U(2),\, {\mathrm{T}^{2}}\}$;
2. if $(p,q)=(0,h)$ for some $h\in{\mathbb{Z}},\,h>1$, then $K_{1}=(SU(2)\times
{\mathbb{Z}}_{2h})/{\mathbb{Z}}_{2}$, except in the case $h=3$, where also $K_{1}=SO(3)\times {\mathbb{Z}}_{3}$ is possible;
3. if $(p,q)\neq(0,h)$ and $\gcd(p,q)=1$ then $K_{1}\in\{{\mathrm{T}^{2}}, \,U(2)\}$;
4. if $(p,q)\neq(0,h)$ and $\gcd(p,q)\neq1$ then $K_{1}^{0}={\mathrm{T}^{2}}$.
------------------------------- --------------------------------- --------------------------------- --------------------------------
$_{M_{2} }\backslash^{M_{1}}$ ${\mathbb{S}}/{\mathbb{Z}}_{h}$ ${\mathbb{L}}/{\mathbb{Z}}_{3}$ ${\mathbb{F}}(l,m)$
\[1pt\] ${\mathbb{F}}(p,q)$ $\delta^{p}_{0}\delta^{h}_{q}$ $\delta^{p}_{0}\delta^{3}_{q}$ $\delta^{p}_{l}\delta^{m}_{q}$
\[1pt\]
------------------------------- --------------------------------- --------------------------------- --------------------------------
\[nonconn\]
The first statement follows from the same general position argument as in Proposition \[propstab1\]. As proved in Lemma \[lemmastabtorus\], the principal stabilizer is of the form $U_{k,l}\times {\mathbb{Z}}_{h}$, so we need to determine which of the singular stabilizers contain this subgroup.
In the first case we have $h=1$ and $U_{k,l}=U_{1,-1}$, which appears as a principal stabilizer associated to any of the connected stabilizers above, with the appropriate slice representation ${V}$, as already shown in Theorem \[classu3\]. In this case $K_{1}$ is connected, because the sphere $S^{r}\subset{V}$ is.
For the second case we argue as follows: $K_{1}$ must contain a subgroup of type $U_{1,-1}\times {\mathbb{Z}}_{h}$, but we have to exclude $U(2)$, because it allows only connected principal stbilizers ($h=1$). Another possible choice is the subgroup $$\frac{SU(2)\times {\mathbb{Z}}_{2h}}{{\mathbb{Z}}_{2}}$$ where ${\mathbb{Z}}_{2}$ is the center of $SU(2)$; topologically it is the union of $h$ copies of $SU(2)$, and ${\mathbb{Z}}_{2h}$ should be regarded as a subgroup of the singular $U(1)$ centralizing $SU(2)$ (for instance $U_{1,1}$ for the standard immersion). We observe that the singular orbit in this case is isomorphic to ${\mathbb{S}}/{\mathbb{Z}}_{h}$. Suppose instead that $K_{1}^{0}=SO(3)$: then $K_{1}$ must be a subgroup of the normalizer $N(SO(3))=SO(3)\times {\mathbb{Z}}_{3}$, which are $SO(3)$ itself or the whole $N(SO(3))$. The latter case in this situation corresponds for instance to the weight $(p,q)=(0,3)$ for ${\mathrm{T}^{2}}$. As observed after Lemma \[normu1nc\], in both cases the two gluing maps $\phi^{e},\psi^{\tau}$ give rise to isomorphic $SU(3)$-spaces.
For the third case, the connected component $K_{1}^{0}$ must contain ${\mathrm{T}^{2}}$, because only then the corresponding Lie algebra does contain the correct ${\mathfrak{u}}(1)$.
Finally, in the fourth case we have to exclude $U(2)$ because, as observed in case $2$, it allows only connected principal stabilizers. $\blacksquare$
*Observation.* Two of the manifolds that are new with respect to the classification given in Theorem \[classu3\] come from case $2$. We note that in these cases the singular stabilizer $(SU(2)\times {\mathbb{Z}}_{2h})/{\mathbb{Z}}_{2}$ admits as a slice representation ${V}$ only the standard ${\mathbb{R}}^{3}\cong [{\Sigma}^{2}\otimes A^{0}]$: in fact any ${\mathbb{Z}}_{h}$ representation can be extended to a $U(1)$ representation $A^{m}$ with $0\leq m\leq h-1$, hence ${V}$ is the restriction of a $U(2)$ representation; therefore $${V}_{{\mathbb{C}}}\cong \sum {\Sigma}^{l}\otimes A^{m}$$ as seen in Proposition \[su2u1\]; for dimensional reasons ${\Sigma}^{2}\otimes A^{0}$ is the only possible choice. Analogous considerations hold for $SO(3)\times {\mathbb{Z}}_{3}$.
Let us consider the case in which $K_{i}\neq K_{i}^{0}={\mathrm{T}^{2}}$, so that $K_{i}\subset N({\mathrm{T}^{2}})$. Here, the two stabilizers must have the same number of connected components, otherwise the two tubular neighborhoods could not be glued together, as the principal orbits would not be isomorphic. In this case $K_{1}=K_{2}$ and $\pi_{1}(SU(3)/K_{i})\neq0$ for $i=1,2$; moreover in the long exact sequence $$\xymatrix{\cdots\, \pi_{1}(SU(3)/H)\ar[r] &\pi_{1}(SU(3)/K_{i})\ar[r]&\pi_{0}(S^{1}) \,\cdots}$$ the bundle projections induce surjections on the respective fundamental groups. The Seifert–van Kampen Theorem tells us that this is incompatible with the simply connectedness of the manifold $M$, so we have to exclude this case.
Examples {#examplesandapplications}
========
In order to present some familar examples, the notation ${\mathscr{M}}(M_{1},\,M_{2})$ indicates an $8$-dimensional $SU(3)$-manifold obtained by gluing appropriate disk bundles over singular orbits $M_{1},\,M_{2}$ with a map $\phi$ which may or may not be the identity.
Then we have the following remarkable identifications:
- the complex Grassmannian ${\mathbb{G}\mathrm{r}_{2}({\mathbb{C}}^{\,4})}$ is ${\mathscr{M}}({\mathbb{P}},\,{\mathbb{P}})$;
- the quaternionic projective plane ${\mathbb{H}}{\mathbb{P}}^{2}$ is ${\mathscr{M}}({\mathbb{P}},\,{\mathbb{S}})$
- the exceptional Wolf space $G_{2}/SO(4)$ is ${\mathscr{M}}({\mathbb{P}},\,{\mathbb{L}})$
- the product ${\mathbb{C}}{\mathbb{P}}^2\times{\mathbb{C}}{\mathbb{P}}^2={\mathbb{P}}\times{\mathbb{P}}$ is ${\mathscr{M}}({\mathbb{P}},{\mathbb{F}})$
- the Lie group $SU(3)$ is itself ${\mathscr{M}}({\mathbb{L}},\,{\mathbb{S}})$.
We describe these $SU(3)$ spaces in a bit more detail. Recall that ${\mathbb{L}}=SU(3)/SO(3)$. The first three examples are obtained by standard inclusions of $SU(3)$ in $SU(4)$, $Sp(3)$ and $G_{2}$, and in these cases, the normal bundle over each ${\mathbb{C}}{\mathbb{P}}^2={\mathbb{P}}$ is ${\mathbb{P}}(1)$.
The fourth (product) case is given by the diagonal action of $SU(3)$, where the first singular orbit consists of all couples $([z],[z])$ of identical complex lines in ${\mathbb{C}}^{3}$, and the second consists of couples $([z],[w])$ with $[w]\subset [z]^{\perp}$. In this case, the slice representation ${V}$ is isomorphic to the isotropy representation at ${\mathbb{P}}$: in fact if $(v,v)$ is a tangent vector at $([z],[z])$, with $v$ generated by an elelement in ${\mathfrak{su}(3)}/({\mathfrak{u}}(2)\oplus{\mathbb{R}})$, then normal vectors must be of the form $(v,-v)$ and give rise to the same $U(2)$ representation. It is straightforward to check that this tubular neighborhood is of type ${\mathbb{P}}(3)$.
The final case is given by a modification of the Adjoint action of $SU(3)$ on itself, discussed in more detail in Subsection \[theconsimilarityaction\].
The case in which the two tubular neighborhoods are isomorphic and the gluing map is the identity is particularly simple. We can identify the singular orbits $M_{1}=M_{2}=M$ and call the unique normal representation ${V}$; the resulting manifold ${\mathscr{D}}(M)={\mathscr{M}}(M_1,M_2)$ is then the “double” of the disk bundle associated to $V$. This manifold is obtained by the one-point compactification ${\mathbb{R}}^{n}\leadsto S^{n}$ of the ${V}$ fibres over $M$: $$\label{dou}
\xymatrix{S^{n}\>\ar@{^{(}->}[r]& {\mathscr{D}}(M)\ar[d] \\ & M.}$$ The other singular orbit becomes the section at infinity of this new bundle.
The manifolds ${\mathscr{D}}({\mathbb{S}})$ and ${\mathscr{D}}( {{\mathbb{L}}})$ do not admit any $SU(3)$-invariant metric of positive sectional curvature.
This is just a consequence of [@podestaverdiani2 Lemma 3.2], which asserts that any even dimensional cohomogeneity-one $G$-manifold $M$ with an invariant metric of positive sectional curvature has $\chi(M)>0$, and of the observation $$\chi({\mathscr{D}}({\mathbb{S}}))=\chi({\mathscr{D}}({\mathbb{L}}))=\chi(S^{3}) \chi({\mathbb{S}})=0$$ (recall that $\chi({\mathbb{L}})=\chi({\mathbb{S}})$). $\blacksquare$
Consimilarity {#theconsimilarityaction}
-------------
We are going now to consider a group action $\mathbf{c}$ of $GL(n,{\mathbb{C}})$ on itself, called *consimilarity*, defined by $$\label{consim}
{\mathbf{c}}(A)B := A\kern1pt B\kern1pt \overline{A}^{-1}.$$ This action naturally occurs when considering *anti*-linear mappings between a given vector space, of relevance in quantum theory. It also occurs in various geometrical situations (see, for example, [@finopartsala]), and is intimately related to *similarity*. The mapping $$\label{AA}
{\Gamma}\colon A\mapsto A\kern1pt \overline A$$ induces a mapping between consimilarity classes and similarity classes (i.e. orbits under (\[consim\]) and orbits under conjugation). Although this mapping is not in general a bijection between the respective classes, it is true that ${\Gamma}^{-1}(I)$ coincides with the consimilarity orbit $$\label{AAminus}
\{A\kern1pt \overline A^{-1}:A\in GL(n,{\mathbb{C}})\}$$ of the identity. This fact is not entirely obvious, but has an easy proof [@hornjohnson].
Consimilarity can be restricted to $SU(n)\subset GL(n,{\mathbb{C}})$,so that $SU(n)$ acts on itself, as in this case $$A\kern1pt B\kern1pt \overline{A}^{-1}=AB A^{t}$$ is in $SU(n)$ if $A,\,B$ are. It is straightforward to prove that the consimilarity action of $SU(n)$ on itself is isometric with respect to the Killing metric. The resulting action is in fact a special case of a family of actions of a Lie group $G$ on itself, constructed using an automorphism $\sigma$ of $G$ (see [@hptt1], [@conlon] and [@hornjohnson]).
Let us return to the case $n=3$.
\[consimlemma\] Consimilarity is a cohomogeneity-one action of $SU(3)$ on itself with singular orbits ${\mathbb{L}}$ and ${\mathbb{S}}=S^5$. The former is the orbit containing the identity matrix $I$ and coincides with ${\Gamma}^{-1}(I)\cap SU(3)$.
It can be shown that ${\Gamma}^{-1}(I)\cap SU(3)$ coincides with the set $$\mathcal{S}=\{A\in SU(3):A=A^t\}$$ of symmetric matrices. The map $\xi:{\mathbb{L}}\to\mathcal{S}$ defined by $$A \,SO(3)\xymatrix{\ar@{|->}[r] &} AA^{t}$$ is well defined and surjective. It is also injective as if $AA^{t}=CC^{t}$ then $$C^{-1}A=C^{t}(A^{t})^{-1}=\big((C^{-1}A)^{t}\big)^{-1}$$ so that $C^{-1}A\in SO(3)$. This shows that the ${\mathbf{c}}$-orbit through the identity is ${\mathbb{L}}$.
Consider the point $I$ and its stabilizer $SO(3)$. The isotropy and the slice representations are determined by the decomposition $${\mathfrak{su}(3)}=T{\mathbb{L}}\oplus{V}={\mathfrak{so}(3)}^{\perp}\oplus{\mathfrak{so}(3)}=[{\Sigma}^4]\oplus [{\Sigma}^2]$$ as $SO(3)$ representations. The slice representation is sphere transitive (see Table 1), and this shows that the cohomogeneity of the action is $1$. For instance we can choose the normal direction determined by the matrix $$w=
\begin{pmatrix}
0&1&0\\
-1&0&0\\
0&0&0
\end{pmatrix}
\in {\mathfrak{so}(3)}={\Sigma}^2\,.$$ The corresponding geodesic $B(t)=\exp(tw)$ intersects orthogonally all the ${\mathbf{c}}$-orbits (see [@hptt1]): the second singular one is reached at $B_{s}=B(\pi/4)$. In fact, an explicit calculation shows that the stabilizer at $B_{s}$ is $SU(2)$; therefore the corresponding orbit is indeed ${\mathbb{S}}$. $\blacksquare$
Observe that $$\label{imtrz}
\overline{{\mathrm{Tr}}(A\kern1pt \overline{A})}={\mathrm{Tr}}(\overline A\kern1pt{A})
={\mathrm{Tr}}(A\kern1pt \overline{A})\;;$$ this implies that the image ${\Gamma}(SU(3))$ is contained in the hypersurface $$\label{hyper}
{\mathcal{H}}:=\{B\in SU(3):{\mathrm{Tr}}\,B\in{\mathbb{R}}\}$$ of $SU(3)$. We shall investigate the resulting mapping ${\Gamma}\colon
SU(3)\to {\mathcal{H}}$ in the next section.
Quotients by circle subgroups {#quotientsbycirclesubgroups}
=============================
An analogous classification of $SU(3)$ actions is possible in dimension $7$, and partial results can be found in [@podestaverdiani]. Restricting attention here to the case in which both singular orbits are ${\mathbb{P}}={\mathbb{C}}{\mathbb{P}}^2$, and both tubular neighborhoods are isomorphic to the rank 3 vector bundle $\Lambda^{2}_{-}{\mathbb{C}}{\mathbb{P}}^2$, it is not hard to show the existence of only two classes of cohomogeneity-one $SU(3)$-spaces with this data. There is a choice of gluing map between the generic orbits ${\mathbb{F}}$: the identity in one case, and a map $\phi^{\tau}$ associated to a non-trivial element $\tau\in{W}$ in the other. With the latter choice, we obtain the sphere $S^{7}\subset{\mathfrak{su}(3)}$ with the action induced by the Adjoint representation.
We now exhibit a model for the manifold obtained in the former case, denoted here by $N^{7}$, involving the Grassmannian ${\widetilde{\mathbb{G}\mathrm{r}}_{3}}({\mathfrak{su}(3)})$ of oriented 3-dimensional subspaces of the Lie algebra ${\mathfrak{su}(3)}$, which is an $SU(3)$-space under the action induced by $Ad_{SU(3)}$.
\[N7\] The manifold $N^{7}$ is a submanifold of ${\widetilde{\mathbb{G}\mathrm{r}}_{3}}({\mathfrak{su}(3)})$ with the $SU(3)$ action induced by the Adjoint action on ${\mathfrak{su}(3)}$.
Following [@swann98], we consider the function $f\colon
{\widetilde{\mathbb{G}\mathrm{r}}_{3}}({\mathfrak{su}(3)})\to{\mathbb{R}}$ induced by the standard $3$-form on ${\mathfrak{su}(3)}$. Thus $$f(U)=\langle x,[y,z]\rangle,$$ where $\{x,y,z\}$ is an orthonormal basis of the $3$-dimensional subspace $U\subset{\mathfrak{su}(3)}$. The absolute maxima and minima of $f$ are each attained on a copy of ${\mathbb{C}}{\mathbb{P}}^2$ corresponding to the highest root embedding ${\mathfrak{su}(2)}\subset{\mathfrak{su}(3)}$ and a choice of orientation for ${\mathfrak{su}(2)}$. The tangent space $T_U{\widetilde{\mathbb{G}\mathrm{r}}_{3}}({\mathfrak{su}(3)})$ has the form $U\otimes U^{\perp}$; for $U={\mathfrak{su}(2)}$ it can be decomposed as $$T_{{\mathfrak{su}(2)}}{\widetilde{\mathbb{G}\mathrm{r}}_{3}}({\mathfrak{su}(3)})={\mathfrak{su}(2)}\otimes ({\Sigma}^0 +2{\Sigma}^1)\cong {\Sigma}^2+ 2({\Sigma}^1+{\Sigma}^3).$$ The subspace $2{\Sigma}^{1}={\Sigma}^{1}\oplus{\Sigma}^1$ represents the tangent space to the critical manifold ${\mathbb{C}}{\mathbb{P}}^2$; if we choose instead the summand ${\Sigma}^2$, we obtain the bundle $\Lambda^{2}_{-}{\mathbb{C}}{\mathbb{P}}^2$, which is therefore a subbundle of the normal bundle at both ${\mathbb{C}}{\mathbb{P}}^{2}$. In the two cases it turns out to be a stable or an unstable subbundle respectively.
The manifold $N^7$ is obtained from the two $\Lambda^{2}_{-}{\mathbb{C}}{\mathbb{P}}^2$ over the two extremal ${\mathbb{C}}{\mathbb{P}}^2$. To see this, denote by $\tilde{N}^{7}$ the manifold obtained by considering the union of the flow lines of the vector field ${\mathrm{grad}}\,f$ with limit points in the two copies of ${\mathbb{C}}{\mathbb{P}}^{2}$ and tangent directions corresponding to the respective ${\Sigma}^{2}$. Such a flow line (without caring about the parametrization) is given by $$V(t)={\mathrm{span}}\{{\mathbf{u}}\cos{t}+{\mathbf{v}}\sin{t},\,{\mathbf{u}}_{2},\,{\mathbf{u}}_{3}\}\,,$$ with ${\mathbf{u}},{\mathbf{v}}$ as in (\[princu1\]) and ${\mathfrak{su}(2)}={\mathrm{span}}\{{\mathbf{u}},{\mathbf{u}}_{2},{\mathbf{u}}_{3}\}$. It is straightforward to see that the stabilizer for $t\neq k\pi$ under the $Ad_{SU(3)}$ action is ${\mathrm{T}^{2}}$, and for $t=\pi$ the integral curve intersects the minimal critical submanifold at the same subalgebra ${\mathfrak{su}(2)}$ with opposite orientation. In both cases the tangential direction of $V(t)$ belongs to the summand ${\Sigma}^2$ at the critical points: these facts imply that the gluing map for the two tubular neighborhoods must be the identity, so $\tilde{N}^{7}\cong N^{7}$. $\blacksquare$
*Remark.* We point out that $N^{7}$ is *not* homeomorphic to $S^{7}$. As the double ${\mathscr{D}}({\mathbb{C}}{\mathbb{P}}^2)$, it can be regarded as a $3$-sphere bundle over ${\mathbb{C}}{\mathbb{P}}^2$ as in (\[dou\]), in contrast to $S^7$. Now, $\pi_{2}({\mathbb{C}}{\mathbb{P}}^2)=H_{2}({\mathbb{C}}{\mathbb{P}}^2,{\mathbb{Z}})={\mathbb{Z}}$, and writing the homotopy exact sequence for a fibration we obtain $$\xymatrix{\cdots\, \pi_{2}(S^{3})\ar[r] &\pi_{2}(N^{7})\ar[r]&\pi_{2}({\mathbb{C}}{\mathbb{P}}^2)\ar[r]&\pi_{1}(S^{3}) \,\cdots}.$$ This implies $\pi_{2}(N^{7})=\pi_{2}({\mathbb{C}}{\mathbb{P}}^2)={\mathbb{Z}}$, whilst $\pi_{2}(S^{7})=0$.
It is shown in [@podestaverdiani] that $N^{7}$ cannot be equipped with an invariant metric of positive curvature. Indeed, $S^{7}$ is the unique $7$-dimensional positively curved cohomogeneity-one $G$-manifold, if the semisimple part of $G$ has dimension greater then $6$.
The above example is linked to the $8$-dimensional case by a moment map $\mu$ associated to the action of $SU(3)$ on the Wolf spaces ${\mathbb{H}}{\mathbb{P}}^{2}$ and ${\mathbb{G}\mathrm{r}_{2}({\mathbb{C}}^{\,4})}$ (see [@galaw1]). Denoting by $M$ either of these space, it is possible to construct from $\mu$ an equivariant map $$\label{mmapsu3}
\Psi: M_{0} \xymatrix{\ar[r]&}{\widetilde{\mathbb{G}\mathrm{r}}_{3}}({\mathfrak{su}(3)})$$ defined on an open dense subset $M_{0}\subset M$. This construction was used in [@gambioli1] in order to relate the geometry of a quaternion-Kähler manifold with the geometry of the Grassmannian ${\widetilde{\mathbb{G}\mathrm{r}}_{3}}({\mathfrak{g}})$, but we cannot use the same techniques here since in the two cases considered, the differential $\Psi_{*}$ is nowhere injective. Moreover the subset $M_{0}$ is strictly contained in $M$: indeed $M_{0}= {\mathbb{H}}{\mathbb{P}}^{2}\setminus {\mathbb{C}}{\mathbb{P}}^2$ and $M_{0}={\mathbb{G}\mathrm{r}_{2}({\mathbb{C}}^{\,4})}\setminus
{\mathbb{C}}{\mathbb{P}}^2$ respectively, and $$\Psi(M_{0})\subset N^{7}\subset {\widetilde{\mathbb{G}\mathrm{r}}_{3}}({\mathfrak{su}(3)})\,.$$
One may ask if the map $\Psi$ could be extended equivariantly to the whole $W$ in both cases, as this happens in other significant cases (for instance $Sp(n)Sp(1)$ acting on ${\mathbb{H}}{\mathbb{P}}^{n}$ or $Sp(n)$ acting on ${\mathbb{G}\mathrm{r}_{2}({\mathbb{C}}^{\,n})}$). In fact the generic fibre $\Psi^{-1}(x)$ is a circle $S^{1}$: the resulting $S^{1}$ action on ${\mathbb{H}}{\mathbb{P}}^{2}$ was described in [@batt], and $${\mathbb{H}}{\mathbb{P}}^{2}/S^{1}\cong S^{7}$$ (see [@atiyahberndt] and [@atiyahwitten]). For the same reason we have a topological quotient $${\mathbb{G}\mathrm{r}_{2}({\mathbb{C}}^{\,4})}/S^{1}\cong S^{7}.$$ However, as observed above, $S^{7}$ is different from $N^{7}$, and it is easy to check that $\Psi$ cannot be extended equivariantly to the whole Wolf spaces ${\mathbb{H}}{\mathbb{P}}^{2}$ and ${\mathbb{G}\mathrm{r}_{2}({\mathbb{C}}^{\,4})}$.
The fact that $S^7$ is a compactification of $\Lambda^2_{-}{\mathbb{C}}{\mathbb{P}}^2$ was used in [@miyaoka] to unify the construction of various Ricci-flat metrics on complements of homogeneous spaces inside spheres. There are analogous constructions on $G_2/SO(4)$ and $SU(3)$. The descriptions at the start of Section \[examplesandapplications\] show that dense open subsets of thse two manifolds can be $SU(3)$-equivariantly identified. However, the respective singular orbits ${\mathbb{P}}$ and ${\mathbb{S}}$ are not directly related by the Hopf fibration ${\mathbb{S}}\to{\mathbb{P}}$; indeed passing from the ${\mathbb{P}}$ of $G_2/SO(4)$ to the ${\mathbb{S}}$ of $SU(3)$ requires a “flip” of the type considered in [@gukospartong]. This is made possible by the existence of three distinct mappings ${\mathbb{F}}\to{\mathbb{P}}$, similarly exploited in the theory of harmonic maps [@sal85].
To conclude the paper, we identify an analogue for $SU(3)$ of the map $\Psi$ described in (\[mmapsu3\]).
The image ${\Gamma}(SU(3))$ of (\[AA\]) is the hypersurface (\[hyper\]), and is homeomorphic to the Thom space of the vector bundle $\Lambda^2_-{\mathbb{C}}{\mathbb{P}}^2$. The restriction of ${\Gamma}$ to $SU(3)\setminus{\mathbb{L}}$ is a principal $S^1$ bundle over $\Lambda^2_-{\mathbb{C}}{\mathbb{P}}^2$.
Let $SU(2)\subset SU(3)$; then $$\label{eqimtr}
{\mathcal{H}}=\bigcup_{g\in SU(3)} Ad_{g}SU(2).$$ In fact, consider the Lie algebra ${\mathfrak{su}(3)}$; it is well known that any element $x\in {\mathfrak{su}(3)}$ belongs to the standard ${\mathfrak{t}}={\mathrm{span}}\{{\mathbf{u}},{\mathbf{v}}\}$ (see (\[basiscartan\])), up to conjugation. It is sufficient therefore to solve the equation $$\mathrm{Im}\,{\mathrm{Tr}}\big(\exp{(t\,{\mathbf{u}}+s\,{\mathbf{v}})}\big)=0,$$ equivalent to $$\sin(t+s)+\sin(s-t)-\sin(2s)=0\,;$$ this has solutions $\{s=0+k\pi\} \cup \{s=\pm t +2k\pi\}$. These are nothing other than the three lines corresponding to ${\mathfrak{u}\lower1.5pt\hbox{$_{1,-1}$}}$, ${\mathfrak{u}\lower1.5pt\hbox{$_{1,0}$}}$, ${\mathfrak{u}\lower1.5pt\hbox{$_{0,1}$}}$ and their translates. However, when we exponentiate, all the solutions are sent to the triplet $$\label{toruscaph}
U_{1,-1}\cup U_{1,0}\cup U_{0,1}={\mathrm{T}^{2}}\cap {\mathcal{H}}$$ which are the intersections of ${\mathrm{T}^{2}}$ with three conjugate copies of $SU(2)$; the equality in (\[eqimtr\]) follows by noting that both sides are $Ad$-invariant. Consider again any subgroup $SU(2)\subset SU(3)$ and let $g\in
SU(3)$: then $$\label{sudint}
SU(2) \cap Ad_{g}SU(2)= SU(2)\hbox{ or }\{e\}.$$ In fact let us consider a point $x\in SU(2)\cap Ad_{g}SU(2)$; if $x$ is regular then it is contained in a unique maximal torus ${\mathrm{T}^{2}}$; on the other hand $x$ belongs to one of the connected components of (\[toruscaph\]), say $U_{1,-1}$, which therefore belongs entirely to $SU(2)\cap Ad_{g}SU(2)$. This implies that $g\in N(U_{1,-1})$, which is contained in $N(SU(2))$, hence we fall in the first case of (\[sudint\]). Suppose now that $x$ is singular: there exists only one singular point for each copy of $SU(2)$, namely $$\label{antipsud}
x=
\begin{pmatrix}
-1&0&0\\
0&-1&0\\
0&0&1
\end{pmatrix}\,$$ for the standard embedding of $SU(2)$; singular elements are preserved by the adjoint action, therefore $g$ is in the stabilizer of $x$, which is $U(2)=N(SU(2))$, and we are again in in the first case of (\[sudint\]). If $g\not\in N(SU(2))$ then the intersection consists of just $e$.
This discussion proves that we can realize ${\mathcal{H}}$ as the union of copies of $SU(2)$ which share only the identity $e$ inside $SU(3)$. On the other hand, the singular orbit ${\mathbb{C}}{\mathbb{P}}^2$ parametrizes this union; our conclusion is that ${\mathcal{H}}$ is therefore isomorphic to the total space of a fibre bundle ${\mathcal{P}}$ over ${\mathbb{C}}{\mathbb{P}}^2$ with fibre $SU(2)$ and with one point for each fibre identified: $$\xymatrix{ {\mathcal{P}}\ar[d]& \> SU(2)\ar@{_{(}->}[l] \\
{\mathbb{C}}{\mathbb{P}}^{2}&}$$ and ${\mathcal{H}}={\mathcal{P}}/\sim$, with $e\sim e'$ if and only if $e$ and $e'$ are the identity of two fibres $SU(2)$ and $SU(2)'$ (the identity is well defined as it is fixed by the isotropy subgroup of ${\mathbb{C}}{\mathbb{P}}^2$ acting on the fibres).
The Thom space of a vector bundle $E\to M$ is obtained by a $1$-point compactification of the total space $E$. Our construction shows that ${\mathcal{H}}$ is indeed the Thom space of the bundle $\Lambda^{2}_{-}{\mathbb{C}}{\mathbb{P}}^2$: in fact the fibre of this vector bundle is isomorphic to ${\mathfrak{su}(2)}$ as a representation of the stabilizer $U(2)$; then, consider the closed disk $D_{\!\sqrt{2}\pi}\subset {\mathfrak{su}(2)}$: we can identify $$SU(3)=\exp D_{\!\sqrt{2}\pi}$$ where the spheres $S^{2}_{r}$ of radius $r<\sqrt{2}\pi$ are sent $Ad_{SU(2)}$-equivariantly to $2$-spheres, whilst the boundary $S^{2}_{\sqrt{2}\pi}$ is collapsed to a point $x$ antipodal to $e$ (see (\[antipsud\])). We have therefore a corresponding disk subbndle $D$, and a bundle with fibre $SU(2)\cong S^{3}$ obtained from the former by collapsing the boundary of each fibre to a point. The Thom space can be therefore obtained by additionally identifying all the antipodal points of the various fibres. This is precisely what happens for the hypersurface ${\mathcal{H}}$, but this time identifying the identities $e$ instead of the antipodal points. This is not a real difference: in fact the antipodal element $x$ belongs to the center $C(SU(2))={\mathbb{Z}}_{2}$, and the automorphism $SU(2)\to xSU(2)$ is $Ad_{SU(2)}$ equivariant and swaps $e$ and $x$, giving rise to isomorphic bundles with fibre $SU(2)$. The hypersurface ${\mathcal{H}}$ can be shown to be smooth everywhere excepted at $e$.
The image ${\Gamma}(SU(3))$ is contained in ${\mathcal{H}}$, as seen in (\[imtrz\]); the surjectivity of ${\Gamma}$ can be established in the following way by equivariance: the normal geodesic $B(t)$ used in Lemma \[consimlemma\] intersects all the ${\mathbf{c}}$ orbits; its image is given by $${\Gamma}(B(t))=B(2t)$$ which intersects all the $Ad_{SU(3)}$ orbits orthogonally, joining the two singular orbits $e$ and ${\mathbb{C}}{\mathbb{P}}^2$. We observe that the singular orbit ${\mathbb{L}}\subset SU(3)$ is collapsed to $e$.
We pass now to the last statement of the theorem: we will use an argument which is a bundle version of that discussed in Section \[preliminaryresults\] (see (\[equivhom\])). We can describe the tubular neighborhood $D_{{\mathbb{S}}}\cong SU(3) \setminus {\mathbb{L}}$ around ${\mathbb{S}}$ as the $[{\Sigma}^{2}]={\mathbb{R}}^{3}$ bundle obtained by the twisted product $$SU(3)\times _{\scriptscriptstyle SU(2)} [{\Sigma}^{2}]\,;$$ in other words the couples $(g,v)\in SU(3)\times {\mathbb{R}}^{3}$ are identified by the relation $(g,v)\sim (g',v')$ if and only if $ g'=g h\;,\,v= h^{-1}v'$ for some $h\in SU(2)$. The space of classes $[g,v]$ is naturally a left $SU(3)$-space under the action $g'[g,v]=[g'g,v]$. Observe now that the $SU(2)$ representation ${\mathbb{R}}^{3}$ can be extended to a $U(2)$ representation of the form $[{\Sigma}^{2}]\otimes A^{0}$, so that the $U(1)$ centralizer of $SU(2)$ acts trivially. This implies that $D_{{\mathbb{S}}}$ becomes also a *right* $U(2)$-space in the following way: an element $k\in U(2)$ acts by $k[g,v]=[gk,k^{-1}v]$. This action is well defined because $U(2)=N (SU(2))$, and it is equivariant with respect to the left $SU(3)$ action. Clearly $SU(2)\subset U(2)$ is precisely the non-effectivity kernel, so we can just consider this action a $U(2)/SU(2)=U(1)$ effective action. The quotient space $D_{{\mathbb{S}}}/U(1)$ turns out to be a twisted product of the form $SU(3)\times _{\scriptscriptstyle U(2)} {V}$, with ${V}=[{\Sigma}^{2}]\otimes A^{0}$, which is nothing other than $\Lambda^{2}_{-}{\mathbb{C}}{\mathbb{P}}^2$. The projection $\pi_{U(1)}$ is therefore an equivariant map $$\xymatrix{SU(3)\setminus {\mathbb{L}}\ar[r]& {\mathcal{H}}\setminus\{e\}}$$ as is the map ${\Gamma}$; the restriction to each orbit is an equivariant projection of homogeneous spaces in both cases, and an inspection of the normalizers of $SU(2)$ and $U_{1,-1}$ shows that the choice is unique, hence ${\Gamma}=\pi_{ U(1)}$. $\blacksquare$
*Observation.* The proof above has identified ${\Gamma}$ with the quotient $$SU(3)\setminus{\mathbb{L}}\>\cong\>
\xymatrix{{\mathbb{H}}{\mathbb{P}}^{2}\setminus {\mathbb{C}}{\mathbb{P}}^2\ar[r]&
S^{7}\setminus {\mathbb{C}}{\mathbb{P}}^{2}\cong \Lambda ^{2}_{-}{\mathbb{C}}{\mathbb{P}}^{2}}\,.$$ induced by the $U(1)$ action described in [@atiyahwitten], [@miyaoka].
Complete metrics of holonomy $Spin(7)$, invariant under a $Spin(5)$ action, have been discovered on the positive spin bundle over $S^{4}$ [@brysal]; more recently other metrics of this type have been constructed on $4$-dimensional vector bundles over ${\mathbb{C}}{\mathbb{P}}^{2}$ (see [@gukovsparks]). These bundles belong to the family we have denoted by ${\mathbb{P}}(l)$ (see Proposition \[su2u1\]). In a future article, we hope to use the examples of this paper to construct new special geometries in dimensions 7 and 8, by gluing together tubular neighborhoods that arise in our classification, adapting invariant structures to appropriate conditions at the boundaries.
[ The author wishes to thank S. Salamon for his constant support and encouragment. He is also grateful to Y. Nagatomo and F. Podestà for essential help in getting this paper underway, and to F. Lonegro, M. Pontecorvo and A. Di Scala for additional input. The paper was written whilst the author was a recipient of a grant within the research projects “Geometria Riemanniana e strutture differenziabili" (University of Rome *La Sapienza*) and “Geometria delle variet[à ]{}differenziabili" (University of Florence).]{}
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<span style="font-variant:small-caps;">Dipartimento di Matematica “G. Castelnuovo", Universit[à ]{}“La Sapienza", Piazzale A. Moro, 2 - 00185 Roma - Italy</span>\
*E-mail address:* `gambioli@mat.uniroma1.it`
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abstract: |
In this paper, we study the Khovanov homology of an alternating virtual link $L$ and show that it is supported on $g+2$ diagonal lines, where $g$ equals the virtual genus of $L$. Specifically, we show that $Kh^{i,j}(L)$ is supported on the lines $j=2i-\sigma_{\xi}+2k-1$ for $0\leq k\leq g+1$ where ${\sigma}_{\xi^*}(L)+2g= {\sigma}_{\xi}(L)$ are the signatures of $L$ for a checkerboard coloring $\xi$ and its dual $\xi^*$. Of course, for classical links, the two signatures are equal and this recovers Lee’s $H$-thinness result for $Kh^{*,*}(L)$. Our result applies more generally to give an upper bound for the homological width of the Khovanov homology of any checkerboard virtual link $L$. The bound is given in terms of the *alternating genus* of $L$, which can be viewed as the virtual analogue of the Turaev genus. The proof rests on associating, to any checkerboard colorable link $L$, an alternating virtual link diagram with the same Khovanov homology as $L$.
In the process, we study the behavior of the signature invariants under vertical and horizontal mirror symmetry. We also compute the Khovanov homology and Rasmussen invariants in numerous cases and apply them to show non-sliceness and determine the slice genus for several virtual knots. Table \[ras-table\] at the end of the paper lists the signatures, Khovanov polynomial, and Rasmussen invariant for alternating virtual knots up to six crossings.
address: 'Mathematics & Statistics, McMaster University, Hamilton, Ontario'
author:
- Homayun Karimi
title: The Khovanov homology of alternating virtual links
---
Introduction {#introduction .unnumbered}
============
Khovanov homology was introduced in [@Khovanov-00]. It is a powerful invariant that is known to detect the unknot by deep results of Kronheimer and Mrowka ([@KM-2011]). In [@Lee], Lee modified the differential to define a new homology. The Lee homology of a knot is equivalent to an even integer, which surprisingly yields a powerful concordance invariant, as shown by Rasmussen in [@Rasmussen]. Lee proved that the Khovanov homology of alternating links is supported in two lines. Her result implies that for an alternating knot, Rasmussen’s invariant is equal to minus the signature of that knot. She also proved that for alternating knots, Khovanov homology is determined by the signature and the Jones polynomial. To see a generalization of Lee’s theorem to tangles see [@Bar14].
Manturov extended Khovanov homology to virtual knots and links, first with ${{\mathbb Z}}/{2}$ coefficients [@Manturov-kh] and later for arbitrary coefficients [@Manturov-2007]. In [@DKK], Dye, Kaestner and Kauffman reformulated Manturov’s approach and extended Lee homology theory to the virtual setting. They also used it to define a Rasmussen invariant for virtual knots. As we shall see, for virtual knots and links, Khovanov homology is not as powerful an invariant as it is for classical knots. For instance, there exist nontrivial virtual knots with trivial Khovanov homology (see Example \[ex37\]).
Signatures were extended to checkerboard colorable virtual links in [@Im-Lee-Lee-2010], and they depend not only on the link $L$ but also on the choice of checkerboard coloring $\xi$. In particular, instead of a single signature, we have a pair of signatures $(\sigma_\xi,\sigma_{\xi^*})$, where $\xi^*$ is the dual coloring. In Theorem \[mirror images\], we examine how the signatures $(\sigma_\xi,\sigma_{\xi^*})$ change under taking the vertical and horizontal mirror images. If $D$ is an alternating link diagram for $L$ with supporting genus equal to $g$, then by Theorem 5.19 in [@Karimi], we have $\sigma_\xi-\sigma_{\xi^*}=2g$. In this paper, we prove that the Khovanov homology of $L$ is supported in $g + 2$ lines, where $g$ is the virtual genus of $L$.
\[theorem1\] If $D$ is a connected alternating virtual link diagram with genus $g$, and signatures $\sigma_{\xi},\sigma_{\xi^*}$, then its Khovanov homology is supported in $g+2$ lines: $$j=2i-\sigma_{\xi^*}+1,j=2i-\sigma_{\xi^*}-1,\ldots,j=2i-\sigma_{\xi}-1.$$
This theorem is the analogue for virtual links of Lee’s result on $H$-thinness of Khovanov homology for an alternating classical links [@Lee]. It is proved in Section \[four\] (cf. Theorem \[prop\]) by induction on the number of crossings of $D$.
For each non-split link diagram, one can construct a surface called the Turaev surface, such that the diagram is alternating on that surface. Given a non-split link $L$, the minimum genus over all diagrams and all surfaces is denoted $g_T(L)$ and called the *Turaev genus*. The Turaev genus equals zero if and only if the link is alternating. In other words, the Turaev genus measures how far a given link is from being alternating.
For classical links, the Turaev genus provides an upper bound for the homological width of the Khovanov homology [@Kofman07]. For a given checkerboard diagram $D$ of a virtual link $K$, we can associate an alternating diagram, $D_{alt}$, to $D$, without changing the Khovanov homology. Suppose the new diagram $D_{alt}$ has supporting genus $g$, then by the previous theorem, its Khovanov homology, which is the same as the Khovanov homology of $D$, is supported in $g+2$ lines.
For a checkerboard colorable virtual link $L$, we define the alternating genus $g_{alt}(L)$ to be the minimum, over all checkerboard diagrams $D$ for $L$, of the supporting genus of $D_{alt}$. Corollary \[cor3\] shows that the alternating genus provides an upper bound for the homological width of the Khovanov homology. When $L$ is classical and non-split, Lemma \[lem-alt-Turaev\] implies that $g_{alt}(L) \leq g_T(L)$; thus our result recovers and generalizes the Turaev genus bound for the homological width of Khovanov homology of classical links (cf. Corollary 3.1 of [@Kofman07]).
We also give new computations of the Rasmussen invariant, and we apply it to show non-sliceness of several virtual knots.
In Section \[sec-1\], we introduce the basic notions of virtual knot theory. In Section \[sec-sign\], we define checkerboard colorability and the signatures for virtual links. In Section \[two\], we recall the definition of the Khovanov homology for classical knots and links. In Section \[three\], we review the extension of Khovanov homology to virtual knots and links. The main result is proved in Section \[four\], and in Section \[five\], we present computations of signatures, Khovanov polynomials, and Rasmussen invariants of alternating virtual knots up to six crossings.
Basic Notions {#sec-1}
=============
In this section we recall some basic definitions from virtual knot theory, including virtual link diagrams, abstract link diagrams, supporting genus, virtual genus, virtual knot concordance, and the slice genus. We then introduce the notion of checkerboard coloring for virtual knots and links. We also recall the notion of the boundary property for virtual link diagrams (see [@Boundary]), and relate it to checkerboard colorability.
Virtual link diagrams {#virtual-link-diagrams .unnumbered}
---------------------
A *virtual link diagram* is a collection of immersed closed curves in the plane, with a finite number of intersection points all of which are double points. Each double point is either classical or virtual. At classical crossings, we record extra information by specifying which of the two strands goes over the other, and at virtual crossings we place a small circle around the double point. A virtual link is an equivalence class of virtual link diagrams modulo the *generalized Reidemeister moves* and planar isotopy. The combination of classical and virtual Reidemeister moves is called the generalized Reidemeister moves. See Figure \[grms\].
![The virtual Reidemeister moves.[]{data-label="grms"}](grm.pdf){height="20mm"}
Abstract link diagrams {#abstract-link-diagrams .unnumbered}
----------------------
Suppose $S$ is a compact oriented surface with boundary. Let $D$ be a link diagram on $S$ with no virtual crossings. We denote by $|D|$, the graph obtained by replacing each classical crossing in $D$ by a tetravalent vertex. We say $P=(S,D)$ is an *abstract link diagram* (ALD) if $|D|$ is a deformation retract of $S$.
Let $\Sigma$ be a closed, connected and oriented surface and $f:S\to \Sigma$ be an orientation preserving embedding. We call $(\Sigma,f(D))$ a *realization* of $P$.
Given a virtual link diagram, we can construct an abstract link diagram (see [@KK00]). We review that construction here.
Let $D$ be a virtual link diagram with $n$ classical crossings and $U_{1},U_{2},\ldots,U_{n}$ regular neighborhoods of the crossings of $D$. Put $W=\text{cl}({{\mathbb R}}^2-\cup_{i=1}^{n}U_{i})$. Thickening the arcs and loops of $D\cap W$, we obtain immersed bands and annuli in $W$ whose cores are $D\cap W$. Their union together with $U_{1},U_{2},\ldots,U_{n}$ forms an immersed disk-band surface $N(D)$ in the plane. Modifying $N(D)$ as shown below at each virtual crossing of $D$, we obtain a compact oriented surface $S_{D}$ embedded in ${{\mathbb R}}^3$, and a diagram $\widetilde{D}$ on $S_{D}$ corresponding to $D$. We call the pair $P=(S_{D},\widetilde{D})$ the *abstract link diagram associated to $D$*.
{height="20mm"}
The *supporting genus* of an ALD $P=(\Sigma,\widetilde{D})$ is denoted by $sg(P)$, and is defined to be the minimal genus among the realization surfaces $F$ of $P$. The supporting genus of a virtual link diagram $D$ is defined to be the supporting genus of the ALD $P=(S_{D},\widetilde{D})$ associated with $D$ and denoted by $sg(D)$. The *virtual genus* of a virtual link $L$ is denoted by $g_v(L)$ and defined to be the minimum number among the supporting genus $sg(D)$, where $D$ runs over all virtual link diagrams representing $L$.
Let $L$ be a virtual link. A virtual link diagram $D$ representing $L$ such that $sg(D)=g_v(L)$ is called a *minimal diagram* of $L$.
Virtual knot concordance {#virtual-knot-concordance .unnumbered}
------------------------
We recall the notions of cobordism and concordance of virtual knots.
[**(i)**]{} Two knots $K_0\subset \Sigma_0\times I$ and $K_1\subset \Sigma_1\times I$ in thickened surfaces are called *virtually cobordant* if there exists a compact connected oriented $3$-manifold $W$ with $\partial W\cong-\Sigma_0\sqcup\Sigma_1$ and an oriented surface $S \subset W\times I$ with $\partial S = -K_0\sqcup K_1$.\
[**(ii)**]{} The knots $K_0, K_1$ in part (i) are called *virtually concordant* if the surface $S$ can be chosen to be an annulus.
Given a knot $K$ in the thickened surface $\Sigma\times I$, an elementary argument shows that there exists a compact oriented $3$-manifold $W$ and compact oriented surface $S \subset W\times I$ with $\partial S =K.$ Consequently, every virtual knot is cobordant to the unknot. This motivates the following definition.
Suppose $K$ is a knot in a thickened surface $\Sigma \times I.$\
[**(i)**]{} The *slice genus* of $K$, denoted $g_{s}(K)$, is the minimum genus of $S$, over all 3-manifolds $W$ with $\partial W =\Sigma$ and over all surfaces $S \subset W \times I$ with $\partial S =K.$\
[**(ii)**]{} The knot $K$ is called *virtually slice* if $g_s(K)=0.$ Equivalently, $K$ is virtually slice if it bounds a disk $\Delta\subset W\times I$.
Signatures of checkerboard colorable virtual links {#sec-sign}
==================================================
Checkerboard colorings {#checkerboard-colorings .unnumbered}
----------------------
We review checkerboard colorings for virtual knots and links and recall the construction of the Goeritz matrices, which are used to define signatures for checkerboard colorable virtual links.
Given $P=(F,D)$, where $F$ is a compact, connected, oriented surface and $D$ is a link diagram on $F,$ a *checkerboard coloring* $\xi$ is an assignment to each region of $F \smallsetminus |D|$ one of two colors, say black and white, such that any two adjacent regions sharing an edge of $|D|$ have different colors. Define the dual checkerboard coloring $\xi^*$ to be the one obtained from $\xi$ by interchanging black and white regions.
For a crossing $c$, we have the following pictures:
{height="24mm"} {height="24mm"} {height="24mm"}
They are a positive crossing and a negative crossing, a type $A$ crossing with $\eta(c)=+1$ and a type $B$ crossing with $\eta(c)=-1$ in a checkerboard colored link diagram, and a type I and a type II crossing in an oriented, colored link diagram, respectively. We call ${\varepsilon}(c)$ the *writhe* of the crossing and $\eta(c)$ the *incidence* of the crossing $c$. An elementary calculation shows that a crossing $c$ has type II if ${\varepsilon}(c) \eta(c) = 1,$ otherwise it has type I.
Let $\xi$ be a checkerboard coloring for a pair $P=(F,D)$, where $F$ is a closed, oriented and connected surface and $D$ is a link diagram on $F$. We enumerate the white regions of $F \smallsetminus |D|$ by $X_{0},X_{1},\ldots,X_{m}$. Let $C(D)$ denote the set of all classical crossings of $D$ on $F$. For each pair $i,j\in\{1,2,\ldots,m\}$, let $$C_{ij}(D)=\{c\in C(D)\ \mid\ c\ \text{is adjacent to both}\ X_{i}\ \text{and}\ X_{j}\},$$ and define $$g_{ij}=\begin{cases}
-\sum\limits_{c\in C_{ij}(D)}\eta(c),\ \ \text{for}\ i\neq j,\\
-\sum\limits_{k=0; k\neq i}^m g_{ik},\ \ \text{for}\ i=j.
\end{cases}$$
The pre-Goeritz matrix of $D$ is defined to be the symmetric integral matrix $G_\xi'(D)=(g_{ij})_{0\leq i,j\leq m}$, and the Goeritz matrix of $D$ is the principal minor $G_\xi(D)=(g_{ij})_{1\leq i,j\leq m}$ obtained by removing the first row and column of $G_\xi'(D).$
The correction term is defined by setting $\mu_{\xi}(D)=\sum\limits_{c\ \text{is type II}}\eta(c)$.
Suppose $D$ is a checkerboard colorable link diagram on a surface $F$ with a checkerboard coloring $\xi$, we define the signature as follows: $$\sigma_{\xi}(D)={\operatorname{sig}}(G_\xi(D))-\mu_{\xi}(D).$$
The following result is proved in [@Im-Lee-Lee-2010].
If $L$ is a non-split checkerboard link represented by a diagram $D$ of minimal genus and with coloring $\xi$, then the pair $\{\sigma_{\xi}(D), \sigma_{\xi^*}(D)\}$ of signatures is independent of the choice of virtual link diagram and gives a well-defined invariant of the virtual link $L$.
Given a virtual link diagram $D$, for each classical crossing, we can resolve the crossing into a $0$-smoothing or a $1$-smoothing (see Figure \[resolving\]).
![The $0$- and $1$-smoothing of a crossing.[]{data-label="resolving"}](crossing-smoothing.pdf){height="25mm"}
If we resolve all the classical crossings, the resulting diagram is called a *state*. A state is a virtual link diagram with only virtual crossings, i.e. it is a diagram of the unlink. For a link diagram with $n$ classical crossings, we have $2^{n}$ states. In fact, once an ordering of the crossings $\{c_1, \ldots,c_n\}$ has been fixed, the states are in one-to-one correspondence with binary strings of length $n$. For a given state $s$, the dual state is denoted ${ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}$ and it is obtained from $s$ by changing all 0-smoothings to 1-smoothings, and vice versa. In other words, if $s$ corresponds to the binary word with $i$-th entry $s_i \in \{0,1\}$, then ${ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}$ corresponds to the binary word with $i$-th entry ${ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_i = 1-s_i.$
\[boundary\] Let $D$ be a virtual link diagram, and $(S_{D},\widetilde{D})$ be the abstract link diagram associated with $D$. Then $D$ has the *boundary property* if there exists a state $s_{\partial}$ such that $\partial S_{D}=s_{\partial}\cup { \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}$, where ${ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}$ is the dual state of $s_{\partial}$.
The following lemma relates the boundary property to checkerboard colorability.
\[lem-bound-prop\] A virtual link diagram $D$ has the boundary property if and only if it is checkerboard colorable.
Suppose $D$ has the boundary property and define a checkerboard coloring $\xi$ as follows. Let the white regions be those with boundary a component of $s_{\partial}$, and let the black regions be those with boundary a components of ${ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}$. This gives a checkerboard coloring $\xi$ for $D$.
Conversely, suppose $\xi$ is a checkerboard coloring of the abstract link diagram $(S_D, \widetilde{D}).$ Let $s_\partial$ be the state obtained by performing 0-smoothing to all crossings $c$ with $\eta(c) = +1$ and 1-smoothing to all crossings $c$ with $\eta(c) = -1$, and let ${ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}$ be the dual state. Then it can be easily checked that $\partial S_D = s_\partial \cup { \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_\partial,$ therefore $D$ has the boundary property.
Alternating virtual links {#alternating-virtual-links .unnumbered}
-------------------------
A virtual link is called *alternating* if it admits a virtual link diagram whose crossings alternate between over and under crossing as one travels around each component of the link. Since every alternating virtual link diagram is checkerboard colorable (see [@Kamada-2002]), it follows that every alternating virtual link diagram has the boundary property.
Let $|s_{\partial}|$ and $|{ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}|$ be the number of components of $s_{\partial}$ and ${ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}$, respectively.
\[boundary lemma\] Suppose $D$ is a virtual link diagram with $n$ classical crossings. If $S_{D}$ has genus $g$ and $D$ has the boundary property, then $$|s_{\partial}|+|{ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}|=n+2-2g.$$
Attach disks to the boundary components of $S_{D}$ to get a closed surface $\Sigma$. There is a cell decomposition on $\Sigma$, defined as follows: There is a one-to-one correspondence between the classical crossings of $D$ and $0$-cells, bands of $S_{D}$ and $1$-cells, and $2$-disks that we attached to $S_{D}$ and $2$-cells. The Euler characteristic of $\Sigma$ is $2-2g$. And the number of $0$, $1$ and $2$-cells are $n$, $2n$ and $|s_{\partial}|+|{ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}|$, respectively. The lemma now follows.
The proof of the next result is elementary and is left to the reader.
\[eta\] If $D$ is a connected checkerboard colorable link diagram, then $D$ is alternating if and only if all its crossing have the same incidence number.
Unknotting operations {#unknotting-operations .unnumbered}
---------------------
For a virtual link, we introduce the operations of *orientation reversal* ([**or**]{}), *sign change* ([**sc**]{}), and *crossing change* ([**cc**]{}) at a given crossing. The operations [**or**]{} and [**sc**]{} are shown in Figure \[fig3\], and [**cc**]{} is the result of applying [**or**]{} and [**sc**]{}. As is well-known, crossing change [**cc**]{} is an unknotting operation for classical knots and links, but this is no longer true for virtual links. Together with the operation of *chord deletion* ([**cd**]{}), which replaces a classical crossing with a virtual one, these moves form a complete set of unknotting operations for virtual knots (see [@Boden2]). Note that each of [**cc**]{}, [**or**]{} and [**sc**]{} can be achieved in a genus one cobordism.
![The operations [**or**]{} and [**sc**]{}.[]{data-label="fig3"}](or.pdf "fig:"){height="15mm"} ![The operations [**or**]{} and [**sc**]{}.[]{data-label="fig3"}](sc.pdf "fig:"){height="15mm"}
Given a checkerboard colored diagram, if we apply any one of $\{{\bf cc}, {\bf sc}, {\bf or}\}$ to a crossing, then the new diagram is again checkerboard colored. We will examine the effect of these operations on the writhe, incidence, and type of the crossing.
If we apply [**or**]{} to a crossing $c$, then it is elementary to see that $\eta(c)$ changes sign and the writhe ${\varepsilon}(c)$ remains the same. If we instead apply [**sc**]{}, then the writhe ${\varepsilon}(c)$ changes sign and $\eta(c)$ remains the same. In particular, under either operation, the type of the crossing changes. On the other hand, if we apply [**cc**]{} to a crossing $c$, then both the writhe ${\varepsilon}(c)$ and $\eta(c)$ change signs, but the type remains the same. These facts are summarized in Table \[effect\].
writhe incidence type
------------ -------------------- ------------ -------------------
[**sc**]{} $ -\varepsilon(c)$ $\eta(c)$ $-\text{type}(c)$
[**or**]{} $\varepsilon(c)$ $-\eta(c)$ $-\text{type}(c)$
[**cc**]{} $-\varepsilon(c)$ $-\eta(c)$ $\text{type}(c)$
: The effect of applying [**sc**]{}, [**or**]{}, [**cc**]{} to the crossing $c$.[]{data-label="effect"}
Mirror images of virtual knots {#mirror-images-of-virtual-knots .unnumbered}
------------------------------
We define the vertical and horizontal mirror image of a virtual knot and relate their signatures.
Given an oriented virtual knot diagram $D$, let $-D$ be the diagram obtained by reversing the orientation of $D$. The *vertical mirror image* of $D$ is denoted $D^{*}$ and is the diagram obtained by applying [**cc**]{} to all crossings of $D$. The *horizontal mirror image* of $D$ is denoted $D^{\dag}$ and is the diagram obtained by applying [**sc**]{} to all crossings of $D$.
If $D$ is a minimal genus diagram for a virtual knot $K$, then so are $-D, D^*$ and $D^\dag$.
It is obvious that if $D$ is a minimal genus diagram for $K$, then $-D$ is a minimal genus diagram for $-K$.
Suppose that $P=(S_D,\widetilde{D})$ is the abstract link diagram associated with $D$. Place it inside $\{(x,y,z)\in{{\mathbb R}}^3\mid y<0\}$ in such a way that the projection of $\widetilde{D}$ on the $xy$-plane is $D$. Now reflect $P$ with respect to the plane $y=0$. The result is an abstract link diagram associated with $D^{\dag}$. This shows that if $D$ is a minimal genus diagram, then $D^{\dag}$ is also minimal genus.
On the other hand, switching all the over-crossings and under-crossings in $\widetilde{D}$, we obtain an abstract link diagram for $D^{*}$. It follows that if $D$ is a minimal genus diagram, then $D^{*}$ is also minimal genus.
Suppose $\xi$ is a checkerboard coloring of $D$ and $\xi^*$ is its dual coloring. Notice that a coloring is determined by the underlying flat knot. Therefore we can use the same notation for the colorings of the diagrams of the mirror images and the inverse knot.
The following picture shows a colored crossing in $D$ (left) and $-D$ (right).
{height="25mm"}
Therefore, at each crossing, the incidence number and type of that crossing are unchanged. Notice that black and white regions are also unchanged. Thus the two signatures for $D$ and $-D$ are the same.
For $D^{*}$, at each crossing, the type is unchanged but the incidence number changes sign. As a result, both the Goeritz matrix and the correction term are multiplied by $-1$. Thus $\sigma_{\xi}(D^{*})=-\sigma_{\xi}(D)$ and $\sigma_{\xi^*}(D^{*})=-\sigma_{\xi^*}(D)$.
For $D^{\dag}$, we use the dual coloring $\xi^*$. Thus at each crossing, the type is unchanged but the incidence number changes by a negative sign. Therefore $\sigma_{\xi^*}(D^{\dag})=-\sigma_{\xi}(D)$, and $\sigma_{\xi}(D^{\dag})=-\sigma_{\xi^*}(D)$.
Since ${\bf or} = {\bf sc} \circ {\bf cc},$ it follows that $D^{*\dag}$ is obtained by applying [**or**]{} to all crossings of $D$. For $D^{*\dag}$, we find that $\sigma_{\xi^*}(D^{*\dag})=\sigma_{\xi}(D)$ and $\sigma_{\xi}(D^{*\dag})=\sigma_{\xi^*}(D)$.
The next theorem summarizes these observations.
\[mirror images\] If $K$ is a virtual knot with checkerboard coloring $\xi$, then the signatures of the inverse and mirror images of $K$ satisfy: $$\begin{aligned}
(\sigma_{\xi}(-K),\sigma_{\xi^*}(-K))&=&(\sigma_{\xi}(K),\sigma_{\xi^*}(K)),\\
(\sigma_{\xi}(K^{*}),\sigma_{\xi^*}(K^{*}))&=&(-\sigma_{\xi}(K),-\sigma_{\xi^*}(K)),\\
(\sigma_{\xi}(K^{\dag}),\sigma_{\xi^*}(K^{\dag}))&=&(-\sigma_{\xi^*}(K),-\sigma_{\xi}(K)),\\
(\sigma_{\xi}(K^{*\dag}),\sigma_{\xi^*}(K^{*\dag}))&=&(\sigma_{\xi^*}(K),\sigma_{\xi}(K)).\end{aligned}$$
Khovanov Homology for Classical Knots {#two}
=====================================
In this section, we briefly introduce the Khovanov homology for classical knots and links. For more details see [@Turner] and [@Bar-natan-02].
Khovanov homology is a $(1+1)$-TQFT (*topological quantum field theory*), i.e. it is a functor from the category of compact $1$-dimensional manifolds (a collection of circles) with morphisms, compact and orientable $2$-dimensional cobordisms (surfaces) between them, into the category of graded vector spaces and graded linear maps.
Khovanov introduced the invariant for classical links in [@Khovanov-00]. It is a bigraded homology theory which is defined and computed in a purely combinatorial way. Khovanov homology is a categorification of the Jones polynomial, in that its graded Euler characteristic is equal to the unnormalized Jones polynomial. For a link $L$, we denote its Khovanov homology by $Kh^{*,*}(L)$, and we have $$\widehat{\chi}(Kh^{*,*}(D))=\sum_{i,j\in {{\mathbb Z}}}(-1)^iq^j\text{dim}Kh^{i,j}(D)=\widehat{V}_L(q).$$
Suppose $D$ is a link diagram with $n_{+}$ positive crossings and $n_{-}$ negative crossings. Let $n=n_{+}+n_{-}$ and enumerate the crossings by $c_1,\ldots,c_n$. With ${{\mathbb Q}}$ as the coefficient ring, we set $V={{\mathbb Q}}{{\bf 1}}\oplus{{\mathbb Q}}X$ to be the $2$ dimensional vector space with basis $\{{{\bf 1}},X\}$. Setting the degree of ${{\bf 1}}$ to be $+1$ and the degree of $X$ to be $-1$ gives $V^{\otimes n}$ the structure of a graded vector space. This grading will be denoted $j$ and called *vertical* or *quantum grading*.
If $W=\bigoplus_{m\in {{\mathbb Z}}}W_{m}$ is a graded vector space, then a *vertical grading shift* of $W$ by $\ell$ is defined as $W\{\ell\}=\bigoplus_{m\in{{\mathbb Z}}}W_{m}'$, where $W_{m}'=W_{m-\ell}$.
We consider the cube of resolutions of $D$, which is an $n$-dimensional cube with $2^n$ vertices, one for each state. Here we denote states by $\alpha\in\{0,1\}^n$, which is a binary sequence of length $n$ that indicates how each crossing has been resolved. Let $r_{\alpha}$ and $k_{\alpha}$ be the number of ${{\bf 1}}$’s and cycles in $\alpha$, respectively. Let $\mathcal{C} ^{i,*}(D)$ be $\bigoplus V^{\otimes k_{\alpha}}\{r_{\alpha}+n_{+}-2n_{-}\}$, where we take the direct sum over all the states $\alpha$ with $r_{\alpha}=i+n_{-}$. Here $i$ is called *horizontal* or *homological grading*.
If $\mathcal{C}(D)=\bigoplus_{i}\mathcal{C}^i(D)$, then a *horizontal grading shift* of $\mathcal{C}(D)$ by $l$ is defined as $\mathcal{C}(D)[l]=\bigoplus_{i}\mathcal{C}'^i(D)$, where $\mathcal{C}'^i(D)=\mathcal{C}^{i-l}(D)$.
We define the Khovanov complex as $\mathcal{C}Kh(D)=\bigoplus_{i,j}\mathcal{C}^{i,j}(D)$. To define the differential $d$, we introduce the product and coproduct maps. Note that henceforth we will suppress the symbol $\otimes$ in writing elements of $V^{\otimes k}$. $$\begin{aligned}
\Delta:V\to V\otimes V,&\ \ \ &m:V\otimes V\to V.\\
{{\bf 1}}\mapsto {{\bf 1}}X+X{{\bf 1}}&\ \ \ &{{\bf 1}}{{\bf 1}}\mapsto{{\bf 1}}\\
X\mapsto XX&\ \ \ & {{\bf 1}}X\mapsto X \\
&\ \ \ & X{{\bf 1}}\mapsto X\\
&\ \ \ & XX\mapsto 0\end{aligned}$$
We only define a map from a state $\alpha$ to a state $\alpha'$ if $\alpha'$ obtained from $\alpha$ by changing one $0$ to $1$. In that case, either two cycles of $\alpha$ merge into one cycle of $\alpha'$, or one cycle of $\alpha$ splits into two cycles of $\alpha'$. In the first case, we use the product map $m$, and in the second, we use the coproduct map $\Delta$. For all other cycles of $\alpha$, we apply the identity. In order to write down all the maps, we fix once and for all an enumeration of the cycles in each state, and these are not changed throughout the calculations.
The last step is to assign negative signs to some of the maps. There are many ways to do that, but the homology groups for the different choices of signs are all isomorphic. Here we follow the sign convention of [@Bar-natan-02].
Suppose we change $0$ to $1$ in the $m$-th spot to obtain $\alpha'$ from $\alpha$. In $\alpha$, we count how many $1$’s we have before the $m$-th spot. If it is an odd number, we assign a negative sign to the associated map.
For a fixed $j$ the map $d^2:\mathcal{C}^{i,j}\to \mathcal{C}^{i+2,j}$ is zero and we obtain a bigraded homology theory denoted by $Kh^{*,*}(D)$.
In [@Lee], Lee constructs a new complex by modifying the maps $\Delta$ and $m$: $$\begin{aligned}
\Delta':V\to V\otimes V,&\ \ \ &m':V\otimes V\to V.\\
{{\bf 1}}\mapsto {{\bf 1}}X+X{{\bf 1}}&\ \ \ &{{\bf 1}}{{\bf 1}}\mapsto{{\bf 1}}\\
X\mapsto {{\bf 1}}{{\bf 1}}+XX&\ \ \ & {{\bf 1}}X\mapsto X \\
&\ \ \ & X{{\bf 1}}\mapsto X\\
&\ \ \ & XX\mapsto {{\bf 1}}\end{aligned}$$
This results in a new homology theory called Lee homology and denoted $\text{Lee}(D)$. Notice the maps no longer preserve the quantum degree, thus Lee homology is only graded rather being bigraded.
It turns out that $\text{Lee}(K)\cong {{\mathbb Q}}\oplus{{\mathbb Q}}$ for all knots, nevertheless as we will see the Lee homology contains a nontrivial and powerful invariant $s(K)$ called the *Rasmussen invariant*. This invariant was introduced by Rasmussen in [@Rasmussen], and we briefly recall its definition.
The quantum degree defines a decreasing filtration on $\mathcal{C}Kh(K)$. This induces a filtration on $\text{Lee}(K)$, $$H_{*}(\mathcal{C})=\mathcal{F}^nH_{*}(\mathcal{C})\supset \mathcal{F}^{n+1}H_{*}(\mathcal{C})\supset\ldots\supset \mathcal{F}^mH_{*}(\mathcal{C}).$$ For $x\in \text{Lee}(K)$, let $s(x)$ be the filtration degree of $x$, i.e. $s(x)=k$ if $x\in\mathcal{F}^kH_{*}(\mathcal{C})$ but $x$ does not belong to $\mathcal{F}^{k+1}H_{*}(\mathcal{C})$. We define $$\begin{aligned}
s_{min}(K)&=&\min\{s(x)\in\text{Lee}(K)\mid x\neq0 \}, \\
s_{max}(K)&=&\max\{s(x)\in\text{Lee}(K)\mid x\neq0 \}.\end{aligned}$$
Rasmussen proves that $s_{max}(K) = s_{min}(K)+2$ for all knots, and the *Rasmussen invariant* is defined to be $s(K)=s_{min}(K)+1=s_{max}(K)-1$.
For a link $L$, the filtration on $\mathcal{C}Kh(L)$ induces a spectral sequence with $E_{0}$ term the Khovanov complex and $d_0=d_{Kh}$. It follows that the $E_1$ term is $Kh^{*,*}(L)$. For every $m$, $d_m=0$ unless $m$ is a multiple of $4$. As a result, for any $m\geq 0$, $E_{4m+1}\cong E_{4m+2}\cong E_{4m+3}\cong E_{4m+4}$. The $E_{\infty}$ page is isomorphic to the Lee homology. For a knot $K$, it has two copies of ${{\mathbb Q}}$ which are placed on the $y$-axis. Their location indicates the filtration degree of the generators of the Lee homology. In particular the average of their $y$-coordinates is equal to $s(K)$.
In [@Lee], Lee proves that, for any alternating link $L$, its Khovanov homology $Kh^{*,*}(L)$ is supported in the two lines $j=2i-\sigma(L)\pm1$. As a result, in the spectral sequence $d_{m}=0$ for every $m\geq 5$ and $E_{\infty}=E_5$. If $K$ is an alternating knot, then the $y$-coordinates of the two surviving copies of ${{\mathbb Q}}$ are $-\sigma(K)\pm 1$. This implies that $s(K)=-\sigma(K)$.
Lee homology is a functor, and if we have a cobordism $S$ between two links $L_0$ and $L_{1}$, then $S$ induces a map $\varphi_{S}':\text{Lee}(L_0)\to \text{Lee}(L_1)$ with filtration degree equal to $\chi(S)$. We will describe the map $\varphi'_S$ in a moment, but first notice that this implies that if $K$ is a knot, then $|s(K)|\leq 2g_{4}(K)$, where $g_{4}(K)$ denotes the classical 4-ball genus. The same inequality holds for the knot signature $\sigma(K)$, and Lee’s theorem tells us that, for alternating knots, the Rasmussen invariant $s(K)$ gives the same bound on the $4$-ball genus as the knot signature. However, for non-alternating knots, it is no longer true that $s(K) = -\sigma(K)$, and sometimes the Rasmussen invariant provides a better bound. It should further be noted that Rasmussen’s invariant gives a lower bound on the smooth 4-ball genus, whereas the knot signature gives a bound on the topological 4-ball genus.
Let $K$ be the classical knot $9_{42}$. (For classical knots, we adopt the notation of [@Knotinfo].) Then this knot has Rasmussen invariant $s(K)=0$ and signature $\sigma(K)=2$. Thus the signature provides a better bound on the 4-ball genus than the Rasmussen invariant for this knot. On the other hand, for the knot $K=10_{132}$, we find that it has Rasmussen invariant $s(K)=-2$ and signature $\sigma(K)=0$. Thus, we see that the Rasmussen invariant gives a better bound on the $4$-ball genus in this case.
We now describe the map $\varphi_{S}'$. Since any cobordism decomposes into a sequence of elementary cobordisms, it suffices to define $\varphi'_S$ for births, deaths, and saddles. In doing that, we will use the maps $\iota:{{\mathbb Q}}\to V$ ($1\mapsto{{\bf 1}}$) and $\varepsilon:V\to{{\mathbb Q}}$ (${{\bf 1}}\mapsto 0$ and $X\mapsto 1$).
Note that an elementary cobordism is either a birth, a death, or a saddle. For a birth, we set $\varphi_{S}'=\iota$. For a death, we set $\varphi_{S}'=\varepsilon$. For a saddle $S$, $\varphi_{S}'$ is either $m'$ or $\Delta'$.
In general, the Rasmussen invariant is difficult to compute. However, the calculation simplifies for positive (or negative) knots, as we now explain.
A link is called *positive* if it admits a diagram with only positive crossings. Similarly, a link is *negative* if it admits a diagram with only negative crossings.
If $K$ is a positive knot with diagram $D$ with $n$ positive crossings, then the Rasmussen invariant is given by $$s(K)=-k+n+1,$$ where $k$ is the number of cycles in the all $0$-smoothing state of $D$ [@Rasmussen].
If $K$ is a negative knot with diagram $D$ with $n$ negative crossings, then the mirror image $D^{*}$ has $n$ positive crossings, and the all $0$-smoothing state of $D^{*}$ is the all $1$-smoothing state of $D$. Since $s(K^{*})=-k+n+1$, and since the Rasmussen invariant satisfies $s(K^{*})=-s(K)$ under taking mirror images, it follows that $s(K)=k-n-1$.
Next, we recall the definition of the Turaev genus for classical links from [@Tur]. Let $D$ be a connected classical link diagram with $c(D)$ crossings, and suppose $s_0$ and $s_1$ are the all $0$ and all $1$ smoothing states, respectively.
\[def-Turaev\] The *Turaev genus* of the link diagram $D$ is defined by setting $$g_{T}(D)=\frac{1}{2}(c(D)+2-|s_0(D)|-|s_1(D)|).$$ For a non-split classical link $L$, the Turaev genus, denoted $g_T(L)$, is defined to be the minimal of $g_T(D)$ over all connected classical link diagrams $D$ for $L$.
For more on Turaev genus, see [@SurTur].
Khovanov Homology for Virtual Knots {#three}
===================================
In this section, we briefly introduce the Khovanov homology for virtual knots and links.
When one attempts to define a Khovanov theory for virtual knots the major problem is the presence of the *single cycle smoothing* (see Figure \[single-cycle\]). We need to assign a map to a single cycle smoothing, which we can do by assigning the zero map. In classical Khovanov theory, the signs of maps are chosen in a way to make each face of the cube of resolutions to be anti-commutative. Then this fact enables us to define a differential $d$ satisfying $d^2=0$. For virtual knots, the existence of single cycle smoothings makes it more difficult to assign signs.
![A single cycle smoothing.[]{data-label="single-cycle"}](single-cycle.pdf){height="28mm"}
Tubbenhauer in [@Tubbenhauer], used un-oriented TQFT’s to define a Khovanov homology for virtual knots and links. In what follows, we describe Tubbenhauer’s method. We will cover the combinatorial definitions. For a discussion about un-oriented TQFTs, see [@Tubbenhauer].
Let ${{\mathbb Q}}$ be the coefficient ring and $V={{\mathbb Q}}{{\bf 1}}\oplus {{\mathbb Q}}X$. Start with a virtual link diagram $D$ with $n$ classical crossings. Resolve all the crossings in both ways to obtain $2^n$ states, leaving virtual crossings untouched. The Khovanov chain complex $\mathcal{C}(D)$ is defined as before, i.e. we assign $V^{\otimes k}$ to a state with $k$ components. The degree of each element and the grading shifts are defined as before. Whenever two vertices of an edge in the cube of resolutions have the same number of states, then that indicates the presence of a single cycle smoothing. In that case, we assign the zero map to the edge. It remains to define the joining and splitting maps and the signs.
Choose orientations for the cycles of each state. Although we can do this in an arbitrary way, to have less complicated maps at the end, we use a spanning tree argument. Choose a spanning tree for the cube of the resolution and start with the rightmost vertices and choose orientations for the cycles of corresponding states. Now remove those vertices and again choose orientations for the rightmost vertices, in a compatible way. That means we compare the two vertices which are joined by an edge of the spanning tree, and orient the cycles of the left vertex as follows. For cycles which are not involved in the join, split or the single cycle smoothing, orient each cycle of the left vertex exactly like the corresponding cycle in the right vertex. For other cycles try to orient them in a way to have the most compatibility.
Choose an $x$-marker for each crossing and the corresponding $0$- and $1$-smoothings, as in Figure \[x-marker\]. We choose either $x$ or $x'$ and notice that up to rotating the diagram and the corresponding states, there are only these two ways to assign the $x$-markers.
![An $x$-marker for a crossing and the corresponding smoothings.[]{data-label="x-marker"}](crossing-smoothing-1.pdf){height="26mm"}
We define the sign of the non-zero maps as follows. By a spanning tree argument, number the cycles of each state. Suppose we have a joining map from a state $s$ to another state $s'$, and suppose $s$ has $m+1$ cycles. Let the joining map, merges the cycles number $a$ and $b$ in $s$ and the resulting cycle in $s'$ has number $c$, and let the cycle number $a$ has the $x$-marker. In an (m+1)-tuple, put $a$ first, then $b$ and then place the remaining numbers in an ascending order. Let $\tau_{1}$ be the permutation which takes $(1,2,\cdots,m+1)$ to this $(m+1)$-tuple. Now in an $m$-tuple, put $c$ first, and place the remaining numbers in an ascending order, and let $\tau_{2}$ be the permutation which takes $(1,2,\cdots,m)$ to this $m$-tuple. Define the sign of the joining map to be $\text{sign}(\tau_1)\text{sign}(\tau_2)$. The sign of the splitting map is defined similarly.
Next we define the maps. They are defined between the two vertices of an edge of the cube of resolutions. For each edge the smoothing of only one of the crossings is different, and we define a map from the state with $0$-smoothing to the state with $1$-smoothing. If a cycle of the source state is disjoint from the smoothing change, assign the identity map to it, if its orientation agrees with the orientation of the corresponding cycle in the target state, otherwise assign negative of the identity map.
At a small neighborhood of the crossing, there are two parallel strings in each state. Notice that each cycle is oriented. Now if the map looks like $\downarrow\uparrow\ \to\ \rightleftarrows$, we decorate the four strings in the source and target state with a $+$ sign, and we call this decoration *standard*. We always rotate the states so the two strings in the source state are vertical, and the two strings in the target state are horizontal. Then we compare the orientation of each string with the orientation of the corresponding one in the standard decoration, if they agree, decorate that string with a $+$ sign, otherwise decorate it with a $-$ sign. We record the possible cases in Table \[decoration\].
string splitting map string joining map
---------------------------------------------- ------------------- ----------------------------------------------- --------------
$\downarrow\uparrow\ \to\ \rightleftarrows$ $\Delta_{++}^{+}$ $\downarrow\uparrow\ \to\ \rightleftarrows$ $m_{+}^{++}$
$\downarrow\uparrow\ \to\ \rightrightarrows$ $\Delta_{-+}^{+}$ $\uparrow\uparrow\ \to\ \rightleftarrows$ $m_{+}^{-+}$
$\downarrow\uparrow\ \to\ \leftleftarrows$ $\Delta_{+-}^{+}$ $\downarrow\downarrow\ \to\ \rightleftarrows$ $m_{+}^{+-}$
$\downarrow\uparrow\ \to\ \leftrightarrows$ $\Delta_{--}^{+}$ $\uparrow\downarrow\ \to\ \rightleftarrows$ $m_{+}^{--}$
$\uparrow\downarrow\ \to\ \rightleftarrows$ $\Delta_{++}^{-}$ $\downarrow\uparrow\ \to\ \leftrightarrows$ $m_{-}^{++}$
$\uparrow\downarrow\ \to\ \rightrightarrows$ $\Delta_{-+}^{-}$ $\uparrow\uparrow\ \to\ \leftrightarrows$ $m_{-}^{-+}$
$\uparrow\downarrow\ \to\ \leftleftarrows$ $\Delta_{+-}^{-}$ $\downarrow\downarrow\ \to\ \leftrightarrows$ $m_{-}^{+-}$
$\uparrow\downarrow\ \to\ \leftrightarrows$ $\Delta_{--}^{-}$ $\uparrow\downarrow\ \to\ \leftrightarrows$ $m_{-}^{--}$
: String decoration and corresponding maps.[]{data-label="decoration"}
Other cases occur only when we have a single cycle smoothing. We describe the map $\Delta_{bc}^{a}(v)$ as follows. Multiply $v$ by $a$, apply $\Delta$, then multiply the first component of the resulting tensor product by $b$ and the second component by $c$. Notice that the first component of the tensor product, corresponds to the lower string or the string with the $x$-marker on it. Similarly, we define the map $m_{a}^{bc}$. First multiply the first component of the tensor product by $b$ and the second component by $c$, then apply $m$, at the end multiply the result by $a$.
We know that every checkerboard colorable diagram admits a source-sink orientation ([@Kamada-skein Proposition 6]). We can use this orientation to make all the decorations standard. In that case we only need the maps $\Delta_{++}^{+}$ and $m_{+}^{++}$.
\[ex37\] We compute the Khovanov homology for the virtual knot $K=3.7$; here the decimal number refers to the virtual knots in Green’s tabulation [@Green]. Figure \[3-7\] is a diagram for $K$ and Figure \[resolution\] is the cube of resolutions.
![The virtual knot $3.7$.[]{data-label="3-7"}](3-7.pdf){height="32mm"}
We enumerate the components of a state in a way that the one which has more $x$-markers in it be the first component. All the $m$ maps are $m_{+}^{++}$, and $\Delta$ maps are $\Delta_{++}^{+}$. A red arrow means the associated map has negative sign. All the maps are a single splitting or joining map except for the state which has 3 components in it. For this state the incoming map is $\Delta\otimes id$, and for the outgoing maps, the upper one is $\varphi$ defined as $\varphi(a,b,c)=-m(a,c)\otimes b$, and the lower one is $-id\otimes m$.
![The cube of resolutions for $K=3.7$.[]{data-label="resolution"}](res.pdf){height="94mm"}
The Khovanov complex is as follows: $$V\otimes V\{-3\}\to V\oplus V\oplus \left(V^{\otimes 3}\right)\{-2\}\to
\left(V^{\otimes 2}\right)\oplus\left(V^{\otimes 2}\right)\oplus\left(V^{\otimes 2}\right)\{-1\}
\to V.$$
We record the basis elements of the chain complex in Table \[basis elements\].
$j\setminus i$ $-2$ $-1$ 0 1
---------------- ---------------------- ------------------------------------- ---------------------------- -------------
$({{\bf 1}}{{\bf 1}},0,0)$
1 $(0,0,{{\bf 1}}{{\bf 1}}{{\bf 1}})$ $(0,{{\bf 1}}{{\bf 1}},0)$ ${{\bf 1}}$
$(0,0,{{\bf 1}}{{\bf 1}})$
$({{\bf 1}},0,0)$ $({{\bf 1}}X,0,0)$
$(0,{{\bf 1}},0)$ $(X{{\bf 1}},0,0)$
$-1$ ${{\bf 1}}{{\bf 1}}$ $(0,0,{{\bf 1}}{{\bf 1}}X)$ $(0,{{\bf 1}}X,0)$ $X$
$(0,0,{{\bf 1}}X{{\bf 1}})$ $(0,X{{\bf 1}},0)$
$(0,0,X{{\bf 1}}{{\bf 1}})$ $(0,0,{{\bf 1}}X)$
$(0,0,X{{\bf 1}})$
$(X,0,0)$
${{\bf 1}}X$ $(0,X,0)$ $(XX,0,0)$
$-3$ $(0,0,{{\bf 1}}XX)$ $(0,XX,0)$
$X{{\bf 1}}$ $(0,0,X{{\bf 1}}X)$ $(0,0,XX)$
$(0,0,XX{{\bf 1}})$
$-5$ $XX$ $(0,0,XXX)$
: The basis elements for the chain complex.[]{data-label="basis elements"}
The image of each basis element is in Table \[image of the basis elements\].
$j\setminus i$ $-2$ $-1$ 0 1
---------------- ----------------------------------------------------------------- ----------------------------------------------------- --------------- ------
$ {{\bf 1}}$
1 $(0,-{{\bf 1}}{{\bf 1}},-{{\bf 1}}{{\bf 1}})$ $ -{{\bf 1}}$ $ 0$
$ {{\bf 1}}$
$({{\bf 1}}X+X{{\bf 1}},{{\bf 1}}X+X{{\bf 1}},0)$ $ X$
$ (-{{\bf 1}}X-X{{\bf 1}},0,{{\bf 1}}X+X{{\bf 1}})$ $ X$
$-1$ $({{\bf 1}},{{\bf 1}},{{\bf 1}}X{{\bf 1}}+X{{\bf 1}}{{\bf 1}})$ $ (0,-X{{\bf 1}},-{{\bf 1}}X)$ $ -X$ $ 0$
$ (0,-{{\bf 1}}X,-{{\bf 1}}X)$ $ -X$
$ (0,-X{{\bf 1}},-X{{\bf 1}})$ $ X$
$ X$
$ (XX,XX,0)$
$ (X,X,{{\bf 1}}XX+X{{\bf 1}}X)$ $ (-XX,0,XX)$ $ 0$
$-3$ $ (0,-XX,0)$ $ 0$
$(X,X,XX{{\bf 1}})$ $ (0,0,-XX)$ $ 0$
$ (0,-XX,-XX)$
$-5$ $ (0,0,XXX)$ $ (0,0,0)$
: The image of the basis elements.[]{data-label="image of the basis elements"}
It is easy to check $d^2=0$. When we take the homology, two copies of ${{\mathbb Q}}$ survive, both in homological degree $0$, one in quantum degree $1$ and the other in quantum degree $-1$. Therefore the Khovanov homology of $K$ is isomorphic to the Khovanov homology of the unknot.
In [@DKK], Dye, Kaestner and Kauffman define Lee homology and the Rasmussen invariant for virtual knots, and they show that the Rasmussen invariant is an invariant of virtual knot concordance.
Khovanov homology and the Rasmussen invariant are invariants of unoriented virtual knots. If $D$ is a virtual knot diagram, then under mirror symmetry, we have $$Kh^{i,j}(D^{*})=Kh^{i,j}(D^{\dag})=Kh^{-i,-j}(D)\quad \text{ and } \quad s(D^{*}) = s(D^{\dag}) = -s(D).$$ The statement about $D^\dag$ follows from the one about $D^*$ since $D^\dag$ is obtained by applying [**or**]{} to all the crossings in $D^*$. More generally, applying [**or**]{} to a crossing does not change the cube of resolutions (see [@DKK]), nor does it change any of the quantities needed to compute the Khovanov homology, such as the number of positive and negative crossings. Hence the Khovanov homology and the Rasmussen invariant are unchanged under the [**or**]{} move.
Connected sum is not a well-defined operation on virtual knots; it depends on the diagrams used and the choice of where to form the connected sum. The Rasmussen invariant is independent of these choices, and it is, in fact, additive under connected sum. For a proof, we refer the reader to [@Rushworth]. This fact and invariance under concordance imply that it induces a homomorphism from the virtual concordance group to ${{\mathbb Z}}$.
Table \[ras-table\] lists the Rasmussen invariant for the alternating virtual knots up to six crossings. The three virtual knots 6.90115, 6.90150 and 6.90170 appearing in Figure \[fig-6-90170\] all have Rasmussen invariant equal to $-2$, and as a result we conclude that none of these virtual knots are slice.
In [@BCG17], Boden et al. define slice obstructions in terms of signatures of symmetrized Seifert matrices for almost classical knots, and as an application, they show that neither 6.90115 nor 6.90150 are slice. The Rasmussen invariant provides an alternative proof of non-sliceness for these two almost classical knots, and it also gives a new result by showing that 6.90170 is not slice.
Since each of these knots can be unknotted using two crossing changes, it follows that their slice genus satisfies $1 \leq g_{s}(K) \leq 2$.
![The alternating virtual knots $6.90115,$ $6.90150$ and $6.90170$, from left to right.[]{data-label="fig-6-90170"}](6-90115.pdf "fig:") ![The alternating virtual knots $6.90115,$ $6.90150$ and $6.90170$, from left to right.[]{data-label="fig-6-90170"}](6-90150.pdf "fig:") ![The alternating virtual knots $6.90115,$ $6.90150$ and $6.90170$, from left to right.[]{data-label="fig-6-90170"}](6-90170.pdf "fig:")
Khovanov Homology and Alternating Virtual Links {#four}
===============================================
Let $D$ be a checkerboard virtual link diagram. Apply [**or**]{} to all crossings with $\eta=-1$. The result is a diagram in which $\eta = +1$ for each crossing. Hence, by Lemma \[eta\], the new diagram, which we call $D_{alt}$, is alternating. The diagrams $D$ and $D_{alt}$ have isomorphic Khovanov homology groups.
In particular, starting with any classical diagram, we can change it to an alternating virtual diagram with the same Khovanov homology. We can do the same, starting with any checkerboard colorable diagram.
Following [@Lee], we seek a relation between the Rasmussen invariant and the signatures of alternating virtual knots. If $i$ is the homological degree and $j$ is the quantum degree for Khovanov homology, then $H$-thinness for classical alternating knots means, $j=2i-\sigma\pm 1$, where $\sigma$ is the signature. This implies that $s=-\sigma$, where $s$ is the Rasmussen invariant. On the other hand, not all virtual alternating knots are $H$-thin. For example the Khovanov polynomial for the knot $K=5.2426$ depicted in Figure \[fig-5-2426\] is as follows: $$\frac{1}{q^{11}t^{3}}+\frac{1}{q^{9}t^{3}}+\frac{1}{q^{7}t^{2}}+\frac{1}{q^{5}t^{2}}+\frac{1}{q^{5}}+\frac{1}{q^{3}},$$ which is supported in three lines $j=2i-1$, $j=2i-3$ and $j=2i-5$. Notice that from Table \[ras-table\] $(\sigma_{\xi^{*}},\sigma_{\xi})=(2,4),$ and we can write the three lines as: $$j=2i-\sigma_{\xi^*}+1,j=2i-\sigma_{\xi^*}-1,j=2i-\sigma_{\xi}-1.$$
In fact, instead of $H$-thinness we have the following theorem. Here the coloring $\xi$ is the one for which every crossing of $D$ has $\eta=-1$.
\[prop\] If $D$ is a connected alternating virtual link diagram with genus $g$ and signatures $\sigma_{\xi},\sigma_{\xi^*}$, then its Khovanov homology is supported in $g+2$ lines: $$j=2i-\sigma_{\xi^*}+1,j=2i-\sigma_{\xi^*}-1,\ldots,j=2i-\sigma_{\xi}-1.$$
Following [@Lee], we apply induction on the number of crossings. The base case is trivial. Let $D$ be an alternating virtual link diagram with $n$ crossings. $0$ and $1$ smooth the last crossing to obtain $D(*0)$ and $D(*1)$, respectively. We can easily see that they are alternating link diagrams. Shift the Khovanov complex by $n_{-}$ horizontally, and $2n_{-}-n_{+}$ vertically. Denote the resulting complex by ${ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{C} \kern-0.05em } } \kern0.05em}}(D)$ and its homology by ${ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{H} \kern-0.05em } } \kern0.05em}}(D)$. We denote this shift by ${ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{C} \kern-0.05em } } \kern0.05em}}(D)=C(D)[n_{-}]\{2n_{-}-n_{+}\}$. We have the following short exact sequence: $$0\to { \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{C} \kern-0.05em } } \kern0.05em}}(D(*1))[+1]\{+1\}\to { \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{C} \kern-0.05em } } \kern0.05em}}(D)\to { \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{C} \kern-0.05em } } \kern0.05em}}(D(*0))\to 0,$$ which gives a long exact sequence involving ${ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{H} \kern-0.05em } } \kern0.05em}}(D), { \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{H} \kern-0.05em } } \kern0.05em}}(D(*0))$ and ${ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{H} \kern-0.05em } } \kern0.05em}}(D(*1))[+1]\{+1\}$, which encodes ${ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{H} \kern-0.05em } } \kern0.05em}}(D)$ is supported on ${ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{H} \kern-0.05em } } \kern0.05em}}(D(*0))$ and ${ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{H} \kern-0.05em } } \kern0.05em}}(D(*1))$.
It suffices to show that ${ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{H} \kern-0.05em } } \kern0.05em}}(D)$ is supported in $g+2$ lines with $y$-intercepts of $$-|s_{\partial}|+2,-|s_{\partial}|,\cdots,-|s_{\partial}|-2g,$$ because after shifting back ${ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{H} \kern-0.05em } } \kern0.05em}}(D)$, the result follows.
The all $0$ state of $D$ is the same as the all $0$ state of $D(*0)$. Also the all $1$ state of $D$ is the same as the all $1$ state of $D(*1)$. In the all $0$ state of $D$, if we change the resolution of the last crossing from a $0$-smoothing to a $1$-smoothing, we obtain the all $0$ state for $D(*1)$. Similarly, in the all $1$ state of $D$, if we change the resolution of the last crossing from a $1$-smoothing to a $0$-smoothing, we obtain the all $1$ state for $D(*0)$.
These three diagrams, all have the boundary property. $D(*0)$ and $D(*1)$, both have $n-1$ crossings. Thus we have: $$\begin{aligned}
|s_{\partial}(D)|+|{ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}(D)|&=&n+2-2g(D),\\
|s_{\partial}(D(*0))|+|{ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}(D(*0))|&=&n+1-2g(D(*0)),\\
|s_{\partial}(D(*1))|+|{ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}(D(*1))|&=&n+1-2g(D(*1)).\end{aligned}$$ Using the above observations, we can rewrite the last two equations as: $$\begin{aligned}
|s_{\partial}(D)|+|{ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}(D(*0))|&=&n+1-2g(D(*0)),\\
|s_{\partial}(D(*1))|+|{ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}(D)|&=&n+1-2g(D(*1)).\end{aligned}$$
Since the genus is an integer, the first equation implies that $|{ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}(D(*0))|$ cannot be equal to $|{ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}(D)|$, so it is either one more, or one less. Similarly, $|s_{\partial}(D(*1))|$ is either one more, or one less than $|s_{\partial}(D)|$. Thus we have four different cases:
[**Case 1:**]{} $|{ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}(D(*0))|=|{ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}(D)|-1\ ,\ |s_{\partial}(D(*1))|=|s_{\partial}(D)|-1\ \to\ g(D)=g(D(*0))=g(D(*1)).$
We use the induction hypothesis. Since $|s_{\partial}(D(*0))|=|s_{\partial}(D)|$ and $g(D(*0))=g(D)$, the $y$-intercepts of the lines for $D(*0)$, are:
$$-|s_{\partial}(D)|+2,-|s_{\partial}(D)|,\cdots,-|s_{\partial}(D)|-2g(D).$$
The $y$-intercepts of the lines for $D(*1)[+1]\{+1\}$ are the $y$-intercepts of the lines for $D(*1)$ minus $1$. Since $|s_{\partial}(D(*1))|=|s_{\partial}(D)|-1$, the $y$-intercepts of the lines for $D(*1)[+1]\{+1\}$ and $D(*0)$ agree, and they are precisely the numbers that we are looking for. Thus the result follows in this case.
[**Case 2:**]{} $|{ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}(D(*0))|=|{ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}(D)|+1\ ,\ |s_{\partial}(D(*1))|=|s_{\partial}(D)|-1\ \to\ g(D)=g(D(*0))+1=g(D(*1)).$
In this case, there are $g(D)+1$ lines for $D(*0)$, and their $y$-intercepts are: $$-|s_{\partial}(D)|+2,-|s_{\partial}(D)|,\cdots,-|s_{\partial}(D)|-2g(D)+2.$$
On the other hand for $D(*1)$, the $y$-intercepts are as before. Hence the union of the supports of $D(*0)$ and $D(*1)[+1]\{+1\}$ is again the desired $g(D)+2$ lines.
[**Case 3:**]{} $|{ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}(D(*0))|=|{ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}(D)|-1\ ,\ |s_{\partial}(D(*1))|=|s_{\partial}(D)|+1\ \to\ g(D)=g(D(*0))=g(D(*1))+1.$
In this case, there are $g(D)+1$ lines for $D(*1)[+1]\{+1\}$, and their $y$-intercepts are: $$-|s_{\partial}(D)|,-|s_{\partial}(D)|,\cdots,-|s_{\partial}(D)|-2g(D).$$
For $D(*0)$, we have the same $g(D)+2$ line, as in case 1. As before, their union is the $g(D)+2$ lines with the desired $y$-intercepts.
[**Case 4:**]{} $|{ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}(D(*0))|=|{ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}(D)|+1\ ,\ |s_{\partial}(D(*1))|=|s_{\partial}(D)|+1\ \to\ g(D)=g(D(*0))+1=g(D(*1))+1$.
Combining case 2 and 3, we see that the result follows.
Classical alternating links are $H$-thin.
If $D$ is a connected alternating virtual link diagram with genus $g$ and signatures $\sigma_{\xi},\sigma_{\xi^*}$, then $$-\sigma_{\xi}\leq s(D)\leq -\sigma_{\xi^{*}}.$$
This follows from the previous theorem, and Lee’s spectral sequence.
For the classical knot $K=9_{42}^{*}$ shown on the left of Figure \[fig-9-42-alt\], the Khovanov polynomial is as follows $$\frac{1}{q^7t^4}+ \frac{1}{q^3t^3}+ \frac{1}{q^3t^2}+ \frac{1}{qt}+\frac{q}{t}+ \frac{1}{q}+q+ q^3+ q^3t+ q^7t^2,$$ which is supported in three lines.
Observe that, given a 2-strand classical tangle with $n$ half-twists, applying the [**or**]{} move to each of the crossings has the effect of adding two virtual crossings at either end of the tangle (see Figure \[last\]).
![The effect of applying [**or**]{} to a 2-strand classical tangle with $n$ half-twists.[]{data-label="last"}](l2.pdf){height="16mm"}
![The effect of applying [**or**]{} to a 2-strand classical tangle with $n$ half-twists.[]{data-label="last"}](l3.pdf){height="16mm"}
Let $\xi$ be the coloring in which the unbounded region is white. Then the five crossings in the top half of $9_{42}^*$ of Figure \[fig-9-42-alt\] have $\eta =-1$ and the four crossings in the bottom half have $\eta=1.$ The five crossings with $\eta=-1$ occur in two 2-strand classical tangle, one with three positive crossings and the other with two negative crossings. Thus, the above observation implies that applying the [**or**]{} move to the five crossings with $\eta=-1$ results in the alternating virtual knot shown on the right of Figure \[fig-9-42-alt\]. Then $(\sigma_{\xi^{*}},\sigma_{\xi})=(-2,0)$, and by Theorem \[prop\], its Khovanov homology, which coincides with the Khovanov homology of $K=9_{42}^{*}$ is supported in the following three lines: $$j=2i+3,j=2i+1,j=2i-1.$$
![The classical knot $9_{42}^{*}$ (left) and the virtual knot $(9_{42}^{*})_{alt}$ (right).[]{data-label="fig-9-42-alt"}](9-42.pdf "fig:"){height="46mm"} ![The classical knot $9_{42}^{*}$ (left) and the virtual knot $(9_{42}^{*})_{alt}$ (right).[]{data-label="fig-9-42-alt"}](9-42-alt.pdf "fig:"){height="46mm"}
Similar to the Turaev genus (cf. Definition \[def-Turaev\]), we have the following definition.
Let $K$ be a non-split checkerboard colorable virtual link. The *alternating genus* of $K$ is $$g_{alt}(K)=\min\{g(D_{alt}) \mid D\ \text{is\ a\ checkerboard\ diagram\ for\ }K\},$$ where $g(D_{alt})$ is the supporting genus of $D_{alt}$.
\[lem-alt-Turaev\] If $L$ is a non-split classical link, then $g_{alt}(L)\leq g_T(L)$.
Let $D$ be a classical diagram for $L$. We obtain $s_{\partial}$ by $0$-smoothing crossings $c$ with $\eta(c)=+1$, and $1$-smoothing crossings $c$ with $\eta(c)=-1$. To obtain $D_{alt}$ we apply [**or**]{} exactly to crossings $c$ with $\eta(c)=-1$. In $D_{alt}$ black and white disks are obtained by switching $0$ and $1$ smoothing and vice versa for crossings $c$ with $\eta(c)=-1$. This means $s_{\partial}(D_{alt})=s_0(D)$ and ${ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}(D_{alt})=s_1(D)$, and we have: $$\begin{aligned}
g(D_{alt})&=&\frac{1}{2}\left(c(D_{alt})+2-|s_{\partial}(D_{alt})|-|{ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}(D_{alt})|\right),\\
&=&\frac{1}{2}\left(c(D)+2-|s_{0}(D)|-|s_{1}(D)|\right)=g_T(D).\end{aligned}$$ This shows $g_{alt}(L)\leq g_T(L)$.
The following corollary is immediate, and is a generalization of Corollary 3.1 in [@Kofman07], which was first obtained by Manturov in [@Man].
\[cor3\] For any checkerboard colorable (in particular, classical) link $K$, the Khovanov homology of $K$ is supported in $g_{alt}+2$ lines, i.e. the homological width of the Khovanov homology is less than or equal to $g_{alt}+2$.
Here we have another proof for Corollary 3.1 in [@Kofman07].
For any classical non-split link L with Turaev genus $g_T(L)$, the thickness of the (unreduced) Khovanov homology of K is less than or equal to $g_T(L)+2$.
We have $g_{alt}(L)\leq g_T(L)$, and $g_{alt}(L)+2$ is an upper bound for the thickness of the Khovanov homology. The result is immediate.
Suppose $D$ is a positive alternating virtual knot. Then $s(D)=-\sigma_{\xi^*}(D)$.
Since $D$ is alternating, $\sigma_{\xi^*}=\beta-1-n_{+}$, where $\beta$ is the number of all $0$-smoothing state (see [@Karimi]). For any positive knot $K$, we have $s(K)=1-\beta+n_{+}$ (see [@DKK]).
For a negative virtual knot (for example $K=5.2426$ depicted in Figure \[fig-5-2426\]), the Rasmussen invariant and the signatures are the negatives of the corresponding invariants for its vertical mirror image (a positive virtual knot). It follows that $s(K)=-\sigma_{\xi}(K)$. In general it is not true that the Rasmussen invariant is the negative of one of the signatures for alternating virtual knots. For example, the virtual knot $5.2427$ is alternating (see Figure \[fig-5-2426\]), has Rasmussen invariant $s(K)=-2$, and signatures $\sigma_\xi(K)=4$ and $\sigma_{\xi^*}(K)=0$.
![Alternating virtual knots $5.2426$ (left) and $5.2427$ (right).[]{data-label="fig-5-2426"}](5-2426.pdf "fig:"){height="46mm"} ![Alternating virtual knots $5.2426$ (left) and $5.2427$ (right).[]{data-label="fig-5-2426"}](5-2427.pdf "fig:"){height="46mm"}
Ordinarily, for Khovanov homology of virtual links, one assigns the zero map to each single cycle smoothing. Alternative theories can be constructed using different maps for the single cycle smoothings, and as virtual knot homologies, these are generally stronger than the usual Khovanov homology for virtual knots. For example, in [@Rushworth] Rushworth uses this approach to define a variant theory called doubled Khovanov homology. For virtual links whose cube of resolutions has no single cycle smoothings, the doubled Khovanov homology is the direct sum of two copies of ordinary Khovanov homology. In that case, the doubled Khovanov homology is completely determined by the ordinary Khovanov homology and thus it contains no new information.
We will show that when the underlying virtual link diagram is checkerboard colorable, there are no single cycle smoothings in its cube of resolutions. This was first proved by Rushworth [@Rushworth], and here we provide a different proof.
Let $D$ be an alternating link diagram, and $s_{\partial}$ be the all $0$-smoothing state, and ${ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}$ the all $1$-smoothing state. If we change one $0$-smoothing to obtain the state $s$, the number of components of $s_{\partial}$ and $s$, differs by one. A similar result holds for ${ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}$.
Assume we change the smoothing in the last crossing. We consider $D(*1)$, which is an alternating diagram and has the boundary property. If $D$ has $c$ crossings, and $g$ and $g_{1}$ are the genera for $D$ and $D(*1)$ respectively, we have: $$\begin{aligned}
|s_{\partial}|+|{ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}| &=& c+2-2g,\\
|s|+|{ \hbox{ \kern-0.2em \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.15em \ensuremath{s} \kern-0.05em } } \kern0.05em}}_{\partial}| &=& c-1+2-2g_{1},\\
|s_{\partial}|-|s| &=& 1+2(g_{1}-g).\end{aligned}$$
Thus the difference is an odd number, and the result follows. The proof for the other case is similar.
Let $D$ be an alternating link diagram. Then there is no single cycle smoothing in the cube of resolutions for $D$.
Assume we change a $0$-smoothing of the state $s$ to a $1$-smoothing at the crossing $c_{i}$. If for all the other crossings, we have $0$-smoothing in $s$, then this is the previous lemma. Otherwise, we apply [**sc**]{} to the crossings of $D$, which have been resolved to $1$-smoothings in $s$. Call the new diagram $D'$. Since the state $s$ is the all $0$-smoothing state for $D'$, the result follows from the previous lemma.
Let $D$ be a checkerboard colorable link diagram. Then there is no single cycle smoothing in the cube of resolutions for $D$.
Assume we change one $0$-smoothing of the state $s$ to a $1$-smoothing at the crossing $c_{i}$, and call the resulting state $s'$. First we consider $D_{alt}$. Let $C'=\{c_{i_1},\ldots,c_{i_k}\}$ be the set of crossings of $D$ which are changed to obtain $D_{alt}$. There are two cases. If $c_{i}$ does not belong to $C'$, then the edge with vertices $s$ and $s'$ corresponds exactly to an edge in the cube of resolutions for $D_{alt}$, and the result follows.
If $c_{i}\in C'$, then the same thing happens. The only difference is the direction of the map in $D_{alt}$ is reversed, going from $s'$ to $s$. The result still holds.
If $D$ is a checkerboard colorable link diagram, then the doubled Khovanov homology for $D$ is the direct sum of two copies of the ordinary Khovanov homology for $D$.
We have calculated the Rasmussen invariant for all alternating virtual knots up to 6 crossings, and the results are listed in Table \[ras-table\]. Note that the Rasmussen invariants are calculated for virtual knots up to 4 crossings in [@rush].
In [@Boden2019], Boden and Chrisman list the number of all virtual knots up to 6 crossings with unknown slice status (cf. [@MTables]). The following example concerns three non-alternating virtual knots whose slice status was previously unknown. We use Rasmussen invariants to show they are not slice and deduce that they have slice genus equal to one.
Consider the three non-alternating virtual knots 6.31460, 6.52378, and 6.66907 depicted in Figure \[Fig:non-slice\]. They all have isomorphic Khovanov homology, and the Khovanov polynomial is as follows $$\frac{1}{q^{3}t^{2}}+\frac{1}{qt}+\frac{q}{t}+2+q+2q^{3}+2q^{2}t+2q^{4}t+q^{3}t^{2}+2q^{6}t^{2}+q^{7}t^{3}.$$ From this polynomial, it is not difficult to see that the two surviving copies of ${{\mathbb Q}}$ in Lee’s spectral sequence are in degrees $1$ and $3$. It follows these three knots have Rasmussen invariant equal to $2$.
Since each of these virtual knots has nonzero Rasmussen invariant, none of them are slice. Further, notice that, for each of 6.31460, 6.52378, and 6.66907, performing a crossing change to one of the crossings in the clasp produces a diagram of the unknot. Since a crossing change can be achieved in a genus one cobordism, it follows that each of the three virtual knots has slice genus equal to one.
![The non-alternating virtual knots $6.31460,$ $6.52378$ and $6.66907,$ from left to right.[]{data-label="Fig:non-slice"}](6-31460a.pdf "fig:") ![The non-alternating virtual knots $6.31460,$ $6.52378$ and $6.66907,$ from left to right.[]{data-label="Fig:non-slice"}](6-52378a.pdf "fig:") ![The non-alternating virtual knots $6.31460,$ $6.52378$ and $6.66907,$ from left to right.[]{data-label="Fig:non-slice"}](6-66907.pdf "fig:")
Alternating Virtual Knots and Their Invariants {#five}
==============================================
The signatures are computed using a Mathematica program written by Micah Chrisman, and the Khovanov homology (unlisted) is computed using online Mathematica program written by Daniel Tubbenhauer. The Rasmussen invariants are then computed by hand using Lee’s spectral sequence. Boldface font is used to indicate that the knot is classical. Here the decimal numbers refer to the virtual knots in Green’s tabulation [@Green]. For a list of the associated Gauss words, see [@Karimi].
In the following example, we outline how to calculate the Rasmussen invariant of a virtual knot once its Khovanov homology has been determined.
For the alternating virtual knot $K=6.90170$ depicted in Figure \[fig-6-90170\], the Khovanov homology is recorded in Table \[690170\]. In the spectral sequence, $E_{\infty}\cong E_5$, and $E_4\cong E_2$. This fact dictates the exact location of the one nontrivial $d_4$, which is from the $(-3,-9)$-entry to the $(-2,-5)$-entry of Table \[690170\]. Now the surviving copies of ${{\mathbb Q}}$ in $E_{\infty}$, are in the entries $(0,-1)$ and $(0,-3)$, which shows that this knot has Rasmussen invariant $s=-2$.
$j\setminus i$ $-3$ $-2$ $-1$ 0
---------------- ----------------- ----------------- ------ -----------------
$-1$ ${{\mathbb Q}}$
$-3$ ${{\mathbb Q}}$
$-5$ ${{\mathbb Q}}$
$-7$
$-9$ ${{\mathbb Q}}$
: The Khovanov homology for the alternating virtual knot $K=6.90170$.[]{data-label="690170"}
[**Acknowledgements.**]{} I would like to thank my adviser, Hans U. Boden, for all of his support. I would also like to thank Micah Chrisman for providing the Mathematica package used to compute the signatures.
[cccc]{}
[|l|c|c|l|]{}
[Virtual]{} & & &\
& $(\sigma_\xi^*,\sigma_{\xi})$ & &\
& $(2,2)$ & $-2$& $1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
3.7& $(0,2)$ & 0 & $1/q+q$\
4.105& $(0,2)$ &$-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
4.106& $(0,2)$ &0 &$1/q+q$\
4.107& $(-2,2)$ &0&$1/q+q$\
& $(0,0)$ &0 &$ 1/q^5t^2 + 1/qt+1/q + q + qt + q^5t^2$\
5.2426 & $(2,4)$ &$-4$ &$ 1/q^{11} t^3+1/q^9 t^3+1/q^7 t^2+1/q^5 t^2 +1/q^5 + 1/q^3 $\
5.2427 & $(0,4)$ & $-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
5.2428 & $(0,2)$ & 0 &$1/q^7 t^3+ 1/q^3 t^2 +2/q + q + q^3 t$\
5.2429 & $(2,4)$ & $-2$ &$1/q^{11} t^4+1/q^9 t^3+1/q^7 t^3+1/q^5 t^2 +1/q^5t + 1/q^3+2/q$\
5.2430 & $(0,2)$ & 0 &$1/q+q$\
5.2431 & $(-2,2)$ & 0 &$1/q+q$\
5.2432 & $(-2,2)$ & 0 &$1/q+q$\
5.2433 & $(0,4)$ & $-4$&$ 1/q^{11} t^3+1/q^9 t^3+1/q^7 t^2+1/q^5 t^2 +1/q^5 + 1/q^3 $\
5.2434 & $(0,4)$ & $-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
5.2435 & $(0,2)$ & 0 &$1/q^7 t^3+ 1/q^3 t^2 +2/q + q + q^3 t$\
5.2436 & $(-2,2)$ & 0 &$1/q+q$\
& $(2,2)$ & $-2$ &$1/q^{13}t^{5}+1/q^{9}t^{4}+ 1/q^{9} t^3+1/q^7 t^2+1/q^5 t^2 +1/q^3 t+1/q^3 + 1/q $\
5.2438 & $(0,2)$ & $-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
5.2439& $(0,2)$ & 0 &$1/q+q$\
5.2440& $(0,2)$ & 0 &$1/q+q$\
5.2441 & $(-2,2)$ & 0& $1/q+q$\
5.2442 & $(-2,2)$ & 0 &$1/q+q$\
5.2443& $(-2,0)$ & 0 &$ 1/q^5t^2 + 1/qt+1/q + q + qt + q^5t^2$\
5.2444& $(-2,0)$ & 0 &$1/q+q$\
& $(4,4)$ & $-4$ &$1/q^{15}t^{5}+1/q^{11}t^{4}+ 1/q^{11} t^3+1/q^7 t^2+1/q^5 +1/q^3 $\
5.2446 & $(2,4)$ & $-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
5.2447& $(0,2)$ & 0 &$1/q+q$\
5.2448 & $(0,2)$ & 0 &$1/q+q$\
& $(4,4)$ &$-4$ &$1/q^{17}t^{6}+1/q^{15}t^{5}+1/q^{13}t^{5}+1/q^{11}t^{4}+ 2/q^{11} t^3+2/q^7 t^2+1/q^5 +1/q^3 $\
6.89188 & $(2,4)$ &$-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
6.89189 & $(0,2)$ & $-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
& $(0,0)$ & 0 &$ 1/q^7t^3 +1/q^3t^2 + 1/q^3t+2/q + 2q + q^3t + q^3t^2+ q^7t^3$\
6.90101 & $(0,4)$ & 0 &$1/q+q$\
6.90102 & $(0,4)$ & 0 &$1/q+q$\
6.90103 & $(-2,2)$ & 0 &$1/q+q$\
6.90104 & $(0,4)$ & 0& $1/q+q$\
6.90105 & $(-2,2)$ &0 &$1/q+q$\
6.90106 & $(-2,2)$ & 0 &$1/q+q$\
6.90107 & $(0,4)$ & 0 &$1/q+q$\
6.90108& $(-2,2)$ & 0 &$1/q+q$\
6.90109 & $(2,4)$ & $-4$&$ 1/q^{11} t^3+1/q^9 t^3+1/q^7 t^2+ 1/q^5 t^2 +1/q^5 + 1/q^3 $\
6.90110 & $(2,4)$ & $-2$ & $1/q^{13}t^{5}+1/q^{11}t^{4}+1/q^{9}t^{4}+ 1/q^{9} t^3 +1/q^7 t^3+1/q^7 t^2+1/q^5 t^2 $\
&&& $+1/q^5 t +1/q^3 t+1/q^3 + 2/q$\
6.90111 & $(0,4)$ & $-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
6.90112 & $(0,4)$ & $-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
6.90113 & $(0,4)$ & $-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
6.90114 & $(0,4)$ & $-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
6.90115 & $(0,2)$ & $-2$ &$1/q^9 t^3+1/q^7 t^2+ 1/q^5 t^2+ 1/q^3 t +1/q^3+1/q + t/ q + q^3 t^2$\
6.90116 & $(0,2)$ & $0$ &$1/q^7 t^3+1/q^3 t^2+ 2/q + q + q^3 t$\
6.90117 & $(0,4)$ & $-2$ &$1/q^{11}t^{4}+1/q^{9}t^{3}+1/q^{7}t^{3}+ 1/q^{5} t^2+1/q^5 t+1/q^3 +2/q $\
6.90118 & $(0,2)$ & $0$ &$1/q^7 t^2+1/q^3 t+ t/q +1/q+ q + q^3 t^2$\
6.90119 & $(0,4)$ & 0 &$1/q+q$\
6.90120 & $(-2,2)$& 0 &$1/q+q$\
6.90121 & $(0,4)$ & 0 &$1/q+q$\
6.90122 & $(-2,2)$ &0 &$1/q+q$\
6.90123 & $(0,2)$ & $0$& $1/q^7 t^3+1/q^5 t^2+1/q^3 t^2+1/q t+ 2/q + q +qt+ q^3 t+q^5 t^2$\
[cccc]{}
[|l|c|c|l|]{}
[Virtual]{} & & &\
& $(\sigma_\xi^*,\sigma_{\xi})$ & &\
6.90124 & $(-2,2)$ & $0$& $1/q^3 t+ 1/q + 2q + q^3 t^2+q^7 t^3$\
6.90125 & $(0,4)$ & $-2$&$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
6.90126 & $(0,2)$ & 0&$1/q+q$\
6.90127 & $(-2,4)$ & 0 &$1/q+q$\
6.90128 & $(-2,2)$ & 0 &$1/q+q$\
6.90129 & $(-2,4)$ & 0 &$1/q+q$\
6.90130 & $(-2,2)$ & 0 &$1/q+q$\
6.90131 & $(-2,2)$ & $0$ &$1/q^5 t^2+ 1/qt + 1/q+q + q t+q^5 t^2$\
6.90132 & $(-2,2)$ & 0 &$1/q+q$\
6.90133 & $(-2,2)$ & 0 &$1/q+q$\
6.90134 & $(-2,2)$ & 0 &$1/q+q$\
6.90135 & $(-4,2)$ & 0 &$1/q+q$\
6.90136 & $(-2,2)$ & 0 &$1/q+q$\
6.90137 & $(-4,2)$ & 0 &$1/q+q$\
6.90138 & $(-4,0)$ & $2$&$q+q^{3}+q^{5}t^{2}+q^{9}t^{3}$\
6.90139 & $(2,4)$ & $-4$ &$1/q^{15}t^{6}+1/q^{11}t^{5}+1/q^{11}t^{3}+ 2/q^{9} t^3+1/q^7 t^2+2/q^5 t^2 +1/q^5 +1/q^3 $\
6.90140 & $(0,4)$ & $-2$ &$1/q^{11}t^{4}+1/q^{9}t^{3}+ 1/q^{7} t^3+1/q^5 t^2+1/q^5 t +1/q^3 +2/q $\
6.90141 & $(-2,4)$ & 0&$1/q+q$\
6.90142 & $(-2,2)$ & $0$&$1/q^7 t^2+1/q^3 t+ 1/q + q + t/q+q^3 t^2$\
6.90143 & $(-2,2)$ &0 &$1/q+q$\
6.90144 & $(0,4)$ & $0$&$1/q^{9}t^{4}+1/q^{5}t^{3}+1/q^{3}t+1/q+2q$\
6.90145 & $(-2,2)$ & $0$&$1/q^{5}t^{2}+1/qt+1/q+q+qt+q^5t^2$\
6.90146 & $(-2,2)$ & $0$ &$2/q^{5}t^{2}+2/qt+1/q+q+2qt+2q^5t^2$\
6.90147& $(0,4)$ & $-4$ &$1/q^{11}t^{3}+ 1/q^{9} t^3+1/q^7 t^2+1/q^5 t^2 +1/q^5 +1/q^3 $\
6.90148& $(0,4)$ & $-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
6.90149 & $(0,4)$ & $-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
6.90150 & $(0,2)$ &$-2$& $1/q^{9}t^{3}+1/q^{7}t^{2}+1/q^{5}t^{2}+1/q^{3}t+1/q^3+1/q+t/q+q^3t^2$\
6.90151 & $(0,4)$ & $-2$&$1/q^{11}t^{4}+1/q^{9}t^{3}+ 1/q^{7} t^3+1/q^5 t^2+1/q^5 t +1/q^3 +2/q $\
6.90152 & $(0,4)$ & 0 &$1/q+q$\
6.90153 & $(0,4)$ & 0 &$1/q+q$\
6.90154 & $(0,2)$ & $0$&$1/q^7 t^3+1/q^5 t^2+1/q^3 t^2+1/q t+ 2/q + q + qt+q^3t+ q^5t^2$\
6.90155 & $(0,2)$ & $-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
6.90156 & $(0,4)$ & 0 &$1/q+q$\
6.90157 & $(-2,4)$ & 0&$1/q+q$\
6.90158 & $(-2,2)$ & $0$ &$1/q^5 t^2+1/q t+ 1/q + q + qt+ q^5t^2$\
6.90159 & $(-2,2)$ & 0 &$1/q+q$\
6.90160 & $(-2,2)$ & 0 &$1/q+q$\
6.90161 & $(-2,2)$ & $0$ &$1/q+q$\
6.90162 & $(-2,2)$ & 0 &$1/q+q$\
6.90163 & $(-2,2)$ & 0 &$1/q+q$\
6.90164 & $(-2,2)$ &$0$ &$1/q^3 t+ 1/q + 2q + q^3t^2+ q^7t^3$\
6.90165 & $(0,2)$ & $0$ &$1/q^{9}t^{4}+1/q^{5}t^{3}+1/q^{3}t+1/q+2q$\
6.90166 & $(-2,2)$ & $0$ &$ 1/q^5t^2 + 1/qt+1/q + q + qt + q^5t^2$\
6.90167 & $(2,4)$ & $-4$ &$1/q^{15} t^5+1/q^{11} t^4+1/q^{11} t^3 +1/q^7 t^2+1/q^5 + 1/q^3 $\
6.90168 & $(2,4)$ & $-2$ &$1/q^{13} t^5+1/q^{9} t^4+1/q^{9} t^3+1/q^{7} t^2+1/q^{5} t^2 +1/q^3 t+1/q^3 + 1/q $\
6.90169 & $(0,4)$ & $-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
6.90170 & $(0,2)$ & $-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
6.90171 & $(0,2)$ & 0 &$1/q+q$\
& $(0,0)$ & $0$ &$1/q^{7}t^{3}+1/q^{5}t^{2}+ 1/q^{3} t^2+1/q^3 t+1/qt +2/q +2q + qt+ q^3t$\
&&& $ + q^3t^2+ q^5t^2+ q^7t^3 $\
6.90173& $(0,4)$ & 0 &$1/q+q$\
6.90174 & $(-2,4)$ & 0 &$1/q+q$\
6.90175 & $(-2,2)$ & 0 &$1/q+q$\
6.90176 & $(-2,4)$ & 0 &$1/q+q$\
6.90177 & $(-2,2)$ & 0 &$1/q+q$\
6.90178 & $(-2,2)$ & 0 &$1/q+q$\
6.90179 & $(0,4)$ & 0 &$1/q+q$\
[cccc]{}
[|l|c|c|l|]{}
[Virtual]{} & & &\
& $(\sigma_\xi^*,\sigma_{\xi})$ & &\
6.90180 & $(-2,4)$ & 0 &$1/q+q$\
6.90181 & $(-2,2)$ & 0 &$1/q+q$\
6.90182 & $(-2,2)$ & 0 &$1/q+q$\
6.90183 & $(-2,2)$ & $0$&$ 1/q^5t^2 + 1/qt+1/q + q + qt + q^5t^2$\
6.90184 & $(-4,0)$ & $2$ &$q+q^{3}+q^{5}t^{2}+q^{9}t^{3}$\
6.90185 & $(0,4)$ & $-4$ &$ 1/q^{11} t^3+1/q^9 t^3+1/q^7 t^2+1/q^5 t^2 +1/q^5 + 1/q^3 $\
6.90186 & $(0,4)$ & $-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
6.90187 & $(0,4)$ & $-2$ &$ 1/q^{11} t^4+1/q^9 t^3+1/q^7 t^3+1/q^5 t^2 +1/q^5t + 1/q^3+2/q $\
6.90188 & $(-2,4)$ & 0 &$1/q+q$\
6.90189 & $(-2,2)$ & 0 &$1/q+q$\
6.90190 & $(-2,2)$ & 0 &$1/q+q$\
6.90191 & $(0,2)$ & $0$ &$ 1/q^{7} t^2+1/q^3 t+ 1/q+q+t/q +1/q^3 t^2 $\
6.90192 & $(0,4)$ & $0$ &$1/q^{9}t^{4}+1/q^{5}t^{3}+1/q^{3}t+1/q+2q$\
6.90193 & $(-2,2)$ & $0$ &$1/q^{5}t^{2}+1/qt+1/q+q+qt+q^5t^2$\
6.90194 & $(0,2)$ & $0$ &$1/q+q$\
6.90195 & $(2,4)$ & $-4$ &$1/q^{15} t^5+1/q^{11} t^4+1/q^{11} t^3 +1/q^7 t^2+1/q^5 + 1/q^3 $\
6.90196 & $(0,4)$ & $-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
6.90197 & $(0,4)$ & $-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
6.90198 & $(-2,2)$ & 0 &$1/q+q$\
6.90199 & $(-2,2)$ & 0 &$1/q+q$\
6.90200 & $(-2,0)$ & 0 &$1/q+q$\
6.90201 & $(2,4)$ & $-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
6.90202 & $(0,4)$ &0 &$1/q+q$\
6.90203 & $(0,4)$ & 0 &$1/q+q$\
6.90204 & $(-2,2)$ & 0 &$1/q+q$\
6.90205 & $(0,4)$ &0 &$1/q+q$\
6.90206 & $(-2,2)$ & 0&$1/q+q$\
6.90207 & $(-2,2)$ & 0 &$1/q+q$\
6.90208 & $(-4,0)$ & $2$&$q+q^{3}+q^{5}t^{2}+q^{9}t^{3}$\
& $(2,2)$ & $-2$& $ 1/q^{11} t^4+1/q^9 t^3+1/q^7 t^3+1/q^7 t^2+1/q^5 t^2 +1/q^5t+1/q^3t + 1/q^3$\
&&& $+2/q+t/q+q^3t^2$\
6.90210 & $(0,2)$ & $0$ &$ 1/q^5t^2 + 1/qt+1/q + q + qt + q^5t^2$\
6.90211 & $(-2,0)$ & 0 &$1/q+q$\
6.90212 & $(-2,0)$ & 0 &$1/q+q$\
6.90213 & $(-4,-2)$ & $2$& $q+q^{3}+q^3t+q^{5}t^{2}+q^{7}t^{2}+q^{9}t^{3}+q^{9}t^{4}+q^{13}t^{5}$\
6.90214 & $(0,2)$ & $-2$ &$1/q^{13} t^5+1/q^{9} t^4+1/q^{9} t^3+1/q^{7} t^2+1/q^{5} t^2 +1/q^3 t+1/q^3 + 1/q $\
6.90215 & $(0,2)$ & $-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
6.90216 & $(0,2)$ & $-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
6.90217 & $(0,2)$ & 0 &$1/q+q$\
6.90218 & $(0,2)$ & 0 &$1/q+q$\
6.90219 & $(0,2)$ & 0 &$1/q+q$\
6.90220 & $(0,2)$ & $0$&$ 1/q^5t^2 + 1/qt+1/q + q + qt + q^5t^2$\
6.90221 & $(0,2)$ & 0 &$1/q+q$\
6.90222 & $(-2,2)$ & 0 &$1/q+q$\
6.90223 & $(-2,2)$ & 0 &$1/q+q$\
6.90224 & $(-2,2)$ & 0 &$1/q+q$\
6.90225 & $(-2,2)$ & 0 &$1/q+q$\
6.90226 & $(-2,2)$ & 0 &$1/q+q$\
& $(0,0)$ & $0$ &$ 1/q^{9} t^4+1/q^5 t^3+1/q^5 t^2+1/q^3 t+1/qt + 1/q+2q+qt+q^5t^2$\
6.90228 & $(0,2)$ & $-2$ &$2/q^{13} t^5+2/q^{9} t^4+1/q^{9} t^3+2/q^{7} t^2+1/q^{5} t^2 +2/q^3 t+1/q^3 + 1/q $\
6.90229 & $(0,2)$ & $-2$ &$1/q^{9}t^{3}+1/q^{5}t^{2}+1/q^{3}+1/q$\
6.90230 & $(-2,2)$ & 0 &$1/q+q$\
6.90231 & $(-2,2)$ & 0 &$1/q+q$\
6.90232 & $(0,2)$ & $0$ &$1/q^{9}t^{4}+1/q^{5}t^{3}+1/q^{3}t+1/q+2q$\
6.90233 & $(-2,2)$ & $0$& $ 1/q^5t^2 + 1/qt+1/q + q + qt + q^5t^2$\
6.90234 & $(-2,2)$ & 0 &$1/q+q$\
6.90235 & $(-2,2)$ & 0 &$1/q+q$\
|
---
abstract: 'A multi-epoch X-ray spectral and variability analysis is conducted for the narrow-line Seyfert 1 (NLS1) active galactic nucleus (AGN) Mrk 478. All available X-ray data from and satellites, spanning from 2001 to 2017, are modelled with a variety of physical models including partial covering, soft-Comptonisation, and blurred reflection, to explain the observed spectral shape and variability over the 16 years. All models are a similar statistical fit to the data sets, though the analysis of the variability between data sets favours the blurred reflection model. In particular, the variability can be attributed to changes in flux of the primary coronal emission. Different reflection models fit the data equally well, but differ in interpretation. The use of [reflionx]{} predicts a low disc ionisation and power law dominated spectrum, while [relxill]{} predicts a highly ionised and blurred reflection dominated spectrum. A power law dominated spectrum might be more consistent with the normal X-ray-to-UV spectral shape ($\alpha_{\rm ox}$). Both blurred reflection models suggest a rapidly spinning black hole seen at a low inclination angle, and both require a sub-solar ($\sim0.5$) abundance of iron. All physical models require a narrow emission feature at $6.7\kev$ likely attributable to emission, while no evidence for a narrow $6.4\kev$ line from neutral iron is detected.'
author:
- |
S. G. H. Waddell,$^{1}$[^1] L. C. Gallo,$^{1}$ A. G. Gonzalez,$^{1}$ S. Tripathi$^{1}$ and A. Zoghbi$^{2}$\
$^{1}$Department of Astronomy & Physics, Saint Mary’s University, 923 Robie Street, Halifax, Nova Scotia, B3H 3C3, Canada\
$^{2}$Department of Astronomy, University of Michigan, 1085 South University Avenue, Ann Arbor, MI 48109-1107, US\
bibliography:
- 'refs.bib'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: 'Multi-epoch X-ray spectral analysis of the narrow-line Seyfert 1 galaxy Mrk 478'
---
\[firstpage\]
galaxies: active – galaxies: nuclei – galaxies: individual: Mrk 478 – X-rays: galaxies
Introduction {#sect:intro}
============
Active galactic nuclei (AGN) are supermassive black holes that are actively accreting material. These objects are responsible for some of the most energetic phenomena in the Universe, and are typically highly variable across all wavelengths, from radio to $\gamma$-ray emission. The spectrum of these AGN will depend on the viewing angle through the obscuring torus (@Antonucci+1993 [@Urry+1995]). Seyfert 1 galaxies allow direct viewing of the central engine, whereas Seyfert 2 galaxies are viewed through the dusty torus.
Narrow-line Seyfert 1 (NLS1) galaxies are a sub-classification of Seyfert 1s that exhibit narrow Balmer lines and strong emission lines in the optical region (@Osterbrock+1985 [@Goodrich+1989]). They are thought to contain lower mass black holes which are accreting material at a high rate, close to the Eddington limit (e.g. @Mathur2000 [@Gallo+2018]). In the X-ray regime, where the hottest photons located in the innermost regions are emitted, spectra for different objects have distinct similarities. Above $\sim2\kev$, the spectrum is usually dominated by a power law, produced by Compton up-scattering of UV accretion disc photons within a hot corona. Other spectral features include a soft excess below energies of $2\kev$ of disputed origin, and broadened emission lines attributed to the emission line at $6.4\kev$ (e.g. @Fabian+1989).
Despite the common appearance in AGN spectra, the physical origin of the X-ray soft excess is uncertain, and a variety of mechanisms may be responsible. A variety of models have been suggested, including partial covering (e.g. @Tanaka+2004), soft-Comptonisation (e.g. @Done+2012) and blurred reflection (e.g. @RossFabian+2005). In the partial covering explanation, the soft excess could be an artifact of some obscuring material along the line-of-sight that absorbs some part of the intrinsic emission produced by the corona. As photons pass through this material, they are absorbed, producing deep absorption edges. Some of these photons are trapped by the Auger effect, but others are released, producing emission lines. The strengths of these lines are governed by the fluorescent yield of each element and the geometry of the absorbers around the primary emitter. Spectral variability can be explained by changes in the column density, covering fraction and ionisation of the absorbers without invoking changes in the intrinsic emission. The partial covering scenario has been used to explain the spectrum and variability found in low mass X-ray binaries (e.g. @Brandt+1996 [@Tanaka+2003]) and AGN (e.g. @Tanaka+2004 [@Miyakawa+2012; @Gallo+2015]). This model typically includes a strong absorption edge at $\sim7\kev$, a weak line at $6.4\kev$, and absorption at low energies. A variety of column densities, ionisation states (from neutral to highly ionised) structures, (e.g. opaque, partially transparent, patchy) and geometries can be invoked to explain the spectral curvature.
In the soft-Comptonisation model, while Comptonisation from the hot, primary corona produces the hard X-ray power law at energies above $\sim2\kev$, the soft excess can also be produced from Compton up-scattering of UV seed photons by a secondary, cooler corona. This secondary corona exists as a thin layer above the accretion disc, and is optically thick (e.g. @Done+2012). The resulting spectrum features a smooth soft excess due to the blending of the two power law components. This feature is seen in some AGN including Ark 120 (e.g. @Vaughan+2004 [@Porquet+2018]), Mrk 530 (@Ehler+2018), and Zw 229.015 (@zw229). A separate source of emission is then required to explain the presence of lines and other emission features.
In the blurred reflection model, some of the photons emitted from the corona will shine on the inner accretion disc, producing a reflection spectrum (e.g. @Fabian+2005). When the coronal photons strike the accretion disc, they are absorbed by atoms in the top layer, which produces absorption features. As the atoms de-excite, they undergo fluorescence, which produces a multitude of emission lines. This includes an emission line at $6.4\kev$ associated with emission. As the material in the innermost regions of the accretion disc rotates, it is subjected to the effects of general relativity, which broaden the intrinsically narrow emission lines and produces an excess of photons below $2\kev$. The resulting X-ray spectrum can show different levels of contributions from the reflection and primary emission components. If the X-ray emitting corona is sufficiently close to the black hole, more photons will be pulled back towards the black hole itself due to its extreme gravity in a process known as light bending (e.g. @Miniutti+2004), producing a reflection dominated spectrum. On the other hand, if the corona is moving away from the central region, the primary emission may be beamed towards the observer and produce a power law dominated spectrum. In a more typical system, the isotropic emission from the corona implies comparable contributions from the reflection and power law components, and reflection fraction (R) values are $\simeq1$.
Mrk 478 (PG1440+356; z=0.079) is a luminous, nearby NLS1 (@Zoghbi+2008). The source appears in some larger AGN surveys given its Palomar-Green and Markarian classifications (e.g. @Boroson+1992 [@Leighly1999; @Porquet+2004; @Grupe+2010]). [@Leighly1999] analyzed the (@ascains) data as part of a large sample of NLS1s, and found that the spectrum can be characterised with a power law of slope $\sim2$, a black body with temperature of $\sim120\eV$, and a weak, broad ($0.9\kev$) feature at $\sim6\kev$ attributed to emission. Mrk 478 was not detected in the BAT 105 month catalogue (@Oh+2018), indicating a low X-ray flux at high energies.
An 80ks (@chandrains) LETG exposure was obtained in 2000. The data were well fitted with a power law and Galactic absorption, and no significant narrow emission or absorption lines were found, suggesting a lack of obscuring material in the vicinity (@Marshall+2003). Comparing this result with a near simultaneous Medium Energy Concentrator Spectrometer (MECS; @saxins) observation, [@Marshall+2003] concluded that the soft excess is likely produced by Comptonisation of thermal emission from the accretion disc by a thin corona on top of the disc.
Mrk 478 was also the subject of four short ($\sim20\ks$) (@xmmins) observations over 13 months between 2001 and 2003. [@Guainazzi2004] found changes in $0.35-15\kev$ flux by a factor of $\sim6$ and that most spectral changes occurred above $3\kev$. They also suggested that the emission from the AGN could be explained with a double Comptonisation scenario. The soft and hard-band light curves from these observations were examined, and it was found that on short (hourly) timescales, the hard and soft bands varied together, while on longer timescales the soft flux varies more dramatically. [@Guainazzi2004] also noted that the while the soft X-ray flux is variable, the spectral shape does not significantly change between observations.
[@Zoghbi+2008] studied the same data, noting the same large flux variations as [@Guainazzi2004]. They also noted that the ratio of hard to soft photons was not dependent on changes in flux. In contrast to [@Marshall+2003], however, [@Zoghbi+2008] concluded that the spectrum is dominated by a blurred reflection component, which accounted for $\sim90$ per cent of the total observed X-ray emission in each observation. They found that a highly blurred spectrum originating from the inner region of the accretion disc around a Kerr black hole produced an excellent fit to the smooth soft excess. Curiously, they found that the model also requires a sub-solar iron abundance.
In this work, a multi-epoch analysis is conducted using all available data from and from 2001 to 2017. Only the average properties of these spectra are considered. For the longest exposure, obtained in 2017, flux-resolved spectral and timing analysis will be presented in a future work (Waddell in prep). A variety of spectral models are compared to attempt to explain the X-ray emission and describe the long-term spectral variations. In Section \[sect:data\], the observations used in the work are tabulated, and data reduction and modelling techniques are outlined. In Section \[sect:var\], long and short term light curves are discussed. Section \[sect:model\] outlines the different models used to fit the spectra (partial covering, soft-Comptonisation, and blurred reflection). Section \[sect:discussion\] presents a discussion of the results, and final conclusions are drawn in Section \[sect:conclusion\].
Observations and Data Reduction {#sect:data}
===============================
{#sect:xmm}
Mrk 478 has been observed a total of five times between 2001 and 2017 with (@xmmins) and all observations are shown in Table \[tab:obs\]. Spectra are referred to as XMM1 - XMM5. The Observation Data Files (ODFs) were processed to produce calibrated event lists using the Science Analysis System ([sas v15.0.0]{}). Light curves created from these event lists were then checked for evidence of background flaring. This was significant in XMM1 and XMM3, so good time intervals (GTI) were created and applied. Further corrections were applied to XMM3 after examining the $0.3-10\kev$ background light curve and finding high count rates in the second half of the observation. Only the first $12\ks$ are used for this observation, and this spectrum is only considered up to $9.0\kev$.
Evidence for pile-up on the order of $10$ per cent was also found in XMM1. Attempting to correct for this effect resulted in a loss of $\sim60$ per cent of detected photons, but did not noticeably affect the shape of the spectrum, so no corrections were made.
Source photons were extracted from a circular region with a $35\arcs$ radius centred on the source, and background photons were extracted from an off-source circular region with a $50\arcs$ radius on the same CCD. Single and double events were selected for the EPIC-pn (@pnins) detector products, while single through quadruple events were selected for the EPIC-MOS (@mosins) detectors. The [sas]{} tasks [rmfgen]{} and [arfgen]{} were used to generate response files. Light curves using the same source and background regions were extracted, and background corrected using [epiclccorr]{}.
[@Zoghbi+2008] determined that the spectra from MOS1 and MOS2 were comparable to the pn data for observations XMM1 through XMM4. We also compared the spectra from MOS1 and MOS2 to the pn spectra from XMM5, and determined these to be consistent. However, the effective areas of MOS1 and MOS2 are smaller than those of the pn detector, resulting in fewer counts from these instruments. Therefore, for simplicity, only data from the pn detector are considered for the remainder of this work.
------------- ---------------- ------ -------------- ---------- ---------- -------- --------------
(1) (2) (3) (4) (5) (6) (7) (8)
Observatory Observation ID Name Start Date Duration Exposure Counts Energy Range
(yyyy-mm-dd) \[s\] \[s\] \[keV\]
EPIC-pn 0107660201 XMM1 2001-12-23 32616 20390 113966 0.3-10.0
0005010101 XMM2 2003-01-01 27751 17230 94079 0.3-10.0
0005010201 XMM3 2003-01-04 29435 8371 39293 0.3-9.0
0005010301 XMM4 2003-01-07 26453 18180 49822 0.3-10.0
0801510101 XMM5 2017-06-30 135300 93710 291574 0.3-10.0
XIS0+3 706041010 SUZ 2011-07-14 170600 85323 42336 0.7-8.0
------------- ---------------- ------ -------------- ---------- ---------- -------- --------------
Data from the RGS instrument (@rgsins) were also obtained during this time. The data were reduced using the [sas]{} task [rgsproc]{}. First order spectra were obtained from RGS1 and RGS2, and were merged using the task [rgscombine]{}.
Optical Monitor (OM) (@omins) imaging mode data were obtained during all observations, including data from UVW2 in all epochs, and UVW1 in all epochs except XMM5. Data were reduced using the [sas]{} task [omichain]{}, from which the average count rate in each filter was obtained. Count rates were then converted to fluxes using standard conversion tables. These were then corrected for reddening using $E(B-V) = 0.011$ (@Willingale+2013).
{#sect:suz}
(@suzins) observed Mrk 478 in HXD (Hard X-ray Detector) nominal mode using the front illuminated (FI) CCDs XIS0 and XIS3, the back illuminated (BI) CCD XIS1, and the HXD-PIN detector. Cleaned event files from the HXD-PIN detector were processed using the tool [hxdpinxbpi]{}, yielding $72\ks$ of good time exposure. After considering both the instrumental and cosmic ray backgrounds, the observation resulted in a null detection, with a detection significance of only a few per cent.
Cleaned event files from the FI and BI CCDs were used for the extraction of data products in [xselect v2.4d]{}. For each instrument, source photons were extracted using a $240\arcs$ region centred around the source, while background photons were extracted from a $180\arcs$ off-source region. Calibration zones located in the corners of the CCDs were avoided in the background extraction. Response matrices for each detector were generated using the tasks [xisrmfgen]{} and [xissimarfgen]{}. The XIS0 and XIS3 detectors were first checked for consistency. Spectra were then merged using the task [addascaspec]{}. The resulting FI spectrum was found to be background dominated at $E > 8\kev$, so only data from $0.7-8.0\kev$ are considered. The regions $1.72-1.88\kev$ and $2.19-2.37\kev$ are also excluded from spectral analysis because of calibration uncertainties (@Nowak+2011). The merged FI data were checked for consistency against the BI spectrum and found to be consistent. Only the FI data are presented for simplicity.
{#sect:swift}
Mrk 478 has been observed with (@swiftins) XRT (@xrtins) a total of fifteen times between 2006 and 2017, with exposures ranging from $0.1-5\ks$. Data products are extracted using the web tool XRT data products generator[[^2]]{} (@Evans+2009) and the average count-rate from each observation is presented. The averaged, background subtracted spectrum created using all observations is modelled with a power law + black body, which over-fits the data ($\redchi = 0.74$). These parameters are used to obtain the $2-10\kev$ flux based on the count rate at each epoch using the [webpimms]{}[[^3]]{} tool. The spectrum has poor signal-to-noise due to the short exposure, and is not used in further spectral modelling.
Spectral Fits {#sect:fits}
-------------
All spectral fits are performed using [xspec]{} v12.9.0n (@xspec) from [heasoft]{} v6.26. Spectra from the EPIC-pn and FI detectors have been grouped using optimal binning (@optbin), using the [ftool]{} [ftgrouppha]{}. All spectra are background subtracted. Model fitting was done using C-statistic (@Cash1979). All model parameters are reported in the rest frame of the source, but figures are shown in the observed frame. The Galactic column density in the plane of Mrk 478 is kept frozen at $1.08\times10^{20}\cm^{-2}$ (@Willingale+2013) for all models, and elemental abundances used for this parameter only are taken from [@Wilms+2000].
All models were originally fit with parameters free to vary between all data sets. However, because of the low exposure, high background, and overall poor data quality of XMM3, many parameters could not be well constrained at this epoch. Consequently, all parameters for XMM3 are linked to those of XMM1, since the two spectra are almost identical in shape (see Fig. \[fig:eeuf\_plext\]). A constant is left free to vary between the data sets to account for the small flux differences. This technique does not result in any significant change in fit quality or parameters for any model.
To determine parameter errors from spectral fits, Monte Carlo Markov Chain (MCMC) techniques are employed using [xspec\_emcee]{}[[^4]]{}. After obtaining the best-fitting parameters, MCMC calculations are run. The Goodman-Weare (@GW+2010) algorithm is used, and each chain is run with at least twice as many walkers (64 walkers) as there are free parameters to ensure sufficient sampling. Chain lengths are set to be at least 10000. A burn-in phase of 1000 is selected to ensure that no bias is introduced based on the starting parameters. All errors on parameters are quoted at the $90$ per cent confidence level.
Characterising the Variability {#sect:var}
==============================
Light curves for the $2001-2003$ (XMM1–XMM4) observations in the $0.2-10\kev$ band using $200\s$ bins were presented in [@Zoghbi+2008]. Here, light curves for all the observations (XMM1–XMM5) are created in the $0.3-10\kev$ band, to match energy ranges used in spectral modelling, using $1000\s$ bins. Additionally, the light curve is created between $0.7-10\kev$, using a bin size of $5760\s$ corresponding to the orbit.
The shapes of light curves are similar to those of [@Zoghbi+2008]. Variations in the count rate are on the order of $15-20$ per cent about the average during the short exposures (< $20\ks$, XMM1 through XMM4), and on the order of $40$ per cent for the longer XMM5. The light curve shows significant deviations of $\sim80$ per cent from the mean, over the course of days.
Normalized hardness ratios were presented by [@Zoghbi+2008] for XMM1 and XMM4 using $S=0.3-2\kev$ and $H=3-10\kev$, with $200\s$ binning. The ratios were found to be consistent with a constant over the course of the observations. Hardness ratios are calculated here using $HR = H/S$, where $S=0.3-2\kev$ and $0.7-2\kev$ for and , respectively, and $H=2-10\kev$. The same binning as in the light curves is used for the hardness ratios. For all observations, variations in $HR$ are on the order of $10-20$ per cent. Given the modest hardness ratio variability within each observation, the average SUZ and XMM5 spectra are used in the analysis, despite their longer length.
The $2-10\kev$ light curve for Mrk 478 spanning from 1997 to 2017 is presented in Fig. \[fig:lcdiv\] making use of data from , , , and the Rossi X-ray Timing Explorer (). The data were taken from the AGN Timing & Spectral Database[[^5]]{} (@Breedt+2009 [@Rivers+2013]). A total of nine short exposures were taken between August 16-21, 1997, lasting between $1-10\ks$ each.
The average flux between $1997-2017$ is approximately $2.88\times10^{-12}\ergpscmps$. Between the dimmest and brightest observations, variations by a factor of $\sim5$ are seen, and deviations from the mean are on the order of $80$ per cent at the extremes. By comparing each observation to the average flux, it is apparent that XMM2 is found at a slightly higher than average flux state, XMM1, XMM3 and SUZ are at average flux states, and XMM4 and XMM5 are at a dimmer flux level. Overall, the variability is less significant than is seen in more extreme NLS1s, like Mrk 335 (e.g. @Gallo+2019m335 [@Wilkins+2015]) and IRAS 13224-3809 (e.g. @Alston+2019).
To assess the UV-to-X-ray shape at each epoch, the hypothetical power law between $2500$ angstrom and $2\kev$ ($\alpha_{\rm ox}$; @Tananbaum+1979) is calculated. Comparing the measured $\alpha_{\rm ox}$ to the expected value given the UV luminosity ($\alpha_{\rm ox}(L_{2500}$); @Vagnetti+2013) reveals the X-ray weakness parameter ($\Delta \alpha_{\rm ox} = \alpha_{\rm ox} - \alpha_{\rm ox}(L_{2500}$)), where negative values indicate X-ray weak sources and positive values imply X-ray strong sources relative to the UV luminosity.
The resulting $\Delta \alpha_{\rm ox}$ values range from $-0.07$ for XMM2 to $-0.17$ for XMM4, but are comparable within uncertainties. All values are negative, perhaps implying that all epochs are slightly X-ray weak. [@Gallo2006] suggests that more X-ray weak sources (negative $\Delta \alpha_{\rm ox}$ values) are likely to be accompanied with increased spectral complexity and are more reflection dominated or more highly absorbed. However, the distribution on the expected $\alpha_{\rm ox}$ is large (@Vagnetti+2013), so all values measured for Mrk 478 agree with one another and with $\Delta \alpha_{\rm ox}\approx0$. This implies that Mrk 478 is in an X-ray normal state (e.g. @Gallo2006). Therefore, extreme spectra that are highly absorbed or highly dominated by reflection are not expected.
![Long term light curve between $1997-2017$ for Mrk 478 with data from (black diamonds), (green triangles), (red circles) and (blue squares). The $2-10\kev$ flux is shown on the y-axis, and the average flux is shown as a dashed black line.[]{data-label="fig:lcdiv"}](jun13_longlc.png){width="\columnwidth"}
Interestingly, both the short and long term light curves, as well as the calculated $\alpha_{\rm ox}$ values, can be interpreted in similar fashion. First, all are suggestive of a relatively X-ray “normal” spectrum, without extreme flux or spectral changes. This is true on timescales of days (within observations) and throughout the 20 years of observation. This suggests a lack of excessive spectral complexity produced by complicated absorption or blurred reflection. Secondly, the variability on short and long time scales is likely simple, produced mainly by normalisation changes. The multi-epoch analysis should provide good constraint on parameters that are not expected to vary between observations.
Spectral Modelling {#sect:model}
==================
Spectral Characterisation {#sect:phen}
-------------------------
To obtain an initial assessment of differences between spectra, data from each epoch are unfolded against a power law with $\Gamma=0$. This allows data from different instruments to be compared on the same plot. The result is shown in the top panel of Fig. \[fig:eeuf\_plext\], and data have been re-binned in [xspec]{} for clarity. The colours and symbols used in Fig. \[fig:eeuf\_plext\] are adopted throughout the remainder of this work. The shape of the spectra do vary between observations, however, most of the changes are likely attributable to normalisation changes between lower flux level spectra (XMM4 and XMM5) and higher levels (XMM1, XMM2, XMM3 and SUZ), in agreement with the findings of [@Guainazzi2004].
![**Top panel:** Data from each epoch unfolded against a power law with $\Gamma=0$. The shape of all spectra are comparable, with XMM4 and XMM5 being dimmer than other epochs. **Bottom panel:** Residuals from a power law fit from $2-4$ and $7-10\kev$, extrapolated over the $0.3-10\kev$ band. All data sets are fit with the average power law, and scaled by a constant to account for flux variations between observations. A strong soft excess below $2\kev$ and excess residuals between $5-7\kev$ are evident in all data sets.[]{data-label="fig:eeuf_plext"}](jul4_eeuf_plext.png){width="\columnwidth"}
In an attempt to characterise the and spectra, the data are fit from $2-10\kev$, excluding the $4-7\kev$ band where iron emission is typically detected, with a single average power law. A constant factor is applied to account for changes in flux between the different spectra. The model is then extrapolated over the full usable energy range for each data set and the ratio (data/model) is shown in the bottom panel of Fig. \[fig:eeuf\_plext\]. A smoothly rising soft excess below $\sim2\kev$ is evident in all data sets, as well as some evidence for excess residuals in the $4-7\kev$ band. This also shows some spectral variability between observations, as the spectra appear softer when brighter, which is typical of X-ray binaries and AGN.
To model the soft excess, a black body is added to the spectrum from each epoch. The soft excess can be characterised by a black body with temperatures of $\sim90\eV$ at all epochs, and a steep photon index of $\sim2.4$. However, the data are not well fitted by the model, with significant curvature around $1\kev$ and in the $4-10\kev$ band. Replacing the black body component with a secondary power law improves the fit, but is still unable to explain the overall shape and does not fit the spectrum well. For either model, adding a Gaussian emission feature results in a good fit to the residuals in the $4-7\kev$ band. The feature has a best fit energy of $\sim6.7\kev$ and a width of $\sim0.5\kev$. This may indicate a broad line profile, however the high best fit energy may also indicate the presence of ionised iron emission lines at $6.7\kev$ () and $6.97\kev$ (). More physical models will be examined to explain the observed spectrum.
The RGS data from the long exposure in 2017 (XMM5) are also modelled with a power law and Galactic absorption. The best fit photon index for the power law component is $\Gamma=2.83\pm0.03$. No strong emission or absorption lines are detected in the $0.3-2\kev$ range. The lack of any strong emission or absorption features in the RGS data are consistent with the LETG observation in 2000 (@Marshall+2003).
{width="\textwidth"}
Partial Covering {#sect:pc}
----------------
Partial covering has been used to successfully describe the soft excess and high energy curvature in NLS1 X-ray spectra (e.g. @Tanaka+2004 [@Gallo+2015]). To find a partial covering model to explain the observed spectral shape and variability, all data sets are first modelled with a single, constant power law modified by neutral absorption ([zpcfabs]{}). The redshift of the absorber is the same as the host galaxy. To describe the spectral changes, the column density and covering fraction of the absorber are free to vary between data sets. This gives $C = 1998$ for 557 degrees of freedom (dof). The model underestimates the data at $\sim4\kev$ and between $7-10\kev$. Allowing the photon index to vary between epochs improves the fit; $\Delta C = 129$ for 4 additional free parameters, but $\Gamma$ remains comparable ($\sim2.9$) for all data sets.
To explain the residuals left by the single partial covering model, a second [zpcfabs]{} component is added. The column density and covering fraction are again left free to vary between observations. This improves the fit, with $\Delta C = 1244$ for 10 additional free parameters. The analysis suggests two distinct absorbers, both with very different covering fractions and column densities. One absorber has a column density of $\sim4-7\times10^{22}$ cm$^{-2}$ and covering fraction of $\sim0.4$, while the other has column densities of $\sim80\times10^{22}$ cm$^{-2}$ and high covering fraction of $\sim0.7$. This higher density absorber produces a deep edge, which fits the data much better at high energies; however, some excess residuals are still visible at low energies in all data sets.
One neutral component is then replaced with an ionised absorber ([zxipcf]{}). This model includes the column density and covering fraction of the absorber, as well as an ionisation parameter ($\xi = 4\pi F/n$, where F is the illuminating flux and n is the hydrogen number density of the absorber). The redshift of this absorber is again fixed to that of the host galaxy. Fitting the data gives $C = 676$ for 544 dof. Although the ionisation parameter is low, the use of [zxipcf]{} results in much lower column densities for the secondary absorber, causing the large change in fit statistic. Replacing the remaining neutral absorber with a second ionised absorber does not improve the fit, so the combination of one neutral and one ionised absorber is maintained.
To describe the variability, various parameter combinations are examined. The best fit is obtained when the ionisation parameter is linked between epochs, but the column density and covering fractions of both absorbers are left free to vary. The slope and normalisation of the power law component are also kept linked between epochs.
A close examination of the residuals reveals some emission in the $6-7\kev$ range. A narrow emission line is added. The width is kept fixed at $1\eV$, and the line energy and normalisation are left free to vary, but kept linked between data sets. This again improves the fit, $\Delta C = 9$ for 2 additional free parameters. No signatures of a $6.4\kev$ emission feature are detected. The best fit energy is at $\sim6.7\kev$, indicating the presence of ionised iron emission as suggested in Section \[sect:phen\].
![Correlations between best fit parameters using the partial covering model. While no linear trends are observable between data sets, the column density and covering fraction are higher for the dimmer flux XMM4 and XMM5 than for other data sets.[]{data-label="fig:pcfcorr"}](jun26_pcfabs_corr.png){width="\columnwidth"}
The best fit parameters and MCMC errors for this final model are presented in Table \[tab:covering\], and the model and residuals are shown in the top-left corner of Fig. \[fig:model\]. Parameter correlations for the best fit model are presented in Fig. \[fig:pcfcorr\]. The intrinsic power law (shown in Fig. \[fig:model\]) is extremely steep, with $\Gamma = 2.99\pm0.02$. The ionisation parameter of [zxipcf]{} is low, poorly constrained, and consistent with neutral material. The neutral absorber ([zpcfabs]{}) has a low column density and low covering fraction, and displays only limited variability between epochs.
The ionised absorber displays more significant variability; in particular, the covering fraction and density are both higher for the dimmer XMM4 and XMM5 observations, and are lower for the brightest data set (XMM2). The higher overall densities and covering fractions suggest that this component is driving the shape and variability of the observed spectra rather than changes to the intrinsic power law emission or neutral absorption.
[c c c c c c c c]{} (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8)\
Model & Parameter & XMM1 & XMM2 & XMM3 & XMM4 & XMM5 & SUZ\
\
Constant & scale factor & $1^{f}$ & - & $0.96\pm0.02$ & - & - & -\
Intrinsic & $\Gamma$ & $2.99\pm0.02$ & - & - & - & - & -\
Power Law & norm ($\times10^{-2}$) & $1.1\pm0.1$ & - & - & - & - & -\
Neutral Absorber & nH ($\times10^{22}$ cm$^{-2}$) & $7\pm2$ & $3\pm1$ & $7^{l}$ & $4\pm1$ & $3.6\pm0.5$ & $7\pm2$\
([zpcfabs]{}) & CF & $0.43\pm0.07$ & $0.36\pm0.10$ & $0.43^{l}$ & $0.47\pm0.07$ & $0.50\pm0.02$ & $0.49\pm0.06$\
Ionised Absorber & nH ($\times10^{22}$ cm$^{-2}$) & $39\pm12$ & $19\pm6$ & $39^{l}$ & $43\pm9$ & $43\pm7$ & $38\pm11$\
([zxipcf]{}) & log($\xi$) (erg cm s$^{-1}$) & $0.4\pm0.5$ & - & - & - & - & -\
& CF & $0.62\pm0.06$ & $0.65\pm0.06$ & $0.62^{l}$ & $0.78\pm0.04$ & $0.75\pm0.03$ & $0.62\pm0.06$\
Ionised Iron & E () & $6.7\pm0.3$ & - & - & - & - & -\
Emission & $\sigma (\eV)$ & $1^{f}$ & - & - & - & - & -\
& Flux ($\times10^{-14}$ erg cm$^{-2}$ s$^{-1}$) & $1.4^{+1.5}_{-1.4p}$ & - & - & - & - & -\
Flux & F$_{\rm 0.3-10}$ ($\times10^{-11}$ erg cm$^{-2}$ s$^{-1}$) & $1.23\pm0.02$ & $1.33\pm0.03$ & $1.18\pm0.02$ & $0.72\pm0.02$ & $0.76\pm0.01$ & $1.11\pm0.02$\
Fit Statistic & C/dof & 667/544 & - & - & - & - & -\
Another aspect of the absorption models is that they do not include the emission lines associated with the included absorption edges. The RGS data are consistent with the best-fitting model, but none of the expected emission lines are present in the RGS spectrum. Most notably, the $6.4\kev$ emission line associated with the edge at $\sim7\kev$ is not included in the model. Assuming the obscuring sources are spherically symmetric, the predicted strength of the line can be calculated by measuring the absorption strength of the edge (i.e., the drop in flux in the $7-20\kev$ range), and multiplying it by the fluorescent yield of iron (e.g. @Reynolds+2009). If one assumes that only the neutral absorber is responsible for producing the $6.4\kev$ feature, the resulting emission line is very weak, with an equivalent width of $4\eV$ using the absorption of the brightest spectrum (XMM2) and $5\eV$ using the dimmest spectrum (XMM4). No negative residuals are seen, suggesting that this line strength is consistent with a null detection.
It is also, however, interesting to consider the ionisation on the other absorber. This value is in agreement with neutral, so this absorber would also contribute to the line profile. This increases the equivalent width of the line, to $90\eV$ using the absorption of the brightest observations, and to $200\eV$ using the absorption of the dimmest. Both cases show significant negative residuals compared to the best-fitting model, implying that these lines should have been easily detectable in the observed spectrum. This suggests that non-spherically symmetric absorption is required to produce the observed spectra.
Comptonisation {#sect:optx}
--------------
The soft-Comptonisation (e.g. @Done+2012) is characterised by a smooth soft excess as observed in the residuals in Fig. \[fig:eeuf\_plext\]. This model was also suggested to explain the lack of features in the soft excess observed with the LETG detector by [@Marshall+2003], and for the early observations by [@Guainazzi2004].
To test this physical interpretation, the model [optxagnf]{} (@Done+2012) is used. This model describes the X-ray spectrum above $2\kev$ with a power law from a hot, spherical corona centred around the black hole. The soft X-ray spectrum is described with a second, cooler, optically thick corona located on top of the accretion disc. Emitted power is supplied by the energy released through accretion. The mass and accretion rate are fixed at values from [@Porquet+2004], and the co-moving distance is set to 347 Mpc. Other free parameters include the radius of the primary, spherical corona (r$_{\rm cor}$), the temperature (kT) and opacity ($\tau$) of the soft X-ray corona, the slope of the hard Compton power law ($\Gamma$), and the fraction of power emitted in the hard Comptonisation component below r$_{\rm cor}$ (fpl).
A variety of combinations of these parameters are tested to attempt to explain the variability between data sets. Allowing only the soft corona parameters (kT and $\tau$) or only the hard corona parameters ($\Gamma$ and fpl) to vary did not reproduce the spectral shape, with $C = 1045$ and $C = 1858$ for 554 dof respectively. Therefore, all four parameters are left free to vary between epochs. Allowing the radius of the primary corona to be free between data sets causes it to become unconstrained, so it is linked between epochs. Spin values ($a = cJ/GM^2$, where $M$ is the black hole mass and $J$ is the angular momentum) fixed at 0, 0.5 and 0.998 are tested, with a maximum spin giving the best fit.
As with the partial covering model, excess residuals are visible in the $6-7\kev$ band, so a narrow ($\sigma = 1\eV$) Gaussian emission line is added. The best fit line energy and normalisation are free to vary, but are again linked between epochs. This improves the fit by $\Delta C = 30$ for 2 additional free parameters, for a final fit statistic of $C = 609$ for 546 dof. Again, no signatures of $6.4\kev$ emission are detected - the feature has a best fit energy of $6.6\pm0.1\kev$, consistent with the line energy found in the partial covering model.
[c c c c c c c c]{} (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8)\
Model & Parameter & XMM1 & XMM2 & XMM3 & XMM4 & XMM5 & SUZ\
\
Constant & scale factor & $1^{f}$ & - & $0.96\pm0.02$ & - & - & -\
Soft-Comptonisation & Mass ($\times10^{7}$ ) & $1.99^{f}$ & - & - & - & - & -\
([optxagnf]{}) & Distance (Mpc) & $347^{f}$ & - & - & - & - & -\
& log(L$_{\rm edd}$) & $-0.027^{f}$ & - & - & - & - & -\
& a & $0.998^{f}$ & - & - & - & - & -\
& r$_{\rm cor}$ (r$_{\rm g}$) & $64\pm5$ & - & - & - & - & -\
& log(r$_{\rm out}$) & $3^{f}$ & - & - & - & - & -\
& kT () & $0.26\pm0.03$ & $0.23\pm0.02$ & $0.26^{l}$ & $0.6\pm0.2$ & $0.57\pm0.12$ & $0.28\pm0.05$\
& $\tau$ & $12.1\pm0.8$ & $13.2\pm0.8$ & $12.1^{l}$ & $7.2\pm1.2$ & $7.1\pm0.8$ & $11.5\pm1.3$\
& $\Gamma$ & $2.23\pm0.05$ & $2.18\pm0.03$ & $2.23^{l}$ & $2.0\pm0.1$ & $2.01\pm0.06$ & $2.12\pm0.05$\
& fpl & $0.13\pm0.01$ & $0.137\pm0.008$ & $0.13^{l}$ & $0.056\pm0.005$ & $0.061\pm0.003$ & $0.100\pm0.008$\
Ionised Iron & E () & $6.6\pm0.1$ & - & - & - & - & -\
Emission & $\sigma (\eV)$ & $1^{f}$ & - & - & - & - & -\
& Flux & $1.5\pm0.8$ & - & - & - & - & -\
&($\times10^{-14}$ erg cm$^{-2}$ s$^{-1}$) & & & & & &\
Flux & F$_{\rm 0.3-10}$ & $1.23\pm0.04$ & $1.33\pm0.03$ & $1.18\pm0.03$ & $0.7\pm0.3$ & $0.8\pm0.3$ & $\sim1.12$\
& ($\times10^{-11}$ erg cm$^{-2}$ s$^{-1}$) & & & & & &\
Fit Statistic & C/dof & 609/546 & - & - & - & - & -\
The best fit model is shown in the top right panel of Fig. \[fig:model\], where the effects of Galactic absorption have been removed for display. Best fit parameters are listed in Table \[tab:optxagnf\], and correlations between parameters are shown in Fig. \[fig:optxcorr\]. All parameters appear highly correlated with one another, and all change according to the flux of each spectrum. For the dimmer XMM4 and XMM5, the temperature of the secondary corona increases, while the opacity, slope of the hard power law, and fpl all decrease compared to the brighter data sets. All parameters are well constrained and vary significantly between the brighter and dimmer flux epochs. Despite the AGN being at a bright flux during the SUZ epoch, the parameter values are more intermediate to the high and low states. This could be due to the more limited band pass studied with .
![Correlations between best fit parameters using the Comptonisation model. Clear correlations are present between all parameters, and changes between brighter and dimmer flux epochs are evident in all parameters.[]{data-label="fig:optxcorr"}](jun30_optx_corr.png){width="\columnwidth"}
Another interesting test of the [optxagnf]{} model is to take the best fit model to the X-ray data and apply it to optical/UV data, as this model is intended for broad SED fitting. To do so, the tool [ftflx2xsp]{} is used to build dummy response files suitable for use in [xspec]{} for the available UVW1 and UVW2 data. Only these filters are used to avoid potential host galaxy contamination. We then extrapolate the best-fitting models for both a maximum spin ($a=0.998$) and non-spinning ($a=0$) black hole to the UV data and examine the fit.
![All available UVW1 and UVW2 data for all epochs compared to the a = 0 (dashed) and a = 0.998 (solid) best fit Comptonisation models. Colours and shapes match those of the corresponding X-ray data sets. Y-axis error bars on the UV data are shown, but are smaller than the symbols. The best fit models to the X-ray data underestimate the data when extrapolated to UV energies.[]{data-label="fig:optxuv"}](jun26_optx_uv.png){width="\columnwidth"}
The result is shown in Fig. \[fig:optxuv\]. The colours and symbols used on the optical data match those of the corresponding X-ray spectra. No simultaneous optical data is available for the data. The best fit models at spins of 0.998 and 0 are shown as black solid and dashed lines, respectively. Models for all data sets are the same at this low energy range, so only one model line is shown for each spin value. Neither model is able to reproduce the shape or flux of the data at these low energies. Both underestimate the UV flux, by a factor of $\sim5$ for the UVW2 data and $\sim10$ for UVW1, assuming maximum spin, and a more modest factor of $\sim3.5$ for UVW2 and $\sim5$ for UVW1 assuming no spin. Allowing the Eddington luminosity to go free results in a best-fitting super-Eddington accretion rate (log(L/L$_{\rm edd}$)$ \simeq0.14$) and does not improve the fit to the UV data. Similarly, simultaneous modelling of the UV and X-ray data fails to find a fit which explains the UV flux. This shows that the soft-Comptonisation model is unable to explain the observed SED for Mrk 478.
Blurred Reflection {#sect:refl}
------------------
In the blurred reflection model, the intrinsic power law is seen alongside a reflection spectrum, produced when X-ray photons from the corona strike the inner accretion disc. This model has been used successfully to explain the spectral properties and variability of numerous NLS1 galaxies (e.g. @Fabian+2004 [@Ponti+2010; @Gallo+2019m335]). It has also been used to explain time domain variability and lags (e.g. @Wilkinslags). This interpretation was also discussed in detail by [@Zoghbi+2008], who found that a highly blurred, highly ionised, reflection dominated model explained the spectra of XMM1 - XMM4.
To test the blurred reflection interpretation, two different models are used; [relxill]{} version 1.2.0 (@relxillmodel) and [reflionx]{} (@Ross+1999 [@RossFabian+2005]). [reflionx]{} was convolved by the blurring model [kerrconv]{} (@kerrmodel). [reflionx]{} was combined with a more simplistic blurring model [kdblur]{} for analysis in [@Zoghbi+2008]. Both of these models are combined with a power law to model the intrinsic coronal emission.
To measure the reflection fraction, the convolution model [cflux]{} was applied to both the reflection and power law components. The flux parameter was linked between the two models. The relative fluxes of the models were then determined by adding a constant between the reflection and power law components. This constant measures the reflection fraction, R, the ratio of emitted flux in the reflection and power law components in the $0.1-100\kev$ band.
Parameters are the same between models. Both include two emissivity index values q$_{\rm in}$ and q$_{\rm out}$, separated at a break radius r$_{\rm br}$. These parameters define the illumination pattern, which goes as $\epsilon \propto r^{-q}$. The inner emissivity index is free to vary between epochs, while the break radius and outer emissivity index are kept fixed at 6 r$_g$ and 3, respectively. The inner radius of the accretion disc is kept fixed at the innermost stable circular orbit (ISCO), while the outer radius is fixed at 400 r$_{g}$, as little emission is expected to originate from outside this radius. The spin ($a$), inclination and iron abundance are left free to vary, but linked between epochs, as none are expected to vary within the given timescales. The [xspec]{} function [steppar]{} is also run on these parameters to ensure that none are confined to local minima.
The flux produced by the power law component is left free to vary between epochs, as is the photon index ($\Gamma$). The photon index of the reflection component is linked to that of the power law, as leaving it free to vary does not improve the fit. Various combinations between other parameters are tested, and it is found that the best fit is produced when the reflection fraction and ionisation (again defined as $\xi = 4\pi F/n$) are left free to vary between observations, although the changes in these parameters are limited within uncertainties between epochs.
As in the other models, a narrow feature is added to fit the residuals apparent between $6-7\kev$. The best fit line energies are identical for both models, at $6.6\pm0.1\kev$. Once again, no evidence for narrow $6.4\kev$ emission is detected. As such, no distant reflection model to account for reflection off of the neutral torus is added to the model.
The best fit parameters are shown in Table \[tab:reflection\], and the models and residuals are shown in the bottom two panels of Fig. \[fig:model\]. Both models produced comparable fits and have the same degrees of freedom (539). The C-statistic is $593$ and $606$ for the [relxill]{} and [reflionx]{} models, respectively. However, each model gives a different interpretation of the X-ray emitting region.
[c c c c c c c c]{} (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8)\
Model & Parameter & XMM1 & XMM2 & XMM3 & XMM4 & XMM5 & SUZ\
\
Constant & scale factor & $1^{f}$ & - & $0.89\pm0.06$ & - & - & -\
Power Law & $\Gamma$ & $2.73\pm0.06$ & $2.7\pm0.1$ & $2.73^{l}$ & $2.6\pm0.1$ & $2.65\pm0.06$ & $2.8\pm0.1$\
& E$_{\rm cut} (\kev)$ & $300^{f}$ & - & - & - & - & -\
& log(F$_{0.1-100}$) & $-10.67\pm0.05$ & $-10.66\pm0.09$ & $-10.67^{l}$ & $-11.0\pm0.1$ & $-10.91\pm0.04$ & $-10.8\pm0.1$\
Blurring & R$_{\rm in}$ (ISCO) & 1$^{f}$ & - & - & - & - & -\
([kerrconv]{}) & R$_{out}$ (r$_{g}$) & 400$^{f}$ & - & - & - & - & -\
& R$_{br}$ (r$_{g}$) & 6$^{f}$ & - & - & - & - & -\
& q$_{\rm in}$ & $8\pm1$ & $9.0^{+0.8}_{-1.4}$ & $8^{l}$ & $9.2^{+0.8p}_{-1.4}$ & $8.9\pm1.0$ & $9.4^{+0.6p}_{-1.2}$\
& q$_{\rm out}$ & 3$^{f}$ & - & - & - & - & -\
& a & $0.94\pm0.02$ & - & - & - & - & -\
& i () & $<22$ & - & - & - & - & -\
Reflection & $\xi$ (erg cm s$^{-1}$) & $63\pm35$ & $55\pm40$ & $63^{l}$ & $<165$ & $40\pm22$ & $85\pm63$\
([reflionx]{}) & A$_{\rm Fe}$ (Fe/solar) & $0.44\pm0.28$ & - & - & - & - & -\
& R$_{0.1-100}$ & $0.7\pm0.3$ & $0.8\pm0.3$ & $0.7^{l}$ & $1.0\pm0.5$ & $0.7\pm0.1$ & $1.3\pm0.6$\
Ionised Iron & E () & $6.6\pm0.1$ & - & - & - & - & -\
Emission & $\sigma (\kev)$ & $0.001^{f}$ & - & - & - & - & -\
& Flux ($\times10^{-14}$ erg cm$^{-2}$ s$^{-1}$) & $1.3\pm0.8$ & - & - & - & - & -\
Flux & F$_{\rm 0.3-10}$ ($\times10^{-11}$ erg cm$^{-2}$ s$^{-1}$) & $1.23\pm0.02$ & $1.35\pm0.03$ & $1.18\pm0.03$ & $0.7\pm0.1$ & $0.76\pm0.02$ & $1.2\pm0.3$\
Fit Statistic & C/dof & 606/539 & - & - & - & - & -\
\
Constant & scale factor & $1^{f}$ & - & $0.96\pm0.02$ & - & - & -\
Power Law & $\Gamma$ & $2.51\pm0.04$ & $2.47\pm0.06$ & $2.51^{l}$ & $2.5\pm0.1$ & $2.42\pm0.03$ & $2.6\pm0.2$\
& E$_{\rm cut} (\kev)$ & $300^{f}$ & - & - & - & - & -\
& log(F$_{0.1-100}$) & $-11.1\pm0.2$ & $-11.0\pm0.1$ & $-11.1^{l}$ & $-11.2\pm0.1$ & $-11.25\pm0.08$ & $-11.0\pm0.2$\
Blurred & R$_{\rm in}$ (ISCO) & $1^{f}$ & - & - & - & - & -\
Reflection & R$_{\rm out}$ (r$_{g}$) & $400^{f}$ & - & - & - & - & -\
([relxill]{}) & R$_{br}$ (r$_{g}$) & $6^{f}$ & - & - & - & - & -\
& q$_{\rm in}$ & $6.9\pm0.6$ & $7.3\pm0.8$ & $6.9^{l}$ & $8.0\pm1.6$ & $7.8\pm0.7$ & $8.7^{+1.3p}_{-1.8}$\
& q$_{\rm out}$ & $3^{f}$ & - & - & - & - & -\
& a & $0.98\pm0.01$ & - & - & - & - & -\
& i () & $31\pm8$ & - & - & - & - & -\
& log($\xi$) (erg cm s$^{-1}$) & $3.1\pm0.1$ & $2.8\pm0.2$ & $3.1^{l}$ & $2.8\pm0.3$ & $2.9\pm0.1$ & $2.7\pm0.4$\
& A$_{\rm Fe}$ (Fe/solar) & $0.84\pm0.14$ & - & - & - & - & -\
& R$_{0.1-100}$ & $3.5\pm1.5$ & $2.3\pm0.9$ & $3.5^{l}$ & $2.0\pm0.9$ & $2.4\pm0.6$ & $3.1\pm1.7$\
Ionised Iron & E & $6.6\pm0.1$ & - & - & - & - & -\
Emission & $\sigma \kev$ & $0.001^{f}$ & - & - & - & - & -\
& Flux ($\times10^{-14}$ erg cm$^{-2}$ s$^{-1}$) & $1.5\pm0.8$ & - & - & - & - & -\
Flux & F$_{\rm 0.3-10}$ ($\times10^{-11}$ erg cm$^{-2}$ s$^{-1}$) & $1.2\pm0.4$ & $1.33\pm0.06$ & $1.2\pm0.4$ & $0.71\pm0.04$ & $0.8\pm0.2$ & $1.2\pm0.1$\
Fit Statistic & C/dof & 593/539 & - & - & - & - & -\
To begin comparing the two models, correlation plots are made using the best fitting free parameters. The results are presented in Fig. \[fig:refcorr\], with [relxill]{} parameters shown on the left and [reflionx]{} parameters shown on the right. Axes between figures are not identical, but corresponding cells in each panel show correlations between the same parameters. The colours and styles of the data points match those used throughout to represent data sets. Points for XMM3 are not shown, as these parameters have all been linked to XMM1 and are not independently measured.
The inner emissivity is in agreement, although poorly constrained, between both models for all data sets. For the power law component, photon indices are generally steeper using [reflionx]{} than for [relxill]{}, and do not agree within error. This is also true for the power law fluxes, which are generally higher (brighter) for [reflionx]{}. For XMM4 and SUZ, however, both of these values agree between models, as these data sets most poorly constrain the parameters. None of the spectra are as steep as the value measured using partial covering, and are significantly steeper than when using soft-Comptonisation.
Despite the fact that not all variable parameters agree between the two reflection models represented here, the parameters kept constant between data sets are in agreement with one another; notably black hole spin ($a$), inclination and iron abundance. This is demonstrated in Fig. \[fig:relvsref\_corner\]. Data for [relxill]{} are shown in blue and data from [reflionx]{} are in pink. Contours are produced using 68, 90 and 99 percent of MCMC fits. The histograms reveal that all parameters are evenly distributed around the measured mean. The contours also reveal that all parameters are in agreement, and confirm that the reflection interpretation requires a high spin, low inclination AGN.
It is in the interpretation of the reflection fraction and ionisation parameters that the two reflection models are mostly extremely in disagreement. [relxill]{} suggests a highly ionised spectrum, with $\xi\simeq1000\ergpscmps$, while [reflionx]{} predicts a lower value of $\xi\simeq50\ergpscmps$. Setting the ionisation parameters of [relxill]{} to the ones found by [reflionx]{} and vice versa degrade the fit quality by $\Delta C = 146$ and $\Delta C = 40$, respectively. For the reflection fraction, use of [relxill]{} suggests a reflection dominated spectrum, with all reflection fractions band constrained to be larger than one, and the largest being less than 3.5. However, using [reflionx]{} suggests a power law dominated spectrum, or one with equal contributions from the intrinsic power law and reflection spectra, with R values constrained between $\sim0.7-1.3$ for all spectra.
As seen in Fig. \[fig:refcorr\], using [relxill]{} suggests that parameters for all data sets are mostly in agreement within error, and no correlations between parameters are apparent. This is not the case with [reflionx]{}. There is a clear positive correlation between the photon index $\Gamma$ and the flux of the power law component. This relationship is expected; as $\Gamma$ increases, more emission is expected in the soft band, increasing the overall flux. Interestingly, it is also for [reflionx]{} that we see clear changes in flux state; with [relxill]{}, the flux between epochs does not change within error. For [reflionx]{}, weaker evidence for correlations are also visible between the reflection fraction, inner emissivity index and ionisation, but these parameters are not well constrained.
The two reflection models presented here ([relxill]{} and [reflionx]{}) are both dominated by a single spectral component in the $0.3-10\kev$ band. For [relxill]{}, this is the reflection spectrum, while [reflionx]{} suggests the power law dominates. Changes in spectra are caused primarily by flux variations of these components. Both interpretations seem consistent with the approximately constant hardness ratio found in Section \[sect:var\] (see also Section \[sect:pca\]).
{width="\columnwidth"}
{width="\columnwidth"}
![Comparison between the best fit spin, inclination and iron abundance using [relxill]{} (blue) and [reflionx]{} (pink). Contours are shaded using 68, 90 and 99 percent of MCMC fits.[]{data-label="fig:relvsref_corner"}](reflection_jun26_4.png){width="\columnwidth"}
Principal Component Analysis {#sect:pca}
----------------------------
To obtain a model-independent assessment of the spectral variations between observations, principal component analysis (PCA) is used. This technique is implemented using the method and code[[^6]]{} described by [@Parker+2014]. By finding the eigenvalues corresponding to the maximum spectral variability, it allows for the detection of correlated variability between energy bands.
To investigate long-term variability trends, data from all observations are combined and the full $0.3-10\kev$ band is used. The long exposure (XMM5) is broken into seven $20\ks$ segments so all spectra have similar exposures. Only one significant principal component (PC1) is revealed in this analysis, accounting for $\sim90$ per cent of the variability. This component is shown as a function of energy in Fig. \[fig:pca\]. The shape is mostly flat, with some curvature in the $4-7\kev$ region. There is also an upturn above $\sim8.5\kev$, which may be attributable to variations in the background spectrum between epochs. All PC values are positive, indicating that the changes in each energy band are correlated.
[@Parkerpca] and [@Gallant+2018] are able to reproduce this shape with a single variable model component (e.g. a power law) that varies in normalisation. This implies that the spectral changes between epochs can mostly be attributed to flux changes of a single model component, such as the power law. This interpretation is consistent with the minimal changes in hardness ratio within and between observations, and lack of suggested spectral state change based on $\Delta \alpha_{\rm ox}$ (see Section \[sect:var\]). Significant changes in spectral parameters in Mrk 478 are not expected based on the PCA. Such changes typically yield more complex principal component shapes, as outlined in [@Parkerpca]. It is important to note that more complex physical scenarios, such as multiple variable absorption zones, can in some cases produce an overall flat PCA shape (see @Miller+2008), however, the available data are insufficient to model with more complex scenarios.
PCA also provides a unique way to assess the capability of the models above to reproduce the observed variability. To do so, 100 fake data sets with $20\ks$ exposures are simulated for each model. Parameters which are linked between data sets are frozen for all simulations. Free parameters are varied between each simulated data set, with ranges based on model constraints. Correlations between parameters are not apparent using partial covering or [relxill]{} and very weak using [reflionx]{}, but all parameters are highly correlated for [optxagnf]{}. A PCA is then produced for these simulated data sets.
The results are shown in Fig. \[fig:pca\]. The width of the bands represent the calculated error bars from the PCA analysis. All models predict correlated variability between all energy bands, but differ in shape of the first principal component. The PCA results for partial covering and [optxagnf]{} exhibit significant curvature, turning sharply downwards towards high energies. Both reflection models, however, produce first principle components which are fairly constant across all energies. This is likely due to the fact that most variations in these models are in the flux of the dominant emission component. For [relxill]{}, this corresponds to flux changes in the reflection spectrum, whereas for [reflionx]{}, this is the power law component. This is consistent with the simulations presented in [@Parkerpca] and [@Gallant+2018], where normalisation changes of a single spectral component produced the PCA shapes.
To evaluate the statistical fit of each model, $\redchi$ values are obtained for each simulated PCA compared to the data. For 50 degrees of freedom, the $\redchi$ values are 17, 39, 4 and 6 for partial covering, soft-Comptonisation, [relxill]{} and [reflionx]{}, respectively. The measured statistics, as well as the overall flatter shapes produced by the blurred reflection models, suggest that this interpretation best fits the observed PCA. Although the varied parameters cannot exactly reproduce the observed variability, the overall shape is very close. The other models are clearly unable to reproduce the almost flat shape of the PCA calculated using all observations.
Discussion {#sect:discussion}
==========
The X-ray nature of Mrk 478 {#sect:discussionxray}
---------------------------
The spectra of Mrk 478 at all epochs are similar, characterised by a strong, smooth soft excess and excess residuals in the $5-7\kev$ band, attributable to some form of iron emission. The variation between epochs is primarily due to normalisation changes, and little evidence for shape changes are found. The data also do not require any additional emission or absorption features produced in a warm absorber. This is supported by an examination of the RGS data from XMM5, and is consistent with the LETG results found by [@Marshall+2003].
A variety of physical models; partial covering, soft-Comptonisation and blurred reflection, all provide similar statistical fits to the data. Each of the models suggests a different physical interpretation for the variability. In the partial covering model, Fig. \[fig:pcfcorr\] shows that the lower flux observed in XMM4 and XMM5 is explained by an increase in column density and covering fraction in the ionised absorber. No change in the intrinsic power law spectrum is required, and changes in the neutral absorber show no evident correlations. For the soft-Comptonisation model, both the hard and soft Comptonisation components must vary to produce the observed spectra. For the dimmer XMM4 and XMM5, the fraction of power emitted below r$_{\rm cor}$ in the hard Compton component (fpl), the index of the hard power law, and the opacity of the soft corona all decrease, while the temperature of the soft component increases. These parameters increase and decrease accordingly as the spectra rise in flux. Finally, both blurred reflection models attribute the variability to normalisation changes in the coronal emission.
While the best fit partial covering model was able to explain the observed spectral shape, it is the poorest fit of the four spectral models. It also cannot easily explain the observed changes in the spectra, as seen in the PCA of Section \[sect:pca\]. At these high column densities, more variability in the absorption spectrum is present at lower energies than at higher energies, so the entire model changes shape between data sets. This is also contrary to the findings from the variability analysis of Section \[sect:var\], in which the flatness of the hardness ratios between epochs are more suggestive of a single spectral feature changing in normalisation.
Additionally, from the fact that $\Delta \alpha_{\rm ox}$ values agreed with 0, it was hypothesized that the X-ray spectra would not be significantly absorbed. In the best-fitting partial covering model, however, the absorbers dim the source by a factor of $\sim5-10$. This significant amount of absorption in an apparently X-ray normal AGN is difficult to explain. This model also requires a complex, non-spherical symmetry of the absorbers to explain the lack of $6.4\kev$ emission expected from the measured absorption.
A soft excess produced by soft-Comptonisation in a secondary, warm corona was proposed in previous works (@Marshall+2003 [@Guainazzi2004]). However, similarly to the partial covering interpretation, the variations in this model fail to reproduce the observed PCA. This is likely due to the fact that the soft-Comptonisation and hard Comptonisation components are almost independently responsible for the variability at low and high energies, which is contrary to the changes in normalisation of a single component suggested by the variability analysis and shape of the PCA. As seen in Section \[sect:optx\], the model is also unable to explain the high fluxes observed in the UV band, even when testing different black hole spins. The $\Delta \alpha_{\rm ox}$ values suggest an X-ray normal state for Mrk 478 at all epochs, so it is unlikely that the UV data is intrinsically extreme.
One crucial disagreement between models is on the shape of the primary continuum. In the partial covering model, the power law slope is extremely steep, with $\Gamma = 2.99\pm0.02$ for all epochs. Although NLS1 galaxies typically feature steeper spectra than other AGN (e.g. @Boller+1996 [@Brandt+1997; @Grupe+2001]), this value is high. In the soft-Comptonisation model, the spectrum is much flatter, with the index ranging between $2.0-2.2$ between epochs. These values are average among NLS1 AGN. The blurred reflection models require intermediate slopes, steeper than those of typical NLS1 galaxies, but not so extreme as the partial covering model. [reflionx]{} suggests slopes in the $2.6-2.8$ range, and [relxill]{} measures this to be lower, with slopes around $2.4-2.6$.
Although the variations in slopes between models are all able to explain the $0.3-10\kev$ spectra, they differ at $E > 10\kev$. The partial covering and blurred reflection models predict $15-100\kev$ fluxes below the BAT survey limit. Owing to the very shallow power law measured in the [optxagnf]{} model, the predicted flux for this model is the highest ($\sim2\times10^{-12}$ ) and just at the threshold limit of the BAT survey (@Oh+2018). All spectra are consistent with null detections using PIN.
One of the most curious features revealed by the spectral analysis is the unique iron line profile of Mrk 478. AGN typically show evidence of both narrow and broad $6.4\kev$ lines, with the broad line likely produced by reflection of the primary emission off of neutral material (e.g. the torus). However, no evidence for such a feature is detected in any of the presented models. There are some plausible explanations for this - the spectrum is very steep, meaning that a narrow line may be buried below the continuum at these high energies if the reflection off of neutral material was sufficiently weak. Additionally, only XMM5 has high signal-to-noise at these high energies; the other data sets suffer from short exposure and high background making it difficult to model narrow features at high energies.
The $\sim6.7\kev$ emission line, corresponding to emission, is required by all models. For 2 additional free parameters, the fit statistic improves by $\Delta C=9$ for partial covering, up to $\Delta C=36$ using [relxill]{}. Examining the residuals for all data before adding the line consistently shows an excess at this energy. The line energy is well constrained in all models, and is always in agreement with $6.7\kev$. An examination of the background of each spectrum does not suggest that the feature arises from improper background subtraction. The $6.7\kev$ emission line is narrow, and therefore likely does not originate in the inner accretion disc, as no evidence for relativistic broadening is seen. Instead, the feature may be the result of fluorescent emission lines from ionised layers in the inner region of the torus (@Matt+1996 [@Bianchi+2002; @Costantini+2010]). It may also be produced in the broad or narrow line regions.
Typically, the detection of $6.7\kev$ emission is accompanied by an emission line at $6.97\kev$. However, no evidence for this feature is seen in the partial covering model. For the other models, the data are consistent with an emission feature at $6.97\kev$. Attempting to add the emission line to each data set only results in a significant detection for the dim flux, high quality XMM5 data. For this data set, the feature is particularly prominent when modelling the data with [relxill]{}, where adding a narrow $6.97\kev$ feature improves the fit by $\Delta C = 12$ for one additional free parameter. However, additional higher quality data is required to confirm the detection of emission. If present, the emission likely originates from the same region as the emission.
![PCA using 20ks segments for all observations (black). Only PC1 is significant, and accounts for $90$ per cent of the variability. The shape is relatively flat, with some curvature around $4-7\kev$. The PCA results from simulated data sets are also shown, with partial covering in green, [optxagnf]{} in red, [relxill]{} in blue and [reflionx]{} in pink. The blurred reflection models best reproduce the flat shape of PC1.[]{data-label="fig:pca"}](jul22_pca.png){width="\columnwidth"}
The Blurred Reflection Model {#sect:discussionref}
----------------------------
Based on the low variability in hardness ratio throughout and between all epochs as well as the predicted and observed shapes of the PCA, the reflection model is the most likely physical explanation for the X-ray emission of Mrk 478. This reflection component was modelled using both [relxill]{} and [reflionx]{} convolved with the blurring model [kerrconv]{}. The results from these spectral fits differ in interpretation. [relxill]{} predicts a highly ionised spectrum dominated by reflection off the inner accretion disc, while [reflionx]{} suggests a low ionisation and power law dominated spectrum. However, based on the calculated $\alpha_{\rm ox}$ values (Section \[sect:var\]), Mrk 478 is behaving like a normal AGN and is not expected to display extreme spectral properties, like high values of R. In this sense, the [reflionx]{} interpretation seems more consistent.
The different measurements of ionisation and reflection fraction may be driven by intrinsic differences in the models themselves, discussed in detail in [@Garcia+2013]. In the reflection model, the soft excess is produced by a multitude of narrow emission lines which are then relativistically blurred. Between the two models, however, there are significant differences in the abundances of the elements responsible for these narrow emission lines, including O, Ne and Fe. This results in significant differences in the shape of the broad line and, more notably, the soft excess. [@Garcia+2013] also note significant deviations between models at high ionisations, which although hard to explain, may also be attributable to the different abundances as well as different ionisation states of each element included in each model.
Contrary to the differences in ionisation and reflection fraction, the measured values for spin and inclination between reflection models are comparable at the $90$ per cent level. The uncertainties on these values are only measurement errors. [@Bonson+2016] perform extensive simulations based on [relxill]{} parameters, and test the reproducibility of each model parameter. Although they consider a narrower energy range of $2.5-10\kev$, they find that the inclination can be reliably constrained to within $10\degree$, and spin to within $\sim10$ per cent above $a=0.9$. If these systematic error bars are adopted, spin and inclination values are in close agreement. Therefore, both reflection models suggest a maximum spinning black hole viewed at a low inclination.
The preference of the blurred reflection model to explain the observed spectrum of Mrk 478 also requires an explanation for the under-abundance of iron found by both [reflionx]{} and [relxill]{}. The discrepancy in best fit abundance between models can be explained by the different element abundances used in the two models. The iron abundance used in [relxill]{} is $\sim30$ per cent lower than that of [reflionx]{} (@Garcia+2013), and when this is taken into account, the values agree within error. The low iron abundance is also in agreement with the results of [@Zoghbi+2008].
While many AGN require super-solar abundances to explain the reflection spectrum, very few require sub-solar abundances. For HE 0436–4717, [@Bonson+2015] find an iron abundance of $\sim0.4$ times solar value, and list possible causes such as low star formation rates yielding more pristine and metal-poor gas or a lack of type Ia supernovae (e.g. @Groves+2006). Another interpretation presented by [@Skibo+1997] is cosmic ray spallation, wherein cosmic rays strike iron nuclei and cause it to lose nucleons, resulting in the creation of lower mass elements including Ti, V, Mn and Cr (@Skibo+1997 [@Gallo+2019spal]). [@Turner+2010] also propose this explanation for NGC 4051, another source that can be identified as having low iron (e.g. @Patrick+2012).
Conclusion {#sect:conclusion}
==========
A variability and spectral analysis for the NLS1 Mrk 478 is presented, using all available data from and spanning from 2001 to 2017. These spectra differ in $0.3-10\kev$ flux by a factor of $\sim2$, while spectral shape does not appear to vary between observations. Data are well fitted by a variety of physical models; partial covering including one ionised and one neutral absorber, soft-Comptonisation using [optxagnf]{}, and blurred reflection using both [relxill]{} and [reflionx]{} reflection models. However, through an analysis of the variability between data sets, it is revealed that the blurred reflection model best explains changes between epochs. This is especially apparent using PCA, when only the blurred reflection models can reproduce the flat shape of the first principal component.
Although the two blurred reflection models disagree in some measurements, they both predict a rapidly rotating black hole seen at a shallow viewing angle. Both reflection models also suggest an under-abundance of iron, with values of $\sim0.5$ times solar abundances. All models support the existence of a narrow $6.7\kev$ emission line, which is attributable to emission. XMM5 also shows some evidence for a narrow emission line at $6.97\kev$ using both reflection models and soft-Comptonisation.
Mrk 478 has not been detected by the BAT instrument, and a $72\ks$ PIN observation resulted in a null-detection. There is some disagreement in the $10-100\kev$ fluxes predicted by the presented models, with soft-Comptonisation suggesting a much flatter and brighter spectrum at high energies than partial covering or blurred reflection models. Hard X-ray observations of Mrk 478 with may help distinguish further between spectral models, as well as provide another look at the $6-7\kev$ iron profile.
More consideration to the blurred reflection interpretation, as well as the iron profile, will be given in a companion to this work, in which a more detailed timing and time-resolved spectroscopic analysis will be given to the long exposure (XMM5). X-ray spectra obtained with higher resolution instruments such as (@xrismins) and (@athenains) will allow for more detailed mapping of the iron profile, and possibly allow for the a clearer detection of ionised iron emission, narrow $6.4\kev$ emission, and confirm an under-abundance of iron in this unique NLS1.
Acknowledgements {#acknowledgements .unnumbered}
================
The project is an ESA Science Mission with instruments and contributions directly funded by ESA Member States and the USA (NASA). This research has made use of data obtained from the satellite, a collaborative mission between the space agencies of Japan (JAXA) and the USA (NASA). We thank the referee for their helpful comments and suggestions which improved the original manuscript. AZ was partially supported by NASA under award 80NSSC18K0377. SGHW and LCG acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC).
\[lastpage\]
[^1]: E-mail: swaddell@ap.smu.ca
[^2]: http://www.swift.ac.uk/user\_objects/
[^3]: https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3pimms/w3pimms.pl
[^4]: Made available by Jeremy Sanders (http://github.com/jeremysanders/xspec\_emcee)
[^5]: https://cass.ucsd.edu/$\sim$rxteagn/
[^6]: http://www-xray.ast.cam.ac.uk/$\sim$mlparker/
|
---
abstract: 'In this paper, we consider two dynamical systems associated to the nearest integer continued fraction, and show that both of them have full Hausdorff dimension spectrum.'
author:
- 'A. Ghenciu, S. Munday, M. Roy'
title: The Hausdorff dimension spectrum of Conformal Graph Directed Markov Systems and applications to Nearest Integer continued fractions
---
Introduction and statement of results
=====================================
It is well known that every irrational real number $x$ can be written uniquely as an infinite fraction $$\begin{aligned}
x=a_0+\frac{1}{a_1 + \frac{1}{a_2+\frac{1}{a_3 + \cdots}}},\end{aligned}$$ where $a_0\in \Z$ and each $a_i\in \N$, for $i\geq1$. This is the regular continued fraction expansion of $x$. This classical and extremely well-studied expansion is far from being the only interesting one that has been introduced. Another class of expansions, a generalisation of the regular continued fraction (RCF), are the semi-regular continued fraction (SRCF) expansions. These are expansions which improve the approximation properties of the regular continued fraction (for more information on this see [@DK] and references therein) and they are defined as follows. A SRCF expansion is a finite or infinite fraction $$[b_0; {\varepsilon}_1b_1, {\varepsilon}_2b_2, {\varepsilon}_3b_3, \ldots]:=b_0+\frac{{\varepsilon}_1}{b_1+\frac{{\varepsilon}_2}{b_2+\frac{{\varepsilon}_3}{b_3+\ldots}}},$$ with ${\varepsilon}_n=\pm1$, $b_0\in \Z$ and $b_n\in \N$ for all $n\geq1$, subject to the conditions that ${\varepsilon}_{n+1}+b_n\geq1$ for all $n\geq1$, and, if the fraction is infinite, we have infinitely often that ${\varepsilon}_{n+1}+b_n\geq2$.
In this paper, we are interested in a particular example of a SCRF, namely, the nearest integer continued fraction (NICF). This expansion was introduced by Minnigerode in 1873 [@min73], and has been studied quite intensively by several authors, starting with Hurwitz [@hur89]. The NICF is a SRCF satisfying $b_n\geq2$ and $b_n+{\varepsilon}_{n+1}\geq2$ for all $n\geq1$. The NICF is intimately related to the regular continued fraction, via the process of singularization, which we now describe (see [@Kra] for more details and further references). First, for any two positive integers $a$ and $b$, and $\xi \in (0, 1)$, observe that $$a+\frac{1}{1+\frac{1}{b+\xi}} = (a+1)+\frac{-1}{b+1+\xi}.$$ Then, if we have a SRCF expansion $$\begin{aligned}
\label{star1}
[b_0; {\varepsilon}_1 b_1, {\varepsilon}_2 b_2, {\varepsilon}_3 b_3, \ldots]\end{aligned}$$ with $b_{k+1}={\varepsilon}_{k+1}={\varepsilon}_{k+2} = 1$ for some $k\geq0$, we can replace (\[star1\]) by $$\begin{aligned}
\label{star2}
[b_0; {\varepsilon}_1 b_1, {\varepsilon}_2 b_2, \ldots, {\varepsilon}_{k-1}b_{k-1}, {\varepsilon}_k (b_{k}+1), -(b_{k+2}+1), {\varepsilon}_{k+3}b_{k+3}, \ldots].\end{aligned}$$ Now consider the RCF expansion of an irrational number $x$ and the following algorithm. Suppose that we have $a_{n+1}=\cdots = a_{n+m}=1$, for $m\in \N\cup\{\infty\}$, $n\geq0$, $a_{n+m+1}\neq1$ and $a_n\neq1$ (assuming $n>0$). Then singularize $a_{n+1}$, $a_{n+3}$, $a_{n+5}$, and so on, in turn. One immediately verifies that the expansion obtained in this way is the NICF expansion of $x$. Notice that this implies, in particular, that every irrational number admits an infinite NICF expansion. Moreover, this expansion is unique.
Let $[b_0; {\varepsilon}_1 b_1, {\varepsilon}_2 b_2, \ldots]$ be an infinite SRCF (we ignore finite expansions from here on, as they are only countably many). Then it is shown in [@Kra Theorem 1.7] that there exist sequences $(p_n)_{n\geq-1}$ and $(q_n)_{n\geq-1}$ in $\Z$ that satisfy the recurrence relations $$\left\{
\begin{array}{ll}
p_{-1}:=1,\ \ \ p_0:=b_0 ,& \hbox{$p_n=b_np_{n-1}+{\varepsilon}_np_{n-1}$,} \\
q_{-1}:=0,\ \ \ q_0:=1, & \hbox{$q_n=b_nq_{n-1}+{\varepsilon}_nq_{n-1}$.}
\end{array}
\right.$$ It is also shown that for all $n\geq-1$, we have $\mathrm{gcd}(p_n, q_n)=1$ and $\mathrm{gcd}(q_{n}, q_{n+1})=1$. Then, if for each $n\geq0$ we define $p_n/q_n:=[b_0; {\varepsilon}_1 b_1, \ldots, {\varepsilon}_n b_n]$, the continued fraction $[b_0; {\varepsilon}_1 b_1, {\varepsilon}_2 b_2, \ldots]$ is said to be convergent if and only if $\lim_{n\to\infty}p_n/q_n$ exists and is finite. It turns out that every SRCF converges to an irrational number (see [@Kra] again, and references therein), so it makes sense to refer to $(p_n/q_n)_{n\geq-1}$ as the sequence of convergents to the number $x= [b_0; {\varepsilon}_1 b_1, {\varepsilon}_2 b_2, \ldots]$. For the NICF example, we have that $|q_{n-1}|\leq |q_n|$ for all $n\geq1$ (see Corollary 1.9 in [@Kra]).
Much of the work done on the NICF has concentrated on its Diophantine approximation properties (see, for instance, [@jag85], [@jagkra], [@rock80]). We instead will focus on the question of its Hausdorff dimension spectrum, which we define shortly below. For this, it will be helpful to have a more dynamical representation of the NICF. Let $[\cdot]$ denote the integer part function. Then the nearest integer continued fraction expansions are determined by the discontinuous transformation $T:[-1/2,1/2]\to[-1/2,1/2]$ which is defined by setting $$T(x):=\left\{
\begin{array}{ll}
\frac{1}{x}-\left[\frac{1}{x}-\frac{1}{2}\right], & \hbox{if $x\neq0$;} \\
0, & \hbox{if $x=0$.}
\end{array}
\right.$$ By “determined by”, we mean that the digits of the NICF can be found using the map $T$ as follows: For all $n\geq1$, $$b_n=b_n(x)=\left[\frac{1}{T^{n-1}(x)}\right].$$ Note that the digits are now integers, instead of natural numbers coupled with a sign. That is, our $b_n$ generated by the map $T$ is equal to ${\varepsilon}_nb_n$ from above. The inverse branches of $T$ are the conformal[^1] maps $$\varphi_b(x)=\frac{1}{b+x},\,\,\, |b|>1,$$ whose respective domains are $$\varphi_2:\left[0,\frac{1}{2}\right]
\to
\left[\frac{2}{5},\frac{1}{2}\right]\subset\left[0,\frac{1}{2}\right],$$ $$\varphi_{-2}:\left[-\frac{1}{2},0\right]
\to
\left[-\frac{1}{2},-\frac{2}{5}\right]\subset\left[-\frac{1}{2},0\right],$$ and $$\varphi_b:\left[-\frac{1}{2},\frac{1}{2}\right]
\to
\left[\frac{1}{b+1/2},\frac{1}{b-1/2}\right]
\subset
\left\{
\begin{array}{cll}
\left[0,\frac{1}{2}\right] & \mbox{ if } & b>2 \\
\left[-\frac{1}{2},0\right] & \mbox{ if } & b<-2
\end{array}
\right\}
\subset\left[-\frac{1}{2},\frac{1}{2}\right].$$
Now, let $E=\{b\in\Z:|b|\geq2\}$. Let $F\subset E$, and let $J_F$ be the set of all numbers in $[-1/2,1/2]$ which can be represented by an infinite NICF with all digits belonging to the set $F$. If $F=E$, then $J_E$ is the set of all irrational numbers in the interval $[-1/2,1/2]$. This set has Lebesgue measure $1$. However, if $F$ is a proper subset of $E$, then the set $J_F$ has Lebesgue measure $0$. Therefore, to distinguish between these sets, we use the Hausdorff dimension, which we will denote by $\dim_H(\cdot)$. (We will assume basic familiarity with properties of the Hausdorff dimension throughout, and refer to [@Fal].)
The problem we are interested in is this: Given $0\leq t\leq1$, does there exist a set $F\subset E$ such that $\dim_H(J_F)=t$? For the RCF expansion, this was an open problem for several years, known as the Texan Conjecture. It was answered affirmatively for $0\leq t\leq1/2$ by Mauldin and Urbański [@mutr]. Later, it was answered positively for all $0\leq t\leq1$ by Kesseböhmer and Zhu [@KZ]. It is then said that the standard continued fraction expansion has full Hausdorff dimension spectrum. Similar results were obtained by Ghenciu for the backward continued fraction expansions [@AGpara] and the Gauss-like continued fraction expansions [@G]. To solve these problems, these authors associated to each continued fraction expansion an infinite conformal iterated function system (cIFS), which, very briefly, is a finite or infinite set of conformal contracting similarities of a compact metric space.
In this paper, we consider questions related to the Hausdorff dimension spectrum of the NICF. The observant reader will have already spotted the main difficulty - the NICF cannot be associated to an IFS, since the domains of the inverse branches of the map $T$ are not all the same space. To get around this problem, we need to introduce graph directed Markov systems. Then, there are two natural IFSs that can be associated to the NICF. The first is the IFS obtained by restricting the digits of the NICF to the set $F:=\{b\in\Z: |b|\geq3\}$, which we shall denote by $\Phi_F$. The second is an IFS associated to one of the vertices of the graph directed Markov system we will use to describe the NICF; we will denote this IFS by $\Phi^{(v)}$, but for the details of how it is defined we defer to Section 5. Our main results concern the dimension spectra of these two systems.
\[mainthm1\] $\Phi_F$ has full Hausdorff dimension spectrum.
\[mainthm2\] $\Phi^{(v)}$ has full Hausdorff dimension spectrum.
The paper is organised as follows. In Section \[sec\_prelim\], we will introduce much of the preliminary material needed for the rest of the paper, beginning with the definition of a conformal graph directed Markov system. Section 3 contains a collection of lemmas needed for the proof of Theorem \[mainthm1\]; the proof itself can be found in Section 4. Section 5 contains the details necessary to construct an IFS associated to the vertex of a GDMS and the proof of Theorem \[mainthm2\]. Finally, we add an appendix containing further background results on CGDMSs, mostly these results are given simply to clear up small inaccuracies in previously available proofs.
Preliminaries {#sec_prelim}
=============
Graph directed Markov systems
-----------------------------
Let us first introduce graph directed Markov systems. To do this, we need a directed multigraph $(V,E,i,t)$ and an associated incidence matrix $A$, i.e., a matrix containing only 0s and 1s. The multigraph consists of a finite set $V$ of vertices, a (possibly infinitely) countable set $E$ of directed edges and two functions $i,t:E\to V$, where $i(e)$ is the initial vertex of edge $e$ and $t(e)$ is its terminal vertex. The incidence matrix $A$ of size $\#E\times \# E$ indicates which edge(s) may follow any given edge. In other words, $A_{ef}=1$ if and only if $t(e)=i(f)$. For later use, let us also introduce some more notation. The set $E_A^\infty$ of one-sided infinite $A$-admissible words is defined to be $$E_A^\infty:=\left\{\omega=\omega_1\omega_2\ldots\in
E^\infty:A_{\omega_i\omega_{i+1}}=1,\ \forall i\geq1\right\}.$$ The set of all finite subwords of $E_A^\infty$ will be denoted by $E_A^*$. The *length* of any word $\omega$ is defined to be the number of letters it is made up of, and will be denoted by $|\omega|$. For each $n\geq1$, the set of all subwords of $E_A^\infty$ of length $n$ shall be denoted by $E^n_A$. There is a unique word of length $0$ in $E_A^*$ called the *empty word*. If $\omega \in E_A^\infty$ and $n\geq1$, then we write $\omega|_n$ for the initial $n$-block of the word $\omega$, that is, $$\omega|_n=\omega_1\omega_2\ldots\omega_n.$$ A *Graph Directed Markov System* (GDMS) consists of a directed multigraph $(V,E,i,t)$, an incidence matrix $A$, a set of non-empty compact metric spaces $\{X_v\}_{v\in V}$ and a set of $1$-to-$1$ contractions $\{\varphi_e:X_{t(e)}\to X_{i(e)}\}_{e\in E}$ with Lipschitz constant $s$, where $0<s<1$. Sometimes, in a slight abuse of notation, we will refer to this set of contractions as a GDMS, but only when the context is clear. The matrix $A$ tells us which contractions can be applied after each other, in the following way. For each $\omega\in E_A^*$, the map coded by $\omega$ is defined to be $$\varphi_\omega:=\varphi_{\omega_1}\circ\ldots\circ\varphi_{\omega_{|\omega|}}
:X_{t(\omega)}\to X_{i(\omega)},$$ where $t(\omega):=t(\omega_{|\omega|})$ and $i(\omega):=i(\omega_1)$.
\[defn\_IFS\] If the set of vertices in the GDMS is a singleton and all the entries in the incidence matrix are 1, then the GDMS is an [*iterated function system*]{}, abbreviated to IFS. More concretely, an IFS is a countable set of contraction maps with Lipschitz constant $0<s<1$ which map a compact metric space into itself. Iterated function systems were well-studied before GDMSs were introduced, particularly in terms of generating fractal sets (see [@Fal]).
Returning to our GDMS, for each $\omega\in E_A^\infty$, the sets $\{\varphi_{\omega|_n}(X_{t(\omega_n)})\}_{n\geq1}$ form a decreasing sequence of non-empty compact subsets of $X_{i(\omega_1)}$. Also, since for every $n\geq1$ we have that $$\mbox{diam}(\varphi_{\omega|_n}(X_{t(\omega_n)})) \leq s^n\mbox{diam}(X_{t(\omega_n)})\leq s^n\max\{\mbox{diam}(X_v):v\in V\},$$ the intersection $$\bigcap_{n\geq1}\varphi_{\omega|_n}\left(X_{t(\omega_n)}\right)$$ is a singleton whose element is denoted by $\pi(\omega)$. If we set $X$ to be the disjoint union of the sets $\{X_v\}_{v\in V}$, then the map $$\pi:E_A^\infty\to X$$ defined in this way is called the *coding map*. The set $$J:=J_{E,A}=\pi(E_A^\infty)$$ is called the *limit set* of the GDMS $S$.
From this point on in the paper, we make two simplifying assumptions about the directed graph. First, we assume that for all $e\in E$ there exists $f\in E$ so that $A_{ef}=1$. Otherwise, if there were $e\in E$ so that $A_{ef}=0$ for every $f\in E$, then the limit set $J_{E,A}$ would be the same as the limit set $J_{E\setminus\{e\},A}$ (in the construction of this latter set, $A$ is restricted to $(E\setminus\{e\})^2$). Second, we assume that for every vertex $v\in V$ there exists $e\in E$ so that $i(e)=v$. Otherwise, if there existed $v\in V$ such that no edge has for initial vertex $v$, then the limit set $J$ would be the same if the vertex set were $V\setminus\{v\}$.
We emphasize that we have two directed graphs that play an important role in our study. The first one is the given multigraph $(V,E,i,t)$. The second one, $G_{E,A}$, is determined by the matrix $A$. The vertices of $G_{E,A}$ are the edges of the first one, and $G_{E,A}$ has a directed edge from $e$ to $f$ if and only if $A_{ef}=1$. Therefore $G_{E,A}$ has infinitely many vertices and edges if and only if $E$ is an infinite set.
We will also need the following properties of the incidence matrix $A$. Firstly, $A$ is said to be *irreducible* if for any two edges $e,f\in E$ there exists a word $\omega\in E_A^*$ so that $e\omega f\in E_A^*$. This is equivalent to saying that the directed graph $G_{E,A}$ is *strongly connected*, i.e. for any two vertices there exists a path starting from one and ending at the other. The matrix $A$ is said to be *finitely irreducible* if there exists a finite set $\Omega\subset E_A^*$ so that for any two edges $e,f\in E$ there is a word $\omega\in\Omega$ so that $e\omega f\in E_A^*$.
The matrix $A$ is called *primitive* if there exists $p\geq1$ such that all the entries of $A^p$ are positive (written $A^p>0$) or, in other words, for any two edges $e,f\in E$ there exists a word $\omega\in E_A^{p-1}$ so that $e\omega f\in E_A^{p+1}$. Similarly, the matrix $A$ is called *finitely primitive* if there exist $p\geq1$ and a finite set $\Omega\subset E_A^{p-1}$ such that for any two edges $e,f\in E$ there is a word $\omega\in\Omega$ so that $e\omega f\in E_A^{p+1}$.\
\
A GDMS is called *conformal*, and hence a CGDMS, if the following conditions are satisfied:
1. For every $v\in V$, the set $X_v$ is a compact connected subset of a Euclidean space $\mathbf{R}^d$ (the dimension $d$ common for all vertices) and $X_v=\overline{\mbox{Int}(X_v)}$.
2. (Open Set Condition (OSC)) For every $e,f\in E$, $e\neq f$, $$\varphi_e\left(\mbox{Int}(X_{t(e)})\right)\bigcap\varphi_f\left(\mbox{Int}(X_{t(f)})\right)={\varnothing}.$$
3. For every vertex $v\in V$ there exists an open connected set $W_v\supset X_v$ so that for every $e\in E$ with $t(e)=v$, the map $\varphi_e$ extends to a $C^1$ conformal diffeomorphism of $W_v$ into $W_{i(e)}$.
4. (Cone property) There exists $\gamma,l>0$, such that for every $x\in X$ there exists an open cone $\mbox{Con}(x,\gamma,l)\subset\mbox{Int}(X)$ with vertex $x$, central angle of measure $\gamma$, and altitude $l$.
5. There are two constants $L\geq1$ and $\alpha>0$ so that $$\left||\varphi_e'(y)|-|\varphi_e'(x)|\right|\leq L\|(\varphi_e')^{-1}\|^{-1}
\|y-x\|^\alpha$$ for every $e\in E$ and for every pair of points $x,y\in W_{t(e)}$, where $|\varphi_e'(x)|$ represents the norm of the derivative of $\varphi_e$ at $x$. This says that the norms of the derivative maps are all Hölder continuous functions of order $\alpha$ with Hölder constant depending on the map.
As explained in [@gdms], condition (5) plays a central role in dimension $d=1$. If $d\geq2$ and we are given a GDMS which satisfies conditions (1) and (3), then it automatically fulfills condition (5) with $\alpha=1$. In this paper, we will only be considering GSMSs in dimension $d=1$. This also means that the property (4) will not concern us, as it is always satisfied for $d=1$.
As a straightforward consequence of (5), we obtain the famous Bounded Distortion Property (BDP):
1. There exists $K\geq1$ such that for all $\omega\in E_A^*$ and for all $x,y\in W_{t(\omega)}$, $$\label{BDP}
|\varphi_\omega'(y)|\leq K|\varphi_\omega'(x)|.$$
GDMS for the NICF
-----------------
As mentioned already in the introduction, the NICF cannot be described by an iterated function system, as the inverse branches are not all defined upon the same domain. Let us recall the definition of these branches: $$\varphi_b(x)=\frac{1}{b+x},\,\,\, |b|\geq2,$$ where ${\varphi}_2$ is defined on the interval $[0, 1/2]$, ${\varphi}_{-2}$ on $[-1/2, 0]$ and ${\varphi}_b$, for $|b|\geq3$, is defined upon $[-1/2, 1/2]$. Therefore the composition of these inverse branches are subject to some restrictions. We shall describe the restrictions by means of an incidence matrix $A$ and by identifying the branch $\varphi_b$ with the letter $b$. Thus, the composition $\varphi_{e}\circ\varphi_f$ shall be allowed if and only if $A_{ef}=1$, that is, if and only if the word $ef$ is $A$-admissible. Let $E=\{b\in\Z:|b|\geq2\}$ and $A:E^2\to\{0,1\}$ be the matrix defined by setting $$A_{ef}:=\left\{\begin{array}{llcll}
1 & \mbox{ if } & |e|>2 & & \\
1 & \mbox{ if } & e=2 & \mbox{ and } & f>0 \\
0 & \mbox{ if } & e=2 & \mbox{ and } & f<0 \\
1 & \mbox{ if } & e=-2 & \mbox{ and } & f<0 \\
0 & \mbox{ if } & e=-2 & \mbox{ and } & f>0.
\end{array}
\right.$$
We introduce an infinite conformal graph directed Markov system which reflects the backward trajectories of $T$, that is, the composition of the inverse branches $\{\varphi_b\}_{|b|\geq2}$ of $T$. As these inverse branches have three different domains, we shall need three vertices. Let the set of vertices and attached spaces be $V=\{v,w,z\}$ and $$X_v=\left[-\frac{1}{2},\frac{1}{2}\right],
\hspace{1cm}
X_w=\left[0,\frac{1}{2}\right]
\hspace{1cm} \mbox{ and } \hspace{1cm}
X_z=\left[-\frac{1}{2},0\right].$$ Note that the alphabet $E$ is not sufficient to construct a graph directed Markov system. In [@gdms], page 1, the authors state: “the incidence matrix $A$ determines which edge(s) may follow any given edge. In other words, if $A_{ef}=1$ then $t(e)=i(f)$” . We need copies of some of the letters in $E$.
- Draw a graph with the three vertices $v$, $w$ and $z$;
- Draw a self-loop based at vertex $v$ for each $|e|>2$;
- Draw an edge from vertex $v$ to vertex $w$ and identify it by the letter $2$.
- Draw an edge from $w$ to $v$ for each $e>2$ and identify it by $\overline{e}$. (These edges are identified by $\overline{e}$ to distinguish them from the self-loops $e$. However, their corresponding generators $\varphi_{\overline{e}}$ have for codomain $X_w$, whereas the generators $\varphi_e$ corresponding to the self-loops have for codomain $X_v$. Hence, they are maps given by the same expression, having the same domain but different codomains.);
- Draw a self-loop based at $w$ and identify it as $\overline{2}$;
- Draw an edge from vertex $v$ to vertex $z$ and identify it by $-2$;
- Draw an edge from $z$ to $v$ for each $e<-2$ and identify it by $\overline{e}$. (These edges are identified by $\overline{e}$ to differentiate them from the self-loops $e$. Note that their corresponding generators $\varphi_{\overline{e}}$ have for codomain $X_z$, whereas the generators corresponding to the self-loops $\varphi_e$ have for domain $X_v$. Thus, they are maps given by the same expression, with the same domain but different codomains.);
- Draw a self-loop based at $z$ and identify it by $\overline{-2}$.
We hence obtain a graph directed system $\Phi$. Define a matrix $\overline{A}$ that exactly reflects that graph. This means that the new alphabet is $\overline{E}=\{e:|e|\geq 2\}\cup\{\overline{e}:|\overline{e}|\geq 2\}$. Observe that the matrix $\overline{A}$ contains essentially the same information as the original matrix $A$. As mentioned earlier, the generators of this system are $$\varphi_e(x)=\varphi_{\overline{e}}(x)=\frac{1}{e+x}$$ with domains and codomains reflecting the above graph.
Let $\om\in E_{\overline{A}}^*$. Then $$\varphi_\om(x)=\frac{p_{|\om|}+xp_{|\om|-1}}{q_{|\om|}+xq_{|\om|-1}},$$ where the $p_n=p_n(\om)$’s and $q_n=q_n(\om)$’s are as defined in the introduction. Therefore, $$\begin{aligned}
\label{derivative}
|\varphi_\om'(x)|=\frac{1}{(q_{|\om|}+xq_{|\om|-1})^2}\end{aligned}$$ since $$p_{n-1}q_n-q_{n-1}p_n=(-1)^n$$ for all $1\leq n\leq|\om|$.
Lemmas for later
================
In this section, we give a series of Lemmas that will be used in the proof of Theorem \[mainthm1\]. So, let us recall that for this theorem we are using the alphabet $F:=\{b\in \Z: |b|\geq3\}$ and the full shift space associated to $F$. We will also make extensive use of the recurrence relations for the NICF which were given in the introduction. Here, though, we are using them directly on the symbolic alphabet. Note that this really means we are taking a word consisting of letters from $F$, applying the inverse coding map to it, then calculating the $q_n$s for the NICF. To save complicated notation, we will simply write it directly for the letters of $\omega$. Remember that these recurrence relations are given, for $1\leq n\leq|\omega|$, by $$\label{p equation}
p_n = \omega_n p_{n-1} + p_{n-2}.$$ $$\label{q equation}
q_n = \omega_n q_{n-1} + q_{n-2}.$$ where: $q_0=1$ and $q_1=\omega_1.$\
We begin with a series of estimates on the size of the denominators of the convergents.
\[2.1\]Let $\alpha = \frac{3 - \sqrt{5}}{2}$. Then for every $n \geq 2$ we have that $$\frac{|q_{n-1}|}{|q_n|} \leq \alpha.$$
Using \[q equation\], for every $n \geq 2$, we obtain $$\frac{q_n}{q_{n-1}}=\omega_n + \frac{q_{n-2}}{q_{n-1}}.$$ Thus, inductively, we have that $$\left|\frac{q_{n-1}}{q_n}\right|=\frac{1}{\left|\omega_n + \frac{q_{n-2}}{q_{n-1}}\right|} \leq \frac{1}{|\omega_n| - \left|\frac{q_{n-2}}{q_{n-1}}\right|} \leq \frac{1}{3-\alpha}=\alpha.$$
From this, we obtain the following immediate corollary:
\[2.2\] For every $n \geq 2$, we have: $$\frac{|q_{n-1}|}{|q_n|} \leq \frac{1}{|\omega_n|-\alpha}.$$
The next lemma provides an estimate in the other direction.
\[2.3\] For every $n \geq 2$, we have: $$\frac{|q_{n-1}|}{|q_n|} \geq \frac{1}{|\omega_n|+\alpha}.$$
Using (\[q equation\]) as in Lemma \[2.1\], we first obtain that $$\begin{aligned}
\label{formerLem2.4}
\left|\frac{q_{n-1}}{q_n}\right|=\frac{1}{\left|\omega_n + \frac{q_{n-2}}{q_{n-1}}\right|} \geq \frac{1}{|\omega_n| + \left|\frac{q_{n-2}}{q_{n-1}}\right|}.\end{aligned}$$ At this point we use Lemma \[2.1\] to conclude that: $$\frac{|q_{n-1}|}{|q_n|} \leq \frac{1}{|\omega_n|+\alpha}.$$
Finally, we fix $\omega\in F^{n+1}$ and study the behaviour of the function $G_\om:\left[\frac{-1}{2},\frac{1}{2}\right] \to \left(-\infty,\infty\right)$, where we define $G_\om(x):=\left|\frac{q_n+xq_{n-1}}{q_{n+1} + x q_n}\right|$. This function will be used in the following section.
\[2.4\] For each $n\in \N$, we have $$G_\om(x) \leq \frac{3}{2} \frac{1}{|\omega_{n+1}|-\alpha}.$$
First we make the simple observation that $$G_\om(x) \leq \frac{|q_n| + |x| |q_{n-1}|}{|q_{n+1}| - |x||q_n|} \leq \frac{|q_n| + \frac{1}{2} |q_{n-1}|}{|q_{n+1}| - \frac{1}{2}|q_n|}.$$ Then, using first Lemma \[2.1\] and then Corollary \[2.2\] we obtain the desired estimate: $$G_\om(x) \leq \frac{|q_n| + \frac{1}{2}\frac{2}{5} |q_n|}{|q_{n+1}| - \frac{1}{2}\frac{2}{5}|q_{n+1}|} \leq \frac{\frac{6}{5}|q_n|}{\frac{4}{5}|q_{n+1}|} \leq \frac{3}{2} \frac{1}{|\omega_{n+1}| - \alpha}.$$
\[2.5\] For each $n\in \N$, we have $$G_\om(x) \geq \frac{2}{3} \frac{1}{|\omega_{n+1}|+\alpha}.$$
Similarly to the proof of Lemma \[2.4\], we first notice that $$G_\om(x) \geq \frac{|q_n| - |x| |q_{n-1}|}{|q_{n+1}| + |x||q_n|} \geq \frac{|q_n| - \frac{1}{2} |q_{n-1}|}{|q_{n+1}| + \frac{1}{2}|q_n|}.$$ Thus, in light of Lemmas \[2.1\] and \[2.3\], $$G_\om(x) \geq \frac{|q_n| - \frac{1}{2}\frac{2}{5} |q_n|}{|q_{n+1}| + \frac{1}{2}\frac{2}{5}|q_{n+1}|} \geq \frac{\frac{4}{5}|q_n|}{\frac{6}{5}|q_{n+1}|} \geq \frac{2}{3} \frac{1}{|\omega_{n+1}| + \alpha}.$$
Proof of Theorem \[mainthm1\]
=============================
In order to prove our first main theorem, we will need several results which originate (in slightly different form) in [@AGthesis] and [@KZ]. To state them, we must make two further definitions: The topological pressure of $P_F$ of the IFS $\Phi_F$ is defined for each $t\in \R$ by $$P_F(t):= \lim_{n\to\infty} \frac{1}{n}\log (Z_n),$$ where $Z_n:= \sum_{\om\in F^n} ||{\varphi}_\om'||^t$. Also, for any subset $G\subseteq F$, we write $\lambda_G:=\exp (P_G)$.
\[Me1\] Let $\Phi$ be a conformal iterated function system. Let $F\subset E$ and $e\in E$. If $M_e>0$ is such that $$\|\varphi_{\om e\overline{\om}}'\|\leq M_e\|\varphi_{\om\overline{\om}}'\|$$ for all words $\om\in F^*$ and $\overline{\om}\in(F\cup\{e\})^*$, then $$\l_F(t)\leq\l_{F\cup\{e\}}(t)\leq\l_F(t)+M_e^t.$$
\[me1\] Let $\Phi$ be a conformal iterated function system. Let $F\subset E$ and $e\in E$. If $m_e>0$ is such that $$\|\varphi_{\om e\overline{\om}}'\|\geq m_e\|\varphi_{\om\overline{\om}}'\|$$ for all words $\om\in F^*$ and $\overline{\om}\in(F\cup\{e\})^*$, then $$\l_{F\cup\{e\}}(t)\geq\l_F(t)+m_e^t.$$
\[distort\] Under the hypothesis of Theorems \[Me1\] and \[me1\], the existence of $M_e$ and $m_e$ is guaranteed by the bounded distortion property of the system. In particular, $M_e$ can be taken to be $K\|\varphi_e'\|$, whereas $m_e$ can be taken to be $K^{-1}\inf_{x\in X}|\varphi_e'(x)|$ or $K^{-2}\|\varphi_e'\|$, where $K$ is the constant appearing in (\[BDP\]).
The following theorem is a weakening of Theorem 2.2 in [@KZ].
\[fullspec\] Let $\Phi_\N$ be a conformal iterated function system indexed by the natural numbers $\N$. For every $b\in\N$, let $L_b=\{b+1,b+2,\ldots\}$. If for every $b\in\N$, every $F\subset\{1,2,\ldots,b-1\}$ and $0<t\leq\dim_H(J_{\Phi_\N})$ we have $$\label{full}
\l_{F\cup\{b\}}(t)\leq\l_{F\cup L_b}(t),$$ then $\Phi_\N$ has full Hausdorff dimension spectrum.
\[Mme\] Let $\Phi_\N$ be a conformal iterated function system such that $\dim_H(J_{\Phi_\N})\leq1$. For every $b\in\N$, let $S_b=\{1,2,\ldots,b-1\}$ and $L_b=\{b+1,b+2,\ldots\}$. If for some $b\in\N$ there are positive constants $M_b$ and $\{m_c\}_{c>b}$ such that $$m_b\|\varphi_{\om\overline{\om}}'\|
\leq
\|\varphi_{\om b\overline{\om}}'\|
\leq
M_b\|\varphi_{\om\overline{\om}}'\|$$ for all words $\om\in S_b^*$ and $\overline{\om}\in(S_b\cup\{b\})^*$ and so that $$M_b<\sum_{c=b+1}^\infty m_c,$$ then $$\l_{F\cup\{b\}}(t)\leq\l_{F\cup L_b}(t)$$ for all $F\subset\{1,2,\ldots,b-1\}$ and all $0<t\leq\dim_H(J_{\Phi_\N})$, so (due to Theorem 7), $\Phi_\N$ has full Hausdorff dimension spectrum.
Fix $b\in\N$ as in the statement of the theorem. Pick any $F\subset\{1,2,\ldots,b-1\}$ and $0\leq t\leq\dim_H(J_{\Phi_\N})$. Choose $n\geq b+1$ so that $M_b\leq\sum_{c=b+1}^n m_c$. Using Theorems \[Me1\] and \[me1\] (the second repeatedly), we obtain that $$\begin{aligned}
\l_{F\cup\{b\}}(t)
\leq\l_F(t)+M_b^t
\leq\l_F(t)+\left(\sum_{c=b+1}^n m_c\right)^t
&\leq&\l_F(t)+\sum_{c=b+1}^n m_c^t \\
&\leq&\l_{F\cup\{b+1\}}(t)+\sum_{c=b+2}^n m_c^t \\
&\leq&\l_{F\cup\{b+1,b+2\}}(t)+\sum_{c=b+3}^n m_c^t \\
&\leq&\ldots \\
&\leq&\l_{F\cup\{b+1,b+2,\ldots,n\}}(t) \\
&\leq&\l_{F\cup L_b}(t).\end{aligned}$$
\[useG\] In light of Lemmas \[2.4\] and \[2.5\], we can easily obtain some better constants $M_b$ and $m_b$ than were given in Remark \[distort\]. To see this, let $\om$ and $\overline{\om}$ be two admissible words and let $b$ be a letter from our alphabet. We have the following: $$\varphi'_{\om b \overline{\om}}(x) = \varphi'_{\om b}(\varphi_{\overline{\om}}(x))\varphi'_{\overline{\om}}(x)\leq \left(\frac{3}{2} \frac{1}{|b|-\alpha}\right)^2
\varphi'_{\om }(\varphi_{\overline{\om}}(x)) \varphi'_{\overline{\om}}(x) \leq \left(\frac{3}{2} \frac{1}{|b|-\alpha}\right)^2
\varphi'_{\om {\overline{\om}}}(x)$$ Thus we can take $M_b = \left(\frac{3}{2} \frac{1}{|b|-\alpha}\right)^2$ and, similarly, $m_b = \left(\frac{2}{3} \frac{1}{|b|+\alpha}\right)^2$.
Now we are ready to prove our first main result, with the aid of the following lemma.
\[2.6\] For every $k \geq 4$, we have: $$\label{*lem2.6}
\left(\frac{9}{4}\right) \frac{1}{(k - \alpha)^2} \leq 2 \left(\frac{4}{9}\right) \sum_{j \geq k+1} \frac{1}{(j+\alpha)^2}$$
Using the Integral Test yields that $$\sum_{j \geq k+1} \frac{1}{(j+\alpha)^2} \geq \frac{1}{k+1+\alpha}.$$ On the other hand, for every $k \geq 4$ we have that $$\frac{9}{4}\frac{1}{(k - \alpha)^2} \leq \frac{8}{9} \frac{1}{k+1+\alpha}.$$ This finishes the proof.
Now, combining Lemma \[2.6\] with Remark \[useG\] completes the proof of Theorem \[mainthm1\].
Proof of Theorem \[mainthm2\] {#sectionIFS2}
=============================
We will shortly describe in detail the IFS associated to the vertex $v$ of the GDMS introduced in Section 2 for the NICF. We refer back to that section for the definition of the alphabet $\overline{E}$. First, we give the general construction.
Suppose we have a CGDMS $\Phi= (V, E, i, t, A, \{X_v\}_{v\in V}, \{{\varphi}_e\}_{e\in E})$. For every vertex $v\in V$ we define the alphabet $E_v\subset E_A^*$ by induction as the union $\cup_{n=1}^\infty E_{v,n}$ as follows. To begin, define $$E_{v,1}:=\{e\in E:i(e)=t(e)=v\}.$$ Suppose now that all the sets $E_{v,k}\subset E_A^k$, for $k=1,\ldots,n$, have been defined. We then say that $\om\in E_A^{n+1}$ belongs to $E_{v,n+1}$ if $i(\om)=t(\om)=v$ and $\om$ is not the concatenation of words from $\cup_{k=1}^n E_{v,k}$. In other words, $E_{v,n}$ is the set of all $A$-admissible first-return loops of length $n$ originating from the vertex $v$. By construction, no element of $E_v$ is a concatenation of other elements of $E_v$. We further define the matrix $A^{(v)}:E_v\times E_v\to\{0,1\}$ by $A^{(v)}_{\om\overline{\om}}=1$ if and only if $\om\overline{\om}\in E_A^*$, that is, if and only if $A_{\om_{|\om|}\overline{\om}_1}=1$, where $\om,\overline{\om}\in E_v$. The system $\Phi_v$ is a CIFS whenever $\Phi$ satisfies $A_{ef}=1$ if and only if $t(e)=i(f)$. We then say that $\Phi_v$ is the CIFS associated with the CGDMS $\Phi$ via vertex $v$.
For our example, the associated iterated function system based at vertex $v$, which we will call $\Phi^{(v)}$, has for alphabet the first-return loops based at $v$, i.e., all the loops $|e|>2$, all the $(n+1)$-loops $2(\overline{2})^{n-1}\overline{e}$ with $e>2$, and all the $(n+1)$-loops $(-2)(\overline{-2})^{n-1}\overline{f}$ with $f<-2$, where $n\geq1$. Thus, $$\overline{E}^{(v)}=\left\{e:|e|>2\right\}
\bigcup
\left\{2(\overline{2})^{n-1}\overline{e}:e>2,n\geq1\right\}
\bigcup
\left\{(-2)(\overline{-2})^{n-1}\overline{f}:f<-2,n\geq1\right\}.$$ Recall from [@gdms] that the Hausdorff dimension of the limit set $J_{\overline{E}^{(v)}}$ of the associated iterated function system $\Phi^{(v)}$ is equal to the Hausdorff dimension of the limit set $J$ of the original graph directed system $\Phi$.
To shorten the notation, we shall replace $\overline{e}$ by $e$ whenever it is clear from the context which of $e$ and/or $\overline{e}$ is meant. For instance, in order to respect the graph, the word $(2)^n3$ is really the word $2(\overline{2})^{n-1}\overline{3}$.
Next, we impose the following order on the alphabet $\overline{E}^{(v)}$: $$\begin{array}{l}
-3, 3, -4, 4, \\
(-2)(-3), (-2)^2(-3), (-2)^3(-3), (2)(3), (2)^2(3), (2)^3(3), \\
(-2)(-4), (-2)^2(-4), (-2)^3(-4), (-2)^4(-4), (-2)^4(-3), \\
(2)(4), (2)^2(4), (2)^3(4), (2)^4(4), (2)^4(3), \\
-5, 5, \\
(-2)(-5), (-2)^2(-5), (-2)^3(-5), (-2)^4(-5), (-2)^5(-5), (-2)^5(-4), (-2)^5(-3), \\
(2)(5), (2)^2(5), (2)^3(5), (2)^4(5), (2)^5(5), (2)^5(4), (2)^5(3), \\
-6, 6, \\
(-2)(-6), (-2)^2(-6), (-2)^3(-6), (-2)^4(-6), (-2)^5(-6), (-2)^6(-6), (-2)^6(-5), (-2)^6(-4), \\
(-2)^6(-3), (2)(6), (2)^2(6), (2)^3(6), (2)^4(6), (2)^5(6), (2)^6(6), (2)^6(5), (2)^6(4), (2)^6(3), \ldots \\
\end{array}$$
For the calculations to follow, we will also need the following slightly different estimate of distortion.
\[distortword\] For any $\om,\tau\in E_A^*$ we have $$K_\om^{-1}\inf_{y\in X}|\varphi_\tau'(y)|\cdot\|\varphi_{\om\overline{\om}}'\|
\leq\|\varphi_{\om\tau\overline{\om}}'\|
\leq K_\om\|\varphi_\tau'\|\cdot\|\varphi_{\om\overline{\om}}'\|$$ where $K_\om$ is a constant of distortion for $\varphi_\om$, i.e., $$K_\om:=\sup_{x,y\in X}\frac{|\varphi_\om'(x)|}{|\varphi_\om'(y)|}\leq K,$$ where $$K=\sup_{\om\in E_A^*}K_\om$$ is a constant of distortion for the entire system.
Fix $\om,\tau\in E_A^*$. Then $$\begin{aligned}
\|\varphi_{\om\tau\overline{\om}}'\|
&=&\sup_{x\in X}|\varphi_{\om\tau\overline{\om}}'(x)| \\
&=&\sup_{x\in X}\left(|\varphi_\om'(\varphi_{\tau\overline{\om}}(x))|\cdot
|\varphi_\tau'(\varphi_{\overline{\om}}(x))|\cdot
|\varphi_{\overline{\om}}'(x)|\right) \\
&\leq&\|\varphi_\tau'\|
\sup_{x\in X}\left(|\varphi_\om'(\varphi_{\tau\overline{\om}}(x))|\cdot
|\varphi_{\overline{\om}}'(x)|\right) \\
&=&\|\varphi_\tau'\|
\sup_{x\in X}\left(|\varphi_\om'(\varphi_{\overline{\om}}(x))|\cdot
|\varphi_{\overline{\om}}'(x)|\cdot
\frac{|\varphi_\om'(\varphi_{\tau\overline{\om}}(x))|}
{|\varphi_\om'(\varphi_{\overline{\om}}(x))|}
\right) \\
&\leq&\|\varphi_\tau'\|K_\om
\sup_{x\in X}\left(|\varphi_\om'(\varphi_{\overline{\om}}(x))|\cdot
|\varphi_{\overline{\om}}'(x)|\right) \\
&=&K_\om\|\varphi_\tau'\|
\sup_{x\in X}|\varphi_{\om\overline{\om}}'(x)| \\
&\leq&K_\om\|\varphi_\tau'\|\cdot\|\varphi_{\om\overline{\om}}'\|.\end{aligned}$$ On the other hand, $$\begin{aligned}
\|\varphi_{\om\tau\overline{\om}}'\|
&=&\sup_{x\in X}|\varphi_{\om\tau\overline{\om}}'(x)| \\
&=&\sup_{x\in X}\left(|\varphi_\om'(\varphi_{\tau\overline{\om}}(x))|\cdot
|\varphi_\tau'(\varphi_{\overline{\om}}(x))|\cdot
|\varphi_{\overline{\om}}'(x)|\right) \\
&\geq&\inf_{y\in X}|\varphi_\tau'(y)|\cdot
\sup_{x\in X}\left(|\varphi_\om'(\varphi_{\tau\overline{\om}}(x))|\cdot
|\varphi_{\overline{\om}}'(x)|\right) \\
&=&\inf_{y\in X}|\varphi_\tau'(y)|\cdot
\sup_{x\in X}\left(|\varphi_\om'(\varphi_{\overline{\om}}(x))|\cdot
|\varphi_{\overline{\om}}'(x)|\cdot
\frac{|\varphi_\om'(\varphi_{\tau\overline{\om}}(x))|}
{|\varphi_\om'(\varphi_{\overline{\om}}(x))|}
\right) \\
&\geq&\inf_{y\in X}|\varphi_\tau'(y)|\cdot K_\om^{-1}
\sup_{x\in X}\left(|\varphi_\om'(\varphi_{\overline{\om}}(x))|\cdot
|\varphi_{\overline{\om}}'(x)|\right) \\
&=&K_\om^{-1}\inf_{y\in X}|\varphi_\tau'(y)|\cdot
\sup_{x\in X}|\varphi_{\om\overline{\om}}'(x)| \\
&\geq&K_\om^{-1}\inf_{y\in X}|\varphi_\tau'(y)|\cdot\|\varphi_{\om\overline{\om}}'\| \\
&\geq&K_\om^{-2}\|\varphi_\tau'\|\cdot\|\varphi_{\om\overline{\om}}'\|.
$$
In order to apply this result, we will need the following.
\[lem\_2s\] Let $\om\in \overline{E}_{\overline{A}}^*$ and suppose that $\om$ has the form $\om=\om_1\ldots \om_{n-k-1} \underbrace{2\ldots 2}_{k \ \text{times}} \om_n$, where $|\om_{n-k-1}|$ and $|\om_n|$ are both at least equal to 3. Then, $$\left|\frac{q_{n-1}}{q_n}\right| \leq \frac{k+2}{2k+5}.$$
We will use the estimate from the proof of Lemma \[2.1\] repeatedly: $$\begin{aligned}
\left|\frac{q_{n-1}}{q_n}\right|&\leq& \frac{1}{3- \left|\frac{q_{n-2}}{q_{n-1}}\right|}\\
&\leq& \frac{1}{3s- \frac{1}{2-\left|\frac{q_{n-3}}{q_{n-2}}\right|}}\\
&\leq& \ldots\\
&\leq& \frac{1}{3-\frac{1}{2- \frac{1}{2 - \ddots \frac{1}{2-\left|\frac{q_{n-k-2}}{q_{n-k-1}}\right| }}}}.\end{aligned}$$ Now, since $|\om_{n-k-1}|\geq3$, the last ratio satisfies $\left|\frac{q_{n-k-2}}{q_{n-k-1}}\right|\leq \frac12$. A simple calculation then finishes the proof. (Note that if the word $\om = 2\ldots 2\om_{k+1}$, the calculation stops one step earlier and we obtain a slightly better estimate.)
\[lem\_est\_distort\] For the system $\Phi^{(v)}$, we may take $K=25/9$.
Let $\om \in (\overline{E}^{(v)})_{\overline{A}}^*$ and $x,y\in X_v$. Recalling that $|\om|$ refers to the length of the word $\om$ considered as consisting of letters from $\overline{E}$, we note from the proof of Lemma \[2.1\] that $$\left|\frac{q_{k-1}}{q_{k}}\right|\leq \frac{1}{|\om_{|\om|}|-\left|\frac{q_{k-2}}{q_{k-1}}\right|}$$ Note that $|\om_{|\om|}|\geq3$, by the definition of the alphabet $\overline{E}^{(v)}$. It then follows from the fact that $|q_{n-1}|\leq |q_n|$ for all $n\geq1$ and from the above observation that $$\begin{aligned}
\label{*p17}
\left|\frac{q_{|\omega|-1}}{q_{|\omega|}}\right|\leq \frac{1}{3-1}=\frac12.\end{aligned}$$ Using (\[derivative\]) and (\[\*p17\]), we obtain that $$\begin{aligned}
\frac{|\varphi_\om'(x)|}{|\varphi_\om'(y)|}
&=&\left|\frac{q_{|\om|}+yq_{|\om|-1}}{q_{|\om|}+xq_{|\om|-1}}\right|^2
\leq\left(\frac{|q_{|\om|}|+|y||q_{|\om|-1}|}{|q_{|\om|}|-|x||q_{|\om|-1}|}\right)^2 \\
&\leq&\left(\frac{|q_{|\om|}|+\frac{1}{2}|q_{|\om|-1}|}{|q_{|\om|}|-\frac{1}{2}|q_{|\om|-1}|}\right)^2 \leq\left(\frac{|q_{|\om|}|+\frac{1}{2}\cdot\frac{1}{2}|q_{|\om|}|}
{|q_{|\om|}|-\frac{1}{2}\cdot\frac{1}{2}|q_{|\om|}|}\right)^2 \\
&=&25/9.\end{aligned}$$ Hence $25/9$ is a constant of bounded distortion for our system.
We are now ready to begin the proof of Theorem \[mainthm2\].
We will mostly use Theorem \[Mme\] to establish that the associated iterated function system $\Phi^{(v)}$ has full spectrum, and the proof is split into several different cases. Note that we can express the letters in $\overline{E}^{(v)}$ in the following general form: $2^jk$ and $(-2)^j(-k)$, where $j\geq0$ and $k>2$. The calculations below involve the derivatives of the generators. Due to the symmetry in the system, we have $\|\varphi_b'\|=\|\varphi_{-b}'\|$ for all $|b|\geq2$. Consequently, the letters $2^jk$ and $(-2)^j(-k)$ can be treated in the same manner. Without loss of generality, we will restrict our attention to the letters $2^jk$.
[**Case of the letters $2^jk$, where $j>k$.**]{}
According to our ordering of the letters of $\overline{E}^{(v)}$, if $j>k\geq3$ then the letter $2^jk$ precedes the letters $\pm l$, where $l\geq j+1$. It is thus sufficient to prove that $$\begin{aligned}
\label{*p18}
M_{2^jk}\leq2\sum_{l=j+1}^\infty m_l,\end{aligned}$$ where, according to Remark \[distort\], we may take $$m_l=K^{-1}\inf_{x\in X_v}|\varphi_l'(x)|
=\frac{9}{25}\frac{1}{(l+\frac{1}{2})^2},$$ and $$M_{2^jk}
=K\|\varphi_{2^jk}'\|
\leq K\|\varphi_2'\|^j\|\varphi_k'\|
=\frac{25}{9}\left(\frac{1}{2^2}\right)^j\frac{1}{(k-\frac{1}{2})^2}
\leq\frac{25}{9}\left(\frac{1}{2^2}\right)^j\frac{1}{(3-\frac{1}{2})^2}
=\frac{4}{9}\left(\frac{1}{2^2}\right)^j.$$ Substituting these values into (\[\*p18\]), we see that it suffices to prove that $$\frac{4}{9}\left(\frac{1}{2^2}\right)^j
\leq2\cdot\frac{9}{25}\sum_{l=j+1}^\infty \frac{1}{(l+\frac{1}{2})^2},$$ or, in other words, that $$\frac{50}{81}\leq 2^{2j+1}\sum_{l=j+1}^\infty \frac{1}{(l+\frac{1}{2})^2}.$$ Using the integral test yields that $$\sum_{l=j+1}^\infty \frac{1}{(l+\frac{1}{2})^2}
\geq\frac{1}{j+1+\frac{1}{2}}.$$ Consequently, proving (\[\*p18\]) boils down to proving that $$\frac{50}{81}(j+1+\frac{1}{2})\leq 2^{2j+1},$$ which is certainly satisfied for $j>k\geq3$.
[**Case of the letters $2^jk$, where $1\leq j\leq k$.**]{}
According to our ordering of the letters of $\overline{E}^{(v)}$, if $1\leq j\leq k$ then the letter $2^jk$ precedes the letters $\pm l$, where $l\geq k+2$. It is thus sufficient to prove that $$M_{2^jk}\leq2\sum_{l=k+2}^\infty m_l,$$ where, as above, we may take $$m_l=\frac{9}{25}\frac{1}{(l+\frac{1}{2})^2}\ \text{ and } \ M_{2^jk}\leq \frac{25}{9} \left(\frac{1}{2^2}\right)^j\frac{1}{(k-\frac{1}{2})^2}.$$ Using the integral test again, we have that $$\sum_{l=k+2}^\infty \frac{1}{(l+\frac{1}{2})^2}
\geq\frac{1}{k+2+\frac{1}{2}}.$$ Consequently, exactly analogously to the first case, it is sufficient to show that $$\left(\frac{25}{9}\right)^2\frac{(k+\frac{5}{2})}{(k-\frac{1}{2})^2}\leq 2^{2j+1}.$$ It is then easy to show that the left-hand side is a decreasing function of $k$ when $k\geq3$. Therefore it suffices to show that $$\left(\frac{25}{9}\right)^2\frac{(3+\frac{5}{2})}{(3-\frac{1}{2})^2}\leq 2^{2j+1}.$$ One immediately verifies that this is true for all $j\geq1$.
[**Case of the letters $k$, where $k\geq6$.**]{}
According to our ordering of the letters of $\overline{E}^{(v)}$, the letter $k$, for $k\geq6$, precedes the letters $\pm l$, where $l\geq k+1$. It is thus sufficient to prove that $$M_k\leq2\sum_{l=k+1}^\infty m_l,$$ where once again using Remark \[distort\] we may take $$m_l=\frac{9}{25}\frac{1}{(l+\frac{1}{2})^2} \ \text{ and } \
M_k =K\|\varphi_k'\ |=\frac{25}{9}\frac{1}{(k-\frac{1}{2})^2}.$$ It therefore suffices to prove that $$\left(\frac{25}{9}\right)^2\frac{1}{(k-\frac{1}{2})^2}
\leq 2\sum_{l=k+1}^\infty \frac{1}{(l+\frac{1}{2})^2}.$$ Using the integral test again gives $$\sum_{l=k+1}^\infty \frac{1}{(l+\frac{1}{2})^2}
\geq\frac{1}{k+1+\frac{1}{2}}.$$ Consequently, it is sufficient to show that $$\label{esti}
\left(\frac{25}{9}\right)^2\frac{(k+\frac{3}{2})}{(k-\frac{1}{2})^2}\leq 2.$$ One immediately verifies that the left-hand side is a decreasing function of $k$ when $k\geq3$. The smallest value of $k$ for which relation (\[esti\]) holds is $k=6$.
[**Case of the letters $\pm5$.**]{}
We have just proved the case $k\geq6$. To prove the result for smaller values of $k$, we need better estimates on the distortion and to consider more of the letters following $k$. Since the words $\om$ in Theorems \[Me1\] and \[me1\] can be taken to be composed of letters that precede $k$, according to Lemma \[distortword\] we may always replace $K$ by $\max_\om K_\om$, where the maximum is taken over all words comprising only letters that precede $k$. Moreover, according to our ordering of the letters of $\overline{E}^{(v)}$, the letter $k$ precedes the letters $\pm l$, where $l\geq k+1$, as well as the letters $2^rl$ and $(-2)^r(-l)$ for all $r\geq1$ and $l\geq k+1$. It is thus sufficient to prove that $$\begin{aligned}
\label{*p19}
M_k
\leq2\sum_{l=k+1}^\infty m_l+2\sum_{l=k+1}^\infty\sum_{r=1}^\infty m_{2^rl}
=2\sum_{l=k+1}^\infty\left[m_l+\sum_{r=1}^\infty m_{2^rl}\right].\end{aligned}$$ Using Remark \[distort\] once again, we may take $$m_l=\frac{9}{25}\frac{1}{(l+\frac{1}{2})^2} \ \text{ and }\ m_{2^rl}=K^{-1}\inf_{x\in X_v}|\varphi_{2^rl}'(x)|
\geq\frac{9}{25}\frac{1}{\left[(\frac{3}{2}+\sqrt{2})(1+\sqrt{2})^{r-1}\right]^2}
\frac{1}{(l+\frac{1}{2})^2}.$$ The latter inequality above comes from the following calculation: First observe that for the letter $2^r$ a straightforward induction argument shows that $1\leq q_n\leq(1+\sqrt{2})^n$ for all $0\leq n\leq r$. Then $$\begin{aligned}
\inf_{x\in X_v}|\varphi_{2^r}'(x)|
&=&\inf_{x\in[0,1/2]}\frac{1}{(q_r+xq_{r-1})^2} \\
&=&\frac{1}{(q_r+\frac{1}{2}q_{r-1})^2} \\
&\geq&\frac{1}{\left[(1+\sqrt{2})^r+\frac{1}{2}(1+\sqrt{2})^{r-1}\right]^2} \\
&\geq&\frac{1}{\left[(\frac{3}{2}+\sqrt{2})(1+\sqrt{2})^{r-1}\right]^2}.\end{aligned}$$
For the left-hand side of (\[\*p19\]), according to Lemma \[distortword\], we may choose $$M_k
=\left(\sup_{\om\prec5}K_\om\right)\|\varphi_k'\|
=\left(\frac{27}{17}\right)^2\frac{1}{(k-\frac{1}{2})^2},$$ where the supremum is taken over all words $\om\in(\overline{E}^{(v)})_{\overline{A}}^*$ comprising only letters that precede $5$. Moreover, in light of Lemma \[lem\_2s\], we have that $\sup_{\om\prec5}K_\om=\left(\frac{8}{5}\right)^2$ (this follows from a calculation identical to that in Lemma \[lem\_est\_distort\]). It therefore suffices to prove that $$\left(\frac{8}{5}\right)^2\frac{1}{(k-\frac{1}{2})^2}
\leq
2\cdot\frac{9}{25}\sum_{l=k+1}^\infty \frac{1}{(l+\frac{1}{2})^2}
\left\{1+\sum_{r=1}^\infty\frac{1}{\left[(\frac{3}{2}+\sqrt{2})(1+\sqrt{2})^{r-1}\right]^2}\right\},$$ Using the integral test, it is sufficient to show that $$\label{ineq}
\frac{25}{18}\left(\frac{8}{5}\right)^2\frac{(k+\frac{3}{2})}{(k-\frac{1}{2})^2}
\leq
1+\frac{1}{(\frac{3}{2}+\sqrt{2})^2}\sum_{r=1}^\infty\frac{1}{[(1+\sqrt{2})^2]^{r-1}}
$$ It is again a straightforward calculation to show that the left-hand side is a decreasing function of $k$ when $k\geq3$. It is then easy to establish that $$\frac{25}{18}\left(\frac{8}{5}\right)^2\frac{(5+\frac{3}{2})}{(5-\frac{1}{2})^2}
\leq1+\frac{1+\sqrt{2}}{2(\frac{3}{2}+\sqrt{2})^2}.$$ Relation (\[ineq\]) thus holds for all $k\geq5$.
[**Case of the letters $\pm4$.**]{}
We have so far proved the case $k\geq5$. To prove the result for smaller values of $k$, we need an even better estimate on the distortion and to take all the letters following $k$. Since the words $\om$ in Theorems \[Me1\] to \[me1\] can be taken to be composed of letters that precede $k$, in light of Lemma \[distortword\] we may always replace $K$ by $\max_\om K_\om$, where the maximum is taken over all words comprising only letters that precede $k$. Moreover, according to our ordering of the letters of $\overline{E}^{(v)}$, the letter $k$ precedes the letters $\pm l$, where $l\geq k+1$, as well as the letters $2^rl$ and $(-2)^r(-l)$ for all $r\geq1$ and $l\geq3$. It is thus sufficient to prove that $$M_4\leq2\sum_{l=5}^\infty m_l+2\sum_{l=3}^\infty\sum_{r=1}^\infty m_{2^rl},$$ where, according to Lemma \[distortword\], we may take $$m_l=(\sup_{\om\prec l}K_\om)^{-1}\inf_{x\in X_v}|\varphi_l'(x)|
=\left(\frac{3l+5}{5l+7}\right)^2\frac{1}{(l+\frac{1}{2})^2},$$ where the supremum is taken over all words $\om\in(\overline{E}^{(v)})_{\overline{A}}^*$ comprising only letters that precede $l$ and $$m_{2^rl}=K^{-1}\inf_{x\in X_v}|\varphi_{2^rl}'(x)|
\geq\frac{9}{25}\frac{1}{\left[(\frac{3}{2}+\sqrt{2})(1+\sqrt{2})^{r-1}\right]^2}\frac{1}{(l+\frac{1}{2})^2}.$$ Here we obtain that $\sup_{\om\prec l}K_\om= ((5l+7)/(3l+5))^2$ by applying Lemma \[lem\_2s\] and making a calculation as in the previous case.
We also have that $$M_4
=\left(\sup_{\om\prec4}K_\om\right)\|\varphi_4'\|
=\left(\frac{1+\frac{3-\sqrt{5}}{4}}{1-\frac{3-\sqrt{5}}{4}}\right)^2\frac{1}{(4-\frac{1}{2})^2},$$ where the supremum is taken over all words $\om\in(\overline{E}^{(v)})_{\overline{A}}^*$ comprising only letters that precede $4$. Indeed, $$\sup_{\om\prec4}K_\om
=\left(\frac{1+\frac{3-\sqrt{5}}{4}}{1-\frac{3-\sqrt{5}}{4}}\right)^2 = \left(\frac{7-\sqrt 5}{1+\sqrt 5}\right)^2,$$ as one can show that for any word $\om\in(\overline{E}^{(v)})_{\overline{A}}^*$ comprising only letters that precede $4$, we have $|q_{n-1}|\leq\frac{3-\sqrt{5}}{2}|q_n|$ for all $0\leq n\leq|\om|$ since in this case we have $3$ repeated any finite number of times, and the solution in $[-1/2, 1/2]$ to the equation $x=1/(3-x)$ is $\frac{3-\sqrt{5}}{2}$. Then we again make a calculation as in Lemma \[lem\_est\_distort\].
It therefore suffices to prove that $$\left(\frac{7-\sqrt5}{1+\sqrt5}\right)^2
\left(\frac{4}{49}\right)^2
\leq
2\sum_{l=5}^\infty \left(\frac{3l+5}{5l+7}\right)^2\frac{1}{(l+\frac{1}{2})^2}
+2\cdot\frac{9}{25}\sum_{l=3}^\infty \sum_{r=1}^\infty
\frac{1}{\left[(\frac{3}{2}+\sqrt{2})(1+\sqrt{2})^{r-1}\right]^2}\frac{1}{(l+\frac{1}{2})^2},$$ i.e. $$\left(\frac{7-\sqrt5}{1+\sqrt5}\right)^2
\left(\frac{4}{49}\right)^2
\leq
2\sum_{l=5}^\infty \left(\frac{3l+5}{5l+7}\right)^2\frac{1}{(l+\frac{1}{2})^2}
+\frac{18/25}{(\frac{3}{2}+\sqrt{2})^2}\sum_{r=1}^\infty\frac{1}{[(1+\sqrt{2})^2]^{r-1}}
\sum_{l=3}^\infty \frac{1}{(l+\frac{1}{2})^2}.$$ Using the integral test, we have that $$\sum_{l=k}^\infty \frac{1}{(l+\frac{1}{2})^2}
\geq\frac{1}{k+\frac{1}{2}}.$$ Consequently, it is sufficient to show that $$\left(\frac{7-\sqrt5}{1+\sqrt5}\right)^2
\left(\frac{4}{49}\right)^2
\leq
2\sum_{l=5}^\infty \left(\frac{3l+5}{5l+7}\right)^2\frac{1}{(l+\frac{1}{2})^2}
+\frac{18/25}{(\frac{3}{2}+\sqrt{2})^2}\frac{1}{1-\frac{1}{(1+\sqrt{2})^2}}
\frac{1}{3+\frac{1}{2}}.$$ Hence it is sufficient to show that $$\left(\frac{7-\sqrt5}{1+\sqrt5}\right)^2
\left(\frac{4}{49}\right)^2
\leq2\sum_{l=5}^\infty \left(\frac{3l+2}{5l+2}\right)^2\frac{1}{(l+\frac{1}{2})^2}
+\frac{18}{175}\frac{1+\sqrt{2}}{(\frac{3}{2}+\sqrt{2})^2}.$$ Numerical calculations using Mathematica show that this relation is true.
[**Case of the letter $-3$.**]{}
Rather than using Theorem \[Mme\], we shall show directly that relation (\[full\]) holds. Since $-3$ is the first letter in the alphabet $\overline{E}^{(v)}$, relation (\[full\]) holds as $-3$ is followed by $3$ and $$\l_{\{-3\}}(t)=\l_{\{3\}}(t)\leq\l_{\overline{E}^{(v)}\backslash\{-3\}}(t)$$ for all $t\geq0$.
[**Case of the letter $3$.**]{}
Again, we shall show directly that relation (\[full\]) holds. Since $3$ is the second letter in the alphabet $\overline{E}^{(v)}$, relation (\[full\]) holds as $$\l_{\{-3\}}(t)\leq\l_{\{-3,3\}}(t)
\leq\l_{\{l:|l|\geq4\}}(t)
\leq\l_{\overline{E}^{(v)}\backslash\{-3,3\}}(t)
\leq\l_{\overline{E}^{(v)}\backslash\{-3\}}(t)$$ for all $0\leq t\leq1$. Indeed, let us prove that $\l_{\{-3,3\}}(t)\leq\l_{\{l:|l|\geq4\}}(t)$ for all $0\leq t\leq1$. On the one hand, we have that $$\label{z3}
\l_{\{-3,3\}}(t)
\leq Z_{1,\{-3,3\}}(t)
=\|\varphi_{-3}'\|^t+\|\varphi_3'\|^t
=\frac{2}{(3-\frac{1}{2})^{2t}}
=\frac{2}{\left(\frac{5}{2}\right)^{2t}}.$$ On the other hand, we have that $$\l_{\{l:|l|\geq4\}}(t)\geq K_4^{-1}Z_{1,\{l:|l|\geq4\}}(t)
=2K_4^{-1}\sum_{l=4}^\infty\|\varphi_l'\|^t
=2K_4^{-1}\sum_{l=4}^\infty\frac{1}{(l-\frac{1}{2})^{2t}},$$ where $K_4$ is a constant of bounded distortion for the subsystem $\{\varphi_l:|l|\geq4\}$. Thus, since $\l_{\{-3,3\}}(t)$ is finite and $\l_{\{l:|l|\geq4\}}(t)$ is infinite whenever $t\leq1/2$, we have $\l_{\{-3,3\}}(t)<\l_{\{l:|l|\geq4\}}(t)$. When $t>1/2$, it follows from the integral test that $$\label{z4}
\l_{\{l:|l|\geq4\}}(t)
\geq2K_4^{-1}\frac{1}{(2t-1)(4-\frac{1}{2})^{2t-1}}
=2K_4^{-1}\frac{1}{(2t-1)(\frac{7}{2})^{2t-1}}.$$ One can show that for any word $\om\in\{l:|l|\geq4\}^*$, we have $|q_{n-1}|\leq(2-\sqrt{3})|q_n|$ for all $0\leq n\leq|\om|$ as the worst case scenario is to have the letters $-4$ and/or $4$ repeated any finite number of times, and $2-\sqrt3$ is the solution of $x=1(4-x)$ in $[-1/2, 1/2]$. Then calculating as in Lemma \[lem\_est\_distort\] again, we can take $$\label{k4}
K_4=\left(\frac{1+\frac{2-\sqrt{3}}{2}}{1-\frac{2-\sqrt{3}}{2}}\right)^2= \left(\frac{4-\sqrt3}{1+\sqrt3}\right)^2.$$ According to (\[z3\]), (\[z4\]) and (\[k4\]), to prove $\l_{\{-3,3\}}(t)\leq\l_{\{l:|l|\geq4\}}(t)$ when $t>1/2$ it suffices to show that $$\frac{1}{(2t-1)}\left(\frac{5}{7}\right)^{2t}
\geq\frac{2}{7}\left(\frac{4-\sqrt3}{1+\sqrt3}\right)^2.$$ By looking at its first derivative, it is easy to see that the left-hand side is a decreasing function of $t$ on $1/2<t\leq1$. Thus, we only need to show that $$\left(\frac{5}{7}\right)^2
\geq\frac{2}{7}\left(\frac{4-\sqrt3}{1+\sqrt3}\right)^2.$$ Numerical calculations show that this relation is true.
We have hence demonstrated that relation (\[full\]) holds for all letters of $\overline{E}^{(v)}$ under the ordering we chose. Therefore the system $\Phi^{(v)}$ has full spectrum according to Theorem \[fullspec\].
Appendix
========
In this appendix we follow the ideas from [@gdms], and add some explanatory examples.
The following is a restatement of Proposition 4.7.2 in [@gdms] with an annotated proof.
\[jivjv\] Suppose that $\Phi$ is a CGDMS with an irreducible matrix. For every vertex $v\in V$ the limit set $J_{E_v}:=\pi(E_v^\infty)$ of $\Phi_v$ is contained in the subset $J_v:=\pi(\{\om\in E_A^\infty:i(\om)=v\})$ of the limit set of $\Phi$. Moreover, $\overline{J_{E_v}}=\overline{J_v}$.
Since $E_v^\infty\subset\{\om\in E_A^\infty:i(\om)=v\}$, we have $J_{E_v}\subset J_v$. Hence $\overline{J_{E_v}}\subset\overline{J_v}$. In order to prove the opposite inclusion it suffices to demonstrate that each element of $J_v$ is the limit of elements of $J_{E_v}$. Indeed, let $x=\pi(\om)$, where $\om\in E_A^\infty$ with $i(\om)=v$. Since $A$ is irreducible, for every $n\in \N$ there exist $\a^{(n)}\in E_A^*$ and $\b^{(n)}\in
E_v^\infty$ such that $\om|_n\a^{(n)}\b^{(n)}\in E_A^\infty$. Since $\b^{(n)}\in E_v^\infty$ and $i(\om)=v$, we have $\om|_n\a^{(n)}\b^{(n)}\in E_v^\infty$. Hence $\pi(\om|_n\a^{(n)}\b^{(n)})\in J_{E_v}$ for every $n\in\N$ and thus $\lim_{n\to\infty}\pi(\om|_n\a^{(n)}\b^{(n)})
=\pi(\lim_{n\to\infty}\om|_n\a^{(n)}\b^{(n)})=\pi(\om)=x$. Consequently, $x\in\overline{J_{E_v}}$. Since $x$ was chosen arbitrarily in $J_v$, we deduce that $J_v\subset\overline{J_{E_v}}$. Hence $\overline{J_v}\subset\overline{J_{E_v}}$.
We shall now compare the pressures of the original and the associated systems.
\[pressass\] If $\Phi$ is a CGDMS with a finitely irreducible matrix, then $$P(t)\leq\min_{v\in V}P_{E_v}(t) \mbox{ whenever } P(t)>0$$ and $$P(t)=\max_{v\in V}P_{E_v}(t) \mbox{ whenever } P(t)\leq0,$$ where $P_{E_v}(t)$ is the pressure of the system $\Phi_v$.
First, we prove that $P(t)\leq\min_{v\in V}\max\{P_{E_v}(t),0\}$. Fix $v\in V$. If $P_{E_v}(t)=\infty$, then clearly $P(t)\leq\max\{P_{E_v}(t),0\}$. If $P_{E_v}(t)<\infty$, then let $u>\max\{P_{E_v}(t),0\}$. Let $W\subset E_A^*$ be a finite set witnessing the irreducibility of $A$. Let $\rho:=\min\{\|\varphi_\tau'\|:\tau\in W\}$ and $\lambda:=\max\{|\tau|:\tau\in W\}$. Then $\rho>0$ and $\lambda<\infty$. For every $e\in E$ let $\a(e),\b(e)\in W$ be such that $i(\a(e))=v$, $t(\b(e))=v$ and $\a(e)e\b(e)\in E_A^*$. Set $\a(\om):=\a(\om_1)$ and $\b(\om):=\b(\om_{|\om|})$ for every $\om\in E_A^*$. Observe that the function $\om\mapsto\a(\om)\om\b(\om)$ is at most $\lambda$-to-one. Indeed, suppose that $\om,\tau\in E_A^*$ are such that $\a(\om)\om\b(\om)=\a(\tau)\tau\b(\tau)$. If $|\a(\om)|=|\a(\tau)|$, then $\a(\om)\om\b(\om)=\a(\tau)\tau\b(\tau)$ forces $\a(\om)=\a(\tau)$. This in turn imposes that $\om\b(\om)=\tau\b(\tau)$. Without loss of generality, we may assume that $|\om|\leq|\tau|$. Then $\tau=\om\star\b(\om)|_{|\tau|-|\om|}$. There are at most $\l$ such $\tau$ since $\b(\cdot)\in W\subset\bigcup_{k=1}^\lambda E_A^k$. As $\a(\cdot)\in W\subset\bigcup_{k=1}^\lambda E_A^k$, there are at most $\l$ of the $\a(\cdot)$’s that are of different lengths, and for each of these there are at most $\l$ preimages. Thus, the function $\om\mapsto\a(\om)\om\b(\om)$ is at most $\lambda^2$-to-one. Furthermore, notice that $\a(\om)\om\b(\om)\in\bigcup_{k=|\om|}^{2\lambda+|\om|}E_A^k$. Moreover, recall that $|\om|_v\leq|\om|$ for every $\om\in E_v^*$. Then $$\begin{aligned}
\sum_{\om\in E_A^*}\|\varphi_\om'\|^t e^{-u|\om|}
&\leq&(K\rho^{-1})^{2t}\sum_{\om\in E_A^*}
\|\varphi_{\a(\om)\om\b(\om)}'\|^t e^{-u|\om|} \\
&\leq&(K\rho^{-1})^{2t}e^{2\lambda u}\sum_{\om\in E_A^*}
\|\varphi_{\a(\om)\om\b(\om)}'\|^t e^{-u|\a(\om)\om\b(\om)|} \\
&\leq&\lambda^2(K\rho^{-1})^{2t}e^{2\lambda u}\sum_{\tau\in E_v^*}\|\varphi_\tau'\|^t e^{-u|\tau|_v} \\
&<&\infty.\end{aligned}$$ The first inequality is a direct repercussion of the bounded distortion of the system. The second inequality follows from the fact that $|\a(\om)\om\b(\om)|\leq2\lambda+|\om|$ for every $\om\in E_A^*$ and that $u>0$. The third inequality is a consequence of the fact the function $\om\mapsto\a(\om)\om\b(\om)$ is at most $\lambda^2$-to-one, that $|\tau|_v\leq|\tau|$ for every $\tau\in E_v^*$, and that $u>0$. Finally, the last inequality follows from Theorem 2.1.3 in [@gdms] since $u>P_{E_v}(t)$. Since $\sum_{\om\in E_A^*}\|\varphi_\om'\|^t e^{-u|\om|}<\infty$, Theorem 2.1.3 in [@gdms] affirms that $u>P(t)$. Since this is true for every $u>\max\{P_{E_v}(t),0\}$, we deduce that $\max\{P_{E_v}(t),0\}\geq P(t)$. Since this holds for every $v\in V$, we conclude that $P(t)\leq\min_{v\in V}\max\{P_{E_v}(t),0\}$.
In particular, note that if $P(t)>0$, then $P(t)\leq\min_{v\in V}P_{E_v}(t)$.
Secondly, we show that $\max_{v\in V}P_{E_v}(t)\leq P(t)$ whenever $P(t)\leq0$. Let $v\in V$. Let $t$ be such that $P(t)<0$ and $P(t)<u\leq0$. Then $\sum_{\om\in E_A^*}\|\varphi_\om'\|^t e^{-u|\om|}<\infty$ according to Theorem 2.1.3 in [@gdms] since $u>P(t)$. Since $u\leq0$, we deduce that $$\sum_{\tau\in E_v^*}\|\varphi_\tau'\|^t e^{-u|\tau|_v}
\leq\sum_{\om\in E_A^*}\|\varphi_\om'\|^t e^{-u|\om|}<\infty.$$ Thus, $P_{E_v}(t)\leq u$ by Theorem 2.1.3 in [@gdms]. Since this holds for every $P(t)<u\leq0$, we conclude that $P_{E_v}(t)\leq P(t)$ whenever $P(t)<0$. The right-continuity of the pressure function ensures that $P_{E_v}(t)\leq P(t)$ if $P(t)=0$ for some $t$. Hence $P_{E_v}(t)\leq P(t)$ whenever $P(t)\leq0$. Since the vertex $v$ was chosen arbitrarily, we conclude that $\max_{v\in V}P_{E_v}(t)\leq P(t)$ whenever $P(t)\leq0$.
Thirdly, we prove that $P(t)\leq\max_{v\in V}P_{E_v}(t)$ for all $t\geq0$. To do this, fix $t\geq0$. Let $W\subset E_A^*$ be a finite set witnessing the irreducibility of $A$. Let $\rho:=\min\{\|\varphi_\tau'\|:\tau\in W\}$ and $\lambda:=\max\{|\tau|:\tau\in W\}$. Then $\rho>0$ and $\lambda<\infty$. For every $v\in V$ and every $e\in E$ there exist $\a_v(e),\b_v(e)\in W$ such that $i(\a_v(e))=v$, $t(\b_v(e))=v$ and $\a_v(e)e\b_v(e)\in E_A^*$. Set $\a_v(\om):=\a_v(\om_1)$ and $\b_v(\om):=\b_v(\om_{|\om|})$ for every $\om\in E_A^*$. As previously, note that the function $\om\mapsto\a_v(\om)\om\b_v(\om)$ is at most $\lambda^2$-to-one and that $\a_v(\om)\om\b_v(\om)\in\cup_{k=|\om|}^{2\lambda+|\om|}E_A^k$, that is, every word $\om$ generates words $\a_v(\om)\om\b_v(\om)$ whose lengths are at most $2\lambda+|\om|$ in the alphabet $E$ and thereby whose $v$-lengths, i.e. as a concatenation of letters of the alphabets $E_v$, are at most $2\lambda+|\om|$. Moreover, there is a vertex $v(\om)\in V$ such that the word $\om$ visits $v(\om)$ at least $[|\om|/|V|]+1$, where $[\cdot]$ denotes the integer part function. This means that the $v(\om)$-length of $\a_{v(\om)}(\om)\om\b_{v(\om)}(\om)$ is at least $[|\om|/|V|]$. For each $v\in V$ and each $n\in\N$, let $[n/|V|]\leq k(v,n)\leq 2\lambda+n$ be such that $$\sum_{\tau\in E_v^{k(v,n)}}\|\varphi_\tau'\|^t
=\max_{[n/|V|]\leq k\leq 2\lambda+n} \sum_{\tau\in E_v^k}\|\varphi_\tau'\|^t.$$ Thereafter, let $v(n)\in V$ be such that $$\sum_{\tau\in E_{v(n)}^{k(v(n),n)}}\|\varphi_\tau'\|^t
=\max_{v\in V} \sum_{\tau\in E_v^{k(v,n)}}\|\varphi_\tau'\|^t.$$ Then for every $n\in\N$, we have $$\begin{aligned}
\sum_{\om\in E_A^n}\|\varphi_\om'\|^t
&\leq&(K\rho^{-1})^{2t}
\sum_{\om\in E_A^n}\|\varphi_{\a_{v(\om)}(\om)\om\b_{v(\om)}(\om)}'\|^t \\
&\leq&\lambda^2(K\rho^{-1})^{2t}\sum_{v\in V}
\sum_{\tau\in\bigcup_{k=[n/|V|]}^{2\lambda+n}E_v^k}\|\varphi_\tau'\|^t \\
&=&\lambda^2(K\rho^{-1})^{2t}\sum_{v\in V}
\sum_{k=[n/|V|]}^{2\lambda+n}
\sum_{\tau\in E_v^k}\|\varphi_\tau'\|^t \\
&\leq&\lambda^2(K\rho^{-1})^{2t}\sum_{v\in V}
(2\lambda+n)\max_{[n/|V|]\leq k\leq 2\lambda+n} \sum_{\tau\in E_v^k}\|\varphi_\tau'\|^t \\
&=&\lambda^2(K\rho^{-1})^{2t}(2\lambda+n)\sum_{v\in V}\sum_{\tau\in E_v^{k(v,n)}}\|\varphi_\tau'\|^t \\
&\leq&\lambda^2(K\rho^{-1})^{2t}(2\lambda+n)|V|\max_{v\in V}\sum_{\tau\in E_v^{k(v,n)}}\|\varphi_\tau'\|^t \\
&=&\lambda^2(K\rho^{-1})^{2t}|V|(2\lambda+n)\sum_{\tau\in E_{v(n)}^{k(v(n),n)}}\|\varphi_\tau'\|^t.\end{aligned}$$ Since $|V|<\infty$, there exists $v\in V$ and a strictly increasing subsequence $\{n_m\}_{m\in\N}$ of natural numbers such that $v(n_m)=v$ for all $m\in\N$. Therefore $$\begin{aligned}
P(t)
&=&\lim_{m\to\infty}\frac{1}{n_m}\log\sum_{\om\in E_A^{n_m}}\|\varphi_\om'\|^t \\
&\leq&\lim_{m\to\infty}\frac{1}{n_m}\log\left(\lambda^2(K\rho^{-1})^{2t}|V|\right)
+\lim_{m\to\infty}\frac{1}{n_m}\log(2\lambda+n_m)
+\lim_{m\to\infty}\frac{1}{n_m}\log\hspace{-0.25cm}
\sum_{\tau\in E_{v(n_m)}^{k(v(n_m),n_m)}}\hspace{-0.5cm}\|\varphi_\tau'\|^t \\
&=&0+0+\lim_{m\to\infty}\frac{1}{n_m}\log
\sum_{\tau\in E_v^{k(v,n_m)}}\|\varphi_\tau'\|^t \\
&=&\lim_{m\to\infty}\frac{k(v,n_m)}{n_m}\frac{1}{k(v,n_m)}\log
\sum_{\tau\in E_v^{k(v,n_m)}}\|\varphi_\tau'\|^t \\
&\leq&\lim_{m\to\infty}\frac{2\lambda+n_m}{n_m}\frac{1}{k(v,n_m)}\log
\sum_{\tau\in E_v^{k(v,n_m)}}\|\varphi_\tau'\|^t \\
&=&\lim_{m\to\infty}\frac{2\lambda+n_m}{n_m}
\cdot\lim_{m\to\infty}\frac{1}{k(v,n_m)}\log
\sum_{\tau\in E_v^{k(v,n_m)}}\|\varphi_\tau'\|^t \\
&=&1\cdot P_{E_v}(t),\end{aligned}$$ where it is important to remember that $\lim_{m\to\infty}k(v,n_m)\geq\lim_{m\to\infty}[n_m/|V|]=\infty$. Thus, $P(t)\leq\max_{v\in V}P_{E_v}(t)$ for all $t\geq0$. Taken together, the second and third parts allow us to deduce that $P(t)=\max_{v\in V}P_{E_v}(t)$ when $P(t)\leq0$.
The relationship between the pressures of the original and the associated systems ensures that the limit sets of these systems have the same Hausdorff dimension.
\[hdimass\] If $\Phi$ is a CGDMS with a finitely irreducible matrix, then $\theta_{E_v}\geq\theta$ and $\dim_H(J_{E_v})=\dim_H(J_v)=\dim_H(J)$ for every vertex $v\in V$.
Fix $v\in V$. If $t<\dim_H(J)$, then $P(t)>0$ and hence we deduce from the first part of Theorem \[pressass\] that $P_{E_v}(t)>0$. If $t>\dim_H(J)$, then $P(t)<0$ and hence we deduce from the second part of Theorem \[pressass\] that $P_{E_v}(t)<0$. Thus, $\dim_H(J_{E_v})=\dim_H(J)=\dim_H(J_v)$.
Similarly, if $t<\theta$ then $P(t)=\infty>0$ and hence we deduce from the first part of Theorem \[pressass\] that $P_{E_v}(t)\geq P(t)=\infty$. Thus, $t\leq\theta_{E_v}$. Since this is true for all $t<\theta$, we conclude that $\theta\leq\theta_{E_v}$.
The relationship between the pressures further indicates that the original and the associated systems sometimes have similar natures. Before our next corolarry, we recall several definitions from [@gdms].
A CGDMS is strongly regular iff there exists $t \geq 0$ such that $0 < P(t) < \infty$.
If a CGDMS $S$ is not regular,we call it irregular.
A CGDMS $S$ is called critically regular if $P(\theta)=0$.
A CGDMS $S$ is absolutely regular if every non-empty subsystem is regular.
\[natureass\] Let $\Phi$ be a CGDMS with a finitely irreducible matrix. Then we have the following.
- If $\Phi$ is strongly regular, then each $\Phi_v$ may have any nature;
- If $\Phi$ is critically regular, then $\Phi_v$ is either critically regular or irregular and $\theta_{E_v}=\theta$ for each $v$;
- If $\Phi$ is irregular, so is every $\Phi_v$ and $\theta_{E_v}=\theta$ for each $v$.
The relationship between the pressures also reveals that (at least) one of the associated systems eventually has the same pressure as the original system.
There is some $v\in V$ such that $P(t)=P_{E_v}(t)$ for all $t\geq\dim_{H}(J)$.
Theorem \[pressass\] affirms that $P(t)=\max_{v\in V}P_{E_v}(t)$ for all $t\geq\dim_{H}(J)$. Since all the pressure functions $P_{E_v}(t)$, $v\in V$, and $P(t)$ are real-analytic, there is $v\in V$ and an interval $I\subset[\dim_{H}(J),\infty)$ such that $P(t)=P_{E_v}(t)$ for all $t\in I$. The real-analyticity then ensures that $P(t)=P_{E_v}(t)$ for all $t\geq\dim_{H}(J)$.
However, the following example shows that $P(t)=P_{E_v}(t)$ on $[\dim_H(J),\infty)$ may not hold for all $v\in V$.
Let $\Phi$ be a SGDS (that is, a CGDS whose generators are all similarities) whose set of vertices is $V=\{v,w,z\}$ and whose set of edges is $E=\{1,2,3,4\}$, where $$\begin{aligned}
i(1)=v, & t(1)=w & \\
i(2)=w, & t(2)=z & \\
i(3)=z, & t(3)=v & \\
i(4)=z, & t(4)=w. &\end{aligned}$$ Observe that $$E_v=\left\{1(24)^n23:n\geq0\right\},$$ $$E_w=\left\{231,24\right\},$$ and $$E_z=\left\{312,42\right\}.$$ Let $r:=|\varphi_{24}'|=|\varphi_{42}'|$ and $s:=|\varphi_{123}'|=|\varphi_{231}'|=|\varphi_{312}'|$. Because all the generators are similarities, $$P_{E_v}(t)
=\log\sum_{\tau\in E_v}|\varphi_\tau'|^t
=\log\left(|\varphi_{123}'|^t\sum_{n=0}^\infty|\varphi_{24}'|^{nt}\right)
=\log\left(s^t\sum_{n=0}^\infty r^{nt}\right)
=\log\left(\frac{s^t}{1-r^t}\right),$$ whereas $$P_{E_w}(t)=P_{E_z}(t)
=\log\left(|\varphi_{123}'|^t+|\varphi_{24}'|^t\right)
=\log(s^t+r^t).$$ The Hausdorff dimension of the limit sets of the original system and the associated systems is the unique $h>0$ such that $s^h+r^h=1$. When $t<h$, we have $s^t+r^t>1$ and it follows that $P_{E_v}(t)>P_{E_w}(t)=P_{E_z}(t)\geq P(t)$. When $t>h$, we have $s^t+r^t<1$ and it ensues that $P_{E_v}(t)<P_{E_w}(t)=P_{E_z}(t)=P(t)$. Since $P(t)$ and $P_{E_z}(t)$ are real-analytic functions which coincides on a non-empty interval, they must coincide everywhere on their finiteness set, which is $[0,\infty)$. We conclude that $P(t)=\log(s^t+r^t)$ for all $t\geq0$.
We shall now show that the inequality $P(t)\leq\min_{v\in V}P_{E_v}(t)$ when $t<\dim_H(J)$ may be strict. Indeed, there exist finite CGDSs with (finitely) irreducible matrices whose associated systems $\Phi_v$ are all infinite. Then $P(0)<\infty=\min_{v\in V}P_{E_v}(0)$. Moreover, $P(t)<\infty=\min_{v\in V}P_{E_v}(t)$ for all $t\in[0,\min_{v\in V}\theta_{E_v})$. The strict inequality $P(t)<\min_{v\in V}P_{E_v}(t)$ extends to the right of $\min_{v\in V}\theta_{E_v}$ in some cases.
Take any SGDS which consists of three vertices and one edge in each direction between every pair of vertices. Such a finite system has finite pressure. However, each of its associated systems is infinite and absolutely regular. To be more precise, let $V=\{v,w,z\}$ be the set of vertices. Let $E=\{a,b,c,d,e,f\}$ be the set of edges, where $$\begin{aligned}
i(a)=v, & t(a)=w & \\
i(b)=w, & t(b)=v & \\
i(c)=w, & t(c)=z & \\
i(d)=z, & t(d)=w & \\
i(e)=v, & t(e)=z & \\
i(f)=z, & t(f)=v. &\end{aligned}$$ Because of the obvious symmetry, we may concentrate our efforts on any given vertex, say $v$. Observe that $$E_v=\left\{a(cd)^nb,a(cd)^ncf,e(dc)^nf,e(dc)^ndb:n\geq0\right\}.$$ As all the generators are similarities, we obtain $$\begin{aligned}
P_{E_v}(t)
=\log\sum_{\tau\in E_v}|\varphi_\tau'|^t
&=&\log\left(
\left(|\varphi_{ab}'|^t+|\varphi_{acf}'|^t+|\varphi_{ef}'|^t+|\varphi_{edb}'|^t\right)
\sum_{n=0}^\infty|\varphi_{cd}'|^{nt}
\right) \\
&=&\log\left(|\varphi_{ab}'|^t+|\varphi_{acf}'|^t+|\varphi_{ef}'|^t+|\varphi_{edb}'|^t\right)
+\log\frac{1}{1-|\varphi_{cd}'|^t}.\end{aligned}$$ In particular, this shows that every associated system is absolutely regular, i.e. $\theta_{E_v}=0$ for all $v\in V$. The continuity of the pressure functions then asserts that there is some interval $[0,L)$, with $L>0$, on which $P(t)<\min_{v\in V}P_{E_v}(t)$ for every $t<L$.
All of the above examples show that Theorem \[pressass\] is the best general result one can achieve.
Section 4.7 in [@gdms] contains some inaccuracies. First, the system generated by a strictly Markov system is strictly Markov, and thus not an iterated function system, as claimed. Moreover, the proof of Theorem 4.7.4 in [@gdms] contains a mistake. Indeed, the correct argument is: By Proposition \[jivjv\], we have $\dim_H(J_{E_v})\leq\dim_H(J_v)
=\dim_H(J)$ for every vertex $v\in V$. Since $\Phi_v$ and $\Phi$ are irreducible, Theorem 4.2 in [@AGGDMS] shows that to prove that $\dim_H(J)\leq\dim_H(J_{E_v})$ it suffices to demonstrate that $P(t)\leq\max\{P_{E_v}(t),0\}$ for every $t\geq0$ and every $v\in V$. To do this, fix $v\in V$ and $t\geq0$. Let $W\subset E_A^q$ be a set witnessing the finite primitivity of $A$. Then for every $e\in E$ there exist $\a(e),\b(e)\in W$ such that $i(\a(e))=v$, $t(\b(e))=v$ and $\a(e)e\b(e)\in E_A^*$. Set $\a(\om):=\a(\om_1)$ and $\b(\om):=\b(\om_{|\om|})$ for every $\om\in E_A^*$. Let $u>\max\{P_{E_v}(t),0\}$. Then $$\begin{aligned}
\sum_{\om\in E_A^*}\|\varphi_\om'\|^t e^{-u|\om|}
&\leq& (K\rho)^{2t}
\sum_{\om\in E_A^*}\|\varphi_{\a(\om)\om\b(\om)}'\|^t e^{-u|\om|} \\
&=& (K\rho)^{2t}e^{2qu}
\sum_{\om\in E_A^*}\|\varphi_{\a(\om)\om\b(\om)}'\|^t e^{-u|\a(\om)\om\b(\om)|} \\
&\leq& (K\rho)^{2t}e^{2qu}
\sum_{\tau\in E_v^*}\|\varphi_\tau'\|^t e^{-u|\tau|} \\
&\leq& (K\rho)^{2t}e^{2qu}
\sum_{\tau\in E_v^*}\|\varphi_\tau'\|^t e^{-u|\tau|_v} \\
&<&\infty,\end{aligned}$$ where the second inequality sign follows from the facts that $\a(\om)\om\b(\om)\in E_v^*$ and the function $\om\mapsto\a(\om)\om\b(\om)$ is one-to-one; the third inequality sign follows from the facts that $u>0$ and $|\tau|_v\leq|\tau|$; the last inequality follows from Theorem 4.2 in [@AGGDMS] since $u>P_{E_v}(t)$. The resulting inequality $\sum_{\om\in E_A^*}\|\varphi_\om'\|^t e^{-u|\om|}<\infty$ implies that $u>P(t)$ according to Theorem 4.2 in [@AGGDMS]. Since this is true for every $u>\max\{P_{E_v}(t),0\}$, we deduce that $\max\{P_{E_v}(t),0\}\geq P(t)$. This implies $\dim_H(J_{E_v})\geq\dim_H(J)$.
Though this argument confirms the equality of the Hausdorff dimensions of the limit sets of the original and its associated systems, it does not provide as strong information about their pressures as in Theorem \[pressass\].
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[^1]: Recall that a map is conformal is the derivative at every point is a rotation.
|
---
abstract: 'We give all the polynomials functions of degree 20 which are APN over an infinity of field extensions and show they are all CCZ-equivalent to the function $x^5$, which is a new step in proving the conjecture of Aubry, McGuire and Rodier.'
author:
- |
Florian Caullery\
Institut of Mathematiques of Luminy\
C.N.R.S.\
France\
florian.caullery@etu.univ-amu.fr
title: 'Polynomial functions of degree 20 which are APN infinitely often.'
---
[[**Keywords:** ]{}]{}vector Boolean functions, almost perfect nonlinear functions, algebraic surface, CCZ-equivalence.
Introduction
============
Modern private key crypto-systems, such as AES, are block cipher. The security of such systems relies on what is called the S-box. This is simply a Boolean function $f : \mathbbm{F}_{2^n} \rightarrow \mathbbm{F}_{2^n}$ where $n$ is the size of the blocks. It is the only non linear operation in the algorithm.
One of the best known attack on these systems is differential cryptanalysis. Nyberg proved in \[13\] that the S-boxes with the best resistance to such attacks are the one who are said to be Almost Perfectly Non-linear (APN).
Let $q = 2^n$. A function $f : \mathbbm{F}_q \rightarrow \mathbbm{F}_q$ is said APN on $\mathbbm{F}_q$ if the number of solutions in $\mathbbm{F}_q$ of the equation $$f \left( x + a \right) + f \left( x \right) = b$$ is at most 2 for all $a, b \in \mathbbm{F}_q$, $a \neq 0$. The fact that $\mathbbm{F}_q$ has characteristic 2 implies that the number of solutions is even for any function $f$ on $\mathbbm{F}_q$.
The study of APN functions has focused on power functions and it was recently generalized to other functions, particularly polynomials (Carlet, Pott and al. \[5, 7, 8\]) or polynomials on small fields (Dillon \[6\]). On the other hand, several authors (Berger, Canteaut, Charpin, Laigle-Chapuy \[2\], Byrne, McGuire \[4\] or Jedlicka \[10\]) showed that APN functions did not exist in certain cases. Some also studied the notion of being APN on other fields than $\mathbbm{F}_{2^n}$ (Leducq \[12\]).
Toward a full classification of all APN functions, an approach is to show that certain polynomials are not APN for an infinity of extension of $\mathbbm{F}_2$.
Hernando and McGuire showed a result on classification of APN functions which was conjectured for 40 years : the only exponents such that the monomial $x^d$ is APN over an infinity of extension of $\mathbbm{F}_2$ are of the form $2^i +
1$ or $4^i - 2^i + 1$. Those exponents are called [[[*exceptional exponents*]{}]{}]{}.
It lead Aubry, McGuire and Rodier to formulate the following conjecture:
[[**Conjecture:**]{}]{} (Aubry, McGuire and Rodier) a polynomial can be APN for an infinity of ground fields $\mathbbm{F}_q$ if and only if it is CCZ-equivalent (as defined by Carlet, Charpin and Zinoviev in \[5\]) to a monomial $x^d$ where $d$ is an exceptional exponent.
A way to prove this conjecture is to remark that being APN is equivalent to the fact that the rational points of a certain algebraic surface $X$ in a 3-dimensional space linked to the polynomial $f$ defining the Boolean function are all in a surface $V$ made of 3 planes and independent of $f$. We define the surface $X$ in the 3-dimensional affine space $\mathbbm{A}^3$ by $$\phi \left( x, y, z \right) = \frac{f \left( x \right) + f \left( y \right)
+ f \left( z \right) + f \left( x + y + z \right)}{\left( x + y \right)
\left( x + z \right) \left( y + z \right)}$$ which is a polynomial in $\mathbbm{F}_q \left[ x, y, z \right]$. When the surface is irreducible or has an irreducible component defined over the field of definition of $f$, a Weil’s type bound may be used to approximate the number of rational points of this surface. When it is too large it means the surface is too big to be contained in the surface $V$ and the function $f$ cannot be APN.
This way enabled Rodier to prove in \[14\] that when the degree of $f$ is equal to $4 e$ with $e \equiv 3 \left( {\ensuremath{\operatorname{mod}}} 4 \right)$ and $\phi$ is not divisible by a certain form of polynomial then $f$ is not APN for an infinity of extension of $\mathbbm{F}_q$. He also found all the APN function of degree 12 and proved they are all CCZ-equivalent to $x^3$.
To continue in this way, let’s get interested in the APN functions of degree 20 which were the next ones on the list. The main difference in this case is that $e \equiv 1 \left( {\ensuremath{\operatorname{mod}}} 4 \right)$. We got inspired by the proof of Rodier in \[14\] but we had an other approach using divisors of the surface $\bar{X}$. This was due to the fact that some of the components of $\bar{X}$ are no longer irreducible in our case.
Then we were able to obtain all the APN functions of degree 20 by calculation. The conditions of divisibility by the polynomials we obtained made the first part of our work, we had to work on the quotient after to obtain the final forms of the functions.
The second part was to prove that all were CCZ-equivalent to $x^5$.
[[[*This work has been done with François Rodier as adviser.*]{}]{}]{}
The state of the art
====================
The best known APN functions are the Gold functions $x^{2^i + 1}$ and the Kasami-Welch functions by $x^{4^i - 2^i + 1}$. These 2 functions are defined over $\mathbbm{F}_2$ and they are APN on any field $\mathbbm{F}_{2^m}$ if $\gcd \left( m, i \right) = 1$. Aubry, McGuire and Rodier obtained the following results in \[1\].
[[**(Aubry, McGuire and Rodier, \[1\])**]{}]{} If the degree of the polynomial function f is odd and not an exceptional number then f is not APN over $\mathbbm{F}_{q^n}$ for all n sufficiently large.
[[**(Aubry, McGuire and Rodier \[1\])**]{}]{} If the degree of the polynomial function f is 2e with e odd and if f contains a term of odd degree, then f is not APN over $\mathbbm{F}_{q^n}$ for all n sufficiently large.
There are some results in the case of Gold degree $d = 2^i + 1$:
[[**(Aubry, McGuire and Rodier \[1\])**]{}]{} Suppose $f \left( x \right) =
x^d + g \left( x \right)$ where $\deg \left( g \right) \leqslant 2^{i - 1} +
1$. Let $g \left( x \right) = \sum_{j = 0}^{2^{i - 1} + 1} a_j x^j$. Suppose moreover that there exists a nonzero coefficient $a_j$ of g such that $\phi_j \left( x, y, z \right)$ is absolutely irreducible (where $\phi_j
\left( x, y, z \right)$ denote the polynomial $\phi \left( x, y, z \right)$ associated to $x^j$). Then f is not APN over $\mathbbm{F}_{q^n}$ for all n sufficiently large.
And for Kasami degree as well:
[[**(Férard, Oyono and Rodier \[9\])**]{}]{} Suppose $f \left( x \right) =
x^d + g \left( x \right)$ where $d$ is a Kasami exponent and $\deg \left( g
\right) \leqslant 2^{2 k - 1} - 2^{k - 1} + 1$. Let $g \left( x \right) =
\sum_{j = 0}^{2^{2 k - 1} - 2^{k - 1} + 1} a_j x^j .$ Suppose moreover that there exist a nonzero coefficient $a_j$ of g such that $\phi_j \left( x, y,
z \right)$ is absolutely irreducible. Then $\phi \left( x, y, z \right)$ is absolutely irreducible.
Rodier proved the following results in \[14\]. We recall that for any function $f : \mathbbm{F}_q \rightarrow \mathbbm{F}_q$ we associate to $f$ the polynomial $\phi \left( x, y, z \right)$ defined by: $$\phi \left( x, y, z \right) = \frac{f \left( x \right) + f \left( y \right)
+ f \left( z \right) + f \left( x + y + z \right)}{\left( x + y \right)
\left( x + z \right) \left( y + z \right)} .$$
[[**(Rodier \[14\])**]{}]{} If the degree of a polynomial function f is even such that $\deg \left( f \right) = 4 e$ with $e \equiv 3 \left( {\ensuremath{\operatorname{mod}}} 4
\right)$, and if the polynomials of the form $$\left( x + y \right) \left( x + z \right) \left( y + z \right) + P,$$ with $$P \left( x, y, z \right) = c_1 \left( x^2 + y^2 + z^2 \right) + c_4
\left( xy + xz + zy \right) + b_1 \left( x + y + z \right) + d,$$ for $c_1, c_4, b_1, d \in \mathbbm{F}_{q^3}$, do not divide $\phi$ then $f$ is APN over $\mathbbm{F}_{q^n}$ for $n$ large.
There are more precise results for polynomials of degree 12.
[[**(Rodier \[14\])**]{}]{} If the degree of the polynomial f defined over $\mathbbm{F}_q$ is 12, then either f is not APN over $\mathbbm{F}_{q^n}$ for large n or f is CCZ equivalent to the Gold function $x^3$.
New Results
===========
We have been interested in the functions defined by a polynomial of degree 20.
The main difference with the case already studied is that, when $e = 5$, $\phi_e \left( x, y, z \right)$ (where $\phi_e \left( x, y, z \right)$ denote the polynomial $\phi \left( x, y, z \right)$ associated to $x^e$) is not irreducible. So we had to detail more cases in the proof and use divisors on the surface $X$. And then obtained the following results :
If the degree of a polynomial function defined over $\mathbbm{F}_q$ is 20 and if the polynomials of the form $$(x + y) (x + z) (y + z) + P_1$$ with $P_1 \in \mathbbm{F}_{q^3} \left[ x, y, z \left] \right. \right.$ and $P_1 (x, y, z) = c_1 (x^2 + y^2 + z^2) + c_4 (xy + xz + yz) + b_1 (x + y +
z) + d$
or $$\phi_5 + P_2$$ with $P_2 = a (x + y + z) + b$
do not divide $\phi$ then $f$ is APN over $\mathbbm{F}_{q^n}$ for $n$ large.
If the degree of the polynomial f defined over $\mathbbm{F}_q$ is 20, then either f is not APN over $\mathbbm{F}_{q^n}$ for large n or f is CCZ equivalent to the Gold function $x^5$.
Preliminaries
=============
The following results are needed to prove the theorem 7 All the proofs are in \[14\].
[[**[\[14\]]{}**]{}]{}The class of APN functions is invariant by adding a q-affine polynomial.
[[**[\[14\]]{}**]{}]{}The kernel of the map $$f \rightarrow \frac{f \left( x \right) + f \left( y \right) + f \left( z
\right) + f \left( x + y + z \right)}{\left( x + y \right) \left( x + z
\right) \left( y + z \right)}$$ is made of q-affine polynomials.
We define the surface $X$ in the 3-dimensional affine space $\mathbbm{A}^3$ by $$\phi \left( x, y, z \right) = \frac{f \left( x \right) + f \left( y \right)
+ f \left( z \right) + f \left( x + y + z \right)}{\left( x + y \right)
\left( x + z \right) \left( y + z \right)}$$ and we call $\bar{X}$ its projective closure.
[[**[\[14\]]{}**]{}]{}If the surface X has an irreducible component defined over the field of definition of $f$ which is not one of the planes $\left( x + y
\right) \left( x + z \right) \left( y + z \right) = 0$, the function $f$ cannot be APN for infinitely many extensions of $\mathbbm{F}_q$.
[[**[\[11\]]{}**]{}]{}Let $H$ be a projective hyper-surface. If $\bar{X} \cap H$ has a reduced absolutely irreducible component defined over $\mathbbm{F}_q$ then $\bar{X}$ has an absolutely irreducible component defined over $\mathbbm{F}_q$.
[[**[\[1\]]{}**]{}]{}Suppose $d$ is even and write $d = 2^j e$ where $e$ is odd. In $\bar{X} \cap H$ we have $$\phi_d = \phi_e \left( x, y, z \right)^{2^j} \left( \left( x + y \right)
\left( x + z \right) \left( y + z \right) \right)^{2^j \text{-} 1}$$
[[**[\[14\]]{}**]{}]{}The function $x + y$ (and therefore $A$) does not divide $\phi_i \left( x, y, z \right)$ for $i$ an odd integer.
[[[*$\phi_5$*]{}]{}]{} is not irreducible and we have $$\phi_5 = \left( x + \alpha y + \alpha^2 z \right) \left( x + \alpha^2 y +
\alpha z \right)$$ with $\alpha \in \mathbbm{F}_4 \text{-} \mathbbm{F}_2$.
Calculus is sufficient to prove this.
Proof of theorem 7
==================
Let $f : \mathbbm{F}_q \rightarrow \mathbbm{F}_q$ be a function which is APN over infinitely many extensions of $\mathbbm{F}_q$. As a consequence of proposition 11 no absolutely irreducible component of $X$ is defined over $\mathbbm{F}_q$, except perhaps $x + y = 0$, $x + z = 0$ or $y + z = 0$.
If some component of $X$ is equal to one of these planes then by symmetry in $x {}$, $y$, and $z$, all of them are component of $X$, which implies that $A = \left( x + y \right) \left( x + z \right) \left( y + z \right)$ divides $\phi$. Let us suppose from now on that this is not the case.
Let $H_{\infty}$ is the plane at infinity of $\mathbbm{A}^3$ and $X_{\infty} =
\bar{X} \cap H_{\infty}$. The equation of $X_{\infty}$ is $\phi_{20} = 0$ which gives, using lemma 13 and 14 $$A^3 \left( x + \alpha y + \alpha^2 z \right)^4 \left( x + \alpha^2 y +
\alpha z \right)^4 = 0$$ As the curve $X_{\infty}$ does not contain any irreducible component defined over $\mathbbm{F}_q$, $\alpha \notin \mathbbm{F}_q$ and then $q = 2^n$ with $n$ odd.
Let $X_0$ be a reduced absolutely irreducible component of $\bar{X}$ which contains the line $x + y = 0$ in $H_{\infty}$. The cases where $X_0$ contains 2 or 3 copies of the line $x + y = 0$ in $H_{\infty}$ and where $X_0$ contains one copy of the line $x + y = 0$ and is of degree $1$ are treated in \[14\] and do not differ in our case. So from now on we assume that $X_0$ contains only one copy of the line $x + y = 0$ and is at least of degree 2.
Let $d_1$ be the plane of equation $\left( x + \alpha y + \alpha^2 z \right) =
0$, $d_2$ the plane of equation $\left( x + \alpha^2 y + \alpha z \right) = 0$ we denote $C_i = d_i \cap H_{\infty}$ for $i = 1 {}, 2$. Let $A_0$ be the line of equation $x + y = 0$ in $H_{\infty}$ , $A_1$ the line of equation $y + z = 0$ in $H_{\infty}$ and $A_2$ the line of equation $x + z = 0$ in $H_{\infty}$.
Let us consider $D$ the divisor associated to the hyperplane section $\bar{X}
\cap H_{\infty}$, so $$D = 4 C_1 + 4 C_2 + 3 A_0 + 3 A_1 + 3 A_2$$ We now denote $\mathfrak{X}_0$ the divisor associated to the hyperplane section of $X_0$ which is a sub-divisor of $D$ of degree at least 2. We will denote $\mathfrak{X}_1$ the divisor obtained from $\mathfrak{X}_0$ by applying the permutation $\left( x, y, z \right)$, $\mathfrak{X}_2$ the divisor obtained from $\mathfrak{X}_0$ by applying the permutation $\left( x, z, y
\right)$, $\mathfrak{X}_3$ the divisor obtained from $\mathfrak{X}_0$ by applying the transposition $\left( x, y \right)$, $\mathfrak{X}_4$ the divisor obtained from $\mathfrak{X}_0$ by applying the transposition $\left( x, z
\right)$ and $\mathfrak{X}_5$ the divisor obtained from $\mathfrak{X}_0$ by applying the transposition $\left( y, z \right)$. As $\phi \left( x, y, z
\right)$ is symmetrical in $x, y$ and $z$ we know that $\mathfrak{X}_i$ is a subdivisor of $D$ for $i = 1, \ldots, 5$. The cases where $\mathfrak{X}_0
\geqslant 2 A_0$ or $\mathfrak{X}_0 = A_0$ are already treated in \[14\] so we have to study the cases below.
Case where $\mathfrak{X}_0$ is of degree 2.
-------------------------------------------
If $\mathfrak{X}_0 = A_0 + A_1$ therefore from \[14\] 5.7 we have a contradiction with the fact that $\mathfrak{X}_0$ is at most of degree 2.
If $\mathfrak{X}_0 = A_0 + C_i$, then $\mathfrak{X}_1 = A_1 + C_i$, $\mathfrak{X}_2 = A_2 + C_i$, $\mathfrak{X}_3 = A_0 + C_j$, $\mathfrak{X}_4
= A_1 + C_j$, $\mathfrak{X}_5 = A_2 + C_j$ with $j \neq i$. As seen in \[14\] the group $< \rho > = {\ensuremath{\operatorname{Gal}}} \left( \mathbbm{F}_{q^3} /\mathbbm{F}_q
\right)$ acts on $X_0$ and as $X_0$ is not defined over $\mathbbm{F}_q$ there exist sub-varieties $X_6$, $X_7$ and $X_8$ which have, respectively the associated divisor $\mathfrak{X}_6$, $\mathfrak{X}_7$ and $\mathfrak{X}_8$. We have $\mathfrak{X}_6 = A_0 + C_i$, $\mathfrak{X}_7 =
A_1 + C_i$ and $\mathfrak{X}_8 = A_2 + C_i$. Finally we have $\sum
\mathfrak{X}_i \geqslant D$ which is a contradiction.
Case where $\mathfrak{X}_0$ is of degree 3.
-------------------------------------------
The case where $\mathfrak{X}_0 = A_0 + A_1 + A_2$ has already been treated in \[14\], this is the case where $A + P_1$ divides $\phi$.
If $\mathfrak{X}_0$ contains 2 of the $A_i$ from \[14\] 5.7 it contains the 3 and it is the same case than previously.
If $\mathfrak{X}_0 = A_0 + 2 C_i$, then $\mathfrak{X}_1 = A_1 + 2
C_i$ and $\mathfrak{X}_2 = A_2 + 2 C_i$, in this case $\mathfrak{X}_0
+\mathfrak{X}_1 +\mathfrak{X}_2 \geqslant D$ which is a contradiction.
If $\mathfrak{X}_0 = A_0 + C_1 + C_2$, then $\mathfrak{X}_1 = A_1 +
C_1 + C_2$ and $\mathfrak{X}_2 = A_2 + C_1 + C_2$, $\mathfrak{X}_3 = A_0 +
C_1 + C_2$, $\mathfrak{X}_4 = A_1 + C_1 + C_2$ and $\mathfrak{X}_5 = A_2 +
C_1 + C_2$. Then $\sum \mathfrak{X}_i \geqslant D$ which is a contradiction.
Case where $\mathfrak{X}_0$ is of degree 4.
-------------------------------------------
If $\mathfrak{X}_0 = A_0 + A_1 + A_2 + C_i$, then $\mathfrak{X}_1 =
A_0 + A_1 + A_2 + C_i$, $\mathfrak{X}_2 = A_0 + A_1 + A_2 + C_i$, $\mathfrak{X}_3 = A_0 + A_1 + A_2 + C_j$, $\mathfrak{X}_4 = A_0 + A_1 + A_2
+ C_j$ and $\mathfrak{X}_5 = A_0 + A_1 + A_2 + C_j$. Then $\sum
\mathfrak{X}_i \geqslant D$ which is a contradiction.
If $\mathfrak{X}_0$ contains 2 of the $A_i$ from \[14\] 5.7 it contains the 3 and we are in the same case than in i).
If $\mathfrak{X}_0 = A_0 + 3 C_i$, then $\mathfrak{X}_1 = A_1 + 3 C_i$ and $\mathfrak{X}_2 = A_2 + 3 C_i$. Then $\sum \mathfrak{X}_i \geqslant D$ which is a contradiction.
If $\mathfrak{X}_0 = A_0 + 2 C_i + C_j$ then $\mathfrak{X}_1 = A_1 + 2
C_i + C_j$ and $X_2 = A_2 + 2 C_i + C_j$, with $j \neq i$. Then $\sum
\mathfrak{X}_i \geqslant D$ which is a contradiction.
Case where $\mathfrak{X}_0$ is of degree 5.
-------------------------------------------
If $X_0 = A_0 + 2 \left( C_1 + C_2 \right)$, then $\mathfrak{X}_1 =
A_1 + 2 \left( C_1 + C_2 \right)$ and $\mathfrak{X}_2 = A_2 + 2 \left( C_1 +
C_2 \right)$. Then $\sum \mathfrak{X}_i \geqslant D$ which is a contradiction.
If $\mathfrak{X}_0 = A_0 + 3 C_i + C_j$, $j \neq i$, $\mathfrak{X}_1 =
A_1 + 3 C_i + C_j$ and $\mathfrak{X}_2 = A_2 + 3 C_i + C_j$. Then $\sum
\mathfrak{X}_i \geqslant D$ which is a contradiction.
If $\mathfrak{X}_0 = A_0 + 4 C_i$, then $\mathfrak{X}_1 = A_1 + 4 C_i$ then $\mathfrak{X}_0 +\mathfrak{X}_1 \geqslant D$ which is a contradiction.
If $\mathfrak{X}_0$ contains 2 of the $A_i$ from \[14\] 5.7 it contains the 3 and we will treat those cases in the following points.
If $\mathfrak{X}_0 = A_0 + A_1 + A_2 + 2 C_i$ then $\mathfrak{X}_1 =
A_0 + A_1 + A_2 + 2 C_i$ and $\mathfrak{X}_2 = A_0 + A_1 + A_2 + 2 C_i$. Then $\sum \mathfrak{X}_i \geqslant D$ which is a contradiction.
The only case left is when $X_0 = A_0 + A_1 + A_2 + C_1 + C_2$. As seen in \[14\] the group $< \rho > = {\ensuremath{\operatorname{Gal}}} \left( \mathbbm{F}_{q^3}
/\mathbbm{F}_q \right)$ acts on $X_0$ and as $X_0$ is not defined over $\mathbbm{F}_q$ there exist sub-varieties $X_6$, $X_7$ and $X_8$ which have, respectively the associated divisor $\mathfrak{X}_6$, $\mathfrak{X}_7$ and $\mathfrak{X}_8$. We have $\mathfrak{X}_6 = A_0 + A_1 + A_2 + C_1 + C_2$, $\mathfrak{X}_7 = A_0 + A_1 + A_2 + C_1 + C_2$ and $\mathfrak{X}_8 = A_0 +
A_1 + A_2 + C_1 + C_2$. It remains the sub-divisor $\mathfrak{X}_9 = C_1 +
C_2$. Therefore $\sum \mathfrak{X}_i = D$ and the form of $\phi$ is : $$\phi = \left( \phi_5 + R \right) \left( A \phi_5 + Q \right) \left( A
\phi_5 + \rho \left( Q \right) \right) \left( A \phi_5 + \rho^2 \left( Q
\right) \right)$$ with $R$ a polynomial of degree 1 such as $\phi_5 + R$ is not irreducible, $Q$ a polynomial of degree 4 and $\rho$ the generator of ${\ensuremath{\operatorname{Gal}}} \left(
\mathbbm{F}_{q^3} /\mathbbm{F}_q \right)$.
It is useless to consider the cases where $X_0$ is of degree more than 5 as we obtain 2 other divisors of the same degree from $X_0$ and $D$ is of degree 17. Therefore it is sufficient to prove the theorem 7.
Proof of theorem 8
==================
We have the two following cases to study:
Case where $A + P_1$ divides $\phi$.
------------------------------------
If $P_1$ divides $\phi$ then $\left( A + P_1 \right) \left( A + \rho \left(
P_1 \right) \right) \left( A + \rho^2 \left( P_1 \right) \right)$ divides $\phi$ too (see \[14\]. By calculus (see Appendix 1) we can state that:
$P_1 = c_1 \phi_5 + c_1^3$.
The trace of $c_1$ in $\mathbbm{F}_{q^3}$ is 0.
$\left( A + P_1 \right) \left( A + \rho \left( P_1 \right) \right)
\left( A + \rho^2 \left( P_1 \right) \right)$ is the polynomial $\phi$ associated to $L \left( x \right)^3$ where $L \left( x \right) = x \left( x
+ c_1 \right) \left( x + \rho \left( c_1 \right) \right) \left( x + \rho^2
\left( c_1 \right) \right)$.
${\ensuremath{\operatorname{We}}} {\ensuremath{\operatorname{have}}} f = L \left( x \right)^3 \left( L \left( x
\right)^2 + a \right) + a_{16} x^{16} + a_8 x^8 + a_4 x^4 + a_2 x^2 + a_1 x
+ a_0$ where $a, a_0, a_1, a_2, a_4, a_8, a_{16} \in \mathbbm{F}_q$.
By proposition 9 $f$ is equivalent to $L \left( x \right)^5 + aL \left( x
\right)^3$. As ${\ensuremath{\operatorname{tr}}} \left( c_1 \right) = 0$, $L \left( x \right)$ is a $q$-affine permutation hence $f$ is CCZ-equivalent to $x^5 + ax^3$.
By theorem 3 $f$ cannot be APN over infinitely many extensions of $\mathbbm{F}_q$ if $a \neq 0$. Hence $a = 0$ and $f$ is CCZ-equivalent to $x^5$, which is a gold function.
Case where $P_2$ divides $\phi$.
--------------------------------
If $P_2$ divides $\phi$ then, by calculus (see Appendix 2), we obtain that $f
= \left( x^{20} + ax^{10} + bx^5 \right) + a_{16} x^{16} + a_8 x^8 + a_4 x^4 +
a_2 x^2 + a_1 x + a_0$, where $a, b, a_0, a_1, a_2, a_4, a_8, a_{16} \in
\mathbbm{F}_q$. By proposition 9 $f$ is equivalent to $\left( x^{5} + ax^2 +
bx \right)^4$. Therefore $f$ can be written $f \left( x \right) = L \left( x^5
\right)$ with $L \left( x \right) = x^{4} + ax^2 + bx$ which is a permutation. Hence, $f$ is CCZ-equivalent to $x^5$.
In conclusion, we proved that if $f \left( x \right)$ is a polynomial of $\mathbbm{F}_q$ of degree 20 which is APN over infinitely many extensions of $\mathbbm{F}_q$, then $f \left( x \right)$ is CCZ-equivalent to $x^5$.
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Appendix
========
In this part we give the details of the calculus we made in order to state the theorem 8 We just use the fact that $P_1$ or $P_2$ divides $\phi$ and it gives us conditions on the coefficients of $P_1$ or $P_2$ and $\phi$. As $\phi$ is a symmetrical polynomial in $x, y, z$ we can write it using symmetrical functions $s_1 = x + y + z$, $s_2 = xy + xz + yz$ and $s_3 = xyz$. We recall that $\phi_i$ is the polynomial $\phi$ associated to $x^i$ and therefore $\phi
\left( x, y, z \right)$ the polynomial associated to $f \left( x \right) =
\sum_{i = 0}^d a_i x^i$ can be written $\phi = \sum a_i \phi_i$. Denoting $p_i
= x^i + y^i + z^i$, we have $p_i = s_1 p_{i - 1} + s_2 p_{i - 2} + s_3 p_{i -
3} {}$. We remark that $\phi_i = \frac{p_i + s_1^i}{A}$ and that $A =
s_1 s_2 + s_3$.
The calculus were made on the software Sage and you can find the sheet at the following adress: [http://sagenb.org/home/pub/5035]().
Case where $A + P_1$ divides $\phi$.
------------------------------------
We will write $P$ for $P_1$ in this section in order to make the calculus more readable.
If $A + P$ divides $\phi$ then $\left( A + P \right) \left( A + \rho \left( P
\right) \right) \left( A + \rho^2 \left( P \right) \right)$ is of degree 9 and divides $\phi$ too (see \[14\]). We write $$\left( A + P \right) \left( A + \rho \left( P \right) \right) \left( A +
\rho^2 \left( P \right) \right) = \sum_{i = 0}^9 P_i,$$ where $P_i$ is the term of degree $i$ of $\left( A + P_{} \right) \left( A +
\rho \left( P \right) \right) \left( A + \rho^2 \left( P \right) \right)$.
As $\left( A + P \right) \left( A + \rho \left( P \right) \right) \left( A +
\rho^2 \left( P \right) \right)$ divides $\phi$ there exists a polynomial $Q$ of degree 8 such as $$\phi = \left( A + P \right) \left( A + \rho \left( P \right) \right)
\left( A + \rho^2 \left( P \right) \right) Q$$ and we write $$Q = \sum_{i = 0}^8 Q_i,$$ where $Q_i$ is the term of degree $i$ of $Q$.
### Degree 17
We put $a_{20} = 1$ and we have : $$\phi_{20} = P_9 Q_8 .$$ As $P_9 = A^3$ we have $Q_8 = \phi_5^4$.
### Degree 16.
We have $$a_{19} \phi_{19} = P_9 Q_7 + P_8 Q_8 .$$ As $P_8 = A^2 (s_1^2 {\ensuremath{\operatorname{tr}}} (c_1) + s_2 {\ensuremath{\operatorname{tr}}} (c_4))$, where ${\ensuremath{\operatorname{tr}}}
\left( c_i \right)$ is the trace of $c_1$, it gives us $$a_{19} \phi_{19} = A^3 Q_7 + A^2 \phi_5^4 (s_1^2 {\ensuremath{\operatorname{tr}}} (c_1) + s_2
{\ensuremath{\operatorname{tr}}} (c_4)) .$$ As $\phi_{19}$ is not divisible by $A$ (by lemma 13) so $a_{19} = 0$ and $$AQ_7 = \phi_5^4 (s_1^2 {\ensuremath{\operatorname{tr}}} (c_1) + s_2 {\ensuremath{\operatorname{tr}}} (c_4)) .$$ We know that $A$ is prime with $s_1^2 {\ensuremath{\operatorname{tr}}} (c_1) + s_2 {\ensuremath{\operatorname{tr}}} (c_4)$ because $(x + y)$ does not divide this polynomial, and $A$ does not divide either $\phi_5^4$ which implies $Q_7 = P_8 = 0$ and ${\ensuremath{\operatorname{tr}}} (c_1) =
{\ensuremath{\operatorname{tr}}} (c_4) = a_{19} = 0$.
### Degree 15.
We have $$a_{18} \phi_{18} = a_{18} (A \phi_9^2) = P_9 Q_6 + P_8 Q_7 + P_7 Q_8 .$$ Knowing that $P_8 = Q_7 = 0$ we obtain $$a_{18} (A \phi_9^2) = P_9 Q_6 + P_7 Q_8 = A^3 Q_6 + \phi_5^4 P_7 .$$ We also know that
$$\phi_5^4 = \left( s_1^2 + s_2 \right)^4 = s_1^8 + s_2^4$$
and
$P_7 = A \left( s_1^4 q_1 \left( c_1 \right) +
s_2^2 q_1 \left( c_4 \right) + s_1^2 s_2 q_5 \left( c_1, c_4 \right) \right) +
A^2 s_1 {\ensuremath{\operatorname{tr}}} \left( b_1 \right)$
denoting
$q_1 \left( c_i \right) = c_i \rho \left( c_i \right) + c_i \rho^2 \left( c_i
\right) + \rho \left( c_i \right) \rho^2 \left( c_i \right)$ and
$q_5 (c_1, c_4) = c_1 \left( \rho (c_4) + \rho^2 (c_4) \right) + c_4 \left(
\rho (c_1) + \rho^2 (c_1) \right) + \rho (c_1) \rho^2 (c_4) + \rho (c_4)
\rho^2 \left( c_1 \right)$.
So $$a_{18} \phi_9^2 = A^2 Q_6 + \phi_5^4 \left( s_1^4 q_1 \left( c_1 \right) +
s_2^2 q_1 \left( c_4 \right) + s_1^2 s_2 q_5 \left( c_1, c_4 \right)
\right. + As_1 \left. {\ensuremath{\operatorname{tr}}} \left( b_1 \right) \right),$$ hence $A$ divide $a_{18} \phi_9^2 + \phi_5^4 \left( s_1^4 q_1 \left( c_1
\right) + s_2^2 q_1 \left( c_4 \right) + s_1^2 s_2 q_5 \left( c_1, c_4 \right)
\right)$. As $A = s_1 s_2 + s_3$ the polynomial $a_{18} \phi_9^2 + \phi_5^4
\left( s_1^4 q_1 \left( c_1 \right) + s_2^2 q_1 \left( c_4 \right) + s_1^2 s_2
q_5 \left( c_1, c_4 \right) \right)$ cannot contain monomial in $s_1^{12}$ or $s_2^6$, therefore $a_{18} = q_1 \left( c_1 \right) = q_1 \left( c_4 \right)$.
Then $A$ divides $a_{18} \left( \phi_9^2 + \phi_5^6 \right) + \phi_5^4 s_1^2
s_2 q_5 \left( c_1, c_4 \right)$. As $\phi_9^2 + \phi_5^6 = A^4$ and $A$ does not divide $\phi_5^4$ we have $q_5 \left( c_1, c_4 \right) = 0$. Replacing in the first equation we have $$a_{18} A^4 = A^2 Q_6 + A \phi_5^4 s_1 \left( {\ensuremath{\operatorname{tr}}} \left( b_1 \right)
\right) .$$ So $$a_{18} A^3 + AQ_6 = \phi_5^4 s_1 \left( {\ensuremath{\operatorname{tr}}} \left( b_1 \right)
\right),$$ as $A$ does not divide $\phi_5^4 s_1$, ${\ensuremath{\operatorname{tr}}} \left( b_1 \right) = 0$ and $Q_6 = a_{18} A^2$.
### Degree 14.
[[**[We first prove that $c_1 = c_4$.]{}**]{}]{}
We have $$a_{17} \phi_{17} = P_9 Q_5 + \ldots + P_6 Q_8 = P_9 Q_5 + P_6 Q_8 .$$ We know that $$P_6 = A^2 N (d) + A \left( s_1^3 q_5 (c_1, b_1) + s_1 s_2 q_5 (c_1, b_1)
\right) + s_1^6 N (c_1) + s_1^4 s_2 q_4 (c_1, c_4) + s_1^2 s_2^2 q_4 (c_4,
c_1) + s_2^3 N (c_4)$$ where $$N (a) = a \rho (a) \rho^2 (a) \text{which is the norm of} a \text{in} \mathbbm{F}_q .\\
q_4 (a, b) = a \rho (a) \rho^2 (b) + a \rho (b) \rho^2 (a) + b \rho (a)\rho^2 (a). \\
q_5 (a, b) = a (\rho (b) + \rho^2 (b)) + b (\rho (a) + \rho^2 (a)) + \rho(a) \rho^2 (b) + \rho (b) \rho^2 (a) .$$ for all $a, b$ in $\mathbbm{F}_{q^3}$.
We can write $$P_6 = A^2 {\ensuremath{\operatorname{tr}}} (d) + A \left( s_1^3 q_5 (c_1, b_1) + s_1 s_2 q_5
(c_1, b_1) \right) + P_6^{\ast} \rho (P_6^{\ast}) \rho^2 (P_6^{\ast}),$$ where $P_6^{\ast} = c_1 s_1^2 + c_4 s_2$. So we can deduce that $$a_{17} \phi_{17} = A^3 Q_5 + \phi_5^4 \left( A^2 {\ensuremath{\operatorname{tr}}} (d) + A \left(
s_1^3 q_5 (c_1, b_1) + s_1 s_2 q_5 (c_1, b_1) \right) + P_6^{\ast} \rho
(P_6^{\ast}) \rho^2 (P_6^{\ast}) \right) .$$ We now have $A$ divides $a_{17} \phi_{17} + \phi_5^4 P_6^{\ast} \rho
(P_6^{\ast}) \rho^2 (P_6^{\ast})$. In addition, denoting $s = x + y$, $$(x + z)^2 \phi_5 = (x + z)^4 + s (x^2 y + x^2 z + yz^2 + z^3) = (x + z)^4 +
sR_1$$ and $$(x + z)^2 \phi_{17} = (x + z)^{16} + sR_2,$$ where $R_1$ is a polynomial of degree 3 and $R_2$ is a polynomial of degree 15. As $(x + z)^8 A = (x + z)^9 s (x + z + s)$ divides $a_{17} (x + z)^8
\phi_{17} + P_6^{\ast} \rho (P_6^{\ast}) \rho^2 (P_6^{\ast}) (x + z)^8
\phi_5^4 $ which is equal to $$a_{17} (x + z)^6 \left( x^{16} + z^{16} + sR_1 \right) + P_6^{\ast} \rho
(P_6^{\ast}) \rho^2 (P_6^{\ast}) \left( x^4 + z^4 + sR_2 \right) .$$ Therefore we have $P_6^{\ast} = c_1 (s^2 + z^2) + c_4 (x^2 + s (x + z)) = c_1
z^2 + c_4 x^2 + sR_3 = P_6^{\ast \ast} + sR_3$.
As $s$ divides (1) the constant term in $s$ vanishes : $$(x + z)^{16} \left( a_{17} (x + z)^6 + P_6^{\ast \ast} \rho (P_6^{\ast
\ast}) \rho^2 (P_6^{\ast \ast}) \right) = 0,$$ then $$a_{17} (x + z)^6 + P_6^{\ast \ast} \rho (P_6^{\ast \ast}) \rho^2 (P_6^{\ast
\ast}) = 0,$$ hence $$a_{17} (x + z)^6 + (c_4 x^2 + c_1 z^2) (\rho (c_4) x^2 + \rho (c_1) z^2)
(\rho^2 (c_4) x^2 + \rho^2 (c_1) z^2) = 0,$$ so $$a_{17} (x + z)^3 + ( \sqrt{c_4} x + \sqrt{c_1} z) (\rho ( \sqrt{c_4}) x +
\rho ( \sqrt{c_1}) z) (\rho^2 ( \sqrt{c_4}) x + \rho^2 ( \sqrt{c_1}) z) =
0.$$ The polynomial $x + z$ divides $( \sqrt{c_4} x + \sqrt{c_1} z) (\rho (
\sqrt{c_4}) x + \rho ( \sqrt{c_1}) z) (\rho^2 ( \sqrt{c_4}) x + \rho^2 (
\sqrt{c_1}) z)$ so it divides one component and then $c_1 = c_4$.
[[**[We now calculate $P_6$ and $Q_5$.]{}**]{}]{}
As $c_1 = c_4$ we have $P_6 = A^2 {\ensuremath{\operatorname{tr}}} \left( d \right) + A \phi_5 s_1
q_5 \left( c_1, b_1 \right) + \phi_5^3 N \left( c_1 \right)$, so from $a_{17}
\phi_{17} = P_9 Q_5 + P_6 Q_8$ we can deduce that $A$ divides $a_{17}
\phi_{17} + q_3 \left( c_1 \right) \phi_5^7$. Hence the coefficient of the monomials $s_1^{14}$ in $a_{17} \phi_{17} + N \left( c_1 \right) \phi_5^7$, which is $a_{17} + N \left( c_1 \right)$, must be equal to 0, so $a_{17} = N
\left( c_1 \right)$.
Remarking that $\phi_{17} + \phi_5^7 = A^2 \phi_5 \phi_9$ we have $A$ divides $\phi_5 s_1 q_5 \left( c_1, b_1 \right)$. As $\phi_5 s_1$ is not divisible by $A$ we have $q_5 \left( c_1, b_1 \right)$=0. So now we have $$A^3 Q_5 = A^2 \phi_5^4 {\ensuremath{\operatorname{tr}}} \left( d \right) + a_{17} A^2 \phi_5
\phi_9,$$ which gives $$\begin{aligned}
& AQ_5 = \phi_5^4 {\ensuremath{\operatorname{tr}}} \left( d \right) + a_{17} \phi_5 \phi_9 . & \end{aligned}$$ Using the same argument as precedent we have ${\ensuremath{\operatorname{tr}}} \left( d \right) = N
\left( c_1 \right)$ and then $Q_5 = a_{17} \frac{\phi_5 \phi_9 +
\phi_5^4}{A^2} = a_{17} A^2 \phi_5$ and $P_6 = a_{17} \left( A^2 + \phi_5^3
\right)$.
### Degree 13
We use $$0 = a_{16} \phi_{16} = P_9 Q_4 + P_8 Q_5 + P_7 Q_6 + P_6 Q_7 + P_5 Q_8$$ $$= A^3 Q_4 + a^2_{18} A^3 \phi_5^2 + \phi_5^4 P_5,$$ with $$P_5 = q_4 (c_1, b_1) \left( s_1^5 + s_2^2 s_1 \right) + A \left( q_1 (b_1)
s_1^2 + q_5 (c_1, d) \left( s_1^2 + s_2 \right) \right) .$$ As $A^3$ does not divide $P_5 \phi_5^4$, so $P_5 = 0$ and $q_4 (c_1, b_1) =
q_1 (b_1) = q_5 (c_1, d) = 0$. We deduce $$Q_4 = a^2_{18} \phi_5^2 .$$
### Degree 12
We have $$a_{15} \phi_{15} = P_9 Q_3 + P_8 Q_4 + P_7 Q_5 + P_6 Q_6 + P_5 Q_7 + P_4
Q_8$$ $$= A^3 Q_3 + a_{18} a_{17} A^2 \phi_5^3 + a_{18} a_{17} A^2 \left( A^2 +
\phi_5^3 \right) + P_4 \phi_5^4,$$ with $$P_4 = q_4 (b_1, c_1) (s_1^4 + s_1^2 s_2) + q_4 (c_1, d) (s_1^4 + s_2^2) +
q_5 (b_1, d) As_1 = H_4 + AG_4,$$ where $H_4 = q_4 (b_1, c_1) (s_1^4 + s_1^2 s_2) + q_4 (c_1, d) (s_1^4 +
s_2^2)$ and $G_4 = q_5 (b_1, d) s_1$. So $A | H_4 \phi_5^4 + a_{15}
\phi_{15}$. As $$H_4 \phi_5^4 + a_{15} \phi_{15} = s_1^{12} \left( a_{15} + q_4 (b_1, c_1) +
q_4 (c_1, d) \right) + s_1^{10} s_2 q_4 (b_1, c_1) + a_{15} s_1^9 s_3 +
s_1^8 s_2^2 \left( a_{15} + q_4 (c 1, d) \right) + s_1^4 s_2^4 \left( q_4
(b_1, c_1) + q_4 (c_1, d) \right) + s_1^2 s_2^5 q_4 (b 1, c 1) + a_{15}
\left( s_1^3 s_3^3 + s_1 s_2^4 s_3 + s_3^4 \right) + s_2^6 \left( a_{15} +
q_4 (c 1, d) \right),$$ the coefficients of $s_1^{12}$ and $s_2^6$ must be 0 and so $$a_{15} + q_4 (b 1, c 1) + q_4 (c 1, d) = 0 \text{ and}\\
a_{15} + q_4 (c 1, d) = 0 {\ensuremath{\operatorname{so}}} q_4 (b 1, c 1) = 0.$$ Replacing in the equation we now have $$H_4 \phi_5^4 + a_{15} \phi_{15} = a_{15} \left( s_1^9 s_3 + s_1^4 s_2^4 +
s_1^3 s_3^3 + s_1 s_2^4 s_3 + s_3^4 \right) = a_{15} \left( \phi_{15} +
\phi_5^6 \right),$$ but $A$ does not divide $\phi_{15} + \phi_5^6$ so $a_{15} = 0$ so $H_4 = 0$.
Hence $$0 = A^3 Q_3 + a_{18} a_{17} A^2 \phi_5^3 + a_{18} a_{17} A^2 \left( A^2 +
\phi_5^3 \right) + AG_4 \phi_5^4 .$$ So $A$ divides $G_4$, but the degree of $G_4$ is less than or equal to 1 so $G_4 = 0$ it implies $q_5 (b_1, d) = 0$ so $P_4 = 0$.
We conclude $$Q_3 = a_{18} a_{17} A.$$
### Degree 11.
We have $$a_{14} \phi_{14} = P_9 Q_2 + P_8 Q_3 + P_7 Q_4 + P_6 Q_5 + P_5 Q_6 + P_4
Q_7 + P_3 Q_8,$$ so $$a_{14} A (\phi_5^4 + s_1^2 s_3^2) = A^3 Q_2 + a_{18}^3 A \phi_5^4 +
a_{17}^2 A \phi_5 \left( A^2 + \phi_5^3 \right) + P_3 \phi_5^4 . \left(
\ast \right)$$ So $A$ divides $P_3$. But $P_3 = N (b_1) s_1^3 + q_6 (c_1, b_1, d)) s_1 \phi_5
+ q_1 (d) A$ so $N (b_1) = q_6 (c_1, b_1, d) = 0$ with $$q_6 (c_1, b_1, d) = b_1 \rho (c_1) \rho^2 (d) + b_1 \rho (d) \rho^2 (c_1) +
c_1 \rho (b_1) \rho^2 (d) + c_1 \rho (d) \rho^2 (b_1) + d \rho (c_1) \rho^2
(b_1) + d \rho (c_1) \rho^2 (b_1) .$$
As $N \left( b_1 \right) = 0, b_1 = 0$.
When we replace in the equation $\left( \ast \right)$ we have $$A^3 (Q_2 + a_{17}^2 \phi_5) = A \left( \phi_5^4 \left( a_{14} + a_{18}^3 +
a_{17}^2 + q_1 (d) \right) + a_{14} s_1^2 s_3^2 \right),$$ so $A$ divides $\phi_5^4 \left( a_{14} + a_{18}^3 + a_{17}^2 + q_1 (d) \right)
+ a_{14} s_1^2 s_3^2 = (s_1^8 + s_2^4) \left( a_{14} + a_{18}^3 + a_{17}^2 +
q_1 (d) \right) + a_{14} s_1^2 s_3^2$, then $a_{14} + a_{18}^3 + a_{17}^2 +
q_1 (d) = 0$, with the same argument as before on the coefficients of the monomials $s_1^8$ and $s_2^4$, therefore $a_{14} = 0$ because $A$ does not divide $s_1^2 s_3^2$.
We obtain $$Q_2 = a^2_{17} \phi_5,$$ and $$P_3 = (a_{17}^2 + a_{18}^3) A.$$
### Degree 10.
We have $$a_{13} \phi_{13} = P_9 Q_1 + P_8 Q_2 + P_7 Q_3 + P_6 Q_4 + P_5 Q_5 + P_4
Q_6 + P_3 Q_7 + P_2 Q_8$$ $$= A^3 Q_1 + a_{17} a^2_{18} A^2 \phi_5^2 + a_{17} a^2_{18} \phi_5^2 \left(
A^2 + \phi_5^3 \right) + \phi^4_5 \left( s_1^2 q_4 \left( d, c 1 \right) +
s_2 q_4 \left( d, c 1 \right) \right),$$ so $A$ divides $a_{13} \phi_{13} + \phi_5^5 \left( a_{17} a^2_{18} + q_4
\left( d, c 1 \right) \right) = a_{13} \left( s_1^4 s_3^2 + s_1^3 s_2^2 s_3 +
s_1^2 s_2 s_3^2 + s_1 s_3^3 \right) + \phi_5^5 \left( a_{13} + a_{17} a_{18}^2
+ q_4 \left( d, c 1 \right) \right)$. with the same argument as before on the coefficients of the monomials $s_1^8$ and $s_2^4$ we have $$a_{13} + a_{17} a_{18}^2 + q_4 \left( d, c 1 \right) = 0.$$ in addition, $A$ does not divide $s_1^4 s_3^2 + s_1^3 s_2^2 s_3 + s_1^2 s_2
s_3^2 + s_1 s_3^3$ so $a_{13} = 0$ and $q_4 (d, c 1) = a_{17} a^2_{18}$.
Now we have $$AQ_1 = 0.$$ So $Q_1 = 0$ and $P_2 = a_{17} a_{18}^2 \phi_5$.
### Degree 9.
We have $$a_{12} \phi_{12} = P_9 Q_0 + P_8 Q_1 + P_7 Q_2 + P_6 Q_3 + P_5 Q_4 + P_4
Q_5 + P_3 Q_6 + P_2 Q_7 + P_1 Q_8,$$ but $\phi_{12} = A^3$ and as $b_1 = 0$ we have $P_1 = 0$. So $$a_{12} A^3 = A^3 Q_0 + a_{17}^2 a_{18}^{} A \phi_5^3 + a_{17}^2 a_{18}^{}
A \left( A^2 + \phi_5^3 \right) + a_{18} (a^2_{17} + a_{18}^3) A^3,$$ so $Q_0 = a_{12} + a_{18}^4$.
### Degree 8.
We have $$a_{11} \phi_{11} = P_8 Q_0 + P_7 Q_1 + P_6 Q_2 + P_5 Q_3 + P_4 Q_4 + P_3
Q_5 + P_2 Q_6 + P_1 Q_7 + P_0 Q_8,$$ which gives $$a_{11} \phi_{11} = (P_0 + a_{17}^3) \phi_5^4 .$$ But $\phi_5$ does not divide $\phi_{11}$ so $a_{11} = 0$ et $P_0 = a_{17}^3$.
### Conclusion.
We now have the following systems: $$\left\{ \begin{array}{l}
{\ensuremath{\operatorname{tr}}} \left( c_1 \right) = 0\\
N (c_1) + {\ensuremath{\operatorname{tr}}} (d) = 0\\
q_5 (c_1, d) = 0\\
q_4 (c_1, d) = 0\\
q_1 (d) = q_1^3 (c_1) + N (c_1)^2\\
q_4 (d, c_1) = N (c_1) q_1^2 (c_1)\\
N (d) = N (c_1)^3
\end{array} \right.$$ and $$a_{18} = q_1 (c_1), a_{17} = N (c_1) = {\ensuremath{\operatorname{tr}}} (d) .$$ Solving the system formed by the linear equations in $d, \rho (d), \rho^2
(d)$, we obtain $d = c_1^3$. We also have $b_1 = 0$ as $b_1 \rho \left( b_1
\right) \rho^2 \left( b_1 \right) = 0$. Therefore $$P = c_1 \phi_5 + c_1^3,$$ and $$Q = \phi_5^4 + q_1 (c_1) A^2 + N (c_1) A \phi_5 + q_1 (c_1)^2 \phi_5^2 +
q_1 (c_1) N (c_1) A + q_3 (c_1)^2 \phi_5 + a_{12} + q_1 (c_1)^4,$$ therefore $$f \left( x \right) = x^{20} + a_{18} x^{18} + a_{17} x^{17} + a_{16}
x^{16} + a_{12} x^{12} + a_{18} a_{12} x^{10} + a_{17} a_{12} x^9 + a_8 x^8
+ \left( a_{18}^7 + a_{18}^4 a_{17}^2 + a_{18}^3 a_{12} + a_{18} a_{17}^4 +
a_{17}^2 a_{12} \right) x^6 + \left( a_{18}^6 a_{17} + a_{18}^2 a_{17}
a_{12} + a_{17}^5 \right) x^5 + a_4 x^4 + \left( a_{18}^4 a_{17}^3 +
a_{17}^3 a_{12} \right) x^3 + a_2 x^2 + a_1 x + a_0 .$$ Putting $L \left( x \right) = x \left( x + c_1 \right) \left( x + \rho \left(
c_1 \right) \right) \left( x + \rho^2 \left( c_1 \right) \right)$ we have that $\left( A + P \right) \left( A + \rho \left( P \right) \right) \left( A +
\rho^2 \left( P \right) \right)$ is the polynomial $\phi$ associated to $L
\left( x \right)^3$ wich leads us to study the divisibility of $f$ by $L
\left( x \right)^3$. We have in our case $f = L \left( x \right)^3 \left( L
\left( x \right)^2 + a_{12} \right) + a_{16} x^{16} + a_8 x^8 + a_4 x^4 + a_2
x^2 + a_1 x + a_0$.
Case where $P_2$ divides $\phi$.
--------------------------------
We will write $P$ for $P_2$ in this section in order to make the calculus more readable.
From theorem 7 we have $$\phi = \left( \phi_5 + R \right) \left( A \phi_5 + Q \right) \left( A
\phi_5 + \rho \left( Q \right) \right) \left( A \phi_5 + \rho^2 \left( Q
\right) \right),$$ where $R$ is a symmetrical polynomial of $\mathbbm{F}_q$ of degree 1 and $Q$ is a symmetrical polynomial of $\mathbbm{F}_{q^3}$ of degree 4. We will denote $R = as_1 + b$ and $\left( A \phi_5 + Q \right) \left( A \phi_5 + \rho \left(
Q \right) \right) \left( A \phi_5 + \rho^2 \left( Q \right) \right) = \sum_{i
= 0}^{15} Q_i$. We will identify degree by degree the expression of $\phi$.
### Degree 17.
We have $$\phi_{20} = A^3 \phi_5^4 = \phi_5 Q_{15},$$ so $Q_{15} = A^3 \phi_5^3$.
### Degree 16.
We have $$a_{19} \phi_{19} = \phi_5 Q_{14} + as_1 Q_{15} = \phi_5 Q_{14} + as_1 A^3
\phi_5^3,$$ which implies $\phi_5$ divides $\phi_{19}$ but this is not the case hence $a_{19} = 0$ and $Q_{14} = as_1 A^3 \phi_5^2$.
### Degree 15.
We have $$a_{18} \phi_{18} = \phi_5 Q_{13} + as_1 Q_{14} + bQ_{14} = \phi_5 Q_{13} +
as_1^2 A^3 \phi_5^2 + bA^3 \phi_5^3,$$ which implies $\phi_5$ divides $\phi_{18}$ but this is not the case hence $a_{18} = 0$ and $Q_{13} = A^3 \left( a^2 s_1^2 \phi_5 + b \phi_5^2 \right)$.
### Degree 14 and 13
We have $$a_{17} \phi_{17} = \phi_5 Q_{12} + as_1 Q_{13} + bQ_{14},$$ and $$a_{16} \phi_{16} = 0 = \phi_5 Q_{11} + as_1 Q_{12} + bQ_{13} = \phi_5 Q_{11}
+ as_1 Q_{12} + bA^3 \left( a^2 s_1^2 \phi_5 + b \phi_5^2 \right),$$ (2) implies that $Q_{12}$ is divisible by $\phi_5$ or $a = 0$. Lets assume $a
\neq 0$. From (2) we have $$a_{17} \frac{\phi_{17}}{\phi_5} = Q_{12} + a^3 s_1^3 A^3 .$$ (we can show easily that $\phi_5$ divides $\phi_{17}$ by calculus). As [[[*$\phi_5$*]{}]{}]{} divides $Q_{12}$ it divides $a_{17} \frac{\phi_{17}}{\phi_5}
+ a^3 s_1^3 A^3$ too. But $$a_{17} \frac{\phi_{17}}{\phi_5} + a^3 s_1^3 A^3 = a_{17} s_3^4 + R_1,$$ so $a_{17} = 0$. As $\phi_5$ does not divide $s_1^3 A^3$ it means $a = 0$ and $Q_{12} = 0$. We now have, in both case $$\phi = \left( \phi_5 + b \right) \left( \sum_{i = 0}^{15} Q_i \right) .$$ We know that $\phi_5 + b$ is irreducible if $b \neq 0$ (\[11\]), which is in contradiction with the fact that $f$ is APN over infinitely many extension of $\mathbbm{F}_q$ and then $b = 0$.
We now have $Q_{15} = A^3 \phi_5^3 {}, Q_{14} = Q_{13} = Q_{12} = Q_{11}
= 0.$
### Degree 12 to 8.
We have $$a_{15} \phi_{15} = \phi_5 Q_{10},$$ as $\phi_5$ does not divide $\phi_{15}$ we have $a_{15} = 0$ and $Q_{10} = 0$. The same method can be applied until the degree 8. It gives $a_{14} = a_{13} =
a_{12} = a_{11} = 0$ and $Q_9 = Q_8 = Q_7 = Q_6 = 0$.
### Degree 7.
We have $$a_{10} \phi_{10} = a_{10} A \phi_5^2 = Q_5 \phi_{10},$$ so $Q_5 = a_{10} A \phi_5$.
### Degree 6.
The same argument than in section 7.2.5 gives $a_9 = 0$ and $Q_4 = 0$.
### Degree 5.
We have $$a_8 \phi_8 = 0 = Q_3 \phi_5,$$ therefore $Q_3 = 0$.
### Degree 4 and 3.
The same argument than in section 7.2.5 gives $a_7 = a_6 = 0$ and $Q_2 = Q_1 =
0$.
### Degree 2.
We have $$a_5 \phi_5 = Q_0 \phi_5,$$ therefore $Q_0 = a_5$.
### Conclusion.
In conclusion we have $$\phi = \phi_5 \left( A^3 \phi_5^3 + a_{10} A \phi_5 + a_5 \right) =
\phi_{20} + a_{10} \phi_{10} + a_5 \phi_5,$$ which gives $f \left( x \right) = x^{20} + a_{16} x^{16} + a_{10} x^{10} + a_8
x^8 + a_5 x^5 + a_4 x^4 + a_2 x^2 + a_1 x + a_0$.
|
---
address:
- |
${}^1$Center for the Fundamental Laws of Nature\
Jefferson Physical Laboratory, Harvard University,\
Cambridge, MA 02138 USA
- |
${}^2$School of Natural Sciences, Institute for Advanced Study,\
Princeton, NJ 08540 USA
author:
- 'Simone Giombi$^{1,a}$ and Xi Yin$^{1,2,b}$'
title: Higher Spins in AdS and Twistorial Holography
---
\
\
Introduction
============
The conjectured duality between Vasiliev’s minimal bosonic higher spin gauge theory in $AdS_4$ and the free/critical $O(N)$ vector model [@Klebanov:2002ja] (for earlier closely related work, see [@Sezgin:2002rt]) is an example of AdS/CFT duality [@Maldacena:1997re; @Gubser:1998bc; @Witten:1998qj] which is remarkable for a number of reasons. Firstly, the bulk higher spin gauge theory is analogous to the tensionless limit of string field theories in AdS space, but has explicitly known classical equations of motion. Secondly, the conjecture provides the first explicit holographic dual of a free (gauge) theory. Thirdly, the conjecture provides the first precise holographic dual of a CFT that can be realized in the real world, namely the critical $O(N)$ vector model (for small values of $N$). In a recent work by the authors [@Giombi:2009wh], concrete evidence in support of this conjecture was found by computing tree level three point functions of currents from the bulk theory, specialized to the case where one of the currents is a scalar operator, and comparing to the boundary CFT, for both $\Delta=1$ and $\Delta=2$ boundary conditions. However, the method of computation in [@Giombi:2009wh] was laborious and difficult to extend to more complicated correlation functions. It was also difficult to recognize the rather simple structures of the boundary CFT in the messy details of the bulk computation.
In this paper, we will compute the holographic correlation functions in a different gauge [@Vasiliev:1990bu] (see also [@Bolotin:1999fa; @Bekaert:2005vh; @Sezgin:2005pv]), in which the spacetime dependence of Vasiliev’s master fields are eliminated completely, and one only needs to work with the internal twistor-like variables. We will refer to it as the “$W=0$ gauge". We will find drastic simplification in the computation of three point functions. In fact, one no longer needs to explicitly perform the integration over the bulk $AdS_4$, which is entirely encoded in the star product of the master fields in the new gauge. The boundary-to-bulk propagators for the higher spin fields are essentially given by delta functions on the twistor space, and the resulting correlation function is represented as a contour integral on the twistor variables. We will find a completely explicit answer for the three point functions of all higher spin currents, which precisely agrees with that of the free $O(N)$ vector theory.
We would like to emphasize that this agreement is a highly nontrivial test of the structure of higher derivative couplings of Vasiliev theory. For instance, the three point function of the stress energy tensor $\langle TTT\rangle$ of a three dimensional CFT is constrained by conformal symmetry up to a linear combination of two tensor structures [@Osborn:1993cr], corresponding to that of a free massless scalar field and a free fermion, respectively. In particular, the tree level contribution to $\langle TTT\rangle$ from Einstein-Hilbert Lagrangian in the bulk is a linear combination of both [@Arutyunov:1999nw]. A holographic dual of free scalars therefore must involve higher derivative couplings in the graviton sector. Our result confirms that Vasiliev theory has precisely the higher derivative couplings to produce the correct three point functions.
The $W=0$ gauge
===============
Vasiliev’s minimal bosonic higher spin gauge theory in $AdS_4$ [@Vasiliev:1992av; @Vasiliev:1995dn; @Vasiliev:1999ba] is formulated in terms of the master fields $W=W_\mu dx^\mu$, $S=S_{{\alpha}}dz^{{\alpha}}+ S_{{\dot\alpha}}d\bar z^{{\dot\alpha}}$ and $B$, which are functions of the spacetime coordinates $x^\mu$ and the internal variables $(Y,Z)=(y_{{\alpha}}, \bar y_{{\dot\alpha}}, z_{{\alpha}}, \bar z_{{\dot\alpha}})$.[^1] The classical equation of motion takes the form \[cleom\] & d\_xW+W\*W=0,\
& d\_Z W + d\_x S + W\*S+S\*W=0,\
& d\_ZS+S\*S = B\*(K dz\^2 + |K d|z\^2),\
& d\_x B + W\*B-B\*|(W)=0,\
& d\_Z B + S\*B - B\*|(S)=0. Here $K=e^{z^{{\alpha}}y_{{\alpha}}}$ and $\bar K=e^{\bar z^{{\dot\alpha}}\bar y_{{\dot\alpha}}}$ are Kleinians, and we define $dz^2=\frac{1}{2}dz^{\alpha} dz_{\alpha}$, $d\bar z^2=\frac{1}{2}d\bar z^{\dot \alpha} d\bar z_{\dot \alpha}$; $\bar\pi$ is the operation $(y,\bar y,z,\bar z,dz,d\bar z)\mapsto (y,-\bar y,z,-\bar z,dz,-d\bar z)$. We shall refer the reader to [@Giombi:2009wh] and references therein for a review of Vasiliev’s theory and the detailed conventions. Throughout most of this paper we will be working with the type A model of [@Sezgin:2003pt], where the bulk scalar is chosen to be parity even, while commenting on the type B model (with a parity odd scalar) briefly towards the end. The minimal bosonic type A model can be defined by projecting the fields onto the components invariant under the symmetry \[proja\] &W(x|y,|y,z,|z) -W(x|iy,i|y,-iz,-i|z),\
&S(x|y,|y,z,|z,dz,d|z) -S(x|iy,i|y,-iz,-i|z,-idz,-id|z),\
&B(x|y,|y,z,|z) B(x|iy,-i|y,-iz,i|z),\
and consequently only the even integer spin fields are retained.
Because $W$ is a flat connection in spacetime, at least locally we can always go to a gauge in which $W$ is set to zero. We will denote by $S'$ and $B'$ the corresponding master fields in this gauge. The equations of motion then states that $S'$ and $B'$ are independent of the spacetime coordinates $x^\mu$, and are functions of $Y,Z$ only. Explicitly, we can write &W(x|Y,Z) = g\^[-1]{}(x|Y,Z)\*d\_xg(x|Y,Z),\
&S(x|Y,Z) = g\^[-1]{}(x|Y,Z)\*d\_Zg(x|Y,Z)+ g\^[-1]{}(x|Y,Z)\*S’(Y,Z)\*g(x|Y,Z),\
&B(x|Y,Z) = g\^[-1]{}(x|Y,Z)\*B’(Y,Z)\*(g(x|Y,Z)). Here $g^{-1}$ stands for the $*$-inverse of $g$. The equations for $S'$ and $B'$ & d\_ZS’+S’\*S’ = B’\*(K dz\^2 + |K d|z\^2),\
& d\_Z B’+ S’\*B’ - B’\*(S’)=0, are now much simpler to solve. In order to extract holographic correlation functions, however, we must go back to the standard “physical" gauge in the end, and extract boundary expectation of the fields.
As in [@Giombi:2009wh], our strategy of computing the $n$-point correlation functions is to take $n-1$ higher spin currents (inserted at $n-1$ points $\vec x_1, \vec x_2, \cdots, \vec x_{n-1}$ on the boundary) as sources for the bulk fields, and solve for the $(n-1)$-th order field in the bulk by sewing together the boundary-to-bulk propagators using the equation of motion. The tree-level correlation function of the higher spin currents will then be extracted from the expectation value of $B(x|y_{{\alpha}},\bar y_{{\dot\alpha}}=z_{{\alpha}}=\bar z_{{\dot\alpha}}=0)$ near the boundary, say at a point $\vec x_n$.[^2]
Working in perturbation theory, we start by writing the $AdS_4$ vacuum solution as W\_0(x|y,|y) = L\^[-1]{}(x|y,|y)\*dL(x|y,|y), for a gauge function $L(x|Y)$. One begins with the linearized field $B^{(1)}(x,Y)$, and transform it to the $W=0$ gauge $B'^{(1)}(Y)$. We can then solve the linearized field $S'^{(1)}$ from d\_Z S’\^[(1)]{}= B’\^[(1)]{}\*(Kdz\^2+|Kd|z\^2). Explicitly, the solution is S’\^[(1)]{} &=-z\_dz\^\_0\^1 dtt (B’\^[(1)]{}\*K)|\_[ztz]{} + c.c.\
&= -z\_dz\^\_0\^1 dtt B’\^[(1)]{}(-tz,|y)K(t) + c.c. Here we have made the gauge choice $S'|_{Z=0}=0$, following [@Vasiliev:1995dn; @Vasiliev:1999ba]. Next, the quadratic order fields $B'^{(2)}$ and $S'^{(2)}$ can be solved from & d\_ZB’\^[(2)]{} = -S’\^[(1)]{}\*B’\^[(1)]{}+B’\^[(1)]{}\*(S’\^[(1)]{}),\
& d\_Z S’\^[(2)]{} = -S’\^[(1)]{}\*S’\^[(1)]{} + B’\^[(2)]{}\*(Kdz\^2+|K d|z\^2). For tree-level three point functions it suffices to solve for $B'^{(2)}(Y,Z)$ only, which is explicitly given in terms of the linearized fields by &B’\^[(2)]{}(y,|y, z,|z) = -z\^\_0\^1 dt \_[ztz]{} + c.c.\
&= z\^\_0\^1 dt \_[ztz]{} + c.c.\
&= z\^\_0\^1 dt\_0\^1 d\_[ztz]{} + c.c.\
&= z\^\_0\^1 dt\_0\^1 ddu dv e\^[uv+|u|v]{}\_B’\^[(1)]{}(y+v,|y+|v)\
& + c.c.\
&= -2 z\^\_0\^1 dt\_0\^1 dd\^4u d\^4v \_B’\^[(1)]{}(v,|v)B’\^[(1)]{}(-u,|u) e\^[(u-tz)(v-y)+ |u(|v-|y)+u y]{} ( |y|v+t uz ) +c.c.\
&= -2 z\^\_0\^1 dt\_0\^1 d\^[-1]{} d\^4u d\^4v B’\^[(1)]{}(u,|u)\_B’\^[(1)]{}(v,|v) e\^[([u]{}+tz)(y-v)+ |u(|v-|y)- u y]{} ( |y|v- t uz ) +c.c.\
&= 2 \_0\^1 dt\_0\^dd\^4u d\^4v e\^[-uv+|u|v]{} B’\^[(1)]{}(u,|u) B’\^[(1)]{}(v,|v) (zu) e\^[(u+tz)(y-v) + |y|u]{} ( |y|v- t uz ) +c.c.\
& = 2 d\^4u d\^4v e\^[-uv+|u|v]{} B’\^[(1)]{}(u,|u) B’\^[(1)]{}(v,|v) f(y,|y,z;U,V) +c.c. \[Bprime2\] In the steps above we have made several redefinitions on the variables $u,\bar u,v,\bar v$ and $\eta$. In the last step we defined the function f(y,|y,z;U,V) = \_0\^1 dt\_0\^d(zu) e\^[(u+tz)(y-v) + |y|u]{} ( |y|v+ t zu ). Finally, we will be able to recover the second order $B$ field in the standard “physical" gauge by \[btwo\] B\^[(2)]{}(x|Y,Z)& = L\^[-1]{}(x,Y) \* B’\^[(2)]{}(Y,Z) \* (L(x,Y))\
& - \^[(1)]{}(x,Y,Z)\*B\^[(1)]{}(x,Y) + B\^[(1)]{}(x,Y)\*(\^[(1)]{}(x,Y,Z)), and then take $x= (\vec x,z\to 0)$ while restricting to $\bar y = \hat z=\hat {\bar z}=0$ to extract the three point function. Here $\epsilon^{(1)}(x,Y,Z)$ is a first order correction to the gauge function $L(x,Y)$. To understand its effect, let us consider the linearized fields &W\^[(1)]{}(x|Y,Z) = D\_0 \^[(1)]{}(x,Y,Z) ,\
&S\^[(1)]{}(x|Y,Z) = L\^[-1]{}(x,Y)\*S’\^[(1)]{}(Y,Z)\*L(x,Y) + d\_Z \^[(1)]{}(x|Y,Z). Near the boundary $z\to 0$, the spin-$s$ component of $\Omega(\vec x,z|Y)$ falls off like $z^s$, whereas the spin-$s$ component of $B(\vec x,z|Y)$ falls off like $z^{s+1}$. It is then natural to impose the $z^s$ fall-off condition on the spin-$s$ component of the gauge function $\epsilon^{(1)}(x|Y,Z)$. So generically we expect the “gauge correction" in (\[btwo\]) to fall off like $z^{s_1+s_2+1}$, which does not affect the leading boundary behavior of the spin-$s'$ component of $B^{(2)}$, if $s'<s_1+s_2$. Given three spins $s_1, s_2, s_3$ (not all zero), we can always choose two sources, say $s_1, s_2$, so that $s_3<s_1+s_2$. In this case, we can drop the linear gauge function in (\[btwo\]), for the purpose of extracting the boundary correlation function.
Note that in going back to the physical gauge, $\epsilon^{(1)}(x|Y,Z)$ should be chosen so that the gauge condition $S|_{Z=0}=0$ is preserved. There are additional gauge ambiguities of the form $\tilde\epsilon^{(1)}(x|Y)$, under which $\Omega(x|Y)$ transforms by $\delta \Omega^{(1)}(x|Y) = D_0\tilde\epsilon^{(1)}(x|Y)$. For the purpose of extracting three point functions from the boundary expectation value, it suffices to consider the second order $B$-field, restricted to $\bar y_{{\dot\alpha}}=z_{{\alpha}}=\bar z_{{\dot\alpha}}=0$ (which contains the self-dual part of the higher spin Weyl curvature tensor). Its gauge variation under $\tilde\epsilon^{(1)}(x|Y)$ is given by \[bvar\] B\^[(2)]{}(x|y,|y=z=|z=0) = -\^[(1)]{}(x|Y) \* B\^[(1)]{}(x|Y) + B\^[(1)]{}(x|Y)\*(\^[(1)]{}(x|Y)) The spin $s_1$-components of $\tilde\epsilon^{(1)}$ consists of terms of the form $\tilde\epsilon^{(s_1-1+k,s_1-1-k)}$, $1-s_1\leq k\leq s_1-1$, where the superscripts indicate the degrees in $y$ and $\bar y$ respectively. The spin $s_2$-components of $B^{(1)}$ consists of terms of the form $B^{(2s_2+n,n)}$ and $B^{(n,2s_2+n)}$, $n\geq 0$. It is then easily seen that after contracting all the $\bar y$’s under the $*$ product on the RHS of (\[bvar\]), $\delta B^{(2)}(x|y,\bar y=z=\bar z=0)$ may be nonzero only for components of spin $s_3<s_1+s_2$ (i.e. terms of degree $2s_3$ in $y$). We have argued previously that the falloff behavior of the gauge functions near the boundary of $AdS$ is such that the leading boundary behavior of $B^{(2)}$ is not affected when $s_3<s_1+s_2$. Therefore, there is no ambiguity due to $\tilde\epsilon^{(1)}$ in extracting the boundary correlators for all spins.
The gauge function and boundary-to-bulk propagator
==================================================
To carry out the computation in $W=0$ gauge explicitly, first we shall write down the gauge function L(x,Y) = [**P**]{} \_\*(-\^[x\_0]{}\_x W\_0\^(x’|Y) dx’\_) where the $*$-exponential is path ordered, from $x=(\vec x,z)$ to a base point $x_0=(\vec x_0,z_0)$. The $AdS_4$ vacuum solution is given by $W_0=e_0+\omega_0^L$, where $e_0$ and $\omega_0^L$ are the vielbein and spin connection of $AdS_4$, which in our conventions [@Giombi:2009wh] take the form (in Poincaré coordinates) &\_0\^L = [18]{} [dx\^iz]{} ,\
& e\_0 = [14]{} [dx\_z]{} \^\_y\^|y\^. If we choose the straight contour $x(t) = (1-t)x_0+t x$, then the value of $W_0$ along different points on the contour $*$-commute with one another, and we can write simply L(x,Y) &= \_\*\
&= \_\*\
&= \_\*\
&= \_\* . Here we have introduced the notation ${\bf x}=x^{\mu} \sigma_{\mu}=x^i \sigma_i + z \sigma^z$.
Generally, given a symmetric matrix $M$, one can calculate the $*$-exponential \_\*( [t2]{} Y M Y ) = where the symmetric matrix $\Omega(t)$ and function $f(t)$ satisfy &[d(t)dt]{} = (1-(t)) M (1+(t)),\
&[df(t)dt]{} = -[12]{}[Tr]{}(M(t)). The solution is (t) = (tM), f(t) =-[12]{}[Tr]{} (tM). So the result for the $*$-exponential is \_\*([12]{}YMY) = \^[-[12]{}]{} . Applying this formula to $L(x,Y)$, working in the basis $(y,\bar y)$, we can write $M$ as M(x) = -[14]{}[(z/z\_0)z-z\_0]{} (
(-\_0)\^z & [**x**]{}-[**x\_0**]{}\
[**x**]{}-[**x\_0**]{} & (-\_0)\^z
) It is convenient to choose the base point to be $\vec x_0 =0,z_0=1$, so that M(x) = -[z4(z-1)]{} (
\^z & +(z-1)\^z\
+(z-1)\^z &\^z
) and then &L(x,Y) =\^[-[12]{}]{} ,\
&L\^[-1]{}(x,Y) = \^[-[12]{}]{} . Our goal will be to extract the correlation function from the expectation value of a bulk field near a boundary point, given a number of boundary sources. By translation invariance we can choose the boundary point to be at $\vec x=0$, near which the bulk field will be evaluated. In other words, we are choosing the $\vec x$ Poincaré coordinate of the boundary point to coincide with that of the base point in the definition of $L(x,Y)$. At $\vec x=0$ and nonzero values of $z$, we have &M = - [z4]{} (
0 & \^z\
\^z & 0
),\
&M = (z/4) = [z\^[14]{}+z\^[-[14]{}]{}2]{}[**1**]{},\
&M = [1-z\^[12]{}1+z\^[12]{}]{} (
0 & \^z\
\^z & 0
). So & L\^[1]{}(x=0,z, Y) = [4z\^[-[12]{}]{}+2+z\^[12]{}]{} ( y\^z |y ).
By definition, $L(x_0,Y)=1$, at the base point $x_0^\mu=(\vec x_0,z)$,. So the linearized field in the $W=0$ gauge is simply B’\^[(1)]{}(Y) = B\^[(1)]{}(x\_0|Y). Explicitly, using the formulae derived in [@Giombi:2009wh] (see eq. (3.31) and eq. (3.33) of [@Giombi:2009wh]), the boundary-to-bulk propagator for the spin-$s$ component of $B'$ corresponding to a boundary source located at $\vec x =0$ with a null polarization vector $\vec\varepsilon$ is given by B’\^[(1)]{}\_[(s)]{}(y,|y) = [(y(\_0 +\^z )[/]{}\^z(\_0 +\^z) y)\^s (x\_0\^2+1)\^[2s+1]{}]{}e\^[-y(\^z-2[\_0 +\^zx\_0\^2+1]{})|y]{}+c.c. Alternatively, if we fix the base point to be at $(0,z=1)$ and the source at $\vec x_0$, then in the $W=0$ gauge we have B’\^[(1)]{}\_[(s)]{}(y,|y) = [(y(\_0 -\^z )[/]{}\^z(\_0 -\^z) y)\^s (x\_0\^2+1)\^[2s+1]{}]{}e\^[-y(\^z+2[\_0 -\^zx\_0\^2+1]{})|y]{}+c.c. It will be useful to express the null polarization vector as a spinor bilinear $({\slash\!\!\!\varepsilon}\sigma^z)_{{{\dot\alpha}}{{\dot\beta}}} =\bar \lambda_{{\dot\alpha}}\bar\lambda_{{\dot\beta}}$. In our conventions, we can also write $\bar\lambda = \sigma^z \lambda$. We can then construct a generating function for the boundary-to-bulk propagator associated with currents of all spins as B’\^[(1)]{}(y,|y) &= [1x\_0\^2+1]{} + c.c.\
&= [1x\_0\^2+1]{} + c.c. Keep in mind that we should in fact only select the part of this generating function which is even in $\lambda$, because the theory describe all the [*integer*]{} spins.[^3]
Once we solve the second order field in the $W=0$ gauge, the expectation value of $B$ in the standard gauge at $\vec x=0$, near $z=0$, is recovered from &B\^[(2)]{}(x =0,z0,y,|y,z,|z) = L\^[-1]{}(x =0,z0,y,|y)\*B’\^[(2)]{}(y,|y,z,|z)\*L(x =0,z0,y,-|y) \[B2phys\] Given a function $f(Y,Z)$, let us consider the twisted adjoint action by $L$, evaluated near the boundary of $AdS_4$, &F(z,Y,Z)=L\^[-1]{}(x=0,z0, Y)\*f(Y,Z)\*(L(x=0,z0, Y))\
&16 z (- [1-z\^[1/2]{}1+z\^[1/2]{}]{} y\^z |y )\* f(y,|y,z,|z) \*(- [1-z\^[1/2]{}1+z\^[1/2]{}]{} y\^z |y )\
&=16z d\^4 u d\^4 v d\^4u’ d\^4v’ (uv+|u|v+u’v’+|u’|v’) (- [1-z\^[1/2]{}1+z\^[1/2]{}]{} (y+u+u’)\^z (|y+|u+|u’) )\
& f(y+v+u’,|y+|v+|u’,z-v+u’,|z-|v+|u’) (- [1-z\^[1/2]{}1+z\^[1/2]{}]{} (y+v’)\^z (|y+|v’) ) In the second line we have dropped subleading terms in $z$ in the overall factor, which do not affect the boundary expectation value of fields of various spins. For the purpose of extracting the three-point functions of the currents, we may restrict to $\bar y=\hat z=\bar z=0$ while keeping the dependence on $y$ only. Denote by $F(z,y_{{\alpha}})\equiv F(z,y_{{\alpha}},\bar y_{{\dot\alpha}}=z_{{\alpha}}=\bar z_{{\dot\alpha}}=0)$, we have &F(z,y\_) =16z d\^4 u d\^4 v d\^4u’ d\^4v’ (u(v-[y2]{})+|u|v+(u’-[y2]{})v’+|u’|v’)\
& (- [1-z\^[1/2]{}1+z\^[1/2]{}]{} ([y2]{}+u+u’)\^z (|u+|u’) ) (- [1-z\^[1/2]{}1+z\^[1/2]{}]{} (y+v’)\^z |v’ )\
& f(v+u’,|v+|u’,-v+u’,-|v+|u’)\
&=16z d\^4 u d\^4 v d\^4u’ d\^4v’\
& (uv+|u|v+u’v’+|u’|v’ - u’ v-|u’ |v) f(v+u’,|v+|u’,-v+u’,-|v+|u’)\
&=z d\^4 u d\^4v d\^4 p d\^4q\
& ([u(p-q)+|u(|p-|q)+(p+q)v+(|p+|q)|v +pq+|p|q2]{}) f(p,|p,q,|q). The functions $f(Y,Z)$ that shows up in the computation of $B^{(2)}$ depend either only on $z_{{\alpha}}$ or only on $\bar z_{{\dot\alpha}}$ (see eq. (\[Bprime2\])). We will treat the two cases separately. First, consider the case where $f(y,\bar y,z,\bar z)=f(y,\bar y,z)$ is independent of $\bar z_{{\dot\alpha}}$. Then &F(z,y\_) =4 z d\^4 u d\^2v d\^4 p d\^2q ([(u-v)p-(u+v)q+2|u |p +pq2]{}) f(p,|p,q)\
&=z d\^2 p d\^2q { + } f(p,-[1+z\^[12]{}1-z\^[12]{}]{}\^z q,q)\
&z d\^2 p d\^2q e\^[ (1+)pq - 2 yq ]{} f(p,-\^z q,q) In the last step, we have taken the limit $z\to 0$ while keeping $\sqrt{z} y$ fixed. $\epsilon\sim \sqrt{z}$ is understood as a small positive number that will be taken to zero at the end. For the moment, we need to keep it nonzero to regularize integrals that appear in $*$-products.
Now consider the other case, where $f(y,\bar y,z,\bar z) = \bar f(y,\bar y,\bar z)$ is independent of $z_{{\alpha}}$. Then &F(z,y\_) = 4z d\^4 u d\^2|v d\^4 p d\^2|q\
& ([2up+(|u-|v)|p-(|u+|v)|q +|p|q2]{}) |f(p,|p,|q)\
&z d\^2|p d\^2|q { [1+z\^[12]{}1-z\^[12]{}]{} y\^z |p + } |f(-y-[1+z\^[12]{}1-z\^[12]{}]{}\^z|q,|p, |q)\
&z d\^2|p d\^2|q e\^[(1+)|p|q+2y\^z |p ]{} |f(-\^z|q,|p, |q-\^z y) In the last step, we again take $z\to 0$ while keeping $\sqrt{z}y$ fixed. It may seem that this limit is not well defined, because of the $y$ dependence in $\bar f\left(-\sigma^z\bar q,\bar p, \bar q-\sigma^z y\right)$. We will see below that in fact this is not the case. To be more precise, let us define \_[z0\^+]{} z\^[-1]{}F(z, z\^[-[12]{}]{}y\_) = F(2y\_) whose order ${\cal O}(y^{2s})$ term contains the boundary expectation value of the spin-$s$ component of $B$ field, with the power of $z$ stripped off. $\tilde F$ is then computed from[^4] & F(w\_) = \_[0\^+]{}Although not obvious from this expression, the $\xi\to +\infty$ limit of the second integral is expected to be well defined, as shown below. Recall that f(y,-\^z z,z;U,V) = \_0\^1 dt\_0\^d(zu) e\^[(u+tz)(y-v) + z \^z|u]{} ( z(tu+\^z|v) ). We can compute the integrals & \_[0\^+]{}d\^2y d\^2 z e\^[(1+)yz+zw]{}f(y,-\^z z,z;U,V)\
&= \_[0\^+]{} d\^2z(zu) \_0\^1 dt\_0\^d(u+(t-1-)z) e\^[z(w-v+\^z|u) ]{} (z(tu+\^z|v))\
&=0, and &\_[0\^+]{} \_[+]{}d\^2|y d\^2|z e\^[(1+)|y|z+|y\^z w]{}|f(-\^z |z,|y, |z-\^z w;U,V)\
&= \_[0\^+]{}\_[+]{} d\^2|z((|z+w\^z)|u) \_0\^1 dt\_0\^d(|u+t(|z-\^z w)-((1+)|z+\^z w))\
& e\^[-((1+)|z-w\^z)|v+|z\^z u ]{} (t(|z+w\^z)|u+|z\^z v)\
&= [sgn]{}(w\^z|u) d\^2|z ([w\^z |zw\^z |u]{}) ([(|z-w\^z)|uw\^z |u]{}) e\^[|z(-|v+\^z u)+ w \^z |v ]{} (|z(|u+\^z v)-w\^z|u). To obtain the last line, we have used the two-dimensional $\delta$-function to integrate over $t,\eta$. The step functions come from requiring that the value of $t,\eta$ which solve the $\delta$-function constraint lie inside the corresponding integration domains.
Writing $\bar z=-(\tau_1+1) \sigma^z w + \tau_2 \bar u$, ${{\alpha}}_\pm = (\bar v-\sigma^z u)\pm (\bar u+\sigma^z v)$, we can express the above integral as \[inttmp\] & -[12]{}(w\^z |u)\_0\^d\_1 \_0\^d\_2\
&=-[12]{}(w\^z|u). From (\[Bprime2\]) and (\[B2phys\]), we then obtain the result \[ftd\] &\_[z0]{} z\^[-1]{}B\^[(2)]{}(x=0,z|y=z\^[-[12]{}]{},|y=Z=0)\
&=-d\^4u d\^4v e\^[uv-|u|v]{}B’\^[(1)]{}(u,|u)B’\^[(1)]{}(v,|v) (w\^z|u). The integration over $(u,\bar u,v,\bar v)$ should be understood as a contour integral, and the choice of contour is now important. The need for this choice of contour is possibly due to the slightly singular nature of the $W=0$ gauge. In the next section, we will see that the three point function is essentially a twistor transform of \[uvuv\] e\^[uv-|u|v]{} (w\^z|u). Namely if we regard $(u,\bar u,v,\bar v)$ as independent holomorphic variables, and Fourier transform two of them, then we obtain (a generating function of) the three-point functions in terms of polarization spinors (see eq. (\[twprop\]) and the paragraph thereafter). $w=2y$ will be identified with the polarization spinor of the third (outcoming) operator. The question of contour prescription now amounts to choosing a 4-dimensional contour (on two of $(u,\bar u,v,\bar v)$) for the twistor transform. We will demand that $w\sigma^z{{\alpha}}_\pm$ and $\bar u{{\alpha}}_\pm$ encircle the origin in the complex plane with opposite orientation, so that \[rational\] picks up residue $\mp 1$ when integrated in ${{\alpha}}_\pm$.[^5] Consequently, (\[ftd\]) can be replaced by the residue contribution &\_[z0]{} z\^[-1]{}B\^[(2)]{}(x=0,z|y=z\^[-[12]{}]{},|y=Z=0)\
&=d\^4u d\^4v e\^[uv-|u|v]{}B’\^[(1)]{}(u,|u)B’\^[(1)]{}(v,|v)\
& . \[B2-final\]
Three point functions from twistor space
========================================
In this section we show that a drastic simplification occurs if we consider a twistor transform of the correlation functions on the polarization spinors $\lambda_1,\lambda_2$ of the boundary sources (recall that these are related to the null polarization vectors by $({\slash\!\!\!\varepsilon}\sigma^z)_{{{\dot\alpha}}{{\dot\beta}}} =\bar \lambda_{{\dot\alpha}}\bar\lambda_{{\dot\beta}}$, and $\bar\lambda = \sigma^z \lambda$). To see this, let us perform the Fourier transform of the boundary-to-bulk propagator for $B'$ in the $W=0$ gauge, with boundary source located at $\vec{x}_0$ B’\^[(1)]{}(y,|y;) = [1x\_0\^2+1]{} e\^[-y(\^z+ 2 [\_0 -\^zx\_0\^2+1]{})|y]{} { + }, \[b-t-b-generating\] whose Fourier transform is given by &B\_[tw]{}\^[(1)]{}(y,|y;) = [14]{} d\^2e\^[2]{} B’\^[(1)]{}(y,|y;)\
&= [1x\_0\^2+1]{}\
&= [1x\_0\^2+1]{}(+ \^z[\_0 -\^zx\_0\^2+1]{} y )\
& + [1x\_0\^2+1]{} (- [\_0 -\^zx\_0\^2+1]{} |y )\
&=( y + (\_0 \^z-1)) e\^[ -( \_0 -\^z )|y ]{} + (|y-(\_0 -\^z) ) e\^[ -y(\^z\_0 +1)]{}. \[lam-to-mu\] As remarked earlier, the fact that we only have integer spins in the spectrum implies that we should actually take the contribution even in $\lambda$ in (\[b-t-b-generating\]), or even in $\mu$ in (\[lam-to-mu\]).
We can also write $\bar\mu=-\sigma^z\mu$ and &B\_[tw]{}\^[(1)]{}(y,|y;) = (y- (\_0 -\^z)|) e\^[-( \_0 -\^z )|y ]{} + (|y-(\_0 -\^z) ) e\^[- |(\_0 -\^z)y ]{}. Let us further define &= (\_0 -\^z)|,\
&|= (\_0 -\^z), \[chi-def\] so we end up with simply \[twprop\] &B\_[tw]{}\^[(1)]{}(y,|y;,|) = (y- ) e\^[||y ]{} + (|y-|) e\^[y ]{}. We could regard $y,\bar y$ as independent holomorphic variables, and interpret the two terms in $B_{tw}^{(1)}$ as delta functions in the corresponding twistor space, where one of $y$ and $\bar y$ is Fourier transformed. This explains our earlier claim that the generating function of three point functions can be viewed as a twistor transform of (\[uvuv\]) over two of $(u,\bar u,v,\bar v)$.
Assuming the choice of contour as explained in the previous section, we can now easily compute the rescaled expectation value of the outcoming higher spin fields near the boundary (more precisely, the generalized Weyl curvature of the HS fields). Denoting the position of the two boundary sources by $\vec{x}_1$ and $\vec{x}_2$, with $\chi_{1,2}$ defined as in (\[chi-def\]), we have &\_[z0]{} z\^[-1]{}B\^[(2)]{}(x=0,z|z\^[-[12]{}]{}y,|y=Z=0;\_1,\_2)\
& = d\^4u d\^4v e\^[uv-|u|v]{} B’\^[(1)]{}\_[tw]{}(u,|u,\_1,|\_1)|\_[\_1-[even]{}]{} B\_[tw]{}’\^[(1)]{}(v,|v,\_2,|\_2)|\_[\_2-[even]{}]{}\
& + (12)\
&= d\^4u d\^4v e\^[uv-|u|v]{} \_[\_1-[even]{}]{}\_[\_2-[even]{}]{}\
& + (12)\
&= 2 ([\_1\_2+|\_1|\_2]{})\
& +(\_1-\_1)+(\_2-\_2)+(\_1-\_1,\_2-\_2). \[solve-deltas\] In terms of $\mu_1, \mu_2$, it is \[resa\] &\_[z0]{} z\^[-1]{}B\^[(2)]{}(x=0,z|z\^[-[12]{}]{}y,|y=Z=0;\_1,\_2)\
&=[12]{} ([2\_1\^z[**x\_[12]{}**]{}\_2]{} )\
& +(\_1-\_1)+(\_2-\_2)+(\_1-\_1,\_2-\_2). Here and in what follows, we will use for convenience the notation ${\bf x_{1,2}}$ instead of $\vec{x}_{1,2}\cdot \vec{\sigma}$, which is equivalent since $x_{1,2}$ are by definition three dimensional vectors.
Now let us Fourier transform back in the polarization spinors $\lambda_1,\lambda_2$. For example, in the case when the outcoming field is a scalar, we can set $y=0$ in (\[resa\]). The result is then the $(\lambda_1,\lambda_2)$-even part of &4d\^2\_1 d\^2\_2 e\^[2\_1\^z [**x\_[12]{}**]{}\_2 -2\_1\_1-2\_2\_2]{}([**x\_1**]{}\_1+[**x\_2**]{}\_2) + (12)\
&=[4x\_2\^2]{}d\^2\_1 + (12)\
&= [2|x\_1||x\_2||x\_[12]{}|]{} + (12)\
&= [4|x\_1||x\_2| |x\_[12]{}|]{} where we redefined \_i = [[**x\_i**]{}\_ix\_i\^2]{}, x\_i = [x\_i|x\_i|]{}. In terms of the polarization vectors $\varepsilon_1,\varepsilon_2$, or the corresponding hatted variables, it is given by For general spin we need to keep the $y$ dependence of the outcoming field in (\[resa\]), and hence the three point function receives two contributions, from the two terms in the second line of (\[resa\]). The Fourier transform of the first term into $(\lambda_1,\lambda_2)$ is &[14|x\_1||x\_2| |x\_[12]{}|]{} + (12)\
&= [14|x\_1||x\_2| |x\_[12]{}|]{} + (12)\
&= [14|x\_1||x\_2| |x\_[12]{}|]{} + (12) where we defined ${\bf \check X_{12}} = {{\bf x_1}\over x_1^2}-{{\bf x_2}\over x_2^2}$. Now we replace $y$ by $\lambda_3$, and replace the origin by $\vec x_3$ where the third operator is inserted. The resulting contribution to the (generating function of) three point functions is &[14|x\_[12]{}||x\_[23]{}| |x\_[31]{}|]{} + (12). On the other hand, the Fourier transform of the second term in the second line of (\[resa\]) is given by &[14|x\_1||x\_2| |x\_[12]{}|]{} + (12)\
& + (12). where we have again made the substitution of $y$ by $\lambda_3$ so that the crossing symmetry in the three currents is manifest. Together with the terms related by flipping the sign of $\lambda_1$ and $\lambda_2$ respectively, the total contribution to the generating function of all three point functions is $$\begin{aligned}
\label{ttm}
&&{4\over |x_{12}||x_{23}| |x_{31}|}
\cosh\left( {x_{23}^2\lambda_1\sigma^z {\bf x_{12}x_{23}x_{13}}\lambda_1
+x_{13}^2\lambda_2\sigma^z{\bf x_{23} x_{31} x_{21}}\lambda_2 + x_{12}^2\lambda_3\sigma^z{\bf x_{31}x_{12}x_{32}}\lambda_3
\over 2x_{12}^2x_{23}^2x_{31}^2} \right)\cr
&&~~~\times \cosh\left(\lambda_1 \sigma^z{{\bf x_{12}}\over x_{12}^2}\lambda_2\right)\cosh\left( \lambda_1\sigma^z {{\bf x_{13}}\over x_{13}^2}\lambda_3 \right)\cosh\left( \lambda_2\sigma^z {{\bf x_{23}}\over x_{23}^2}\lambda_3 \right).\end{aligned}$$ A given three point function of higher spin currents $\langle J_{s_1}(x_1;\lambda_1) J_{s_2}(x_2;\lambda_2) J_{s_3}(x_3;\lambda_3)\rangle$ can be now obtained from this generating function by simply extracting the contribution which goes like $\lambda_1^{2s_1} \lambda_2^{2s_2} \lambda_3^{2s_3}$. In the conjectured dual free scalar theory, using free field Wick contractions, one may derive the following generating function of $n$-point functions [@Giombi:2009wh] (here we assume null polarization vectors as above) \_[S\_n]{} P\_ \[free-gen\] where $P_\sigma$ stands for the permutation on $(\vec x_i;{\vec \varepsilon}_i)$ by $\sigma$, and the product is understood to be of cyclic order ($\overleftarrow\partial$ and $\overrightarrow\partial$ act on their neighboring propagators only). The $n$-point function for given spins is obtained by extracting the appropriate powers of the polarization vectors $\varepsilon_i$. Our bulk result (\[ttm\]) in fact generates exactly the same set of three point functions as the $n=3$ case of (\[free-gen\]).[^6] A proof is given in the appendix. Thus we have found complete agreement of the bulk tree-level three-point functions with the three point functions of higher spin currents in the free $O(N)$ scalar CFT. In the following we describe some simple checks in special cases.
Without loss of generality, we can fix the positions $x_1, x_2, x_3$ by conformal symmetry to $x_1=e_1,x_2=-e_1,x_3=0$, so that (\[ttm\]) reduces to &2( [-\_1\^z [**e\_1**]{}\_1 -\_2\^z[**e\_1**]{}\_2 -4 \_3\^z[**e\_1**]{}\_3 4]{}) ([\_1 \^z[[**e\_1**]{}]{}\_22]{})( \_1\^z [[**e\_1**]{}]{}\_3 )( \_2\^z [[**e\_1**]{}]{}\_3 ) As an example, let us extract the three point function of the stress energy tensor $\langle TTT\rangle$, from the ${\cal O}(\lambda_1^4\lambda_2^4\lambda_3^4)$ term. If we further use the remaining 1 conformal transformation to set $e_1\cdot \varepsilon_1=0$, we end up with the following simple expression for $\langle TTT\rangle$, \[tttsp\] &[124]{} Let us compare this with the stress energy tensor of a free massless scalar in 3d, contracted with a null polarization vector $\varepsilon$, T\_ = ()\^2 - [18]{}()\^2\^2. We have &T\_[\_1]{}(x\_1)T\_[\_2]{}(x\_2)T\_[\_3]{}(x\_3) = (\_1(x\_1))\^2(\_2(x\_2))\^2(\_3(x\_3))\^2\
& -[18]{}\
& +[164]{}\
& -[1512]{} (\_1\_1)\^2(\_2\_2)\^2(\_3\_3)\^2(x\_1)\^2 (x\_2)\^2 (x\_3)\^2\
&= 8 (\_1\_1 \_2\_2 [1|x\_[12]{}|]{}) (\_1\_1 \_3\_3 [1|x\_[13]{}|]{}) (\_3\_3 \_2\_2 [1|x\_[23]{}|]{})\
& -{ (\_1\_1)\^2 + cyclic }\
& +[18]{}{ (\_1\_1)\^2(\_2\_2)\^2 + cyclic }\
& -[164]{} (\_1\_1)\^2(\_2\_2)\^2(\_3\_3)\^2 [1|x\_[12]{}||x\_[13]{}||x\_[23]{}|]{} Of course, we could also extract this result directly from the generating function (\[free-gen\]), but we have repeated the derivation for clarity. Without loss of generality, we can now specialize to the case $x_1=e_1,x_2=-e_1,x_3=0$ and $e_1\cdot \varepsilon_1=0$ using conformal symmetry, and the result exactly matches (\[tttsp\]) (up to the overall normalization constant).
Another check of (\[ttm\]) is in the limit $\vec x_{12}=\vec\delta\to 0$, $\vec x_{13}\simeq \vec x_{23}\simeq \vec x$. This can be compared to the limit of “colliding sources" which was studied in [@Giombi:2009wh]. We have ( [\_1\^z[/]{}\_1+\_2\^z[/]{}\_22\^2]{} + [\_3\^z[**x**]{}[/]{}[**x**]{}\_32x\^4]{} ) . There are two special cases that we studied before in the “physical gauge": $\lambda_2=0$ and $\lambda_3=0$. In the $\lambda_2=0$ case, the three point function in the $\delta\to 0$ limit is ( [\_1\^z[/]{}\_12\^2]{} + [\_3\^z[**x**]{}[/]{}[**x**]{}\_32x\^4]{} ) \[collide1\] whereas in the $\lambda_3=0$ case, it is given by &[1x\^2]{} ( [\_1\^z[/]{}\_1+\_2\^z[/]{}\_22\^2]{} ) ( [\_1\^z[/]{}\_2\^2]{} )\
&=[1x\^2]{} . \[collide2\] These indeed agree with the results we found in [@Giombi:2009wh]. [^7]
Finally, let us turn to the type B model of [@Sezgin:2003pt]. Instead of (\[twprop\]), the boundary-to-bulk propagator for the $B$ master field in the type B model, after the Fourier transform in polarization spinors, is given by[^8] \[twb\] &B\_[tw;B]{}\^[(1)]{}(y,|y;,|) = i(y- ) e\^[||y ]{} - i(|y-|) e\^[y ]{}. Note that the scalar field component has disappeared from (\[twb\]). The bulk scalar is parity odd in the type B model, and the “standard" boundary condition assigns scaling dimension 2 to its dual operator. Therefore the scalar has to be treated separately, and we will only consider HS currents for now. The generating function for $\langle JJJ\rangle$ is now the $(\lambda_1,\lambda_2,\lambda_3)$-even part of \[ttmf\] &[4|x\_[12]{}||x\_[23]{}| |x\_[31]{}|]{} ( [x\_[23]{}\^2\_1\^z [**x\_[12]{}x\_[23]{}x\_[13]{}**]{}\_1 +x\_[13]{}\^2\_2\^z[**x\_[23]{} x\_[31]{} x\_[21]{}**]{}\_2 + x\_[12]{}\^2\_3\^z[**x\_[31]{}x\_[12]{}x\_[32]{}**]{}\_3 2x\_[12]{}\^2x\_[23]{}\^2x\_[31]{}\^2]{} )\
& (\_1 \^z[[**x\_[12]{}**]{}x\_[12]{}\^2]{}\_2) ( \_1\^z [[**x\_[13]{}**]{}x\_[13]{}\^2]{}\_3 ) ( \_2\^z [[**x\_[23]{}**]{}x\_[23]{}\^2]{}\_3 ). This is conjectured to be dual to the free $O(N)$ fermion theory in three dimensions [@Sezgin:2003pt]. As a check let us consider $\langle TTT\rangle$. As before, by conformal symmetry we can fix $x_1=e_1,x_2=-e_1,x_3=0$ and $e_1\cdot\varepsilon_1=0$, and the three point function of the stress energy tensor from Vasiliev theory in this case is given by \[testa\] -[13]{}( e\_1\_3\_1\_2+e\_1\_2\_1\_3 )\^2 The stress energy tensor of the free fermion theory, with null polarization vector $\varepsilon$, is T\_\^F = (). It is straightforward to check that that (\[testa\]) indeed produces exactly $\langle T^F_{\varepsilon_1}(x_1)T^F_{\varepsilon_2}(x_2)T^F_{\varepsilon_3}(x_3)\rangle$, up to the overall normalization constant.
Concluding remarks
==================
In this paper we have shown that the tree level three point functions of Vasiliev’s minimal bosonic higher spin gauge theory in $AdS_4$ exactly agree with the three point functions of higher spin currents in the free theory of $N$ massless scalars in the $O(N)$ singlet sector in 3 dimensions. The bulk computation is made possible by the remarkable simplification in the $W=0$ gauge, where the integration over the $AdS_4$ is replaced by the $*$-product of twistor-like internal variables of Vasiliev’s master fields.
The agreement of the three point functions $\langle JJJ\rangle$ with the complete position and polarization dependence included is a nontrivial check of the conjecture of Sezgin-Sundell-Klebanov-Polyakov. As a special case, the three point function of the stress energy tensor $\langle TTT\rangle$ in a three dimensional CFT is constrained by conformal symmetry up to a linear combination of two possible structures, one corresponding to that of a free massless scalar, the other corresponding to that of a free massless fermion [@Osborn:1993cr]. From the perspective of the bulk Lagrangian, the tree level $\langle TTT\rangle$ is sensitive to the higher derivative terms in the graviton. Indeed, computing $\langle TTT\rangle$ from pure Einstein gravity in $AdS_4$ would produce a linear combination of the two tensor structures [@Arutyunov:1999nw]. The agreement we found is therefore a test of the precise higher derivative structure of Vasiliev’s theory.
We have also seen that the three point functions in type B model matches that of free fermions, verifying a conjecture of [@Sezgin:2003pt]. In fact, our result also applies to the nonminimal Vasiliev theory, without imposing the projection (\[proja\]) and so both even and odd integer spins are included. The result then matches the free CFT of $N$ complex scalars in the $SU(N)$ singlet sector. In this theory, we may choose alternative boundary conditions for the bulk scalar field as well as the vector gauge field [@Witten:2003ya; @Leigh:2003ez; @Petkou:2004wb], which would lead to conjectured dual critical scalar QED (with $N$ flavors) or critical $\mathbb{CP}^{N-1}$ models in 2+1 dimensions. It would be very interesting if one can learn about these CFTs from Vasiliev theory.
It is now technically feasible to generalize our computation to higher point functions as well as to loop corrections in the bulk. An extremely interesting problem is to understand the HS symmetry breaking in the critical $O(N)$ model from corrections by scalar loops with $\Delta=2$ boundary condition in the bulk. We hope to report on results toward these directions in the near future.
Acknowledgments {#acknowledgments .unnumbered}
---------------
We are grateful to N. Boulanger, V.E. Didenko, C. Iazeolla, I. Klebanov, J. Maldacena for very useful discussions, and especially to P. Sundell for pointing out to us the importance of the $W=0$ gauge in Vasiliev theory. X.Y. would like to thank Université de Mons and University of Crete for their hospitality during the course of this work. This work is supported in part by the Fundamental Laws Initiative Fund at Harvard University. S.G. is supported in part by NSF Award DMS-0244464. X.Y. is supported in part by NSF Award PHY-0847457.
The equivalence of two generating functions
===========================================
In this appendix we will show that (\[ttm\]) and (\[free-gen\]) generate the same three-point functions of higher spin currents. In terms of the null polarization vectors $\vec\varepsilon_i$, (\[ttm\]) can be written as &[4|x\_[12]{}||x\_[23]{}||x\_[31]{}|]{}\
&\_[i=1]{}\^3 . We can use the conformal group to fix $\vec\varepsilon_i=t_i \vec\varepsilon$, $i=1,2,3$, where $t_i$ is a scale factor and $\vec\varepsilon$ is a common polarization vector. The expression then simplifies to \[simp\] [12|x\_[12]{}||x\_[23]{}||x\_[31]{}|]{} \_[\_i=1]{}On the other hand, (\[free-gen\]) for $n=3$ can be written as &[18]{}\_[\_i=1]{} [1|x\_[i,i+1]{}|]{} + (12)\
&\_[\_i=1]{} [1|x\_[i,i+1]{}|]{}+ (12) where in the second step we have restricted to the case $\vec\varepsilon_i=t_i\vec\varepsilon$. Expanding the exponential, we have &[18]{}\_[\_i=1]{}\_[s\_1,s\_2,s\_3]{}[t\_1\^[s\_1]{}t\_2\^[s\_2]{}t\_3\^[s\_3]{}s\_1!s\_2!s\_3!]{} \_[n\_1,n\_2,n\_3]{} [2s\_1n\_1]{}[2s\_2n\_2]{}[2s\_3n\_3]{} \_1\^[n\_1]{}\_2\^[n\_2]{}\_3\^[n\_3]{}\
& + (12)\
&= [1|x\_[12]{}||x\_[23]{}||x\_[31]{}|]{} \_[s\_1,s\_2,s\_3]{}[t\_1\^[s\_1]{}t\_2\^[s\_2]{}t\_3\^[s\_3]{}s\_1!s\_2!s\_3!]{} \_[m\_1,m\_2,m\_3]{} [2s\_12m\_1]{}[2s\_22m\_2]{}[2s\_32m\_3]{} 2\^[s\_1+s\_2+s\_3]{}\
& \
& + (12) Redefining $s_1-m_1+m_2=k_1$, $s_2-m_2+m_3=k_2$, $s_3-m_3+m_1=k_3$, we can write it as & [1|x\_[12]{}||x\_[23]{}||x\_[31]{}|]{} \_[s\_1,s\_2,s\_3]{}[t\_1\^[s\_1]{}t\_2\^[s\_2]{}t\_3\^[s\_3]{}s\_1!s\_2!s\_3!]{} \_[m\_1,m\_2,m\_3]{} [2s\_12m\_1]{}[2s\_22m\_2]{}[2s\_32m\_3]{} 2\^[k\_1+k\_2+k\_3]{}\
& + (12)\
& = \_[s\_1,s\_2,s\_3]{}[t\_1\^[s\_1]{}t\_2\^[s\_2]{}t\_3\^[s\_3]{}]{} A\_[s\_1,s\_2,s\_3]{}(x\_i, ) The term $A_{s_1,s_2,s_3}(\vec x_i,\varepsilon)$ gives the three-point function of currents of spin $(s_1,s_2,s_3)$, $\langle J_{s_1} J_{s_2} J_{s_3} \rangle$, up to a normalization factor. Now consider a sum with a different normalization factor on the currents, &\_[s\_1,s\_2,s\_3]{} [2\^[s\_1+s\_2+s\_3]{}s\_1!s\_2!s\_3!(2s\_1)!(2s\_2)!(2s\_3)!]{}[t\_1\^[s\_1]{}t\_2\^[s\_2]{}t\_3\^[s\_3]{}]{} A\_[s\_1,s\_2,s\_3]{}(x\_i, )\
&= [18|x\_[12]{}||x\_[23]{}||x\_[31]{}|]{}\_[\_i=1]{} \_[k\_1,k\_2,k\_3]{} (t\_1\^[12]{}+\_3 t\_2\^[12]{})\^[2k\_1]{} (t\_2\^[12]{}+\_1 t\_3\^[12]{})\^[2k\_2]{} (t\_3\^[12]{}+\_2 t\_1\^[12]{})\^[2k\_3]{} 4\^[k\_1+k\_2+k\_3]{}\
& + (12)\
&= [18|x\_[12]{}||x\_[23]{}||x\_[31]{}|]{}\_[\_i=1]{}\
& + (12)\
&= [14|x\_[12]{}||x\_[23]{}||x\_[31]{}|]{}\_[\_i=1]{}This indeed agrees with (\[simp\]), thus proving the equivalence of the generating functions.
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S. Giombi and X. Yin, “Higher Spin Gauge Theory and Holography: The Three-Point Functions,” arXiv:0912.3462 \[hep-th\]. M. A. Vasiliev, “More On Equations Of Motion For Interacting Massless Fields Of All Spins In (3+1)-Dimensions,” Phys. Lett. B [**285**]{}, 225 (1992). M. A. Vasiliev, “Higher-spin gauge theories in four, three and two dimensions,” Int. J. Mod. Phys. D [**5**]{}, 763 (1996) \[arXiv:hep-th/9611024\]. M. A. Vasiliev, “Higher spin gauge theories: Star-product and AdS space,” arXiv:hep-th/9910096. M. A. Vasiliev, “Nonlinear equations for symmetric massless higher spin fields in (A)dS(d),” Phys. Lett. B [**567**]{}, 139 (2003) \[arXiv:hep-th/0304049\]. M. A. Vasiliev, “Algebraic aspects of the higher spin problem,” Phys. Lett. B [**257**]{}, 111 (1991). K. I. Bolotin and M. A. Vasiliev, Phys. Lett. B [**479**]{}, 421 (2000) \[arXiv:hep-th/0001031\]. X. Bekaert, S. Cnockaert, C. Iazeolla and M. A. Vasiliev, “Nonlinear higher spin theories in various dimensions,” arXiv:hep-th/0503128. E. Sezgin and P. Sundell, “Holography in 4D (super) higher spin theories and a test via cubic scalar couplings,” JHEP [**0507**]{}, 044 (2005) \[arXiv:hep-th/0305040\]. E. Sezgin and P. Sundell, “An exact solution of 4D higher-spin gauge theory,” Nucl. Phys. B [**762**]{}, 1 (2007) \[arXiv:hep-th/0508158\]. H. Osborn and A. C. Petkou, “Implications of Conformal Invariance in Field Theories for General Dimensions,” Annals Phys. [**231**]{}, 311 (1994) \[arXiv:hep-th/9307010\]. G. Arutyunov and S. Frolov, “Three-point Green function of the stress-energy tensor in the AdS/CFT correspondence,” Phys. Rev. D [**60**]{}, 026004 (1999) \[arXiv:hep-th/9901121\]. E. Witten, “SL(2,Z) action on three-dimensional conformal field theories with Abelian symmetry,” arXiv:hep-th/0307041. R. G. Leigh and A. C. Petkou, “SL(2,Z) action on three-dimensional CFTs and holography,” JHEP [**0312**]{}, 020 (2003) \[arXiv:hep-th/0309177\]. A. C. Petkou, “Double-trace deformations and SL(2,Z) action on three-dimensional CFTs,” Fortsch. Phys. [**52**]{}, 636 (2004).
[^1]: Sometimes we will use the notation $\hat z_{{\alpha}}$ instead of $z_{{\alpha}}$ for the internal variables to avoid confusion with the Poincaré radial coordinate $z$.
[^2]: Note that $B$ contains the generalized Weyl curvatures of the higher spin gauge fields. Nevertheless, to leading order in the Poincaré radial coordinate $z$ near the boundary $z=0$, the generalized Weyl curvature is proportional to the gauge field itself (in the gauge of [@Giombi:2009wh], see in particular eq. (3.60) of [@Giombi:2009wh]), and the correlation function of the current can be directly extracted from $B(x|y_{{\alpha}},\bar y_{{\dot\alpha}}=z_{{\alpha}}=\bar z_{{\dot\alpha}}=0)$.
[^3]: Equivalently, recall that consistency of the purely bosonic Vasiliev’s equations require the constraints $W(x|-Y,-Z)=W(x|Y,Z)$, $S_{\alpha}(x|-Y,-Z)=-S_{\alpha}(x|Y,Z)$, $S_{\dot\alpha}(x|-Y,-Z)=-S_{\dot\alpha}(x|Y,Z)$ and $B(x|Y,Z)=B(x|-Y,-Z)$.
[^4]: The limit $\xi \rightarrow \infty$ arises from the $y$-dependence in $\bar f\left(-\sigma^z\bar q,\bar p, \bar q-\sigma^z y\right)$, when taking the $z\rightarrow 0$ limit with $\sqrt{z}y$ fixed. Alternatively, one may also denote $\xi=\frac{1}{2\epsilon}$ and take a single limit $\epsilon \rightarrow 0^+$.
[^5]: One might contemplate the alternative possibility of choosing the orientation of the contour so that (\[rational\]) picks up residue $+1$ (or $-1$) when integrated in ${{\alpha}}_\pm$. In this case, however, one finds that (\[B2-final\]) vanishes identically, see eq. (\[solve-deltas\]).
[^6]: Note however that the two generating functions are defined with different normalizations on the currents.
[^7]: To see this, compare (\[collide1\]) and (\[collide2\]) to respectively eq. (6.23) and eq. (4.88) of [@Giombi:2009wh].
[^8]: In the type B model, the third equation of (\[cleom\]) is modified to $d_Z S+S*S=B*(-iK dz^2+i\bar K d\bar z^2)$. This leads to the extra factors of $i$ and $-i$ in the boundary-to-bulk propagator for $B$.
|
---
abstract: 'A statistical theory is presented of the magnesium ion interacting with lysozyme under conditions where the latter is positively charged. Temporarily assuming magnesium is not noncovalently bound to the protein, I solve the nonlinear Poisson-Boltzmann equation accurately and uniformly in a perturbative fashion. The resulting expression for the effective charge, which is larger than nominal owing to overshooting, is subtle and cannot be asymptotically expanded at low ionic strengths that are practical. An adhesive potential taken from earlier work together with the assumption of possibly bound magnesium is then fitted to be in accord with measurements of the second virial coefficient by Tessier [*et al.*]{} The resulting numbers of bound magnesium ions as a function of MgBr$_2$ concentration are entirely reasonable compared with densimetry measurements.'
author:
- |
Theo Odijk\
Lorentz Insitute for Theoretical Physics\
Leiden University\
The Netherlands\
E-mail: odijktcf@online.nl
title: 'Theory of the Interaction of the Magnesium Ion with Biopolymers: The Case of Lysozyme'
---
Introduction
============
We have a fairly good understanding of the way biopolymers interact with monovalent ions like Na$^+$ and Cl$^-$ (although there is now evidence that the electric fields of polyelectrolytes are so high that they influence the quantum mechanical properties of water [@1] which has obvious implications for charged biopolymers). On the other hand, the interaction with multivalent ions remains elusive. The magnesium ion at small concentrations, for instance, has a strong influence on the thermodynamic properties of DNA solutions as was established by Lerman [*et al.*]{} a long time ago [@2]. In the case of lysozyme, the Mg$^{2+}$ ion binds noncovalently to the positively charged protein but this happens at high concentrations of the cation [@3]. The binding appears to be corroborated in studies of the second virial coefficient $B_2$ of lysozyme in MgBr$_2$ solutions where a minimum was found at around 0.3 M [@4; @5].
The second virial of lysozyme in NaCl solutions was measured thoroughly by many experimental groups which allowed Prinsen and myself to establish the two parameters of the purported adhesive potential $U_{\rm A}$ between lysozyme spheres quite unambiguously [@6]. The potential is independent of protein charge and ionic strength so there is sound reason to hypothesize that it remains valid even when the salt is divalent like MgBr$_2$. As in Prinsen and Odijk [@6], I solve the Poisson-Boltzmann equation perturbatively in order to compute the effective charge of the protein based solely on electrostatics. The magnesium ion is excluded from the lysozyme surface so that the effective charge is larger than nominal as has already been discussed by Tellez and Trizac [@7]. The object of this paper is to set up a self-consistent theory of the interaction of the magnesium ion with positively charged lysozyme. Because the binding constant of the ion is unknown a priori, I evaluate an actual effective charge as an adjustable parameter via the measurements of $B_2$ [@4; @5] by letting the lysozyme spheres interact via the Poisson-Boltzmann equation and the adhesive potential $U_{\rm A}$. The resulting values of bound Mg$^{2+}$ as a function of Mg$^{2+}$ concentration are then compared with those established by densimetry [@3].
Tellez and Trizac already presented interesting numerical and analytical computations for spherical and cylindrical colloids in a 2–1 electrolyte (the cation is divalent whereas the counterion is monovalent) [@7]. Their analysis extends the previous multiscale method of Shkel [*et al.*]{} [@8] and is useful when $a \kappa \!>\! 1$ where $a$ is the radius of curvature and $\kappa^{-1}$ is the Debye screening length, as they showed numerically. Here, the objective is different: I solve the nonlinear Poisson-Boltzmann equation perturbatively for all $a \kappa$ where $a$ is the radius of the spherical colloid (lysozyme in our case). This results in a uniformly valid expression for the effective charge. The overshooting effect discussed in [@7] can then be understood at all $a \kappa$ for positively charged proteins or nanoparticles. The fully computed expression turns out to be subtle.
Solution of the Poisson-Boltzmann equation for a 2–1 electrolyte
================================================================
The nanosphere bears a charge $Zq$ were $q$ is the elementary charge and $Z \!>\! 1$. The electrostatic potential $\varphi(r)$ between two spheres separated by a distance $r$ is scaled by $k_{\rm B} T$ where $k_{\rm B}$ is Boltzmann’s constant and $T$ is the temperature: $\psi(r) \equiv q \, \varphi(r) / k_{\rm B} T$. If the concentration of 2–1 salt (MgBr$_2$ in the experiments to be discussed below) is $n$, the Debye screening length $\kappa ^{-1}$ is given by $\kappa^2 = 8 \pi Q I$ with ionic strength $I \!=\! 3 n$, the Bjerrum length $Q \!=\! q^{2} / D \, k_{\rm B} T$ where the permittivity $D$ is assumed to be uniform. The Poisson-Boltzmann equation then reads $$\Delta \Psi = \tfrac{1}{3} \, \kappa^2 \, \left( e^{\Psi} - e^{-2 \Psi} \right) \,,$$ with boundary conditions $$\begin{aligned}
\frac{d \Psi}{dr} |_{r=a} &=& - \frac{Z \, Q}{a^2} \,, \\
\lim_{r \rightarrow \infty} \Psi(r) &=& 0 \,.\end{aligned}$$ The linearized version of eq (1) has the usual Debye-Hückel solution $$\Psi_0(r) = \frac{Z \, Q}{(1 + \kappa a)} \, \frac{e^{-\kappa(r-a)}}{r} \,.$$ A pertubative solution to eq (1) is derived as follows (see Appendix 1 in ref 6; an error was made there – the zero-order screening term was deleted – but this is corrected here; fortunately, it turns out that errors incurred in the tables of ref 6 are within the margin of error). I seek a solution $\Psi = \Psi_0 + \Psi_1$ where $\Psi_1$ is uniformly smaller than $\Psi_0$ though $\Psi_0$ now has a higher effective charge $Z_ {\rm eff}$ instead of $Z$ owing to the divalent ion being substantially suppressed by the particle surface (see eq (1)). If we next scale the distance $r$ between the spheres by $\kappa$, $R \!=\! \kappa r$, we have $$\frac{1}{R^2} \, \frac{d}{dR} \left( R^2 \, \frac{d\Psi_1}{dR} \right) = \Psi_1 + S(R) \,,$$ with a source term $$S(R) \equiv - \tfrac{1}{2} \, \Psi_0^2 \,,$$ and $$\begin{aligned}
\Psi_0 &\equiv& \frac{B \, e^{-R}}{R} \,, \\
B &\equiv& \frac{Z_{\rm eff} \, Q \, \kappa \, e^{\mu}}{(1 + \mu)} \,, \\
\mu &\equiv& \kappa \, a \,.\end{aligned}$$ Eq (5) is readily solved by quadrature. First, set $\Psi_1 \!\equiv\! f(R) / R$ which leads to $$f^{\prime \prime} - f = R \, S(R) \,.$$ Then, set $f(R) \!\equiv\! w(R) \, e^{-R}$ yielding $$w^{\prime \prime} -2 \, w^{\prime} = R \, e^R \, S(R) \,.$$ Noting that $w^{\prime} \rightarrow$ 0 as $R \rightarrow \infty$, we can integrate eq (11) to get $$w^{\prime}(R) = \tfrac{1}{2} \, B^2 \, e^{2R} \, \int\limits_{R}^{\infty} \!\! dR^{\prime} \; \frac{e^{-3R^{\prime}}}{R^{\prime}} \,.$$ Another integration gives $$w(R) = - \tfrac{1}{2} \, B^2 \, \int\limits_{R}^{\infty} \!\! dR^{\prime\prime} \; e^{2R^{\prime\prime}} \,
\int\limits_{R^{\prime\prime}}^{\infty} \!\! dR^{\prime} \; \frac{e^{-3R^{\prime}}}{R^{\prime}} \,,$$ since $w \rightarrow$ 0 as $R \rightarrow \infty$. I note that $w$ is negative as it should be. Eq (13) may be reexpressed in terms of the exponential integral $$E_1(u) \equiv \int\limits_{u}^{\infty} \!\! dt \; \frac{e^{-t}}{t} \,,$$ yielding $$w(R) = - \tfrac{1}{4} \, B^2 \, \left[ E_1(R) - e^{2R} \, E_1(3R) \right] \,.$$ We thus have that $\Psi_1(R) \!=\! w(R) \, \exp(-R) / R$ so eq (2) becomes $$\left. \frac{d\Psi}{dR} \right|_{R=\mu} = - \frac{Z \, Q}{\kappa \, a^2} = \left. \frac{d\Psi_0}{dR} \right|_{R=\mu} + \left. \frac{d\Psi_1}{dR} \right|_{R=\mu}
= - \frac{Z_{\rm eff} \, Q}{\kappa \, a^2} + \left. \frac{d\Psi_1}{dR} \right|_{R=\mu}$$ The second term on the right-hand side of eq (16) is rewritten in terms of exponential integrals with the help of eqs (13) and (15) $$\left. \frac{d\Psi_1}{dR} \right|_{R=\mu} = \tfrac{1}{4} \, B^2 \, \left[ \frac{e^{\mu}}{\mu} \, E_1(3\mu)
- \frac{e^{\mu}}{\mu^2} \, E_1(3\mu) + \frac{(\mu+1) \, e^{-\mu}}{\mu^2} \, E_1(\mu) \right] \,,$$ which is always positive. Hence, we have $Z \!=\! Z_{\rm eff} - \lambda \, Z_{\rm eff}^2$ or $Z_{\rm eff} \!=\! Z + \lambda \, Z^2$ correct to ${\cal O}(Z^2)$ where $$\begin{aligned}
\lambda &\equiv& \tfrac{1}{4} \, Q \, \kappa \, \left[ \frac{(\mu-1) \, e^{3 \mu}}{(\mu + 1)^2} \, E_1(3\mu)
+ \frac{e^{\mu}}{(\mu+1)} \, E_1(\mu) \right] \nonumber \\
&\equiv& \tfrac{1}{4} \, Q \, \kappa \, g(\mu) \,.\end{aligned}$$ The asymptotic expansion of $E_1(\mu)$ at large $\mu$ is well known to be virtually useless [@9]. In effect, it is only as $\mu$ becomes exceedingly large ($\mu \!=\! {\cal O}(100)$) that eq (18) agrees with eq (3.16) computed by Tellez and Trizac [@7]. Obviously we need to use the full expression for $g(\mu)$ in practical calculations.
Developing a series expansion of the effective charge at low ionic strength ($a \kappa \!\ll\! 1$) is, however, straightforward. The leading term is of interest for it does not depend on $a$ $$Z_{\rm eff} = Z \, \left( 1 + \frac{Z \, Q \, \kappa \, \ln(3)}{4} \right) \,.$$ This is a useful estimate for proteins and nanocolloids in the case $Z = {\cal O}(10)$ and $Q \kappa = {\cal O}(0.1)$ say.
Application to lysozyme in MgBr$_2$
===================================
As in previous work [@6], the radius of the lysozyme is set $a \!=$ 1.7 nm and the Bjerrum length $Q \!=$ 0.71 nm at room temperature ($T \!=$ 298 K). Koehner [*et al.*]{} established the charge $Z \!=$ 7 at pH = 7.5 for lysozyme in a NaCl solution by titration [@10]. I assume this is also the bare charge for lysozyme in the case at hand. The theoretical second virial coefficient $$B_2 = 2 \pi \, \int\limits_{0}^{\infty} \!\! dr \; r^2 \, \left( 1 - e^{-U(r) / k_{\rm B} T} \right) \,,$$ is connected to the experimental second virial $B_{\rm exp}$ via $B_{\rm exp} = N_{\rm av} B_2 / M^2$ [@11] where $N_{\rm av}$ is Avogadro’s number and $M$ is the molar mass (14.3 kg / mol for lysozyme). The hard sphere coefficient $B_{\rm HS} = 16 \pi a^{3} / 3 \!=$ 82 nm$^3$. Hence, if the lysozyme molecule were a sphere, the experimental hard-sphere value would be $B_{\rm exp, HS} \!=$ 2.41 $\times$ 10$^{-4}$ mol ml/g$^2$.
Tessier [*et al.*]{} measured $B_{\rm exp}$ of lysozyme in MgBr$_2$ solutions at pH = 7.8 by self-interaction chromatography [@4]. By contrast, Guo [*et al.*]{} had already determinded $B_{\rm exp}$ by static light scattering in 1999 [@5]. The two sets are shown in Table 1. The second virial is probably difficult to measure accurately when the protein molecules attract each other, which may rationalize the disparity between the two methods. Both methods, however, establish that there is a minimum at about 0.3 M MgBr$_2$.
-------------- ------- -------
MgBr$_2$ (M)
SIC SLS
0.10 -2.30 -2.40
0.20 -5.00 -
0.30 -6.14 -4.50
0.43 -5.24 -4.40
0.53 -4.25 -3.70
0.70 -2.70 -3.20
1.00 0.00 -
-------------- ------- -------
: Experimental second virial coefficient $B_{\rm exp}$ as a function of the magnesium bromide concentration. Self-interaction chromatography (SIC) [@4]; Static light scattering (SLC) [@5].
Next, I shall use the comprehensive chromatograpy data to compute the number of magnesium ions using $$\frac{U(r)}{k_{\rm B} T} = \left\{
\begin{array}{cl}
\infty & \hspace*{10pt} 0 \leq r \leq 2a \\
U_{\rm DH} - U_{\rm A} & \hspace*{5pt} 2a \leq r < 2a + \delta a \\
U_{\rm DH} & \hspace*{11pt} r \geq 2a + \delta a \\
\end{array}
\right.$$ which is analogous to the total interaction potential introduced in [@7] with Debye-Hückel potential $$U_{\rm DH}(r) = 2 \, a \, \xi \, \frac{e^{- \kappa (r-2a)}}{r} \,,$$ where $$\xi \equiv \frac{Q \, Z^2_{\rm eff}}{2 a \, (1 + \mu)^2} \,,$$ is the coupling parameter of the renormalized nonlinear Poisson-Boltzmann interaction. The depth of the adhesive well is $U_{\rm A}$ and it’s thickness is $\delta a$. An accurate estimate of the second virial $B_{2}$ is computed in ref [@6]: $$\begin{aligned}
\frac{B_2}{B_{\rm HS}} &=& 1 + \tfrac{3}{8} \, J \,, \\
J &=& J_1 - \left( e^{U_{\rm A}} - 1 \right) \, J_2 \,, \\
J_1 &\simeq& \frac{4 \, (\mu + \tfrac{1}{2}) \, \xi}{\mu^2} \, \left( 1 - \tfrac{1}{2} \, \alpha \, \xi \right) \,, \\
J_2 &\simeq& 2 \delta \left[ e^{-\xi} + (1 + \tfrac{\delta}{2})^2 \, \exp(- \frac{\xi}{1+\frac{\delta}{2}} \, e^{-\mu \, \delta}) \right] \,, \\
\alpha &\equiv& \left( e^{- \xi} - 1 + \xi \right) / \xi^2 \,.\end{aligned}$$ In the case of lysozyme in a NaCl solution $\delta \!=$ 0.079, $U_{\rm A} \!=$ 3.70 and $\delta \, e^{U_{\rm A}} \!=$ 3.20 [@6]. It is assumed that these values pertain to lysozyme – MgBr$_2$ solutions also.
Table 2 is derived as follows. First, the effective charge $Z_{\rm eff}$ is computed from Poisson-Boltzmann electrostatics by numerically evaluating the function $g(\mu)$ (eq (18)). This pertains to the case when no magnesium ions are assumed to be bound to the lysozyme. The numerical calculation of the exponential integral is well known to be notoriously nontrivial but a powerful representation has been presented by applied mathematicians [@12].
-------------------- -------- -------- -------- -------- --------- --------- ---------
MgBr$_2$ (M) 0.10 0.20 0.30 0.43 0.53 0.73 1.00
$\mu$ 3.06 4.33 5.31 6.35 7.05 8.10 9.69
$g(\mu)$ 0.0759 0.0466 0.0320 0.0237 0.01985 0.01559 0.01135
$Z_{\rm eff}$ 8.2 8.0 7.9 7.8 7.7 7.6 7.6
$\xi_{\rm eff}$ 0.852 0.471 0.328 0.235 0.191 0.146 0.106
$B_2 / B_{\rm HS}$ -0.95 -2.07 -2.55 -2.17 -1.76 -1.12 0.0
$\xi_{\rm num}$ 0.839 0.490 0.357 0.508 0.691 1.945 2.15
$Z_{\rm eff, num}$ 8.1 8.2 8.2 11.5 14.6 27.7 34.2
bound Mg$^{2+}$ 0 0 0 2 3-4 10 13
-------------------- -------- -------- -------- -------- --------- --------- ---------
: Number of magnesium ions bound to lysozyme computed as outlined in the text. The function $g(\mu)$ is calculated numerically with the help of the procedure in ref.[@12] and $Z_{\rm eff} \!=\! Z + \lambda \, Z^2$. The purely electrostatic coupling constant $\xi_{\rm eff}$ is given by eq (23). When the attractive potential between two lysozyme spheres is switched on, $\xi_{\rm num}$ is computed from eqs (24)-(28) and $Z_{\rm eff, num}$ from eq (23).
Next, the coupling parameter $\xi$ is supposed adjustable in view of Mg$^{2+}$ binding to the surface and is computed numerically by imposing the experimental values of $B_2 / B_{\rm HS}$ [@4] in eq (24). Eq (23) then yields the corresponding numerically adjusted $Z_{\rm eff, num}$. The number of bound Mg ions is simply ($Z_{\rm eff, num} - Z_{\rm eff}) / 2$. The number of bound ions in the densitometery experiments at 1 M MgCl$_2$ was 4 at pH = 3.0 and 6 at pH = 4.5, which would lead to a tentative estimate of 10 at pH = 7.5. At 0.5 M MgCl$_2$, this number was 3 at pH = 3.0.
Concluding remarks
==================
It is expected that no magnesium ions are bound to the lysozyme at low concentrations and this is well borne out by the first three entries in Table 2. Beyond the minimum in $B_2$, the second virial coefficient measured by Tessier [*et al.*]{} imposes a value of 13 bound magnesium ions at 1 M MgBr$_2$ compared with a tentative extrapolation of 10 bound magnesium ions by densimetry [@3]. Accordingly, it would be of interest to perform new measurements at the appropriate pH in a full range of ionic strengths to see how well the current theory applies.
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abstract: |
By investigating in detail discontinuities of the first kind of real-valued functions and the analysis of unordered sums, where the summands are given by values of a positive real-valued function, we develop a measure-theoretical framework which in particular allows us to describe *rigorously* the representation and meaning of sums of jumps of type $\sum_{0 < s \leq t} \Phi \circ \vert \Delta
X_s \vert$, where $X : \Omega \times {\mathbb R}_+ \longrightarrow {\mathbb R}$ is a stochastic process with regulated trajectories, $t \in {\mathbb R}_+$ and $\Phi : {\mathbb R}_+ \longrightarrow {\mathbb R}_+$ is a strictly increasing function which maps $0$ to $0$ (cf. Proposition \[prop:sum of jumps on R+ with invertible function\]). Moreover, our approach enables a natural extension of the jump measure of càdlàg and adapted processes to an integer-valued random measure of optional processes with regulated trajectories which need not necessarily to be right- or left-continuous (cf. Theorem \[thm:optional random measures\]). In doing so, we provide a detailed and constructive proof of the fact that the set of all discontinuities of the first kind of a given real-valued function on ${\mathbb R}$ is at most countable (cf. Lemma \[lemma:right limits and left limits\], Theorem \[thm:at most countably many jumps on compact intervals\] and Theorem \[thm:at most countably many jumps on R+\]).
By using the powerful analysis of unordered sums, we hope that our contributions fill an existing gap in the literature, since neither a detailed proof of (the frequently used) Theorem \[thm:at most countably many jumps on compact intervals\] nor a precise definition of sums of jumps seems to be available yet.[^1] [^2]
---
[ ]{}
[<span style="font-variant:small-caps;">Frank Oertel</span>]{}\
*Department of Mathematics\
University College Cork*
Preliminaries and notations
===========================
In this section, we introduce the basic notation and terminology which we will throughout in this paper. To perpetuate the lucidity of the main ideas, we only consider ${\mathbb R}$-valued functions and ${\mathbb R}$-valued trajectories of stochastic processes, although a transfer to the (finite) multi-dimensional case is easily possible. Most of our notations and definitions including those ones originating from the general theory of stochastic processes and stochastic analysis are standard. We refer the reader to the monographs [@ChWi90], [@HeWaYa92], [@Ja79], [@JaSh03], [@Kl98], [@Me82] and [@Pr03]. Concerning a basic introduction to the the powerful theory of unordered sums, we recommend the monographs [@Di60] and [@PrMo91]. Since at most countable unions of pairwise disjoint sets play an important role in this paper, we use a symbolic abbreviation. For example, if $A : = \bigcup_{n=1}^{\infty} A_n$, where $(A_n)_{n \in {\mathbb N}}$ is a sequence of sets such that $A_i \cap A_j = \emptyset$ for all $i
\not= j$, we write shortly $A : = {\bigcup_{n=1}^{\infty}\hspace{-8.8mm\cdot}\hspace{7.5mm}}A_n$.
Throughout this paper, $(\Omega , \mathcal{F}, {\bf{F}}, {\Bbb{P}})$ denotes a fixed probability space, together with a fixed filtration ${\bf{F}}$. Even if it is not explicitly emphasized, the filtration ${\bf{F}} = (\mathcal{F}_t)_{t \geq 0}$ always is supposed to satisfy the usual conditions[^3]. A real-valued (stochastic) process $X : \Omega \times {\mathbb R}_+
\longrightarrow {\mathbb R}$ (which may be identified with the family of random variables $(X_t)_{t \geq 0}$, where $X_t(\omega) : =
X(\omega, t)$) is called *adapted* (with respect to ${\bf{F}}$) if $X_t$ is ${\mathcurl F}_t$-measurable for all $t \in {\mathbb R}_+$. $X$ is called *right-continuous* (respectively *left-continuous*) if for all $\omega \in \Omega$ the trajectory $X_{\bullet}(\omega) : {\mathbb R}_+ \longrightarrow {\mathbb R}, t \mapsto
X_t(\omega)$ is a right-continuous (respectively left-continuous) real-valued function. If all trajectories of $X$ do have left-hand limits (respectively right-hand limits) everywhere on ${\mathbb R}_+$, $X_{-}
= (X_{t -})_{t \geq 0}$ (respectively $X_{+} = (X_{t +})_{t \geq
0})$ denotes the *left-hand* (respectively *right-hand*) *limit process*, where $X_{0 -} : =
X_{0+}$ by convention. If all trajectories of $X$ do have left-hand limits and right-hand limits everywhere on ${\mathbb R}_+$, the *jump process* $\Delta X = (\Delta X_t)_{t \geq 0}$ is well-defined on $\Omega \times {\mathbb R}_+$. It is given by $\Delta X : = X_{+} - X_{-}$ (cf. also Section $2$). A right-continuous process whose trajectories do have left limits everywhere on ${\mathbb R}_+$, is known as a *càdlàg* process. If $X$ is $\mathcal{F} \otimes
\mathcal{B}({\Bbb{R}}_+)$-measurable, $X$ is said to be *measurable*. $X$ is said to be *progressively measurable* (or simply *progressive*) if for each $t \geq 0$, its restriction $X\vert_{\Omega \times [0, t]}$ is $\mathcal{F}_t
\otimes \mathcal{B}([0, t])$-measurable. Obviously, every progressive process is measurable and (thanks to Fubini) adapted.
A random variable $T : \Omega \longrightarrow [0, \infty]$ is said to be a *stopping time* or *optional time* (with respect to ${\bf{F}}$) if for each $t \geq 0$, $\{T \leq t\} \in
{\mathscr{F}}_t$. Let $\mathcal{T}$ denote the set of all stopping times, and let $S, T \in \mathcal{T}$ such that $S \leq T$. Then ${[\![}S, T {[\![}: = \{ (\omega , t) \in \Omega \times {\mathbb R}_+ :
S(\omega) \leq t < T(\omega)\}$ is an example for a *stochastic interval*. Similarly, one defines the stochastic intervals ${]\!]}S, T {]\!]}$, ${]\!]}S, T {[\![}$ and ${[\![}S, T {]\!]}$. Note again that ${[\![}T {]\!]}: = {[\![}T, T {]\!]}=
\textup{Gr}(T)\vert_{\Omega \times {\mathbb R}_+}$ is simply the graph of the stopping time $T : \Omega \longrightarrow [0, \infty]$ restricted to $\Omega \times {\mathbb R}_+$. $\mathcal{O} = \sigma\big\{[\![T,\infty [\![
\hspace{1mm}: T \in \mathcal{T}\big\}$ denotes the *optional $\sigma$-field* which is generated by all càdlàg adapted processes. The *predictable $\sigma$-field* $\mathcal{P}$ is generated by all left-continuous adapted processes. An $\mathcal{O}$- (respectively $\mathcal{P}$-) measurable process is called *optional* or *well-measurable* (respectively *predictable*). All optional or predictable processes are adapted. For the convenience of the reader, we recall and summarise the precise relation between those different types of processes in the following
\[thm:POPA\] Let $(\Omega , \mathcal{F}, {\bf{F}}, {\Bbb{P}})$ be a filtered probability space such that ${\bf{F}}$ satisfies the usual conditions. Let $X$ be a stochastic process on $\Omega \times
{\Bbb{R}}_+$. Consider the following statements:
- $X$ is predictable;
- $X$ is optional;
- $X$ is progressive;
- $X$ is adapted.
Then the following implications hold: $$\textstyle{(i)} \Rightarrow \textstyle{(ii)} \Rightarrow
\textstyle{(iii)} \Rightarrow \textstyle{(iv)}.$$ If $X$ is right-continuous, then the following implications hold: $$\textstyle{(i)} \Rightarrow \textstyle{(ii)} \iff \textstyle{(iii)}
\iff \textstyle{(iv)}.$$ If $X$ is left-continuous, then all statements are equivalent.
The general chain of implications $\textstyle{(i)} \Rightarrow
\textstyle{(ii)} \Rightarrow \textstyle{(iii)} \Rightarrow
\textstyle{(iv)}$ is well-known (for a detailed discussion cf. e.g. [@ChWi90], Chapter 3). If $X$ is left-continuous and adapted, then $X$ is predictable. Hence, in this case, all four statements are equivalent. If $X$ is right-continuous and adapted, then $X$ is optional (cf. [@ChWi90], Remark following Theorem 3.4. and [@HeWaYa92], Theorem 4.32). In particular, $X$ is progressive. [$\Box$]{}
By identifying processes that are almost everywhere identical, there is no difference between adapted *measurable* processes, optional processes, progressive processes and predictable processes (cf. [@Ni99]). In particular, since every adapted right-continuous process is optional, hence measurable, it is therefore almost everywhere identical to a predictable process.
Let $A \subseteq \Omega \times {\Bbb{R}}_{+}$ and $\omega \in
\Omega$. Consider $$D_A(\omega) : = \inf \{t \in {\Bbb{R}}_{+} : (\omega , t) \in A\}
\in [0, \infty]$$ $D_A$ is said to be the *début* of $A$. Recall that $\inf(\emptyset) = + \infty$ by convention. $A$ is called a *progressive set* if ${1\hspace{-2.5mm}1}_{A}$ is a progressively measurable process. For a better understanding of the main ideas in the proof of Theorem \[thm: jumps of optional processes and stopping times\], we need the following non-trivial result (a detailed proof of this statement can be found in e.g. [@Ba98] or [@HeWaYa92]):
\[thm:optional debut is stopping time\] Let $(\Omega , \mathcal{F}, {\bf{F}}, {\Bbb{P}})$ be a filtered probability space such that ${\bf{F}}$ satisfies the usual conditions. Let $A \subseteq \Omega \times {\Bbb{R}}_{+}$. If $A$ is a progressive set, then $D_A$ is a stopping time.
Discontinuities of the first kind
=================================
In the following, let us denote by $I$ an arbitrary (bounded or non-bounded) closed interval in ${\mathbb R}$, containing at least two elements. In other words, let $I$ be precisely one of the following sets: $$[a, b], [a, \infty), (-\infty, a], {\mathbb R},$$ where $a, b\in {\mathbb R}$, $a < b$. Let $f : I \longrightarrow {\Bbb{R}}$ be a real-valued function and $t \in I$ such that $(t, \infty) \cap
I \not= \emptyset$.[^4] Recall that the real value $f(t+)$ is the *right-hand limit of $f$* at $t$, if for every $\varepsilon > 0$ there exists a $\delta > 0$ such that $\vert f(t+) - f(s) \vert < \varepsilon $ whenever $s \in (t, t + \delta)$.[^5] Let $t
\in I$ such that $I \cap (-\infty, t) \not= \emptyset$. The real value $f(t-)$ is said to be the *left-hand limit of $f$* at $b$, if for every $\varepsilon
> 0$ there exists a $\delta > 0$ such that $\vert f(t-) - f(s)
\vert < \varepsilon $ whenever $s \in (t - \delta, t)$. Let us denote by $$L^+(f) : = \{t \in I : (t, \infty) \cap I \not= \emptyset \mbox{ and
} f(t+) \textup{ exists}\}$$ the set of all finite right-hand limits of $f$, and by $$L^-(f) : = \{t \in I : I \cap (-\infty, t) \not= \emptyset \mbox{
and } f(t-) \textup{ exists} \}$$ the set of all finite left-hand limits of $f$. Let $t \in L(f) : =
L^+(f) \cap L^-(f)$. Then $\Delta f(t) : = f(t+) - f(t-) \in {\mathbb R}$ denotes the *jump of $f$ at $t$*, leading to the well-defined function $\Delta f : L(f) \longrightarrow {\mathbb R}$, the associated *function of jumps of $f$*. Let $\textup{int}(I)$ denote the interior of $I$. An easy calculation shows that $$\textup{int}(I) = \{t \in I : (t, \infty) \cap I \not= \emptyset\}
\cap \{t \in I : I \cap (-\infty, t) \not= \emptyset\},$$ and it follows that $$\label{eq:L(f)}
L(f) = \textup{int}(I) \cap \{t \in I : f(t-) \textup{ exists and }
f(t+) \textup{ exists}\}$$ is a subset of $\textup{int}(I)$. The set $L(f)$ is known as the set of *discontinuities of the first kind* of $f$ or *jump points* of $f$. $I \setminus L(f)$ is called the set of *discontinuities of the second kind* of $f$ (cf. [@Kl98]).
Fix an arbitrary $\varepsilon>0$ and consider the set $J(f;
\varepsilon)$ of all jumps of $f$ of size at least $\varepsilon$, i.e., $$J(f; \varepsilon) := \{t \in L(f) : \vert \Delta f(t)\vert \geq
\varepsilon\}.$$ The set of *all* jumps of the function $f$ is then given by $$J(f) : = \{t \in L(f) : \Delta f(t) \not= 0\} = \{t \in L(f) : \vert
\Delta f(t) \vert > 0\} = \bigcup\limits_{n \in {\mathbb N}}J(f;
\frac{1}{n}).$$ Consider the function $\widetilde{f} : -I \longrightarrow
{\Bbb{R}}$, defined by $\widetilde{f}(s) : = f(-s)$. $\widetilde{f}$ simply describes the vertical reflection of $f$. Since the right-hand limit of $f$ (respectively the left-hand limit of $f$) is uniquely determined, vertical reflection of $f$ immediately implies the following important[^6]
\[prop:reflection\] Let $f : I \longrightarrow {\Bbb{R}}$ be a real-valued function. Let $\widetilde{f} : -I \longrightarrow {\Bbb{R}}$, defined by $\widetilde{f}(s) : = f(-s)$ for all $s \in -I$. Then
- $L^+(f) = -L^-(\widetilde{f})$, and $f(t+)= \widetilde{f}((-t)-)$ for all $t \in L^+(f)$;
- $L^-(f) = -L^+(\widetilde{f})$, and $f(t-)= \widetilde{f}((-t)+)$ for all $t \in L^-(f)$.
In particular, $L(f) = -L(\widetilde{f})$ and $$J(f; \varepsilon) = - J(\widetilde{f}; \varepsilon)$$ for all $\varepsilon >0$.
Clearly, there exists a direct link to the well-known and rich class of regulated functions (cf. [@Di60], 7.6. and [@Od04]). By using our notation, recall that $f : I \longrightarrow {\mathbb R}$ is said to be *regulated on $I$* if and only if if $\textup{int}(I)
\subseteq \{t \in I : f(t-) \textup{ exists and } f(t+) \textup{
exists}\}$, if the left endpoint of $I$ belongs to $L^+(f)$, and if the right endpoint of $I$ belongs to $L^-(f)$ (if the latter exist). Consequently, due to , we may state the following
\[remark: regulated fct\] Let $f : I \longrightarrow {\Bbb{R}}$ be a real-valued function. Then the following statements are equivalent:
- f is regulated on $I$;
- $L(f) = \textup{int}(I)$, the left endpoint of $I$ belongs to $L^+(f)$, and the right endpoint of $I$ belongs to $L^-(f)$ (if the latter exist).
Let $t \in {\mathbb R}$ and $(t_n)_{n\in {\mathbb N}}$ be an arbitrary sequence of elements in a given non-empty subset $A$ of ${\mathbb R}$. If $\lim\limits_{n
\to \infty}t_n = t$ and $t < t_{n+1} < t_n$ (respectively $t_n <
t_{n+1} < t$) for all $n \in {\mathbb N}$, as usual, we make use of the shorthand notation $t_n \downarrow t$ (respectively $t_n \uparrow
t$). Since compact intervals will play an important role later on, the next statement is given for $I : = [a, b]$ only, where $a < b$. However, as the proof clearly shows, our arguments are of local nature, so that we actually may choose every interval $I$ of the above type (including ${\mathbb R}_+$).
\[lemma:right limits and left limits\] Let $a<b$, $f : [a, b] \longrightarrow {\Bbb{R}}$ be an arbitrary real-valued function and $t \in [a, b]$.
- Let $(t_n)_{n \in {\mathbb N}} \subseteq L(f)$ such that $t_n \downarrow t$. If $t \in L^+(f)$, then $$\lim\limits_{n \to \infty} f(t_n-) = f(t+) = \lim\limits_{n \to
\infty} f(t_n+).$$
- Let $(t_n)_{n \in {\mathbb N}} \subseteq L(f)$ such that $t_n \uparrow t$. If $t \in L^-(f)$, then $$\lim\limits_{n \to \infty} f(t_n-) = f(t-) = \lim\limits_{n \to
\infty} f(t_n+).$$
In each of these cases, we have $$\lim\limits_{n \to \infty}\vert f(t_n+) - f({t_n-})\vert = 0.$$
To verify $(i)$, let $t \in L^+(f)$ and $(t_n)_{n \in {\mathbb N}} \subseteq
L(f)$ such that $t_n \downarrow t$ and $n, m \in {\mathbb N}$ arbitrary. Put $\tau_{mn} : = t_n - \xi_{mn}$, where $0 < \xi_{mn} : =
\frac{t_n-t_{n+1}}{2^m}$. Then $$t < t_{n+1} < \tau_{mn} < t_n
$$ for all $m, n \in {\mathbb N}$, and $\tau_{mn} \uparrow t_n$ (as $m \to
\infty$) for all $n \in {\mathbb N}$. Thus, using the definition of left-hand limits, we have $$\label{eqn:LHL} f({t_n}-) = \lim\limits_{m \to
\infty}f(\tau_{mn})
$$ for all $n \in {\mathbb N}$. Let $\varepsilon
> 0$. Since by assumption $t \in L^+(f)$, there exists a $\delta
> 0$ such that $$f\big((t, t + \delta)\big) \subseteq \big(f(t+) - \varepsilon, f(t+)
+ \varepsilon\big).$$ Since $t_n \downarrow t$, it follows that $\lim\limits_{n \to
\infty}f(t_n) = f(t+)$ and that there exists $N_{\delta} \in {\mathbb N}$ such that $t_n - t < \delta$ for all $n \geq N_{\delta}$. Consequently, $\tau_{mn} \in (t, t + \delta)$ for all $m \in {\mathbb N}$ and $n \geq N_{\delta}$, implying that $\vert f(t+) - f(\tau_{mn})\vert
< \varepsilon$ for all for all $m, n \geq N_{\delta}$. In other words, if $t \in L^+(f)$, then the double-sequence limit $\lim\limits_{m,n \to \infty}f(\tau_{mn}) = f(t+)$ exists! Thanks to a further epsilon-delta argument, we therefore obtain $$f(t+)
= \lim\limits_{n \to \infty}\big(\lim\limits_{m \to
\infty}f(\tau_{mn})\big) \stackrel{\eqref{eqn:LHL}}{=}
\lim\limits_{n \to \infty}f(t_n-).$$ Now we use the same method to approach each $t_n$ decreasingly from the right side. More precisely, let $m, n \in {\mathbb N}$ arbitrary and put $\rho_{mn} : = t_n + \xi_{m,n-1}$, where $\xi_{m,0} := 0$ and $0 < \xi_{mn} : = \frac{t_n-t_{n+1}}{2^m}$. Then $$t < t_n < \rho_{mn} < t_{n-1}
$$ for all $m \in {\mathbb N}, n \in {\mathbb N}\cap [2, \infty)$, and $\rho_{mn}
\downarrow t_n$ (as $m \to \infty$) for all $n \in {\mathbb N}$. Thus, using the definition of right-hand limits, we have $$\label{eqn:RHL} f({t_n}+) = \lim\limits_{m \to
\infty}f(\rho_{mn})
$$ for all $n \in {\mathbb N}$. Again, since $t \in L^+(f)$, we obtain the existence of a double-sequence limit, namely $\lim\limits_{m,n \to
\infty}f(\rho_{mn}) = f(t+)$. Hence, $$f(t+)
= \lim\limits_{n \to \infty}\big(\lim\limits_{m \to
\infty}f(\rho_{mn})\big) \stackrel{\eqref{eqn:RHL}}{=}
\lim\limits_{n \to \infty}f(t_n+).$$ To complete the proof, we only have to consider the remaining case $(ii)$. So, let $t \in L^-(f)$ and $(t_n)_{n \in {\mathbb N}} \subseteq L(f)$ such that $t_n \uparrow t$. Then $s_n \downarrow s$, where $s_n : =
-t_n$ and $s : = -t$. Consider the function $\widetilde{f} : [-b,
-a] \longrightarrow {\Bbb{R}}$, defined by $\widetilde{f}(s) : =
f(-s)$. Due to Proposition \[prop:reflection\], it follows that $s_n = -t_n \in -L(f) = L(\widetilde{f})$ for all $n \in {\mathbb N}$ and $s
= -t \in -L^-(f) = L^+(\widetilde{f})$. Therefore, we precisely obtain the situation of part $(i)$, but now related to the function $\widetilde{f}$! Consequently, $$\lim\limits_{n \to \infty} \widetilde{f}(s_n-) = \widetilde{f}(s+) =
\lim\limits_{n \to \infty} \widetilde{f}(s_n+),$$ and the claim follows by Proposition \[prop:reflection\]. [$\Box$]{}
If $f : {\mathbb R}_+ \longrightarrow {\mathbb R}$ is a regulated function, it follows that $L^+(f) = {\mathbb R}_+$ and $L^{-}(f) = (0, \infty)$. Hence, we may define $f_{+}(t) : = f(t+) \in {\mathbb R}$ for all $t \in {\mathbb R}_+$ and $f_{-}(t) : = f(t-) \in {\mathbb R}$ for all $t \in (0, \infty)$, implying the existence of well-defined functions $f_{+} : {\mathbb R}_+
\longrightarrow {\mathbb R}$ and $f_{-} : (0, \infty) \longrightarrow {\mathbb R}$. A first immediate non-trivial implication of Lemma \[lemma:right limits and left limits\] is the following statement which will be used in the proof of Lemma \[lemma:optional and regulated paths\].
\[cor:f+ and f- for regulated f\] Let $f : {\mathbb R}_+ \longrightarrow {\mathbb R}$ be a regulated function. Then $f_{+}$ is right-continuous on ${\mathbb R}_+$ and $f_{-}$ is left-continuous on $(0, \infty)$.
\[thm:at most countably many jumps on compact intervals\] Let $f : [a, b] \longrightarrow {\Bbb{R}}$ be an arbitrary real-valued function, where $a<b$. Then
- $J(f; \varepsilon)$ is finite for all $\varepsilon >0$.
- $J(f)$ is at most countable.
Since $J(f) = \bigcup\limits_{n \in {\mathbb N}}J(f; \frac{1}{n})$, we only have to prove $(i)$. Assume by contradiction that $J(f;
\varepsilon)$ is not finite. Due to the Bolzano-Weierstrass Theorem the bounded and infinite set $J(f; \varepsilon)$ has at least one accumulation point $t \in [a, b]$ (cf. e.g. [@Bu98]). Then there exists a sequence $(t_n)_{n\in{\mathbb N}} \subseteq J(f; \varepsilon)$ such that $t_n \to t$ (as $n \to \infty$), $t_k \not= t_l$ for all $k \not= l$, and $t_n \not= t$ for all $n \in {\mathbb N}$ (since $J(f;
\varepsilon)$ is not finite). We therefore can select a monotone subsequence of $(t_n)_{n\in{\mathbb N}}$ which then also converges to $t$. To avoid some cumbersome notation, WLOG, we may assume that the original sequence $(t_n)_{n\in{\mathbb N}}$ is already the monotone one. Consequently, we arrived exactly at either scenario $(i)$ or scenario $(ii)$ of Lemma \[lemma:right limits and left limits\]. Since $t_n \in J(f; \varepsilon)$ for all $n \in {\mathbb N}$, we clearly obtain a contradiction, and the claim follows. [$\Box$]{}
The next result shows that at most countability of the jumps even can be guaranteed for all real-valued functions which are defined on the whole of ${\mathbb R}_+$ (respectively ${\mathbb R}$).
\[thm:at most countably many jumps on R+\] Let $f : J \longrightarrow {\Bbb{R}}$ be an arbitrary real-valued function, where $J\ \in \{{\mathbb R}_+, {\mathbb R}\}$. Then
- $J(f; \varepsilon)$ is at most countable for all $\varepsilon >0$.
- There exists a partition $\{D_k : k \in {\mathbb N}\}$ of $J(f)$ such that each $D_k$ is a finite subset of $J(f)$. In particular, $J(f)$ is at most countable.
First, consider the case $J = {\mathbb R}_+$. Let $M : = \{t \in {\mathbb R}_+ : f(t-)
\textup{ exists and } f(t+) \textup{ exists}\}$. Since $\textup{int}({\mathbb R}_+) = (0, \infty) = {\bigcup_{n=1}^{\infty}\hspace{-9mm\cdot}\hspace{7mm}}(n-1, n]$, representation therefore implies that $$L(f) = {\bigcup_{n=1}^{\infty}\hspace{-4.2mm\cdot}\hspace{2.2mm}} \big((n-1, n] \cap M \big) =
{\bigcup_{n=1}^{\infty}\hspace{-4.2mm\cdot}\hspace{2.2mm}} \big( (n-1, n) \cap M \big) \cup ({\mathbb N}\cap M)
\stackrel{\eqref{eq:L(f)}}{=} {\bigcup_{n=1}^{\infty}\hspace{-4.2mm\cdot}\hspace{2.2mm}}L(f\vert_{[n-1, n]})
\cup ({\mathbb N}\cap M).$$ Hence, $$\label{eq:jump size rep for functions on R+ II}
J(f) = {\bigcup_{n=1}^{\infty}\hspace{-4.2mm\cdot}\hspace{2.2mm}}J(f\vert_{[n-1, n]}) \cup ({\mathbb N}\cap J(f))$$ and $$\label{eq:jump size rep for functions on R+} J(f;
\varepsilon) = {\bigcup_{n=1}^{\infty}\hspace{-4.2mm\cdot}\hspace{2.2mm}}J(f\vert_{[n-1, n]}; \varepsilon)
\cup ({\mathbb N}\cap J(f; \varepsilon))$$ for all $\varepsilon > 0$. Thus, (i) follows by Theorem \[thm:at most countably many jumps on compact intervals\]. To prove (ii), fix $n \in {\mathbb N}$ and consider $f_n : = f\vert_{[n-1, n]}$. Due to Theorem \[thm:at most countably many jumps on compact intervals\], the set $J(f_n; \frac{1}{m})$ is finite for each $m \in {\mathbb N}$. Since $J(f_n;
\frac{1}{m}) \subseteq J(f_n; \frac{1}{m+1})$ for all $m \in {\mathbb N}$, it therefore follows that $J(f_n)$ can be written as an at most countable union of disjoint *finite* sets, namely as $$\label{eq:standard jump partition}
J(f_n) = \bigcup_{m = 1}^{\infty} J(f_n; \frac{1}{m}) = {\bigcup_{m =
1}^{\infty}\hspace{-4.5mm\cdot}\hspace{2.2mm}} A_{m,n},$$ where $A_{1, n} : = J(f_n; 1) = (\Delta f_n)^{-1}\big([1,
\infty)\big)$ and $A_{m+1, n} := J(f_n; \frac{1}{m+1}) \setminus
J(f_n; \frac{1}{m}) = (\Delta f_n)^{-1}\big([\frac{1}{m+1},
\frac{1}{m})\big)$ for all $m \in {\mathbb N}$. Hence, implies that $$J(f) = {\bigcup_{n=1}^{\infty}\hspace{-4.2mm\cdot}\hspace{2.2mm}}{\bigcup_{m = 1}^{\infty}\hspace{-4.5mm\cdot}\hspace{2.2mm}} A_{m,n} \cup
({\mathbb N}\cap J(f)) = {\bigcup_{k=1}^{\infty}\hspace{-4.2mm\cdot}\hspace{2.2mm}} B_{k} \cup ({\mathbb N}\cap J(f)),$$ where $\{B_k : k \in {\mathbb N}\} = \{A_{m,n} : (n,m) \in {\mathbb N}\times {\mathbb N}\}$. Consequently, $$J(f) = {\bigcup_{l=1}^{\infty}\hspace{-3.8mm\cdot}\hspace{2.2mm}} D_l,$$ where $D_l : = B_l \cup (\{l\} \cap J(f))$ is a finite set for all $l \in {\mathbb N}$. [$\Box$]{}
Since partitions of this type will play a fundamental role, we introduce the following
Let $D$ be an at most countable subset of ${\mathbb R}$ which is not empty. A partition $\{D_k : k \in {\mathbb N}\}$ of $D$ is called a finitely layered partition of $D$ if $D = {\bigcup_{k = 1}^{\infty}\hspace{-8.8mm\cdot}\hspace{6.5mm}} D_k$, where $D_k$ is a finite subset of $D$ for all $k \in {\mathbb N}$.
Unordered sums
==============
Using the previous results about the structure of the sets $L(f)$ and $J(f)$, we can introduce sum of jumps functions like e.g. $\mathscr{P}(L(f)) \ni B \mapsto \sum_{s \in B} (\Delta f(s))^2 \in
[0, \infty]$ in a mathematically concise manner. Our aim is to provide an exact description of such sums which is independent of the choice of the partition of $J(f)$ (cf. Theorem \[thm:unordered sum as measure\] and Proposition \[prop:sum of jumps on R+ with invertible function\]). In particular, we will show that finitely layered jump partitions provide a natural frame for integer valued random measures which are a *special case* of such a (randomised) sum (cf. Theorem \[thm:optional random measures\]).
To this end, let $L$ be an arbitrary non-empty set and $h : L
\longrightarrow {\mathbb R}_+$ a *positive* real-valued function. Consider the set $\Bbb{F}(L) : = \{F : F \mbox { is a finite subset
of } L\}$. Clearly, $(\Bbb{F}(L), \subseteq \nolinebreak)$ is an ordered set, and we may therefore consider the well-defined net $s_h
: \Bbb{F}(L) \longrightarrow {\mathbb R}_+$, defined by $s_h(F) : = \sum_{s
\in F} h(s)$, where $F \in \Bbb{F}(L)$. If the net $s_h$ converges to a limit point $p \in {\mathbb R}_+$, $\sum_{s \in L} h(s) : = p$ is called the *unordered sum over $L$*. If the net $s_h$ converges, $s_h$ is called *summable*. Let us recall the following
\[thm:summability\] Let $L$ be an arbitrary non-empty set and $h : L \longrightarrow
{\mathbb R}_+$ a positive real-valued function. Then the following statement s are equivalent:
- $\sum_{s \in L} h(s)$ exists;
- The set $\{s_h(F) : F \in \Bbb{F}(L)\}$ is bounded in ${\mathbb R}_+$.
If the net $s_h$ converges, then $\sum_{s \in L} h(s) = \sup\{s_h(F)
: F \in \Bbb{F}(L) \}$.
Since we have to include the case that the net $s_h$ is not convergent, Theorem \[thm:summability\] justifies the following natural extension of the unordered sum above:
Let $L$ be an arbitrary non-empty set and $h : L \longrightarrow
{\mathbb R}_+$ an arbitrary positive real-valued function. Define $$\sum_{s \in L} h(s) : = \sup\Big\{\sum_{s \in F} h(s) : F \in
\Bbb{F}(L) \Big\} $$ If $\emptyset \not= A \subseteq L$, put $\sum_{s \in A} h(s) : =
\sum_{s \in A} h\vert_A(s)$. Put $\sum_{s \in \emptyset} h(s) : =
0$.
First note that in general, $\sum_{s \in L} h(s) \in [0, \infty]$ and that $\sum_{s \in E} h(s) \leq \sum_{s \in F} h(s)$ for all subsets $E \subseteq F \subseteq L$. If $L = \{s_1, \ldots, s_n\}$ itself is a finite set, then obviously $\sum_{s \in L} h(s) =
\sum_{i=1}^{n}h(s_i) = \sum_{i=1}^{n}h(s_{\sigma(i)})$ for all permutations $\sigma \in S_n$, which justifies the notation. However, the following important fact, which we will use later on, requires a proof.
\[lemma:indicator function in sum\] Let $L$ be an arbitrary non-empty set and $h : L \longrightarrow
{\mathbb R}_+$ an arbitrary positive real-valued function. Let $A$ and $B$ be arbitrary subsets of $L$. Then the following statements hold:
- $\sum_{s \in L} h(s) {1\hspace{-2.5mm}1}_A(s) < +\infty$ if and only if $\sum_{s
\in A} h(s) < +\infty$. Moreover, $$\sum_{s \in A} h(s) = \sum_{s \in L} h(s) {1\hspace{-2.5mm}1}_A(s).$$
- $\sum_{s \in A} h(s) {1\hspace{-2.5mm}1}_B(s) < +\infty$ if and only if $\sum_{s
\in A \cap B} h(s) < +\infty$. Moreover, $$\sum_{s \in A \cap B} h(s) = \sum_{s \in A} h(s){1\hspace{-2.5mm}1}_{B}(s).$$
These sums may be finite or infinite.
Since (ii) obviously follows by (i) (by applying (i) to the function $h {1\hspace{-2.5mm}1}_B$), we only have to prove (i). If $A = \emptyset$, nothing is to prove. So, let $A \not= \emptyset$. Assume first that $\sum_{s
\in A} h(s) < +\infty$. Let $F$ be an arbitrary finite subset of $L$. Since $F$ equals the disjoint union of the (finite) sets $A
\cap F$ and $(L\setminus A) \cap F$, standard associative and commutative summation of finitely many numbers immediately gives $$\sum_{s \in F} h(s) {1\hspace{-2.5mm}1}_A(s) = \sum_{s \in A \cap F} h(s) {1\hspace{-2.5mm}1}_A(s)
= \sum_{s \in A \cap F} h(s).
$$ Since the finite subset $F$ of $L$ was arbitrarily chosen, it therefore follows that $$\sum_{s \in L} h(s) {1\hspace{-2.5mm}1}_A(s) \leq \sum_{s \in A} h(s) < +\infty.$$ Now let $\sum_{s \in L} h(s) {1\hspace{-2.5mm}1}_A(s)$ be finite. Then, if $G$ is an arbitrary finite subset of $A \subseteq L$, we obviously have $$\sum_{s \in G} h(s) = \sum_{s \in G} h(s) {1\hspace{-2.5mm}1}_A(s) \leq \sum_{s \in
L} h(s) {1\hspace{-2.5mm}1}_A(s) < +\infty,$$ which proves the other inequality. Consequently, we have shown that the equality holds if $\sum_{s \in A} h(s)$ is finite or if $\sum_{s
\in L} h(s) {1\hspace{-2.5mm}1}_A(s)$ is finite. Hence, it must be true if these sums are finite or if this is not the case. [$\Box$]{}
\[prop:linearity of unordered sums\] Let $L$ be an arbitrary non-empty set, $\alpha, \beta \geq 0$ and $h, g : L \longrightarrow {\mathbb R}_+$ arbitrary positive real-valued functions. Then $\sum_{s \in L} g(s) < +\infty$ and $\sum_{s \in L}
h(s) < +\infty$ if and only if $\sum_{s \in L} (\alpha g(s) + \beta
h(s)) < +\infty$. Moreover, $$\sum_{s \in L} (\alpha g(s) + \beta h(s)) = \alpha \sum_{s \in L}
g(s) + \beta \sum_{s \in L} h(s).$$ These sums may be finite or infinite.
First, let $\sum_{s \in L} g(s) < +\infty$ and $\sum_{s \in L} h(s)
< +\infty$. Since addition is associative and commutative, the equality obviously is true for every finite subset $F$ of $L$. Consequently, we already obtain the inequality $$\sum_{s \in L} (\alpha g(s) + \beta h(s)) \leq \alpha \sum_{s \in L}
g(s) + \beta \sum_{s \in L} h(s) < +\infty.$$ Now let $\sum_{s \in L} (\alpha g(s) + \beta h(s)) < +\infty$. Let $E$ be an arbitrary finite subset of $L$. Then, $$\max\Bigg\{\alpha \sum_{s \in E} g(s), \beta \sum_{s \in E}
g(s)\Bigg\} \leq \sum_{s \in E} (\alpha g(s) + \beta h(s)) \leq
\sum_{s \in L} (\alpha g(s) + \beta h(s)) < +\infty,$$ and it follows that both, $\Gamma : = \sum_{s \in L} g(s)$ and $\Delta : = \sum_{s \in L} h(s)$ are finite. To prove the other inequality, let $\varepsilon > 0$ be given. Then there exist finite subsets $F$ and $G$ of $L$ such that $$\alpha \Gamma + \beta \Delta
< \sum_{s \in F} \alpha g(s) + \sum_{s \in G} \beta h(s) +
\varepsilon.$$ Since we currently are working with summation of finitely many elements only, we obviously may conclude that $$\alpha \Gamma + \beta \Delta - \varepsilon < \sum_{s \in F \cup G}
\alpha g(s) + \sum_{s \in F \cup G} \beta h(s) = \sum_{s \in F \cup
G} (\alpha g(s) + \beta h(s)).$$ Since $F \cup G$ is a finite subset of $L$, we have arrived at the other inequality. Hence, similarly as in the proof of Lemma \[lemma:indicator function in sum\], we have shown that the equality holds if $\sum_{s \in L} (\alpha g(s) + \beta h(s)) <
+\infty$ or if $\sum_{s \in L} g(s) < +\infty$ and $\sum_{s \in L}
h(s) < +\infty$. [$\Box$]{}
\[cor:disjoint sums\] Let $L$ be an arbitrary non-empty set and $h : L \longrightarrow
{\mathbb R}_+$ an arbitrary positive real-valued function. Let $C$ and $D$ be arbitrary subsets of $L$. If $C \cap D = \emptyset$, then $$\sum_{s \in C \cup D} h(s) = \sum_{s \in C} h(s) + \sum_{s \in D}
h(s).$$ These sums may be finite or infinite.
Since $C \cap D = \emptyset$, we have ${1\hspace{-2.5mm}1}_{C \cup D} = {1\hspace{-2.5mm}1}_{C} +
{1\hspace{-2.5mm}1}_{D}$. Hence, by Lemma \[lemma:indicator function in sum\] and Proposition \[prop:linearity of unordered sums\], it therefore follows that $$\sum_{s \in C \cup D} h(s) = \sum_{s \in L} h(s){1\hspace{-2.5mm}1}_{C \cup D}(s) =
\sum_{s \in L} (h(s){1\hspace{-2.5mm}1}_{C}(s) + h(s){1\hspace{-2.5mm}1}_{D}(s)) = \sum_{s \in C}
h(s) + \sum_{s \in D} h(s).$$ [$\Box$]{}
\[thm:Fubini for unordered sums\] Let $L$ be an arbitrary non-empty set and $h : L \longrightarrow
{\mathbb R}_+$ a real-valued function. Let $\{D_n : n\in {\mathbb N}\}$ be an arbitrary partition of a set $D \subseteq L$. Then the following statements are equivalent:
- $\sum_{s \in D} h(s) < + \infty$;
- $\sum_{s \in D_n} h(s) < +\infty$ for all $n \in {\mathbb N}$ and $\Big( \sum_{s \in D_n} h(s)\Big)_{n \in {\mathbb N}} \in l_1$.
Moreover, $$\sum_{s \in D} h(s) = \sum_{n = 1}^{\infty}\sum_{s \in D_n} h(s).$$ These sums may be finite or infinite.
Nothing is to show if $D = \emptyset$. So, let $D \not= \emptyset$, and assume first that (i) holds. Since $D_n \subseteq D$ for all $n
\in {\mathbb N}$, each finite subset of each $D_n$ is already a finite subset of $D$, implying that $\sum_{s \in D_n} h(s) \leq \sum_{s \in D}
h(s) < +\infty$ for all $n \in {\mathbb N}$. Let $n \in {\mathbb N}$ arbitrary and consider the set $C_n : = {\bigcup_{k = 1}^{n}\hspace{-9mm\cdot}\hspace{7mm}} D_k \subseteq D$. Due to Corollary \[cor:disjoint sums\], we have $$0 \leq \sum_{k = 1}^{n}\sum_{s \in D_k} h(s) = \sum_{s \in C_n} h(s)
\leq \sum_{s \in D} h(s) < +\infty.$$ Since $n \in {\mathbb N}$ was arbitrarily chosen, we may conclude that $$0 \leq \sum_{k = 1}^{\infty}\sum_{s \in D_k} h(s) \leq \sum_{s \in
D} h(s) < +\infty.$$ Hence, $\Big( \sum_{s \in D_k} h(s)\Big)_{k \in {\mathbb N}} \in l_1$, and statement (ii) follows. Now assume that (ii) holds. Then $0 \leq
\sum_{s \in D_n} h(s) < +\infty$ for all $n \in {\mathbb N}$ and $0 \leq
\sum_{n = 1}^{\infty}\sum_{s \in D_n} h(s) < + \infty$. Let F be an arbitrary finite subset of $D$. Choose a sufficiently large number $n \in {\mathbb N}$ such that $F \subseteq {\bigcup_{k = 1}^{n}\hspace{-9mm\cdot}\hspace{7mm}}D_k$, implying that $F = {\bigcup_{k = 1}^{n}\hspace{-9mm\cdot}\hspace{7mm}}F_k$, where $F_k : = F \cap D_k$ for all $k \in
\{1, 2, \cdots , n\}$. Consequently, we have $$\sum_{s \in F} h(s) = \sum_{k = 1}^{n}\sum_{s \in F_k} h(s).$$ Since each $F_k$ is a finite subset of $D_k$, assumption (ii) further implies that $$\sum_{k = 1}^{n}\sum_{s \in F_k} h(s) \leq \sum_{k = 1}^{n}\sum_{s
\in D_k} h(s) \leq \sum_{k = 1}^{\infty}\sum_{s \in D_k} h(s) <
+\infty.$$ Since the finite subset $F$ of $D$ was arbitrarily chosen, it follows that statement (i) is true, and we have $$\sum_{s \in D} h(s) \leq \sum_{n = 1}^{\infty}\sum_{s \in D_n} h(s)
< +\infty.$$ Clearly, we have shown that the equality holds if the case (i) or the case (ii) is given. Since (i) is equivalent to (ii), the equality necessarily also must hold if one of the both unordered sums is not finite. [$\Box$]{}
Since ${\mathbb N}\times {\mathbb N}= {\bigcup_{m=1}^{\infty}\hspace{-9.8mm\cdot}\hspace{7.5mm}}{\bigcup_{n=1}^{\infty}\hspace{-9mm\cdot}\hspace{7mm}}
\{(m, n)\} = {\bigcup_{n=1}^{\infty}\hspace{-9mm\cdot}\hspace{7mm}}{\bigcup_{m=1}^{\infty}\hspace{-9.8mm\cdot}\hspace{7.8mm}} \{(m,
n)\}$, Theorem \[thm:Fubini for unordered sums\] immediately recovers a well-known result concerning the rearrangement of the terms in a double series (cf. e.g. [@Bu98]):
\[cor:double series\] Let $(a_{mn})_{(m, n) \in {\mathbb N}\times {\mathbb N}}$ be an arbitrary double-sequence in ${\mathbb R}_+$. Then $$\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} a_{mn} = \sum_{n \in {\mathbb N}\times {\mathbb N}} a_{mn} = \sum_{n=1}^{\infty} \sum_{n=1}^{\infty} a_{mn}$$
By using the language of measure theory, we have proven the following important result:
\[thm:unordered sum as measure\] Let $L$ be an arbitrary non-empty set and $h : L \longrightarrow
{\mathbb R}_+$ an arbitrary positive real-valued function. Then $$\begin{aligned}
\mu_h : \mathscr{P}(L) & \longrightarrow & [0,
\infty]\\
A & \mapsto & \sum_{s \in A} h(s),\end{aligned}$$ is a well-defined measure on the measurable space $(L,
\mathscr{P}(L))$.
\[measure theoretic version of indicator fct in sum\] Let $A$ and $B$ be arbitrary subsets of $L$. Then Lemma \[lemma:indicator function in sum\] implies that $$\label{eq:mu of A versus mu of L}
\mu_h(A) = \sum_{s \in L} h(s){1\hspace{-2.5mm}1}_A(s) =
\mu_{h{1\hspace{-2mm}1}_A}(L)$$ and $$\label{eq:mu of B cap D}
\mu_h(A \cap B) = \sum_{s \in A} h(s){1\hspace{-2.5mm}1}_B(s) =
\mu_{h{1\hspace{-2mm}1}_B}(A).$$
Dependent on the choice of the function $h$, we recognise two special and well-known cases:
- If $h(s) : = 1 = {1\hspace{-2.5mm}1}_L(s)$ for all $s \in L$, then $\mu_{{1\hspace{-2mm}1}_{L}}$ is precisely the [*counting measure*]{}.
- If $h : = {1\hspace{-2.5mm}1}_{\{s_0\}}$, where $s_0 \in L$, we obtain exactly the [*Dirac measure*]{} at $s_0$, since $$\mu_{{1\hspace{-2mm}1}_{\{s_0\}}}(A) \stackrel{\eqref{eq:mu of B cap D}}{=}
\mu_{{1\hspace{-2mm}1}_{L}}(\{s_0\} \cap A) \stackrel{\eqref{eq:mu of B cap
D}}{=} \mu_{{1\hspace{-2mm}1}_{A}}(\{s_0\}) = {1\hspace{-2.5mm}1}_A(s_0) = \delta_{s_0}(A)
$$ for all $A \in \mathscr{P}(L)$.
\[cor:countable sums\] Let $L$ be an arbitrary non-empty set and $h : L \longrightarrow
{\mathbb R}_+$ an arbitrary positive real-valued function.
- If $A \in \mathscr{P}(L)$ is finite, then $$\mu_h(A) = \sum_{\nu = 1}^{n}h(\nu),$$ where $n = card(A)$.
- If $A \in \mathscr{P}(L)$ is countable, then $$\mu_h(A) = \sum_{n = 1}^{\infty}h(\varphi(n))$$ for all bijective mappings $\varphi : {\mathbb N}\longrightarrow A$.
First note that $\mu_h(\{a\}) = h(a)$ for all $a \in A \subseteq L$. Statement (i) now follows directly by Theorem \[thm:unordered sum as measure\]. To prove (ii), let $a_n : = \varphi(n)$, where $n \in
{\mathbb N}$ is arbitrary. Then, by Theorem \[thm:unordered sum as measure\] again, we have $$\mu_h(A) = \mu_h({\bigcup_{n=1}^{\infty}\hspace{-4.2mm\cdot}\hspace{2.2mm}} \{a_n\}) = \sum_{n=1}^{\infty}
\mu_h(\{a_n\}) = \sum_{n=1}^{\infty} h(a_n),$$ and the proof is finished. [$\Box$]{}
We have developed all necessary tools which now allow us to give a lucid and short proof of the following non-trivial result.
\[thm:countably many positive values\] Let $L$ be an arbitrary non-empty set and $h : L \longrightarrow
{\mathbb R}_+$ an arbitrary positive real-valued function. Put $P : = \{s \in
L : h(s) > 0\}$. If $\sum_{s \in L}h(s) < +\infty$, then $P$ is at most countable, and $$\sum_{s \in L}h(s) = \sum_{s \in P}h(s) = \sum_{n =
1}^{\infty}h(\varphi(n))$$ for all bijective mappings $\varphi : {\mathbb N}\longrightarrow P$.
By assumption, $\Sigma : = \sum_{s \in L}h(s) < +\infty$. Since $P =
\bigcup_{n = 1}^{\infty}P_n$, where $P_n : = \{s \in L : h(s) >
\frac{1}{n}\}$, we only have to show that each subset $P_n$ of $L$ is at most countable. We even show more and claim that $$\label{eq:P_n is finite}
P_n \mbox{ is finite and consists of at most } \lfloor n\Sigma
\rfloor \mbox{ elements for all } n \in {\mathbb N},$$ where ${\mathbb R}\ni x \mapsto \lfloor x \rfloor : = \max\{m \in {\mathbb Z}: m
\leq x\}$ describes the assignment rule of the floor function. We assume by contradiction that is false. Then there would exist $m \in {\mathbb N}$ and a *finite* subset $G_m$ of $P_m$ such that $\mbox{card}({G_m}) = \lfloor m\Sigma \rfloor + 1$. But then, due to the definition of the floor function, we would have $$+\infty > \lfloor m\Sigma \rfloor + 1 > m\Sigma = \sum_{s \in L}m
h(s) \geq \sum_{s \in G_m}m h(s) > \mbox{card}(G_m)\cdot 1 = \lfloor
m\Sigma \rfloor + 1,$$ which obviously is a contradiction. Hence, statement is true, implying that the set $P$ is at most countable.
Clearly, we have $\sum_{s \in L\setminus P}h(s) = 0$. Hence, $\sum_{s \in L}h(s) = \sum_{s \in P}h(s) = \mu_h(P)$ (due to Corollary \[cor:disjoint sums\]), and Corollary \[cor:countable sums\] finishes the proof. [$\Box$]{}
By linking Lemma \[lemma:indicator function in sum\] and (the proof of) Theorem \[thm:Fubini for unordered sums\], we can characterise the finiteness of the measure $\mu_h$ in the following way:
\[prop:when is mu\_h finite?\] Let $L$ be an arbitrary non-empty set and $h : L \longrightarrow
{\mathbb R}_+$ an arbitrary positive real-valued function. Then the following statements are equivalent:
- $\mu_h : \mathscr{P}(L) \longrightarrow {\mathbb R}_+$ is a finite measure.
- If $\{L_n : n \in {\mathbb N}\}$ is an arbitrary partition of $L$ such that $\sum_{s \in L_n} h(s) < +\infty$ for all $n \in {\mathbb N}$ and $\Big(\sum_{s \in L_n} h(s)\Big)_{n \in {\mathbb N}} \in l_1$, then $$\mu_h(A) = \sum_{n = 1}^{\infty}\sum_{s \in L_n} h(s){1\hspace{-2.5mm}1}_{A}(s)$$ for all $A \in \mathscr{P}(L)$.
- There exists a partition $\{C_l : l \in {\mathbb N}\}$ of $L$ such that $\sum_{s \in C_l} h(s) < +\infty$ for all $l \in {\mathbb N}$, $\Big(\sum_{s \in C_l} h(s)\Big)_{l \in {\mathbb N}} \in l_1$ and $$\mu_h(A) = \sum_{l = 1}^{\infty}\sum_{s \in C_l} h(s){1\hspace{-2.5mm}1}_{A}(s)$$ for all $A \in \mathscr{P}(L)$.
We have arrived at a point now, where we can apply our general framework to discontinuities of the first kind. In particular, we can easily provide a representation of unordered sums over all jumps of $f$; a fact, which frequently is used in the literature on general semimartingales including references on Lévy processes, but which seemingly hasn’t been *rigorously* proven yet, very similar to the case of the proof of Theorem \[thm:at most countably many jumps on compact intervals\] (cf. e.g. [@Ap04], [@Kl98], [@Pa67]).
\[prop:sum of jumps on R+ with invertible function\] Let $f : {\mathbb R}_+ \longrightarrow {\mathbb R}$ be an arbitrary function, and assume that $\emptyset \not= J(f)$. Let $\{D_n : k \in {\mathbb N}\}$ be an arbitrary partition of $J(f)$. Let $\Phi : {\mathbb R}_+ \longrightarrow
{\mathbb R}_+$ be strictly increasing and continuous such that $\Phi(0) = 0$. Let $B$ be a non-empty subset of $L(f)$. Then $$\sum_{s \in B} \Phi\big(\vert \Delta f(s) \vert\big) = \mu_{\Phi
\circ \vert \Delta f \vert}\big(B \cap J(f) \big) = \sum_{n =
1}^{\infty} \sum_{s \in D_n} \Phi\big(\vert \Delta f(s) \vert\big)
{1\hspace{-2.5mm}1}_{B}(s) = \sum_{n = 1}^{\infty} \Phi\big(\vert \Delta
f(\varphi(n)) \vert\big)$$ for all bijective mappings $\varphi: {\mathbb N}\longrightarrow B \cap
J(f)$. If in addition $f$ is regulated, then $$\sum_{0 < s \leq t} \Phi\big(\vert \Delta f(s) \vert\big) =
\mu_{\Phi \circ \vert \Delta f \vert}\big((0, t] \cap J(f)\big) =
\sum_{n = 1}^{\infty} \sum_{s \in D_n} \Phi\big(\vert \Delta f(s)
\vert\big) {1\hspace{-2.5mm}1}_{(a, t)}(s) = \sum_{n = 1}^{\infty} \Phi\big(\vert
\Delta f(\varphi(n)) \vert\big)$$ for all $t \in (0, \infty)$ and bijective mappings $\varphi: {\mathbb N}\longrightarrow (0, t] \cap J(f)$.
Let $B$ be an arbitrary non-empty subset of $L(f)$. Due to Theorem \[thm:at most countably many jumps on R+\], $D : = J(f)$ is at most countable. Since $\Phi : {\mathbb R}_+ \longrightarrow {\mathbb R}_+$ is strictly increasing and continuous, it is invertible, and $\Phi^{-1} : {\mathbb R}_+
\longrightarrow {\mathbb R}_+$ is strictly increasing as well (due to the Inverse Function Theorem). Hence, the set $\{s \in B: \Phi(\vert
\Delta f(s) \vert) > 0 \} = \{s \in B: \vert \Delta f(s) \vert > 0
\} = B \cap D$ is an at most countable subset of $D \subseteq L(f)$. Consider the the function $h : = \Phi \circ \vert \Delta f \vert$. Then $\mu_h(B \cap D) \stackrel{\eqref{eq:mu of B cap D}}{=}
\mu_{h{1\hspace{-2mm}1}_B}(D).$ Since $\Phi(0) = 0$, equality implies that $\mu_h(B \cap (L\setminus D)) = \mu_{h{1\hspace{-2mm}1}_B}(L\setminus D) =
0$, and it follows that $$\sum_{s \in B}h(s) = \mu_{h}(B) = \mu_{h}(B \cap D) =
\mu_{h{1\hspace{-2mm}1}_B}(D) = \sum_{n = 1}^{\infty}\sum_{s \in D_n} h(s)
{1\hspace{-2.5mm}1}_{B}(s).$$ Since the set $B \cap D$ is an at most countable subset of $L(f)$, the first statement follows by Corollary \[cor:countable sums\]. If in addition $f$ is regulated, then $L(f) = (0, \infty)$ (due to Remark \[remark: regulated fct\]), implying that $B : = (0, t]
\subseteq L(f)$ for all $t \in (0, \infty)$. Now, the second statement follows immediately from the first one. [$\Box$]{}
If $f : {\mathbb R}_+ \longrightarrow {\mathbb R}$ were regulated, a natural question would be to ask for the representation of the function of jumps $\Delta g : L(g) \longrightarrow {\mathbb R}$, where $g(t) : = \sum_{s \in
(0, t]} \Phi\big(\vert \Delta f(s) \vert\big)$, $t \in (0, \infty)$. To this end, let $h : (0,\infty) \longrightarrow {\mathbb R}_+$ be an arbitrary positive real-valued function, and assume that $\mu_h((0,
\infty)) = \sum_{s \in (0, \infty)}h(s) < \infty$. Then $g(t) : =
\sum_{s \in (0, t]}h(s) = \mu_h((0, t]) \cap P) < +\infty$ for all $t \in {\mathbb R}_+$, where $P : = \{s \in (0, \infty) : h(s) > 0\}$. Let $t
\in (0, \infty)$. Since $\mu_h : \mathscr{P}((0,\infty))
\longrightarrow {\mathbb R}_+$ is a (finite) measure, it follows that $g(t +
\frac{1}{n}) - g(t - \frac{1}{n}) = \mu_h(I_n)$ for sufficiently large $n \in {\mathbb N}$, where $I_n : = (t - \frac{1}{n}, t +
\frac{1}{n}]$. Since $I_n \downarrow \{t\}$ as $n \to \infty$, we obviously have $$\Delta g(t) = \lim_{n \to \infty} \mu_h(I_n) = \mu_h(\{t\}) = h(t).$$ Hence, $L(g) = (0, \infty)$, and $\Delta \Big(\sum_{s \in (0, \cdot
)}h(s)\Big)= \Delta g = h$ on $(0, \infty)$. Moreover, since $g(\frac{1}{n}) = \mu_h((0, \frac{1}{n}]) \to \mu_h(\emptyset)= 0$ as $n \to \infty$, it follows that $0 \in L^+(g)$ and $g(0+) = 0 =
g(0)$. Consequently, Remark \[remark: regulated fct\] implies the following
Let $h : (0,\infty) \longrightarrow {\mathbb R}_+$ be an arbitrary positive real-valued function. If $\sum_{s \in (0, \infty)}h(s) < \infty$, then the function $$\begin{aligned}
g : {\mathbb R}_+ & \longrightarrow & {\mathbb R}_+\\
t & \mapsto & \sum_{s \in (0, t]} h(s),\end{aligned}$$ is regulated, $L(g) = (0, \infty)$, and $$\Delta g = h.$$
Random measures induced by optional processes
=============================================
Next, we transfer the main results of our previous investigations to (trajectories of) stochastic processes. Again, let $(\Omega ,
\mathcal{F}, {\bf{F}}, {\Bbb{P}})$ be a given filtered probability space such that ${\bf{F}}$ satisfies the usual conditions. If $X$ is an adapted and càdlàg process, then it is well-known that the left limit process $X_{-}$ is predictable. Recall that every adapted and right-continuous process is optional (cf. Theorem \[thm:POPA\]). Consequently, we deal with a special case of the slightly more general
\[lemma:optional and regulated paths\] Let $(\Omega , \mathcal{F}, {\bf{F}}, {\Bbb{P}})$ be a filtered probability space such that ${\bf{F}}$ satisfies the usual conditions. Let $X : \Omega \times {\mathbb R}_+ \longrightarrow {\mathbb R}$ be a stochastic process such that all trajectories of $X$ are regulated. Then all trajectories of the left limit process $X_{-}$ $($respectively of the right limit process $X_{+}$$)$ are left-continuous $($respectively right-continuous$)$. If in addition $X$ is optional, then $X_{-}$ is predictable.
Fix $\omega \in \Omega$ and consider the (fixed) trajectory $f : =
X_{\bullet}(\omega) : {\mathbb R}_+ \ \longrightarrow {\mathbb R}$. Since $f$ is a regulated function, it follows that that $L^{-}(f) = (0, \infty)$. Consequently, due to Corollary \[cor:f+ and f- for regulated f\], it clearly follows that the trajectory $Y_{\bullet}(\omega)$ of the left limit process $Y : = X_{-}$ is left-continuous on $(0,
\infty)$. Similarly, it follows that the trajectory of the right limit process $X_{+}$ is right-continuous on ${\mathbb R}_+ = L^{+}(f)$. Now assume that in addition $X$ is optional and therefore adapted. Then $X_{-}$ is an adapted process too. Consequently it follows that $Y =
X_{-}$ is adapted and left-continuous, and the definition of predictability finishes the proof.[$\Box$]{}
Now, we return to *finitely* layered partitions and start with the following observation. Despite its seemingly clear context, it will be of high importance for our further investigations.
\[lemma:recursive construction of finite sets\] Let $\emptyset \not= D$ be a finite subset of ${\mathbb R}$, consisting of $\kappa_D$ elements. Consider $$s_1^D : = \min(D)$$ and, if $\kappa_D \geq 2$, $$s_{n+1}^D := \min(D \cap (s_n^D, \infty)\big),$$ where $n \in \{1, 2, \ldots, \kappa_D - 1\}$. Then $D \cap
(s_{n}^D, \infty) \not= \emptyset$ for all $n \in \{1, 2, \ldots,
\kappa_D - 1\}$ and $s_{n}^D < s_{n+1}^D$ for all $n \in \{1, 2,
\ldots, \kappa_D - 1\}$. Moreover, we have $$D = {\bigcup_{n=1}^{\kappa_D}\hspace{-4.2mm\cdot}\hspace{2.2mm}}\{s_n^D\}.$$
Obviously, nothing is to prove if $\kappa_D \in \{1, 2\}$. Let $\kappa_D \geq 3$. Obviously, we have $D \cap (s_{1}^D, \infty)
\not= \emptyset$. Now assume by contradiction that there exists $n
\in \{2, \ldots, \kappa_D - 1\}$ such that $D \cap (s_{n}^D,
\infty) = \emptyset$. Choose the *minimal* $m \in \{2,
\ldots, \kappa_D - 1\}$ such that $D \cap (s_{m}^D, \infty) =
\emptyset$. Then $s_{k}^D : = \min(D \cap (s_{k-1}^D, \infty)\big)
\in D$ is well-defined for all $k \in \{2, \ldots, m\}$, and we obviously have $s_{1}^D < s_{2}^D < \ldots < s_{m}^D$. Moreover, by construction of $m$, it follows that $$\label{eq:induction statement}
s \leq s_{m}^D \textup{ for all } s \in D.$$ Assume now that there exists $s^\ast \in D$ such that $s^\ast
\not\in \{s_{1}^D, s_{2}^D, \ldots, s_{m}^D\}$. Then, by , there must exist $l \in \{1, 2,
\ldots, m -1 \}$ such that $s_{l}^D < s^\ast < s_{l+1}^D$, which is a contradiction, due to the definition of $s_{l+1}^D$. Hence, such a value $s^\ast$ cannot exist, and it consequently follows that $D =
\{s_{1}^D, s_{2}^D, \ldots, s_{m}^D\}$. But then $m =
\textup{card}(D) \leq \kappa_D - 1 < \kappa_D$, which is a contradiction. Hence, $s_{k}^D \in D$ is well-defined for all $k \in
\{1, 2, \ldots, \kappa_D\}$. Since $\textup{card}(D) = \kappa_D$, the proof is finished. [$\Box$]{}
\[thm: jumps of optional processes and stopping times\] Let $(\Omega , \mathcal{F}, {\bf{F}}, {\Bbb{P}})$ be a filtered probability space such that ${\bf{F}}$ satisfies the usual conditions. Let $X : \Omega \times {\mathbb R}_+ \longrightarrow {\mathbb R}$ be an optional process such that all trajectories of $X$ are regulated. Then $\Delta X$ is also optional. Put $X_{0-} : = X_{0+}$. If for each trajectory of $X$ its set of jumps is not finite, then there exists a sequence of stopping times $(T_n)_{n \in {\mathbb N}}$ such that $(T_n(\omega))_{n \in {\mathbb N}}$ is a strictly increasing sequence in $(0,
\infty)$ for all $\omega \in \Omega$ and $$J(X_{\bullet}(\omega)) = {\bigcup_{n=1}^{\infty}\hspace{-4.2mm\cdot}\hspace{2.2mm}}\{T_n(\omega)\} \textup{
for all } \omega \in \Omega,$$ or equivalently, $$\{\Delta X \not= 0\} = {\bigcup_{n = 1}^{\infty}\hspace{-4.2mm\cdot}\hspace{2.2mm}} {[\![}T_n {]\!]}.$$
Since the filtration is right-continuous, a direct calculation shows that the right limit process $X_{+}$ is adapted. Due to Lemma \[lemma:optional and regulated paths\], all paths of $X_{+}$ are right-continuous on ${\mathbb R}_+$. Since the filtration is right-continuous, it follows that $X_{+}$ is also adapted and hence an optional process. Consequently, since the process $X$ was assumed to be optional, and since each predictable process is optional, a further application of Lemma \[lemma:optional and regulated paths\] implies that the jump process $\Delta X = X_{+} - X_{-}$ is the sum of two optional processes, hence optional itself. Fix $\omega \in \Omega$. Consider the trajectory $f : =
X_{\bullet}(\omega)$. Due to statement (ii) of Theorem \[thm:at most countably many jumps on R+\], there exists a finitely layered partition of $J(f) \subseteq L(f) = (0, \infty)$ which now is randomised, and it follows that we may write $J(f)$ as $$J(f) = {\bigcup_{m = 1}^{\infty}\hspace{-4.6mm\cdot}\hspace{2.2mm}} D_m(\omega),$$ where $\kappa_m(\omega) : = \textup{card}(D_m(\omega)) < +\infty$ for all $m \in {\mathbb N}$. Let ${\Bbb{M(\omega)}} : =\{m \in {\mathbb N}: D_{m}(\omega) \not=
\emptyset\}$. Fix an arbitrary $m \in {\Bbb{M(\omega)}}$. Consider $$0 < S_1^{(m)}(\omega) : = \min(D_{m}(\omega))$$ and, if $\kappa_m(\omega) \geq 2$, $$0 < S_{n+1}^{(m)}(\omega) := \min\big(D_{m}(\omega) \cap
(S_n^{(m)}(\omega), \infty)\big),$$ where $n \in \{1, 2, \ldots, \kappa_m(\omega) - 1\}$. Since $\Delta X$ is optional, it follows that $\{\Delta X \in B\}$ is optional for all Borel sets $B \in \mathcal{B}({\mathbb R})$. Moreover, since $\Delta f(0) = \Delta X_0(\omega) : = 0$ (by assumption), it actually follows that $\{s \in {\mathbb R}_+ : (\omega, s) \in \{\Delta X \in
C\}\} = \{s \in (0, \infty) : (\omega, s) \in \{\Delta X \in C\}\}$ for all Borel sets $C \in \mathcal{B}({\mathbb R})$ which do not contain $0$. Hence, as the construction of the sets $D_{m}(\omega)$ in the proof of Theorem \[thm:at most countably many jumps on R+\] clearly shows, $S_1^{(m)}$ is the début of an optional set. Consequently, due to Theorem \[thm:optional debut is stopping time\], it follows that $S_1^{(m)}$ is a stopping time. If $S_n^{(m)}$ is a stopping time, the stochastic interval ${]\!]}S_n^{(m)}, \infty {[\![}$ is optional too (cf. [@HeWaYa92], Theorem 3.16). Thus, by construction, $S_{n+1}^{(m)}$ is the début of an optional set and hence a stopping time. Due to Lemma \[lemma:recursive construction of finite sets\], we have $$\label{eq:rep of the set of jumps of an optional regulated
process} J(f) = {\bigcup_{m \in {\Bbb{M(\omega)}}}^{}\hspace{-7mm\cdot}\hspace{2.2mm}}
D_{m}(\omega) = {\bigcup_{m \in {\Bbb{M(\omega)}}}^{}\hspace{-7mm\cdot}\hspace{4.2mm}} {\bigcup_{n =
1}^{\kappa_m(\omega)}\hspace{-5.5mm\cdot}\hspace{2.2mm}}\{S_n^{(m)}(\omega)\}.$$ Hence, since for each trajectory of $X$ its set of jumps is not finite, the at most countable set ${\Bbb{M(\omega)}}$ is not finite, hence countable, and a simple relabelling of the stopping times $S_n^{(m)}$ finishes the proof. [$\Box$]{}
We will see now how the choice of finitely layered jump partitions enables a natural approach for recovering the jump measure of a càdlàg and adapted stochastic process, and how we can transfer the structure of the jump measure to more general classes of optional processes which need not necessarily to be right-continuous.
\[prop:random measure for general processes X\] Let $(\Omega , \mathcal{F}, {\bf{F}}, {\Bbb{P}})$ be a filtered probability space such that ${\bf{F}}$ satisfies the usual conditions. Let $X : \Omega \times {\mathbb R}_+ \longrightarrow {\mathbb R}$ be an optional process such that all trajectories of $X$ are regulated and $\Delta X_0 : = 0$. Consider $$M_X(\omega) : = \Big\{(s, \Delta X_{s}(\omega)) : s \in
J(X_{\bullet}(\omega)) \Big\},$$ where $\omega \in \Omega$. Then $$\textup{card}(M_X(\omega) \cap G) = \sum_{s \in
J(X_{\bullet}(\omega))} {1\hspace{-2.5mm}1}_{G}\big(s, \Delta X_{s}(\omega) \big) =
\sum_{s > 0} {1\hspace{-2.5mm}1}_{G}\big(s, \Delta X_{s}(\omega) \big)
{1\hspace{-2.5mm}1}_{\{\Delta X \not= 0\}}(\omega, s)
$$ for all $(\omega, G) \in \Omega \times \mathcal{B}({\mathbb R}_+) \otimes
\mathcal{B}({\mathbb R})$.
For simplicity reasons, we may assume that the set of jumps of each trajectory of $X$ is not finite (due to Theorem \[thm:Fubini for unordered sums\] and representation ). Fix $(\omega, G) \in \Omega \times \mathcal{B}({\mathbb R}_+) \otimes
\mathcal{B}({\mathbb R})$. Consider $D_X(\omega) : = J(X_{\bullet}(\omega))
\subseteq L(X_{\bullet}(\omega)) = (0, \infty)$. Theorem \[thm:at most countably many jumps on R+\] implies that the $$\begin{aligned}
M_X(\omega) : = \Big\{(s, \Delta X_{s}(\omega)) : s \in
D_X(\omega) \Big\} = \mbox{Gr}\Big(\Delta
X_{\bullet}(\omega)\vert_{D_X(\omega)}\Big)\end{aligned}$$ . Put $j{_{{}_{X}}}(\omega, \textcolor{owngreen2}{G}) : =
\textup{card}(M_X(\omega) \cap G)$. Due to Theorem \[thm: jumps of optional processes and stopping times\], it follows that there exists a sequence of stopping times $\{T_n : n \in {\mathbb N}\}$ such that $$M_X(\omega) \cap \textcolor{owngreen2}{G} = {\bigcup_{m =
1}^{\infty}\hspace{-4.5mm\cdot}\hspace{2.2mm}} \big\{\big(T_m(\omega), \Delta X_{T_m}(\omega)
\big)\big\} \cap G,$$ where $\Delta X_{T_m}(\omega) : = \Delta X_{T_m(\omega)}(\omega) =
\Delta X(\omega, T_m(\omega))$. Since all unions are disjoint ones, the respective cardinals are additive. Consequently, $$j{_{{}_{X}}}(\omega, \textcolor{owngreen2}{G}) = \sum_{m = 1}^\infty
\mbox{card}\big( \big\{\big(T_m(\omega), \Delta X_{T_m}(\omega)
\big)\big\} \cap G \big) = \sum_{m = 1}^\infty
{1\hspace{-2.5mm}1}_{G}\big(T_m(\omega), \Delta X_{T_m}(\omega) \big),$$ and Theorem \[thm:Fubini for unordered sums\] together with Lemma \[lemma:indicator function in sum\] imply that $$j{_{{}_{X}}}(\omega, \textcolor{owngreen2}{G}) = \sum_{s \in
D_X(\omega)} {1\hspace{-2.5mm}1}_{G}\big(s, \Delta X_{s}(\omega) \big) = \sum_{s >
0} {1\hspace{-2.5mm}1}_{G}\big(s, \Delta X_{s}(\omega) \big) {1\hspace{-2.5mm}1}_{\{\Delta X \not=
0\}}(\omega, s).
$$ [$\Box$]{}
\[thm:optional random measures\] Let $(\Omega , \mathcal{F}, {\bf{F}}, {\Bbb{P}})$ be a filtered probability space such that ${\bf{F}}$ satisfies the usual conditions. Let $X : \Omega \times {\mathbb R}_+ \longrightarrow {\mathbb R}$ be an optional process such that all trajectories of $X$ are regulated and $\Delta X_0 : = 0$. Then the function $$\begin{aligned}
j{_{{}_{X}}} : \Omega \times \mathcal{B}({\mathbb R}_+) \otimes
\mathcal{B}({\mathbb R}) & \longrightarrow & {\mathbb Z}_+ \cup \{+ \infty\}\\
(\omega, G) & \mapsto & \sum_{s > 0} {1\hspace{-2.5mm}1}_{G}\big(s, \Delta
X_{s}(\omega) \big) {1\hspace{-2.5mm}1}_{\{\Delta X \not= 0\}}(\omega, s)\end{aligned}$$ is an integer-valued random measure.
We only have to combine Theorem \[thm: jumps of optional processes and stopping times\] and [@HeWaYa92], Theorem 11.13.[$\Box$]{}
Put $G : = B \times \Lambda$, where $B \in \mathcal{B}({\mathbb R}_+)$ and $\Lambda \in \mathcal{B}({\mathbb R})$. Since each trajectory of $X$ is regulated, $B\setminus\{0\} \subseteq (0, \infty) =
L(X_\bullet(\omega))$. Hence, Lemma \[lemma:indicator function in sum\] directly leads to the following representation: $$j{_{{}_{X}}}(\omega, B \times \Lambda) = j{_{{}_{X}}}(\omega,
(B\setminus\{0\}) \times \Lambda) = \sum_{s \in B\setminus\{0\}}
{1\hspace{-2.5mm}1}_\Lambda(\Delta X_s(\omega)){1\hspace{-2.5mm}1}_{\{\Delta X \not= 0\}}(\omega,
s)$$ for all $B \in \mathcal{B}({\mathbb R}_+)$ and $\Lambda \in
\mathcal{B}({\mathbb R})$.
To sum up, $j{_{{}_{X}}}(\omega, G)$ in general counts, $\omega$-by-$\omega$, the number of all $s > 0$ such that $\Delta
X{_{{}_{s}}}(\omega) \not= 0$ and $(s, \Delta X_s(\omega)) \in G$. In other words, $$j{_{{}_{X}}}(\omega, (\mbox{d}t, \mbox{d}x)) = \sum_{s
>0}{1\hspace{-2.5mm}1}_{\{\Delta X \not= 0\}}(\omega, s)\cdot\delta_{\big(s, \Delta
X_{s}(\omega)\big)}(\mbox{d}t,\mbox{d}x).$$ We finish this paper by considering right-continuous trajectories again and note the following
Let $(\Omega , \mathcal{F}, {\bf{F}}, {\Bbb{P}})$ be a filtered probability space such that ${\bf{F}}$ satisfies the usual conditions. If $X : \Omega \times {\mathbb R}_+ \longrightarrow {\mathbb R}$ is an adapted and càdlàg process such that $\Delta X_0 : = 0$, then $j_X$ is the jump measure of $X$.
**Acknowledgements**: The author gratefully thanks Dave Applebaum and Finbarr Holland for highly fruitful discussions including Finbarr Holland’s information about the very useful M. Sc. Thesis [@Od04].
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[^1]: [*AMS 2000 subject classifications.*]{} 28A05, 40G99, 60G05, 60G57
[^2]: [*Key Words and Phrases.*]{} Regulated functions, unordered sums, at most countable sets, jumps, optional stochastic processes, stopping times, random measures
[^3]: $\mathcal{F}_0$ contains all $\mathbb{P}$-null sets and ${\bf{F}}$ is right-continuous.
[^4]: Any interior point of $I$ satisfies that condition.
[^5]: Due to the choice of $t$ and the structure of $I$, we obviously may choose $\delta >0$ sufficiently small such that $(t, t + \delta) \subseteq I$.
[^6]: Note that the set $-I$ belongs to the same class as the given interval $I$.
|
---
abstract: 'Given a graded complete intersection ideal $J = (f_1, \dots, f_c) \subseteq k[x_0, \dots, x_n] = S$, where $k$ is a field of characteristic $p > 0$ such that $[k:k^p] < \infty$, we show that if $S/J$ has an isolated non-F-pure point then the Frobenius action on top local cohomology $H^{n+1-c}_\mathfrak{m}(S/J)$ is injective in sufficiently negative degrees, and we compute the least degree of any kernel element. If $S/J$ has an isolated singularity, we are also able to give an effective bound on $p$ ensuring the Frobenius action on $H^{n+1-c}_\mathfrak{m}(S/J)$ is injective in all negative degrees, extending a result of Bhatt and Singh in the hypersurface case.'
address: 'Department of Mathematics, University of Nebraska – Lincoln, 203 Avery Hall, Lincoln NE 68588'
author:
- Eric Canton
title: |
A Note on Injectivity of Frobenius on Local Cohomology\
of Global Complete Intersections
---
Introduction
============
Let $k$ be a field of characteristic $p > 0$ such that $[k:k^p] < \infty$. We study the kernel of the Frobenius action on local cohomology of a graded complete intersection $R = k[x_0, \dots, x_n]/(f_1, \dots, f_c)$ under the assumption that Frobenius is pure on $R_\mathfrak{p}$ for all primes $\mathfrak{p}$ different from the graded maximal ideal $\mathfrak{m} = (x_0, \dots, x_n)$. Let $J = (f_1, \dots, f_c)$. Recall that $R_\mathfrak{p}$ has a pure Frobenius homomorphism for a prime $\mathfrak{p} \ne \mathfrak{m}$ of $S$ containing $J$ if and only if $(f_1\cdots f_c)^{p-1} \not\in \mathfrak{p}^{[p]}$ [@Fedder], where $\mathfrak{p}^{[p]} = (a^p \,|\, a \in \mathfrak{p})$. The first of our main results is the following.
Suppose $(f_1 \cdots f_c)^{p-1} \not\in \mathfrak{p}^{[p]}$ for any prime $\mathfrak{p} \ne \mathfrak{m}$ with $J \subseteq \mathfrak{p}$, but $(f_1 \cdots f_c)^{p-1} \in \mathfrak{m}^{[p]}$. Set $a(R) = - (n+1) + \sum_1^c \deg(f_i)$. Let $\tau$ be the smallest ideal of $S$ such that $J \subseteq \tau$ and $(f_1\cdots f_c)^{p-1} \in \tau^{[p]}$. Note in this case $\sqrt{\tau} = \mathfrak{m}$; set $\ell = \max\{s \,|\, \mathfrak{m}^s \not\subseteq \tau\} < \infty$. Then the below Frobenius action is injective: $$F: H^{n+1-c}_\mathfrak{m}(R)_{< a(R) - \ell} \to H^{n+1-c}_\mathfrak{m}(R)_{< p(a(R) - \ell)}.$$
The proof also shows that this bound is sharp: there always exists a class $\alpha \in H^{n+1-c}_\mathfrak{m}(R)$ of degree $a(R) - \ell$ such that $F(\alpha) = 0$. The above theorem now yields the following characterization of graded complete intersection quotients such that $\sqrt{\tau} = \mathfrak{m}$. The main advantage of this characterization is that it depends only on $n$, the codimension $c$, and the degree $d := \sum_1^c \deg(f_j)$.
Let $\tau$ be as in Theorem A and suppose $\tau \ne S$. The condition $\sqrt{\tau} = \mathfrak{m}$ is equivalent to the Frobenius action on $H^{n+1-c}_\mathfrak{m}(R)$ being injective in degrees $< -(n+1-c)d$.
An easy way to ensure that a complete intersection $R$ satisfies the hypothesis of theorem A is to assume $R$ has isolated singularity. Under this additional assumption we are able to combine an argument of Fedder [@FedderGradedCI Theorem 2.1] with one of Bhatt and Singh to generalize [@Bhatt-Singh Theorem 3.5].
Assume $R$ has an isolated singularity. If $p \ge (n+1-c)(d-c)$ then the Frobenius action on $H^{n+1-c}_\mathfrak{m}(R)$ is injective in negative degrees.
I would like to thank my advisor, Wenliang Zhang, for numerous helpful discussions. I would also like to thank Tom Marley, Anurag Singh, and Thanh Vu for useful conversations, and the referee for many suggestions which improved the readability of this paper.
Notation and Conventions
========================
The letter $q$ will always denote an integer power $p^e$ of a prime $p > 0$. Let $S$ be a $\mathbb{Z}$-graded Noetherian ring containing a field $k$ of characteristic $p > 0$ with $[k:k^p] < \infty$ and posessing a unique maximal homogeneous ideal $\mathfrak{m}$. For any ideal $I$ of $S$, define $I^{[p]} = (f^p \,|\, f \in I)$. Similarly, we define $I^{[q]} = (f^q \,|\, f\in I)$ for all $q \ge 0$.
\[regularity\] Suppose $M$ is a $\mathbb{Z}$-graded $S$-module such that $M_\ell = 0$ for $\ell \gg 0$. Define the [*Castelnuovo-Mumford regularity*]{} of $M$ to be $$\mathrm{reg}(M) = \max\{\ell \,\,|\,\, (M)_\ell \ne 0\}.$$
\[regularity example\] If $I \subseteq S$ is a homogeneous ideal such that $\sqrt{I} = \mathfrak{m}$ then for all $\ell \gg 0$ we know $\mathfrak{m}^\ell \subseteq I$. Thus $(S/I)_\ell = 0$ for $\ell \gg 0$ and we can define $\mathrm{reg}(S/I)$ as above. The regularity of $S/I$ is the unique integer $a$ such that $\mathfrak{m}^a \not\subseteq I$ but $\mathfrak{m}^{a+1} \subseteq I$.
We introduce the following
Let $0 \ne I \subsetneq S$ be a proper homogeneous ideal. For each $q \ge 0$ define $$M_q(I) = \max\left\{\ell \,\,\left|\,\, (\mathfrak{m}^{[q]} : I) \subseteq \mathfrak{m}^{[q]} + \mathfrak{m}^\ell \right.\right\}.$$
Suppose $S = k[x, y]$ and let $I = (x^a, y^b)$ with $a \le b$. For $q \le \min\{a, b\}$, $(\mathfrak{m}^{[q]} : I) = S$ so $M_q(I) = 0$. For $a < q \le b$ the colon $(\mathfrak{m}^{[q]} : I)$ becomes $\mathfrak{m}^{[q]} + (x^{q - a})$ so $M_q(I) = q - a$. Finally for $q > b$ the colon is $\mathfrak{m}^{[q]} + (x^{q-a}y^{q-b})$ and so $M_q(I) = 2q - (a + b)$ for all $q \ge b$. In particular, $2q - M_q(I) = a + b$ for all large $q$. Note that since $\mathfrak{m}^{a+b-1} \subseteq I$ but $\mathfrak{m}^{a+b-2}\not\subseteq I$, $2q - M_q(I) = \mathrm{reg}(S/I) + 2$. Our next proposition shows this is always the case for $\mathfrak{m}$-primary $I$.
\[colon containment\] Let $I$ be any ideal of $S$. We have a containment $(\mathfrak{m}^{[q]} : I) \subseteq (\mathfrak{m}^{[q]} : \mathfrak{m}^\ell)$ for all $q \gg 0$ if and only if $\mathfrak{m}^\ell \subseteq I$.
Very generally, if $J$ and $L$ are ideals in a zero dimensional Gorenstein ring then we have a containment of colon ideals $(0: J) \subseteq (0 : L)$ if and only if $L \subseteq J$. Since $S/\mathfrak{m}^{[q]}$ is zero dimensional and Gorenstein for all $q \ge 0$ it follows that for a specific $q$ we have $(\mathfrak{m}^{[q]} : I) \subseteq (\mathfrak{m}^{[q]} : \mathfrak{m}^\ell)$ if and only if $\mathfrak{m}^\ell + \mathfrak{m}^{[q]} \subseteq I + \mathfrak{m}^{[q]}$. For large $q$ and any fixed $\ell$ we know $\mathfrak{m}^{[q]} \subseteq \mathfrak{m}^\ell$. Therefore, $\mathfrak{m}^\ell \subseteq \bigcap_{q \gg 0} (I + \mathfrak{m}^{[q]})$ if and only if $(\mathfrak{m}^{[q]} : I) \subseteq (\mathfrak{m}^{[q]} : \mathfrak{m}^\ell)$ for all $q \gg 0$. Krull’s intersection theorem implies $I = \bigcap_{q \gg 0} (I + \mathfrak{m}^{[q]})$, and the statement of the lemma follows.
\[Regularity Proposition\] Let $S = k[x_0, \dots, x_n]$ and $I \subsetneq S$ a proper ideal.
1. $(n+1)q - M_q(I) \le n + \ell$ for $q \gg 0$ if and only if $\mathfrak{m}^\ell \subseteq I$. \[first part\]
2. If $\sqrt{I} = \mathfrak{m}$ then for $q \gg 0$, $$(n+1)q - M_q(I) = \mathrm{reg}(S/I) + (n+1).$$
To establish the first statement, suppose $(n+1)q - M_q(I) \le n + \ell$ for all $q \ge q_0$. Then by definition $$(\mathfrak{m}^{[q]} : I) \subseteq \mathfrak{m}^{[q]} + \mathfrak{m}^{(n+1)q - (n + \ell)}.$$ Bhatt and Singh show [@Bhatt-Singh 3.2] that the ideal on the right is $(\mathfrak{m}^{[q]} : \mathfrak{m}^\ell)$. Since this containment holds for all $q \ge q_0$ we conclude that $\mathfrak{m}^\ell \subseteq I$ using . This argument is easily reversed, and the statement follows.
If $\sqrt{I} = \mathfrak{m}$ then statement shows that for $q \gg 0$, $(n+1)q - M_q(I) = n + \ell$ where $\ell$ is the least integer such that $\mathfrak{m}^\ell \subseteq I$. Using we see that $\ell = \mathrm{reg}(S/I) + 1$.
\[lch identification\] Let $S = k[x_0, \dots, x_n]$ and let $f_1, \dots, f_c \in S$ be homogeneous forms which form a regular sequence. Set $J = (f_1, \dots, f_c)$, $R = S/J$, $f = \prod_1^c f_j$, and $d = \deg(f) = \sum_1^c \deg(f_j)$. Then the Koszul complex $K_\bullet$ on the forms $f_j$ gives the minimal graded free resolution of $R$ over $S$, and after applying $H_\mathfrak{m}^*$ to $K_\bullet$ we may identify $$\tag{$\star$}\label{identify}
H_\mathfrak{m}^{n+1-c}(R) = \mathrm{Ann}_{H_\mathfrak{m}^{n+1}(S)[-d]}(J).$$ If we compute $H_\mathfrak{m}^{n+1}(S)$ via the Čech complex Č$(x_0, \dots, x_n)$, then we can represent classes $\alpha \in H_\mathfrak{m}^{n+1-c}(R)$ by $[g/x^q]$ where $g \in S$, $x^q := (x_0\cdots x_n)^q$, and this class is zero if and only if $g \in \mathfrak{m}^{[q]}$. The identification is degree-preserving, so that classes $\alpha \in H_\mathfrak{m}^{n+1-c}(R)_t \subseteq H_\mathfrak{m}^{n+1}(S)[-d]_t = H_\mathfrak{m}^{n+1}(S)_{t - d}$ are represented by $[g/x^q]$ with $g$ homogeneous, $g \in (\mathfrak{m}^{[q]} : J)$, and $\deg(g) - (n+1)q = t - d$.
\[frobenius action\] Using notation from , the Frobenius homomorphism on $R$ lifts to a chain map $F_\bullet: K_\bullet \to K_\bullet$ which is [*not*]{} $S$-linear, though it is $\mathbb{Z}$-linear. In the case $c = 1$ (so that $f_1 = f$) the Koszul complex and chain map associated to the Frobenius homomorphism take the form $$\begin{CD}
0 @>>> S[-d] @>f>> S @>>> R @>>> 0 \\
@. @Vf^{p-1}F VV @VF VV @VF VV \\
0 @>>> S[-d] @>f>> S @>>> R @>>> 0.
\end{CD}$$ For any $c$ the map $F_c: S[-d] \to S[-d]$ is still given by $f^{p-1}F = (f_1\cdots f_c)^{p-1}F$, so again identifying $H_\mathfrak{m}^{n+1-c}(R)$ as in we can describe the Frobenius action on local cohomology of $R$ as $[g/x^q] \mapsto [f^{p-1}g^p/x^{pq}]$.
Injectivity of Frobenius in codimension $c$ {#Theorem section}
===========================================
Let $I \subsetneq k[x_0, \dots, x_n]$ be a proper homogeneous ideal and denote $k[x_0, \dots, x_n]/I$ by $A$. It is well-known that the local cohomology modules $H_{(x_0, \dots, x_n)}^i(A)$ are graded Artinian modules. We define the [*a-invariant*]{} of $A$, denoted $a(A)$, as $\mathrm{reg}\left(H_{(x_0, \dots, x_n)}^{\dim(A)}(A)\right)$. In the case $A = k[x_0, \dots, x_n]$ it is straightforward to see ([*e.g.*]{}, using graded local duality) that $a(A) = -(n+1)$, and if $I = (f_1, \dots, f_c)$ is generated by a homogeneous regular sequence then $a(A) = -(n+1) + \sum_1^c \deg(f_j)$.
\[Injectivity Theorem\] Let $J = (f_1, \dots, f_c) \subseteq k[x_0, \dots, x_n] = S$ be an ideal generated by a homogeneous regular sequence and set $R = S/J$. Define $\tau$ as the smallest ideal of $S$ such that $J \subseteq \tau$ and $(f_1\cdots f_c)^{p-1} \in \tau^{[p]}$. Assume $\sqrt{\tau} = \mathfrak{m} := (x_0, \dots, x_n)$. Then the below Frobenius action is injective: $$F: H_\mathfrak{m}^{n+1-c}(R)_{< a(R) -\mathrm{reg}(S/\tau) } \to H_\mathfrak{m}^{n+1-c}(R)_{< p(a(R) -\mathrm{reg}(S/\tau)) }.$$
Let $f = f_1\cdots f_c$ and $d = \deg(f)$. Note that $a(R) = d - (n+1)$. We use and to identify $H_\mathfrak{m}^{n+1-c}(R)$ with $\mathrm{Ann}_{H_\mathfrak{m}^{n+1}(S)[-d]}(J) =: T$ and the Frobenius action on $H_\mathfrak{m}^{n+1-c}(R)$ with $f^{p-1}F|_T$. Aiming for a contradiction, suppose there exists nonzero $\alpha \in T$ such that $f^{p-1}F(\alpha) = 0$ but $\deg(\alpha) < (d - (n+1) -\mathrm{reg}(S/\tau)) - d = -\mathrm{reg}(S/\tau) - (n+1)$. We may represent $\alpha$ by $[g/x^{q}]$ for $q \gg 0$ and $g \in (\mathfrak{m}^{[q]} : J)\setminus \mathfrak{m}^{[q]}$. Using this representation, $$\begin{aligned}
f^{p-1}F(\alpha) = 0 &\iff f^{p-1}g^p \in \mathfrak{m}^{[pq]}\\
&\iff f^{p-1} \in (\mathfrak{m}^{[pq]}:g^p) = (\mathfrak{m}^{[q]}:g)^{[p]}.
\end{aligned}$$ Since $[g/x^q] \in T$ we know $J \subseteq (\mathfrak{m}^{[q]} : g)$, so that $f^{p-1} \in (\mathfrak{m}^{[q]} : g)^{[p]}$ is equivalent to $\tau \subseteq (\mathfrak{m}^{[q]} : g)$. This in turn is equivalent to $g \in (\mathfrak{m}^{[q]} : \tau)$. Since $g \not\in \mathfrak{m}^{[q]}$, $\deg(g) \ge M_q(\tau)$. Using we conclude $$-\mathrm{reg}(S/\tau) - (n+1) = M_q(\tau) - (n+1)q \le \deg(g) - (n+1)q = \deg(\alpha) < -\mathrm{reg}(S/\tau) - (n+1),$$ a contradiction.
\[sharpness\] Theorem \[Injectivity Theorem\] is sharp: if we take $g \in (\mathfrak{m}^{[q]} : \tau) \setminus \mathfrak{m}^{[q]}$ of degree $M_q(\tau)$ for $q \gg 0$ then $[g/x^q] \mapsto [f^{p-1}g^p/x^{pq}] = 0$.
\[Degree Bound\] Using notation from , assume $\tau \ne S$. Then $\sqrt{\tau} = \mathfrak{m}$ if and only if the below Frobenius action is injective: $$F: H_\mathfrak{m}^{n+1-c}(R)_{< -(n+1-c)d} \to H_\mathfrak{m}^{n+1-c}(R)_{< -p(n+1-c)d}.$$
If $\sqrt{\tau} \ne \mathfrak{m}$ then the sequence $\{M_q(\tau) - (n+1)q\}_{q \ge 1}$ is unbounded below by . Remark \[sharpness\] shows that there always exists a kernel element of degree $M_q(\tau) - (n+1)q$ for any $q$.
Suppose $\sqrt{\tau} = \mathfrak{m}$. Then by it suffices to show $$-(n+1-c)d \le d - (n+1) -\mathrm{reg}(S/\tau).$$ Towards this end, since $\sqrt{\tau} = \mathfrak{m}$ we know that there must exist generators $\phi_1, \dots, \phi_{n+1-c} \in \tau$ such that $\underline{\phi}, \underline{f}$ is a regular sequence on $S$. Let $\mathfrak{b} = (\underline{\phi}, \underline{f})$. By [@BMS 2.4] we may choose the $\phi_i$ so that $\deg{\phi_i} \le p^{-1} d(p-1) < d$. Thus the Hilbert series $HS(S/\mathfrak{b}, t)$ is $$\frac{\Pi_{i=1}^{n+1-c}(1 - t^{\deg{\phi_i}}) \Pi_{j=1}^c (1-t^{d_i})}{(1 - t)^{n+1}}$$ which is a polynomial of degree $$\left(\sum_1^{n+1-c} \deg(\phi_i) \right) + \left(\sum_1^c \deg{f_j}\right) - (n+1) < (n+1-c)d + d - (n+1).$$ Therefore, $\mathfrak{m}^{(n+1-c)d + d - (n+1)+1}\subseteq \mathfrak{b} \subseteq \tau$ and we conclude $$\begin{aligned}
-\mathrm{reg}(S/\tau) - (n+1) + d &\ge -((n+1-c)d + d - (n+1)) - (n+1) + d\\
&= -(n+1-c)d.
\end{aligned}$$
We conclude this section by extending Bhatt and Singh’s result [@Bhatt-Singh 3.5] using a generalization of their method with an application of the determinant trick used in the proof of [@FedderGradedCI 2.1].
\[Bhatt-Singh style\] Using the notation from , let $\mathrm{Jac}(R)$ be the ideal of $(c\times c)$ minors of the Jacobian matrix $(\partial f_j/\partial x_i)$, $0 \le i \le n$, $1 \le j \le c$. Suppose $\sqrt{\mathrm{Jac}(R) + J} = \mathfrak{m}$. If $p \ge (n + 1 - c)(d-c)$ then the below Frobenius action is injective: $$F: H^{n+1-c}_\mathfrak{m}(R)_{<0} \to H^{n+1-c}_\mathfrak{m}(R)_{<0}.$$
Again using the identification $H_\mathfrak{m}^{n+1-c}(R) = \mathrm{Ann}_{H_\mathfrak{m}^{n+1}(S)[-d]}(J) =: T$ of , suppose there exists $0 \ne \alpha = [g/x^{q/p}] \in T_{\le -1}$ with $f^{p-1}F(\alpha) = [f^{p-1}g^p/x^q] = 0$; then $\deg(g) - (n+1)(q/p) \le -d - 1$. We show this implies $p < (n + 1 - c)(d - c)$, contradicting our assumption on $p$.
Define $(t_1, \dots, t_c) = \mathbf{t}\in \{0, 1, \dots, p-1\}^c$ to be the least vector such that $f^{\mathbf{t}}g^p \in \mathfrak{m}^{[q]}$, in the sense that if $\mathbf{s} = (s_1, \dots, s_c)$ with $s_j \le t_j$ for all $j$ and $s_\ell < t_\ell$ for at least one $\ell$, then $f^\mathbf{s}g^p \not\in \mathfrak{m}^{[q]}$. Note that $\mathbf{t}$ exists since $f^{p-1}g^p \in \mathfrak{m}^{[q]}$, and furthermore at least one $t_j \ne 0$ since $g \not\in \mathfrak{m}^{[q/p]}$ (as $[g/x^{q/p}] \ne 0)$ and thus $g^p \not\in \mathfrak{m}^{[q]}$. Without loss of generality, we assume $t_1 \ne 0$. For $1 \le j \le c$ write $$\hat{f_j} = \prod_{i \in \mathcal{C}} f_i, \;\text{where $\mathcal{C} = \{ i \in \{1, \dots, c\} \,|\, i \ne j \text{ and } t_i > 0\}$.}$$ Let $\mathbf{t'} = (t_1 -1, t_2, \dots, t_c)$; we know $f^\mathbf{t'}g^p \not\in \mathfrak{m}^{[q]}$ by definition of $\mathbf{t}$.
$f^\mathbf{t'}g^p \in (\mathfrak{m}^{[q]}: \mathrm{Jac}(R))$.
Define $\mathbf{t}^*$ by $\mathbf{t}^*_j = \max\{t_j - 1, 0\}$ for $1 \le j \le c$. If $\partial_i$ is the partial derivative with respect to $x_i$, then $\partial_i(\mathfrak{m}^{[q]}) \subseteq \mathfrak{m}^{[q]}$ for all $i$ and so $$\begin{aligned}
\partial_i(f^{\mathbf{t}}g^p) &= g^pf^{\mathbf{t}^*}\left(\sum_{j = 1}^c t_j\hat{f_j} \partial_i(f_j)\right) \equiv 0 \pmod{\mathfrak{m}^{[q]}}.
\end{aligned}$$ For any choice of $\mathbf{i} = (i_1, \dots, i_c)$ with $0 \le i_1 < i_2 < \cdots < i_c \le n$, we have a matrix equation $$f^{\mathbf{t}^*} g^p
\begin{pmatrix}
\partial_{i_1}(f_1) & \cdots & \partial_{i_1}(f_c) \\
\vdots & \ddots & \vdots \\
\partial_{i_c}(f_1) & \cdots & \partial_{i_c}(f_c)\\
\end{pmatrix}
\begin{pmatrix}
t_1 \hat{f_1} \\ \vdots \\ t_c \hat{f_c}
\end{pmatrix}
\equiv
\begin{pmatrix}
0 \\ \vdots \\ 0
\end{pmatrix}
\pmod{\mathfrak{m}^{[q]}}.$$ Calling the determinant of the above matrix of partial derivatives $\Delta_\mathbf{i}$, after multiplying by the adjugate[^1] above we have $$t_j \hat{f_j} f^{\mathbf{t}^*}g^p \Delta_\mathbf{i} \equiv 0 \pmod{\mathfrak{m}^{[q]}} \text{ for all $1 \le j \le c$. }$$ Since $\mathrm{Jac}(R)$ is generated by the determinants $\Delta_\mathbf{i}$ ranging over all choices of $\mathbf{i}$ and $0 < t_1 < p$ we conclude $$f^\mathbf{t'}g^p\mathrm{Jac}(R) = \hat{f_1} f^{\mathbf{t}^*}g^p \mathrm{Jac}(R) \subseteq \mathfrak{m}^{[q]}.$$
We now claim $f^\mathbf{t'}g^p \in (\mathfrak{m}^{[q]} : (f_1, f_2^{p-t_2}, \dots, f_c^{p - t_c}))$. Indeed, we defined $\mathbf{t}$ to have the property $f_1f^\mathbf{t'}g^p = f^\mathbf{t}g^p \in \mathfrak{m}^{[q]}$; if $j > 1$ then $f_j^pg^p$ divides $f_j^{p-t_j}f^\mathbf{t'}g^p$ and we know $f_j^p g^p \in \mathfrak{m}^{[q]}$.
Since $\mathrm{Jac}(R) + J$ is $\mathfrak{m}$-primary, there must exist a sequence of determinants $\underline{\Delta} := \Delta_1, \dots, \Delta_{n+1-c}$ so that $\underline{\Delta}, \underline{f}$ gives a maximal regular sequence on $S$. Then $\underline{\Delta}, f_1, f_2^{p-t_2}, \dots, f_c^{p-t_c}$ is also a regular sequence, and setting $\mathfrak{b} = (\underline{\Delta}, f_1, f_2^{p-t_2}, \dots, f_c^{p - t_c})$ we have $\mathfrak{b} \subseteq \mathrm{Jac}(R) + (f_1, f_2^{p-t_2}, \dots, f_c^{p-t_c})$. If we set $$\ell = (n+1-c)(d-c) + d_1 + \left(\sum_{j=2}^c (p - t_j) d_j\right) - n$$ then a Hilbert series argument similar to the one found in the proof of shows that $\mathfrak{m}^\ell \subseteq \mathfrak{b}$. We will use this containment to bound the degree of $f^\mathbf{t'}g^p$. Towards this end, we know $$\begin{aligned}
f^\mathbf{t'} g^p &\in (\mathfrak{m}^{[q]}: \mathrm{Jac}(R) + (f_1, f_2^{p-t_2}, \dots, f_c^{p-t_c})) \\
&\subseteq (\mathfrak{m}^{[q]}: \mathfrak{b})\\
&\subseteq (\mathfrak{m}^{[q]}: \mathfrak{m}^\ell)\\
&= \mathfrak{m}^{[q]} + \mathfrak{m}^{q(n+1) - n - \ell}.
\end{aligned}$$ Since $f^\mathbf{t'} g^p \not\in \mathfrak{m}^{[q]}$ we conclude $f^\mathbf{t'}g^p \in \mathfrak{m}^{q(n+1)- n - \ell}$. This implies $$\begin{aligned}
q(n+1) - n - \ell &= q(n+1) - n - (n + 1 - c)(d - c) - d_1 - \left(\sum_{j=2}^c (p - t_j)d_j\right) + n\\
&= q(n+1) - (n + 1 - c)(d - c) + (p - 1)d_1 - pd + \left(\sum_{j=2}^c t_jd_j\right) \\
&\le \deg(f^\mathbf{t'}g^p)\\
&= (t_1 - 1)d_1 + \left(\sum_{j=2}^c t_jd_j\right) + p\deg(g).
\end{aligned}$$ Now recalling that $p\deg(g) \le q(n+1) - pd - p$ and $t_1 - 1 < p - 1$, we have $$q(n+1) + (p - 1)d_1 - pd - (n + 1 - c)(d - c) < q(n+1) + (p - 1)d_1 - pd - p$$ which simplifies to $p < (n+1-c)(d-c)$.
Let $f = x^2y^2 + y^2z^2 + z^2x^2 \in k[x,y,z] = S$ with $\mathrm{char}(k) > 2$. Then $\tau = \mathfrak{m}$ but $f$ does not have an isolated singularity. In this case, the Bhatt-Singh result [@Bhatt-Singh Theorem 3.5] does not apply. Theorem \[Injectivity Theorem\] now tells us that the Frobenius action on $H^2_\mathfrak{m}(S/fS)$ is injective in degrees $\le 0$. Note that in this case, $H^2_\mathfrak{m}(S/fS)_1 \ne 0$ but $H^2_\mathfrak{m}(S/fS)_{\ge 2} = 0$ so the Frobenius action on $H^2_\mathfrak{m}(S/fS)_1$ is zero.
A smooth projective variety $X = \mathrm{Proj}(R)$ is a [*Calabi-Yau variety*]{} if $\omega_X = \mathscr{O}_X$. If $R$ is a complete intersection, this is equivalent to $a(R) = 0$. In this case, we can combine theorems \[Injectivity Theorem\] and \[Bhatt-Singh style\] to conclude that for $p \gg 0$ the Frobenius action on local cohomology of $R$ can only fail to be injective in degree $0$. Thus, $\tau \in \{\mathfrak{m}, S\}$ for Calabi-Yau complete intersections and all $p \gg 0$.
[4]{} Manuel Blickle, Mircea Mustata, and Karen E. Smith. Discreteness and rationality of $F$-thresholds. [*The Michigan Math. J.*]{}, 57:43-61, 08 2008.
Bhargav Bhatt and Anurag K. Singh, The $F$-pure threshold of a Calabi-Yau hypersurface. [*Mathematische Annalen*]{}, 362(1):551-567, 2014.
Richard Fedder, $F$-purity and rational singularity. [*Trans. Amer. Math. Soc.*]{}, 278(2):461-480, 1983.
Richard Fedder, $F$-purity and rational singularity in graded complete intersection rings. [*Trans. Amer. Math. Soc.*]{}, 301(1):47-62, 1987.
[^1]: The [*adjugate*]{} $\mathrm{adj}(B)$ of an ($m\times m$) matrix $B$ has the property $\mathrm{adj}(B)B = \det(B)I_m$.
|
---
abstract: 'We generalize a result of Garvan on inequalities and interpretations of the moments of the partition rank and crank functions. In particular for nearly 30 Bailey pairs, we introduce a rank-like function, establish inequalities with the moments of the rank-like function and an associated crank-like function, and give an associated so called higher order smallest parts function. In some cases we are able to deduce inequalities among the rank-like functions. We also conjecture additional inequalities and a large number of congruences for the higher order smallest parts functions.'
address:
- |
The Pennsylvania State University\
State College, Pennsylvania 16802, USA cmb6625@psu.edu
- |
Department of Mathematics, Oregon State University\
Corvallis, Oregon 97331, USA cjenningsshaffer@ufl.edu
- |
University of Miami\
Coral Gables, Florida 33146, USA geoffreysangston@gmail.com
author:
- CATHERINE BABECKI
- 'CHRIS JENNINGS-SHAFFER'
- GEOFFREY SANGSTON
bibliography:
- 'higherOrderSPTBaileyPairsRef.bib'
title: 'Higher Order Smallest Parts Functions and Rank-Crank Moment Inequalities from Bailey Pairs'
---
Introduction
============
The celebrated Rogers-Ramanujan identities state that $$\begin{aligned}
\sum_{n=0}^\infty
\frac{q^{n^2}}{{\left(q\right)_{n}}}
&=
\frac{1}{{\left(q;q^5\right)_{\infty}} {\left(q^4;q^5\right)_{\infty}} }
,&
\sum_{n=0}^\infty
\frac{q^{n^2+n}}{{\left(q\right)_{n}}}
&=
\frac{1}{{\left(q^2;q^5\right)_{\infty}}{\left(q^3;q^5\right)_{\infty}} }
,\end{aligned}$$ where here and throughout the article $$\begin{aligned}
(a)_n
&:=
(a;q)_n = \prod_{j=0}^{n-1}(1-aq^j)
,&
(a)_\infty
&:=
(a;q)_\infty = \prod_{j=0}^{\infty}(1-aq^j)
.\end{aligned}$$ While these identities were stated and proved by Rogers in [@Rogers1], they did not gain attention until they were rediscovered by Ramanujan two decades later. It would be impossible to give here an adequate account of how the Rogers-Ramanujan identities have found their way into various branches of mathematics and related sciences, so we direct the reader to [@Andrews1; @Andrews2; @AndrewsBaxter1; @AndrewsBerndt1; @Askey1; @Berndt1; @GriffinOnoOle1; @LepowskyWilson1]. What we do mention about these identities is that Rogers actually had several more identities of this type, and perhaps most important is that to give uniform proofs of such identities Bailey [@Bailey1; @Bailey2] introduced what would later be known as the Bailey pair machinery. In particular, we recall that a pair of sequences $(\alpha,\beta)$ form a Bailey pair relative to $(a,q)$ if $$\begin{aligned}
\beta_n
&=
\sum_{k=0}^n
\frac{\alpha_k}{{\left(q;q\right)_{n-k}}{\left(aq;q\right)_{n+k}}}
,\end{aligned}$$ and a limiting form of Bailey’s Lemma is that $$\begin{aligned}
\sum_{n=0}^\infty
{\left(x\right)_{n}} {\left(y\right)_{n}}\left(\frac{aq}{xy}\right)^n
\beta_n
&=
\frac{{\left(\frac{aq}{x}\right)_{\infty}} {\left(\frac{aq}{y}\right)_{\infty}}}
{{\left(aq\right)_{\infty}} {\left(\frac{aq}{xy}\right)_{\infty}}}
\sum_{n=0}^\infty
\frac{{\left(x\right)_{n}} {\left(y\right)_{n}} \left(\frac{aq}{xy}\right)^n \alpha_n}
{ {\left(\frac{aq}{x}\right)_{n}} {\left(\frac{aq}{y}\right)_{n}} }
.\end{aligned}$$ By letting $x,y\rightarrow\infty$ in Bailey’s Lemma and plugging in the Bailey pair, relative to $(a,q)$, given by $$\begin{aligned}
\beta_n
&=
\frac{1}{{\left(q\right)_{n}}}
,&
\alpha_n
&=
\frac{(-1)^n (1-aq^{2n}) {\left(a\right)_{n}} a^n q^{\frac{n(3n-1)}{2}} }
{(1-a){\left(q\right)_{n}}}
,\end{aligned}$$ one sees that the right hand side sums to the appropriate products for $a=1$ and $a=q$, according to the Jacobi triple product identity, so that Bailey has given a uniform proof of both Rogers-Ramanujan identities. After Bailey’s success with the above method, Slater [@Slater1; @Slater2] demonstrated the incredible power of Bailey pairs by giving over 100 identities of the Rogers-Ramanujan type by further introducing Bailey pairs where an appropriate choice of $x$ and $y$ would allow the right hand side of Bailey’s Lemma to sum to an infinite product.
Besides identities of the Rogers-Ramanujan type, Bailey pairs have found numerous uses in the study of $q$-series and integer partitions. We discuss two recent uses related to counting the number of smallest parts in integer partitions. For this we recall that a partition of a positive integer $n$ is a non-increasing sequence of positive integers that sum to $n$; we agree that there is a single partition of $0$, which is the empty partition. We let $p(n)$ denote the number of partitions of $n$. As an example, we see $p(5)=7$ as the seven partitions of $5$ are given by $5$, $4+1$, $3+2$, $3+1+1$, $2+2+1$, $2+1+1+1$, and $1+1+1+1+1$. The rank of a partition is given as the largest part minus the number of parts. The crank of a partition is defined as the largest part, if the partition does not contain any ones, and otherwise is the number of parts larger than the number of ones minus the number of ones. Of the partitions of $5$ listed previously we see their respective ranks are $4$, $2$, $1$, $0$, $-1$, $-2$, and $-4$, whereas their respective cranks are $5$, $0$, $3$, $-1$, $1$, $-3$, and $-5$. We let $N(m,n)$ denote the number of partitions of $n$ with rank $m$ and let $M(m,n)$ denote the number of partitions of $n$ with crank $m$. We call the respective generating functions $R(z,q)$ and $C(z,q)$, that is to say, $$\begin{aligned}
R(z,q)
&=
\sum_{n=0}^\infty \sum_{m=-\infty}^\infty
N(m,n)z^mq^n
,&
C(z,q)
&=
\sum_{n=0}^\infty \sum_{m=-\infty}^\infty
M(m,n)z^mq^n
. \end{aligned}$$
In [@Andrews3], Andrews introduced the function ${\mathrm{spt}\left(n\right) }$, which counts the total number of appearances of the smallest part in each partition of $n$. We can think of this as a weighted count on the partitions of $n$, where the weight is the number of times the smallest part appears. From the partitions of $5$ listed above, we see that ${\mathrm{spt}\left(5\right) }=14$. Given a relation with the second moment of the rank function, $\left(z\frac{\partial}{\partial z}\right)^2 R(z,q) \big|_{z=1}$, Andrews proved that ${\mathrm{spt}\left(5n+4\right) }\equiv 0\pmod{5}$, ${\mathrm{spt}\left(7n+5\right) }\equiv 0\pmod{7}$, and ${\mathrm{spt}\left(13n+6\right) }\equiv 0\pmod{13}$. These congruences are reminiscent of Ramanujan’s congruences for the partition function $p(5n+4)\equiv 0\pmod{5}$, $p(7n+5)\equiv 0\pmod{7}$, and $p(11n+6)\equiv 0\pmod{11}$. One can easily verify that a generating function for ${\mathrm{spt}\left(n\right) }$ is given by $$\begin{aligned}
S(q)
&=
\sum_{n=0}^\infty
{\mathrm{spt}\left(n\right) }q^n
=
\sum_{n=1}^\infty
\frac{q^n}{(1-q^n)^2 {\left(q^{n+1}\right)_{\infty}}}
.\end{aligned}$$ With this in mind Andrews, Garvan, and Liang [@AndrewsGarvanLiang1] introduced a so called spt-crank as the series $$\begin{aligned}
S(z,q)
&=
\sum_{n=0}^\infty
\sum_{m=-\infty}^\infty
N_{S}(m,n)z^mq^n
=
\sum_{n=1}^\infty
\frac{q^n {\left(q^{n+1}\right)_{\infty}} }
{ {\left(zq^n\right)_{\infty}}{\left(z^{-1}q^n\right)_{\infty}} }
.\end{aligned}$$ It is trivial to see that $S(1,q)=S(q)$. Three facts that are not so trivial to prove are that each $N_{S}(m,n)$ is a non-negative integer, the $5$-dissection of $S(e^{2\pi i/5},q)$ gives another proof of the congruence ${\mathrm{spt}\left(5n+4\right) }\equiv 0\pmod{5}$, and the $7$-dissection of $S(e^{2\pi i/7},q)$ gives another proof of the congruence ${\mathrm{spt}\left(7n+5\right) }\equiv 0\pmod{7}$. The essential trick for the dissections is to take the following Bailey pair relative to $(1,q)$, $$\begin{aligned}
\beta_n
&=
\frac{1}{{\left(q\right)_{n}}}
,&
\alpha_n
&=
\left\{\begin{array}{ll}
1 & \mbox{ if } n=0,
\\
(-1)^n (1+q^{n}) q^{\frac{n(3n-1)}{2}} &\mbox{ if } n\ge 1,
\end{array}\right.\end{aligned}$$ and apply Bailey’s Lemma with $x=z$ and $y=z^{-1}$ to deduce that $$\begin{aligned}
(1-z)(1-z^{-1})S(z,q)
&=
R(z,q)-C(z,q)
.\end{aligned}$$ Inspired by this identity, Garvan and the second author [@GarvanJenningsShaffer1] investigated smallest parts functions related to overpartitions and partitions without repeated odd parts in terms of spt-cranks and differences between ranks and cranks. Again the essential trick was to apply Bailey’s Lemma with $x=z$ and $y=z^{-1}$, but with a different Bailey pair. In particular, there we used the Bailey pairs $$\begin{aligned}
\beta_n
&=
\frac{1}{{\left(q^2;q^2\right)_{n}}}
,&
\alpha_n
&=
\left\{\begin{array}{ll}
1 & \mbox{ if } n=0,
\\
(-1)^n 2 q^{n^2} &\mbox{ if } n\ge 1,
\end{array}\right.
\\
\beta_n
&=
\frac{1}{2}
\left(\frac{1}{{\left(q^2;q^2\right)_{n}}}+\frac{(-1)^n}{{\left(q^2;q^2\right)_{n}}}\right)
,&
\alpha_n
&=
\left\{\begin{array}{ll}
1 & \mbox{ if } n=0,
\\
(-1)^n (1+q^{n^2}) &\mbox{ if } n\ge 1,
\end{array}\right.
\\
\beta_n
&=
\frac{1}{{\left(-q,q^2;q^2\right)_{n}}}
,&
\alpha_n
&=
\left\{\begin{array}{ll}
1 & \mbox{ if } n=0,
\\
(-1)^n 2 q^{2n^2-n}(1+q^{2n}) &\mbox{ if } n\ge 1,
\end{array}\right.\end{aligned}$$ of which the first two are relative to $(1,q)$ and the third is relative to $(1,q^2)$. Here we note that the same form of Bailey’s Lemma gave four identities for spt-crank functions. It is then natural to ask what would happen if one was to look at all of Slater’s Bailey pairs in this framework of spt-cranks; Garvan and the second author carried this out in a series of articles [@GarvanJenningsShaffer2; @JenningsShaffer1; @JenningsShaffer2]. This can be compared to Slater’s work, but instead of choosing Bailey pairs that would result in a series that sums to a product by the Jacobi triple product identity, we needed to choose Bailey pairs with $a=1$ and such that the associated spt-crank-type function dissected nicely at roots of unity. Altogether this process resulted in over 20 spt-crank-type functions and associated spt-type functions with congruences.
Another use of Bailey’s Lemma related to smallest parts functions arose in [@Garvan1], where Garvan considered the ordinary and symmetrized moments of the rank and crank functions, $$\begin{aligned}
N_k(n)
&=
\sum_{m=-\infty}^{\infty}m^k N(m,n)
,&
\eta_{k}(n)
&=
\sum_{m = -\infty}^{\infty}
\binom{m+\lfloor\frac{k-1}{2}\rfloor}{k}N(m,n)
,\\
M_k(n)
&=
\sum_{m=-\infty}^{\infty}m^kM(m,n)
,&
\mu_{k}(n)
&=
\sum_{m = -\infty}^{\infty}
\binom{m+\lfloor\frac{k-1}{2}\rfloor}{k}M(m,n)
.\end{aligned}$$ Studies of the ordinary moments of the rank and crank function began with the work of Atkin and Garvan in [@AtkinGarvan1], and Andrews introduced $\eta_{2k}$, the symmetrized moment of the rank, in [@Andrews4]. Previous to Garvan’s article, it was conjectured that $M_{2k}(n)>N_{2k}(n)$ for all positive $k$ and $n$ (here only the even moments are of interest as the odd moments are zero). By asymptotics this was known to hold for sufficiently large $n$ for each $k$ [@BringmannMahlburgRhoades1]. Among Garvan’s results, we highlight three. The first is the following form of the generating function of $\mu_{2k}(n)-\eta_{2k}(n)$, $$\begin{aligned}
\sum_{n=1}^\infty
\left(\mu_{2k}(n)-\eta_{2k}(n)\right)q^n
&=
\sum_{n_k\ge\dotsb\ge n_1\ge 1}
\frac{q^{n_1+\dotsb+n_k}}{{\left(q^{n_1+1}\right)_{\infty}}(1-q^{n_1})^2\dotsm(1-q^{n_k})^2}
,\end{aligned}$$ which clearly exhibits that $\mu_{2k}(n)\ge \eta_{2k}(n)$ for all $k$ and $n$, and additionally one can easily determine when the inequality is strict. The second is a formula for writing the ordinary moments as a positive integer linear combination of the symmetrized moments, and in particular $M_{2k}(n)-N_{2k}(n) \ge \mu_{2}(n)-\eta_{2}(n)$. The last is a family of weighted counts of the partitions of $n$, ${\mathrm{spt}^{}_{k}\hspace{-.2em} \left(n\right) }$, the higher order smallest parts function, such that the weighting is clearly non-negative and based on the frequency of the parts of the partitions, ${\mathrm{spt}^{}_{1}\hspace{-.2em} \left(n\right) }={\mathrm{spt}\left(n\right) }$, and ${\mathrm{spt}^{}_{k}\hspace{-.2em} \left(n\right) }=\mu_{2k}(n)-\eta_{2k}(n)$. Additionally Garvan established a large number of congruences for ${\mathrm{spt}^{}_{k}\hspace{-.2em} \left(n\right) }$. The linchpin in establishing the generating function for $\mu_{2k}(n)-\eta_{2k}(n)$ was again a certain form of Bailey’s Lemma applied to the Bailey pair $$\begin{aligned}
\beta_n
&=
\frac{1}{{\left(q\right)_{n}}}
,&
\alpha_n
&=
\left\{\begin{array}{ll}
1 & \mbox{ if } n=0,
\\
(-1)^n (1+q^{n}) q^{\frac{n(3n-1)}{2}} &\mbox{ if } n\ge 1.
\end{array}\right.\end{aligned}$$ Inspired by Garvan’s results, in [@JenningsShaffer3] the second author carried out the same study of differences and inequalities between rank and crank moments related to overpartitions and partitions without repeated odd parts. The upshot was that one needed only choose different Bailey pairs compared to Garvan’s work.
It is now obvious what we are to do next. We are to consider the framework developed by Garvan in [@Garvan1], but applied to all applicable Bailey pairs of Slater. Again this compares with Slater’s original use of Bailey pairs. As we will see shortly, the requirements for our choice of Bailey pairs are just that $a=1$; $\alpha_0=\beta_0=1$; formulaically $\alpha_{-n}=\alpha_n$; and after multiplying by an infinite product, of our own choice, it is clear that $\beta_n$ has non-negative coefficients. Our results will mirror that of Garvan’s study of rank and crank moments. For each Bailey pair considered, we will introduce a rank and crank-like function, obtain a generating function for the difference of symmetrized moments which clearly exhibits non-negative coefficients, deduce an inequality for the associated ordinary moments, and then give a weighted count of partitions that agrees with the difference of the symmetrized moments. To demonstrate that this can be applied to Bailey pairs past those in Slater’s list, we also consider one Bailey pair from [@BowmannMclaughlinSills1]. Altogether we will give 28 instances of this process.
The rest of the article is organized as follows. In Section 2 we give our definitions and main results, which are series identities and inequalities. In Section 3 we prove the series identities and inequalities listed in Section 2. In Section 4 we prove the combinatorial interpretation of the symmetrized rank and crank moment differences, which justifies our definitions of higher order spt functions. In Section 5 we end with a few conjectures and remarks.
Definitions and Statement of Main Results
=========================================
For our series identities and inequalities, we need a small number of general identities, all of which are straightforward to prove. These identities have their origins in a combination of classical works on the rank function as well as [@Andrews4; @Garvan1; @JenningsShaffer3], however here we state and prove them in generality. We combine the main identities into the single following theorem. Given that the proofs primarily already exist in the literature, we will find it takes far longer to state our results than to prove them.
\[TheoremMain\] Suppose $(\alpha,\beta)$ is a Bailey pair relative to $(1,q)$, $\alpha_0=\beta_0=1$, and $\alpha_n=\alpha_{-n}$. Here we note that by $\alpha_n=\alpha_{-n}$, we are treating $\alpha_n$ as a bilateral sequence by extending $\alpha_n$ to negative indices according to whatever general formula is given for $\alpha_n$. Suppose $$\begin{aligned}
R_X(z,q)
:=
P_{X}(q)
\left(
1
+
\sum_{n = 1}^{\infty} \frac{\alpha_{n}q^n(1-z)(1-z^{-1})}{(1-zq^n)(1-z^{-1}q^n)}
\right)
&=
\sum_{n=0}^\infty \sum_{m=-\infty}^\infty
N_{X}(m,n) z^mq^n
,\end{aligned}$$ where $P_{X}(q)$ is a series in $q$, and for $k$ a positive integer let $$\begin{aligned}
N^{X}_{k}(n)
&=
\sum_{m = -\infty}^{\infty}
m^kN_{X}(m,n)
,&
\eta^{X}_{k}(n)
&=
\sum_{m = -\infty}^{\infty}
\binom{m+\lfloor\frac{k-1}{2}\rfloor}{k}N_{X}(m,n)
.\end{aligned}$$ Then $$\begin{aligned}
\label{EqMainTheoremSymmetrizedRankMoment}
\sum_{n=1}^\infty \eta^X_{2k}(n) q^n
&=
-P_X(q)
\sum_{n=1}^\infty
\frac{\alpha_{n}q^{nk}}{(1-q^n)^{2k}}
,\end{aligned}$$ and $$\begin{aligned}
\label{EqMainTheoremSymmetrizexRankCrankDifference}
\sum_{n=1}^\infty
{\mathrm{spt}^{X}_{k}\hspace{-.2em} \left(n\right) }q^n
&:=
P_X(q){\left(q\right)_{\infty}}
\sum_{n=1}^\infty \mu_{2k}(n)q^n
-
\sum_{n=1}^\infty \eta^{X}_{2k}(n)q^n
\nonumber\\
&=
P_X(q)
\sum_{n_k\ge \dotsb \ge n_1\ge 1 }
\frac{{\left(q\right)_{n_1}}^2 \beta_{n_1} q^{n_1+\dotsb +n_k} }
{(1-q^{n_1})^2\dotsm(1-q^{n_k})^2}
.\end{aligned}$$
Furthermore, $N^X_{k}(n)$ and $\eta^{X}_{k}(n)$ are zero if $k$ is odd, and the moments for even $k$ are related by the identities $$\begin{aligned}
\label{EqMainTheoremSymmetrizedRankEquality}
\eta^{X}_{2k}(n)
&=
\frac{1}{(2k)!}\sum_{m=-\infty}^{\infty}g_{k}(m)N_{X}(m,n)
,\\
\label{EqMainTheoremOrdinaryRankEquality}
N^{X}_{2k}(n)
&=
\sum_{j=1}^{k}(2j)!S^*(k,j)\eta^{X}_{2j}(n)
,\end{aligned}$$ where $g_k(x)=\prod_{j=0}^{k-1}(x^2-j^2)$, and the sequence $S^*(k,j)$ is defined recursively by $S^*(k+1,j)=S^*(k,j-1)+j^2S^*(k,j)$ and boundary conditions $S^*(1,1)=1$ and $S^*(k,j)=0 $ if $j\leq 0$ or $j>k$.
We first demonstrate the use of this theorem with a specific Bailey pair. Consider the Bailey pair B(2) of [@Slater1], $$\begin{aligned}
\beta_n
&=
\frac{q^n}{(q)_n}
,&
\alpha_n
&=
\begin{cases}
1 & n=0
,\\
(-1)^nq^{\frac{3n(n-1)}{2}}(1+q^{3n}) & n\geq 1
.\\
\end{cases}\end{aligned}$$ We note that using this formula with negative $n$ does give that $\alpha_{n} = \alpha_{-n}$. Looking to Theorem \[TheoremMain\], if ${\mathrm{spt}^{X}_{k}\hspace{-.2em} \left(n\right) }$ is to be a non-negative integer, then we should choose $P_X(q)=\frac{1}{{\left(q\right)_{\infty}}}$. From this we now know we should define a rank-like function by $$\begin{aligned}
R_{B2}(z,q)
&=
\sum_{n=0}^\infty \sum_{m=-\infty}^\infty
N_{B2}(m,n)z^mq^n
=
\frac{1}{{\left(q;q\right)_{\infty}}}
\left(
1
+
\sum\limits_{n=1}^\infty
\frac{(1-z)(1-z^{-1})(-1)^{n}q^{\frac{n(3n-1)}{2}}(1+q^{3n})}{(1-zq^n)(1-z^{-1}q^n)}
\right)
,\end{aligned}$$ and we have the associated ordinary and symmetrized moments given by $$\begin{aligned}
N^{B2}_{k}(n)
&=
\sum_{m = -\infty}^{\infty}
m^kN_{B2}(m,n)
,&
\eta^{B2}_{k}(n)
&=
\sum_{m = -\infty}^{\infty}
\binom{m+\lfloor\frac{k-1}{2}\rfloor}{k}N_{B2}(m,n)
.\end{aligned}$$ But then $$\begin{aligned}
\sum_{n=1}\left(\mu_{2k}(n)-\eta^{B2}_{2k}(n)\right)q^n
&=
\sum_{n_k\ge \dotsb \ge n_1\ge 1 }
\frac{q^{2n_1+n_2+\dotsb +n_k} }
{{\left(q^{n_1+1}\right)_{\infty}}(1-q^{n_1})^2\dotsm(1-q^{n_k})^2}
,\end{aligned}$$ and clearly $\mu_{2k}(n)\ge \eta^{B2}_{2k}(n)$ for positive $k$ and $n$. Additionally by taking $k=1$ and examining the $n_1=1$ summand, $$\begin{aligned}
\frac{q^2}{{\left(q^2\right)_{\infty}}(1-q)^2}
&=
q^2 + 2q^3+ 4q^4 + 7q^5 + 12q^6 + 19q^7 + 30q^8 + \dotsb
, \end{aligned}$$ we see that $\mu_{2}(n)> \eta^{B2}_{2}(n)$ for $n\ge 2$. To obtain an inequality for the ordinary moments, we make the following observation. The $S^*(k,j)$ are non-negative integers and $S^*(k,1)$ is positive for $k\ge 1$. In particular, we then have that $$\begin{aligned}
M_{2k}(n)-N^{B2}_{2k}(n)
&=
\sum_{j=1}^{k}(2j)!S^*(k,j)( \mu_{2j}(n)-\eta^{B2}_{2j}(n))
\ge
\mu_{2}(n)-\eta^{B2}_{2}(n)
.\end{aligned}$$ Thus $M_{2k}(n)>N^{B2}_{2k}(n)$ for all positive $k$ and $n\ge 2$. Furthermore, we can ask what is the non-negative integer ${\mathrm{spt}^{B2}_{k}\hspace{-.2em} \left(n\right) }:= \mu_{2k}(n)- \eta^{B2}_{2k}(n)$ counting in terms of partitions? This is the question we address in Section 4.
We now repeat this process with the many relevant Bailey pairs from [@Slater1; @Slater2], and tabulate our results in a corollary. In the cases where a Bailey pair would have fractional powers of $q$, we replace $q$ with the appropriate power of $q$. It is worth noting that the Bailey pairs $B1$, $E1$, and the unlabeled Bailey pair on page 468 of [@Slater1] with $\beta_n=\frac{1}{(-q^{1/2})_{n}{\left(q\right)_{n}}}$, (which is also $F1$ with $q^{1/2}\mapsto-q^{1/2}$) correspond respectively to the ordinary rank studied by Garvan in [@Garvan1] and the Dyson rank for overpartitions and the M2-rank for partitions without repeated odd parts studied by the second author studied in [@JenningsShaffer3]. The M2-rank for overpartitions corresponds to a Bailey pair from a specialization of a finite form of the Jacobi triple product identity. As such, we omit these Bailey pairs from our consideration. In labeling our Bailey pairs, we use the existing label in the literature when it exists, otherwise we label the Bailey pairs relative to $(1,q)$ as $X1$, $X2$, $X3$, $X4$, $X5$, $X6$ and the Bailey pairs relative to $(1,q^2)$ as $Y1$, $Y2$, $Y3$, $Y4$. This labeling is not meant to carry any additional semantic value.
To state our corollary, we first introduce the relevant crank-like functions that will appear. In the cases of a crank that has appeared before in the literature, we follow the existing naming conventions. We let $$\begin{aligned}
C(z,q)
&=
\frac{{\left(q\right)_{\infty}}}{{\left(zq\right)_{\infty}}{\left(z^{-1}q\right)_{\infty}}}
=
\sum_{n=0}^\infty\sum_{m=-\infty}^\infty
M(m,n)z^mq^n
,\\
\overline{C}(z,q)
&=
\frac{{\left(-q\right)_{\infty}}{\left(q\right)_{\infty}}}{{\left(zq\right)_{\infty}}{\left(z^{-1}q\right)_{\infty}}}
=
\sum_{n=0}^\infty\sum_{m=-\infty}^\infty
\overline{M}(m,n)z^mq^n
,\\
C^J(z,q)
&=
\frac{{\left(q\right)_{\infty}}}{{\left(q^3;q^3\right)_{\infty}}{\left(zq\right)_{\infty}}{\left(z^{-1}q\right)_{\infty}}}
=
\sum_{n=0}^\infty\sum_{m=-\infty}^\infty
M^J(m,n)z^mq^n
,\\
C^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}(z,q)
&=
\frac{{\left(q\right)_{\infty}}}
{{\left(q^2;q^2\right)_{\infty}}{\left(zq\right)_{\infty}}{\left(z^{-1}q\right)_{\infty}}}
=
\sum_{n=0}^\infty\sum_{m=-\infty}^\infty
M^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}(m,n)z^mq^n
,\\
C^F(z,q)
&=
\frac{{\left(q^2;q^2\right)_{\infty}}}{{\left(zq^2;q^2\right)_{\infty}}{\left(z^{-1}q^2;q^2\right)_{\infty}}}
=
\sum_{n=0}^\infty\sum_{m=-\infty}^\infty
M^F(m,n)z^mq^n
,\\
C^G(z,q)
&=
\frac{{\left(-q;q^2\right)_{\infty}}}
{{\left(zq^2;q^2\right)_{\infty}}{\left(z^{-1}q^2;q^2\right)_{\infty}}}
=
\sum_{n=0}^\infty\sum_{m=-\infty}^\infty
M^G(m,n)z^mq^n
,\\
C^Y(z,q)
&=
\frac{1}
{{\left(zq^2;q^2\right)_{\infty}}{\left(z^{-1}q^2;q^2\right)_{\infty}}}
=
\sum_{n=0}^\infty\sum_{m=-\infty}^\infty
M^Y(m,n)z^mq^n
,\\
C^{L2}(z,q)
&=
\frac{{\left(-q\right)_{\infty}}{\left(q^4;q^4\right)_{\infty}}}
{{\left(zq^4;q^4\right)_{\infty}}{\left(z^{-1}q^4;q^4\right)_{\infty}}}
=
\sum_{n=0}^\infty\sum_{m=-\infty}^\infty
M^{L2}(m,n)z^mq^n
.\end{aligned}$$ We note that $C(z,q)$ is the ordinary crank of partitions, the moments of which Garvan studied in [@Garvan1], and the function $\overline{C}(z,q)$ is known as the (first residual) crank of overpartitions [@BringmannLovejoyOsburn1]. Since all of these functions are directly related to the ordinary crank, upon defining the moments in the obvious way, which we omit, based on [@Garvan1] we have the following, $$\begin{aligned}
\sum_{n=1}^\infty \mu_{2k}(n)q^n
&=
\frac{1}{(q)_{\infty}}
\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}q^{\frac{n(n-1)}{2}+kn}(1+q^n)}{(1-q^n)^{2k}}
,\\
\sum_{n=1}^\infty \overline{\mu}_{2k}(n)q^n
&=
\frac{(-q)_{\infty}}{(q)_\infty}
\sum_{n=1}^{\infty}
\frac{(-1)^{n+1}q^{\frac{n(n-1)}{2}+kn}(1+q^n)}{(1-q^n)^{2k}}
,\\
\sum_{n=1}^\infty \mu^{J}_{2k}(n)q^n
&=
\frac{1}{(q^3;q^3)_{\infty}(q)_\infty}
\sum_{n=1}^{\infty}
\frac{(-1)^{n+1}q^{\frac{n(n-1)}{2}+kn}(1+q^n)}{(1-q^n)^{2k}}
,\\
\sum_{n=1}^\infty \mu^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}_{2k}(n)q^n
&=
\frac{1}{(q)_\infty(q^2;q^2)_{\infty}}
\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}q^{\frac{n(n-1)}{2}+kn}(1+q^n)}{(1-q^n)^{2k}}
,\\
\sum_{n=1}^\infty \mu^{F}_{2k}(n)q^n
&=
\frac{1}{(q^2;q^2)_{\infty}}
\sum_{n=1}^{\infty}
\frac{(-1)^{n+1}q^{n(n-1)+2kn}(1+q^{2n})}{(1-q^{2n})^{2k}}
,\\
\sum_{n=1}^\infty \mu^{G}_{2k}(n)q^n
&=
\frac{(-q;q^2)_\infty}{(q^2;q^2)^2_\infty}
\sum\limits_{n=1}^\infty \frac{(-1)^{n+1}q^{n(n-1)+2kn}(1+q^{2n})}{(1-q^{2n})^{2k}}
,\\
\sum_{n=1}^\infty \mu^{Y}_{2k}(n)q^n
&=
\frac{1}{(q^2;q^2)^2_\infty}
\sum\limits_{n=1}^\infty \frac{(-1)^{n+1}q^{n(n-1)+2kn}(1+q^{2n})}{(1-q^{2n})^{2k}}
,\\
\sum_{n=1}^\infty \mu^{L2}_{2k}(n)q^n
&=
\frac{(-q)_\infty}{(q^4;q^4)_\infty}
\sum_{n=1}^\infty \frac{(-1)^{n+1}q^{2n(n-1)+4kn}(1+q^{4n})}{(1-q^{4n})^{2k}}
.\end{aligned}$$ We find that the ordinary and symmetrized moments satisfy the same relation as for the rank-like functions. In particular, $$\begin{aligned}
M^X_{2k}(n)
&=
\sum_{j=1}^{k}(2j)!S^*(k,j)\mu^X_{2j}(n).\end{aligned}$$ We note that if one wishes to actually compute $\mu^{X}_{2k}(n)-\eta^{X}_{2k}(n)$ numerically, one should do so with the above representation for $\mu^{X}_{2k}(n)$ and that of $\eta^{X}_{2k}(n)$ in (\[EqMainTheoremSymmetrizedRankMoment\]), rather than by (\[EqMainTheoremSymmetrizexRankCrankDifference\]).
\[CorollarySeriesIdentities\] (1) Using the Bailey pair A(1) from [@Slater1], relative to $(1,q)$, $$\begin{aligned}
\beta_n
&=
\frac{1}{(q)_{2n}}
,&
\alpha_n
&=
\begin{cases}
1 & n=0,
\\
-q^{6k^2-5k+1} & n= 3k-1,
\\
q^{6k^2-k}+q^{6k^2+k} &n= 3k,
\\
-q^{6k^2+5k+1} & n=3k+1,
\end{cases} \end{aligned}$$ we define $$\begin{aligned}
R_{A1}(z,q)
&=
\frac{1}{{\left(q\right)_{\infty}}}
\left(
1
-
\sum\limits_{n=1}^\infty
\frac{(1-z)(1-z^{-1})q^{6n^2 - 2n}}{(1-zq^{3n-1})(1-z^{-1}q^{3n-1})}
-
\sum\limits_{n=0}^\infty
\frac{(1-z)(1-z^{-1})q^{6n^2+8n + 2}}{(1-zq^{3n+1})(1-z^{-1}q^{3n+1})}
\right.\\&\left.\quad
+
\sum\limits_{n=1}^\infty
\frac{(1-z)(1-z^{-1})q^{6n^2+2n}(1+q^{2n}) }{(1-zq^{3n})(1-z^{-1}q^{3n})}
\right)
,\end{aligned}$$ and obtain $$\begin{aligned}
&\sum\limits_{n=1}^{\infty} \eta^{A1}_{2k}(n)q^{n}
=
\frac{1}{(q)_{\infty}}
\left(
\sum\limits_{n=1}^\infty \frac{q^{6n^2-5n+1+(3n-1)k}}{(1-q^{3n-1})^{2k}}
+
\sum\limits_{n=0}^\infty \frac{q^{6n^2+5n+1+(3n+1)k}}{(1-q^{3n+1})^{2k}}
-
\sum\limits_{n=1}^\infty \frac{q^{6n^2-n+3nk}(1+q^{2n})}{(1-q^{3n})^{2k}}
\right)
,\\
&\sum_{n = 1}^{\infty} {\mathrm{spt}^{A1}_{k}\hspace{-.2em} \left(n\right) } q^{n}
:=
\sum_{n = 1}^{\infty}\left(\mu_{2k}(n)-\eta_{2k}^{A1}(n)\right)q^{n}
=
\sum\limits_{n_k \geq \cdots \geq n_1 \geq 1}
\frac{q^{n_1+\cdots{}+n_k}}
{(q^{n_1+1})_{n_1}(q^{n_1+1})_{\infty}(1-q^{n_k})^{2}\cdots{}(1-q^{n_1})^2}
. \end{aligned}$$
\(2) Using the Bailey pair A(3) from [@Slater1], relative to $(1,q)$, $$\begin{aligned}
\beta_n
&=
\frac{q^n}{(q)_{2n}}
,&
\alpha_n
&=
\begin{cases}
1 & n=0,
\\
-q^{6k^2-2k} & n= 3k-1,
\\
q^{6k^2-2k}+q^{6k^2+2k} & n= 3k,
\\
-q^{6k^2+2k} & n=3k+1,
\\
\end{cases} \end{aligned}$$ we define $$\begin{aligned}
R_{A3}(z,q)
&=
\frac{1}{{\left(q\right)_{\infty}}}
\Bigg(
1
-
\sum\limits_{n=1}^\infty
\frac{(1-z)(1-z^{-1})q^{6n^2+n-1}}{(1-zq^{3n-1})(1-z^{-1}q^{3n-1})}
-
\sum\limits_{n=0}^\infty
\frac{(1-z)(1-z^{-1})q^{6n^2+5n+1}}{(1-zq^{3n+1})(1-z^{-1}q^{3n+1})}
\\&\quad
+
\sum\limits_{n=1}^\infty
\frac{(1-z)(1-z^{-1})q^{6n^2+n}(1+q^{4n})}{(1-zq^{3n})(1-z^{-1}q^{3n})}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
&\sum\limits_{n=1}^\infty \eta^{A3}_{2k}(n)q^n
=
\frac{1}{(q)_{\infty}}
\left(
\sum\limits_{n=1}^\infty \frac{q^{6n^2-2n+(3n-1)k}}{(1-q^{3n-1})^{2k}}
+
\sum\limits_{n=0}^\infty \frac{q^{6n^2+2n+(3n+1)k}}{(1-q^{3n+1})^{2k}}
-
\sum\limits_{n=1}^\infty \frac{q^{6n^2-2n+3nk}(1+q^{4n})}{(1-q^{3n})^{2k}}
\right)
,\\
&\sum_{n = 1}^{\infty} {\mathrm{spt}^{A3}_{k}\hspace{-.2em} \left(n\right) }q^{n}
:=
\sum_{n = 1}^{\infty}\left(\mu_{2k}(n) - \eta_{2k}^{A3}(n)\right)q^{n}
=
\sum\limits_{n_k \geq \dotsc \geq n_1 \geq 1}
\frac{q^{2n_1+n_2+\dotsb+n_k}}
{(q^{n_1+1})_{n_1}(q^{n_1+1})_{\infty}(1-q^{n_k})^2\dotsm(1-q^{n_1})^2}
.\end{aligned}$$
\(3) Using the Bailey pair A(5) from [@Slater1], relative to $(1,q)$, $$\begin{aligned}
\beta_n
&=
\frac{q^{n^2}}{(q)_{2n}}
,&
\alpha_n
&=
\begin{cases}
1 & n=0
,\\
-q^{3k^2-k} & n= 3k-1
,\\
q^{3k^2-k}+q^{3k^2+k} & n= 3k
,\\
-q^{3k^2+k} & n=3k+1
,\\
\end{cases} \end{aligned}$$ we define $$\begin{aligned}
R_{A5}(z,q)
&=
\frac{1}{{\left(q\right)_{\infty}}}
\Bigg(
1
-
\sum\limits_{n=1}^\infty
\frac{(1-z)(1-z^{-1})q^{3n^2+2n-1}}{(1-zq^{3n-1})(1-z^{-1}q^{3n-1})}
-
\sum\limits_{n=0}^\infty
\frac{(1-z)(1-z^{-1})q^{3n^2+4n+1}}{(1-zq^{3n+1})(1-z^{-1}q^{3n+1})}
\\&\quad
+
\sum\limits_{n=1}^\infty
\frac{(1-z)(1-z^{-1})q^{3n^2+2n}(1+q^{2n})}{(1-zq^{3n})(1-z^{-1}q^{3n})}
\Bigg)
, \end{aligned}$$ and obtain $$\begin{aligned}
&\sum\limits_{n=1}^\infty \eta^{A5}_{2k}(n)q^n
=
\frac{1}{(q)_{\infty}}
\left(
\sum\limits_{n=1}^\infty \frac{q^{3n^2-n+(3n-1)k}}{(1-q^{3n-1})^{2k}}
+
\sum\limits_{n=0}^\infty \frac{q^{3n^2+n+(3n+1)k}}{(1-q^{3n+1})^{2k}}
-
\sum\limits_{n=1}^\infty \frac{q^{3n^2-n+3nk}(1+q^{2n})}{(1-q^{3n})^{2k}}
\right)
,\\
&
\sum_{n = 1}^{\infty} {\mathrm{spt}^{A5}_{k}\hspace{-.2em} \left(n\right) }q^{n}
:=
\sum_{n = 1}^{\infty}\left(\mu_{2k}(n)-\eta_{2k}^{A5}(n)\right)q^{n}
=
\sum\limits_{n_k \geq \dotsb \geq n_1 \geq 1}
\frac{q^{n_1^2+n_1+n_2+\dotsb+n_k}}
{(q^{n_1+1})_{n_1}(q^{n_1+1})_{\infty}(1-q^{n_k})^2\dotsm(1-q^{n_1})^2}
.\end{aligned}$$
\(4) Using the Bailey pair A(7) from [@Slater1], relative to $(1,q)$, $$\begin{aligned}
\beta_n
&=
\frac{q^{n^2-n}}{(q)_{2n}}
,&
\alpha_n
&=
\begin{cases}
1 & n=0, \\
-q^{3k^2-4k+1} & n=3k-1, \\
q^{3k^2-2k}+q^{3k^2+2k} & n=3k, \\
-q^{3k^2+4k+1} & n=3k+1,\\
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{A7}(z,q)
&=
\frac{1}{{\left(q\right)_{\infty}}}
\Bigg(
1
-\sum\limits_{n=1}^\infty
\frac{(1-z)(1-z^{-1})q^{3n^2-n}}{(1-zq^{3n-1})(1-z^{-1}q^{3n-1})}
-
\sum\limits_{n=0}^\infty
\frac{(1-z)(1-z^{-1})q^{3n^2+7n+2}}{(1-zq^{3n+1})(1-z^{-1}q^{3n+1})}
\\&\quad
+
\sum\limits_{n=1}^\infty
\frac{(1-z)(1-z^{-1})q^{3n^2+n}(1+q^{4n})}{(1-zq^{3n})(1-z^{-1}q^{3n})}
\Bigg)
, \end{aligned}$$ and obtain $$\begin{aligned}
&\sum\limits_{n=1}^\infty \eta^{A7}_{2k}(n)q^n
=
\frac{1}{(q)_{\infty}}
\left(
\sum\limits_{n=1}^\infty \frac{q^{3n^2-4n+1+(3n-1)k}}{(1-q^{3n-1})^{2k}}
+
\sum\limits_{n=0}^\infty \frac{q^{3n^2+4n+1+(3n+1)k}}{(1-q^{3n+1})^{2k}}
-
\sum\limits_{n=1}^\infty \frac{q^{3n^2-2n+3nk}(1+q^{4n})}{(1-q^{3n})^{2k}}
\right)
,\\
&
\sum_{n = 1}^{\infty} {\mathrm{spt}^{A7}_{k}\hspace{-.2em} \left(n\right) }q^{n}
:=
\sum_{n = 1}^{\infty}\left(\mu_{2k}(n)-\eta_{2k}^{A7}(n)\right)q^{n}
=
\sum\limits_{n_k \geq \dotsb \geq n_1 \geq 1}
\frac{q^{n_1^2+n_2+\dotsb+n_k}}
{(q^{n_1+1})_{n_1}(q^{n_1+1})_{\infty}(1-q^{n_k})^2\dotsm (1-q^{n_1})^2}
.\end{aligned}$$
\(5) Using the Bailey pair B(2) from [@Slater1], relative to $(1,q)$, $$\begin{aligned}
\beta_n &= \frac{q^n}{(q)_n}
,&
\alpha_n
&=
\begin{cases}
1 & n=0
,\\
(-1)^nq^{\frac{3n(n-1)}{2}}(1+q^{3n}) & n\geq 1
,\\
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{B2}(z,q)
&=
\frac{1}{{\left(q\right)_{\infty}}}
\Bigg(
1
+
\sum\limits_{n=1}^\infty
\frac{(1-z)(1-z^{-1})(-1)^{n}q^{\frac{n(3n-1)}{2}}(1+q^{3n})}{(1-zq^n)(1-z^{-1}q^n)}
\Bigg)
, \end{aligned}$$ and obtain $$\begin{aligned}
&\sum\limits_{n=1}^\infty \eta^{B2}_{2k}(n)q^n
=
\frac{1}{(q)_\infty}
\sum_{n=1}^\infty
\frac{(-1)^{n+1}q^{\frac{3n(n-1)}{2}+nk}(1+q^{3n})}{(1-q^n)^{2k}}
,\\
&
\sum_{n = 1}^{\infty} {\mathrm{spt}^{B2}_{k}\hspace{-.2em} \left(n\right) }q^{n}
:=
\sum_{n = 1}^{\infty}\left(\mu_{2k}(n) -\eta_{2k}^{B2}(n)\right)q^{n}
=
\sum\limits_{n_k \geq \dotsb \geq n_1 \geq 1}
\frac{q^{2n_1+n_2+\dotsb+n_k}}
{(q^{n_1+1})_{\infty}(1-q^{n_k})^2\dotsm (1-q^{n_1})^2}
.\end{aligned}$$
\(6) Using the Bailey pair C(1) from [@Slater1], which is also L(6) from [@Slater2], relative to $(1,q)$, $$\begin{aligned}
\beta_{n}
&=
\frac{1}{{\left(q;q^2\right)_{n}} {\left(q\right)_{n}}}
,&
\alpha_{n}
&=
\begin{cases}
1 & n=0
,\\
(-1)^{k}q^{3k^2 - k}(1 + q^{2k}) & n = 2k
,\\
0 & n = 2k + 1
,
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{C1}(z,q)
&=
\frac{1}{{\left(q\right)_{\infty}}}
\Bigg(
1
+
\sum_{n = 1}^{\infty}
\frac{(1-z)(1-z^{-1})(-1)^{n}q^{3n^2 + n}(1 + q^{2n})}{(1-zq^{2n})(1-z^{-1}q^{2n})}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
\sum_{n = 1}^{\infty}\eta_{2k}^{C1}(n)q^n
&=
\frac{1}{(q)_\infty}
\sum_{n = 1}^{\infty}\frac{(-1)^{n+1}q^{3n^2 - n + 2nk}(1 + q^{2n})}{(1-q^{2n})^{2k}}
,\\
\sum_{n = 1}^{\infty} {\mathrm{spt}^{C1}_{k}\hspace{-.2em} \left(n\right) }q^{n}
&:=
\sum_{n = 1}^{\infty}\left(\mu_{2k}(n)-\eta_{2k}^{C1}(n)\right)q^{n}
=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{q^{n_{1} + \dotsb + n_{k}}}
{(q;q^2)_{n_{1}} (q^{n_{1} + 1})_{\infty} (1-q^{n_{k}})^{2}\dotsm (1-q^{n_{1}})^{2}}
.\end{aligned}$$
\(7) Using the Bailey pair C(2) from [@Slater1], relative to $(1,q)$, $$\begin{aligned}
\beta_{n}
&=
\frac{q^{n}}{{\left(q;q^2\right)_{n}} {\left(q\right)_{n}}}
,&
\alpha_{n}
&=
\begin{cases}
1 & n=0
,\\
(-1)^{k}q^{3k^2 - k}(1 + q^{2k}) & n = 2k
,\\
(-1)^{k+1}q^{3k^2 + k}(1 - q^{4k + 2}) & n = 2k + 1
,
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{C2}(z,q)
&=
\frac{1}{{\left(q\right)_{\infty}}}
\Bigg(
1
+
\sum_{n=1}^{\infty}
\frac{(1-z)(1-z^{-1})(-1)^{n}q^{3n^2 + n}(1 + q^{2n})}{(1-zq^{2n})(1-z^{-1}q^{2n})}
\\&\quad
+
\sum_{n = 0}^{\infty}
\frac{(1-z)(1-z^{-1})(-1)^{n+1}q^{3n^2 + 3n + 1}(1 - q^{4n + 2})}{(1-zq^{2n+1})(1-z^{-1}q^{2n+1})}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
\sum_{n = 1}^{\infty}\eta_{2k}^{C2}(n)q^n
&=
\frac{1}{(q)_\infty}
\left(
\sum_{n = 1}^{\infty}\frac{(-1)^{n+1}q^{3n^2 - n + 2nk}(1 + q^{2n})}{(1-q^{2n})^{2k}}
+
\sum_{n = 0}^{\infty}\frac{(-1)^{n}q^{3n^2 + n + (2n+1)k}(1 - q^{4n + 2})}{(1-q^{2n + 1})^{2k}}
\right)
,\\
\sum_{n = 1}^{\infty} {\mathrm{spt}^{C2}_{k}\hspace{-.2em} \left(n\right) }q^{n}
&:=
\sum_{n = 1}^{\infty}\left(\mu_{2k}(n)-\eta_{2k}^{C2}(n)\right)q^{n}
=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{q^{2n_{1} + n_{2} + \dotsb + n_{k}}}
{(q;q^2)_{n_{1}} (q^{n_{1} + 1})_{\infty} (1-q^{n_{k}})^{2}\dotsm (1-q^{n_{1}})^{2}}
.\end{aligned}$$
\(8) Using the Bailey pair C(5) from [@Slater1], which is also L(4) from [@Slater1], relative to $(1,q)$, $$\begin{aligned}
\beta_{n}
&=
\frac{q^{\frac{n(n-1)}{2}}}{{\left(q;q^2\right)_{n}} {\left(q\right)_{n}}}
,&
\alpha_{n}
&=
\begin{cases}
1 & n=0
,\\
(-1)^{k}q^{k^2 - k}(1 + q^{2k}) & n = 2k
,\\
0 & n = 2k + 1
,
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{C5}(z,q)
&=
\frac{1}{{\left(q\right)_{\infty}}}
\Bigg(
1
+
\sum_{n =1}^{\infty}
\frac{(1-z)(1-z^{-1})(-1)^{n}q^{n^2+ n}(1 + q^{2n})}{(1-zq^{2n})(1-z^{-1}q^{2n})}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
\sum_{n = 1}^{\infty}\eta_{2k}^{C5}(n)q^n
&=
\frac{1}{(q)_\infty}
\sum_{n = 1}^{\infty}\frac{(-1)^{n+1}q^{n^2 - n + 2nk}(1 + q^{2n})}{(1-q^{2n})^{2k}}
,\\
\sum_{n = 1}^{\infty} {\mathrm{spt}^{C5}_{k}\hspace{-.2em} \left(n\right) }q^{n}
&:=
\sum_{n = 1}^{\infty}\left(\mu_{2k}(n)-\eta_{2k}^{C5}(n)\right)q^{n}
=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{q^{ \frac{n_1(n_1 + 1)}{2} + n_{2} + \dotsb + n_{k}}}
{(q;q^2)_{n_{1}} (q^{n_{1} + 1})_{\infty} (1-q^{n_{k}})^{2}\dotsm (1-q^{n_{1}})^{2}}
.\end{aligned}$$
\(9) Using the Bailey pair L(5), upon correcting the formula for $\beta_n$, from [@Slater2], which also appears as the first entry in the second table of page 468 of [@Slater1], relative to $(1,q)$, $$\begin{aligned}
\beta_{n}
&=
\frac{(-1)_n}{(q)_{n}(q;q^2)_n}
,&
\alpha_{n}
&=
\begin{cases}
1 & n=0
,\\
q^{\frac{n(n-1)}{2}}(1+q^n) & n \geq 1
,
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{L5}(z,q)
&=
\frac{1}{{\left(q\right)_{\infty}}}
\Bigg(
1
+
\sum_{n = 1}^{\infty}\frac{(1-z)(1-z^{-1})q^{n(n+1)/2}(1+q^n)}{(1-zq^n)(1-z^{-1}q^n)}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
&\sum\limits_{n=1}^\infty \eta^{L5}_{2k}(n)q^n
=
\frac{-1}{(q)_\infty}
\sum_{n=1}^\infty \frac{q^{\frac{n(n-1)}{2}+nk}(1+q^n)}{(1-q^{n})^{2k}}
,\\
&
\sum_{n = 1}^{\infty} {\mathrm{spt}^{L5}_{k}\hspace{-.2em} \left(n\right) }q^{n}
:=
\sum_{n = 1}^{\infty}\left(\mu_{2k}(n) - \eta_{2k}^{L5}(n)\right)q^{n}
=
\sum\limits_{n_k \geq \dotsb \geq n_1 \geq 1}
\frac{(-1)_{n_1}q^{n_1+\dotsb +n_k}}
{(q;q^2)_{n_1} (q^{n_1+1})_{\infty}(1-q^{n_k})^2\dotsm (1-q^{n_1})^2}
.\end{aligned}$$
\(10) Using the Bailey pair in the seventh entry in the table on page 470 of [@Slater1], relative to $(1,q)$, $$\begin{aligned}
\beta_{n}
&=
\frac{{\left(-1;q^2\right)_{n}}}{{\left(q\right)_{2n}}}
,&
\alpha_{n}
&=
\begin{cases}
1 & n = 0
,\\
(-1)^{k}q^{2k^2 - k}(1 + q^{2k}) & n = 2k
,\\
(-1)^{k}q^{2k^2 + k}(1 - q^{2k+1}) & n = 2k + 1
,
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{{
\IfEqCase{38}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 38}{}]}}(z,q)
&=
\frac{1}{{\left(q\right)_{\infty}}}
\Bigg(
1
+
\sum_{n = 1}^{\infty}
\frac{(1-z)(1-z^{-1})(-1)^{n}q^{2n^2 + n}(1 + q^{2n})}{(1 - zq^{2n})(1-z^{-1}q^{2n})}
\\&\quad
+
\sum_{n = 0}^{\infty}
\frac{(1-z)(1-z^{-1})(-1)^{n}q^{2n^2 +3n+1}(1 - q^{2n+1})}{(1 - zq^{2n+1})(1 - z^{-1}q^{2n+1})}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
&\sum_{n = 1}^{\infty}\eta_{2k}^{{
\IfEqCase{38}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 38}{}]}}(n)q^n
=
\frac{1}{(q)_{\infty}}
\left(
\sum_{n = 1}^{\infty} \frac{(-1)^{n+1}q^{2n^2 - n + 2nk}(1 + q^{2n})}{(1-q^{2n})^{2k}}
+
\sum_{n = 0}^{\infty} \frac{(-1)^{n+1}q^{2n^2 + n + (2n+1)k}(1 - q^{2n + 1})}{(1 - q^{2n + 1})^{2k}}
\right)
,\\
&\sum_{n = 1}^{\infty} {\mathrm{spt}^{{
\IfEqCase{38}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 38}{}]}}_{k}\hspace{-.2em} \left(n\right) }q^{n}
:=
\sum_{n = 1}^{\infty}\left(\mu_{2k}(n)-\eta_{2k}^{{
\IfEqCase{38}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 38}{}]}}(n)\right)q^{n}
=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{(-1;q^{2})_{n_{1}}q^{n_{1} + \dotsb + n_{k}}}
{(q^{n_{1} + 1})_{n_{1}} (q^{n_{1} + 1})_{\infty} (1-q^{n_{k}})^{2}\dotsm (1-q^{n_{1}})^{2}}
.\end{aligned}$$
\(11) Using the Bailey pair in the eighth entry in the table on page 470 of [@Slater1], upon correcting the formula for $\alpha_n$, relative to $(1,q)$, $$\begin{aligned}
\beta_{n}
&=
\frac{q^{n} {\left(-1;q^2\right)_{n}}}{ {\left(q\right)_{2n}}}
,&
\alpha_{n}
&=
\begin{cases}
1 & n = 0
,\\
(-1)^{k}q^{2k^2 - k}(1 + q^{2k}) & n = 2k
,\\
(-1)^{k+1}q^{2k^2 + k}(1 - q^{2k+1}) & n = 2k + 1
,
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{{
\IfEqCase{39}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 39}{}]}}(z,q)
&=
\frac{1}{{\left(q\right)_{\infty}}}
\Bigg(
1
+
\sum_{n = 1}^{\infty}
\frac{(1-z)(1-z^{-1})(-1)^{n}q^{2n^2 + n}(1 + q^{2n})}{(1 - zq^{2n})(1-z^{-1}q^{2n})}
\\&\quad
+
\sum_{n = 0}^{\infty}
\frac{(1-z)(1-z^{-1})(-1)^{n+1}q^{2n^2 +3n+1}(1 - q^{2n+1})}{(1 - zq^{2n+1})(1 - z^{-1}q^{2n+1})}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
&\sum_{n = 1}^{\infty}\eta_{2k}^{{
\IfEqCase{39}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 39}{}]}}(n)q^n
=
\frac{1}{(q)_{\infty}}
\left(
\sum_{n = 1}^{\infty} \frac{(-1)^{n+1}q^{2n^2 - n + 2nk}(1 + q^{2n})}{(1-q^{2n})^{2k}}
+
\sum_{n = 0}^{\infty} \frac{(-1)^{n}q^{2n^2 + n + (2n+1)k}(1 - q^{2n + 1})}{(1 - q^{2n + 1})^{2k}}
\right)
,\\
&\sum_{n = 1}^{\infty} {\mathrm{spt}^{{
\IfEqCase{39}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 39}{}]}}_{k}\hspace{-.2em} \left(n\right) }q^{n}
:=
\sum_{n = 1}^{\infty}\left(\mu_{2k}(n)-\eta_{2k}^{{
\IfEqCase{39}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 39}{}]}}(n)\right)q^{n}
=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{(-1;q^{2})_{n_{1}} q^{2n_{1} + n_{2} + \dotsb + n_{k}}}
{(q^{n_{1} + 1})_{n_{1}} (q^{n_{1} + 1})_{\infty} (1-q^{n_{k}})^{2}
\dotsm (1-q^{n_{1}})^{2}}
.\end{aligned}$$
\(12) Using the Bailey pair in the first entry in the table on page 471 of [@Slater1], relative to $(1,q)$, $$\begin{aligned}
\beta_{n}
&=
\begin{cases}
1 & n=0
,\\
\frac{{\left(-q^{2};q^{2}\right)_{n-1}}}
{{\left(q\right)_{2n}}} & n \geq 1
,
\end{cases}
&
\alpha_{n}
&=
\begin{cases}
1 & n = 0
,\\
0 & n = 4k-2
,\\
-q^{8k^2 - 6k + 1} & n = 4k -1
,\\
q^{8k^2 - 2k}(1 + q^{4k}) & n = 4k
,\\
-q^{8k^2 + 6k + 1} & n = 4k + 1
,\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{{
\IfEqCase{41}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 41}{}]}}(z,q)
&=
\frac{1}{{\left(q\right)_{\infty}}}
\Bigg(
1
-
\sum_{n = 1}^{\infty}
\frac{(1-z)(1-z^{-1})q^{8n^2 - 2n}}{(1-zq^{4n-1})(1-z^{-1}q^{4n-1})}
+
\sum_{n=1}^{\infty}
\frac{(1-z)(1-z^{-1})q^{8n^2 + 2n}(1 + q^{4n})}{(1 - zq^{4n})(1 - z^{-1}q^{4n})}
\\&\quad
-
\sum_{n =0}^{\infty}
\frac{(1-z)(1-z^{-1})q^{8n^2 + 10n + 2}}{(1 -zq^{4n+1})(1 - z^{-1}q^{4n+1})}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
&\sum_{n = 1}^{\infty}\eta_{2k}^{{
\IfEqCase{41}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 41}{}]}}(n)q^n
=
\frac{1}{(q)_{\infty}}
\Bigg(
\sum_{n = 1}^{\infty} \frac{q^{8n^2 - 6n + 1 + (4n-1)k}}{(1-q^{4n-1})^{2k}}
-
\sum_{n = 1}^{\infty} \frac{q^{8n^2 - 2n + 4nk}(1 + q^{4n})}{(1-q^{4n})^{2k}}
\\&\quad
+
\sum_{n = 0}^{\infty} \frac{q^{8n^2 + 6n + 1 + (4n + 1)k}}{(1 - q^{4n + 1})^{2k}}
\Bigg)
,\\
&\sum_{n = 1}^{\infty} {\mathrm{spt}^{{
\IfEqCase{41}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 41}{}]}}_{k}\hspace{-.2em} \left(n\right) }q^{n}
:=
\sum_{n = 1}^{\infty}\left(\mu_{2k}(n)-\eta_{2k}^{{
\IfEqCase{41}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 41}{}]}}(n)\right)q^{n}
=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{(-q^2;q^2)_{n_{1}-1} q^{n_{1} + \dotsb + n_{k}}}
{(q^{n_{1}+1})_{n_{1}} (q^{n_{1} + 1})_{\infty}
(1-q^{n_{k}})^{2}\dotsm (1-q^{n_{1}})^{2}}
.\end{aligned}$$
\(13) Using the Bailey pair in the second entry in the table on page 471 of [@Slater1], relative to $(1,q)$, $$\begin{aligned}
\beta_{n}
&=
\begin{cases}
1 & n=0
,\\
\frac{q^{n} {\left(-q^2;q^2\right)_{n-1}}}
{{\left(q\right)_{2n}}} & n \geq 1
,
\end{cases}
&
\alpha_{n}
&=
\begin{cases}
1 & n = 0
,\\
0 & n = 4k-2
,\\
-q^{8k^2 - 2k} & n = 4k -1
,\\
q^{8k^2 - 2k}(1 + q^{4k}) & n = 4k
,\\
-q^{8k^2 + 2k} & n = 4k + 1
,
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{{
\IfEqCase{42}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 42}{}]}}(z,q)
&=
\frac{1}{{\left(q\right)_{\infty}}}
\Bigg(
1
-
\sum_{n = 1}^{\infty}
\frac{(1-z)(1-z^{-1})q^{8n^2 +2n-1}}{(1-zq^{4n-1})(1-z^{-1}q^{4n-1})}
+
\sum_{n=1}^{\infty}
\frac{(1-z)(1-z^{-1})q^{8n^2 + 2}(1 + q^{4n})}{(1-zq^{4n})(1 - z^{-1}q^{4n})}
\\& \quad
-
\sum_{n =0}^{\infty}
\frac{(1-z)(1-z^{-1})q^{8n^2 + 6n + 1}}{(1 -zq^{4n+1})(1 - z^{-1}q^{4n+1})}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
&\sum_{n = 1}^{\infty}\eta_{2k}^{{
\IfEqCase{42}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 42}{}]}}(n)q^n
=
\frac{1}{(q)_{\infty}}
\Bigg(
\sum_{n = 1}^{\infty} \frac{q^{8n^2 - 2n + (4n-1)k}}{(1-q^{4n-1})^{2k}}
-
\sum_{n = 1}^{\infty} \frac{q^{8n^2 - 2n + 4nk}(1 + q^{4n})}{(1-q^{4n})^{2k}}
+
\sum_{n = 0}^{\infty} \frac{q^{8n^2 + 2n + (4n + 1)k}}{(1 - q^{4n + 1})^{2k}}
\Bigg)
,\\
&\sum_{n = 1}^{\infty} {\mathrm{spt}^{{
\IfEqCase{42}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 42}{}]}}_{k}\hspace{-.2em} \left(n\right) }q^{n}
:=
\sum_{n = 1}^{\infty}\left(\mu_{2k}(n)-\eta_{2k}^{{
\IfEqCase{42}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 42}{}]}}(n)\right)q^{n}
=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{(-q^2;q^2)_{n_{1}-1} q^{2n_{1} + n_{2} + \dotsb + n_{k}}}
{(q^{n_1+1})_{n_1} (q^{n_{1} + 1})_{\infty}
(1-q^{n_{k}})^{2}\dotsm (1-q^{n_{1}})^{2}}
.\end{aligned}$$
\(14) Using the Bailey pair E(4) from [@Slater1], relative to $(1,q)$, $$\begin{aligned}
\beta_n
&=
\frac{q^n}{(q^2; q^2)_n}
,&
\alpha_n
&=
\begin{cases}
1 & n=0
,\\
(-1)^nq^{n^2-n}(1+q^{2n}) & n\geq 1
,\\
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{E4}(z,q)
&=
\frac{{\left(-q\right)_{\infty}}}{{\left(q\right)_{\infty}}}
\Bigg(
1
+
\sum\limits_{n=1}^\infty
\frac{(1-z)(1-z^{-1})(-1)^nq^{n^2}(1+q^{2n})}{(1-zq^n)(1-z^{-1}q^n)}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
\sum_{n = 1}^{\infty}\eta_{2k}^{E4}(n)q^n
&=
\frac{(-q)_\infty}{(q)_\infty}
\sum\limits_{n=1}^\infty \frac{(-1)^{n+1}q^{n^2-n+kn}(1+q^{2n})}{(1-q^n)^{2k}}
,\\
\sum_{n = 1}^{\infty} {\mathrm{spt}^{E4}_{k}\hspace{-.2em} \left(n\right) }q^{n}
&:=
\sum_{n = 1}^{\infty}\left(\overline{\mu}_{2k}(n) - \eta_{2k}^{E4}(n)\right)q^{n}
=
\sum\limits_{n_k \geq \dotsb \geq n_1 \geq 1}
\frac{(-q^{n_1+1})_\infty q^{2n_1+ n_2+\dotsb +n_k}}
{(q^{n_1+1})_\infty (1-q^{n_k})^2\dotsm (1-q^{n_1})^2}
.\end{aligned}$$
\(15) Using the Bailey pair I(14) from [@Slater2], relative to $(1,q)$, $$\begin{aligned}
\beta_{n}
&=
\begin{cases}
1 & n=0
,\\
\frac{{\left(-q^{2};q^{2}\right)_{n-1}}}
{{\left(q;q^{2}\right)_{n}} {\left(q\right)_{n}} {\left(-q\right)_{n-1}}} & n \geq 1
,
\end{cases}
&
\alpha_{n}
&=
\begin{cases}
1 & n = 0
,\\
(-1)^{k}q^{2k^2 - k}(1 + q^{2k}) & n = 2k
,\\
0 & n = 2k + 1
,
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{I14}(z,q)
&=
\frac{{\left(-q\right)_{\infty}}}{{\left(q\right)_{\infty}}}
\Bigg(
1
+
\sum_{n = 1}^{\infty}
\frac{(1-z)(1-z^{-1})(-1)^{n}q^{2n^2 + n}(1 + q^{2n})}{(1-zq^{2n})(1-z^{-1}q^{2n})}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
\sum_{n = 1}^{\infty}\eta_{2k}^{I14}(n)q^n
&=
\frac{(-q)_{\infty}}{(q)_{\infty}}
\sum_{n = 1}^{\infty} \frac{(-1)^{n+1}q^{2n^2 - n + 2nk}(1 + q^{2n})}{(1-q^{2n})^{2k}}
,\\
\sum_{n = 1}^{\infty} {\mathrm{spt}^{I14}_{k}\hspace{-.2em} \left(n\right) }q^{n}
&:=
\sum_{n = 1}^{\infty}\left(\overline{\mu}_{2k}(n)-\eta_{2k}^{I14}(n)\right)q^{n}
=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{(-q^2;q^2)_{n_{1}-1} (-q^{n_{1}})_{\infty} q^{n_{1} + \cdots{} + n_{k}}}
{(q;q^{2})_{n_{1}} (q^{n_{1} + 1})_{\infty}
(1-q^{n_{k}})^{2}\dotsm (1-q^{n_{1}})^{2}}
.\end{aligned}$$
\(16) Using the Bailey pair in Lemma 3.1 from [@BowmannMclaughlinSills1], relative to $(1,q)$, $$\begin{aligned}
\beta_{n}
&=
\begin{cases}
1 & n=0
,\\
\frac{{\left(-q^{3};q^{3}\right)_{n-1}}}
{{\left(-q\right)_{n}} {\left(q\right)_{2n-1}}} & n \geq 1
,
\end{cases}
&
\alpha_{n} =
\begin{cases}
1 & n = 0
,\\
(-1)^{k}q^{\frac{3k(3k-1)}{2}}(1 + q^{3k}) & n = 3k
,\\
-2q^{18k^2 + 9k + 1} & n = 6k + 1
,\\
2q^{18k^2 + 15k + 3} & n = 6k + 2
,\\
2q^{18k^2 + 21k + 6} & n = 6k + 4
,\\
-2q^{18k^2 + 27k + 10} & n = 6k + 5
,
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}(z,q)
&=
\frac{{\left(-q\right)_{\infty}}}{{\left(q\right)_{\infty}}}
\Bigg(
1
+
\sum_{n = 1}^{\infty}
\frac{(1-z)(1-z^{-1})(-1)^{n}q^{\frac{3n(3n+1)}{2}}(1 + q^{3n})}{(1-zq^{3n})(1-z^{-1}q^{3n})}
\\&\quad
-
2\sum_{n = 0}^{\infty}
\frac{(1-z)(1-z^{-1})q^{18n^2 + 15n + 2}}{(1 - zq^{6n+1})(1 - z^{-1}q^{6n+1})}
+
2\sum_{n = 0}^{\infty}
\frac{(1-z)(1-z^{-1})q^{18n^2 + 21n + 5}}{(1-zq^{6n+2})(1 - z^{-1}q^{6n+2})}
\\&\quad
+
2\sum_{n = 0}^{\infty}
\frac{(1-z)(1-z^{-1})q^{18n^2 + 27n + 10}}{(1 -zq^{6n+4})(1 - z^{-1}q^{6n+4})}
-
2\sum_{n = 0}^{\infty}\frac{(1-z)(1-z^{-1})q^{18n^2 + 33n + 15}}{(1 -zq^{6n + 5})(1 - z^{-1}q^{6n+5})}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
&\sum_{n = 1}^{\infty}\eta_{2k}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}(n)q^n
=
\frac{(-q)_{\infty}}{(q)_{\infty}}
\Bigg(
\sum_{n = 1}^{\infty} \frac{(-1)^{n+1}q^{\frac{3n(3n-1)}{2} + 3nk}(1 + q^{3n})}{(1-q^{3n})^{2k}}
+
2\sum_{n = 0}^{\infty} \frac{q^{18n^2 + 9n + 1 + (6n+1)k}}{(1-q^{6n + 1})^{2k}}
\\&\quad
-
2\sum_{n = 0}^{\infty} \frac{q^{18n^2 + 15n + 3 + (6n + 2)k}}{(1 - q^{6n + 2})^{2k}}
-
2\sum_{n = 0}^{\infty} \frac{q^{18n^2 + 21n + 6 + (6n + 4)k}}{(1 - q^{6n + 4})^{2k}}
+
2\sum_{n = 0}^{\infty} \frac{q^{18n^2 + 27n + 10 + (6n + 5)k}}{(1 - q^{6n + 5})^{2k}}
\Bigg)
,\\
&\sum_{n = 1}^{\infty} {\mathrm{spt}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{k}\hspace{-.2em} \left(n\right) }q^{n}
:=
\sum_{n = 1}^{\infty}\left(\overline{\mu}_{2k}(n)-\eta_{2k}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}(n)\right)q^{n}
=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{(-q^3;q^3)_{n_{1}-1} (-q^{n_{1} + 1})_{\infty} q^{n_1+ \dotsb + n_k}}
{(q^{n_{1} + 1})_{n_{1}-1} (q^{n_1+1})_{\infty} (1-q^{n_{k}})^{2} \dotsm (1-q^{n_{1}})^{2}}
.\end{aligned}$$
\(17) Using the Bailey pair J(1) from [@Slater2], which also appears as equation (3.8) in [@Slater1], relative to $(1,q)$, $$\begin{aligned}
\beta_{n}
&=
\begin{cases}
1 & n=0
,\\
\frac{{\left(q^{3};q^{3}\right)_{n-1}}}{{\left(q\right)_{2n-1}}{\left(q\right)_{n}}} & n \geq 1
,
\end{cases}
&
\alpha_{n}
&=
\begin{cases}
1 & n = 0
,\\
0 & n = 3k-1
,\\
(-1)^{k}q^{\frac{3k(3k-1)}{2}}(1 + q^{3k}) & n = 3k
,\\
0 & n = 3k+1
,
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{J1}(z,q)
&=
\frac{1}{{\left(q\right)_{\infty}}{\left(q^3;q^3\right)_{\infty}}}
\Bigg(
1
+
\sum_{n = 1}^{\infty}
\frac{(1-z)(1-z^{-1})(-1)^{n}q^{\frac{3n(3n+1)}{2}}(1+q^{3n})}{(1-zq^{3n})(1-z^{-1}q^{3n})}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
\sum_{n = 1}^{\infty}\eta_{2k}^{J1}(n)q^n
&=
\frac{1}{(q)_{\infty}(q^3;q^3)_\infty}
\sum_{n = 1}^{\infty} \frac{(-1)^{n+1}q^{\frac{3n(3n-1)}{2}+3nk}(1 + q^{3n})}{(1-q^{3n})^{2k}}
,\\
\sum_{n = 1}^{\infty} {\mathrm{spt}^{J1}_{k}\hspace{-.2em} \left(n\right) }q^{n}
&:=
\sum_{n = 1}^{\infty}\left(\mu^J_{2k}(n) - \eta_{2k}^{J1}(n)\right)q^{n}
\\
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{q^{n_{1} + \dotsb + n_{k}}}
{(q)_{2n_1-1} (q^{3n_1};q^3)_\infty (q^{n_{1} + 1})_{\infty} (1-q^{n_{k}})^{2}\dotsm (1-q^{n_{1}})^{2}}
.\end{aligned}$$
\(18) Using the Bailey pair J(2) from [@Slater2], which also appears unlabeled on page 467 of [@Slater1], relative to $(1,q)$, $$\begin{aligned}
\beta_{n}
&=
\begin{cases}
1 & n=0
,\\
\frac{{\left(q^{3};q^{3}\right)_{n-1}}}
{{\left(q\right)_{2n}} {\left(q\right)_{n-1}}} & n \geq 1
,
\end{cases}
&
\alpha_{n}
&=
\begin{cases}
1 & n = 0
,\\
(-1)^{k+1}q^{\frac{9k(k-1)}{2}+1} & n = 3k-1
,\\
(-1)^{k}q^{\frac{3k(3k-1)}{2}}(1 + q^{3k}) & n = 3k
,\\
(-1)^{k+1}q^{\frac{9k(k+1)}{2}+1} & n = 3k+1
,
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{J2}(z,q)
&=
\frac{1}{{\left(q\right)_{\infty}} {\left(q^3;q^3\right)_{\infty}}}
\Bigg(
1
+
\sum\limits_{n=1}^\infty
\frac{(1-z)(1-z^{-1})(-1)^{n+1}q^{\frac{3n(3n-1)}{2}}}{(1-zq^{3n-1})(1-z^{-1}q^{3n-1})}
\\&\quad
+
\sum\limits_{n=0}^\infty
\frac{(1-z)(1-z^{-1})(-1)^{n+1}q^{\frac{3n(3n+5)}{2} + 2}}{(1-zq^{3n+1})(1-z^{-1}q^{3n+1})}
+
\sum\limits_{n=1}^\infty
\frac{(1-z)(1-z^{-1})(-1)^{n}q^{\frac{3n(3n+1)}{2}}(1 + q^{3n})}{(1-zq^{3n})(1-z^{-1}q^{3n})}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
\sum_{n = 1}^{\infty}\eta_{2k}^{J2}(n)q^n
&=
\frac{1}{(q)_{\infty}(q^3;q^3)_\infty}
\Bigg(
\sum_{n=1}^\infty \frac{(-1)^{n}q^{ \frac{9n(n-1)}{2} +1+ (3n-1)k}}{(1-q^{3n-1})^{2k}}
+
\sum_{n=0}^\infty \frac{(-1)^{n}q^{ \frac{9n(n+1)}{2} +1+(3n+1)k}}{(1-q^{3n+1})^{2k}}
\\&\quad
-
\sum_{n=1}^\infty \frac{(-1)^{n}q^{ \frac{3n(3n-1)}{2}+3nk}(1 + q^{3n})}{(1-q^{3n})^{2k}}
\Bigg)
,\\
\sum_{n = 1}^{\infty} {\mathrm{spt}^{J2}_{k}\hspace{-.2em} \left(n\right) }q^{n}
&:=
\sum_{n = 1}^{\infty}\left(\mu^J_{2k}(n) - \eta_{2k}^{J2}(n)\right)q^{n}
\\
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{q^{n_{1} + \dotsb + n_{k}}}
{(q)_{n_{1}-1} (q^{n_{1}+1})_{n_{1}} (q^{3n_1};q^3)_\infty (q^{n_{1} + 1})_{\infty}
(1-q^{n_{k}})^{2}\dotsm (1-q^{n_{1}})^{2}}
.\end{aligned}$$
\(19) Using the Bailey pair J(3) from [@Slater2], which also appears unlabeled on page 467 of [@Slater1], relative to $(1,q)$, $$\begin{aligned}
\beta_{n}
&=
\begin{cases}
1 & n=0
,\\
\frac{q^n{\left(q^{3};q^{3}\right)_{n-1}}}
{{\left(q\right)_{2n}} {\left(q\right)_{n-1}}} & n \geq 1
,
\end{cases}
&
\alpha_{n}
&=
\begin{cases}
1 & n = 0
,\\
(-1)^{k+1}q^{\frac{3k(3k-1)}{2}} & n = 3k-1
,\\
(-1)^{k}q^{\frac{3k(3k-1)}{2}}(1 + q^{3k}) & n = 3k
,\\
(-1)^{k+1}q^{\frac{3k(3k+1)}{2}} & n = 3k+1
,
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{J3}(z,q)
&=
\frac{1}{{\left(q\right)_{\infty}} {\left(q^3;q^3\right)_{\infty}}}
\Bigg(
1
+
\sum\limits_{n=1}^\infty
\frac{(1-z)(1-z^{-1})(-1)^{n+1}q^{\frac{3n(3n+1)}{2}-1}}{(1-zq^{3n-1})(1-z^{-1}q^{3n-1})}
\\&\quad
+
\sum\limits_{n=0}^\infty
\frac{(1-z)(1-z^{-1})(-1)^{n+1}q^{\frac{9n(n+1)}{2}+1}}{(1-zq^{3n+1})(1-z^{-1}q^{3n+1})}
+
\sum\limits_{n=1}^\infty
\frac{(1-z)(1-z^{-1})(-1)^{n}q^{\frac{3n(3n+1)}{2}}(1 + q^{3n})}{(1-zq^{3n})(1-z^{-1}q^{3n})}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
\sum_{n = 1}^{\infty}\eta_{2k}^{J3}(n)q^n
&=
\frac{1}{(q)_{\infty}(q^3;q^3)_\infty}
\Bigg(
\sum\limits_{n=1}^\infty \frac{(-1)^{n} q^{ \frac{3n(3n-1)}{2} +(3n-1)k}}{(1-q^{3n-1})^{2k}}
+
\sum\limits_{n=0}^\infty \frac{(-1)^{n}q^{ \frac{3n(3n+1)}{2} +(3n+1)k}}{(1-q^{3n+1})^{2k}}
\\&\quad
-
\sum\limits_{n=1}^\infty \frac{(-1)^{n}q^{ \frac{3n(3n-1)}{2} +3nk}(1 + q^{3n})}{(1-q^{3n})^{2k}}
\Bigg)
,\\
\sum_{n = 1}^{\infty} {\mathrm{spt}^{J3}_{k}\hspace{-.2em} \left(n\right) }q^{n}
&:=
\sum_{n = 1}^{\infty}\left(\mu^J_{2k}(n) - \eta_{2k}^{J3}(n)\right) q^{n}
\\
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{q^{2n_{1} + n_{2} + \dotsb + n_{k}}}
{(q)_{n_{1}-1} (q^{n_{1}+1})_{n_{1}} (q^{3n_1};q^3)_\infty (q^{n_{1} + 1})_{\infty}
(1-q^{n_{k}})^{2}\dotsm (1-q^{n_{1}})^{2}}
.\end{aligned}$$
\(20) Using the Bailey pair in the ninth entry in the table on page 470 of [@Slater1], relative to $(1,q)$, $$\begin{aligned}
\beta_{n}
&=
\begin{cases}
1 & n=0
,\\
\frac{{\left(q^{2};q^{2}\right)_{n-1}}}
{{\left(q;q^{2}\right)_{n}} {\left(q\right)_{n}} {\left(q\right)_{n-1}}} & n \geq 1
,
\end{cases}
&
\alpha_{n}
&=
\begin{cases}
1 & n = 0
,\\
q^{2k^2 - k}(1 + q^{2k}) & n = 2k
,\\
0 & n = 2k + 1
,
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}(z,q)
&=
\frac{1}{{\left(q\right)_{\infty}} {\left(q^2;q^2\right)_{\infty}}}
\Bigg(
1
+
\sum_{n = 1}^{\infty}
\frac{(1-z)(1-z^{-1})q^{2n^2+n}(1+q^{2n})}{(1-zq^{2n})(1-z^{-1}q^{2n})}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
\sum_{n = 1}^{\infty}\eta_{2k}^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}(n)q^n
&=
\frac{-1}{(q)_{\infty}(q^2;q^2)_{\infty}}
\sum_{n = 1}^{\infty} \frac{q^{2n^2 - n + 2nk}(1 + q^{2n})}{(1-q^{2n})^{2k}}
,\\
\sum_{n = 1}^{\infty} {\mathrm{spt}^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}_{k}\hspace{-.2em} \left(n\right) }q^{n}
&:=
\sum_{n = 1}^{\infty}\left(\mu^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}_{2k}(n)-\eta_{2k}^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}(n)\right)q^{n}
\\
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{q^{n_{1} + \dotsb + n_{k}}}
{(q)_{n_1 - 1} (q;q^2)_{n_1} (q^{2n_1};q^2)_{\infty} (q^{n_1 + 1})_{\infty}
(1-q^{n_{k}})^{2}\dotsm (1-q^{n_{1}})^{2}}
.\end{aligned}$$
\(21) Using the Bailey pair F(3) from [@Slater1], relative to $(1,q^2)$, $$\begin{aligned}
\beta_n
&=
\frac{1}{q^n(q)_{2n}}
,&
\alpha_n
&=
\begin{cases}
1 & n=0
,\\
q^{n}+q^{-n} & n\geq 1
,
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{F3}(z,q)
&=
\frac{1}{{\left(q^2;q^2\right)_{\infty}}}
\Bigg(
1
+
\sum\limits_{n=1}^\infty
\frac{(1-z)(1-z^{-1})q^{n}(1+q^{2n})}{(1-zq^{2n})(1-z^{-1}q^{2n})}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
&
\sum\limits_{n=1}^\infty \eta^{F3}_{2k}(n)q^n
=
\frac{-1}{(q^2;q^2)_\infty}
\sum_{n=1}^\infty \frac{q^{2nk-n}(1+q^{2n})}{(1-q^{2n})^{2k}}
,\\
&
\sum_{n = 1}^{\infty} {\mathrm{spt}^{F3}_{k}\hspace{-.2em} \left(n\right) }q^{n}
:=
\sum_{n = 1}^{\infty}\left(\mu^{F}_{2k}(n) - \eta_{2k}^{F3}(n)\right)q^{n}
=
\sum\limits_{n_k \geq \dotsb \geq n_1 \geq 1}
\frac{q^{n_1+2n_2\dotsb +2n_k}}
{(q;q^2)_{n_1} (q^{2n_1+2};q^2)_{\infty}(1-q^{2n_k})^2 \dotsm (1-q^{2n_1})^2}
.\end{aligned}$$
\(22) Using the Bailey pair G(1) from [@Slater1], which is also L(3) from [@Slater2], relative to $(1,q^2)$, $$\begin{aligned}
\beta_n
&=
\frac{1}{{\left(-q;q^2\right)_{n}} {\left(q^4;q^4\right)_{n}}}
,&
\alpha_n
&=
\begin{cases}
1 & n=0
,\\
(-1)^nq^{\frac{n(3n-1)}{2}}(1+q^{n}) & n\geq 1
,
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{G1}(z,q)
&=
\frac{{\left(-q;q^2\right)_{\infty}}}{{\left(q^2;q^2\right)_{\infty}}^2}
\Bigg(
1
+
\sum_{n = 1}^{\infty}
\frac{(1-z)(1-z^{-1})(-1)^{n}q^{\frac{3n(n+1)}{2}}(1+q^{n})}{(1-zq^{2n})(1-z^{-1}q^{2n})}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
\sum_{n = 1}^{\infty}\eta_{2k}^{G1}(n)q^n
&=
\frac{(-q;q^2)_{\infty}}{(q^2;q^2)^2_{\infty}}
\sum_{n = 1}^{\infty} \frac{(-1)^{n+1}q^{ \frac{n(3n-1)}{2} +2nk}(1+q^{n})}{(1-q^{2n})^{2k}}
,\\
\sum_{n = 1}^{\infty} {\mathrm{spt}^{G1}_{k}\hspace{-.2em} \left(n\right) }q^{n}
&:=
\sum_{n = 1}^{\infty}\left(\mu^G_{2k}(n)- \eta_{2k}^{G1}(n)\right)q^{n}
=
\sum\limits_{n_k \geq \dotsb \geq n_1 \geq 1}
\frac{(-q^{2n_1+1};q^2)_\infty q^{2n_1+\dotsb +2n_k}}
{(q^4;q^4)_{n_1} (q^{2n_1+2};q^2)^2_{\infty}
(1-q^{2n_k})^2\dotsm (1-q^{2n_1})^2}
.\end{aligned}$$
\(23) Using the Bailey pair G(3) from [@Slater1], relative to $(1,q^2)$, $$\begin{aligned}
\beta_n
&=
\frac{q^{2n}}{{\left(-q;q^2\right)_{n}} {\left(q^4;q^4\right)_{n}}}
,&
\alpha_n
&=
\begin{cases}
1 & n=0
,\\
(-1)^nq^{\frac{3n(n-1)}{2}}(1+q^{3n}) & n\geq 1
,
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{G3}(z,q)
&=
\frac{{\left(-q;q^2\right)_{\infty}}}{{\left(q^2;q^2\right)_{\infty}}^2}
\Bigg(
1
+
\sum_{n = 1}^{\infty}
\frac{(1-z)(1-z^{-1})(-1)^{n}q^{\frac{n(3n+1)}{2}}(1+q^{3n})}{(1-zq^{2n})(1-z^{-1}q^{2n})}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
\sum_{n = 1}^{\infty}\eta_{2k}^{G3}(n)q^n
&=
\frac{(-q;q^2)_{\infty}}{(q^2;q^2)^2_{\infty}}
\sum_{n = 1}^{\infty} \frac{(-1)^{n+1}q^{ \frac{3n(n-1)}{2} +2nk}(1+q^{3n})}{(1-q^{2n})^{2k}}
,\\
\sum_{n = 1}^{\infty} {\mathrm{spt}^{G3}_{k}\hspace{-.2em} \left(n\right) }q^{n}
&:=
\sum_{n = 1}^{\infty}\left(\mu^G_{2k}(n)-\eta_{2k}^{G3}(n)\right)q^{n}
=
\sum\limits_{n_k \geq \dotsb \geq n_1 \geq 1}
\frac{(-q^{2n_1+1};q^2)_\infty q^{4n_1+2n_2+\dotsb+2n_k}}
{(q^4;q^4)_{n_1} (q^{2n_1+2};q^2)^2_{\infty}
(1-q^{2n_k})^2\dotsm (1-q^{2n_1})^2}
.\end{aligned}$$
\(24) Using the Bailey pair in the first entry in the table on page 470 of [@Slater1], relative to $(1,q^2)$, $$\begin{aligned}
\beta_{n}
&=
\frac{q^{n^2 - 2n}}{{\left(q^4;q^4\right)_{n}} {\left(q;q^2\right)_{n}}}
,&
\alpha_{n}
&=
\begin{cases}
1 & n = 0
,\\
(-1)^{k}q^{2k^2 - 3k}(1 + q^{6k}) & n = 2k
,\\
(-1)^{k}q^{2k^2 - k - 1}(1 - q^{6k + 3}) & n = 2k + 1
,
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{Y1}(z,q)
&=
\frac{1}{{\left(q^2;q^2\right)_{\infty}}^2}
\Bigg(
1
+
\sum_{n =1}^{\infty}
\frac{(1-z)(1-z^{-1})(-1)^{n}q^{2n^2 + n}(1 + q^{6n})}{(1-zq^{4n})(1-z^{-1}q^{4n})}
\\&\quad
+
\sum_{n = 0}^{\infty}
\frac{(1-z)(1-z^{-1})(-1)^{n}q^{2n^2+3n+1}(1 - q^{6n + 3})}{(1-zq^{4n+2})(1-z^{-1}q^{4n+2})}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
\sum_{n = 1}^{\infty}\eta_{2k}^{Y1}(n)q^n
&=
\frac{1}{(q^2;q^2)_{\infty}^2}
\left(
\sum_{n = 1}^{\infty}\frac{(-1)^{n+1}q^{2n^2 - 3n + 4nk}(1 + q^{6n})}{(1-q^{4n})^{2k}}
\right.\\&\quad\left.
+
\sum_{n = 0}^{\infty}\frac{(-1)^{n+1}q^{2n^2 - n - 1 + (4n+2)k}(1 - q^{6n + 3})}{(1-q^{4n+2})^{2k}}
\right)
,\\
\sum_{n = 1}^{\infty} {\mathrm{spt}^{Y1}_{k}\hspace{-.2em} \left(n\right) }q^{n}
&:=
\sum_{n = 1}^{\infty}\left(\mu^{Y}_{2k}(n) - \eta_{2k}^{Y1}(n)\right)q^{n}
\\&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{q^{n_{1}^2 + 2n_{2} + \dotsb + 2n_{k}}}
{(q;q^2)_{n_{1}} (q^4;q^4)_{n_{1}} (q^{2n_{1} + 2} ; q^2)_{\infty}^{2}
(1-q^{2n_{k}})^{2} \dotsm (1-q^{2n_{1}})^{2}}
.\end{aligned}$$
\(25) Using the Bailey pair in the second entry in the table on page 470 of [@Slater1], relative to $(1,q^2)$, $$\begin{aligned}
\beta_{n}
&=
\frac{q^{n^2}}{{\left(q^4;q^4\right)_{n}} {\left(q;q^2\right)_{n}}}
,&
\alpha_{n}
&=
\begin{cases}
1 & n = 0
,\\
(-1)^{k}q^{2k^2 - k}(1 + q^{2k}) & n = 2k
,\\
(-1)^{k+1}q^{2k^2 + k}(1 - q^{2k+1}) & n = 2k + 1
,
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{Y2}(z,q)
&=
\frac{1}{{\left(q^2;q^2\right)_{\infty}}^2}
\Bigg(
1
+
\sum_{n =1}^{\infty}
\frac{(1-z)(1-z^{-1})(-1)^{n}q^{2n^2 + 3n}(1 + q^{2n})}{(1-zq^{4n})(1-z^{-1}q^{4n})}
\\& \quad
+
\sum_{n = 0}^{\infty}
\frac{(1-z)(1-z^{-1})(-1)^{n+1}q^{2n^2 +5n+2}(1 - q^{2n + 1})}{(1-zq^{4n+2})(1-z^{-1}q^{4n+2})}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
\sum_{n = 1}^{\infty}\eta_{2k}^{Y2}(n)q^n
&=
\frac{1}{(q^2;q^2)_{\infty}^{2}}
\left(
\sum_{n = 1}^{\infty}\frac{(-1)^{n+1}q^{2n^2 - n + 4nk}(1 + q^{2n})}{(1-q^{4n})^{2k}}
+
\sum_{n = 0}^{\infty}\frac{(-1)^{n}q^{2n^2 + n + (4n+2)k}(1 - q^{2n+1})}{(1-q^{4n+2})^{2k}}
\right)
,\\
\sum_{n = 1}^{\infty} {\mathrm{spt}^{Y2}_{k}\hspace{-.2em} \left(n\right) }q^{n}
&:=
\sum_{n = 1}^{\infty}\left(\mu^{Y}_{2k}(n)- \eta_{2k}^{Y2}(n)\right)q^{n}
\\
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{q^{n_{1}^2 + 2n_{1} + 2n_{2} + \dotsb + 2n_{k}}}
{(q;q^2)_{n_{1}} (q^4;q^4)_{n_{1}} (q^{2n_{1} + 2} ; q^2)_{\infty}^{2}
(1-q^{2n_{k}})^{2}\dotsm (1-q^{2n_{1}})^{2}}
.\end{aligned}$$
\(26) Using the Bailey pair in the third entry in the table on page 470 of [@Slater1], relative to $(1,q^2)$, $$\begin{aligned}
\beta_{n}
&=
\frac{1}{{\left(q^4;q^4\right)_{n}} {\left(q;q^2\right)_{n}}}
,&
\alpha_{n}
&=
\begin{cases}
1 & n = 0
,\\
(-1)^{k}q^{6k^2 - k}(1 + q^{2k}) & n = 2k
,\\
(-1)^{k}q^{6k^2 + 5k + 1}(1 - q^{2k+1}) & n = 2k + 1
,
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{Y3}(z,q)
&=
\frac{1}{{\left(q^2;q^2\right)_{\infty}}^2}
\Bigg(
1
+
\sum_{n =1}^{\infty}
\frac{(1-z)(1-z^{-1})(-1)^{n}q^{6n^2 + 3n}(1 + q^{2n})}{(1-zq^{4n})(1-z^{-1}q^{4n})}
\\&\quad
+
\sum_{n = 0}^{\infty}
\frac{(1-z)(1-z^{-1})(-1)^{n}q^{6n^2 + 9n+3}(1 - q^{2n + 1})}{(1-zq^{4n+2})(1-z^{-1}q^{4n+2})}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
\sum_{n = 1}^{\infty}\eta_{2k}^{Y3}(n)q^n
&=
\frac{1}{(q^2;q^2)_{\infty}^2}
\left(
\sum_{n = 1}^{\infty}\frac{(-1)^{n+1}q^{6n^2 - n + 4nk}(1 + q^{2n})}{(1-q^{4n})^{2k}}
\right. \\&\quad \left.
+
\sum_{n = 0}^{\infty}\frac{(-1)^{n+1}q^{6n^2 + 5n + 1 + (4n + 2)k}(1 - q^{2n+1})}{(1-q^{4n+2})^{2k}}
\right)
,\\
\sum_{n = 1}^{\infty} {\mathrm{spt}^{Y3}_{k}\hspace{-.2em} \left(n\right) }q^{n}
&:=
\sum_{n = 1}^{\infty}\left(\mu^{Y}_{2k}(n)-\eta_{2k}^{Y3}(n)\right)q^{n}
\\
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{q^{2n_1 + \dotsb + 2n_k}}
{(q;q^2)_{n_{1}} (q^4;q^4)_{n_{1}} (q^{2n_{1} + 2} ; q^2)_{\infty}^{2}
(1-q^{2n_{k}})^{2} \dotsm (1-q^{2n_{1}})^{2}}
.\end{aligned}$$
\(27) Using the Bailey pair in the fourth entry in the table on page 470 of [@Slater1], relative to $(1,q^2)$, $$\begin{aligned}
\beta_{n}
&=
\frac{q^{2n}}{{\left(q^4;q^4\right)_{n}} {\left(q;q^2\right)_{n}}}
,&
\alpha_{n}
&=
\begin{cases}
1 & n = 0
,\\
(-1)^{k}q^{6k^2 - 3k}(1 + q^{6k}) & n = 2k
,\\
(-1)^{k+1}q^{6k^2 + 3k}(1 - q^{6k+3}) & n = 2k + 1
,
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{Y4}(z,q)
&=
\frac{1}{{\left(q^2;q^2\right)_{\infty}}^2}
\Bigg(
1
+
\sum_{n =1}^{\infty}
\frac{(1-z)(1-z^{-1})(-1)^{n}q^{6n^2 + n}(1 + q^{6n})}{(1-zq^{4n})(1-z^{-1}q^{4n})}
\\&\quad
+
\sum_{n = 0}^{\infty}
\frac{(1-z)(1-z^{-1})(-1)^{n+1}q^{6n^2 +7n+2}(1 - q^{6n + 3})}{(1-zq^{4n+2})(1-z^{-1}q^{4n+2})}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
\sum_{n = 1}^{\infty}\eta_{2k}^{Y4}(n)q^n
&=
\frac{1}{(q^2;q^2)_{\infty}^2}
\left(
\sum_{n = 1}^{\infty}\frac{(-1)^{n+1}q^{6n^2 - 3n + 4nk}(1 + q^{6n})}{(1-q^{4n})^{2k}}
\right.\\&\quad \left.
+
\sum_{n = 0}^{\infty}\frac{(-1)^{n}q^{6n^2 + 3m + (4n + 2)k}(1 - q^{6n+3})}{(1-q^{4n+2})^{2k}}
\right)
,\\
\sum_{n = 1}^{\infty} {\mathrm{spt}^{Y4}_{k}\hspace{-.2em} \left(n\right) }q^{n}
&:=
\sum_{n = 1}^{\infty}\left(\mu^{Y}_{2k}(n)-\eta_{2k}^{Y4}(n)\right)q^{n}
\\
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{q^{4n_{1} + 2n_{2} + \dotsb + 2n_{k}}}
{(q;q^2)_{n_{1}} (q^4;q^4)_{n_{1}} (q^{2n_{1} + 2} ; q^2)_{\infty}^{2}
(1-q^{2n_{k}})^{2}\dotsm (1-q^{2n_{1}})^{2}}
.\end{aligned}$$
\(28) Using the Bailey pair L(2) from [@Slater2], which is also M(1) from [@Slater2], relative to $(1,q^4)$, $$\begin{aligned}
\beta_n
&=
\frac{{\left(q;q^2\right)_{2n}}}{{\left(q^4;q^4\right)_{2n}}}
,&
\alpha_n
&=
\begin{cases}
1 & n=0
,\\
(-1)^nq^{2n^2-n}(1+q^{2n}) & n\geq 1
,
\end{cases}\end{aligned}$$ we define $$\begin{aligned}
R_{L2}(z,q)
&=
\frac{{\left(-q;q\right)_{\infty}}}{{\left(q^4;q^4\right)_{\infty}}}
\Bigg(
1
+
\sum\limits_{n=1}^\infty\frac{(1-z)(1-z^{-1})(-1)^nq^{2n^2+3n}(1+q^{2n})}{(1-zq^{4n})(1-z^{-1}q^{4n})}
\Bigg)
,\end{aligned}$$ and obtain $$\begin{aligned}
\sum\limits_{n=1}^\infty \eta^{L2}_{2k}(n)q^n
&=
\frac{(-q)_\infty}{(q^4;q^4)_\infty}
\sum_{n=1}^\infty
\frac{(-1)^{n+1}q^{2n^2-n+4nk}(1+q^{2n})}{(1-q^{4n})^{2k}}
,\\
\sum_{n = 1}^{\infty} {\mathrm{spt}^{L2}_{k}\hspace{-.2em} \left(n\right) }q^{n}
&:=
\sum_{n=1}^\infty \left(\mu^{L2}_{2k}(n)-\eta^{L2}_{2k}(n)\right)q^n
\\
&=
\sum_{n_k \geq \dotsb \geq n_1 \geq 1}
\frac{q^{4n_1+\dotsb +4n_k}}
{(q^{4n_1+4};q^4)_{n_1} (q^{4n_1+1};q^2)_\infty (q^{4n_1+4};q^4)_{\infty}
(1-q^{4n_k})^2 \dotsm (1-q^{4n_1})^2}
.\end{aligned}$$
As in our example, we see that in each case we have $\mu^{X}_{2k}(n)\ge\eta^{X}_{2k}(n)$. We note that the symmetrized moments $\eta^{X}_{2k}(n)$ for $X=F3$, $L5$, and ${
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}$ are non-positive integers; ${\mathrm{spt}^{{
\IfEqCase{38}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 38}{}]}}_{k}\hspace{-.2em} \left(n\right) }=2{\mathrm{spt}^{{
\IfEqCase{41}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 41}{}]}}_{k}\hspace{-.2em} \left(n\right) }$; and ${\mathrm{spt}^{{
\IfEqCase{39}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 39}{}]}}_{k}\hspace{-.2em} \left(n\right) }=2{\mathrm{spt}^{{
\IfEqCase{42}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 42}{}]}}_{k}\hspace{-.2em} \left(n\right) }$. By determining when $\mu^{X}_{2}(n)>\eta^{X}_{2}(n)$, we also obtain the strict inequalities for $M^{X}_{2k}(n)>N^{X}_{2k}(n)$, upon noting that $$\begin{aligned}
M^X_{2k}(n)-N^{X}_{2k}(n)
&=
\sum_{j=1}^{k}(2j)!S^*(k,j)( \mu^X_{2j}(n)-\eta^{X}_{2j}(n))
\ge
\mu^X_{2}(n)-\eta^{X}_{2}(n)
.\end{aligned}$$ We record these inequalities in the following table.
T[able]{} 1. Strict Inequalities for $M^X_{2k}(n)>N^{X}_{2k}(n)$, for positive $k$.
$$\begin{aligned}
\begin{array}{c|c||c|c||c|c}
X & n & X & n & X & n
\\\hline
A1& n \geq 1&
{
\IfEqCase{39}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 39}{}]}& n \geq 2&
F3& n \geq 1
\\
A3& n \geq 2&
{
\IfEqCase{41}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 41}{}]}& n \geq 1&
G1& n=2,{ } n \geq 4
\\
A5& n \geq 2&
{
\IfEqCase{42}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 42}{}]}& n \geq 1&
G3& n=4,{ } n \geq 6
\\
A7& n \geq 1&
E4& n \geq 2&
Y1& n \geq 1
\\
B2& n \geq 2&
I14& n \geq 2&
Y2& n \geq 3
\\
C1& n \geq 1&
{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}& n \geq 1&
Y3& n \geq 2
\\
C2& n \geq 2&
J1& n \geq 1&
Y4& n \geq 4
\\
C5& n \geq 1&
J2& n \geq 1&
L2& n=4,8,9, n\geq 11
\\
L5& n \geq 1&
J3& n \geq 2&
\\
{
\IfEqCase{38}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 38}{}]}& n \geq 1&
{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}& n \geq 1&
\end{array} \end{aligned}$$
We can actually determine inequalities between some of the ranks that compare against the same crank. In particular, we see we have additional inequalities for two rank moments $\eta^{X}_{2k}(n)$ and $\eta^{X^\prime}_{2k}(n)$ that are compared against the same crank when $$\begin{aligned}
\frac{P_{X}(q) {\left(q\right)_{n}}^2\beta^{X}_n}{(1-q^n)^2}
-
\frac{P_{X^\prime}(q) {\left(q\right)_{n}}^2\beta^{X^\prime}_n}{(1-q^n)^2}\end{aligned}$$ clearly has non-negative coefficients. We record the identities that yield such results in the follow corollary and omit the identities that would lead to an inequality that is already present. While we could also use these identities to deduce strict inequalities between the relevant ordinary moments, we leave that as an exercise to the interested reader.
For positive $k$, we have that $$\begin{aligned}
\sum_{n=1}^\infty \left(\eta^{B2}_{2k}(n)-\eta_{2k}(n)\right)q^n
&=
\sum\limits_{n_k \geq \dotsc \geq n_1 \geq 1}
\frac{q^{n_1+\dotsb+n_k} }
{(q^{n_1})_{\infty}(1-q^{n_k})^{2}\dotsm(1-q^{n_2})^2 }
,\\
\sum_{n=1}^\infty \left(\eta^{B2}_{2k}(n)-\eta^{A3}_{2k}(n)\right)q^n
&=
\sum\limits_{n_k \geq \dotsc \geq n_1 \geq 1}
\frac{q^{2n_1+n_2+\dotsb+n_k} }
{(q^{n_1+1})_{\infty}(1-q^{n_k})^{2}\dotsm(1-q^{n_1})^2 }
\times
\left(
\frac{1}{{\left(q^{n_1+1}\right)_{n_1}}}-1
\right)
,\\
\sum_{n=1}^\infty \left(\eta^{B2}_{2k}(n)-\eta^{C2}_{2k}(n)\right)q^n
&=
\sum\limits_{n_k \geq \dotsc \geq n_1 \geq 1}
\frac{q^{2n_1+n_2+\dotsb+n_k} }
{(q^{n_1+1})_{\infty}(1-q^{n_k})^{2}\dotsm(1-q^{n_1})^2 }
\times
\left(
\frac{1}{{\left(q;q^2\right)_{n_1}}}-1
\right)
,\\
\sum_{n=1}^\infty \left(\eta_{2k}(n)-\eta^{A1}_{2k}(n)\right)q^n
&=
\sum\limits_{n_k \geq \dotsc \geq n_1 \geq 1}
\frac{q^{n_1+\dotsb+n_k} }
{(q^{n_1+1})_{\infty}(1-q^{n_k})^{2}\dotsm(1-q^{n_1})^2 }
\times
\left(
\frac{1}{{\left(q^{n_1+1}\right)_{n_1}}}-1
\right)
,\\
\sum_{n=1}^\infty \left(\eta_{2k}(n)-\eta^{C1}_{2k}(n)\right)q^n
&=
\sum\limits_{n_k \geq \dotsc \geq n_1 \geq 1}
\frac{q^{n_1+\dotsb+n_k} }
{(q^{n_1+1})_{\infty}(1-q^{n_k})^{2}\dotsm(1-q^{n_1})^2 }
\times
\left(
\frac{1}{{\left(q;q^2\right)_{n_1}}}-1
\right)
,\\
\sum_{n=1}^\infty \left(\eta^{A5}_{2k}(n)-\eta^{A3}_{2k}(n)\right)q^n
&=
\sum\limits_{n_k \geq \dotsc \geq n_1 \geq 1}
\frac{q^{n_1+\dotsb+n_k}}
{(q^{n_1+1})_{n_1}(q^{n_1})_{\infty}(1-q^{n_k})^{2}\dotsm(1-q^{n_2})^2}
\sum_{j=0}^{n_1-2}q^{jn_1}
,\\
\sum_{n=1}^\infty \left(\eta^{A5}_{2k}(n)-\eta^{A7}_{2k}(n)\right)q^n
&=
\sum\limits_{n_k \geq \dotsc \geq n_1 \geq 1}
\frac{q^{n_1^2+n_2+\dotsb+n_k} }
{(q^{n_1+1})_{n_1}(q^{n_1})_{\infty}(1-q^{n_k})^{2}\dotsm(1-q^{n_2})^2}
,\\
\sum_{n=1}^\infty \left(\eta^{A3}_{2k}(n)-\eta^{A1}_{2k}(n)\right)q^n
&=
\sum\limits_{n_k \geq \dotsc \geq n_1 \geq 1}
\frac{q^{n_1+\dotsb +n_k}}
{(q^{n_1+1})_{n_1}(q^{n_1})_{\infty}(1-q^{n_k})^{2}\dotsm(1-q^{n_2})^2 }
,\\
\sum_{n = 1}^{\infty}\left(\eta_{2k}^{A3}(n)-\eta_{2k}^{{
\IfEqCase{42}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 42}{}]}}(n)\right)q^{n}
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{q^{2n_{1}+n_2 + \dotsb + n_{k}}\left( (-q^2;q^2)_{n_{1}-1}-1 \right) }
{(q^{n_{1}+1})_{n_{1}} (q^{n_{1}+1})_{\infty}
(1-q^{n_{k}})^{2}\dotsm (1-q^{n_{1}})^2}
,\\
\sum_{n=1}^\infty \left(\eta^{A7}_{2k}(n)-\eta^{A1}_{2k}(n)\right)q^n
&=
\sum\limits_{n_k \geq \dotsc \geq n_1 \geq 1}
\frac{q^{n_1+\dotsb +n_k}}
{(q^{n_1+1})_{n_1}(q^{n_1})_{\infty}(1-q^{n_k})^{2}\dotsm(1-q^{n_2})^2 }
\sum_{j=0}^{n_1-2}q^{jn_1}
,\\
\sum_{n = 1}^{\infty}\left(\eta_{2k}^{A1}(n)-\eta_{2k}^{{
\IfEqCase{41}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 41}{}]}}(n)\right)q^{n}
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{q^{n_{1} + \dotsb + n_{k}}\left( (-q^2;q^2)_{n_{1}-1}-1 \right) }
{(q^{n_{1}+1})_{n_{1}} (q^{n_{1}+1})_{\infty}
(1-q^{n_{k}})^{2}\dotsm (1-q^{n_{1}})^2}
,\\
\sum_{n = 1}^{\infty}\left(\eta_{2k}^{{
\IfEqCase{42}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 42}{}]}}(n)-\eta_{2k}^{{
\IfEqCase{41}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 41}{}]}}(n)\right)q^{n}
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{(-q^2;q^2)_{n_{1}-1} q^{n_{1} + \dotsb + n_{k}}}
{(q^{n_{1}+1})_{n_{1}} (q^{n_{1}})_{\infty}
(1-q^{n_{k}})^{2}\dotsm (1-q^{n_{2}})^2}
,\\
\sum_{n = 1}^{\infty}\left(\eta_{2k}^{{
\IfEqCase{42}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 42}{}]}}(n)-\eta_{2k}^{{
\IfEqCase{39}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 39}{}]}}(n)\right)q^{n}
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{(-q^2;q^2)_{n_{1}-1} q^{2n_{1} +n_2+ \dotsb + n_{k}}}
{(q^{n_{1}+1})_{n_{1}} (q^{n_{1}+1})_{\infty}
(1-q^{n_{k}})^{2}\dotsm (1-q^{n_{1}})^2}
,\\
\sum_{n = 1}^{\infty}\left(\eta^{C2}_{2k}(n)-\eta_{2k}^{C1}(n)\right)q^{n}
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{q^{n_{1} + n_{2} + \dotsb + n_{k}}}
{(q;q^2)_{n_{1}} (q^{n_{1}})_{\infty} (1-q^{n_{k}})^{2}\dotsm (1-q^{n_{2}})^{2}}
,\\
\sum_{n = 1}^{\infty}\left(\eta_{2k}^{{
\IfEqCase{39}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 39}{}]}}(n)-\eta_{2k}^{{
\IfEqCase{38}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 38}{}]}}(n)\right)q^{n}
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{(-1;q^{2})_{n_{1}}q^{n_{1} + \dotsb + n_{k}}}
{(q^{n_{1} + 1})_{n_{1}} (q^{n_{1}})_{\infty} (1-q^{n_{k}})^{2}\dotsm (1-q^{n_{2}})^2}
,\\
\sum_{n = 1}^{\infty}\left(\eta_{2k}^{{
\IfEqCase{41}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 41}{}]}}(n)-\eta_{2k}^{{
\IfEqCase{38}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 38}{}]}}(n)\right)q^{n}
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{(-q^2;q^2)_{n_{1}-1} q^{n_{1} + \dotsb + n_{k}}}
{(q^{n_{1}+1})_{n_{1}} (q^{n_{1}+1})_{\infty}
(1-q^{n_{k}})^{2}\dotsm (1-q^{n_{1}})^2}
,\\
\sum_{n = 1}^{\infty}\left(\eta^{C5}_{2k}(n)-\eta_{2k}^{C1}(n)\right)q^{n}
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{q^{n_{1} + n_{2} + \dotsb + n_{k}} }
{ (q^{n_{1}})_{\infty} (1-q^{n_{k}})^{2}\dotsm (1-q^{n_{2}})^{2}}
\times
\frac{(1-q^{\frac{n_1(n_1-1)}{2}})}{(q;q^2)_{n_{1}}(1-q^{n_1})}
,\\
\sum_{n = 1}^{\infty}\left(\eta^{C1}_{2k}(n)-\eta_{2k}^{L5}(n)\right)q^{n}
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{q^{n_{1} + \dotsb + n_{k}}\left( {\left(-1\right)_{n_1}}-1 \right)}
{(q;q^2)_{n_{1}} (q^{n_{1}+1})_{\infty} (1-q^{n_{k}})^{2}\dotsm (1-q^{n_{1}})^{2}}
,\\
\sum_{n=1}^\infty \left(\eta^{J3}_{2k}(n)-\eta^{J2}_{2k}(n)\right)q^n
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{q^{n_{1} + n_{2} + \dotsb + n_{k}}}
{(q)_{n_{1}-1} (q^{n_{1}+1})_{n_{1}} (q^{3n_1};q^3)_\infty (q^{n_{1}})_{\infty}
(1-q^{n_{k}})^{2}\dotsm(1-q^{n_{2}})^{2} }
,\\
\sum_{n=1}^\infty \left(\eta^{J2}_{2k}(n)-\eta^{J1}_{2k}(n)\right)q^n
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{q^{2n_{1} + n_{2} + \dotsb + n_{k}}}
{(q)_{2n_{1}} (q^{3n_1};q^3)_\infty (q^{n_{1}})_{\infty}
(1-q^{n_{k}})^{2}\dotsm(1-q^{n_{2}})^{2}}
,\\
\sum_{n=1}^\infty \left(\eta^{E4}_{2k}(n)-\overline{\eta}_{2k}(n)\right)q^n
&=
\sum\limits_{n_k \geq \dotsc \geq n_1 \geq 1}
\frac{{\left(-q^{n_1+1}\right)_{\infty}} q^{n_1+\dotsb+n_k} }
{(q^{n_1})_{\infty} (1-q^{n_k})^{2}\dotsm(1-q^{n_2})^2 }
,\\
\sum_{n = 1}^{\infty}\left(\overline{\eta}_{2k}(n)-\eta_{2k}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}(n)\right)q^{n}
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{(-q^{n_{1} + 1})_{\infty} q^{n_1+ \dotsb + n_k}}
{(q^{n_1+1})_{\infty} (1-q^{n_{k}})^{2} \dotsm (1-q^{n_{1}})^{2}}
\times
\left(\frac{(-q^3;q^3)_{n_{1}-1} }{(q^{n_{1} + 1})_{n_{1}-1} }-1\right)
,\\
\sum_{n = 1}^{\infty}\left(\overline{\eta}_{2k}(n)-\eta_{2k}^{I14}(n)\right)q^{n}
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{(-q^{n_{1}+1})_{\infty} q^{n_{1} + \cdots{} + n_{k}}}
{(q^{n_{1} + 1})_{\infty}
(1-q^{n_{k}})^{2}\dotsm (1-q^{n_{1}})^{2}}
\\&\quad
\times
\left(
\frac{(-q^2;q^2)_{n_{1}-1}(1+q^{n_1}) }{(q;q^{2})_{n_{1}} } - 1
\right)
,\\
\sum_{n = 1}^{\infty}\left(\eta^{G3}_{2k}(n)- \eta_{2k}^{G1}(n)\right)q^{n}
&=
\sum\limits_{n_k \geq \dotsb \geq n_1 \geq 1}
\frac{(-q^{2n_1+1};q^2)_\infty q^{2n_1+\dotsb +2n_k}}
{(q^4;q^4)_{n_1} (q^{2n_1+2};q^2)_{\infty} (q^{2n_1};q^2)_{\infty}
(1-q^{2n_k})^2\dotsm (1-q^{2n_2})^2}
,\\
\sum_{n = 1}^{\infty}\left(\eta_{2k}^{Y2}(n) - \eta_{2k}^{Y1}(n)\right)q^{n}
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{q^{n_{1}^2 + 2n_{2} + \dotsb + 2n_{k}}}
{(q;q^2)_{n_{1}} (q^4;q^4)_{n_{1}} (q^{2n_{1} + 2} ; q^2)_{\infty}(q^{2n_{1}} ; q^2)_{\infty} }
\\&\quad
\times
\frac{1}{(1-q^{2n_{k}})^{2} \dotsm (1-q^{2n_{2}})^2}
,\\
\sum_{n = 1}^{\infty}\left(\eta_{2k}^{Y2}(n) - \eta_{2k}^{Y3}(n)\right)q^{n}
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{q^{2n_{1} + \dotsb + 2n_{k}}}
{(q^4;q^4)_{n_{1}} (q^{2n_{1}} ; q^2)_{\infty} (q^{2n_{1} + 2} ; q^2)_{\infty}
(1-q^{2n_{k}})^{2} \dotsm (1-q^{2n_{2}})^2}
\\&\quad
\times
\frac{(1-q^{n_1^2})}
{{\left(q;q^2\right)_{n_1}}(1-q^{2n_1})}
,\\
\sum_{n = 1}^{\infty}\left(\eta_{2k}^{Y4}(n) - \eta_{2k}^{Y3}(n)\right)q^{n}
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{q^{2n_{1} + \dotsb + 2n_{k}}}
{(q;q^2)_{n_{1}} (q^4;q^4)_{n_{1}} (q^{2n_{1} + 2} ; q^2)_{\infty}(q^{2n_{1}} ; q^2)_{\infty}}
\\&\quad
\times
\frac{1}{(1-q^{2n_{k}})^{2} \dotsm (1-q^{2n_{2}})^2}
. \end{aligned}$$ Here $\overline{\eta}_{2k}$ is the symmetrized overpartition rank moment from [@JenningsShaffer3] and satisfies $$\begin{aligned}
\sum_{n = 1}^{\infty}\left(\overline{\mu}_{2k}(n)-\overline{\eta}_{2k}(n)\right)q^{n}
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{(-q^{n_{1}+1})_{\infty} q^{n_{1} + \cdots{} + n_{k}}}
{(q^{n_{1} + 1})_{\infty} (1-q^{n_{k}})^{2}\dotsm (1-q^{n_{1}})^{2}}
. \end{aligned}$$
The Proof of Theorem \[TheoremMain\]
====================================
\[LemmaRankBilateralAndDerivative\] Suppose $\alpha_n$ is a sequence such that $\alpha_{n} = \alpha_{-n}$, then $$\begin{aligned}
\sum_{n = 1}^{\infty} \frac{\alpha_{n}q^n(1-z)(1-z^{-1})}{(1-zq^n)(1-z^{-1}q^n)}
&=
\sum_{n \neq 0}\frac{\alpha_{n}q^n(1-z)}{(1+q^{n})(1-zq^{n})}
.\end{aligned}$$ Furthermore, if $j$ is a positive integer, then $$\begin{aligned}
\left(\frac{\partial}{\partial{}z}\right)^j
\sum_{n = 1}^{\infty} \frac{\alpha_{n}q^n(1-z)(1-z^{-1})}{(1-zq^n)(1-z^{-1}q^n)}
&=
-j!\sum_{n \neq 0}
\frac{\alpha_{n}q^{nj}(1-q^n)}{(1+q^n)(1-zq^n)^{j+1}}
.\end{aligned}$$
We note the first identity is the standard rearrangements used for many rank functions. We have that $$\begin{aligned}
\sum_{n = 1}^{\infty}
\frac{\alpha_{n}q^n(1-z)(1-z^{-1})}{(1-zq^n)(1-z^{-1}q^n)}
&=
\sum_{n = 1}^{\infty}
\frac{\alpha_{n}q^n}{1 + q^n}
\left(\frac{1-z}{1-zq^n} + \frac{1-z^{-1}}{1-z^{-1}q^n}\right)
\\
&=
\sum_{n = 1}^{\infty}
\frac{\alpha_{n}q^n(1-z)}{(1 + q^n)(1-zq^n)}
+
\sum_{n = -\infty}^{-1}
\frac{\alpha_{-n}q^{-n}(1-z^{-1})}{(1 + q^{-n})(1-z^{-1}q^{-n})}
\\
&=
\sum_{n = 1}^{\infty}
\frac{\alpha_{n}q^n(1-z)}{(1 + q^n)(1-zq^n)}
+
\sum_{n = -\infty}^{-1}
\frac{\alpha_{n}q^n(1-z)}{(1 + q^{n})(1-zq^{n})}
\\
&=
\sum_{n \neq 0}\frac{\alpha_{n}q^n(1-z)}{(1+q^{n})(1-zq^{n})}
.\end{aligned}$$ This establishes the first identity. The second identity follows from the first, upon noting that $$\begin{aligned}
\left(\frac{\partial}{\partial{}z}\right)^j
\frac{1-z}{1-zq^n}
&=
\frac{-j!(1-q^n)q^{n(j-1)}}{(1-zq^n)^{j+1}}
.\end{aligned}$$
\[LemmaSymmetrizedMoments\] Suppose $\alpha_n$ is a sequence such that $\alpha_{n} = \alpha_{-n}$ and $$\begin{aligned}
R_{X}(z,q)
:=
P_{X}(q)
\left(
1
+
\sum_{n = 1}^{\infty} \frac{\alpha_{n}q^n(1-z)(1-z^{-1})}{(1-zq^n)(1-z^{-1}q^n)}
\right)
&=
\sum_{n=0}^\infty \sum_{m=-\infty}^\infty
N_{X}(m,n) z^mq^n
,\end{aligned}$$ where $P_{X}(q)$ is some series in $q$. Let $$\begin{aligned}
\eta^{X}_{k}(n)
&=
\sum_{m = -\infty}^{\infty}
\binom{m+\lfloor\frac{k-1}{2}\rfloor}{k}N_{X}(m,n)
.\end{aligned}$$ Then $\eta^{X}_{2k+1}(n)=0$ and $$\begin{aligned}
\sum_{n=1}^\infty \eta^X_{2k}(n) q^n
&=
-P_X(q)
\sum_{n=1}^\infty
\frac{\alpha_{n}q^{nk}}{(1-q^n)^{2k}}
.\end{aligned}$$
We note this is the generalization of Theorems 1 and 2 of [@Andrews4]. The proof that $\eta^{X}_{2k+1}=0$ follows that of Theorem 1 of [@Andrews4] verbatim, as we have $N_X(m,n)=N_X(-m,n)$ due to the symmetry in $z$ and $z^{-1}$. For the even moments, much the same as in the proof of Theorem 2 from [@Andrews3], by Lemma \[LemmaRankBilateralAndDerivative\] we have that $$\begin{aligned}
\sum\limits_{n=1}^\infty \eta^X_{2k}(n)q^n
&=
\frac{1}{(2k)!}
\left(\frac{\partial}{\partial{}z}\right)^{2k}
z^{k-1}R_X(z,q) \Bigr|_{z = 1}
\\
&=
\frac{1}{(2k)!}
\sum_{j = 0}^{k-1}
\binom{2k}{j}(k-1)\cdots{}(k-j)R_X^{(2k-j)}(1,q)
\\
&=
-P_X(q)
\sum_{j=0}^{k-1} \binom{k-1}{j}
\sum_{n \neq 0}
\frac{\alpha_{n}q^{n(2k-j)}(1-q^n)}{(1+q^n)(1-q^n)^{2k-j+1}}
\\
&=
-P_X(q)
\sum_{n \neq 0}\frac{\alpha_{n}q^{2nk}}{(1 - q^n)^{2k}(1+q^n)}
\sum_{j = 0}^{k-1}\binom{k-1}{j}(q^{-n}(1 - q^{n}))^{j}
\\
&=
-P_X(q)
\sum_{n \neq 0}
\frac{\alpha_{n}q^{2nk}}{(1 - q^n)^{2k}(1+q^n)}( 1 + q^{-n}(1 - q^{n}))^{k - 1}
\\
&=
-P_X(q)
\sum_{n \neq 0}
\frac{\alpha_{n}q^{n + nk}}{(1 - q^n)^{2k}(1+q^n)}
\\
&=
-P_X(q)
\sum_{n = 1}^{\infty}
\frac{\alpha_{n}q^{nk}}{(1-q^n)^{2k}}
.\end{aligned}$$
With the above lemma, we have established (\[EqMainTheoremSymmetrizedRankMoment\]) of Theorem \[TheoremMain\]. We then find (\[EqMainTheoremSymmetrizexRankCrankDifference\]) follows immediately from equation (3.3) and Theorem 3.3 of [@Garvan1], both of which we state for completeness. Equation (3.3) is that for positive $k$, $$\begin{aligned}
\sum_{n=1}^\infty \mu_{2k}(n)q^n
&=
\frac{1}{{\left(q\right)_{\infty}}}
\sum_{n_k\geq \dotsb \geq n_1\geq 1}
\frac{q^{n_1+\dotsb+n_k}}{(1-q^{n_k})^2\dotsm (1-q^n_1)^2}
.\end{aligned}$$ Theorem 3.3 states that if $\alpha_{n}$ and $\beta_{n}$ are a Bailey pair relative to $(1,q)$ and $\alpha_{0} = \beta_{0} = 1$. Then $$\begin{aligned}
&\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{(q )_{n_{1}}^{2} q^{n_{1} + \dotsb + n_{k}}\beta_{n_{1}}}
{(1-q^{n_{k}})^{2}(1-q^{n_{k-1}})^{2} \dotsm (1-q^{n_{1}})^{2}}
\\
&=
\sum_{n_{k} \geq \dotsb \geq n_{1} \geq 1}
\frac{q^{n_{1} + \dotsb + n_{k}}}
{(1-q^{n_{k}})^{2}(1-q^{n_{k-1}})^{2} \dotsm (1-q^{n_{1}})^{2}}
+
\sum_{r = 1}^{\infty}\frac{q^{kr}\alpha_{r}}{(1-q^{r})^{2k}}
.\end{aligned}$$ Lastly, for (\[EqMainTheoremSymmetrizedRankEquality\]) and (\[EqMainTheoremOrdinaryRankEquality\]), we find that one can follow verbatim the proof of Theorem 4.3 of [@Garvan1], as the only requirement is that $N_X(m,n)=N_{X}(-m,n)$. Thus Theorem \[TheoremMain\] follows.
Combinatorial Interpretations
=============================
To determine what the functions ${\mathrm{spt}^{X}_{2k}\hspace{-.2em} \left(n\right) }$ are counting, we use the weight from [@Garvan1] as extended to vector partitions in [@JenningsShaffer3]. We recall that a vector partition of $n$ is a vector of partitions, $(\pi_1,\dotsc,\pi_r)$, such that altogether the parts sum to $n$. When $\vec{\pi}$ is a vector partition of $n$, we write $|\vec{\pi}|=n$. For a partition $\pi$ with different parts $n_1 < n_2 < \cdots{} < n_m$, we take $f_{j}(\pi)$ to be the frequency of the part $n_j$. Given a vector partition $\vec{\pi}=(\pi_1,\pi_2,\dotsc,\pi_r)$, we let $f^1_{j} := f^1_{j}(\vec{\pi}) = f_j(\pi_1)$, and have the weight $\omega_k(\vec{\pi})$ given by $$\begin{aligned}
\omega_{k}(\vec{\pi})
&:=
\sum_{\substack{m_1 + m_2 + \cdots{} + m_r = k \\ 1 \leq r \leq k}}
\binom{f_{1}^{1} + m_1 - 1}{2m_1 - 1}
\sum_{2 \leq j_2 < j_3 < \cdots < j_r}
\binom{f_{j_2}^{1} + m_2}{2m_2}\binom{f_{j_3}^{1} + m_3}{2m_3}\cdots{}\binom{f_{j_r}^{1} + m_r}{2m_r}
.\end{aligned}$$ We note that the first sum is over all compositions of $k$. Based on the combinatorial interpretations in [@Garvan1; @JenningsShaffer3], we expect to find that ${\mathrm{spt}^{X}_{2k}\hspace{-.2em} \left(n\right) }$ is the number of vector partitions $\vec{\pi}=(\pi_1,\pi_2,\dotsc,\pi_r)$, weighted by $\omega_{k}$, where $(\pi_2,\dotsc,\pi_r)$ are restricted to the vector partitions enumerated by ${\left(q^{n_1+1}\right)_{\infty}}{\left(q\right)_{n_1}}^2 P_X(q) \beta_{n_1}$, where $n_1$ is the smallest part of $\pi_1$. Upon minor adjustments for the cases where $q\rightarrow q^2$ and $q\rightarrow q^4$, this is correct. In a small number of cases one of the partitions $\pi_k$, for $k\geq 2$, strictly speaking may not be a partition in that we will allow a non-positive part to appear.
We define the follows sets of vector partitions. In all cases we require that $\pi_1$ be non-empty.
- $S^{A1}$ - the set of vector partitions $(\pi_1, \pi_2)$ where $\pi_1$ has smallest part $n$ and the parts $m$ of $\pi_2$ satisfy $n+1 \leq m \leq 2n$.
- $S^{A3}$ - the set of vector partitions $(\pi_1, \pi_2)$ where $\pi_1$ has smallest part $n$; the parts $m$ of $\pi_2$ satisfy $n \leq m \leq 2n$; and the part $n$ appears exactly once in $\pi_2$.
- $S^{A5}$ - the set of vector partitions $(\pi_1, \pi_2)$ where $\pi_1$ has smallest part $n$; the parts $m$ of $\pi_2$ satisfy $n\leq m \leq 2n$; and the part $n$ appears exactly $n$ times in $\pi_2$.
- $S^{A7}$ - the set of vector partitions $(\pi_1, \pi_2)$ where $\pi_1$ has smallest part $n$; the parts $m$ of $\pi_2$ satisfy $n \leq m \leq 2n$; and the part $n$ appears exactly $n-1$ times in $\pi_2$.
- $S^{B2}$ - the set of vector partitions $(\pi_1, \pi_2)$ where $\pi_1$ has smallest part $n$ and $\pi_2$ consists of exactly one copy of the part $n$.
- $S^{C1}$ - the set of vector partitions $(\pi_1, \pi_2)$ where the smallest part of $\pi_1$ is $n$ and the parts $m$ of $\pi_2$ are odd and satisfy $1\leq m\le 2n-1$.
- $S^{C2}$ - the set of vector partitions $(\pi_1, \pi_2)$ where the smallest part of $\pi_1$ is $n$; the parts $m$ of $\pi_2$ satisfy $1\leq m\leq 2n-1$; the parts of $\pi_2$ other than $n$ are odd; if $n$ is odd, then $n$ appears at least once in $\pi_2$; and if $n$ is even, then $n$ appears exactly once in $\pi_2$.
- $S^{C5}$ - the set of vector partitions $(\pi_1, \pi_2, \pi_3)$ where the smallest part of $\pi_1$ is $n$; the parts $m$ of $\pi_2$ are odd and satisfy $1\leq m\leq 2n-1$; and $\pi_3$ consists of exactly one copy of the part $\frac{n(n - 1)}{2}$.
- $S^{L5}$ - the set of vector partitions $(\pi_1, \pi_2, \pi_3)$ where $\pi_1$ has smallest part $n$; the parts $m$ of $\pi_2$ are odd and satisfy $1\leq m\leq 2n-1$; and the parts $m$ of $\pi_3$ are distinct and satisfy $0\leq m \leq n-1$ (so that $\pi_3$ may contain $0$ at most once).
- $S^{{
\IfEqCase{38}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 38}{}]}}$ - the set of vector partitions $(\pi_1, \pi_2, \pi_3)$ where $\pi_1$ has smallest part $n$; the parts $m$ of $\pi_2$ satisfy $n+1\leq m\leq 2n$; and the parts $m$ of $\pi_3$ are distinct even parts and satisfy $0\leq m\leq 2n-2$ (so that $\pi_3$ may contain the part $0$ at most once).
- $S^{{
\IfEqCase{39}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 39}{}]}}$ - the set of vector partitions $(\pi_1, \pi_2, \pi_3)$ where $\pi_1$ has smallest part $n$; the parts $m$ of $\pi_2$ satisfy $n\leq m\leq 2n$; the part $n$ appears exactly once in $\pi_2$; and the parts $m$ of $\pi_3$ are distinct even parts and satisfy $0\leq m\leq 2n-2$ (so that $\pi_3$ may contain the part $0$ at most once).
- $S^{{
\IfEqCase{41}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 41}{}]}}$ - the set of vector partitions $(\pi_1, \pi_2, \pi_3)$ where $\pi_1$ has smallest $n$; the parts $m$ of $\pi_2$ satisfy $n+1\leq m\leq 2n$; and the parts $m$ of $\pi_3$ are distinct even parts and satisfy $1\leq m\leq 2n-2$.
- $S^{{
\IfEqCase{42}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 42}{}]}}$ - the set of vector partitions $(\pi_1, \pi_2, \pi_3)$ where $\pi_1$ has smallest $n$; the parts $m$ of $\pi_2$ satisfy $n\leq m\leq 2n$; the part $n$ appears exactly once in $\pi_2$; and the parts $m$ of $\pi_3$ are distinct even parts and satisfy $1\leq m\leq 2n-2$.
- $S^{E4}$ - the set of vector partitions $(\pi_1, \pi_2)$ where $\pi_1$ has smallest part $n$; the parts $m$ of $\pi_2$ are distinct and satisfy $n\leq m$; and the part $n$ appears exactly once in $\pi_2$.
- $S^{I14}$ - the set of vector partitions $(\pi_1, \pi_2, \pi_3)$ where $\pi_1$ has smallest part $n$; the parts $m$ of $\pi_2$ satisfy $1\leq m\leq 2n-1$; the even parts of $\pi_2$ are distinct; and the parts $m$ of $\pi_3$ are distinct and satisfy $n\leq m$.
- $S^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}$ - the set of vector partitions $(\pi_1, \pi_2, \pi_3, \pi_4)$ where $\pi_1$ has smallest part $n$; the parts $m$ of $\pi_2$ are distinct multiples of $3$ and satisfy $1\leq m\leq 3n-3$; the parts $m$ of $\pi_3$ are distinct and satisfy $n+1\leq m$; and the parts $m$ of $\pi_3$ satisfy $n+1\leq m \leq 2n-1$.
- $S^{J1}$ - the set of vector partitions $(\pi_1, \pi_2)$ where $\pi_1$ has smallest part $n$; $\pi_2$ has no part $m$ such that $2n\leq m\leq 3n-1$; and the parts $m$ of $\pi_2$ satisfying $3n\leq m$ are multiplies of $3$.
- $S^{J2}$ - the set of vector partitions $(\pi_1, \pi_2)$ where $\pi_1$ has smallest part $n$; $\pi_2$ has no part $m$ such that $m=n$ or $2n+1\leq m\leq 3n-1$; and the parts $m$ of $\pi_2$ satisfying $3n\leq m$ are multiples of $3$.
- $S^{J3}$ - the set of vector partitions $(\pi_1, \pi_2)$ where $\pi_1$ has smallest part $n$; $\pi_2$ has no part $m$ such that $2n+1\leq m\leq 3n-1$; the part $n$ appears exactly once in $\pi_2$; and the parts $m$ of $\pi_2$ satisfying $3n\leq m$ are multiples of $3$.
- $S^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}$ - the set of vector partitions $(\pi_1, \pi_2, \pi_3)$ where $\pi_1$ has smallest part $n$; the parts $m$ of $\pi_2$ satisfy $1\leq m\leq n-1$; the parts $m$ of $\pi_3$ satisfying $1\leq m\leq 2n-1$ are odd; and the parts $m$ of $\pi_3$ satisfying $2n\leq m$ are even.
- $S^{F3}$ - the set of vector partitions $(\pi_1, \pi_2)$ where the parts of $\pi_1$ are even and the smallest part is $2n$; $\pi_2$ contains exactly one copy of the negative integer $-n$; and the positive parts $m$ of $\pi_2$ are odd and satisfy $1\leq m\leq 2n-1$.
- $S^{G1}$ - the set of vector partitions $(\pi_1, \pi_2, \pi_3)$ where the parts of $\pi_1$ are even and the smallest part is $2n$; the parts $m$ of $\pi_2$ are multiples of $4$ and satisfy $1\leq m\leq 4n$; the parts $m$ of $\pi_3$ satisfy $2n+1\leq m$; and the odd parts of $\pi_3$ are distinct.
- $S^{G3}$ - the set of vector partitions $(\pi_1, \pi_2, \pi_3)$ where the parts of $\pi_1$ are even and the smallest part is $2n$; the parts $m$ of $\pi_2$ are multiples of $4$ and satisfy $1\leq m\leq 4n$; the parts $m$ of $\pi_3$ satisfy $2n\leq m$; the odd parts of $\pi_3$ are distinct; and the part $2n$ appears exactly once in $\pi_3$.
- $S^{Y1}$ - the set of vector partitions $(\pi_1,\pi_2,\pi_3,\pi_4)$ where the parts of $\pi_1$ are even and the smallest part is $2n$; the parts $m$ of $\pi_2$ are multiples of $4$ and satisfy $1\leq m\leq 4n$; the parts $m$ of $\pi_3$ satisfying $1\leq m\leq 2n-1$ are odd; the parts $m$ of $\pi_3$ satisfying $2n+2\leq m$ are even; the parts $2n$ and $2n+1$ do not appear in $\pi_3$; and $\pi_4$ consists of exactly one copy of the part $n^2-2n$ (which in the case of $n=1$ is $-1$).
- $S^{Y2}$ - the set of vector partitions $(\pi_1,\pi_2,\pi_3,\pi_4)$ where the parts of $\pi_1$ are even and the smallest part is $2n$; the parts $m$ of $\pi_2$ are multiples of $4$ and satisfy $1\leq m\leq 4n$; the parts $m$ of $\pi_3$ satisfying $1\leq m\leq 2n-1$ are odd; the parts $m$ of $\pi_3$ satisfying $2n+2\leq m$ are even; the parts $2n$ and $2n+1$ do not appear in $\pi_3$; and $\pi_4$ consists of exactly one copy of the part $n^2$.
- $S^{Y3}$ - the set of vector partitions $(\pi_1,\pi_2,\pi_3)$ where the parts of $\pi_1$ are even and the smallest part is $2n$; the parts $m$ of $\pi_2$ are multiples of $4$ and satisfy $1\leq m\leq 4n$; the parts $m$ of $\pi_3$ satisfying $1\leq m\leq 2n-1$ are odd; the parts $m$ of $\pi_3$ satisfying $2n+2\leq m$ are even; and the parts $2n$ and $2n+1$ do not appear in $\pi_3$.
- $S^{Y4}$ - the set of vector partitions $(\pi_1,\pi_2,\pi_3)$ where the parts of $\pi_1$ are even and the smallest part is $2n$; the parts $m$ of $\pi_2$ are multiples of $4$ and satisfy $1\leq m\leq 4n$; the parts $m$ of $\pi_3$ satisfying $1\leq m\leq 2n-1$ are odd; the parts $m$ of $\pi_3$ satisfying $2n\leq m$ are even; and the part $2n$ appears exactly once in $\pi_3$.
- $S^{L2}$ - the set of vector partitions $(\pi_1, \pi_2)$ where the parts of $\pi_1$ are multiplies of $4$ and the smallest part is $4n$; the parts $m$ of $\pi_2$ satisfy $m\not\equiv 2\pmod{4}$ and $4n+1\leq m$; and the parts $m$ of $\pi_2$ that are multiples of $4$ additionally satisfy $m\leq 8n$.
Suppose the assumptions and notation of Theorem \[TheoremMain\] and let $$\begin{aligned}
\beta'_{n}(q)
&=
(q^{n + 1})_{\infty}(q)_{n}^{2} P_X(q) \beta_{n}
.\end{aligned}$$ Suppose $k$ is a positive integer and let $A$ denote the set of all compositions of $k$, then $$\begin{aligned}
\sum_{n=1}^{\infty} {\mathrm{spt}^{X}_{k}\hspace{-.2em} \left(n\right) }q^n
&=
\sum_{(m_1,\dotsc,m_r)=\vec{m} \in A}
\hspace{.5em}
\sum_{1 \leq n_{1} < n_{j_2}< \dotsb < n_{j_r}}
\sum_{f_1=m_1}^{\infty} \sum_{f_{j_2}=m_2}^{\infty}
\dotsm \sum_{f_{j_r}=m_r}^{\infty}
\binom{f_{1}+m_1-1}{2m_1-1}
\\&\quad
\times \binom{f_{j_2}+m_2}{2m_2} \dotsm
\binom{f_{j_r}+m_r}{2m_r} q^{n_1f_1+n_{j_2}f_{j_2}+ \dotsb +n_{j_r}f_{j_r}}
\beta_{n_1}'(q) \prod_{\substack{i > n_1 \\ i \notin \{n_{j_2},\dotsc ,n_{j_r}\}}} \frac{1}{1 - q^i}
.\end{aligned}$$
In particular for all of the Bailey pairs considered in Corollary \[CorollarySeriesIdentities\], we have for all positive $k$ and $n$ that $$\begin{aligned}
\sum_{n=1}^\infty
{\mathrm{spt}^{X}_{k}\hspace{-.2em} \left(n\right) }q^n
&=
\sum_{\vec{\pi} \in S^{X}}\omega_{k}(\vec{\pi}) q^{|\vec{\pi}|}
.\end{aligned}$$
The proof is much the same as that of Theorem 5.6 from [@Garvan1] and its generalization to Theorem 3.1 from [@JenningsShaffer3]. We first note that $$\begin{aligned}
\label{EqTheoremCombinatorics}
\sum_{n=1}^{\infty} {\mathrm{spt}^{X}_{k}\hspace{-.2em} \left(n\right) }q^n
&=
\sum_{1 \leq n_1 \leq n_2 \leq \dotsb \leq n_k }
\frac{ \beta'_{n_1}(q) q^{n_1+n_2+\dotsb +n_k}}
{(q^{n_1+1})_{\infty}(1-q^{n_k})^2\dotsm (1-q^{n_1})^2}
.\end{aligned}$$ We will use that $$\begin{aligned}
\frac{x^j}{(1-x)^{2j}}
&=
\sum_{n = j}^{\infty}\binom{n + j - 1}{2j - 1}x^n
,&
\frac{x^j}{(1-x)^{2j+1}}
&=
\sum_{n = j}^{\infty}\binom{n + j}{2j}x^n
.\end{aligned}$$
To illustrate the series rearrangements of the general proof, we first write out in full detail the case when $k = 3$. For $k=3$ we have $$\begin{aligned}
&\sum_{n=1}^{\infty} {\mathrm{spt}^{X}_{3}\hspace{-.2em} \left(n\right) }q^n
=
\sum_{n_{j_3} \geq n_{j_2} \geq n_{1} \geq 1}
\frac{\beta_{n_1}'(q)q^{n_{1} + n_{j_2} + n_{j_3}}}
{(q^{n_{1} + 1})_{\infty}(1-q^{n_{j_3}})^{2}(1-q^{n_{j_2}})^{2}(1-q^{n_{1}})^{2}}
\\
&=
\left(
\sum_{1 \leq n_{1} = n_{j_2} = n_{j_3}}
+
\sum_{1 \leq n_{1} = n_{j_2} < n_{j_3}}
+
\sum_{1 \leq n_{1} < n_{j_2} = n_{j_3}}
+
\sum_{1 \leq n_{1} < n_{j_2} < n_{j_3}}
\right)
\frac{\beta_{n_1}'(q)q^{n_{1} + n_{j_2} + n_{j_3}}}
{(q^{n_{1} + 1})_{\infty}(1-q^{n_{j_3}})^{2}(1-q^{n_{j_2}})^{2}(1-q^{n_{1}})^{2}}
\\
&=
\sum_{1 \leq n_1}
\frac{q^{3n_1}}{(1-q^{n_1})^6}\beta_{n_1}'(q)\prod_{i >n_1}\frac{1}{1-q^i}
+
\sum_{1 \leq n_1 < n_{j_3}}
\frac{q^{2n_1}}{(1-q^{n_1})^4} \frac{q^{n_{j_3}}}{(1-q^{n_{j_3}})^3}
\beta_{n_1}'(q)\prod_{\substack{i > n_1 \\ i \neq n_{j_3}}} \frac{1}{1 - q^i}
\\&\quad
+
\sum_{1 \leq n_1 < n_{j_2}}
\frac{q^{n_1}}{(1-q^{n_1})^2} \frac{q^{2n_{j_2}}}{(1-q^{n_{j_2}})^5}
\beta_{n_1}'(q)\prod_{\substack{i > n_1 \\ i \neq n_{j_2}}} \frac{1}{1 - q^i}
\\&\quad
+
\sum_{1 \leq n_1 < n_{j_2} <n_{j_3}}
\frac{q^{n_1}}{(1-q^{n_1})^2} \frac{q^{n_{j_2}}}{(1-q^{n_{j_2}})^3}\frac{q^{n_{j_3}}}{(1-q^{n_{j_3}})^3}
\beta_{n_1}'(q)\prod_{\substack{i > n_1 \\ i \neq n_{j_2},n_{j_3}}} \frac{1}{1 - q^i}
\\
&=
\sum_{1 \leq n_{1}} \sum_{f_1=3}^{\infty}
\binom{f_1+3-1}{6-1}q^{n_1f_1}\beta_{n_1}'(q)\prod_{i >n_1}\frac{1}{1-q^i}
\\&\quad
+
\sum_{1 \leq n_1 < n_{j_3}} \sum_{f_1=2}^{\infty}
\binom{f_1+2-1}{4-1}q^{n_1f_1}\sum_{f_{j_3}=1}^{\infty}
\binom{f_{j_3}+1}{2}q^{n_{j_3}f_{j_3}}
\beta_{n_1}'(q)\prod_{\substack{i > n_1 \\ i \neq n_{j_3}}} \frac{1}{1 - q^i}
\\&\quad
+
\sum_{1 \leq n_1 < n_{j_2}} \sum_{f_1=1}^{\infty}
\binom{f_1+1-1}{2-1}q^{n_1f_1}\sum_{f_{j_2}=2}^{\infty}
\binom{f_{j_2}+2}{4}q^{n_{j_2}f_{j_2}}
\beta_{n_1}'(q)\prod_{\substack{i > n_1 \\ i \neq n_{j_2}}} \frac{1}{1 - q^i}
\\&\quad
+
\sum_{1 \leq n_1 < n_{j_2}< n_{j_3}} \sum_{f_1=1}^{\infty}
\binom{f_1+1-1}{2-1}q^{n_1f_1}\sum_{f_{j_2}=1}^{\infty}
\binom{f_{j_2}+1}{2}q^{n_{j_2}f_{j_2}}\sum_{f_{j_3}=1}^{\infty}
\binom{f_{j_3}+1}{2}q^{n_{j_3}f_{j_3}}
\beta_{n_1}'(q)
\hspace{-.5em}
\prod_{\substack{i > n_1 \\ i \neq n_{j_2},n_{j_3}}} \frac{1}{1 - q^i}
.\end{aligned}$$ Now the set of compositions of 3 is $A= \{(3),(2,1),(1,2),(1,1,1)\}$, and so we have that $$\begin{aligned}
\sum_{n=1}^{\infty} {\mathrm{spt}^{X}_{3}\hspace{-.2em} \left(n\right) }q^n
&=
\sum_{(m_1,\dotsc ,m_r)=\vec{m} \in A}
\hspace{.5em}
\sum_{1 \leq n_{1} < n_{j_2}<\dotsb < n_{j_r}}
\sum_{f_1=m_1}^{\infty} \sum_{f_{j_2}=m_2}^{\infty} \dotsm \sum_{f_{j_r}=m_r}^{\infty}
\binom{f_{1}+m_1-1}{2m_1-1}
\\&\quad
\times
\binom{f_{j_2}+m_2}{2m_2} \dotsm \binom{f_{j_r}+m_r}{2m_r}
q^{n_1f_1+n_{j_2}f_{j_2}+\dotsb +n_{j_r}f_{j_r}}\beta'(q)
\prod_{\substack{i > n_1 \\ i \notin \{n_{j_2},\dotsc ,n_{j_r}\}}} \frac{1}{1 - q^i}
. \end{aligned}$$
For general $k$ we take the expression for $\sum_{n= 1}^{\infty} {\mathrm{spt}^{X}_{k}\hspace{-.2em} \left(n\right) }q^{n}$ in (\[EqTheoremCombinatorics\]) and split the sum into $2^{k-1}$ sums by turning the index bounds into $<$ or $=$, each of which corresponds to a composition of $k$. In particular we recall that the $2^{k-1}$ compositions of $k$ can be obtained by taking a list of $k$ ones and between each one we put either a plus or a comma. Given the index bounds $n_1 \leq n_2 \leq \dotsb \leq n_k$, we make a choice of each $\leq$ being “$=$” or “$<$”; we associate to this the composition $1^{+}\hspace{-.5em}, 1^{+}\hspace{-.5em},\dotsc {{^{+}}1} \hspace{-1em},\hspace{1em}$ where in order we choose “$+$” when we chose “$=$” and choose “$,$” when we chose “$<$”.
If we let $A$ be the set of all compositions of $k$, with the manipulations discussed above, we have that $$\begin{aligned}
\sum_{n=1}^{\infty} {\mathrm{spt}^{X}_{k}\hspace{-.2em} \left(n\right) }q^n
&=
\sum_{(m_1,\dotsc,m_r)=\vec{m} \in A}
\hspace{.5em}
\sum_{1 \leq n_{1} < n_{j_2}< \dotsb < n_{j_r}}
\sum_{f_1=m_1}^{\infty} \sum_{f_{j_2}=m_2}^{\infty}
\dotsm \sum_{f_{j_r}=m_r}^{\infty}
\binom{f_{1}+m_1-1}{2m_1-1}
\\&\quad
\times \binom{f_{j_2}+m_2}{2m_2} \dotsm
\binom{f_{j_r}+m_r}{2m_r} q^{n_1f_1+n_{j_2}f_{j_2}+ \dotsb +n_{j_r}f_{j_r}}
\beta_{n_1}'(q) \prod_{\substack{i > n_1 \\ i \notin \{n_{j_2},\dotsc ,n_{j_r}\}}} \frac{1}{1 - q^i}
.\end{aligned}$$ This we recognize as the generating function for vector partitions $\vec{\pi}=(\pi_1,...,\pi_r)$ counted according to the weight $\omega_k(\vec{\pi})$ where $\beta_{n_1}'(q)$ determines the types of partitions in $(\pi_2,\dotsc,\pi_r)$.
We note that taking $k=1$ means that for ${\mathrm{spt}^{X}_{1}\hspace{-.2em} \left(n\right) }$ we simply count the number of appearances of the smallest part in $\pi_1$, and many of these functions have been studied elsewhere. In particular, those ${\mathrm{spt}^{X}_{1}\hspace{-.2em} \left(n\right) }$ which possess simple linear congruences were studied by Garvan and the second author in [@GarvanJenningsShaffer2; @JenningsShaffer1; @JenningsShaffer2]; ${\mathrm{spt}^{C1}_{1}\hspace{-.2em} \left(n\right) }$ was studied by Andrews, Dixit, and Yee in [@AndrewsDixitYee1]; and ${\mathrm{spt}^{C1}_{1}\hspace{-.2em} \left(n\right) }$, ${\mathrm{spt}^{C5}_{1}\hspace{-.2em} \left(n\right) }$, and ${\mathrm{spt}^{J1}_{1}\hspace{-.2em} \left(n\right) }$ were studied by Patkowski in [@Patkowski1; @Patkowski2].
To demonstrate these weighted counts, we compute a table of values for ${\mathrm{spt}^{X}_{k}\hspace{-.2em} \left(n\right) }$ with $X=A1,B2$, $k=1,2,3$, and $n=4,5$. We note that the first three weights are given by $$\begin{aligned}
\omega_{1}(\pi)
&=
f_{1}^{1}(\pi)
,\\
\omega_{2}(\pi)
&=
\binom{f_{1}^{1}(\pi) + 1}{3}
+
f_{1}^{1}(\pi)\sum_{2 \leq j}\binom{f_{j}^{1}(\pi) + 1}{2}
,\\
\omega_{3}(\pi)
&=
\binom{f_{1}^{1}(\pi) + 2}{5}
+
\binom{f_{1}^{1}(\pi) + 1}{3}\sum_{2 \leq j}\binom{f_{j}^{1}(\pi) + 1}{2}
+
f_{1}^{1}(\pi)\sum_{2 \leq j}\binom{f_j + 2}{4}
\\&\quad
+
f_{1}^{1}(\pi)\sum_{2 \leq j < k}\binom{f_{j}^{1}(\pi) + 1}{2}\binom{f_{k}^{1}(\pi) + 1}{2}
. \end{aligned}$$ $$\begin{aligned}
\begin{array}{c}
\begin{array}{c|c|c|c|c|c|c}
\multicolumn{7}{c}{\mbox{T{\sc able} 2. }A1\mbox{ Partitions of }4}
\\
&f_{1}^1 & f_{2}^1 & f_{3}^1 & \omega_1 & \omega_2 & \omega_3
\\\hline
(4, \varnothing) & 1 & 0 & 0 & 1 & 0 & 0 \\
(3+1, \varnothing) & 1 & 1 & 0 & 1 & 1 & 0\\
(2 + 2, \varnothing) & 2 & 0 & 0 & 2 & 1 & 0\\
(2+1+1, \varnothing) & 2 & 1 & 0 & 2 & 3 & 1\\
(1+1+1+1, \varnothing) & 4 & 0 & 0 & 4 & 10 & 6\\
(1+1, 2) & 2 & 0 & 0 & 2 & 1 & 0\\
\mbox{Total} &&&&12&16&7
\end{array}
\\\\\\\\\\\\
\end{array}
&\quad\quad
\begin{array}{c|c|c|c|c|c|c}
\multicolumn{7}{c}{\mbox{T{\sc able} 3. }A1\mbox{ Partitions of }5}
\\
& f_{1}^1 & f_{2}^1 & f_{3}^1 & \omega_1 & \omega_2 & \omega_3
\\\hline
(5, \varnothing) & 1 & 0 & 0 & 1 & 0 & 0 \\
(4+1, \varnothing) & 1 & 1 & 0 & 1 & 1 & 0\\
(3+2, \varnothing) & 1 & 1 & 0 & 1 & 1 & 0\\
(3+1+1, \varnothing) & 2 & 1 & 0 & 2 & 3 & 1\\
(2+2+1, \varnothing) & 1 & 2 & 0 & 1 & 3 & 1\\
(2+1+1+1, \varnothing) & 3 & 1 & 0 & 3 & 7 & 5\\
(1+1+1+1+1, \varnothing) & 5 & 0 & 0 & 5 & 20 & 21\\
(2, 3) & 1 & 0 & 0 & 1 & 0 & 0\\
(2+1, 2) & 1 & 1 & 0 & 1 & 1 & 0 \\
(1+1+1, 2) & 3 & 0 & 0 & 3 & 4 & 1\\
(1, 2+2) & 1 & 0 & 0 & 1 & 0 & 0\\
\mbox{Total} &&&&20&40&29
\end{array}
\\
\begin{array}{c|c|c|c|c|c|c}
\multicolumn{7}{c}{\mbox{T{\sc able} 4. }B2\mbox{ Partitions of }4}
\\
& f_{1}^1 & f_{2}^1 & f_{3}^1 & \omega_1 & \omega_2 & \omega_3
\\\hline
(2, 2) & 1 & 0 & 0 & 1 & 0 & 0
\\
(2+1, 1) & 1 & 1 & 0 & 1 & 1 & 0
\\
(1 + 1 + 1, 1) & 3 & 0 & 0 & 3 & 4 & 1
\\
\mbox{Total} &&&&5&5&1
\end{array}
&\quad\quad
\begin{array}{c|c|c|c|c|c|c}
\multicolumn{7}{c}{\mbox{T{\sc able} 5. }B2\mbox{ Partitions of }5}
\\
& f_{1}^1 & f_{2}^1 & f_{3}^1 & \omega_1 & \omega_2 & \omega_3
\\\hline
(3+1, 1) & 1 & 1 & 0 & 1 & 1 & 0\\
(2+1+1, 1) & 2 & 1 & 0 & 2 & 3 & 1 \\
(1+1+1+1, 1) & 4 & 0 & 0 & 4 & 10& 6\\
\mbox{Total} & & & & 7 & 14&7
\end{array}\end{aligned}$$
Conjectures and Concluding Remarks
==================================
As demonstrated, Bailey’s Lemma is well suited to give inequalities between the moments of rank-like Lambert series and corresponding crank functions, as well as supplying the combinatorial interpretation of the difference of the symmetrized moments. In our work, we focused our attention to the $a=1$ Bailey pairs of Slater [@Slater1; @Slater2] with $\alpha_n=\alpha_{-n}$. We have also seen that it is possible to work with Bailey pairs from other sources that satisfy the same conditions. However it is not true that all Bailey pairs, even from Slater’s lists, satisfy $\alpha_n=\alpha_{-n}$. We leave it open to the interested reader to see to what extent one can develop similar identities that lead to similar results.
We have studied these new rank moments are far as possible while handling them in generality. However, we expect each function to possess interesting properties of its own. In particular, based on how the four prototypical examples from [@Garvan1; @JenningsShaffer3] behave, we expect each ordinary rank moment generating function to correspond to a quasi-mock modular form, each ordinary crank moment to correspond to a quasi-modular form, and potentially an equation exists relating the certain partial derivatives of these rank and crank functions. Using these automorphic properties one could hope to derive asymptotic formulas for the moments, such as was done in [@BringmannMahlburgRhoades1; @BringmannMahlburgRhoades2].
As we have already seen, a large number of inequalities exist between the various moments. We conjecture the following diagram gives all of the inequalities. Here a downward path from $A_{2k}$ to $B_{2k}$ indicates $A_{2k}(n)\geq B_{2k}(n)$ for all positive $k$ and $n$. We have split the diagram into two pieces, to decrease the height and handle the large number of crossings. Besides the rank and crank moments defined in this article, we also include the overpartition rank $\overline{\eta}_{2k}$, the the overpartition M2-rank $\overline{\eta2}_{2k}$, the overpartiton M2-crank $\overline{\mu2}_{2k}$, the M2-rank for partitions without repeated odd parts $\eta2_{2k}$, and the M2-crank for partitions without repeated odd parts $\mu2_{2k}$ from [@JenningsShaffer3]. It is likely some of these inequalities can be proved using the identities of this article, but to get the full picture one will need additional techniques. These inequalities have been verified for $1\leq k\leq 10$ and $1\leq n< 1000$.
(MOverpartition) at (11,38) [$\overline{\mu}_{2k}$]{}; (NE4) at (11,35) [$\eta^{E4}_{2k}$]{}; (MX40) at (9,32) [$\mu^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}_{2k}$]{}; (NOverpartition) at (13,32) [$\overline{\eta}_{2k}$]{}; (MJ) at (7,29) [$\mu^{J}_{2k}$]{}; (M) at (5,26) [$\mu_{2k}$]{}; (NJ3) at (9,26) [$\eta^{J3}_{2k}$]{}; (NB2) at (7,23) [$\eta^{B2}_{2k}$]{}; (N) at (7,20) [$\eta_{2k}$]{}; (NX46) at (15,29) [$\eta^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{2k}$]{}; (MOverpartitionM2) at (18,26) [$\overline{\mu2}_{2k}$]{}; (NOverpartitionM2) at (21,23) [$\overline{\eta2}_{2k}$]{}; (NJ2) at (15,20) [$\eta^{J2}_{2k}$]{}; (NI14) at (21,20) [$\eta^{I14}_{2k}$]{}; (NA5) at (3,20) [$\eta^{A5}_{2k}$]{}; (NA3) at (3,17) [$\eta^{A3}_{2k}$]{}; (NX42) at (1,14) [$\eta^{{
\IfEqCase{42}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 42}{}]}}_{2k}$]{}; (NC2) at (1,11) [$\eta^{C2}_{2k}$]{}; (NA1) at (9,11) [$\eta^{A1}_{2k}$]{}; (MG) at (15,11) [$\mu^{G}_{2k}$]{}; (NG3) at (15,8) [$\eta^{G3}_{2k}$]{}; (NA7) at (6,14) [$\eta^{A7}_{2k}$]{}; (NX41) at (9,8) [$\eta^{{
\IfEqCase{41}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 41}{}]}}_{2k}$]{}; (NC5) at (9,4) [$\eta^{C5}_{2k}$]{}; (NG1) at (17,4) [$\eta^{G1}_{2k}$]{}; (MY) at (9,1) [$\mu^{Y}_{2k}$]{}; (M2) at (13,1) [$\mu 2_{2k}$]{}; (NC1) at (17,1) [$\eta^{C1}_{2k}$]{}; (NX39) at (5,8) [$\eta^{{
\IfEqCase{39}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 39}{}]}}_{2k}$]{}; (NJ1) at (23,13) [$\eta^{J1}_{2k}$]{}; (NX40) at (20.5,10) [$\eta^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}_{2k}$]{}; (NX38) at (23.5,10) [$\eta^{{
\IfEqCase{38}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 38}{}]}}_{2k}$]{}; (NF3) at (26.5,10) [$\eta^{F3}_{2k}$]{};
(MOverpartition) – (NE4); (NE4) – (MX40); (NE4) – (NOverpartition); (MX40) – (MJ); (MJ) – (M); (MJ) – (NJ3); (M) – (NB2); (NJ3) – (NB2); (NB2) – (N); (NOverpartition) – (NX46); (NX46) – (MOverpartitionM2); (MOverpartitionM2) – (NOverpartitionM2); (MOverpartitionM2) – (MG); (NX46) – (N); (NX46) – (NJ2); (NOverpartitionM2) – (NI14); (NJ3) – (NJ2); (NJ2) – (NJ1); (NI14) – (NJ1); (NJ1) – (NX40); (NJ2) – (MG); (MG) – (NG3); (MG) – (NC5); (NI14) – (NG1); (NG3) – (NG1); (NG1) – (NC1); (NG3) – (M2); (NB2) – (NA5); (NA5) – (NA3); (NA3) – (NX42); (NA3) – (NA7); (NX42) – (NC2); (NA7) – (NA1); (N) – (NA1); (NA1) – (NX41); (NX42) – (NX41); (NC2) – (NX39); (NX39) – (NC5); (NX41) – (NC5); (MG) – (NC5); (NC5) – (MY); (NC5) – (M2); (NC5) – (NC1); (NX46) – (NA7); (NJ1) – (NX38); (NJ1) – (NF3);
(AMY) at (5+,22+) [$\mu^{Y}_{2k}$]{}; (ANG3) at (9+,22+) [$\eta^{G3}_{2k}$]{}; (AM2) at (5+,30+) [$\mu2_{2k}$]{}; (ANG1) at (19+,22+) [$\eta^{G1}_{2k}$]{}; (AMF) at (1+,13+) [$\mu^{F}_{2k}$]{}; (ANY2) at (5+,18+) [$\eta^{Y2}_{2k}$]{}; (ANY4) at (9+,18+) [$\eta^{Y4}_{2k}$]{}; (ANC1) at (13+,18+) [$\eta^{C1}_{2k}$]{}; (AN2) at (17+,18+) [$\eta2_{2k}$]{}; (ANY3) at (7+,13+) [$\eta^{Y3}_{2k}$]{}; (ANY1) at (11+,13+) [$\eta^{Y1}_{2k}$]{}; (AML2) at (19+,13+) [$\mu^{L2}_{2k}$]{}; (ANX40) at (19+,5+) [$\eta^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}_{2k}$]{}; (ANL2) at (19+,9+) [$\eta^{L2}_{2k}$]{}; (ANX38) at (15+,13+) [$\eta^{{
\IfEqCase{38}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 38}{}]}}_{2k}$]{}; (ANF3) at (5+,5+) [$\eta^{F3}_{2k}$]{}; (ANL5) at (13+,1+) [$\eta^{L5}_{2k}$]{};
(AMY) – (AMF); (AMY) – (ANY2); (AMY) – (ANY4); (ANG3) – (ANY4); (AM2) – (AN2); (ANG1) – (AN2); (AMF) – (ANX40); (ANY2) – (ANY3); (ANY2) – (ANY1); (ANY4) – (ANY3); (ANY4) – (ANY1); (ANC1) – (ANY3); (ANC1) – (ANY1); (ANC1) – (ANX38); (ANC1) – (AML2); (AN2) – (AML2); (ANY3) – (ANX40); (AML2) – (ANL2); (ANL2) – (ANX40); (AM2) – (AMF); (ANG3) – (ANY2); (AMF) – (ANF3); (ANL2) – (ANF3); (ANL2) – (ANL5); (ANY1) – (ANL5); (ANY3) – (ANL5); (ANX38) – (ANL5); (ANF3) – (ANL5);
Additionally based on numerical evidence, it would appear that some of the higher order spt functions satisfy a number of congruences. While a few of these may follow by elementary means, to approach these likely one should start with the automorphic properties of the moments mentioned above, as this is the method that worked for the original examples. We conjecture the following congruences, $$\begin{aligned}
0
&\equiv
{\mathrm{spt}^{B2}_{3}\hspace{-.2em} \left(4n+3\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(31n\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(41n\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(47n\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(16n+1\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(32n+2\right) }
\\&\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(8n+7\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(49n+7\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(49n+14\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(18n+15\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(24n+17\right) }
\\&\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(40n+17\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(36n+21\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(45n+21\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(49n+21\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(32n+28\right) }
\\&\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(49n+28\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(36n+30\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(40n+33\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(45n+33\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(48n+34\right) }
\\&\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(49n+35\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(45n+39\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(45n+42\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(49n+42\right) }
\equiv
{\mathrm{spt}^{E4}_{3}\hspace{-.2em} \left(16n+3\right) }
\\&\equiv
{\mathrm{spt}^{E4}_{3}\hspace{-.2em} \left(32n+10\right) }
\equiv
{\mathrm{spt}^{E4}_{3}\hspace{-.2em} \left(16n+13\right) }
\equiv
{\mathrm{spt}^{E4}_{3}\hspace{-.2em} \left(49n+21\right) }
\equiv
{\mathrm{spt}^{E4}_{3}\hspace{-.2em} \left(32n+22\right) }
\equiv
{\mathrm{spt}^{E4}_{3}\hspace{-.2em} \left(49n+35\right) }
\\&\equiv
{\mathrm{spt}^{E4}_{3}\hspace{-.2em} \left(49n+42\right) }
\equiv
{\mathrm{spt}^{E4}_{4}\hspace{-.2em} \left(31n\right) }
\equiv
{\mathrm{spt}^{E4}_{4}\hspace{-.2em} \left(47n\right) }
\equiv
{\mathrm{spt}^{E4}_{4}\hspace{-.2em} \left(32n+1\right) }
\equiv
{\mathrm{spt}^{E4}_{4}\hspace{-.2em} \left(16n+7\right) }
\equiv
{\mathrm{spt}^{E4}_{4}\hspace{-.2em} \left(32n+14\right) }
\\&\equiv
{\mathrm{spt}^{E4}_{4}\hspace{-.2em} \left(48n+17\right) }
\equiv
{\mathrm{spt}^{E4}_{5}\hspace{-.2em} \left(32n+11\right) }
\equiv
{\mathrm{spt}^{E4}_{5}\hspace{-.2em} \left(32n+13\right) }
\equiv
{\mathrm{spt}^{E4}_{6}\hspace{-.2em} \left(31n\right) }
\equiv
{\mathrm{spt}^{E4}_{6}\hspace{-.2em} \left(32n+17\right) }
\\&\equiv
{\mathrm{spt}^{E4}_{6}\hspace{-.2em} \left(32n+23\right) }
\equiv
{\mathrm{spt}^{E4}_{7}\hspace{-.2em} \left(32n+29\right) }
\equiv
{\mathrm{spt}^{E4}_{8}\hspace{-.2em} \left(31n\right) }
\equiv
{\mathrm{spt}^{E4}_{8}\hspace{-.2em} \left(32n+7\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{41}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 41}{}]}}_{3}\hspace{-.2em} \left(4n+2\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{42}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 42}{}]}}_{3}\hspace{-.2em} \left(44n+28\right) }
\\&\equiv
{\mathrm{spt}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{2}\hspace{-.2em} \left(48n+1\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{2}\hspace{-.2em} \left(24n+7\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{2}\hspace{-.2em} \left(48n+14\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{2}\hspace{-.2em} \left(24n+17\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{2}\hspace{-.2em} \left(24n+23\right) }
\\&\equiv
{\mathrm{spt}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{2}\hspace{-.2em} \left(48n+34\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{2}\hspace{-.2em} \left(48n+46\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{3}\hspace{-.2em} \left(48n+11\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{3}\hspace{-.2em} \left(48n+13\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{3}\hspace{-.2em} \left(48n+29\right) }
\\&\equiv
{\mathrm{spt}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{3}\hspace{-.2em} \left(48n+43\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{4}\hspace{-.2em} \left(48n+7\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{4}\hspace{-.2em} \left(48n+17\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{4}\hspace{-.2em} \left(32n+23\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{4}\hspace{-.2em} \left(48n+23\right) }
\\&\equiv
{\mathrm{spt}^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}_{2}\hspace{-.2em} \left(46n+3\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}_{2}\hspace{-.2em} \left(49n+15\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}_{2}\hspace{-.2em} \left(49n+29\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}_{2}\hspace{-.2em} \left(41n+36\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}_{2}\hspace{-.2em} \left(49n+36\right) }
\\&\equiv
{\mathrm{spt}^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}_{3}\hspace{-.2em} \left(4n\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}_{3}\hspace{-.2em} \left(18n+2\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}_{3}\hspace{-.2em} \left(18n+14\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}_{3}\hspace{-.2em} \left(49n+15\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}_{3}\hspace{-.2em} \left(49n+29\right) }
\\&\equiv
{\mathrm{spt}^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}_{3}\hspace{-.2em} \left(49n+36\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}_{6}\hspace{-.2em} \left(8n\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}_{6}\hspace{-.2em} \left(8n+7\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{40}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 40}{}]}}_{6}\hspace{-.2em} \left(41n+36\right) }
\equiv
{\mathrm{spt}^{F3}_{2}\hspace{-.2em} \left(4n+1\right) }
\equiv
{\mathrm{spt}^{F3}_{3}\hspace{-.2em} \left(4n\right) }
\\&\equiv
{\mathrm{spt}^{F3}_{4}\hspace{-.2em} \left(8n+1\right) }
\equiv
{\mathrm{spt}^{F3}_{5}\hspace{-.2em} \left(8n+4\right) }
\equiv
{\mathrm{spt}^{F3}_{6}\hspace{-.2em} \left(8n+5\right) }
\equiv
{\mathrm{spt}^{F3}_{8}\hspace{-.2em} \left(16n+1\right) }
\equiv
{\mathrm{spt}^{F3}_{10}\hspace{-.2em} \left(16n+5\right) }
\pmod{2}
,\\
0
&\equiv
{\mathrm{spt}^{A3}_{2}\hspace{-.2em} \left(9n\right) }
\equiv
{\mathrm{spt}^{B2}_{4}\hspace{-.2em} \left(3n\right) }
\equiv
{\mathrm{spt}^{E4}_{3}\hspace{-.2em} \left(27n+15\right) }
\equiv
{\mathrm{spt}^{E4}_{4}\hspace{-.2em} \left(27n+6\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{2}\hspace{-.2em} \left(9n\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{2}\hspace{-.2em} \left(24n+11\right) }
\\&\equiv
{\mathrm{spt}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{2}\hspace{-.2em} \left(32n+12\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{8}\hspace{-.2em} \left(27n\right) }
\equiv
{\mathrm{spt}^{F3}_{2}\hspace{-.2em} \left(6n+1\right) }
\equiv
{\mathrm{spt}^{L2}_{3}\hspace{-.2em} \left(27n+26\right) }
\pmod{3}
,\\
0
&\equiv
{\mathrm{spt}^{L5}_{4}\hspace{-.2em} \left(44n+28\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(16n+14\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(36n+33\right) }
\pmod{4}
,\\
0
&\equiv
{\mathrm{spt}^{A1}_{2}\hspace{-.2em} \left(5n\right) }
\equiv
{\mathrm{spt}^{A1}_{2}\hspace{-.2em} \left(5n+1\right) }
\equiv
{\mathrm{spt}^{A1}_{4}\hspace{-.2em} \left(25n+24\right) }
\equiv
{\mathrm{spt}^{A1}_{5}\hspace{-.2em} \left(25n+24\right) }
\equiv
{\mathrm{spt}^{A3}_{2}\hspace{-.2em} \left(5n+1\right) }
\equiv
{\mathrm{spt}^{A3}_{2}\hspace{-.2em} \left(5n+2\right) }
\\&\equiv
{\mathrm{spt}^{A3}_{2}\hspace{-.2em} \left(5n+4\right) }
\equiv
{\mathrm{spt}^{A5}_{2}\hspace{-.2em} \left(5n\right) }
\equiv
{\mathrm{spt}^{A5}_{2}\hspace{-.2em} \left(5n+4\right) }
\equiv
{\mathrm{spt}^{A5}_{3}\hspace{-.2em} \left(25n+9\right) }
\equiv
{\mathrm{spt}^{A5}_{3}\hspace{-.2em} \left(25n+14\right) }
\equiv
{\mathrm{spt}^{A7}_{2}\hspace{-.2em} \left(5n+1\right) }
\\&\equiv
{\mathrm{spt}^{A7}_{2}\hspace{-.2em} \left(5n+4\right) }
\equiv
{\mathrm{spt}^{A7}_{3}\hspace{-.2em} \left(25n+24\right) }
\equiv
{\mathrm{spt}^{B2}_{2}\hspace{-.2em} \left(5n+1\right) }
\equiv
{\mathrm{spt}^{B2}_{2}\hspace{-.2em} \left(5n+2\right) }
\equiv
{\mathrm{spt}^{B2}_{2}\hspace{-.2em} \left(5n+4\right) }
\equiv
{\mathrm{spt}^{B2}_{3}\hspace{-.2em} \left(25n+1\right) }
\\&\equiv
{\mathrm{spt}^{B2}_{3}\hspace{-.2em} \left(25n+9\right) }
\equiv
{\mathrm{spt}^{B2}_{4}\hspace{-.2em} \left(25n+2\right) }
\equiv
{\mathrm{spt}^{B2}_{4}\hspace{-.2em} \left(25n+4\right) }
\equiv
{\mathrm{spt}^{B2}_{8}\hspace{-.2em} \left(25n+12\right) }
\equiv
{\mathrm{spt}^{C5}_{2}\hspace{-.2em} \left(5n\right) }
\equiv
{\mathrm{spt}^{C5}_{2}\hspace{-.2em} \left(5n+1\right) }
\\&\equiv
{\mathrm{spt}^{C5}_{2}\hspace{-.2em} \left(5n+4\right) }
\equiv
{\mathrm{spt}^{C5}_{3}\hspace{-.2em} \left(25n+24\right) }
\equiv
{\mathrm{spt}^{C5}_{4}\hspace{-.2em} \left(25n+3\right) }
\equiv
{\mathrm{spt}^{C5}_{4}\hspace{-.2em} \left(25n+5\right) }
\equiv
{\mathrm{spt}^{C5}_{4}\hspace{-.2em} \left(25n+24\right) }
\\&\equiv
{\mathrm{spt}^{C5}_{5}\hspace{-.2em} \left(25n+4\right) }
\equiv
{\mathrm{spt}^{C5}_{5}\hspace{-.2em} \left(25n+24\right) }
\equiv
{\mathrm{spt}^{C5}_{6}\hspace{-.2em} \left(25n+5\right) }
\equiv
{\mathrm{spt}^{C5}_{6}\hspace{-.2em} \left(25n+10\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(5n\right) }
\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(5n+2\right) }
\\&\equiv
{\mathrm{spt}^{F3}_{3}\hspace{-.2em} \left(25n+3\right) }
\equiv
{\mathrm{spt}^{F3}_{3}\hspace{-.2em} \left(25n+23\right) }
\equiv
{\mathrm{spt}^{Y1}_{5k}\hspace{-.2em} \left(10n+3\right) }
\equiv
{\mathrm{spt}^{Y1}_{5}\hspace{-.2em} \left(25n+8\right) }
\equiv
{\mathrm{spt}^{Y1}_{10}\hspace{-.2em} \left(25n+8\right) }
\\&\equiv
{\mathrm{spt}^{Y2}_{5k}\hspace{-.2em} \left(10n+9\right) }
\pmod{5}
,\\
0
&\equiv
{\mathrm{spt}^{A1}_{2}\hspace{-.2em} \left(49n+12\right) }
\equiv
{\mathrm{spt}^{A1}_{3}\hspace{-.2em} \left(49n+47\right) }
\equiv
{\mathrm{spt}^{A3}_{2}\hspace{-.2em} \left(49n+19\right) }
\equiv
{\mathrm{spt}^{A5}_{2}\hspace{-.2em} \left(7n+1\right) }
\equiv
{\mathrm{spt}^{A5}_{3}\hspace{-.2em} \left(7n\right) }
\equiv
{\mathrm{spt}^{A5}_{3}\hspace{-.2em} \left(7n+1\right) }
\\&\equiv
{\mathrm{spt}^{A5}_{3}\hspace{-.2em} \left(7n+3\right) }
\equiv
{\mathrm{spt}^{A5}_{3}\hspace{-.2em} \left(7n+5\right) }
\equiv
{\mathrm{spt}^{A5}_{6}\hspace{-.2em} \left(7n+5\right) }
\equiv
{\mathrm{spt}^{A7}_{2}\hspace{-.2em} \left(7n\right) }
\equiv
{\mathrm{spt}^{A7}_{2}\hspace{-.2em} \left(7n+1\right) }
\equiv
{\mathrm{spt}^{A7}_{3}\hspace{-.2em} \left(7n\right) }
\\&\equiv
{\mathrm{spt}^{A7}_{3}\hspace{-.2em} \left(7n+1\right) }
\equiv
{\mathrm{spt}^{A7}_{3}\hspace{-.2em} \left(7n+2\right) }
\equiv
{\mathrm{spt}^{A7}_{3}\hspace{-.2em} \left(7n+4\right) }
\equiv
{\mathrm{spt}^{A7}_{5}\hspace{-.2em} \left(49n+47\right) }
\equiv
{\mathrm{spt}^{A7}_{6}\hspace{-.2em} \left(49n+47\right) }
\\&\equiv
{\mathrm{spt}^{B2}_{2}\hspace{-.2em} \left(7n+1\right) }
\equiv
{\mathrm{spt}^{B2}_{2}\hspace{-.2em} \left(7n+5\right) }
\equiv
{\mathrm{spt}^{B2}_{3}\hspace{-.2em} \left(7n\right) }
\equiv
{\mathrm{spt}^{B2}_{3}\hspace{-.2em} \left(7n+1\right) }
\equiv
{\mathrm{spt}^{B2}_{3}\hspace{-.2em} \left(7n+3\right) }
\equiv
{\mathrm{spt}^{B2}_{3}\hspace{-.2em} \left(7n+5\right) }
\\&\equiv
{\mathrm{spt}^{B2}_{4}\hspace{-.2em} \left(49n+1\right) }
\equiv
{\mathrm{spt}^{B2}_{5}\hspace{-.2em} \left(49n+33\right) }
\equiv
{\mathrm{spt}^{B2}_{6}\hspace{-.2em} \left(7n+5\right) }
\pmod{7}
,\\
0
&\equiv
{\mathrm{spt}^{E4}_{2}\hspace{-.2em} \left(32n+30\right) } \pmod{8}
,\\
0
&\equiv
{\mathrm{spt}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{5}\hspace{-.2em} \left(27n\right) }
\equiv
{\mathrm{spt}^{{
\IfEqCase{46}{
{38}{X1}
{39}{X2}
{40}{X6}
{41}{X3}
{42}{X4}
{46}{X5}
}[\PackageError{X}{Undefined option to X: 46}{}]}}_{5}\hspace{-.2em} \left(27n+18\right) }
\pmod{9}
,\\
0
&\equiv
{\mathrm{spt}^{B2}_{2}\hspace{-.2em} \left(11n+1\right) }
\pmod{11}
,\\
0
&\equiv
{\mathrm{spt}^{A5}_{3}\hspace{-.2em} \left(25n+24\right) }
\equiv
{\mathrm{spt}^{B2}_{2}\hspace{-.2em} \left(25n+14\right) }
\equiv
{\mathrm{spt}^{C5}_{2}\hspace{-.2em} \left(25n+24\right) }
\equiv
{\mathrm{spt}^{F3}_{2}\hspace{-.2em} \left(25n+23\right) }
\pmod{25}
,\\
0
&\equiv
{\mathrm{spt}^{B2}_{3}\hspace{-.2em} \left(49n+26\right) }
\pmod{49}
.\end{aligned}$$
Lastly, there is also the concept of positive moments, where $m$ ranges over just the positive integers, rather than all of $\mathbb{Z}$, that is to say, $$\begin{aligned}
N^+_k(n)
&=
\sum_{m=1}^{\infty}m^k N(m,n)
,&
\eta^+_{k}(n)
&=
\sum_{m=1}^{\infty}
\binom{m+\lfloor\frac{k-1}{2}\rfloor}{k}N(m,n)
,\\
M^+_k(n)
&=
\sum_{m=1}^{\infty}m^kM(m,n)
,&
\mu^+_{k}(n)
&=
\sum_{m=1}^{\infty}
\binom{m+\lfloor\frac{k-1}{2}\rfloor}{k}M(m,n)
.\end{aligned}$$ The advantage to these positive moments is that while $N_{2k}(n)=2N^{+}_{2k}(n)$ and $M_{2k}(n)=2M^{+}_{2k}(n)$, it is no longer the case that the odd moments are zero. It is true that $M^{+}_{2k+1}(n)>N^{+}_{2k+1}(n)$, and there have been several studies of the positive moments corresponding to the rank and crank of ordinary partitions as well as overpartitions, see [@AndrewsChanKimOsburn1; @AndrewsChanKim1; @BringmannMahlburg1; @LarsenRustSwisher1; @ZapataRolon]. As such we should expect that analogous results and inequalities hold for the moments of this article, however our methods do not directly apply and it is not clear if one can handle positive moments in the generality we have managed for the original moments.
|
---
abstract: |
We describe automatic procedures for the selection of DA white dwarfs in the Hamburg/ESO objective-prism survey (HES). For this purpose, and the selection of other stellar objects (e.g., metal-poor stars and carbon stars), a flexible, robust algorithm for detection of stellar absorption and emission lines in the digital spectra of the HES was developed. Broad band ($U-B$, $B-V$) and narrow band (Strömgren $c_1$) colours can be derived directly from HES spectra, with precisions of $\sigma_{U-B}=0.092$mag; $\sigma_{B-V}=0.095$mag; $\sigma_{c_1}=0.15$mag.
We describe simulation techniques that allow to convert model or slit spectra to HES spectra. These simulated objective-prism spectra are used to determine quantitative selection criteria, and for the study of selection functions. We present an atlas of simulated HES spectra of DA and DB white dwarfs.
Our current selection algorithm is tuned to yield maximum efficiency of the candidate sample (minimum contamination with non-DAs). DA candidates are selected in the $B-V$ versus $U-B$ and $c_1$ versus $W_\lambda(\mbox{H}\beta
+\mbox{H}\gamma+\mbox{H}\delta)$ parameter spaces. The contamination of the resulting sample with hot subdwarfs is expected to be as low as $\sim
8$%, while there is essentially no contamination with main sequence or horizontal branch stars. We estimate that with the present set of criteria, $\sim 80$% of DAs present in the HES database are recovered. A yet higher degree of internal completeness could be reached at the expense of higher contamination. However, the external completeness is limited by additional losses caused by proper motion effects and the epoch differences between direct and spectral plates used in the HES.
author:
- 'N. Christlieb'
- 'L. Wisotzki'
- 'D. Reimers'
- 'D. Homeier'
- 'D. Koester'
- 'U. Heber'
bibliography:
- 'classification.bib'
- 'datanaly.bib'
- 'HES.bib'
- 'mphs.bib'
- 'ncpublications.bib'
- 'ncastro.bib'
- 'quasar.bib'
- 'statistics.bib'
- 'wd.bib'
date: 'Received 12 September 2000 / Accepted 12 October 2000'
subtitle: 'I. Automated selection of DA white dwarfs'
title: 'The stellar content of the Hamburg/ESO survey[^1]'
---
Introduction
============
The Hamburg/ESO survey (HES) is an objective prism survey primarily targeting bright quasars [@hespaperI; @heshighlights; @hespaperIII]. However, because its spectral resolution is typically 15[Å]{} FWHM at H$\gamma$, it is also possible to efficiently select a variety of interesting *stellar* objects in the HES. These include, e.g., metal-poor halo stars, carbon stars, cataclysmic variable stars, white dwarfs (WDs), subdwarf B stars (sdBs), subdwarf O stars (sdOs), and field horizontal branch A- and B-type stars [@Christlieb:2000]. In a series of papers, we will report on the development of quantitative selection procedures for the systematic exploitation of the stellar content of the HES, and their application to the digital HES data base. In this paper, we report on an automatic algorithm for the selection of DA white dwarfs (DAs).
The aim of the DA selection in the HES described in this paper is to test the double-degenerate (DD) scenario for SN Ia progenitors, in which a binary consisting of two WDs of large enough mass, merges and produces a thermonuclear explosion. If this scenario is correct, SN Ia progenitor systems should be present among DDs, the latter being identifiable by radial velocity (RV) variations. Although several double degenerates have been found, none of them is sufficiently massive to qualify as a viable SN Ia progenitor [see @Maxted/Marsh:1999]. Note however that recently a system consisting of a sdB and a massive white dwarf has been found, the total mass of which exceeds the Chandrasekhar mass [@Maxtedetal:2000a]. In order to increase the sample of DDs, a *Large Programme* was proposed to (and accepted by) the European Southern Observatory (ESO), aiming at observing a large ($\sim 1\,500$) sample of WDs with VLT Unit Telescope 2 (UT2), and its high resolution spectrograph UVES, at randomly chosen epochs, dictated by observing conditions. That is, every time the weather is *too bad* to carry out Service Mode observations for other programs, WDs are observed.
A program like this requires a large catalog of targets spread all over the accessible sky in order to be successfull. From the “Catalog of Spectroscopically Identified White Dwarfs” of [@McCook/Sion:1999] it is evident that the southern sky has not yet been surveyed as extensively for white dwarfs as the northern sky: only 33% of the objects listed are located at $\delta<0^{\circ}$. Therefore, we were aiming at selecting additional targets in the data base of the HES.
White dwarfs have been selected from wide angle surveys in the southern hemisphere before, and also in the HES (see below). “Classical” UV excess surveys, like the Montreal-Cambridge-Tololo survey [MCT; @Demersetal:1986; @Lamontagneetal:2000], or the Edinburgh-Cape survey [EC; @Stobieetal:1997; @Kilkennyetal:1997] can efficiently select complete samples of hot stars, including WDs. However, completeness at the *cool* end is either sacrificed for efficiency, as in the MCT (see Fig. \[UBV\_DA\]), where only objects with $U-B<-0.6$ enter the sample of stars for which follow-up spectroscopy is obtained [@Lamontagneetal:2000], or efficiency is sacrificed for completeness, as in the EC. It has been shown that the EC is 94% complete for objects of $U-B<-0.4$ down to $B=16.5$ [@Stobieetal:1997]. However, an intermediate selection step based on photoelectric $UBV$ photometry has to be used to eliminate the large fraction ($\sim 30$%; see @Kilkennyetal:1997) of “normal” F and G type stars.
In the HES, WDs enter the quasar candidate sample if they have $U-B<-0.18$ [@hespaperIII]. However, HES quasar candidates are inspected manually at the computer screen, and in this process hot stars, and stars clearly exhibiting stellar absorption lines (like e.g. DA white dwarfs, having strong, broad lines over a wide temperature range; see Fig. \[WDmodels\] in Appendix \[Sect:SpectralAtlas\]) are rejected, and follow-up spectroscopy is not obtained for them in the course of the quasar survey. This results in a very efficient quasar selection: typically 70% of the objects for which follow-up spectroscopy is obtained *are* quasars [@hespaperIII]. The remaining 30% are mainly hot subdwarfs, cool ($T_{\mbox{\scriptsize eff}}\lesssim 20\,000$K), helium-rich WDs [DBs, DZs; see @Friedrichetal:2000], a couple of interesting peculiar objects, e.g., magnetic DBs [@magDB1] and magnetic DAs [@Reimersetal:1994; @Reimersetal:1996] have also been discovered in this way.
The selection of white dwarf candidates described in this paper aims at an *efficient* selection; that is, the contamination of the sample with other objects (e.g., hot subdwarfs) shall be as clean as possible, in order not to waste any observing time at the VLT.
The HES data base {#Sect:HESdatabase}
=================
A description of the HES plate material, plate digitisation and data reduction can be found in [@hespaperIII]. In this section we describe some survey properties that are particularily important for stellar work in more detail than it was done in [@hespaperIII], and we briefly repeat a general description of the HES, for better readability.
The HES is based on IIIa-J plates taken with the 1m ESO Schmidt telescope and its 4$^{\circ}$ prism. It covers the magnitude range $13.0 \gtrsim B_J
\gtrsim 17.5$ [@hespaperIII]. The magnitude limits depend on plate quality. Note that the value given for the faint limit is the completeness limit for quasar search, which we define as the magnitude corresponding to average photographic density in the $B_J$ band $>5\sigma$ above the diffuse plate background, where $\sigma$ is the background noise. The detection limit of the HES is approximately one magnitude deeper than the completeness limit. For stellar applications, the survey magnitude range depends on the object type searched for. E.g., in our search for metal-poor stars, we only use spectra that are not affected by saturation effects, which typically start to be noticable at $B_J\sim 14.0$, and we only include spectra with $S/N>10$ (which roughly corresponds to $B_J\sim 16.4$), because at lower $S/N$ an efficient selection of metal-poor stars, by means of a weak or absent Ca K line, is not feasible anymore. However, for most other object types, including DAs, we adopt the $5\sigma$ magnitude limit.
The atmospheric cutoff at the blue end, and the sharp sensitivity cutoff of the IIIa-J emulsion (“red edge”) result in a wavelength range of $3200\,\mbox{\AA} < \lambda < 5300\,\mbox{\AA}$ (see Fig. \[fig:noisedata\_demo\]). The spectral resolution of the HES is primarily seeing-limited. For plates taken during good seeing conditions, the pixel spacings chosen in the digitization process result in an under-sampling, so that in these cases the spectral resolution is also limited by the sampling.
The definition of the HES survey area makes use of the mean star density and average column density of neutral hydrogen for each ESO/SERC field [@hespaperIII]. The adopted criteria roughly correspond to galactic latitudes of $|b|>30^{\circ}$. The declination range covered by the HES is $+2.5^{\circ}>\delta> -78^{\circ}$. In result, the survey area consists of 380 fields. Between 1989 and 1998, objective-prism plates were taken for all of these, and the plates were subsequently digitized and reduced at Hamburger Sternwarte. As one ESO Schmidt plate covers approximately $5\times 5\deg^2$ on the sky, the nominal survey area is $9\,500$deg$^2$, or the total southern extragalactic sky. Note, however, that the *effective* survey area is $\sim 25$% lower, mainly because of overlapping spectra [@hespaperIII].
Detection of overlapping spectra
--------------------------------
Overlapping spectra (hereafter shortly called overlaps) are detected automatically using the direct plate data of the Digitized Sky Survey I (DSS I). For each spectrum to be extracted, it is looked for objects in the dispersion direction on the direct plate. If there is one, the automatic procedure marks the corresponding spectrum, so that it can later be excluded from further processing, if this is desired. [*It is*]{} desired for stellar work, since the feature detection and object selection algorithms would get confused otherwise, and a lot of “garbage” would enter the candidate samples. The digital HES data base for stellar work consists of $\sim 4$ million extracted, overlap-free spectra with average $S/N>5$ in the $B_J$ band.
Photometry
----------
As described in [@hespaperIII], the calibration of HES $B_J$ magnitudes is done plate by plate with individual photometric sequences. The $B_J$ band is formally defined by the spectral sensitivity curve of the Kodak IIIa-J emulsion multiplied with the filter curve of a Schott GG395 filter. The overall errors of the HES $B_J$ magnitudes, including zero point errors, are less than $\pm
0.2$mag. Note that $B_J$ can be converted to $B$ using the formula $$\label{eq:BJtoB}
B = B_J + 0.28 \cdot (B-V),$$ which is valid for main sequence stars in the colour range $-0.1<(B-V)<1.6$ [@Hewettetal:1995].
Wavelength calibration
----------------------
A global dispersion relation for all HES plates was determined by using A-type stars. In HES spectra of these stars the Balmer lines at least up to H$_{10}$ are resolved (see Fig. \[fig:noisedata\_demo\]), so that a dispersion relation can be derived by comparing the $x$-positions (scan length in $\mu$m) of these lines with the known wavelengths. [@Borraetal:1987] used the position of the “red edge” of objective-prism spectra to determine the zero point of the wavelength calibration, but noticed that the position depends on the energy distribution of the object. Therefore, in the HES we decided to use a zero point specified by an astrometric transformation between direct plates and spectral plates. The wavelength calibration is accurate to $\pm 10\,\mu$m. This corresponds to $\pm 4.5$[Å]{} at H$\gamma$ and $\pm
2.3$[Å]{} at $\lambda = 3500$[Å]{}.
Estimation of the amplitude of pixel-wise noise {#sect:NoiseEstimation}
-----------------------------------------------
Following the approach of [@Hewettetal:1985], we determine the amplitude of pixel-wise noise as a function of photographic density $D$ plate by plate using A- and F-type stars. A straight line fit is done to the spectral region between H$\beta$ and H$\gamma$ (see Fig. \[fig:noisedata\_demo\]).
The $1\,\sigma$-scatter around this pseudo-continuum fit is taken as noise amplitude. In this approach we assume that the scatter is mainly due to noise, since in early-type stars the spectral region under consideration includes only very few absorption lines at the spectral resolution of the HES. Moreover, it is expected that the population of A- and F-type stars found at high galactic latitudes is dominated by metal-poor stars, so that metal lines are usually very weak. However, we can not exclude that we overestimate the noise by a few percent due to contributions of metal lines to the scatter about the pseudo-continuum fit.
We measure the noise amplitude for all A- and F-type stars present on each HES plate (typically several hundred per plate), and compute the mean density $D$ in the fit region. This yields data points $(D,\mbox{noise})$, to which a 2nd order polynomial is fitted, i.e. $$\mbox{noise} = a_0 + a_1\cdot D + a_2\cdot D^2. \label{Parabelgleichung}$$
Note that in the HES, $D$ refers to density above diffuse plate background (bgr) in arbitrary units called counts. The relation between counts and photographic densities $D_{\mbox{\scriptsize photo}}$ is $$\label{Dphoto}
D_{\mbox{\scriptsize photo}} = \frac{D\,\mbox{[counts]}
+ D_{\mbox{\scriptsize bgr}}\,\mbox{[counts]}}{800}.$$
For the determination of the coefficients $a_0\dots a_2$ we use a robust fit algorithm which minimizes the sum of absolute deviations, taking into account the following set of boundary conditions:
$a_0 > 0$.
: The noise at $D=0$ is the noise of the plate background, which is always $>0$.
$a_2\ge 0$.
: Since $D \ge 0$, and the noise increases monotonically with $D$, $a_2$ must be positive (or zero).
$a_1\ge 0$.
: From the previous boundary condition follows that the polynomial has a minimum at $D_{\mbox{\scriptsize min}}=-a_1/(2a_2)$. Using again the argument that the noise increases monotonically with $D$, it follows that $D_{\mbox{\scriptsize min}}\le 0$. Since $a_2\ge 0$, $a_1$ must be $\ge 0$.
An example for a such a fit is shown in Fig. \[fig:noisefit\].
\[par:noisdistrib\] For simulations of spectra (see Sect. \[slit2objprism\]) it is very important to know the *form* of the distribution of noise. We investigated this by using 50 spectra of A-type stars from eight plates with high sky background ($D_{\mbox{\scriptsize
bgr}}>1\,500$), and 60 spectra from seven plates with low background ($D_{\mbox{\scriptsize bgr}}<700$). These spectra were chosen by hand from the sample of automatically selected A-type stars, in order to ensure that misclassified spectra, and spectra for which the fit of the continuum between H$\beta$ and H$\gamma$ by a straight line is not fully adequate, do not confuse the results. Five of the original set of 115 spectra were excluded in the manual selection process.
The deviations from the continuum fits were collected for each spectrum, shifted to a median of zero, and divided by the average of the absolute values of the upper and lower $50$% quartile, so that a comparison of the noise distributions measured in different spectra (with different noise amplitude) is possible. The result is that the distribution of pixel-wise noise is almost perfectly Gaussian, independently of plate background (see Fig. \[noisedistrib\]).
Fig. \[SNofBJ\] shows the relation between average pixel-wise $S/N$ in the $B_J$ band and $B_J$ magnitudes for 589 not saturated point sources from many different HES plates. The large scatter of $S/N$ for given $B_J$ shown in this plot is due to varying plate background, and seeing.
Feature detection {#FeatureDetection}
=================
Detection of stellar lines {#sect:StellarLineDetection}
--------------------------
It is critical for all quantitative selection methods to have a set of *reliable* features at hand. The total set of available features should contain as much information of the objects to be classified as possible. Therefore, we implemented a flexible, robust algorithm which allows to detect stellar absorption and/or emission lines in HES spectra. [@Hewettetal:1985] used a template matching technique to automatically detect absorption and emission lines in objective-prism spectra. Tests with HES spectra have shown that a fit algorithm involving a couple of boundary conditions (see below) is much more stable and leads to more accurate measurements of equivalent widths than template matching. The algorithm used in the HES consists of the following steps:
1. Determination of continuum by filtering with a wide median filter and narrow Gaussian filter. A similar filtering technique was also used by [@Hewettetal:1985] and [@Borraetal:1987].
2. Improvement of determination of the wavelength calibration zero point by fitting of 3 sets of stellar lines. The sets contain the strongest stellar absorption lines of early type, solar type, and late type stars, respectively. The individual line depths, and the zero point offset of wavelength calibration are fitted simultaneously. The *relative* positions of the stellar lines are held fixed, and the line *widths* is held fixed at the value of the seeing profile widths, which is measured during spectral extraction. The set of lines giving the strongest signal, i.e. largest average equivalent widths, is selected, and the wavelength calibration zero point determined with that fit is adopted.
3. Improvement of continuum determination:
1. Fitting of *all* stellar lines detectable in HES spectra
2. Subtraction of fitted lines from the original spectrum
3. Computation of improved continuum by filtering the line-subtracted spectrum again with a wide median filter and narrow Gaussian filter
4. Go back to 3a, if $n_{\mbox{\scriptsize iter}}<3$; otherwise compute rectified spectrum with final continuum.
4. Fitting of all stellar lines in the rectified spectrum by Gaussians.
For each spectral line it can be chosen whether it is to be detected in absorption or emission. The output of the fit algorithm are equivalent width, FWHM and $S/N$ of the lines, and shift of the wavelength calibration zero point. Any spectral lines not yet considered can easily be included by just adding its wavelength to the list of lines to be fitted. In this work we make use of the equivalent width of H$\beta$, H$\gamma$ and H$\delta$ only, which are summed to the parameter [balmsum]{}.
Broad band and intermediate band colours
----------------------------------------
As already noticed by [@Hewettetal:1985], relative colours can be determined quite accurately directly from objective-prism spectra. [@Hewettetal:1995] used so-called “half power points (hpps)” to measure relative colours, that is, bisecting points of the photographic density distribution. hpps are equivalent to broad-band colours, but have the advantage of being more robust against noise. [@hespaperIII] introduced half power points that are computed only for a *part* of the spectrum (see Fig. \[hppdemo\]). The half power points `x_hpp1` and `x_hpp2` are well correlated with $U-B$ and $B-V$, respectively.
Since it is helpful in many stellar applications to have not only relative, but calibrated $U-B$ and $B-V$ colours at hand, we established calibrations of `x_hpp1` and `x_hpp2` versus $U-B$ and $B-V$, respectively.
A more precise colour calibration can be achieved when distances `dx` to a cutoff line in a colour-magnitude diagram (see Fig. \[hpp1\_cutoff\]) is used instead of `x` values (scan length in $\mu$m) for the bisecting point, because plate-to-plate variations of the spectral sensitivity curves are compensated in this way. The cutoff line separates the bulk of “normal” stars from UV-excess objects (or objects with unusually low $B-V$ in case of `dx_hpp2`). The cutoff is determined by a break finding algorithm.
Because the blue end of the HES spectra is sensitive to contamination by overlaps, special care must be taken to exclude such spectra from the calibration of `dx_hpp1`. This was done by applying stricter overlap selection criteria. In addition, an iterative $\kappa\sigma$-clipping with $\kappa=3$ was employed to exclude overlaps unrecognized by the automatic detection. 50 of the 623 spectra in the original data set were rejected, so that the calibration uses spectra of 573 objects (see Fig. \[dxhppfit\]; a similar plot was shown in @hespaperIII).
A potential problem for the $B-V$ calibration is that the $V$ band is not fully covered by the HES wavelength range. Therefore, the calibration for very red objects is inaccurate, or even impossible. As calibrators for red objects, 36 carbon stars were used, for which $BV$ photometry was obtained at the ESO 2.2m telescope in April 1999. Carbon stars with $B-V>2.5$ were excluded from the fit. For $B-V\lesssim 1.0$, 778 stars from the HK survey of [@BPSII], 354 FHB and other A-type stars of [@Wilhelmetal:1999b], and 272 objects from the northern galactic cap fields of the EC survey [@Kilkennyetal:1997] present on HES plates were used. Linear fits in three colour regions were done separately, in order to evaluate the scatter independently, and check consistency. Then, a combined fit to all 1256 unique objects was done (see Fig. \[dxhppfit\]).
The results of the fits are summarized in Tab. \[ColourFits\]. Note that a single fit contains objects from a large fraction of the 329 stellar HES plates, and a wide range of object types, e.g. carbon stars, metal-poor stars, solar metallicity F- and G-type stars, field horizontal branch A-type stars, “normal” A-type stars, DAs, DBs, sdBs and quasars. The achieved accuracies are $\sigma_{U-B}=0.092$mag, and $\sigma_{B-V}=0.095$mag for the $B-V$ fit using all calibration objects together. The accuracy in $B-V$ for red ($B-V\gtrsim 1$) and blue ($B-V\lesssim 0.3$) objects is a factor of $\sim 2$ worse ($\sigma=0.15$mag and $0.12$mag, respectively) than for intermediate $B-V$ objects ($\sigma=0.074$mag).
[lcrl]{} Colour & & N$_{\mbox{\scriptsize stars}}$ & $\sigma$ \[mag\]
------------------------------------------------------------------------
\
$B-V$ & $-0.6<B-V<2.0$ & $1259$ & $0.095$
------------------------------------------------------------------------
\
$U-B$ & $-1.4<U-B<0.8$ & $573$ & $0.092$\
$c_1$ & $-0.4<c_1<1.0$ & $79$ & $0.15$\
We obtain Strömgren coefficients $c_1=(u-v)-(v-b)$ directly from HES spectra by averaging the density in the Strömgren $uvb$ bands, and computing internal coefficients $c_{1,\mbox{\tiny HES}}$ from that. $c_{1,\mbox{\tiny
HES}}$ was calibrated using a total of 79 stars not saturated in the HES, from three different sources. 22 metal-poor stars were taken from [@Schusteretal:1996], 43 stars from Beers (2000, priv. comm.), of which 2 were rejected as outliers (see Fig. \[fig:c1calib\]), and 16 hot subdwarfs from an updated version of the catalog of [@Kilkennyetal:1988]. The $1\,\sigma$ error of the calibration is $0.15$mag. $c_1$ can be used as a gravity indicator for early-type stars, since it measures the strength of the Balmer discontinuity.
Simulation of objective-prism spectra {#slit2objprism}
=====================================
In many stellar applications of the HES, it is not possible to generate large enough training and test samples from *real* spectra present on HES plates. This is because usually the target objects are very rare. Therefore, we have developed methods to generate *artificial* learning samples by simulations, using either model spectra, or slit spectra. In this paper we will use the simulations for the development of selection criteria. In later papers we will use sets of simulated spectra as learning samples for selection of e.g. metal-poor stars by automatic spectral classification.
The conversion of model spectra, or slit spectra, to objective-prism spectra consists of five steps:
1. Rebinning to the non-equidistant pixel size according to the global dispersion relation for the HES
2. Multiplication with HES spectral sensitivity curve(s)
3. Smoothing with a Gaussian filter, for simulation of seeing
4. Adding of pixel-wise, normally distributed noise
5. Random shift of the simulated spectrum according to the error distribution of the wavelength calibration zero point ($\pm 10\,\mu$m).
Step (4) ensures that objects of any brightness can be simulated. The $B_J$ magnitude range corresponding to a given $S/N$ can be read from Fig. \[SNofBJ\].
HES spectral sensitivity curves
-------------------------------
Spectral sensitivity curves (SSCs) for HES plates were determined by comparison of WD *model* spectra, rebinned to the wavelength dependent pixel size $\Delta\lambda$ of the objective-prism spectra, with objective-prism spectra of DAs on HES plates. We do not use the slit spectra directly as reference, because slit losses would produce erroneous results. The theoretical DA spectra, taken from a standard grid of WD model atmospheres [see @Finleyetal:1997 for a description], were fitted to the slit spectra with methods described in [@Finleyetal:1997] and [@Homeieretal:1998]. The spectra were taken with the Boller & Chivens spectrograph attached to the ESO 1.52m telescope, and with DFOSC at the ESO-Danish 1.54m telescope. The atmospheric parameters of the twelve objects used in this investigation are listed in Tab. \[tab:ScurveBase\]
By comparing SSCs for plates from different plate batches, with different sky background, and generated with objects spanning a wide brightness range (see Tab. \[tab:SummarizedScurves\]), but below the saturation threshold, we investigated the possible systematic influence of these characteristics on the shape of the SSCs. By comparing the shapes of the twelve resulting SSCs, we found that there is *no* systematic influence of object brightness, plate batch and sky background on SSC shape. The plate material of the HES is surprisingly homogenous; however, a *slight* variation of SSC shape is present, which hence must be attributed to another parameter. Since it is the blue part of the SSCs that varies, it is very likely that the time span between hypersensitization and development of the plate is responsible for the shape variations.
[lccl]{} Name & $T_{\mbox{\scriptsize eff}}$ \[K\]& $\log g$ & McCook & Sion
------------------------------------------------------------------------
\
HE 0004–5403 & $18\,200\pm 300$ & $8.26\pm 0.06$ &
------------------------------------------------------------------------
\
HE 0059–5701 & $30\,400\pm 300$ & $8.08\pm 0.06$ &\
HE 0252–3501 & $17\,400\pm 300$ & $7.35\pm 0.05$ & WD 0252–350\
HE 0358–5127 & $24\,100\pm 300$ & $8.10\pm 0.05$ &\
HE 0409–5154 & $27\,500\pm 300$ & $8.00\pm 0.06$ &\
HE 0412–4744 & $19\,300\pm 300$ & $8.08\pm 0.06$ &\
HE 0418–5326 & $27\,900\pm 200$ & $8.00\pm 0.05$ &\
HE 1049–1552 & $20\,200\pm 200$ & $8.63\pm 0.04$ & WD 1049–158\
HE 1058–1258 & $24\,700\pm 200$ & $8.84\pm 0.04$ & WD 1058–129\
HE 1058–1334 & $15\,900\pm 300$ & $8.00\pm 0.07$ &\
HE 1017–1618 & $28\,600\pm 300$ & $8.30\pm 0.06$ & WD 1017–163\
HE 1017–1352 & $33\,500\pm 200$ & $8.25\pm 0.05$ &\
We grouped the twelve SSCs into four SSC classes of similar shape, and averaged them within these classes (see Fig. \[scurvecat\]). When converting model spectra or slit spectra to objective prism spectra, we use an SSC created by averaging the four averaged SSCs with randomly assigned weights.
[llcrrl]{} \# & Name & $B_J$ & & [bgr]{} & Batch
------------------------------------------------------------------------
\
1 & HE 0004–5403 & 16.2 & 12076 & 1123 & 1D4
------------------------------------------------------------------------
\
1 & HE 1017–1618 & 15.8 & 8402 & 1363 & 1K6\
1 & HE 1017–1352 & 14.4 & 8402 & 1363 & 1K6\
2 & HE 1049–1552 & 14.2 & 9091 & 752 & 1C8
------------------------------------------------------------------------
\
2 & HE 1058–1258 & 14.8 & 9091 & 752 & 1C8\
3 & HE 0252–3501 & 16.0 & 11420 & 1039 & 1D4
------------------------------------------------------------------------
\
3 & HE 0358–5127 & 15.4 & 10844 & 765 & 1I3\
3 & HE 0409–5154 & 16.1 & 10844 & 765 & 1I3\
3 & HE 0412–4744 & 16.5 & 10844 & 765 & 1I3\
4 & HE 0059–5701 & 16.4 & 12052 & 1026 & 1D4
------------------------------------------------------------------------
\
4 & HE 1058–1334 & 16.6 & 9091 & 752 & 1C8\
Adding noise
------------
We add artificial, normally distributed noise to the converted spectra, in order to simulate objective-prism spectra of a desired brightness. We parameterize the $S/N$ of a spectrum by the mean $S/N$ in the $B_J$ band, $$\overline{\left(\frac{S}{N}\right)}_{B_J}
=\frac{1}{n}\sum_{i=1}^{n}\frac{D_i}{a_0+a_1D_i+a_2D_i^2},$$ using the noise model described in Sect. \[sect:NoiseEstimation\]. Since the noise depends on the density $D$, it is important to take care of the density variation throughout the spectrum. We thus scale the simulated spectra with a scaling factor $c$ such that the desired mean $S/N$ in $B_J$ is achieved, when the appropriate amount of pixel-wise Gaussian noise is added. The mean $S/N$ of the scaled spectrum is: $$\label{sn_neu}
\overline{\left(\frac{S}{N}\right)}_{\mbox{\scriptsize new}}
=\frac{1}{n}\sum_{i=1}^{n}\frac{c\cdot D_i}{
a_0+a_1\cdot c\cdot D_i+a_2\cdot c^2\cdot D_i^2},$$ We use one set of typical noise coefficients $a_0\dots a_2$ for our simulations. The appropriate scaling factor $c$ is determined numerically from Eq. (\[sn\_neu\]) using the Newton-Raphson method. A comparison of a simulated DA spectrum with a real HES spectrum is shown in Fig. \[slit2objprismdemo\].
Selection of DA white dwarfs {#Sect:DAsel}
============================
Since we are aiming at a very *efficient* selection of white dwarf candidates, we investigated to what extent different types of hot stars can be distinguished in the HES in a two-colour diagram ($U-B$ versus $B-V$), and in the two-dimensional feature space $c_1$ versus [balmsum]{}. We also explored several other parameter combinations, including e.g. continuum shape parameters determined by principal component analysis, but found them to be less appropriate. Using various catalogs, we identified 521 hot stars in the HES. The catalogs are: [@Kilkennyetal:1997], [@Wilhelmetal:1999b], an updated version of the [@Kilkennyetal:1988] subdwarf catalog, and [@McCook/Sion:1999]. Additionally, 39 HES A-type stars with known stellar parameters were included. It turned out that high gravity stars (WDs, sdBs, and sdOs) can be distinguished quite reliably from lower gravity stars (main sequence and horizontal branch stars). However, the separation between hot subdwarfs and WDs is much more difficult, since they occupy similar regions in the parameter space (see Fig. \[DASelSimple\]). The only way to obtain clean white dwarf samples is to sacrifice completeness, as we will see below. Note however that unlike in the EC, it is possible in the HES to compile candidate samples that have *no* contamination with main sequence or horizontal branch stars.
With the spectral features automatically detected in HES spectra up to now, it is not yet possible to select DB white dwarfs; in order to do this, Helium lines would have to be added to the line list. This is currently being done.
As a first step in developing selection criteria for DAs, selection boxes in the $B-V$ versus $U-B$ and $c_1$ versus Balmer line sum parameter spaces were chosen such that all DAs were included (see Fig. \[DASelSimple\]). A first set of 47 HES DA candidates (hereafter referred to as the “UVES sample”; see Tab. \[tab:UVESsample\]) selected in this way (and 72 additional WDs) were observed with VLT-UT2 and UVES between April 4 and June 6, 2000. This set includes seven re-discovered DAs listed in [@McCook/Sion:1999]. Based on pipeline-reduced spectra, 19 of these were classified as DAs, 26 as hot subdwarfs (e.g., sdB, sdOB, sdO), two as DOs, and *none* as main sequence or horizontal branch star.
In order to decrease the contamination with hot subdwarfs, we examined if “harder” selection criteria could lead to reduced, but acceptable completeness with respect to DA white dwarfs on the one hand, and a cleaner candidate sample on the other hand.
Contamination and completeness in dependence of the cutoff in Balmer line equivalent width sum was evaluated by using the subsample of those 200 stars from the learning sample of 521 stars mentioned above that were located in the selection boxes also applied to the UVES sample. The relative numbers of stars were scaled such that the fraction of hot subdwarfs and WDs did reflect the fractions in the UVES sample. As can be seen from the results displayed in Fig. \[DASelEval\], it is possible to compile a 80% complete sample of DA white dwarfs, which is contamined by only 8% hot subdwarfs, if one confines the sample to objects with $W_\lambda(\mbox{H}\beta
+\mbox{H}\gamma +\mbox{H}\delta) > 13.5$[Å]{}.
Selecting only the objects with very strong Balmer lines leads to incompleteness at the *hot* end of the DA sample (see Fig. \[c1\_balmsum\_DAs\_sn10\]), whereas DAs at $11\,000\,\mbox{K}>T_{\mbox{\scriptsize eff}}>35\,000\,\mbox{K}$ are very likely to be selected even at faint magnitudes (that is, at low $S/N$).
The final criteria, $$\begin{aligned}
W_\lambda(\mbox{H}\beta+\mbox{H}\gamma +\mbox{H}\delta) &>& 13.5\,\mbox{\AA} \\
c_1 &<& 0.3\\
B-V &<& 0.4\\
U-B &<& -0.2,\end{aligned}$$ were *a posteriori* applied to the UVES sample for an independent test. The resulting sample consists of 15 DAs only; that is, the completeness is 78.9%, which is very close to what was predicted. In a sample of 15 objects, 1.2 subdwarfs are expected if the contamination is 8%, but none was found, which is consistent with the prediction.
We evaluated the *raw candidate* selection by visually inspecting a sample of 675 DA candidates at the computer screen. Visual inspection is a mandatory step in the HES in order to identify plate artifacts, the few overlaps that are not recognized by the automatic detection algorithm, and to reject mis-classified and noisy spectra. The 675 candidates were selected from the 1789792 overlap-free spectra with $S/N>5$ present on 225 HES plates. We define the *specifity* of a selection as the fraction of candidate spectra ($N_{\mbox{\scriptsize can}}$) drawn from the original set of spectra ($N_{\mbox{\scriptsize total}}$); i.e., $$\mbox{specifity}=\frac{N_{\mbox{\scriptsize can}}}{N_{\mbox{\scriptsize total}}}.$$ We define the *efficiency* of a selection as the fraction of *correct* candidates among the automatically selected candidates, as judged from visual inspection of the objective-prism spectra. The results are listed in Tab. \[tab:sel\_eff\].
[lr]{} Specifity & 1/2652
------------------------------------------------------------------------
\
DA candidates & 71.6%\
Uncertain DA candidates & 3.3%\
main sequence stars (F or later) & 7.4%\
Subdwarf B stars & 5.0%\
Horizontal branch A- or B-type stars & 0.9%\
Quasar candidates & 0.9%\
Noisy spectra (no classification possible) & 4.7%\
Spectra disturbed by artifacts & 4.1%\
Overlaps & 2.1%\
Discussion
==========
The selection method described above allows a very efficient selection of DAs, with a high completeness ($\sim 80\,\%$). However, this completeness estimate is based on a sample of objects that are part of the current HES data base of spectra. It is expected that a significant fraction of WDs are not part of that data base due to their proper motion (p.m.). This is because the input catalog for extraction of objective-prism spectra is generated by using the DSS I. Large proper motions and/or large epoch differences between HES and DSS I plates (13.5 years on average) currently result in a mis-extraction of spectra of objects having $\mu_{\alpha}\cdot\Delta\,t_{\mbox{\scriptsize
HES-DSS~I}}\gtrsim 4''$ (i.e., $\gtrsim3$ pixels), because the spectrum of the object is looked for at the wrong place. Proper motion in direction of $\delta$ leads to an offset of the wavelength calibration zero point, resulting in wrong equivalent width measurements (i.e., the measured widths are too low).
How large the incompleteness due to the epoch-difference problem is can roughly be estimated from a cross-identification of the HES with the catalog of [@McCook/Sion:1999]. It lists 2187 objects. 1633 have an available $V$ measurement. Of these, 1295 (or $79.3$%) lie in the HES magnitude range ($13\lesssim V\lesssim 17.5$, assuming an average $B-V$ of zero). 606 objects are located in the southern hemisphere ($\delta < {2\mbox{$^\circ\mskip-6.6mu.\,$}5}$), and at high galactic latitude ($|b|>30^{\circ}$). Assuming that $79.3$% of these are in the HES magnitude range, we expect $\sim 480$ WDs to be in the HES area, and detectable in the HES. Taking into account a loss of $25$% due to overlapping spectra, we expect 360 known WDs to be found on all 380 HES plates, and 310 WDs on the 329 plates used so far for the exploitation of the stellar content. However, in a cross-identification procedure using a $10''\times 10''$ search box, to compensate for the sometimes very inaccurate coordinates of [@McCook/Sion:1999], only 151 WDs ($\sim 50$% of the expected 310) were found.
At the chosen search box width a “saturation” of identified objects was reached; by using a larger box, no further WDs were found. However, the experience from an identification of WDs from [@McCook/Sion:1999] in the northern hemisphere sister project of the HES, the Hamburg Quasar Survey [HQS; @Hagenetal:1995], is that 30–40% of the objects were identified at positions more than $10''$ away from the nominal position; in many cases *much* more (up to several arc*minutes*), which explains why the identification saturation sets in already in for a $10''\times 10''$ search box.
Furthermore, we suspect that two effects may lead to an underestimate of the completeness of the present DA selection in the HES: First, many WDs listed in [@McCook/Sion:1999] were discovered in the course of proper motion surveys. Therefore, the catalog is very likely to be biased towards high p.m. objects, which makes the listed WDs less likely to be detected in the HES than the real population of WDs. Second, the peak of the magnitude distribution of the WDs from [@McCook/Sion:1999] is more than one magnitude brighter than that of the HES, so that on average *closer* WDs (with on average higher p.m.) are listed in [@McCook/Sion:1999].
We conclude that the true completeness of our selection is probably higher than indicated by the simple cross-comparison. We conservatively estimate the proper motion-induced losses of DAs in the HES to be not higher than $\sim
20$%. Special techniques to recover also high p.m. DAs are currently under development. Such techniques will permit us to construct flux-limited samples of DA white dwarfs with a very high degree of completeness, useful e.g. to determine luminosity functions.
Another problem of WD selection in the HES might currently result from the fact that in the HES stellar feature detection algorithm the width of the absorption lines is held fixed at the measured width of the instrumental profile, in order to make the algorithm more robust. However, unlike in “normal” stars, the absorption lines of WDs are so broad that they are usually resolved in the HES spectra. Therefore, the equivalent widths of their absorption lines are expected to be underestimated. Apart from more accurately measured equivalent widths, adding the line widths to the parameters to be fitted would result in the opportunity to employ a further criterion for the separation between hot subdwarfs and WDs: if the absorption lines in the spectrum of an object are significantly broader than the instrumental profile, the object is very likely to be a WD. A modification of the feature detection algorithm is in progress.
A further improvement of the selection can likely be achieved if instead of straight lines higher-order selection boundaries are used, as is the case e.g. in quadratic discriminant analysis. However, defining such selection boundaries requires *large*, and *well-defined* learning samples. A lack of model spectra for early-type stars (main sequence as well as horizontal branch stars) so far prevented us from using a learning sample of simulated objective-prism spectra for this purpose.
Conclusions
===========
We described quantitative procedures for the selection of DA white dwarfs in the digital data base of the HES. Algorithms for the detection of stellar emission and absorption lines, broad- and intermediate-band colours were developed. These are not only used for the selection of DAs, but also for a couple of other interesting stars, e.g. metal-poor stars, and carbon stars. Simulation techniques allow us to convert model spectra to HES spectra, which is important for the development of selection criteria, for the compilation of learning samples used for automatic spectral classification, and for the determination of selection functions.
DAs can be selected very efficiently in the HES, as required by a currently ongoing *Large Programme* using VLT-UT2 and UVES, aiming at the detection of *double degenerates*. A first set of 440 DA candidates not listed in the catalog of [@McCook/Sion:1999] was identified on 225 HES plates (i.e., 59% of the survey).
We note that the UVES sample includes a double-lined DA$+$DA binary, and two further DAs with significant RV variations. The results will be reported elsewhere. Further investigation of these systems is in progress.
For investigations requiring *complete* samples of DAs (or other objects having high proper motions), special extraction techniques have to be employed to take into account the epoch differences between HES plates and the DSS I, since the latter is used for object detection in the HES. Such techniques are currently under development.
N.C. and D.H. acknowledge financial support from Deutsche Forschungsgemeinschaft under grants Re 353/40 and Ko 738/10-3, respectively. We thank V. Beckmann for observing a part of the DAs that were used to determine the HES spectral sensitivity curves. Precise, photoelectric $UBV$ and Strömgren $uvby$ photometry for HK survey stars was kindly provided by T. Beers before publication. We thank D. O’Donoghue for making EC survey photometry in digital form available to us. G. Pizarro kindly compiled a list of HES plate batches from the notes he made at the ESO Schmidt telescope. We thank the Paranal staff for carrying out the observations for the ESO Large Programme 165.H-0588 in Service Mode at VLT-UT2.
The UVES sample {#sect:UVESsample}
===============
In Tab. \[tab:UVESsample\] we list coordinates, magnitudes, colours, Balmer line equivalent widths sums and classifications of the UVES sample of 47 DA candidates.
[lllccrrrll]{} HE name & R.A. & dec & $B$ & $U-B$ & & & [balmsum]{} & Type & McCook & Sion
------------------------------------------------------------------------
\
HE 0952$+$0227 & 09 55 34.6 & $+$02 12 48 & 14.7 & $-1.04$ & $-0.17$ & $-0.10$ & 6.6 & sdO &
------------------------------------------------------------------------
\
HE 0956$+$0201 & 09 58 50.4 & $+$01 47 23 & 15.6 & $-0.64$ & $ 0.06$ & $-0.09$ & 24.7 & DA & WD 0956$+$020 (DA3)\
HE 0958$-$1151 & 10 00 42.6 & $-$12 05 59 & 13.7 & $-1.05$ & $-0.11$ & $-0.15$ & 2.9 & DAB &\
HE 1008$-$1757 & 10 10 33.4 & $-$18 11 48 & 14.9 & $-0.91$ & $-0.15$ & $-0.02$ & 2.5 & DAO (sdO hot) & WD 1008$-$179 (DA)\
HE 1012$-$0049 & 10 15 11.7 & $-$01 04 17 & 15.6 & $-0.86$ & $ 0.06$ & $-0.07$ & 20.5 & DA &\
HE 1026$+$0014 & 10 28 34.8 & $-$00 00 29 & 13.9 & $-0.49$ & $ 0.30$ & $-0.05$ & 23.1 & DA+dM & WD 1026$+$002 (DA3)\
HE 1033$-$2353 & 10 36 07.2 & $-$24 08 34 & 16.0 & $-1.08$ & $-0.34$ & $-0.03$ & 12.4 & sdB (sdOB) &\
HE 1038$-$2326 & 10 40 36.9 & $-$23 42 39 & 15.9 & $-0.86$ & $-0.13$ & $0.00$ & 10.7 & sdB+cool star? &\
HE 1047$-$0436 & 10 50 26.9 & $-$04 52 36 & 14.7 & $-0.96$ & $-0.21$ & $-0.01$ & 10.0 & sdB &\
HE 1047$-$0637 & 10 50 28.7 & $-$06 53 26 & 14.3 & $-1.04$ & $-0.17$ & $-0.11$ & 6.0 & sdO &\
HE 1053$-$0914 & 10 55 45.4 & $-$09 30 58 & 16.4 & $-0.74$ & $ 0.07$ & $-0.12$ & 30.3 & DA &\
HE 1059$-$2735 & 11 01 24.9 & $-$27 51 42 & 15.1 & $-1.23$ & $-0.35$ & $-0.19$ & 10.6 & sdB (sdOB) &\
HE 1117$-$0222 & 11 19 34.7 & $-$02 39 05 & 14.3 & $-0.51$ & $ 0.22$ & $-0.10$ & 24.8 & DA &\
HE 1130$-$0620 & 11 32 41.5 & $-$06 36 53 & 15.8 & $-1.08$ & $-0.09$ & $-0.01$ & 5.3 & sdB (sdOB) &\
HE 1136$-$2504 & 11 39 10.2 & $-$25 20 55 & 13.8 & $-1.09$ & $-0.17$ & $-0.15$ & 9.5 & sdO &\
HE 1142$-$2311 & 11 44 50.2 & $-$23 28 18 & 15.4 & $-1.28$ & $-0.34$ & $-0.23$ & 3.6 & sdO &\
HE 1152$-$1244 & 11 54 34.9 & $-$13 01 17 & 15.8 & $-0.47$ & $ 0.08$ & $-0.13$ & 28.4 & DA &\
HE 1200$-$1924 & 12 02 40.1 & $-$19 41 08 & 14.3 & $-0.95$ & $ 0.21$ & $-0.26$ & 3.0 & sdO &\
HE 1204$-$3217 & 12 06 47.7 & $-$32 34 32 & 15.7 & $-0.84$ & $-0.08$ & $-0.21$ & 27.5 & DA & WD 1204$-$322 (DA)\
HE 1221$-$2618 & 12 24 32.7 & $-$26 35 16 & 14.7 & $-0.79$ & $ 0.07$ & $-0.13$ & 6.7 & sdB+cool star &\
HE 1225$+$0038 & 12 28 07.8 & $+$00 22 17 & 15.2 & $-0.45$ & $ 0.34$ & $-0.10$ & 22.5 & DA &\
HE 1225$-$0758 & 12 27 47.5 & $-$08 14 38 & 14.6 & $-0.85$ & $ 0.06$ & $-0.09$ & 1.8 & DAB, DBA? & WD 1225$-$079 (DZA)\
HE 1229$-$0115 & 12 31 34.8 & $-$01 32 09 & 13.9 & $-0.72$ & $ 0.11$ & $-0.08$ & 24.6 & DA & WD 1229$-$012 (DA4)\
HE 1237$-$1408 & 12 39 56.5 & $-$14 24 48 & 16.1 & $-1.13$ & $-0.07$ & $-0.19$ & 12.0 & sdOB &\
HE 1238$-$1745 & 12 41 01.1 & $-$18 01 58 & 14.3 & $-0.95$ & $ 0.11$ & $-0.16$ & 6.1 & sdO &\
HE 1247$-$1738 & 12 50 22.2 & $-$17 54 47 & 16.2 & $-0.77$ & $ 0.26$ & $-0.35$ & 24.6 & DA+dM & WD 1247$-$176 (DA)\
HE 1256$-$2738 & 12 59 01.4 & $-$27 54 19 & 16.1 & $-1.30$ & $-0.31$ & $-0.36$ & 4.7 & sdO &\
HE 1257$-$2021 & 13 00 27.2 & $-$20 37 27 & 16.5 & $-1.08$ & $ 0.06$ & $-0.12$ & 2.8 & DO/PG1159?? &\
HE 1258$+$0113 & 13 00 59.2 & $+$00 57 10 & 16.5 & $-0.95$ & $-0.01$ & $ 0.00$ & 4.2 & sdO &\
HE 1258$+$0123 & 13 01 10.5 & $+$01 07 39 & 16.5 & $-0.24$ & $ 0.24$ & $ 0.26$ & 31.9 & DA &\
HE 1309$-$1102 & 13 12 02.3 & $-$11 18 16 & 16.1 & $-0.80$ & $-0.17$ & $ 0.03$ & 11.4 & sdB &\
HE 1310$-$2733 & 13 12 50.6 & $-$27 49 02 & 14.3 & $-1.23$ & $-0.25$ & $-0.28$ & 8.2 & sdO &\
HE 1314$+$0018 & 13 17 24.7 & $+$00 02 36 & 15.6 & $-1.12$ & $0.00$ & $-0.23$ & 10.8 & DO &\
HE 1315$-$1105 & 13 17 47.4 & $-$11 21 05 & 15.8 & $-0.42$ & $ 0.23$ & $ 0.05$ & 23.3 & DA &\
HE 1318$-$2111 & 13 21 15.6 & $-$21 27 18 & 14.6 & $-0.99$ & $-0.10$ & $-0.14$ & 4.6 & sdOB &\
HE 1325$-$0854 & 13 28 23.9 & $-$09 09 53 & 15.2 & $-0.64$ & $ 0.05$ & $ 0.04$ & 27.7 & DA &\
HE 1333$-$0622 & 13 36 19.7 & $-$06 37 59 & 16.2 & $-0.62$ & $ 0.15$ & $ 0.11$ & 20.6 & DA+dM &\
HE 1349$-$2320 & 13 52 15.0 & $-$23 34 57 & 15.1 & $-1.20$ & $-0.04$ & $-0.33$ & 10.2 & sdO &\
HE 1352$-$1827 & 13 55 26.6 & $-$18 42 09 & 16.0 & $-0.96$ & $-0.17$ & $-0.06$ & 11.3 & sdB+cool star? &\
HE 1355$-$0622 & 13 57 54.3 & $-$06 37 32 & 13.4 & $-1.12$ & $-0.23$ & $-0.05$ & 2.1 & sdO &\
HE 1356$-$1613 & 13 59 12.5 & $-$16 28 01 & 16.1 & $-1.21$ & $-0.35$ & $-0.23$ & 7.9 & sdO &\
HE 1419$-$1205 & 14 22 02.1 & $-$12 19 30 & 16.2 & $-0.98$ & $-0.30$ & $-0.06$ & 11.3 & sdB (sdOB) &\
HE 1502$-$1019 & 15 05 22.7 & $-$10 31 26 & 15.6 & $-1.04$ & $-0.17$ & $-0.25$ & 7.3 & sdOB+cool star &\
HE 1511$-$0448 & 15 14 12.9 & $-$04 59 33 & 15.3 & $-1.14$ & $-0.32$ & $-0.05$ & 6.8 & DA &\
HE 1511$-$1103 & 15 14 17.0 & $-$11 14 13 & 14.7 & $-1.12$ & $-0.25$ & $-0.11$ & 1.7 & sdO &\
HE 1512$-$0331 & 15 14 50.1 & $-$03 42 50 & 16.0 & $-1.22$ & $-0.32$ & $-0.35$ & 0.8 & sdO/DAO &\
Spectral atlas of DA and DB white dwarfs {#Sect:SpectralAtlas}
========================================
In Fig. \[WDmodels\] we display model spectra of DAs and DBs, converted to objective-prism spectra with the method described in Sect. \[slit2objprism\]. The model spectra were computed by using $\log g=8.0$.
[^1]: Based on observations collected at the European Southern Observatory, La Silla and Paranal, Chile.
|
---
bibliography:
- 'REFER.bib'
---
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---
author:
- |
Pierre Walczak$^{1,*}$, Cristina Rimoldi$^{1}$, Francois Gustave$^{2}$, Lorenzo Columbo$^{3,4}$,\
Massimo Brambilla$^{4,5}$,Franco Prati$^{6,7}$,Giovanna Tissoni$^{1}$,Stéphane Barland$^{1}$
bibliography:
- 'article.bib'
date: |
\
$^{*}$Corresponding author: pierre.walczak@inphyni.cnrs.fr\
title: Extreme events induced by collisions in a forced semiconductor laser
---
**We report on the experimental study of an optically driven multimode semiconductor laser with 1 m cavity length. We observed a spatiotemporal regime where real time measurements reveal very high intensity peaks in the laser field. Such a regime, which coexists with the locked state and with stable phase solitons, is characterized by the emergence of extreme events which produce a heavy tail statistics in the probability density function. We interpret the extreme events as collisions of spatiotemporal structures with opposite chirality. Numerical simulations of the semiconductor laser model, showing very similar dynamical behavior, substantiate our evidences and corroborate the description of such interactions as collisions between phase solitons and transient structures with different phase rotations.**
In the last decade, the understanding of extreme events as rogue waves arose considerable interest. These structures, consisting in a giant wave with a low probability of appearance, were first discovered in the context of oceanographic studies on the marine surface [@White:98], and yet, the origin of rogue waves is still a matter of debate[@Ruban:10]. The common tool used to characterize the emergence of extreme events is the Probability Density Function (PDF). If the statistics deviates from the normal law, with a heavy tail PDF, then one speaks about rogue waves[@Onorato:13].
Since the experiment of Solli *et al.*[@Solli:07], optics has become a useful workbench to study extreme events. Thanks to the analogy between optics and hydrodynamics, rogue waves have been studied in the one-dimensional nonlinear Schrödinger equation (1D NLSE) [@Agafontsev:2015; @Chabchoub:15; @Soto:16; @Walczak:15; @Suret:16]. Although they may be hard to identify unequivocally [@Randoux:16; @Soto:16], different analytic solutions of the 1D NLSE such as solitons, rational solutions, solitons on finite background[@Akhmediev:09b; @Dudley:14; @Kibler:10; @Kibler:12] or collisions between them [@frisquet:13] have been proposed as prototypes of rogue waves.
In dissipative systems, the phenomenon of optical rogue waves has been also studied, for example, in the supercontinuum generation [@Dudley:08], in laser diodes with optical injection [@bonatto2011; @Zamora:13] or optical feedback [@KarsaklianDalBosco:13; @Reinoso:2013; @Mercier:2015], lasers with saturable absorber [@bonazzola2013; @bonazzola2015; @selmi2016; @Coulibaly:17; @rimoldi2017] and in resonant parametric oscillators [@Oppo:13]. In these contexts, the mechanism of extreme events formation is more complex to describe because of the presence of higher nonlinear terms, gain and losses. Different theories of rogue waves formation exist [@Kharif:03] where the presence of dissipation on the ocean surface is clearly present, and in fact some works concentrate on dissipative rogue waves [@kovalsky2011; @Lacaplain:12], although others focus more on the underlying spatiotemporal chaos [@Clerc:16; @selmi2016; @Coulibaly:17; @rimoldi2017] within which rogue waves are observed. A means to gain control of such extreme events was proposed by exploiting instabilities and feedback in semiconductor lasers [@Akhmediev:16]. Dissipative optical rogue waves have been also studied in spatial domain (see *e.g.* [@Arecchi:2011; @Marsal:14; @pierangeli2015spatial]).
Recently, in the context of a forced semiconductor laser, we observed a great variety of regimes including frequency locked homogeneous state, turbulent regime and dissipative phase solitons [@Gustave:2015]. Since they emerge in a forced oscillatory medium, these solitons fundamentally consist of $2\pi$ phase rotations (whose direction sets a chirality) embedded in a phase locked background. In a propagative system with non instantaneous nonlinearity, only the counterclockwise direction is stable [@Gustave:16]. Upon collisions, these phase solitons can form complexes [@Gustave:17] with multiple chiral charge. In a recent paper [@Gibson:16], the existence of extreme events has been associated to collisions of optical vortices with opposite charge, in two dimensional transverse systems showing vortex mediated turbulence, described by complex Ginzburg-Landau and Swift-Hohenberg models with external driving. Yet, the experimental observation of extreme events originated by collisions in an injected laser system remains a challenging task, and the present work bridges this gap in a temporal / propagative context.
In this letter, we experimentally show that collisions can create extreme events in a forced semiconductor laser. We also illustrate how these localized events with high amplitude modify the PDF of the power fluctuations of the wave. Furthermore, we performed numerical simulations based on a well-tested semiconductor laser model and provide a theoretical explanation for extreme events formation.
Fig. \[fig1:setup\] shows a schematic representation of the experimental setup implemented to investigate different dynamical regimes of a forced semiconductor laser. The slave laser is a semiconductor laser in a Fabry-Perot configuration where the corresponding power is called $P_{slave}$. The cavity is composed by an active medium which is antireflection coated only on one side. A lens is inserted to focus the output field on a high reflectivity mirror (99%) which closes the one meter-long cavity. This cavity length has been chosen to allow a large number of longitudinal modes within the active medium’s gain linewidth[@Gustave:2015] and a photon life time bigger than the carriers life time while still keeping a reasonably short round-trip time, enabling fully real-time measurements. Two 10% beam splitters are inserted inside the Fabry-Perot resonator to provide both an input for the forcing field and an output for the detection of the emitted field. The master laser is provided by a grating tunable external cavity semiconductor laser. In order to prevent all reflections from the slave laser to the master laser, an optical isolator is inserted. The output is observed by using a high sensitive photodiode (Thorlabs PDA8GS) connected to a fast oscilloscope (Tektronix DPO71254C). Simultaneously, an optical spectrum analyser is used to measure the spectrum.
![Experimental setup. The Fabry Perot (FP) semiconductor laser is forced by a master laser. C1, C2: beam splitter (10%); Am: active medium with antireflection coating on one side, and with highly reflective coating M1 on the other side; L: lens; M2: high reflectivity mirror (99%); OSA: optical spectrum analyser. Cavity length is about 1 m.[]{data-label="fig1:setup"}](setup.pdf)
We use half-waveplate combined with a polarizing cube to control the power $P_{master}$ of the injected field inside the cavity. We obtain different dynamical regimes (stable phase solitons, turbulence or phase locking) depending on the detuning ($\Delta$) between the master and the slave laser. Here we focused on new regimes where an underlying irregular dynamics could couple with the propagation of self-organized structures. In particular we have selected a specific regime where we observe the emergence of a spatiotemporally localized event with a high amplitude.
![(a) Space time diagram showing the different regimes observed in the experiment. Collisions between dissipative phase solitons and counter-propagating structures lead to the emergence of extreme events. Before 7500 the system is stably locked and after 10500 one soliton is stable. (b) Observation of collision in a slightly modified reference frame (6450 ps roundtrip time instead of 6470). The laser threshold current is 18 mA, here its bias current is set at 1.6 times above the threshold. Experimental parameters: $P_{slave} = 700\mu W$ ; $P_{master} = 2.5 mW$ ; $\Delta=\lambda_{slave}-\lambda_{master} = -0.12 nm$. The actual power coupled into the cavity is at most a few % of $P_{master}$.[]{data-label="fig2:diag_spatio_temp"}](fig2_tot2.pdf)
Using the fast oscilloscope, we have recorded long time series of the intensity and constructed in real time a space-time diagram. To do that, we have folded the long time trace as a function of the roundtrip time of the cavity (6.45 ns). In this configuration, the horizontal axis is the roundtrip time, corresponding to the spatial dimension, and the vertical axis is the number of roundtrips. Fig. \[fig2:diag\_spatio\_temp\] (a) represents a spatiotemporal diagram in the regime described above. We can observe mainly three different behaviors. During the 7751 first roundtrips, the system is in synchronized state. Between roundtrips 7850 and 10060, the system exhibits a markedly different and complex dynamics. We can recognize the propagation of dissipative phase solitons, often with double charge. These coherent structures, which are spontaneously nucleated from the synchronized state, propagate and sometimes collide with different structures propagating in the opposite direction in this reference frame [^1]. These collisions induce extreme events (see below for a statistical characterization) localized in time and space as we can see in Fig. \[fig3:coupe\_fig17\].
![Cross sections through the space time diagram localized around the event with the highest amplitude (corresponding to dashed black lines in Fig. \[fig2:diag\_spatio\_temp\]). Left: Horizontal cut at the roundtrip 9764. Right: vertical cut at the time $\tau /\tau_{c}$ = 0.21.[]{data-label="fig3:coupe_fig17"}](coupe_fig17_2.pdf){width="\linewidth"}
In Fig. \[fig2:diag\_spatio\_temp\] (b), we have chosen a slightly different comoving reference frame with respect to Fig. \[fig2:diag\_spatio\_temp\] (a) in order to optimize the representation of the collision. In this case, we can clearly observe that before the collision, the dissipative phase soliton with chiral charge two collides with a different transient structure. After this collision, one hump of the soliton complex seems to be annihilated and a new hump is regenerated about ten roundtrips later. This double structure propagates and eventually collides again with a counterpropagating structure, giving rise to a new extreme event. Finally, after this complex regime (from roundtrip number 10400 to the end of this record), a regular regime is attained, dissipative phase solitons with a chiral charge of two are stable and propagate without collisions.
![Experiment: Probability Density Function (PDF) of the power fluctuations along the recording time trace. The PDF shows the deformation of the statistics with the emergence of localized events with high amplitude. Dashed red line : PDF from 0 to 7751 (locked state); Green line with circles: from 7850 to 10060 (collision regime); Blue line with squares: from 10500 to 15500 (stable soliton propagation); Black line: PDF from 0 to 15500. Dashed black line: often used rogue wave threshold $<P>+8\sigma$ with $\sigma$ the standard deviation (calculated in the collision regime). Inset: kurtosis ($k_{4}$) for each PDF.[]{data-label="fig3:stat_fig17"}](fig4_pdf.pdf)
The statistical impact of these events can be characterized by the computation of the PDF. Different approaches to PDF computation exist, depending on the flavour of the physics and the field of research[@Hammani:10; @Onorato:13; @Coulibaly:17]. In our system, we can have both complex dynamical states and completely coherent locked states. For this reason, we focus on the PDF of the power fluctuations for a long time trace as it is represented in Fig. \[fig3:stat\_fig17\].
The black solid line represents the PDF on all the roundtrips. The PDF is asymmetric and shows a clear heavy tail at the highest power values. The dashed red line, green line with circles and blue line with squares correspond respectively to the synchronized state, the turbulent regime and the soliton propagation regime described in Fig. \[fig2:diag\_spatio\_temp\] (a). In the synchronized state, where the emission is regular and continuous, the PDF results essentially from the physical and detection noise (dominant), which is stochastic. This explains why the PDF has a gaussian form. In the opposite case, in the turbulent regime, the emergence of extreme events modifies strongly the PDF which deviates from the gaussian statistics to exhibit a heavy tail statistics. Finally, the propagation of dissipative phase solitons plays also a significant role in the modification of the PDF. Due to their particular shape including a depression on the trailing edge [@Gustave:2015; @Gustave:16], the probability of the low power values is increased.
We kept fixed the parameters $\alpha=3$, $T=0.3$, $b=10$, and $d=10^{-6}$, and set the pump current well above threshold $\mu=2$ ($\mu_{th}=1$). The detuning $\theta=-2.7$ and the injection amplitude $y=0.11$ were chosen so that the homogenous stationary solution is triple-valued, and the upper state is stable. Futhermore, the value of $y$, $\mu$ and $\theta$ are very close to the experiment.
![Numerical simulation: zoom of the spatiotemporal diagram of the electric field intensity around a collision (a), and trajectories in the Argand plane for the selected roundtrip sections (b). Roundtrip section 3 coincides with the occurrence of a high intensity event and it is preceded by a clockwise rotation at the roundtrip section 2.[]{data-label="fig5:numerics"}](fig5){width="\linewidth"}
In Fig. \[fig5:numerics\] (b) we show a zoom of the spatio-temporal diagram in a simulation where the initial condition was the homogeneous stationary solution with a superimposed phase kink of 4$\pi$ (but the same kind of regime can develop starting from noise). We can notice that the spatiotemporal dynamics present some clear similarities with the experimental data. In particular we can observe the presence of phase solitons and the emergence of some peculiar structures moving at a different velocity that eventually collide with the phase solitons, giving rise to an event of high intensity.
In Fig. \[fig5:numerics\](a) we show the phase diagrams relative to the horizontal cuts highlighted in Fig. \[fig5:numerics\](b), time grows from red to yellow. The points $A$, $B$ and $C$ are the projections in the Argand plane of the the three fixed points [@Gustave:2015; @Gustave:16], respectively a node (stable for our choice of parameters), a saddle and an unstable focus. In the first frame, relative to roundtrip section 1, we can observe a phase soliton complex [@Gustave:17] of charge 3 that propagates inside the cavity: the trajectory of the system consists in three counterclockwise phase rotations in the Argand plane, passing only once close to point $A$. At roundtrip section 2 a new object emerges at the right side of the soliton complex, together with a clockwise phase rotation (charge -1) in the Argand plane. The interaction between the two structures gives rise to a high intensity event at roundtrip section 3, where the clockwise rotation has already been lost. The interaction has also the effect of altering the phase soliton complex velocity inside the cavity, as can be noticed just after roundtrip section 3. At roundtrip section 4 the complex has regained its shape, with one additional charge, coming from the interaction.
The above evidences indicate that the appearance of these short-lived pulses with clockwise phase rotations that collide with stable phase solitons is the basic physical mechanism responsible for the extreme events observed in this system.
In conclusion, we have reported the experimental observation of extreme events due to the collision of localized structures in a forced semiconductor laser in a Fabry-Perot configuration. The appearance of these events localized in time and space changes strongly the PDF which exhibits a heavy tailed statistics. Numerical simulations based on a semiconductor laser model in a dynamical regime very close to the experiment have shown that strong intensity peaks follow the interaction between a phase soliton carrying a counterclockwise phase rotation and a low-intensity transient structure carrying a clockwise phase rotation. However, this transient object is of comparatively weak amplitude and its detection is beyond the capabilities of the phase measurement apparatus used in [@Gustave:2015; @Gustave:16].
Acknowledgments {#acknowledgments .unnumbered}
===============
F. G., G. T., and S. B. acknowledge funding from Agence Nationale de la Recherche through Grant No. ANR-12-JS04-0002-01. P. W. and S. B. acknowledge funding from Région Provence Alpes Côte d’Azur trough Grant No 15-1351.
[^1]: Note that we measure only one direction of propagation of the field. Opposite propagation direction in the chosen reference frame does not mean opposite propagation direction of the field in the laboratory frame
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abstract: 'In this paper, we verify the $L^p$ coarse Baum-Connes conjecture for spaces with finite asymptotic dimension for $p\in[1,\infty)$. We also show that the $K$-theory of $L^p$ Roe algebras are independent of $p\in(1,\infty)$ for spaces with finite asymptotic dimension.'
address:
- 'Shanghai Center for Mathematical Sciences, Fudan University'
- 'Research Center for Operator Algebras, East China Normal University'
author:
- Jianguo Zhang
- Dapeng Zhou
bibliography:
- 'LpCBC.bib'
title: '$L^p$ Coarse Baum-Connes Conjecture and $K$-theory for $L^p$ Roe Algebras'
---
Introduction
============
An elliptic differential operator on a closed manifold is Fredholm. The celebrated Atiyah-Singer index theorem compute the Fredholm index [@AtiyahSingerI] [@AtiyahSingerIII]. In the recent 40 years, the Atiyah-Singer index theorem has been vastly generalized to the higher index theory [@WillettYuBook][@YuICM]. There are two most important cases. For a manifold carrying a proper cocompact group action, the Baum-Connes assembly map defines a higher index in the $K$-theory of group $C^\ast$-algebra [@MischenkoFumenko][@Kasparov88]. For an open manifold without group actions, the coarse Baum-Connes assembly map defines a higher index in the $K$-theory of the Roe algebra of the manifold [@Roe88].
The Baum-Connes conjecture [@BaumConnesHigson] and the coarse Baum-Connes conjecture [@HigsonRoeCBC][@YuCBC] give algorithms to compute the higher indices using $K$-homology. The $K$-homology is local and much more computable. In recent years, the $L^p$ version of the Baum-Connes and the coarse Baum-Connes conjecture are studied. The motivation for using Banach algebras is that they are more flexible than $C^\ast$-algebras. Since the traditional $C^\ast$-algebraic method [@Kasparov88] is very difficult in dealing with groups with property (T) (these groups admits no proper isometric actions on Hilbert spaces). Actually a lot of interesting groups, e.g. hyperbolic group, may have property (T). Lafforgue invented the Banach $KK$-theory and verified the Baum-Connes conjecture for a large class of groups with property (T) [@LafforgueInven]. In [@YuHyperbolicLpAction], Guoliang Yu proved that hyperbolic groups always admit proper isometric actions on $\ell^p$ spaces. In [@KasparovYuPBC], Kasparov and Yu proved that the $L^p$ Baum-Connes conjecture is true for groups with a proper isometric action on $\ell^p$ space.
In [@LiaoYu], Benben Liao and Guoliang Yu proved that the $K$-theory of $L^p$ group algebras are independent of $p$ for a large class of groups, e.g. hyperbolic groups. Their proof relies on the Lafforgue’s results on the Baum-Connes conjecture [@LafforgueInven] and $L^p$ property (RD) for the group.
Yeong-Chyuan Chung developed a quantitative $K$-theory for Banach algebras [@ChungQuantiativeBanach] and applied this theory to compute $K$-theory of $L^p$ crossed products [@ChungDynamical]. Chung showed that the Baum-Connes conjecture for $G$ with coefficient in $C(X)$ is true if the dynamical system $G\curvearrowright X$ has Finite Dynamical Complexity, introduced by Guentner-Willett-Yu [@GuentnerWillettYuFDC]. As a corollary, Chung proved that the $K$-theory of $L^p$ crossed products $B^{p,\ast}(X,G)$ are independent of $p$ provided that $G\curvearrowright X$ has finite dynamical complexity.
Motivated by Liao-Yu and Chung’s result, we ask the following question: Are the $K$-theory of $L^p$ Roe algebras $B^p(X)$ independent of $p$? The main theorem of the paper provides a positive answer to this question for the spaces with finite asymptotic dimension.
Let $X$ be a separable proper metric space. If $X$ has finite asymptotic dimension, then $K_*(B^p(X))$ does not depend on $p$ for $p\in(1,\infty)$
The proof of the theorem relies on the $L^p$ coarse Baum-Connes conjecture. The key ingredient is the Mayer-Vietoris argument. A coarse geometric Mayer-Vietoris sequence in $K$-theory was formulated by Higson-Roe-Yu [@HigsonRoeYuCoarseMV]. In [@YuFAD], Guoliang Yu invented the quantitative $K$-theory and a quantitative Mayer-Vietoris sequence, and he verified the coarse Baum-Connes conjecture for spaces with finite asymptotic dimension. The quantitative $K$-theory is a refined version of the classical operator $K$-theory. It encodes more geometric information, and it is a powerful tool to compute the $K$-theory of Roe algebras or other $C^\ast$-algebras coming from geometry. The quantitative $K$-theory has been generalized to general geometric $C^\ast$-algebras by Oyono-Oyono and Yu [@OyonoYuQuantitative][@OyonoYuPersistence][@OyonoYuKunneth], to Banach algebras by Yeong-Chyuan Chung [@ChungQuantiativeBanach], and to groupoids by Clement Dell’Aiera [@DellAieraControlledGroupoid]. It has many important applications in dynamical systems [@GuentnerWillettYuFDC][@ChungDynamical] and coarse geometry [@LiWillett][@ChungLiLpUniformRoe]. In this paper, by a similar argument of quantitative $K$-theory for $L^p$ algebras, we prove the following result.
For any $p\in [1,\infty)$, the $L^p$ coarse Baum-Connes conjecture holds for separable proper metric spaces with finite asymptotic dimension.
The result is very similar to Chung’s result on Baum-Connes conjecture with coefficient for dynamical systems with finite dynamical decomposition complexity [@ChungDynamical]. His result is for dynamical systems or transform groupoids, while our result is for coarse geometry or coarse groupoids.
We want to emphasize that the results in this paper does not need the condition of bounded geometry. For the similar result for spaces with bounded geometry, we could generalize the result to spaces with finite decomposition complexity, introduced by Erik Guentner, Romain Tessera and Guoliang Yu [@GuentnerTesseraYuRigidity][@GuentnerTesseraYuGeometric]. Our method also works for uniform $L^p$ Roe algebras. We will study the results in a separate paper.
The paper is organized in the following order: In Section 2, we recall the concept of $L^p$ Roe algebras, $L^p$ localization algebras and $L^p$ coarse Baum-Connes conjecture. In Section 3, we study the Quantitative $K$-theory for $L^p$ algebras. In Section 4, we prove that the $L^p$ Baum-Connes conjecture is true for spaces with finite asymptotic dimension. In Section 5, we prove that the $K$-theory of $L^p$ Roe algebras are independent of $p$ for spaces with finite asymptotic dimensions. In the end, we raise some open problems for future study.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The authors would like to thank our Ph.D. advisor, Guoliang Yu, for suggesting us this project and his guidance.
$L^p$ Coarse Baum-Connes Conjecture
===================================
Let $X$ be a separable proper metric space, $p\in [1,\infty)$. Recall that a metric space is called proper if every closed ball is compact.
\[Def:Lp-module\] An *$L^p$-$X$-module* is an $L^p$ space $E_X^p=\ell^p(Z_X)\otimes \ell^p=\ell^p(Z_X,\ell^p)$ equipped with a natural point-wise multiplication action of $C_0(X)$ by restricting to $Z_X$, where $Z_X$ is a countable dense subset in $X$, $\ell^p=\ell^p({\mathbb{N}})$ and $C_0(X)$ is the algebra of all complex-valued continuous functions on $X$ which vanish at infinity.
We notice that this action can be extended naturally to the algebra of all bounded Borel functions on $X$.
$L^p$ Roe algebra
-----------------
Let $E_X^p$ be an $L^p$-$X$-module and $E_Y^p$ be an $L^p$-$Y$-module, and let $T:E_X^p \rightarrow E_Y^p$ be a bounded linear operator. The *support* of $T$, denoted $\mathrm{supp}(T)$, consists of all points $(x,y)\in X\times Y$ such that $\chi_VT\chi_U \not=0$ for all open neighbourhoods $U$ of $x$ and $V$ of $y$, where $\chi_U$ and $\chi_V$ be the characteristic function on $U$ and $V$, respectively.
We give some properties of the support, the proof can be obtained similarly from chapter 4 of [@WillettYuBook].
\[Remark:support\] Let $E_X^p$ be an $L^p$-$X$-module, $E_Y^p$ be an $L^p$-$Y$-module, and $E_Z^p$ be an $L^p$-$Z$-module. Let $R,S:E_X^p\rightarrow E_Y^p$ and $T:E_Y^p\rightarrow E_Z^p$ be bounded linear operators. Then:
1. $\mathrm{supp}(R+S) \subseteq \mathrm{supp}(R) \cup \mathrm{supp}(S)$;
2. $\mathrm{supp}(TS) \subseteq \mathrm{cl}(\mathrm{supp}(S) \circ \mathrm{supp}(T))=\mathrm{cl}(\{(x,z)\in X\times Z: \text{there is}\ y\in Y \ \text{such that } (x,y)\in \mathrm{supp}(S)$, $(y,z)\in \mathrm{supp}(T)\})$, where ‘$\mathrm{cl}$’ means closure;
3. If the coordinate projections $\pi_Y$: $\mathrm{supp}(T)\rightarrow Y$ and $\pi_Z: \mathrm{supp}(T)\rightarrow Z$ are proper maps, or coordinate projections $\pi_X: \mathrm{supp}(S)\rightarrow X$ and $\pi_Y: \mathrm{supp}(S)\rightarrow Y$ are proper maps, then $\mathrm{supp}(TS) \subseteq \mathrm{supp}(T) \circ \mathrm{supp}(S)$;
4. Let $F$ = supp($S$), then for any compact subset $K$ of $X$, respectively $Y$, we have $S\chi_K=\chi_{K\circ F}S\chi_K$, $\chi_KS=\chi_KS\chi_{F\circ K}$, where $K\circ F:=${$y\in Y:$ there is $x\in K$ such that $(x,y)\in F$}, $F\circ K:=${$x\in X:$ there is $y\in K$ such that $(x,y)\in F$}.
Let $E_X^p$ be an $L^p$-$X$-module and $T$ be a bounded linear operator acting on $E_X^p$.
1. The *propagation* of $T$, denoted $\mathrm{prop}(T)$, is defined to be $\sup\{d(x,y):(x,y)\in \mathrm{supp}(T)\}$;
2. $T$ is said to be *locally compact* if $\chi_K T$ and $T \chi_K$ are compact operators for all compact subset $K$ of $X$.
By Remark \[Remark:support\], We have the following properties of propagation.
\[Remark:propagation\] Let $E_X^p$ be an $L^p$-$X$-module and let $T,S:E_X^p\rightarrow E_X^p$ be bounded linear operators. Then:
1. $\mathrm{prop}(T+S) \leq \max\{\mathrm{prop}(T),\mathrm{prop}(S)\}$;
2. $\mathrm{prop}(TS) \leq \mathrm{prop}(T) + \mathrm{prop}(S)$.
Let $E_X^p$ be an $L^p$-$X$-module. The *$L^p$ Roe algebra* of $E_X^p$, denoted $B^p(E_X^p)$, is defined to be the norm closure of the algebra of all locally compact operators acting on $E_X^p$ with finite propagations.
A Borel map $f$ from a proper metric space $X$ to another proper metric space $Y$ is called *coarse* if (1) $f$ is proper, i.e., the inverse image of any bounded set is bounded; (2) for every $R>0$, there exists $R'>0$ such that $d(f(x),f(y))\leq R'$ for all $x,y\in X$ satisfying $d(x,y)\leq R$.
\[Lemma:CoveringIsometry\] Let $f$ be a continuous coarse map, let $E_X^p$ be an $L^p$-$X$-module and $E_Y^p$ be an $L^p$-$Y$-module. Then for any $\varepsilon>0$, there exist an isometric operator $V_f:E_X^p\rightarrow E_Y^p$ and a contractible operator $V_f^+:E_Y^p\rightarrow E_X^p$ with $V_f^+V_f=I$ such that
$\mathrm{supp}(V_f) \subseteq \{(x,y)\in X\times Y:d(f(x),y)\leq \varepsilon\}$\
$\mathrm{supp}(V_f^+) \subseteq\{(y,x)\in Y\times X:d(f(x),y)\leq \varepsilon\}$
Let $Z_X$, $Z_Y$ be the dense subsets of $X$ and $Y$ for defining $E^p_X$ and $E^p_Y$, respectively, as in Definition \[Def:Lp-module\]
There exists a Borel cover $\{Y_i\}_i$ of $Y$ such that:
1. $Y_i\cap Y_j=\emptyset$ if $i\not=j$;
2. $\mathrm{diameter}(Y_i)\leq \varepsilon$ for all $i$;
3. each $Y_i$ has nonempty interior.
Condition (3) implies that $Y_i\cap Z_Y$ is a countable set. Thus if $f^{-1}(Y_i)\cap Z_X\not=\emptyset$, then there exists an isometric operator $V_i:\ell^p(f^{-1}(Y_i)\cap Z_X)\otimes \ell^p\rightarrow \ell^p(Y_i\cap Z_Y)\otimes \ell^p$ and a contractible operator $V_i^+:\ell^p(Y_i\cap Z_Y)\otimes \ell^p\rightarrow \ell^p(f^{-1}(Y_i)\cap Z_X)\otimes \ell^p$ such that $V_i^+V_i=\chi_{f^{-1}(Y_i)\cap Z_X}\otimes I$. If $f^{-1}(Y_i)\cap Z_X=\emptyset$, then let $V_i=V_i^+=0$. Define $$V_f=\bigoplus_{i} V_i:\bigoplus_{i} \ell^p(f^{-1}(Y_i)\cap Z_X)\otimes \ell^p\rightarrow \bigoplus_{i} \ell^p(Y_i\cap Z_Y)\otimes \ell^p$$ $$V_f^+=\bigoplus_{i} V_i^+:\bigoplus_{i} \ell^p(Y_i\cap Z_Y)\otimes \ell^p \rightarrow \bigoplus_{i} \ell^p(f^{-1}(Y_i)\cap Z_X)\otimes \ell^p.$$ Then $V_f$ is an isometric operator, $V_f^+$ is a contractible operator and $V_f^+V_f=I$. Condition (2) together with the construction of $V_f$ and $V_f^+$, implies that $\mathrm{supp}(V_f) \subseteq \{(x,y)\in X\times Y:d(f(x),y)\leq \varepsilon\}$ and $\mathrm{supp}(V_f^+) \subseteq\{(y,x)\in Y\times X:d(f(x),y)\leq \varepsilon\}$.
\[Lemma:CoveringIsometryPair\] Let $f$, $E_X^p$ and $E_Y^p$ be as in Lemma \[Lemma:CoveringIsometry\]. Then pair $(V_f,V_f^+)$ gives rise to a homomorphism $\mathrm{ad}((V_f,V_f^+)):B^p(E_X^p)\rightarrow B^p(E_Y^p)$ defined by: $$\mathrm{ad}((V_f,V_f^+))(T)=V_fTV_f^+$$ for all $T\in B^p(E_X^p)$.
Moreover, the map $\mathrm{ad}((V_f,V_f^+))_*$ induced by $\mathrm{ad}((V_f,V_f^+))$ on $K$-theory depends only on $f$ and not on the choice of pair $(V_f,V_f^+)$.
Obviously, $\mathrm{ad}(V_f,V_f^+)$ be a contractible homomorphism, thus we just need show that if $T$ has finite propagation and is locally compact, then $\mathrm{ad}((V_f,V_f^+))(T)$ has these properties too.
Assume first that $T$ has finite propagation. Let $\varepsilon$ be as in Lemma \[Lemma:CoveringIsometry\], then $d(f(x),y)\leq \varepsilon$ and $d(f(x'),y')\leq \varepsilon$ for any $(x,y)\in \mathrm{supp}(V_f)$ and $(y',x')\in \mathrm{supp}(V_f^+)$. Let $(y_1,y_2)\in \mathrm{supp}(V_fTV_f^+)$, by Remark \[Remark:support\] part (3), we have that
$\mathrm{supp}(V_fTV_f^+) \subseteq \mathrm{supp}(V_f) \circ \mathrm{supp}(T)\circ \mathrm{supp}(V_f^+)$.
Hence there exist $x_1,x_2\in X$ such that $(x_1,y_1)\in \mathrm{supp}(V_f), (x_1,x_2)\in \mathrm{supp}(T)$ and $(y_2,x_2)\in \mathrm{supp}(V_f^+)$, then $$d(y_1,y_2)\leq d(y_1,f(x_1))+d(f(x_1),f(x_2))+d(f(x_2),y_2)\leq 2\varepsilon +d(f(x_1),f(x_2)).$$ Since $f$ is coarse and $T$ has finite propagation, we have that $d(y_1,y_2)$ is smaller than some constant for all $(y_1,y_2)\in \mathrm{supp}(V_fTV_f^+)$, this completes the proof of finite propagation.
Now assume that $T$ is locally compact. Let $K$ be a compact subset of $Y$, and let $F = \mathrm{supp}(V_f)$. By Remark \[Remark:support\] (4), we have that$$\chi_K V_fTV_f^+=\chi_K V_f\chi_{F\circ K}TV_f^+$$ Since $f$ is proper map and $X$ is proper space, we know that $F\circ K$ is compact subset in $X$, then $\chi_{F\circ K}T$ is compact operator, thus $\chi_K V_f\chi_{F\circ K}TV_f^+$ is a compact operator. The case of $V_fTV_f^+\chi_K$ is similar. Thus $\mathrm{ad}((V_f,V_f^+))(T)$ is locally compact.
Let $(V_1,V_1^+)$ and $(V_2,V_2^+)$ be two pair operators satisfying the conditions of Lemma \[Lemma:CoveringIsometry\], then we just need to prove $$\mathrm{ad}((V_1,V_1^+))_*=\mathrm{ad}((V_2,V_2^+))_*: K_*(B^p(E_X^p))\rightarrow K_*(B^p(E_Y^p))$$ Let
$$U=\begin{pmatrix}
I-V_1V_1^+ & V_1V_2^+ \\
V_2V_1^+ & I-V_2V_2^+
\end{pmatrix}$$
then $U^2=I$ and
$$\begin{pmatrix}
\mathrm{ad}((V_1,V_1^+))(T) & 0\\
0 & 0
\end{pmatrix}
=U\begin{pmatrix}
0 & 0\\
0 & \mathrm{ad}((V_2,V_2^+))(T)
\end{pmatrix}U$$
Thus $\mathrm{ad}((V_1,V_1^+))_*=\mathrm{ad}((V_2,V_2^+))_*$
For different $L^p$-$X$-modules $E_X^p$ and $E_X'^{p}$, $B^p(E_X^p)$ is non-canonically isomorphic to $B^p(E_X'^{p})$, and $K_*(B^p(E_X^p))$ is canonically isomorphic to $K_*(B^p(E_X'^{p}))$.
For convenience, we replace $B^p(E_X^p)$ by $B^p(X)$ representing $L^p$ Roe algebra of $X$.
$L^p$ Localization algebra and $L^p$ $K$-homology
-------------------------------------------------
Let $X$ be a separable proper metric space. The *$L^p$ localization algebra* of $X$, denoted $B_L^p(X)$, is defined to be the norm closure of the algebra of all bounded and uniformly norm-continuous function $f$ from $[0,\infty)$ to $B^p(X)$ such that
prop($f(t)$) is uniformly bounded and prop($f(t)$)$\rightarrow 0$ as $t\rightarrow \infty$.
The *propagation* of $f$ is defined to be $\max\{\mathrm{prop}(f(t)):t\in [0,\infty)\}$.
Let $f$ be a uniformly continuous coarse map from a separable proper metric space $X$ to another separable proper metric space $Y$. Let $\{\varepsilon_k\}_k$ be a sequence of positive numbers such that $\varepsilon_k \rightarrow 0$ as $k \rightarrow \infty$. By Lemma \[Lemma:CoveringIsometry\], for each $\varepsilon_k$, there exists an isometric operator $V_k$ from an $L^p$-$X$-module $E_X^p$ to an $L^p$-$Y$-module $E_Y^p$ and a contractible operator $V_k^+$ from an $L^p$-$Y$-module $E_Y^p$ to an $L^p$-$X$-module $E_X^p$ such that $V_k^+V_k=I$ and
$\mathrm{supp}(V_k) \subseteq\{(x,y)\in X\times Y : d(f(x),y)\leq \varepsilon_k\}$\
$\mathrm{supp}(V_k^+) \subseteq\{(y,x)\in Y\times X : d(f(x),y)\leq \varepsilon_k\}$.
For $t\in [0,\infty)$, define
$V_f(t)=R(t-k)(V_k\oplus V_{k+1})R^*(t-k)$\
$V_f^+(t)=R(t-k)(V_k^+\oplus V_{k+1}^+)R^*(t-k)$
for all $k\leq t \leq k+1$, where $$R(t)=\begin{pmatrix}
\cos(\pi t/2) & \sin(\pi t/2)\\
-\sin(\pi t/2) & \cos(\pi t/2)
\end{pmatrix}.$$ $V_f(t)$ is an operator from $E_X^p \oplus E_X^p$ to $E_Y^p \oplus E_Y^p$, and $V_f^+(t)$ is an operator from $E_Y^p \oplus E_Y^p$ to $E_X^p \oplus E_X^p$ such that $||V_f(t)||\leq 4$, $||V^+_f(t)||\leq 4$ and $V_f^+(t)V_f(t)=I$ for all $t\in[0,\infty)$.
\[Lemma:IsometryPairHom\] Let $f$ and $\{\varepsilon_k\}_k$ be as above, then the pair $(V_f(t),V_f^+(t))$ induces a homomorphism $\mathrm{Ad}((V_f,V_f^+))$ from $B_L^p(X)$ to $B_L^p(Y)\otimes M_2(\mathbb{C})$ defined by: $$\mathrm{Ad}((V_f,V_f^+))(u)(t)=V_f(t)(u(t)\oplus 0)V_f^+(t)$$ for any $u\in B_L^p(X)$ and $t\in [0,\infty)$, such that $$\mathrm{prop}(\mathrm{Ad}((V_f,V_f^+))(u)(t))\leq \sup_{(x,y)\in \mathrm{supp}(u(t))}d(f(x),f(y))+2\varepsilon_k + 2\varepsilon_{k+1}$$ for all $t\in [k,k+1]$.
Moreover, the induced map $\mathrm{Ad}((V_f,V_f^+))_*$ on $K$-theory depends only on f and not on the choice of the pairs $\{(V_k,V_k^+)\}$ in the construction of $V_f(t)$ and $V_f^+(t)$.
For any $u\in B_L^p(X)$, $\mathrm{Ad}((V_f,V_f^+))(u)$ is bounded and uniformly norm-continuous in $t$ although $V_f$ and $V_f^+$ is not norm-continuous. By the same ways as the proof of Lemma \[Lemma:CoveringIsometryPair\], we can obtain that $\mathrm{Ad}((V_f,V_f^+))(u)(t)$ is locally compact when $u(t)$ is locally compact for each $t$ and $\mathrm{Ad}((V_f,V_f^+))_*$ does not depend on the choice of the pair $(V_f,V_f^+)$.
Thus we just need to consider prop($\mathrm{Ad}((V_f,V_f^+))(u)(t)$) for which prop($u(t)$) is uniformly finite and prop($u(t)$)$\rightarrow 0$ as $t\rightarrow \infty$. By Lemma \[Remark:support\] (4), we know that $$\begin{aligned}
\mathrm{prop}(V_k u(t) V_k^+) &\le\sup\{d(f(x),f(y)):(x,y)\in \mathrm{supp}(u(t))\}+2\varepsilon_k\\
\mathrm{prop}(V_k u(t) V_{k+1})& \le \sup\{d(f(x),f(y)):(x,y)\in \mathrm{supp}(u(t))\}+\varepsilon_k+\varepsilon_{k+1}.\end{aligned}$$ Thus by Remark \[Remark:propagation\], we have
prop$(\mathrm{Ad}((V_f,V_f^+))(u)(t))$ $\leq$ sup$\{d(f(x),f(y)):(x,y)\in \mathrm{supp}(u(t))\}$ $+$ $2\varepsilon_k + 2\varepsilon_{k+1}$
Therefore, $\mathrm{prop}(\mathrm{Ad}((V_f,V_f^+))(u)(t))$ is uniformly finite since $f$ is a coarse map, and $\mathrm{prop}(\mathrm{Ad}((V_f,V_f^+))(u)(t)$)$\rightarrow 0$ as $t\rightarrow \infty$ since $f$ is uniformly continuous map and $\varepsilon_k\rightarrow 0$.
The $i$-th *$L^p$ $K$-homology* of $X$, is defined to be $K_i(B^p_L(X))$.
Obstruction group
-----------------
Let $X$ be a separable proper metric space, now consider the evaluation-at-zero homomorphism: $$e_0:B_L^p(X)\rightarrow B^p(X)$$ which induces a homomorphism on $K$-theory: $$e_0:K_*(B_L^p(X))\rightarrow K_*(B^p(X))$$
Let $C$ be a locally finite and uniformly bounded cover for $X$. The *nerve space* $N_C$ associated to $C$ is defined to be the simplicial complex whose set of vertices equals $C$ and where a finite subset $\{U_0,\ldots,U_n\}\subseteq C$ spans an $n$-simplex in $N_C$ if and only if $\bigcap_{i=0}^n U_i \not=\emptyset$. Endow $N_C$ with the *$\ell^1$-metric*, i.e., the path metric whose restriction to each simplex $\{U_0,\ldots,U_n\}$ given by $$d(\sum^n_{i=0}t_iU_i,\sum^n_{i=0}s_iU_i)=\sum^n_{i=0}|t_i-s_i|.$$ The metric of two points which in the different connected components is defined to be $\infty$ by convention.
A sequence of locally finite and uniformly bounded covers $\{C_k\}_{k=0}^\infty$ of metric space $X$ is called an *anti-Čech system* of $X$ [@RoeBookCoarseCohomology], if there exists a sequence of positive numbers $R_k\rightarrow \infty$ such that for each $k$,
\(1) every set $U\in C_k$ has diameter less than or equal to $R_k$;
\(2) any set of diameter $R_k$ in $X$ is contained in some member of $C_{k+1}$.
An anti-Čech system always exists [@RoeBookCoarseCohomology]. By the property of the anti-Čech system, for every pair $k_2>k_1$, there exists a simplicial map $i_{k_1 k_2}$ from $N_{C_{k_1}}$ to $N_{C_{k_2}}$ such that $i_{k_1k_2}$ maps a simplex $\{U_0,\ldots,U_n\}$ in $N_{C_{k_1}}$ to a simplex $\{U_0',\ldots,U_n'\}$ in $N_{C_{k_2}}$ satisfying $U_i\subseteq U_i'$ for all $0\leq i\leq n$. Thus, $i_{k_1 k_2}$ gives rise to the following inductive systems of groups:
$\mathrm{ad}((V_{i_{k_1k_2}},V_{i_{k_1k_2}}^+))_*:K_*(B^p(N_{C_{k_1}}))\rightarrow K_*(B^p(N_{C_{k_2}}))$;\
$\mathrm{Ad}((V_{i_{k_1k_2}},V_{i_{k_1k_2}}^+))_*:K_*(B_L^p(N_{C_{k_1}}))\rightarrow K_*(B_L^p(N_{C_{k_2}}))$.
The following conjecture is called the $L^p$ coarse Baum-Connes conjecture.
Let $X$ be a separable proper metric space, $\{C_k\}_{k=0}^\infty$ be an anti-Čech system of $X$, then the evaluation-at-zero homomorphism $$e_0:\lim_{k\rightarrow \infty} K_*(B_L^p(N_{C_k})) \rightarrow \lim_{k\rightarrow \infty} K_*(B^p(N_{C_k})) \cong K_*(B^p(X))$$ is an isomorphism.
One can check that for each $p\in [1,\infty)$, the group $\lim_{k\rightarrow \infty} K_*(B_L^p(N_{C_k}))$ be the *coarse $K$-homology of $X$* [@WillettYuBook]. Moreover, it is not difficult to see that the $L^p$ coarse Baum-Connes conjecture for $X$ does not depend on the choice of the anti-Čech system.
Let $B_{L,0}^p(X)=\{f\in B_L^p(X):f(0)=0\}$. There exists an exact sequence: $$0 \rightarrow B_{L,0}^p(X) \rightarrow B_L^p(X) \rightarrow B^p(X) \rightarrow 0$$ Thus we have the following reduction:
\[Thm:VanishingObstruction\] Let $X$ be a separable proper metric space, $\{C_k\}_{k=0}^\infty$ be an anti-Čech system of $X$, then the $L^p$ coarse Baum-Connes conjecture is ture if and only if $$\lim_{k\rightarrow \infty} K_*(B_{L,0}^p(N_{C_k}))=0$$
For obvious reason $\lim_{k\rightarrow \infty} K_*(B_{L,0}^p(N_{C_k}))$ is called the obstruction group to the $L^p$ coarse Baum-Connes conjecture.
Controlled obstructions: $QP_{\delta,N,r,k}(X),QU_{\delta,N,r,k}(X)$
====================================================================
The controlled obstruction $QP$ and $QU$ for the coarse Baum-Connes conjecture was introduced by Guoliang Yu [@YuFAD]. In this section, we will introduce and study the $L^p$ version of $QP$ and $QU$, which can be considered as a controlled version of $K_0(B^p_{L,0}(X)\otimes C_0((0,1)^k))$ and $K_1(B^p_{L,0}(X)\otimes C_0((0,1)^k))$. We will follow the notation in [@YuFAD]. One may refer to [@OyonoYuQuantitative][@ChungQuantiativeBanach] for more detail about the controlled $K$-theory for $C^\ast$-algebras and $L^p$-algebras.
Fundamental concept and property
--------------------------------
([@ChungQuantiativeBanach]) Let A be a unital Banach algebra, for $0<\delta<1/100,N\geq 1$, we define (1) an element $e$ in $A$ is called *$(\delta,N)$-idempotent*, if $||e^2-e||<\delta$ and max$\{||e||,||I-e||\}\leq N$; (2) an element $u$ in $A$ is called $(\delta,N)$-$invertible$, if $||u||\leq N$, and there exists $v\in A$ with $||v||\leq N$ such that max$\{||uv-I||,||vu-I||\}<\delta$, where $I$ is the unit of $A$. Such $v$ is called the $(\delta,N)$-inverse of $u$.
Let $X$ be a proper metric space, let $B^p_{L,0}(X)^+$ be the Banach algebra obtained from $B^p_{L,0}(X)$ by adjoining an identity $I$.
Let $0<\delta<1/100, N\geq 1, r>0$, $k$ and $n$ be nonnegative integers. Define $QP_{\delta,N,r,k}(B^p_{L,0}(X)^+ \otimes M_n(\mathbb{C}))$ to be the set of all continuous functions $f$ from $[0,1]^k$ to $B^p_{L,0}(X)^+ \otimes M_n(\mathbb{C})$ such that:
\(1) $f(t)$ is an $(\delta,N)$-idempotent and prop$(f(t))$ $\leq r$ for all $t\in [0,1]^k$;
\(2) $||f(t)-e_m||<\delta$ for all $t\in bd([0,1]^k)$, the boundary of $[0,1]^k$ in $\mathbb{R}^k$, where $e_m=I\oplus \ldots \oplus I\oplus 0\oplus \ldots \oplus 0$ with $m$ identities;
\(3) $\pi(f(t))=e_m$, where $\pi$ is the canonical homomorphism from $B^p_{L,0}(X)^+ \otimes M_n(\mathbb{C})$ to $M_n(\mathbb{C})$.
Let $0<\delta<1/100, N\geq 1, r>0$, $QP_{\delta,N,r,k}(X)$ is defined to be the direct limit of $QP_{\delta,N,r,k}(B^P_{L,0}(X)^+ \otimes M_n(\mathbb{C}))$ under the embedding: $p\rightarrow p\oplus 0$.
Let $0<\delta<1/100, N\geq 1, r>0$, $k$ and $n$ be nonnegative integers. Define $QU_{\delta,N,r,k}(B^p_{L,0}(X)^+ \otimes M_n(\mathbb{C}))$ to be the set of all continuous functions $u$ from $[0,1]^k$ to $B^p_{L,0}(X)^+ \otimes M_n(\mathbb{C})$ such that there exists a continuous function $v:[0,1]^k \rightarrow B^p_{L,0}(X)^+ \otimes M_n(\mathbb{C})$ satisfying that:
\(1) $u(t)$ is an $(\delta,N)$-invertible with a $(\delta,N)$-inverse $v(t)$ such that $\max\{\mathrm{prop}(u(t))$, $\mathrm{prop}(v(t))$} $\leq r$ for all $t\in [0,1]^k$;
\(2) $||u(t)-I||<\delta$ and $||v(t)-I||<\delta$ for all $t\in \mathrm{bd}([0,1]^k)$;
\(3) $\pi(u(t))=\pi(v(t))=I$, where $\pi$ is the canonical homomorphism from $B^p_{L,0}(X)^+ \otimes M_n(\mathbb{C})$ to $M_n(\mathbb{C})$.
Such $v$ is called a *$(\delta,N,r)$-inverse* of $u$.
Let $0<\delta<1/100, N\geq 1, r>0$, $QU_{\delta,N,r,k}(X)$ is defined to be the direct limit of $QU_{\delta,N,r,k}(B^P_{L,0}(X)^+ \otimes M_n(\mathbb{C})$ under the embedding: $u\rightarrow u\oplus I$.
Let $e_1,e_2\in QP_{\delta,N,r,k}(B^p_{L,0}(X)^+ \otimes M_n(\mathbb{C}))$, we call $e_1$ is *$(\delta,N,r)$-equivalent* to $e_2$, if there exists a continuous homotopy $a(t')$ in $QP_{\delta,N,r,k}(B^p_{L,0}(X)^+ \otimes M_n(\mathbb{C}))$ for $t'\in [0,1]$, such that $a(0)=e_1$ and $a(1)=e_2$. Such homotopy be called a *$(\delta,N,r)$-homotopy*.
Notice that (1) any $e\in QP_{\delta,N,r,k}(X)$ is $(\delta',N',r)$-equivalent to some $f$ for which $f(t)=\pi(f)$ for all $t\in \mathrm{bd}([0,1]^k)$; (2) if $e_1$ is $(\delta,N,r)$-equivalent to $e_2$ and $e_1(t)=\pi(e_1)$, $e_2(t)=\pi(e_2)$ for all $t\in \mathrm{bd}([0,1]^k)$, then there exists a homotopy $a(t')$ in $QP_{\delta'',N'',r,k}(X)$ such that $a(0)=e_1,a(1)=e_2$ and $a(t')(t)=\pi(a(t'))$ for all $t\in \mathrm{bd}([0,1]^k)$, where $\delta',\delta''$ depend only on $\delta,N$; $N',N''$ depend only on $N$.
Let $u_1,u_2$ are two elements in $QU_{\delta,N,r,k}(B^p_{L,0}(X)^+ \otimes M_n(\mathbb{C}))$, we call $u_1$ is *$(\delta,N,r)$-equivalent* to $u_2$, if there exists a continuous homotopy $w(t')$ in $QU_{\delta,N,r,k}(B^p_{L,0}(X)^+ \otimes M_n(\mathbb{C}))$ for $t'\in [0,1]$ such that $w(0)=u_1$ and $w(1)=u_2$. This equivalence reduce an equivalent relation in $QU_{\delta,N,r,k}(X)$.
The following lemma tell us that $QP_{\delta,N,r,k}(X)$ can be considered as a controlled version of $K_0(B^p_{L,0}(X) \otimes C_0((0,1)^k))$.
\[Lemma:QuasiVSTrueProjection\] Let $0<\delta<1/100$ and $\chi$ is a function such that $\chi(x)=1$ for $\mathrm{Re}(x)>1/2$; $\chi(x)=0$ for $\mathrm{Re}(x)<1/2$,
\(1) for any $e\in QP_{\delta,N,r,k}(X)$, $\chi(e)$ is an idempotent and define an element $[\chi(e)]\in K_0(B^p_{L,0}(X)\otimes C_0((0,1)^k))$;
\(2) for any two elements $e_1,e_2\in QP_{\delta,N,r,k}(X)$ satisfying that $e_1$ is $(\delta,N,r)$-equivalent to $e_2$, then $[\chi(e_1)]=[\chi(e_2)]$ in $K_0(B^p_{L,0}(X)\otimes C_0((0,1)^k))$;
\(3) for any $0<\delta<1/100$, every element in $K_0(B^p_{L,0}(X)\otimes C_0((0,1)^k))$ can be represented as $[\chi(e_1)]-[\chi(e_2)]$, where $e_1,e_2\in QP_{\delta,N,r,k}(X)$ for some $N\geq 1$ and $r>0$.
\(1) and (2) are straightforward by holomorphic function calculus and the definition of $(\delta,N,r,k)$-equivalence. To prove (3), for any $[p]-[q]\in K_0(B^p_{L,0}(X)\otimes C_0((0,1)^k))$ where $p,q\in (B^p_{L,0}\otimes C_0((0,1)^k))^+ \otimes M_n(\mathbb{C})$, Let $N=||p||+||1-p||+1$, by approximation argument there exists $r>0$ and $e_1\in (B^p_{L,0}\otimes C((0,1)^k) )^+\otimes M_n(\mathbb{C})$ such that $\mathrm{prop}(e_1)<r$ and $||e_1-p||<\frac{\delta}{4N}$, then we have $e_1\in QP_{\delta,N,r,k}(X)$, now we just need to prove $[\chi(e_1)]=[p]$, let $e(t')=t'e_1+(1-t')p$ for $t'\in [0,1]$, then $||e^2(t')-e(t')||<\delta$, thus $\chi(e(t'))$ is a continuous homotopy of projections between $\chi(e(0))=p$ and $\chi(e(1))=\chi(e_1)$.
The following lemma tell us that $QU_{\delta,N,r,k}(X)$ can be considered as a controlled version of $K_1(B^p_{L,0}(X) \otimes C_0((0,1)^k))$.
\[Lemma:QuasiVSTrueInvertible\] Let $0<\delta<1/100$,
\(1) for any $u\in QU_{\delta,N,r,k}(X)$, $u$ is invertible element and define a element $[u]$ in $K_1(B^p_{L,0}(X)\otimes C_0((0,1)^k))$;
\(2) if $u_1$ is $(\delta,N,r)$-equivalent to $u_2$ in $QU_{\delta,N,r,k}(X)$, then $[u_1]=[u_2]$ in $K_1(B^p_{L,0}(X)\otimes C_0((0,1)^k))$;
\(3) for any $0<\delta<1/100$, every element in $K_1(B^p_{L,0}(X)\otimes C_0((0,1)^k))$ can be represented as $[u]$, where $u\in QU_{\delta,N,r,k}(X)$ for some $N\geq 1$ and $r>0$.
\(1) is true since the set of invertible elements in Banach algebra is open, (2) is true by the definition of $(\delta,N,r)$-equivalence. To prove (3), assume $[u']\in K_1(B^p_{L,0}(X)\otimes C_0((0,1)^k))$, let $N=||u'||+||u'^{-1}||+1$, then there exists $r>0$ and $u,v\in B^p_{L,0}(X)\otimes C_0((0,1)^k)\otimes M_n(\mathbb{C})$ such that $||u-u'||<\frac{\delta}{2N}, ||v-u'_{-1}||<\frac{\delta}{2N}$ and $\mathrm{prop}(u)<r,\mathrm{prop}(v)<r$, then we have $||u||\leq N, ||v||\leq N$ and $||uv-I||<\delta,||vu-I||<\delta$, thus $u\in QU_{\delta,N,r,k}(X)$, let $w(t)=tu+(1-t)u'$ for $t\in [0,1]$, then we have $||w(t)u'^{-1}-I||<\delta<1/100$, thus $w(t)u'^{-1}$ is an invertible element, so is $w(t)$, therefore $[u]=[u']$ in $K_1(B^p_{L,0}(X)\otimes C_0((0,1)^k))$.
\[Lemma:HomotopyImpliesLipschitz\](Lemma 2.29 in [@ChungQuantiativeBanach]) If $e$ is $(\delta,N,r)$-equivalent to $f$ by a homotopy $e_{t'}(t'\in [0,1])$ in $QP_{\delta,N,r,k}(X)$, then there exists $\alpha_N>0,m\in \mathbb{N}$ such that $e\oplus I_m\oplus 0_m$ is $(2\delta,3N,r)$-equivalent to $f\oplus I_m\oplus 0_m$ by a $\alpha_N$-Lipschitz homotopy, where $\alpha_N$ depends only on $N$ and not on $e,f,\delta,r$; and $m$ depends only on $\delta,N,e_{t'}$.
There exists a partition $0=t'_0<t'_1<\ldots<t'_m=1$ such that $$||e_{t'_i}-e_{t'_{i-1}}||<\inf_{t'\in [0,1]} \frac{\delta-||e^2_{t'}-e_{t'}||}{2N+1}$$ For each $t'$, we have a Lipschitz $(\delta,3N,r)$-homotopy between $I\oplus 0$ and $e_{t'}\oplus (1-e_{t'})$ given by combining the linear homotopy connecting $I\oplus 0$ to $(e_{t'}-e^2_{t'})\oplus 0$ and the homotopy $$(e_{t'}\oplus 0)+R^*(s)((1-e_{t'})\oplus 0)R(s)$$ where $R(s)=\begin{pmatrix}
\cos(\pi s/2) & \sin(\pi s/2)\\
-\sin(\pi s/2) & \cos(\pi s/2)
\end{pmatrix}$. Obviously, the linear homotopy between $e_{t'_{i-1}}$ and $e_{t'_i}$ is Lipschitz for all $i$. Then
$\begin{pmatrix}
e_{t'_0} & & \\
& I_m & \\
& & 0_m
\end{pmatrix}\\
\simeq \begin{pmatrix}
e_{t'_0} & & & & &\\
& I & & & &\\
& & 0 & & &\\
& & & \ddots & &\\
& & & & I & \\
& & & & & 0
\end{pmatrix}
\simeq \begin{pmatrix}
e_{t'_0} & & & & &\\
& I-e_{t'_1} & & & &\\
& & e_{t'_1} & & &\\
& & & \ddots & &\\
& & & & I-e_{t'_m} & \\
& & & & & e_{t'_m}
\end{pmatrix}\\
\simeq \begin{pmatrix}
e_{t'_0} & & & & &\\
& I-e_{t'_0} & & & &\\
& & e_{t'_1} & & &\\
& & & \ddots & &\\
& & & & I-e_{t'_{m-1}} & \\
& & & & & e_{t'_m}
\end{pmatrix}
\simeq \begin{pmatrix}
I & & & & &\\
& 0 & & & &\\
& & I & & &\\
& & & \ddots & &\\
& & & & 0 & \\
& & & & & e_{t'_m}
\end{pmatrix}\\
\simeq \begin{pmatrix}
e_{t'_m} & & \\
& I_m & \\
& & 0_m
\end{pmatrix}$.\
Where $\simeq$ represent $(2\delta,3N,r)$-equivalence by Lipschitz homotopy.
We remark that we have a result for $QU$ similar to above lemma, i.e., homotopy implies Lipschitz homotopy.
The following lemma tells us that homotopy equivalence of two quasi-invertible elements implies homotopy equivalence of their quasi-inverses.
\[Lemma:EquivImplyInverseEquiv\] Let $u_1,u_2$ be two elements in $QU_{\delta,N,r,k}(X)$ with inverse $v_1,v_2$ respectively, if $u_1$ is $(\delta,N,r)$-equivalent to $u_2$, then $v_1$ is $(4\delta,2N,r)$-equivalent to $v_2$ in $QU_{4\delta,2N,r,k}(X)$.
Let $w(t')$ be the homotopy path jointing $u_1$ and $u_2$, for $\varepsilon=\frac{\delta}{N}$, there exists a partition $0=t'_0<t'_1<\ldots<t'_n=1$ such that $$\max_{0\leq i \leq n-1} \left\{||w(l)-w(l')||:t'_i\leq l,l' \leq t'_{i+1} \right\}<\frac{\delta}{N}.$$ Assume $s_{t'_i}$ is the $(\delta,N,r)$-inverse of $w(t'_i)$, we require $s_0=v_1,s_1=v_2$, let $$s(t')=\frac{t'-t'_i}{t'_{i+1}-t'_i}s_{t'_{i+1}}-\frac{t'-t'_{i+1}}{t'_{i+1}-t'_i}s_{t'_i},t'_i\leq t' \leq t'_{i+1}$$ we have $||s_{t'_i}w(t')-I||\leq ||s_{t'_i}||\cdot ||w(t')-w(t'_i)||+||s_{t'_i}w(t'_i)-I||\leq 2\delta$ for $t'_i\leq t'\leq t'_{i+1}$, then $||s(t')w(t')-I||\leq 4\delta$, similarly, $||w(t')s(t')-I||\leq 4\delta$.
Obviously, $||s(t')||\leq 2N$ and $\mathrm{prop}(s(t'))<r$, thus $s(t')$ is a continuous homotopy between $v_1$ and $v_2$ in $QU_{4\delta,2N,r,k}(X)$.
The following two lemmas can be viewed as the controlled version of the classical result in $K$-theory that stably homotopy equivalence of idempotents is the same as stably similarity.
\[Lemma:HomotopyImpliesSimilar\] Let $0<\delta<1/100$, if $e$ is $(\delta,N,r)$-equivalent to $f$ in $QP_{\delta,N,r,k}(X)$, then there exist a positive number $m$ and an element $u$ in $QU_{\delta,C_1(N),C_2(N,\delta)r,k}(X)$ with $(\delta,C_1(N),C_2(N,\delta)r)$-inverse $v$, such that $$||f\oplus I_m\oplus 0_m - v(e\oplus I_m\oplus 0_m)u||<C_3(N)\delta$$ where $C_1(N)$ and $C_3(N)$ depend only on $N$, $C_2(N,\delta)$ depends only on $N$ and $\delta$.
By Lemma \[Lemma:HomotopyImpliesLipschitz\], there exists $\alpha_N>0,m\in \mathbb{N}$ such that $e\oplus I_m\oplus 0_m$ is $(2\delta,3N,r)$-equivalent to $f\oplus I_m\oplus 0_m$ by a $\alpha_N$-Lipschitz homotopy $e_{t'}$, i.e. $||e_{t'}-e_{t''}||\leq \alpha_N|t'-t''|$ for any $t',t''\in [0,1]$. There exists a partition $0=t'_0<t'_1<\ldots<t'_n=1$ such that $$\alpha_N|t'_{i+1}-t'_i|<\frac{1}{2N+1}$$ Let $w_i=((2e_{t'_i}-I)(2e_{t'_{i+1}}-I)+I)/2$, we have $I-w_i=(2e_{t'_i}-I)(e_{t'_i}-e_{t'_{i+1}})+2(e_{t'_i}-e^2_{t'_i})$, then $$||I-w_i||<||2e_{t'_i}-I||\cdot ||e_{t'_i}-e_{t'_{i+1}}||+2||e_{t'_i}-e^2_{t'_i}||<1/2+4\delta<1$$ thus $w_i$ is an invertible element and $w^{-1}_i=\Sigma^{\infty}_{j=0}(1-w_i)^j$, let $v_i=\Sigma^l_{j=0}(I-w_i)^j$ satisfying $||v_i-w^{-1}_i||<\delta/2((\max_{i}\{||w_i||,||w^{-1}_i||\}+1)^n)$, let $$u=w_0w_1\ldots w_{n-1},v=v_{n-1}v_{n-2}\ldots v_0$$ then $\max\{||u||,||v||\} \leq C_1(N)$, $\max\{\mathrm{prop}(u(t)),\mathrm{prop}(v(t))\} \leq C_2(N,\delta)r$ for $t\in[0,1]^k$ and $\max\{||I-uv||,||I-vu||\}<\delta$, where $C_1(N)$ depends only on $N$ and $C_2(N,\delta)$ depends only on $N,\delta$.
By computation, we have $||e_{t'_i}w_i-w_ie_{t'_{i+1}}||<26N\delta$, then $||ue_1-e_0u||<C'N$, where $C'$ depends only on $N$. Thus, $$||e_1-v(e_0)u||=||e_1-vue_1+v(ue_1-e_0u)||<C_3(N)\delta$$ where $C_3(N)$ depends only on $N$.
\[Lemma:SimilarImplyHomotopy\] Let $N\geq 1$, $0<\delta<1/(800N^4), 0<\varepsilon<1/400$, for $e$ and $f$ in $QP_{\delta,N,r,k}(B^p_{L,0}(X)^+\otimes M_n(\mathbb{C}))$, if there exists $u$ in $QU_{\delta,N,r,k}(X)$ with $(\delta,N,r)$-inverse $v$ satisfying $||uev-f||<\varepsilon$, then $e\oplus 0_n$ is $(2\varepsilon+4N^4\delta,2N^3,3r)$-equivalent to $f\oplus 0_n$ in $QP_{2\varepsilon+4N^4\delta,2N^3,3r,k}(X)$.
Let $e_{t'}$ be a homotopy connecting $f\oplus 0_n$ to $e\oplus 0_n$ obtained by combining the linear homotopy connecting $f\oplus 0_n$ to $uev\oplus 0_n$ with the following homotopy connecting $uev\oplus 0_n$ to $e\oplus 0_n$: $$R(t')(u\oplus I_n)R^*(t')(e\oplus 0_n)R(t')(v\oplus I_n)R^*(t')$$ where $R(t')=\begin{pmatrix}
\cos(\pi t'/2) & \sin(\pi t'/2)\\
-\sin(\pi t'/2) & \cos(\pi t'/2)
\end{pmatrix}$. This is not difficult to verify $e_{t'}$ is a $(2\varepsilon+4N^4\delta,2N^3,3r)$-homotopy between $e$ and $f$.
Let $X$ be a separable proper metric space, define $$GQP_{\delta,N,r,k}(X)=\{e-f:e,f\in QP_{\delta,N,r,k}(X), \pi(e)=\pi(f)\}$$ The equivalent relation in $GQP_{\delta,N,r,k}$ is defined by: $e_1-f_1$ is called $(\delta,N,r)$-equivalent to $e_2-f_2$ if $e_1\oplus f_2\oplus I_n\oplus 0_n$ is $(\delta,N,r)$-equivalent to $f_1\oplus e_2\oplus I_n\oplus 0_n$ for some $n$.
For any $u\in QU_{\delta,N,r,k}(X)$ with $(\delta,N,r)$-inverse $v$, let $Z_t(u)$ be a homotopy connecting $I\oplus I$ to $u\oplus v$ obtained by combining the linear homotopy connecting $I\oplus I$ to $uv\oplus I$ with the homotopy $(u\oplus I)R(t)(v\oplus I)R^*(t)$ connecting $uv\oplus I$ to $u\oplus v$, let $Z'_t(u)$ be a homotopy connecting $I\oplus I$ to $v\oplus u$ obtained by combining the linear homotopy connecting $I\oplus I$ to $uv\oplus I$ with the homotopy $R(t)(u\oplus I)R^*(t)(v\oplus I)$ connecting $uv\oplus I$ to $v\oplus u$, where $$R(t)=\begin{pmatrix}
\cos(\pi t/2) & \sin(\pi t/2)\\
-\sin(\pi t/2) &\cos(\pi t/2)
\end{pmatrix}.$$ Let $$e_t(u)=Z_t(u)(I\oplus 0)Z'_t(u)$$ we have (1) $||e^2_t(u)-e_t(u)||<8N^6\delta$; (2) $||e_t(u)||\leq 4N^4$ and $||I-e_t(u)||\leq 5N^4$; (3) $\mathrm{prop}(e_t(u)(t'))\leq 2r$ for $t'\in [0,1]^k$.
Then we can define a map $\theta$ from $QU_{\delta,N,r,k}(X)$ to $GQP_{8N^6\delta,5N^4,2r,k+1}(X)$ by: $$\theta(u)=e_t(u)-(I\oplus 0)$$
It is not difficult to see that the definition of $\theta$ does not depend on the choose of $(\delta,N,r)$-inverse $v$ of $u$ in the sense of equivalence.
The following result can be considered as a controlled version of a classical result in operator $K$-theory $K_1(A)\cong K_0(SA)$.
\[Lemma:ControlledSuspension\] $\theta:QU_{\delta,N,r,k}(X)\rightarrow GQP_{8N^6\delta,5N^4,2r,k+1}(X)$ is an asymptotic isomorphism in the following sense:
\(1) For any $0<\delta<1/100,r>0,N\geq 1$, there exists $0<\delta_1<\delta, N_1\geq N$ and $0<r_1<r$, such that if two elements $u_1$ and $u_2$ in $QU_{\delta_1,N,r_1,k}(X)$ are $(\delta_1,N,r_1)$-equivalent, then $\theta(u_1)$ and $\theta(u_2)$ are $(\delta,N_1,r)$-equivalent, where $\delta_1$ depends only on $\delta$ and $N$; $N_1$ depends only on $N$ and $r_1$ depends only on $r$.
\(2) For any $0<\delta<1/100,r>0,N\geq 1$, there exists $0<\delta_2<\delta, N_2\geq N$ and $0<r_2<r$, such that if $u'$ and $u''$ in $QU_{\delta_2,N,r_2,k}(X)$ satisfying $\theta(u')$ is $(\delta_2,N,r_2)$-equivalent to $\theta(u'')$, then $u'\oplus I_m$ is $(\delta,N_2,r)$-equivalent to $u''\oplus I_m$ for some $m\in \mathbb{N}$, where $\delta_2$ depends only on $\delta$ and $N$; $N_2$ depends only on $N$ and $r_2$ depends only on $r,\delta,N$.
\(3) For any $0<\delta<1/100,r>0,N\geq 1$, there exists $0<\delta_3<\delta, N_3\geq N$ and $0<r_3<r$, such that for each $e-e_m\in GQP_{\delta_3,N,r_3,k+1}(X)$, there exists $u\in QU_{\delta,N_3,r,k}(X)$ for which $\theta(u)$ is $(\delta,N_3,r)$-equivalent to $e-e_m$, where $\delta_3$ depends only on $\delta$ and $N$; $N_3$ depends only on $N$ and $r_3$ depends only on $r,\delta,N$.
\(1) Let $v_i$ be the $(\delta_1,N,r_1)$-inverse of $u_i$ for $i=1,2$, $w(t)$ be the $(\delta_1,N,r_1)$-homotopy between $u_1$ and $u_2$, by Lemma \[Lemma:EquivImplyInverseEquiv\], there exists a $(4\delta_1,2N,r_1)$-homotopy $s(t)$ connecting $v_1$ and $v_2$ such that $||I-s(t)w(t)||$ and $||I-w(t)s(t)||\}$ are less than $4\delta_1$. Let $a(t)$ be a homotopy connecting $I$ to $v_2u_1$ obtained by combining the linear homotopy connecting $I$ to $v_1u_1$ with the homotopy $s(t)u_1$; let $a'(t)$ be a homotopy connecting $I$ to $v_1u_2$ obtained by combining the linear homotopy connecting $I$ to $v_1u_1$ with the homotopy $v_1w(t)$; let $b(t)$ be a homotopy connecting $I$ to $u_2v_1$ obtained by combining the linear homotopy connecting $I$ to $u_1v_1$ with the homotopy $w(t)v_1$; let $b'(t)$ be a homotopy connecting $I$ to $u_1v_2$ obtained by combining the linear homotopy connecting $I$ to $u_1v_1$ with the homotopy $u_1s(t)$. Define $$x_t=Z_t(u_2)(a(t)\oplus b(t))Z'_t(u_1)$$ $$x'_t=Z_t(u_1)(a'(t)\oplus b'(t))Z'_t(u_2)$$ Then (i) $\max\{||x_t||,||x'_t||\}\leq 8N^6$; (ii) $\max\{||I-x_t x'_t||,||I-x'_t x_t||\} < 64N^{10}\delta_1$; (iii) $\max\{\mathrm{prop}(x_t),\mathrm{prop}(x'_t)\}<6r_1$; (iv) $\max\{||x_i-I||,||x'_i-I||<3\delta_1\}$ for $i=0,1$. Thus $x_t,x'_t\in QU_{64N^{10}\delta_1,8N^6,6r_1,k+1}(X)$. And $$||x_te_t(u_1)x'_t-e_t(u_2)||<(184N^{14})\delta_1$$ By Lemma \[Lemma:SimilarImplyHomotopy\], we can select appropriate $\delta_1,N_1$ and $r_1$ satisfying (1).
\(2) Let $v',v''$ be $(\delta_2,N,r_2)$-inverse of $u',u''$ respectively. By Lemma \[Lemma:HomotopyImpliesSimilar\], there exists an element $u$ in $QU_{\delta_2,C_1(N),C_2(N,\delta_2)r_2,k+1}(X)$ with inverse $v$, such that $$||ue_t(u'\oplus I)v-e_t(u''\oplus I)||<C_3(N)\delta_2$$ i.e. $$||u_tZ_t(u'\oplus I)(I\oplus 0)Z'_t(u'\oplus I)v_t-Z_t(u''\oplus I)(I\oplus 0)Z'_t(u''\oplus I)||<C_3(N)\delta_2 \eqno(A)$$ where $t\in[0,1]$,then we have $$||Z'_t(u''\oplus I)u_tZ_t(u'\oplus I)(I\oplus 0)-(I\oplus 0)Z'_t(u''\oplus I)u_tZ_t(u'\oplus I)||<C_4(N)\delta_2 \eqno(B)$$ Let $$Z'_t(u''\oplus I)u_tZ_t(u'\oplus I)=\begin{pmatrix}
b_t & g_t\\
h_t & d_t
\end{pmatrix},$$ then by $(B)$, we obtain $$||g_t||<C_4(N)\delta_2, ||h_t||<C_4(N)\delta_2. \eqno(*)$$ By $(A)$, we also have $$||(I\oplus 0)Z'_t(u'\oplus I)v_tZ_t(u''\oplus I)-Z'_t(u'\oplus I)v_tZ_t(u''\oplus I)(I\oplus 0)||<C_5(N)\delta_2\eqno(C)$$ Let $$Z'_t(u'\oplus I)v_tZ_t(u''\oplus I)=\begin{pmatrix}
b'_t & g'_t\\
h'_t & d'_t
\end{pmatrix},$$ then by $(C)$, we obtain $$||g'_t||<C_5(N)\delta_2, ||h'_t||<C_5(N)\delta_2. \eqno(**)$$ Thus by $(*)$ and $(**)$, we know that $b_t\in QU_{C_6(N)\delta_2,C_7(N),C_8(N,\delta_2)r_2,k+1}(X)$ with $(C_6(N)\delta_2,C_7(N),C_8(N,\delta_2)r_2)$-inverse $b'_t$. And $$||c_0-I||\leq ||u_0-I||<\delta_2$$ $$||c_1-(v''\oplus I)(u'\oplus I)||<C_9(N)\delta_2.$$ Thus we can select appropriate $\delta_2,N_2$ and $r_2$ satisfying (2).
\(3) $e(t)$ can be considered as a homotopy in $QP_{\delta_3,N,r_3,k}(X)$, where $t\in [0,1]$, we can assume $e(0)=e(1)=e_m=I\oplus 0$, by the proof of Lemma \[Lemma:HomotopyImpliesSimilar\], there exists a homotopy $w(t)$ in $QU_{\delta_3,C_1(N),C_2(N,\delta_3)r_3,k}(X)$ with inverse $s(t)$ for which $w(0)=s(0)=I$ such that $$||w(t)(I\oplus 0\oplus I_m\oplus 0_m)s(t)-e(t)\oplus I_m\oplus 0_m||<C_3(N)\delta_3$$ for some $m\in \mathbb{N}$ and all $t\in [0,1]$, then by some minor modifications of $w(t)$ and $s(t)$, we have $$||w(1)(I\oplus 0)-(I\oplus 0)w(1)||<C_4(N)\delta_3 \eqno(A)$$ let $$w(1)=\begin{pmatrix}
u & g\\
h & u'
\end{pmatrix},
s(1)=\begin{pmatrix}
v & g'\\
h' & v'
\end{pmatrix},$$ then by $(A)$, we obtain $$\max\{||g||,||h||,||g'||,||h'||\}<C_4(N)\delta_3$$ thus $u$ and $u'$ are two elements in $QU_{C_5(N)\delta_3,C_6(N),C_7(N,\delta_3)r_3,k}(X)$ with inverse $v$ and $v'$ respectively.
Let $a_t$ be a homotopy connecting $I\oplus I\oplus I$ to $v'v\oplus I\oplus I$ obtained by combining the linear homotopy connecting $I\oplus I\oplus I$ to $v'u'\oplus I\oplus I$ with the rotation homotopy connecting $(v'\oplus I\oplus I)(u'\oplus I\oplus I)$ to $(v'\oplus I\oplus I)(v\oplus u\oplus u')$ with the homotopy $(v'\oplus I\oplus I)(v\oplus w(1-t))$ connecting $(v'\oplus I\oplus I)(v\oplus u\oplus u')$ to $v'v\oplus I\oplus I$, similarly, let $b_t$ be a homotopy connecting $I\oplus I\oplus I$ to $uu'\oplus I\oplus I$. Define $$y_t=(w(t)\oplus I\oplus I)(I\oplus a(t))(Z'_t(u)\oplus I\oplus I)$$ $$y'_t=(Z_t(u)\oplus I\oplus I)(I\oplus b(t))(s(t)\oplus I\oplus I)$$ then we have $$y_0=y'_0=I, \max\{||y_i-I||,||y'_i-I||\}<C_8(N)\delta_3 \eqno(A)$$ and $$||y_t(e_t(u)\oplus 0)y'_t-(e\oplus 0)||<C_9(N)\delta_3 \eqno(B)$$ Now by Lemma \[Lemma:SimilarImplyHomotopy\], we can choice appropriate $\delta_3,N_3,r_3$ on the basis of $(A)$ and $(B)$ satisfying (3).
Remark: we can also let $$y_t=(w(t)\oplus I\oplus I)(I\oplus Z'_t(u')s(t)\oplus I)(Z'_t(u)\oplus I\oplus I)$$ $$y'_t=(Z_t(u)\oplus I\oplus I)(I\oplus w(t)Z_t(u')\oplus I)(s(t)\oplus I\oplus I)$$
Strongly Lipschitz homotopy invariance
--------------------------------------
Let $f,g:X\rightarrow Y$ be two proper Lipschitz maps, a continuous homotopy $F(t,x)(t\in[0,1])$ between $f$ and $g$ is called *strongly Lipschitz* if: (1) $F(t,x)$ is a proper map from $X$ to $Y$ for each $t$; (2) there exists a constant $C$, such that $d(F(t,x),F(t,y))\leq Cd(x,y)$ for all $x,y\in X$ and $t\in [0,1]$, this $C$ is called Lipschitz constant of $F$; (3) $F$ is equicontinuous in $t$, i.e. for any $\varepsilon>0$, there exists $\delta>0$ such that $d(F(t_1,x),F(t_2,x))<\varepsilon$ for all $x\in X$ if $|t_1-t_2|<\delta$; (4) $F(0,x)=f(x),F(1,x)=g(x)$ for all $x\in X$.
$X$ is said to be strongly Lipschitz homotopy equivalent to $Y$, if there exist proper Lipschitz maps $f:X\rightarrow Y$ and $g:Y\rightarrow X$ such that $fg$ and $gf$ are strongly Lipschitz homotopic to $\mathrm{id}_Y$ and $\mathrm{id}_X$, respectively.
\[Lemma:Homotopy\] $f$ and $g$ be two Lipschitz maps from $X$ to $Y$, let $F(t,x)$ be a strongly Lipschitz homotopy connecting $f$ to $g$ with Lipschitz constant $C$, then there exists $C_0>0$, such that for any $u\in QU_{\delta,N,r,k}(X)$, there exists a homotopy $w(t')$ in $QU_{D(N)\delta,N^{100},C_0r,k}(Y)$ for which $$w(0)=\mathrm{Ad}((V_f,V^+_f))(u)\oplus I$$ $$w(1)=\mathrm{Ad}((V_g,V^+_g))(u)\oplus I$$ where $D(N)$ depends only on $N$ and $C_0$ depends only on $C$.
Choose $\{t_{i,j}\}_{i\geq 0,j\geq 0} \subseteq [0,1]$ satisfying
\(1) $t_{0,j}=0, t_{i,j+1}\leq t_{i,j}, t_{i+1,j}\geq t_{i,j}$;
\(2) there exists $N_j\rightarrow \infty$ such that $t_{i,j}=1$ for all $i\geq N_j$ and $N_{j+1}\geq N_j$;
\(3) $d(F(t_{i+1,j},x),F(t_{i,j},x))<\varepsilon_j=r/(j+1), d(F(t_{i,j+1},x),F(t_{i,j},x))<\varepsilon_j$ for all $x\in X$.
Let $f_{i,j}(x)=F(t_{i,j},x)$, by Lemma \[Lemma:CoveringIsometry\], there exist an isometric operator $V_{f_{i,j}}:E_X^p\rightarrow E_Y^p$ and a contractible operator $V^+_{f_{i,j}}:E_Y^p\rightarrow E_X^p$ with $V^+_{f_{i,j}}V_{f_{i,j}}=I$ such that
$\mathrm{supp}(V_{f_{i,j}}) \subseteq \{(x,y)\in X\times Y:d(f_{i,j}(x),y)<r/(1+i+j)\}$;\
$\mathrm{supp}(V^+_{f_{i,j}}) \subseteq\{(y,x)\in Y\times X:d(f_{i,j}(x),y)<r/(1+i+j)\}$.
For each $i>0$, define a family of operators $V_i(t)(t\in [0,\infty))$ from $E^p_X\oplus E^p_X$ to $E^p_Y\oplus E^p_Y$ and a family of operators $V^+_i(t)(t\in [0,\infty))$ from $E^p_Y\oplus E^p_Y$ to $E^p_X\oplus E^p_X$ by $$V_i(t)=R(t-j)(V_{f_{i,j}}\oplus V_{f_{i,j+1}})R^*(t-j), t\in[j,j+1]$$ $$V^+_i(t)=R(t-j)(V^+_{f_{i,j}}\oplus V^+_{f_{i,j+1}})R^*(t-j), t\in[j,j+1]$$ where $$R(t)=\begin{pmatrix}
\cos(\pi t/2) & \sin(\pi t/2)\\
-\sin(\pi t/2) & \cos(\pi t/2)
\end{pmatrix}.$$ Consider: $$u_0(t)=\mathrm{Ad}((V_f,V^+_f))(u)=V_f(t)(u(t)\oplus I)V^+_f(t)+(I-V_f(t)V^+_f(t));$$ $$u_{\infty}(t)=\mathrm{Ad}((V_g,V^+_g))(u)=V_g(t)(u(t)\oplus I)V^+_g(t)+(I-V_g(t)V^+_g(t));$$ $$u_i(t)=\mathrm{Ad}((V_i,V^+_i))(u)=V_i(t)(u(t)\oplus I)V^+_i(t)+(I-V_i(t)V^+_i(t)),$$ let $v$ be the $(\delta,N,r)$-inverse of $u$, similarly, we can define $$u'_i(t)=\mathrm{Ad}((V_i,V^+_i))(v).$$ For each $i$, define $n_i$ by $$n_i=\left\{
\begin{array}{lr}
\max\{j:i\geq N_j\}, & \{j:i\geq N_j\}\not= \emptyset;\\
0, & \{j:i\geq N_j\}= \emptyset.
\end{array}
\right.$$ We can choose $V_{f_{i,j}}$ in such a way that: $u_i(t)=u_{\infty}$ where $t\leq n_i$.
Define $$w_i(t)=\left\{
\begin{array}{ll}
u_i(t)(u'_{\infty}(t)), & t\geq n_i;\\
(n_i-t)I+(t-n_i+1)u_i(t)u'_{\infty}(t), & n_i-1\leq t \leq n_i;\\
I, & 0\leq t \leq n_i-1.
\end{array}
\right.$$ Consider: $$\begin{aligned}
a&=\bigoplus^{\infty}_{i=0}(w_i\oplus I);\\
b&=\bigoplus^{\infty}_{i=0}(w_{i+1}\oplus I);\\
c&=(I\oplus I)\bigoplus^{\infty}_{i=1}(w_i\oplus I).
\end{aligned}$$ By the construction of $\{t_{i,j}\}$, we know that $a,b,c\in QU_{D_1(N)\delta,N^2,C_1r,k}(Y)$ for some constant $C_1$ depends only on $C$. Let $$V_{i,i+1}(t')=R(t')(V_i\oplus V_{i+1})R^*(t'),t'\in [0,1];$$ $$V^+_{i,i+1}(t')=R(t')(V^+_i\oplus V^+_{i+1})R^*(t'), t'\in [0,1].$$ Define $$u_{i,i+1}(t')=V_{i,i+1}(t')((u\oplus I)\oplus I)V^+_{i,i+1}(t')+(I-V_{i,i+1}(t')V^+_{i,i+1}(t')),$$ then $$\begin{aligned}
u_{i,i+1}(0)&=(V_i(u\oplus I)V^+_i+(I-V_iV^+_i))\oplus I;\\
u_{i,i+1}(1)&=(V_{i+1}(u\oplus I)V^+_{i+1}+(I-V_{i+1}V^+_{i+1}))\oplus I\end{aligned}$$ Using $u_{i,i+1}(t')$, we can construct a homotopy $s_1(t')$ in $QU_{D_2(N)\delta,N^{100},C_2r,k}(Y)$ for some $C_2\geq C_1$ depends only on $C$, such that $$s_1(0)=a, s_1(1)=b.$$
We can also construct a homotopy $s_2(t')$ in $QU_{D_3(N)\delta,N^{100},C_3r,k}(Y)$ for some $C_3\geq C_1$ depends only on $C$, such that $$s_2(0)=b\oplus I, s_2(1)=c\oplus I.$$
Finally, we define $w(t')$ to be the homotopy obtained by combining the following homotopies:
\(1) the linear homotopy between $(u_0\oplus I)\bigoplus^{\infty}_{i=1}(I\oplus I)$ and $c'a((u_{\infty}\oplus I)\bigoplus^{\infty}_{i=1}(I\oplus I))$;
\(2) $s'_2(1-t')a((u_{\infty}\oplus I)\bigoplus^{\infty}_{i=1}(I\oplus I)); $
\(3) $s'_1(1-t')a((u_{\infty}\oplus I)\bigoplus^{\infty}_{i=1}(I\oplus I)); $
\(4) the linear homotopy between $a'a((u_{\infty}\oplus I)\bigoplus^{\infty}_{i=1}(I\oplus I))$ and $(u_{\infty}\oplus I)\bigoplus^{\infty}_{i=1}(I\oplus I)$,\
where $a',b',c',s'_1,s'_2$ be the $(D(N)\delta,N^{100},C_0r)$-inverse of $a,b,c,s_1,s_2$ respectively in $QU_{D_4(N)\delta,N^{100},C_4r,k}(Y)$ for some $C_4\geq \max\{C_1,C_2,C_3\}$ depends only on $C$.
Therefore, $w(t')$ is the homotopy connecting $\mathrm{Ad}((V_f,V^+_f))(u)\bigoplus^{\infty}_{i=1}I$ to\
$\mathrm{Ad}((V_g,V^+_g))(u)\bigoplus^{\infty}_{i=1}I$.
By Lemma \[Lemma:ControlledSuspension\], we have the following result:
Let $X,Y,f$ and $g$ be as in Lemma \[Lemma:Homotopy\]. For any $0<\delta<1/100, N\geq 1,r\geq 0$, there exist $0<\delta_1<\delta, N_1\geq N, 0\leq r_1<r$ such that for any $e\in QP_{\delta_1,N,r_1,k}(X)(k>1)$, there exists a homotopy $e(t')(t'\in[0,1])$ in $QP_{\delta,N_1,r,k}(Y)$ satisfying $$e(0)=\mathrm{Ad}((V_f,V^+_f))(e\oplus 0)\oplus (I\oplus 0)$$ $$e(1)=\mathrm{Ad}((V_g,V^+_g))(e\oplus 0)\oplus (I\oplus 0)$$ where $\delta_1$ depends only on $\delta$ and $N$; $N_1$ depends only on $N$ and $r_1$ depends only on $r,\delta,N,C$.
controlled cutting and pasting {#section:cutpaste}
------------------------------
\[Def:ControlledBoundary\] Let $X$ be a separable proper metric space, $X_1$ and $X_2$ be two subspaces. The triple $(X;X_1,X_2)$ is said to satisfy the *strong excision* condition if :
\(1) $X=X_1\cup X_2$, $X_i$ be Borel subset and $\mathrm{int}(X_i)$ is dense in $X_i$ for $i=1,2$;
\(2) there exists $r_0>0,C_0>0$ such that (i) for any $r'\leq r_0$, $\mathrm{bd}_{r'}(X_1)\cap \mathrm{bd}_{r'}(X_2)=\mathrm{bd}_{r'}(X_1\cap X_2)$; (ii) for each $X'=X_1,X_2,X$ and any $r'\leq r_0$, $\mathrm{bd}_{r'}(X')$ is strongly Lipschitz homotopy equivalent to $X'$ with $C_0$ as the Lipschitz constant.
Let the triple $(X;X_1,X_2)$ be as above. Let $0<\delta<1/100$, for any $u\in QU_{\delta,N,r,k}(X)$ with $(\delta,N,r)$-inverse $v$, we take $uX_1=\chi_{X_1}u\chi_{X_1}$, the same as $vX_1$, define $$w_u=\begin{pmatrix}
I & uX_1\\
0 & I
\end{pmatrix}
\begin{pmatrix}
I & 0\\
-vX_1 & I
\end{pmatrix}
\begin{pmatrix}
I & uX_1\\
0 & I
\end{pmatrix}
\begin{pmatrix}
0 & -I\\
I & 0
\end{pmatrix}$$ then $$w^{-1}_u=\begin{pmatrix}
0 & I\\
-I & 0
\end{pmatrix}
\begin{pmatrix}
I & -uX_1\\
0 & I
\end{pmatrix}
\begin{pmatrix}
I & 0\\
vX_1 & I
\end{pmatrix}
\begin{pmatrix}
I & -uX_1\\
0 & I
\end{pmatrix}.$$
We define a homomorphism $\partial_0: QU_{\delta,N,r,k}(X)\rightarrow QP_{4N^4\delta,2N^6,6r,k}(\mathrm{bd}_{5r}(X_1) \cap \mathrm{bd}_{5r}(X_2))$ by $$\partial_0(u)=\chi_{\mathrm{bd}_{5r}(X_1) \cap \mathrm{bd}_{5r}(X_2)} w_u(I\oplus 0)w^{-1}_u \chi_{\mathrm{bd}_{5r}(X_1) \cap \mathrm{bd}_{5r}(X_2)}$$
Now we verify $\partial_0(u)\in QP_{4N^4\delta,2N^6,6r,k}(\mathrm{bd}_{5r}(X_1) \cap \mathrm{bd}_{5r}(X_2))$, firstly, $||\partial_0(u)||$ and $||1-\partial_0(u)||$ are less than $2N^6$, secondly, $\mathrm{prop}(\partial_0(u))<6r$, finally, we estimate $||(\partial_0(u))^2-\partial_0(u)||$, for convenience, we take $Y=\mathrm{bd}_{5r}(X_1) \cap \mathrm{bd}_{5r}(X_2)$: $$||(\partial_0(u))^2-\partial_0(u)||=||\chi_Y w_u(I\oplus 0)w^{-1}_u\chi_{X_1-Y} w_u(I\oplus 0)w^{-1}_u\chi_Y||$$ we now estimate $||\chi_Y w_u(I\oplus 0)w^{-1}_u\chi_{X_1-Y}||$, we have $\chi_{X_1}u\chi_{X_1-Y}=u\chi_{X_1-Y}$, thus we can replace $uX_1$ by $u$ in $w_u(I\oplus 0)w^{-1}_u$, then $$w_u(I\oplus 0)w^{-1}_u=\begin{pmatrix}
(I-uv)uv+uv & (I-uv)u(I-vu)+u(I-vu)\\
(I-vu)v & (I-vu)^2,
\end{pmatrix}$$ thus $$||\chi_Y w_u(I\oplus 0)w^{-1}_u\chi_{X_1-Y}||=||\chi_{Y\cap X_1}((w_u(I\oplus 0)w^{-1}_u)-(I\oplus 0))\chi_{X_1-Y}||<2N^2\delta$$ similarly, $$||\chi_{X_1-Y} w_u(I\oplus 0)w^{-1}_u \chi_Y||<2N^2\delta$$
Assume that $r<r_0/5$, where $r_0$ is as in Definition \[Def:ControlledBoundary\]. Let $f$ be the proper Lipschitz map from $\mathrm{bd}_{5r}(X_1) \cap \mathrm{bd}_{5r}(X_2)$ to $X_1\cap X_2$ realizing the strong Lipschitz homotopy equivalence in Definition \[Def:ControlledBoundary\], by Lemma \[Lemma:IsometryPairHom\], we have the pair $(V_f,V^+_f)$ corresponding to $\{\varepsilon_m\}$ for which $\sup_m(\varepsilon_m)<r/10$.
We define the boundary map $\partial:QU_{\delta,N,r,k}(X)\rightarrow GQP_{4N^4\delta,2N^6,6C_0r,k}(X_1 \cap X_2)$ by $$\partial(u)=\mathrm{Ad}((V_f,V^+_f))(\partial_0(u))-(I\oplus 0)$$
Then we consider the following sequence: $$QU_{\delta,N,r,k}(X_1) \oplus QU_{\delta,N,r,k}(X_2) \xrightarrow{j} QU_{\delta,N,r,k}(X) \xrightarrow{\partial} GQP_{4N^4\delta,2N^6,6C_0r,k}(X_1 \cap X_2)$$ where $j(u_1\oplus u_2)=(u_1+\chi_{X-X_1}) \oplus (u_2+\chi_{X-X_2}), r<r_0/5 $.
\[Lemma:ControlledExact\] Let $(X;X_1,X_2)$ be as in Definition \[Def:ControlledBoundary\] with $r_0,C_0$, then the above sequence is asymptotically exact in the following sense:
\(1) For any $0<\delta<1/100, N\geq 1, r>0$, there exist $0<\delta_1<\delta, N_1\geq N, 0<r_1<\min\{r,r_0/5 \}$, such that $\partial j(u_1\oplus u_2)$ is $(\delta,N_1,r)$-equivalent to $0$ for any $u_i\in QU_{\delta_1,N,r_1,k}(X_i)(i=1,2)$, where $\delta_1$ depends only on $\delta,N$; $N_1$ depends only on $N$ and $r_1$ depends only on $\delta,N,r$.
\(2) For any $0<\delta<1/100,N\geq 1,r>0$, there exists $0<\delta_2<\delta, N_2\geq N, 0<r_2<\min\{r,r_0/5 \}$, such that if $u$ is an element in $QU_{\delta_2,N,r_2,k}(X)$ for which $\partial(u)$ is $(\delta_2,N,r_2)$-equivalent to $0$ in $GQP_{\delta_2,N,r_2,k}(X)$, then there exist $u_i\in QU_{\delta,N_2,r,k}(X_i)(i=1,2)$ such that $j(u_1\oplus u_2)$ is $(\delta,N_2,r)$-equivalent to $u$, where $\delta_2$ depends only on $\delta,N$; $N_2$ depends only on $N$ and $r_2$ depends only on $\delta,N,r,r_0,C_0$.
\(1) follows from the definition of the boundary map and Lemma \[Lemma:SimilarImplyHomotopy\].
\(2) By strong homotopy invariance of $QP$, for any $0<\delta'<\delta, N\geq 1, 0<r'_2<\min\{r,r_0/5 \}$, there exist $\delta_2<\delta', N'>N, 0<r_2<r'_2$ ($\delta_2$ depends only on $\delta',N$; $N'$ depends only on $N$; $r_2$ depends only on $r'_2,\delta',N,r_0,C_0$), such that , for any $u\in QU_{\delta_2,N,r_2,k}(X)$ whose boundary $\partial(u)$ is $(\delta_2,N,r_2)$-equivalent to $0$, then $\partial_0(u)$ is $(\delta',N',r'_2)$-equivalent to $0$. By Lemma \[Lemma:HomotopyImpliesSimilar\] there exists an element $y$ in $QU_{\delta',C_1(N'),C_2(N',\delta')r'_2,k}(\mathrm{bd}_{5r_2}(X_1) \cap \mathrm{bd}_{5r_2}(X_2))$ with $(\delta',C_1(N'),C_2(N',\delta')r'_2)$-inverse $y'$, such that $$||xw(I\oplus 0)w^{-1}x'-(I\oplus 0)||<C_3(N')\delta',$$ where $x=y+\chi_{X-\mathrm{bd}_{5r_2}(X_1) \cap \mathrm{bd}_{5r_2}(X_2)}, x'=y'+\chi_{X-\mathrm{bd}_{5r_2}(X_1) \cap \mathrm{bd}_{5r_2}(X_2)}, w=w_{u\oplus I}$.
This implies that $$||xw(I\oplus 0)-(I\oplus 0)xw||<C_4(N')\delta'.$$ Thus we have $$xw=\begin{pmatrix}
a & b\\
c & d
\end{pmatrix},
||b||\leq C_4(N')\delta',
||c||\leq C_4(N')\delta', \eqno(A)$$ $$w^{-1}x'=\begin{pmatrix}
a' & b'\\
c' & d'
\end{pmatrix},
||b'||\leq C_4(N')\delta',
||c'||\leq C_4(N')\delta', \eqno(B)$$ Define $$v_1=a\chi_{\mathrm{bd}_{5r_2}}(X_1), v'_1=\chi_{\mathrm{bd}_{5r_2}}(X_1)a',$$ $(A)$ and $(B)$ tell us that $v_1\in QU_{(C_4(N')+1)\delta',C_1(N')N^3_2,(C_2(N',\delta')+3)r'_2,k}(bd_{5r_2}(X_1))$ with inverse $v'_1$. $(A)$ and $(B)$ together with the definition of $w$, implies $$||\chi_{X-\mathrm{bd}_{10r_2}(X_2)}(v'_1(u\oplus I-I))||<C_5(N')\delta', \eqno(C)$$ $$||(v'_1(u\oplus I-I))\chi_{X-\mathrm{bd}_{10r_2}(X_2)}||<C_5(N')\delta'. \eqno(D)$$ Define $$v_2=\chi_{\mathrm{bd}_{10r_2}(X_2)}(v'_1(u\oplus I))\chi_{\mathrm{bd}_{10r_2}(X_2)}, v'_2=\chi_{\mathrm{bd}_{10r_2}(X_2)}((u'\oplus I)v_1)\chi_{\mathrm{bd}_{10r_2}(X_2)},$$ where,$u'$ be the $(\delta_2,N,r_2)$-inverse of $u$. $(C)$ and $(D)$ tell us that $v_2\ \text{belongs to} \\ QU_{C_6(N')\delta', C_7(N'),C_8(N',\delta')r'_2,k}(\mathrm{bd}_{10r_2}(X_2))$ with inverse $v'_2$.
We require $0<r_2<r_0/10$, let $f_1$ be the proper strong Lipschitz map from $\mathrm{bd}_{5r}(X_1)$ to $X_1$ realizing the strong Lipschitz homotopy equivalence; let $f_2$ be the proper strong Lipschitz map from $\mathrm{bd}_{10r}(X_2)$ to $X_2$ realizing the strong Lipschitz homotopy equivalence. Define $u_i=\mathrm{Ad}((V_{f_i},V^+_{f_i}))(v_i)$ for $i=1,2$, where the pair $(V_{f_i},V^+_{f_i})$ corresponding to $\{\varepsilon_k\}$ for which $\sup_k(\varepsilon_k)<r'_2$.
By $(C)$ and $(D)$, we have that $(v_1+\chi_{X-\mathrm{bd}(5r_2)(X_1)})\oplus (v_2+\chi_{X-\mathrm{bd}(10r_2)(X_2)})$ is $(C_9(N')\delta',C_{10}(N'),C_{11}(N',\delta')r'_2)$-equivalent to $u\oplus I$.
Where $C_j(N')$ depends only on $N'$ for $j=1,3,4,5,6,7,9,10$, $C_j(N_2,\delta')$ depends only on $N',\delta',C_0$ for $j=2,8,11$.
By Lemma \[Lemma:Homotopy\], we can choose appropriate $\delta',N_2$ and $r'_2$ such that $u_1$ and $u_2$ satisfy the desired properties of (2), where $\delta'$ depends only on $\delta,N$; $N_2$ depends only on $N$; $r'_2$ depends only on $r,\delta,N,r_0,C_0$.
By Lemma $\ref{Lemma:ControlledSuspension}$ and Lemma $\ref{Lemma:ControlledExact}$, we have the following asymptotically exact sequence for $QU$ when $k>1$: $$QU_{\delta,N,r,k}(X_1) \oplus QU_{\delta,N,r,k}(X_2) \rightarrow QU_{\delta,N,r,k}(X) \rightarrow QU_{\delta,N,r,k-1}(X_1 \cap X_2)$$
Spaces with finite asymptotic dimension
=======================================
In this section, we will recall some fact about space with finite asymptotic dimension, and verify the $L^p$ coarse Baum-Connes conjecture for spaces with finite asymptotic dimension.
The *asymptotic dimension* of a metric space $X$ is the smallest integer $m$ such that for any $r>0$, there exists an uniformly bounded cover $C=\{U_i\}_{i\in I}$ of $X$ for which the $r$-multiplicity of $C$ is at most $m+1$; i.e. no ball of radius $r$ in the metric space intersects more than $m+1$ member of $C$. If no such $m$ exists, we call $X$ has infinite asymptotic dimension.
A finitely generated group can be viewed as a metric space with a left-invariant *word-length metric*. To be more precise, for a group $\Gamma$ with a finite symmetric generating set $S$, for any $\gamma\in\Gamma$, we define its length $$l_S(\gamma):=\min\{n:\gamma=s_1 \ldots s_n, s_i\in S\}$$ the word-length metric $d_S$ on $\Gamma$ is defined by $$d_S(\gamma_1,\gamma_2):=l_S(\gamma_1^{-1}\gamma_2)$$ for all $\gamma_1,\gamma_2\in \Gamma$. We remark that for any two finite symmetric generating sets $S_1,S_2$ of $\Gamma$, $(\Gamma,d_{S_1})$ is quasi-isometric to $(\Gamma,d_{S_2})$.
Now we give some fact about asymptotic dimension:
1. The concept of asymptotic dimension is a coarse geometric analogue of the covering dimension in topology;
2. Hyperbolic groups have finite asymptotic dimension as a metric space with word-length metric [@GromovHyperbolic][@RoeHyperbolic];
3. The class of finitely generated groups with finite asymptotic dimension is hereditary (Proposition 6.2 in [@YuFAD]) , i.e., if a finitely generated group $\Gamma$ has finite asymptotic dimension as a metric space with word-length metric, then any finitely generated subgroups of $\Gamma$ also has finite asymptotic dimension as a metric space with word-length metric.
4. If $\Gamma$ is a discrete subgroup of an almost connected Lie group, e.g. $\mathrm{SL}(n,{\mathbb{Z}})$, then $\Gamma$ has finite asymptotic dimension.
5. CAT(0) cube complexes have finite asymptotic dimension. [@WrightCAT0FAD]
6. Certain relative hyperbolic groups have finite asymptotic dimension. [@OsinRelativeHyperbolic]
7. Certain Coxter groups have finite asymptotic dimension. [@DranishnikovFAD]
8. Mapping class groups have finite asymptotic dimension. [@MappingClassGroup]
\[construct:anticech\] Let $X$ be a separable proper metric space with asymptotic dimension $m$. By the definition of asymptotic dimension there exists a sequence of covers $C_k$ of $X$ for which there exists a sequence of positive numbers $R_k\rightarrow \infty$ such that
1. $R_{k+1}>4R_k$ for all $k$;
2. diameter$(U)<R_k/4$ for all $U\in C_k$;
3. the $R_k$-multiplicity of $C_{k+1}$ is at most $m+1$, i.e. no ball with radius $R_k$ intersects more than $m+1$ member of $C_k+1$.
Let $C_k'=\{B(U,R_k):U\in C_{k+1}\}$, where $B(U,R_k)=\{x\in X:d(x,U)<R_k\}$. (1), (2) and (3) imply that $\{C_k'\}$ is an anti-Čech system for $X$.
Fixed a positive integer $n_0$. For each $n>n_0$, let $r_n=\frac{R_n}{2R_{n_0+1}}-4$. By property (1) of the sequence ${R_k}$, there exists $n_1>n_0$ such that $r_n>2$ if $n>n_1$ and there exists a sequence of nonnegative smooth function $\{\chi_n\}_{n>n_1}$ on $[0,\infty)$ for which
1. $\chi_n(t)=1$ for all $0\leq t\leq 2$, and $\chi_n(t)=0$ for all $t\geq r_n$;
2. there exists a sequence of positive numbers $\varepsilon_n\rightarrow 0$ satisfying $|\chi'_n(t)|<\varepsilon_n\leq 1$ for all $n>n_1$.
For each $U\in C_{n+1}(n>n_1)$, define $$U'=\{V\in N_{C'_{n_0}}:V\in C'_{n_0},U \cap V \not= \emptyset\}$$
We define a map $G_n:N_{C'_{n_0}}\rightarrow N_{C'_n}$ by $$G_n(x)=\sum_{U\in C_{n+1}} \frac{\chi_n(d(x,U'))}{\sum_{V\in C_{n+1}} \chi_n(d(x,V'))} B(U,R_n)$$ for all $x\in N_{C'_{n_0}}$.
Let $n>n_1$, we define a map $i_{n_0n}:N_{C'_{n_0}}\rightarrow N_{C'_n}$ in such a way that, for each $V\in C_{n_0+1}$, $$i_{n_0n}(B(V,R_{n_0}))=B(U,R_n)$$ for some $U\in C_{n+1}$ satisfying $U\cap V\not= \emptyset$.
Let $F_t$ be the linear homotopy between $G_n$ and $i_{n_0n}$, i.e. $F_t(x)=tG_n(x)+(1-t)i_{n_0n}(x)$ for all $t\in [0,1]$ and $x\in N_{C'_{n_0}}$.
By the above construction, we have the following important lemma:
\[Lemma:PropagationArbitrarySmall\](Lemma 6.3 in [@YuFAD]) Let $X$ be a separable proper metric space with finite asymptotic dimension $m$, and $G_n,F_t$ and $i_{n_0n}$ be as above, then
1. $G_n$ is a proper Lipschitz map with a Lipschitz constant depending only on $m$;
2. $F_t$ is a strong Lipschitz homotopy between $G_n$ and $i_{n_0n}$ with a Lipschitz constant depending only on $m$;
3. For any $\varepsilon>0,R>0$, there exists $K>0$ such that $d(G_n(x),G_n(y))<\varepsilon$ if $n>K,d(x,y)<R$.
The following lemma plays a crucial role in the proof of Theorem \[Thm:LpBCCFAD\]. Its proof bases on the Eilenberg swindle argument and the controlled cutting and pasting exact sequence in Section \[section:cutpaste\].
\[Lemma:VanishingControlledObstruction\] Let $X$ be a simplicial complex with finite dimension $m$ and endowed with $\ell^1$ metric. For any $k>m+1, 0<\delta<1/100, N\geq 1, r>0$, there exist $0<\delta_1\leq \delta, N_1\geq N, 0<r_1<r$, such that every element $u$ in $QU_{\delta_1,N,r_1,k}(X)$ is $(\delta,N_1,r)$-equivalent to $I$, where $\delta_1$ depends only on $\delta,N$; $N_1$ depends only on $N$ and $r_1$ depends only on $r,\delta,N$.
Let $X^{(n)}$ be the $n$-skeleton of $X$, we will prove our lemma for $X^{(n)}$ by induction on $n$.
When $n=0$, we choose $r_1=\min\{r,2\}$. Let $v$ be the $(\delta_1,N,r_1)$-inverse of $u$. Then $\mathrm{prop}(u(t))=\mathrm{prop}(v(t))=0$. For $t_0\in [0,\infty)$, we define: $$u_{t_0}(t)=\left \{
\begin{array}{ll}
I, & 0\leq t\leq t_0;\\
u(t-t_0), & t_0\leq t<+\infty.
\end{array} \right.$$ Similarly, we can define $v_{t_0}$ for $t_0\in [0,\infty)$. Thus $v_{t_0}$ is the $(\delta_1,N,r_1)$-inverse of $u_{t_0}$.\
Define $$E^{p,\infty}_X=(\oplus^{\infty}_{k=0} E^p_X)\oplus E^p_X$$ Notice that $E^{p,\infty}_X$ is a standard non-degenerated $L^p$-$X$-module.
Let $w_1(t')$ be the linear homotopy between $u\oplus^{\infty}_{k=1}I \oplus I$ and $u\oplus^{\infty}_{k=1}u_kv_k\oplus I$. Let $w_2(t')=(\oplus^{\infty}_{k=0}u_k\oplus I)(I\oplus^{\infty}_{k=1}v_{k-t'}\oplus I)$, where $t'\in [0,1]$.
Let $T,T^*:E^{p,\infty}_X\rightarrow E^{p,\infty}_X$ be a homomorphism defined by $$\begin{aligned}
T((h_0,h_1,\ldots),h))&=(0,h_0,h_1,\ldots),h)\\
T^*((h_0,h_1,\ldots),h))&=(h_1,h_2,\ldots),h)
\end{aligned}$$ Thus $$I\oplus^{\infty}_{k=1}v_{k-1}\oplus I=T(\oplus^{\infty}_{k=0}v_{k}\oplus I)T^*.$$ Hence there exists a homotopy $s_1(t')(t'\in[0,1])$ connecting $I\oplus^{\infty}_{k=1}v_{k-1}\oplus I$ and $\oplus^{\infty}_{k=0}v_{k}\oplus I$.
Let $s_2(t')(t'\in[0,1])$ be the linear homotopy between $\oplus^{\infty}_{k=0}u_kv_k\oplus I$ and $\oplus^{\infty}_{k=0}I\oplus I$.
Define $$w(t')=\left\{
\begin{array}{ll}
w_1(4t'), & 0\leq t'\leq 1/4;\\
w_2(4t'-1), & 1/4\leq t'\leq 1/2;\\
(\oplus^{\infty}_{k=0}u_k\oplus I)s_1(4t'-2), & 1/2\leq t'\leq 3/4;\\
s_2(4t'-3), & 3/4\leq t'\leq 1.\\
\end{array}\right.$$
It is not difficult to see $w(t')$ is the homotopy connecting $u\oplus I$ to $I$, thus we can choose appropriate $\delta_1$ and $N_1$ satisfying lemma.
Assume by induction that the lemma holds for $n=m-1$, next we will prove the lemma holds for $n=m$. For each simplex $\triangle$ of dimension $m$ in $X$, we let $$\triangle_1=\{x\in \triangle: d(x,c(\triangle))\leq 1/100\},\triangle_2=\{x\in \triangle: d(x,c(\triangle))\geq 1/100\},$$ where $c(\triangle)$ is the center of $\triangle$.
Let $$X_1=\bigcup_{\triangle:\text{simplex of dimension $m$ in X}} \triangle_1;$$ $$X_2=\bigcup_{\triangle:\text{simplex of dimension $m$ in X}} \triangle_2.$$ Notice that:\
(1) $X_1$ is strongly Lipschitz homotopy equivalent to $$\{c(\triangle): \text{$\triangle$ is $m$-dimensional simplex in X}\};$$ (2) $X_2$ is strongly Lipschitz homotopy equivalent to $X^{(m-1)}$;\
(3) $X^{(m)}=X_1 \cup X_2$ and $X_1 \cap X_2$ is the disjoint union of the boundaries of all $m$-dimensional $\triangle_1$ in $X^{(m)}$.
\(1) and (2) together with strongly Lipschitz homotopy invariance of $QU$ and the induction hypothesis, imply that our lemma holds for $X_1$ and $X_2$.
By strongly Lipschitz homotopy invariance of $QU$ and the controlled cutting and pasting exact sequence, we also know that our lemma holds for $X_1\cap X_2$.
Obviously, $(X^{(m)},X_1,X_2)$ satisfies the strong excision condition, thus we can complete our induction process by using the controlled cutting and pasting exact sequence and the controlled five lemma.
Now we are ready to prove the main theorem of this section.
\[Thm:LpBCCFAD\] For any $p\in [1,\infty)$, the $L^p$ coarse Baum-Connes conjecture holds for separable proper metric spaces with finite asymptotic dimension.
Let $X$ be a separable proper metric space with asymptotic dimension $m$. By Theorem \[Thm:VanishingObstruction\], it is enough to prove that $$\lim_{n\rightarrow \infty}K_i(B^p_{L,0}(N_{C'_n}))=0,$$ where $C'_n$ is as in Construction \[construct:anticech\].
Lemma \[Lemma:QuasiVSTrueProjection\], \[Lemma:QuasiVSTrueInvertible\] and \[Lemma:ControlledSuspension\] tell us that any element $[q]$ in $K_i(B^p_{L,0}(N_{C'_{n_0}}))$ can be represented as an element $u$ in $QU_{\delta_1,N,r,k}(N_{C'_{n_0}})$ for some $N,r$ and $k>m+1$, where $\delta_1$ is as in Lemma \[Lemma:VanishingControlledObstruction\] for some $0<\delta<1/100$. Let $$u_n=\mathrm{Ad}((V_{G_n},V^+_{G_n}))(u),$$ where $G_n$ is as in Lemma \[Lemma:PropagationArbitrarySmall\], $\mathrm{Ad}((V_{G_n},V^+_{G_n}))$ is defined by $\{\varepsilon_m\}$ for which $\sup(\varepsilon_m)<r_1/10$, where $r_1$ is as in Lemma \[Lemma:VanishingControlledObstruction\].
By Lemma \[Lemma:PropagationArbitrarySmall\] (3), there exists $K>0$ such that
$\mathrm{prop}(u_n)<r_1,$ for $n>K.$
Since the asymptotic dimension of $X$ is $m$, we have $\dim(N_{C'_n})\leq m$ for all $n$. By Lemma $\ref{Lemma:VanishingControlledObstruction}$, we have that $u_n$ is $(\delta,N_1,r)$-equivalent to $I$ in $QU_{\delta,N_1,r,k}(N_{C'_n})$ for $n>K$.
By Lemma $\ref{Lemma:PropagationArbitrarySmall}$ (2), strongly Lipschitz homotopy invariance of $QU$, Lemma \[Lemma:QuasiVSTrueProjection\] (2) and Lemma \[Lemma:QuasiVSTrueInvertible\] (2), we have that $\mathrm{Ad}((V_{i_{n_0n}},V^+_{i_{n_0n}}))(u)$ and $u_n$ correspond to the same element in $K_i(B^p_{L,0}(N_{C'_n}))$.
Thus $[q]=0$ in $\lim_{n\rightarrow \infty}K_i(B^p_{L,0}(N_{C'_n}))$.
$K$-theory of $L^p$ Roe algebras
================================
In this section, we shall use the dual $L^p$ $K$-homology as a bridge to prove that the $L^p$ $K$-homology is independent of $p$. Combining the Theorem \[Thm:LpBCCFAD\], we obtain that the $K$-theory of the $L^p$ Roe algebra does not depend on $p$ for space with finite asymptotic dimension.
Dual $L^p$ Localization algebra and dual $L^p$ $K$-homology
-----------------------------------------------------------
Let $p\in (1,\infty)$, $Z$ and $Z'$ be countable discrete measure spaces, then $\ell^p(Z)$ has a natural Schauder basis $\{e_i\}_{i\in Z}$, where $e_i(z)=1$ for $i=z$ and $e_i(z)=0$ for $i\not=z$. Similarly, $\ell^p(Z')$ has a natural Schauder basis $\{e'_i\}_{i\in Z'}$. Let $T$ be a bounded operator from $\ell^p(Z)$ to $\ell^p(Z')$, $T$ can be considered as a countably dimensional matrix under the Schauder basis $\{e_i\}$ and $\{e'_i\}$. We can define $T^*$ as the transpose of the matrix of $T$. We call $T$ be a *dual-operator*, if $T^*$ is a bounded operator from $\ell^p(Z')$ to $\ell^p(Z)$ under the Schauder basis $\{e'_i\}$ and $\{e_i\}$. We call $T$ be a *compact dual-operator*, if $T$ and $T^*$ are compact operators from $\ell^p(Z)$ to $\ell^p(Z')$ and from $\ell^p(Z')$ to $\ell^p(Z)$, respectively. We define the *maximal norm* of dual-operator $T$ by $||T||_{\max}:=\max\{||T||,||T^*||\}$.
For $p\in(1,\infty)$, let $\mathcal{B}^*(\ell^p(Z),\ell^p(Z'))$ be the Banach algebra of all dual-operators from $\ell^p(Z)$ to $\ell^p(Z')$ with maximal norm. Let $\mathcal{K}^*(\ell^p(Z),\ell^p(Z'))$ be the Banach algebra of all compact dual-operators from $\ell^p(Z)$ to $\ell^p(Z')$. It is easy to see that $\mathcal{K}^*(\ell^p(Z))$ be a closed ideal of $\mathcal{B}^*(\ell^p(Z))$.
For $p\in (1,\infty)$, let $q$ be the dual number of $p$, i.e., $1/p+1/q=1$. If $T$ be a dual-operator acting on $\ell^p(Z)$, then $T$ can be considered as a bounded operator acting on $\ell^q(Z)$ and $||T||_{\ell^q(Z)}=||T^*||_{\ell^p(Z)}$. This is why we call such $T$ a dual-operator. Otherwise, $\mathcal{B}^*(\ell^p(Z))$ is isomorphic to $\mathcal{B}^*(\ell^q(Z))$ for $p,q\in (1,\infty)$ and $1/p+1/q=1$.
\[Lemma:LpCompact\] Let $p\in (1,\infty)$, $Z$ be a countable discrete measure space, if we fixed a bijection between $Z$ and $\mathbb{N}$, then $\ell^p(Z)$ has a natural Schauder basis $\{e_i\}_{i\in \mathbb{N}}$, for any $K\in \mathcal{K}^*(\ell^p(Z))$, we have $$\lim_{n\rightarrow \infty}F_nKF_n=K$$ in $\mathcal{K}^*(\ell^p(Z))$, where $F_n$ be the coordinate projection from $\ell^p(Z)$ to the subspace generated by $e_1,\cdots,e_n$.
We just need to prove $\lim_{n\rightarrow \infty}||F_nKF_n-K||_{\max}=0$, i.e. $$\lim_{n\rightarrow \infty}||F_nKF_n-K_j||_{l^p(Z)}=0 \text{ and }\lim_{n\rightarrow \infty}||F^*_nK^*F^*_n-K^*_j||_{\ell^p(Z)}=0.$$ These are true by the Proposition 1.8 in [@PhillipsLp].
This lemma is false for $p=1$, N.C. Phillips construct a rank one operator without this property in [@PhillipsLp].
\[Cor:KtheoryLpCompact\] Let $p\in (1,\infty)$, $Z$ be a countable discrete measure space, then $K_1(\mathcal{K}^*(\ell^p(Z)))=0$ and $K_0(\mathcal{K}^*(\ell^p(Z)))=\mathbb{Z}$ generated by a rank one idempotent.
Lemma \[Lemma:LpCompact\] imply that $\mathcal{K}^*(\ell^p(Z))$ can be represented as the direct limit of matrix algebra. By the continuous property of $K$-group, we complete the proof.
Let $X$ be a separable proper metric space, $p\in (1,\infty)$. Recall that an $L^p$-$X$-module is an $L^p$-space $E_X^p=\ell^p(Z_X)\otimes \ell^p=\ell^p(Z_X,\ell^p)$ equipped with a natural point-wise multiplication action of $C_0(X)$ by restricting to $Z_X$, where $Z_X$ is a countable dense subset in $X$.
Let $X$ be separable proper metric spaces, $T$ be an element in $\mathcal{B}^*(E^p_X)$, we call $T$ a *locally compact dual-operator* if $\chi_K T$ and $T \chi_K$ are both compact dual-operators for any compact subset $K$ in $X$.
The *dual $L^p$ Roe algebra* of $E_X^p$, denoted $B^{p,\ast}(E_X^p)$, is defined to be the maximal-norm closure of the algebra of all locally compact dual-operators acting on $E_X^p$ with finite propagations.
Let $X,Y$ be two separable proper metric spaces, and $f$ be a continuous coarse map from $X$ to $Y$. Let $V_f$ and $V^+_f$ be an isometric dual-operator and a contractible dual-operator, respectively, constructed in Lemma \[Lemma:CoveringIsometry\]. Thus we have the following lemma.
Let $f$, $E_X^p$ and $E_Y^p$ be as above. Then pair $(V_f,V_f^+)$ give rise to a homomorphism $\mathrm{ad}((V_f,V_f^+)):B^{p,\ast}(E_X^p)\rightarrow B^{p,\ast}(E_Y^p)$ defined by: $$\mathrm{ad}((V_f,V_f^+))(T)=V_fTV_f^+$$ for all $T\in B^{p,\ast}(E_X^p)$.
Moreover, the map $\mathrm{ad}((V_f,V_f^+))_*$ induced by $\mathrm{ad}((V_f,V_f^+))$ on $K$-theory depends only on $f$ and not on the choice of pair $(V_f,V_f^+)$.
The proof of this lemma is same as the proof of Lemma \[Lemma:CoveringIsometryPair\].
For different $L^p$-$X$-modules $E_X^p$ and $E_X'^{p}$, $B^{p,\ast}(E_X^p)$ is non-canonically isomorphic to $B^{p,\ast}(E_X'^{p})$, and $K_*(B^{p,\ast}(E_X^p))$ is canonically isomorphic to $K_*(B^{p,\ast}(E_X'^{p}))$.
For convenience, we replace $B^{p,\ast}(E_X^p)$ by $B^{p,\ast}(X)$ representing the dual $L^p$ Roe algebra of $X$.
Let $X$ be a separable proper metric space. The *dual $L^p$ localization algebra* of $X$, denoted $B^{p,\ast}_L(X)$, is defined to be the norm closure of the algebra of all bounded and uniformly norm-continuous function $f$ from $[0,\infty)$ to $B^{p,\ast}(X)$ such that
prop($f(t)$) is uniformly finite and prop($f(t)$)$\rightarrow 0$ as $t\rightarrow \infty$.
The *propagation* of $f$ is defined to be $\sup\{\mathrm{prop}(f(t)):t\in [0,\infty)\}$.
We have the following lemma for dual $L^p$ localization algebra just like Lemma \[Lemma:IsometryPairHom\].
Let $X,Y$ be two separable proper metric spaces, $f$ be a uniformly continuous coarse map from $X$ to $Y$ and $\{\varepsilon_k\}_k$ be a sequence of positive numbers such that $\varepsilon_k \rightarrow 0$ as $k \rightarrow \infty$, then pair $(V_f(t),V_f^+(t))$ constructed in Lemma \[Lemma:IsometryPairHom\] induces a homomorphism $\mathrm{Ad}((V_f,V_f^+))$ from $B^{p,\ast}_L(X)$ to $B^{p,\ast}_L(Y)\otimes M_2(\mathbb{C})$ defined by: $$\mathrm{Ad}((V_f,V_f^+))(u)(t)=V_f(t)(u(t)\oplus 0)V_f^+(t)$$ for any $u\in B^{p,\ast}_L(X)$ and $t\in [0,\infty)$, such that $$\mathrm{prop}(\mathrm{Ad}((V_f,V_f^+))(u)(t))\leq \sup_{(x,y)\in \mathrm{supp}(u(t))}\{d(f(x),f(y))\}+2\varepsilon_k + 2\varepsilon_{k+1}$$ for all $t\in [k,k+1]$.
Moreover, the map $\mathrm{Ad}((V_f,V_f^+))_*$ induced by $\mathrm{Ad}((V_f,V_f^+))$ on $K$-theory depends only on f and not on the choice of the pairs $(V_k,V_k^+)$ in the construction of $V_f(t)$ and $V_f^+(t)$.
The proof of this lemma is similar to the proof of Lemma \[Lemma:IsometryPairHom\].
The $i$-th *dual $L^p$ $K$-homology* is defined to be $K_i(B^{p,\ast}_L(X))$.
Strongly Lipschitz homotopy invariance of (dual) $L^p$ $K$-homology
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In this section, we will prove that (dual) $L^p$ $K$-homology is strongly Lipschitz homotopic invariant. In the following, we just discuss the case of dual $L^p$ $K$-homology, similarly, we can obtain the same result for $L^p$ $K$-homology.
\[Lemma:LipschitzHomotopy\] $f$ and $g$ be two Lipschitz maps from $X$ to $Y$, let $F(t,x)$ be a strongly Lipschitz homotopy connecting $f$ and $g$, then $$\mathrm{Ad}((V_f,V^+_f))_*=\mathrm{Ad}((V_g,V^+_g))_*: K_*(B^{p,\ast}_L(X)) \rightarrow K_*(B^{p,\ast}_L(Y))$$
We just prove this lemma for $K_1$ group, and by suspension, we can obtain the same result for $K_0$ group. Choose $\{t_{i,j}\}_{i\geq 0,j\geq 0} \subseteq [0,1]$ satisfying
\(1) $t_{0,j}=0, t_{i,j+1}\leq t_{i,j}, t_{i+1,j}\geq t_{i,j}$;
\(2) there exists $N_j\rightarrow \infty$ such that $t_{i,j}=1$ for all $i\geq N_j$ and $N_{j+1}\geq N_j$;
\(3) $d(F(t_{i+1,j},x),F(t_{i,j},x))<\varepsilon_j=1/(j+1), d(F(t_{i,j+1},x),F(t_{i,j},x))<\varepsilon_j$ for all $x\in X$.
Let $f_{i,j}(x)=F(t_{i,j},x)$, by Lemma \[Lemma:CoveringIsometry\], there exist an isometric dual-operator $V_{f_{i,j}}:E_X^p\rightarrow E_Y^p$ and a contractible dual-operator $V^+_{f_{i,j}}:E_Y^p\rightarrow E_X^p$ with $V^+_{f_{i,j}}V_{f_{i,j}}=I$ such that $$\mathrm{supp}(V_{f_{i,j}}) \subseteq \{(x,y)\in X\times Y:d(f_{i,j}(x),y)<1/(1+i+j)\};$$ $$\mathrm{supp}(V^+_{f_{i,j}}) \subseteq\{(y,x)\in Y\times X:d(f_{i,j}(x),y)<1/(1+i+j)\}.$$ For each $i>0$, define a family of operators $V_i(t)(t\in [0,\infty))$ from $E^p_X\oplus E^p_X$ to $E^p_Y\oplus E^p_Y$ and a family of operators $V^+_i(t)(t\in [0,\infty))$ from $E^p_Y\oplus E^p_Y$ to $E^p_X\oplus E^p_X$ by $$V_i(t)=R(t-j)(V_{f_{i,j}}\oplus V_{f_{i,j+1}})R^*(t-j), t\in[j,j+1]$$ $$V^+_i(t)=R(t-j)(V^+_{f_{i,j}}\oplus V^+_{f_{i,j+1}})R^*(t-j), t\in[j,j+1]$$ where $$R(t)=\begin{pmatrix}
\cos(\pi t/2) & \sin(\pi t/2)\\
-\sin(\pi t/2) & \cos(\pi t/2)
\end{pmatrix}.$$ For any $[u]\in K_1(B^{p,\ast}_L(X))$, consider: $$u_0(t)=\mathrm{Ad}((V_f,V^+_f))(u)=V_f(t)(u(t)\oplus I)V^+_f(t)+(I-V_f(t)V^+_f(t));$$ $$u_{\infty}(t)=\mathrm{Ad}((V_g,V^+_g))(u)=V_g(t)(u(t)\oplus I)V^+_g(t)+(I-V_g(t)V^+_g(t));$$ $$u_i(t)=\mathrm{Ad}((V_i,V^+_i))(u)=V_i(t)(u(t)\oplus I)V^+_i(t)+(I-V_i(t)V^+_i(t)),$$ For each $i$, define $n_i$ by $$n_i=\left\{
\begin{array}{lr}
\max\{j:i\geq N_j\}, & \{j:i\geq N_j\}\not= \emptyset;\\
0, & \{j:i\geq N_j\}= \emptyset.
\end{array}
\right.$$ We can choose $V_{f_{i,j}}$ in such a way that: $u_i(t)=u_{\infty}$ where $t\leq n_i$.
Define $$w_i(t)=u_i(t)(u^{-1}_{\infty}(t))$$ Consider $$\begin{aligned}
a&=\bigoplus^{\infty}_{i=0}(w_i\oplus I);\\
b&=\bigoplus^{\infty}_{i=0}(w_{i+1}\oplus I);\\
c&=(I\oplus I)\bigoplus^{\infty}_{i=1}(w_i\oplus I).
\end{aligned}$$ By the construction of $\{t_{i,j}\}$, we know that $a,b,c\in (B^{p,\ast}_L(X) \otimes M_2(\mathbb{C}))^+$. It is not difficult to see that $a$ is equivalent to $b$ and $b$ is equivalent to $c$ in $K_1(B^{p,\ast}_L(X))$. Thus $u_0u^{-1}_{\infty} \oplus_{i\geq 1} I$ is equivalent to $\oplus_{i\geq 0} I$ in $K_1(B^{p,\ast}_L(X))$. This means that $\mathrm{Ad}((V_f,V^+_f))_*=\mathrm{Ad}((V_g,V^+_g))_*$.
Cutting and pasting of the (dual) $L^p$ $K$-homology
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Let $X$ be a simplicial complex endowed with the $\ell^1$-metric, and let $X_1$ be a simplicial subcomplex of $X$. For $p\in (1,\infty)$ define $B^{p,\ast}_L(X_1;X)$ to be the closed subalgebra of $B^{p,\ast}_L(X)$ generated by all elements $f$ such that there exists $c_t>0$ satisfying $\lim _{t\rightarrow \infty} c_t=0$ and $\mathrm{supp}(f(t)) \subset \{(x,y)\in X \times X: d((x,y),X_1 \times X_1)\leq c_t\}$ for all $t\in [0,\infty)$.
\[Lemma:LocalizationIndepStar\] The inclusion homomorphism $i$ from $B^{p,\ast}_L(X_1)$ to $B^{p,\ast}_L(X_1;X)$ induces an isomorphism from $K_*(B^{p,\ast}_L(X_1))$ to $K_*(B^{p,\ast}_L(X_1;X))$.
For any $\varepsilon>0$, let $B_{\varepsilon}(X_1)=\{x \in X : d(x,X_1) \leq \varepsilon\}$. There exists a small $\varepsilon_0>0$ such that $B_{\varepsilon_0}(X_1)$ is strongly Lipschitz homotopy equivalent to $X_1$. Any element in $K_1(B^{p,\ast}_L(X_1;X))$ can be represented by an invertible element $a\in (B^{p,\ast}_L(X_1;X))^+$ such that $a=a'+I$ and there exists $c_t>0$ satisfying $\lim_{t\rightarrow \infty} c_t=0$, $\mathrm{supp}(a'(t))\subset \{(x,y)\in X \times X: d((x,y),X_1 \times X_1)\leq c_t\}$. Uniform continuity of $a(t)$ implies that $a(t+st_0)(s\in [0,1])$ is norm continuous in $s$ for all $t_0>0$. Thus $[a(t)]$ is equivalent to $[a(t+st_0)]$ in $K_1(B^{p,\ast}_L(X_1;X))$ for any $t_0$. We can choose $t_0$ large enough so that $\mathrm{supp}(a'(t+t_0)) \subset B_{\varepsilon_0}(X_1) \times B_{\varepsilon_0}(X_1)$ for all $t$. By Lemma \[Lemma:LipschitzHomotopy\], we know that $i_*$ is surjective.
A similar argument can be used to show that $i_*$ is injective. The case for $K_0$ can be similarly dealt with by a suspension argument.
Let $X$ be a simplicial complex endowed with the $\ell^1$-metric, let $X_1,X_2$ be its two simplicial subcomplexes. We have the following six term exact sequence:
$$\begin{CD}
K_0(B^{p,\ast}_L(X_1 \cap X_2)) @>>> K_0(B^{p,\ast}_L(X_1))\oplus K_0(B^{p,\ast}_L(X_2)) @>>> K_0(B^{p,\ast}_L(X_1 \cup X_2))\\
@AAA & & @VVV \\
K_1(B^{p,\ast}_L(X_1 \cup X_2)) @<<< K_1(B^{p,\ast}_L(X_1))\oplus K_0(B^{p,\ast}_L(X_2)) @<<< K_1(B^{p,\ast}_L(X_1 \cap X_2))
\end{CD}$$
Let $Y=X_1 \cup X_2$, observe that $B^{p,\ast}_L(X_1;Y)$ and $B^{p,\ast}_L(X_2;Y)$ be the ideals of $B^{p,\ast}_L(Y)$ such that $B^{p,\ast}_L(X_1;Y)+B^{p,\ast}_L(X_2;Y)=B^{p,\ast}_L(Y)$. Then by the Mayer-Vietoris sequence for Banach algebra and Lemma \[Lemma:LocalizationIndepStar\], we can obtain this lemma.
By similar argument as above, we have the following six term exact sequence for $L^p$ localization algebra:
$$\begin{CD}
K_0(B^p_L(X_1 \cap X_2)) @>>> K_0(B^p_L(X_1))\oplus K_0(B^p_L(X_2)) @>>> K_0(B^p_L(X_1 \cup X_2))\\
@AAA & & @VVV \\
K_1(B^p_L(X_1 \cup X_2)) @<<< K_1(B^p_L(X_1))\oplus K_0(B^p_L(X_2)) @<<< K_1(B^p_L(X_1 \cap X_2))
\end{CD}$$
Main result and proof
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Let $X$ be a finite dimensional simplicial complex endowed with $\ell^1$-metric. Recall that $E^p_X$ be the $L^p$-$X$-module and $\mathcal{B}^*(E^p_X)$ be the Banach algebra of all dual-operators on $E_X^p$ for $p\in (1,\infty)$. Every $T\in \mathcal{B}^*(E^p_X)$ can be viewed as an element in $B(E^p_X)$. This induces a contractible homomorphism $$\phi:B^{p,\ast}_L(X)\rightarrow B^p_L(X)$$
Next we use the Riesz-Thorin interpolation theorem to build a connection between the dual $L^p$ localization algebra and the localization $C^*$-algebra. Firstly, let us recall this interpolation theorem.
\[Riesz-Thorin\] Let $(X,\mu)$ and $(Y,\nu)$ be two measure spaces. Let $T$ be a linear operator defined on the set of all simple functions on $X$ and taking values in the set of measurable functions on $Y$. Let $1 \leq p_0,p_1,q_0,q_1 \leq \infty$ and assume that $$||T(f)||_{L^{q_0}}\leq M_0||f||_{L^{p_0}},$$ $$||T(f)||_{L^{q_1}}\leq M_1||f||_{L^{p_1}},$$ for all simple functions $f$ on $X$. Then for all $0<\theta<1$ we have $$||T(f)||_{L^{q'}}\leq M^{1-\theta}_0M^{\theta}_1||f||_{L^{p'}}$$ for all simple functions $f$ on $X$, where $1/p'=(1-\theta)/p_0+\theta/p_1$ and $1/q'=(1-\theta)/q_0+\theta/q_1$.\
By density, $T$ has a unique extension as a bounded operator from $L^{p'}(X,\mu)$ to $L^{q'}(Y,\nu)$.
For any $p\in (1,\infty)$, let $q$ be the dual number of $p$, i.e. $1/p+1/q=1$. Let $p_0=q_0=p$, $p_1=q_1=q$ and $\theta=1/2$ in the above, by Riesz-Thorin interpolation theorem, we have that each element $T\in\mathcal{B}^*(E^p_X)$ can be consider as an element in $B(E^2_X)$. This correspondence induces a contractible homomorphism $$\psi:B^{p,\ast}_L(X) \rightarrow C^*_L(X),$$ where $C^*_L(X)$ be the localization $C^*$-algebra of $X$.
\[Thm:pIndependentLocalization\] Let $X$ be a finite dimensional simplicial complex endowed with $\ell^1$-metric, then for any $p\in (1,\infty)$, $\psi$ induces an isomorphism between $K_*(B^{p,\ast}_L(X))$ and $K_*(C^*_L(X))$.
Let $X^{(n)}$ be the $n$-skeleton of $X$, we shall prove this theorem for $X^{(n)}$ by induction on $n$.
When $n=0$, $K_*(B^{p,\ast}_L(X))$ equals to the direct product of $K_*(\mathcal{K}^*(\ell^p))$ and $K_*(C^*_L(X))$ equals to the direct product of $K_*(\mathcal{K}^*(\ell^2))$ following the fact that the algebra of all bounded and uniformly continuous functions from $[0,\infty)$ to a Banach algebra has the same $K$-theory as this Banach algebra. Then by Corollary \[Cor:KtheoryLpCompact\], $\psi_*$ is an isomorphism in this case.
Assume by induction that the theorem holds when $n=m-1$. Next we shall prove the theorem holds when $n=m$. For each simplex $\triangle$ of dimension $m$ in $X$, we let $$\triangle_1=\{x\in \triangle: d(x,c(\triangle))\leq 1/100\},\triangle_2=\{x\in \triangle: d(x,c(\triangle))\geq 1/100\},$$ where $c(\triangle)$ is the center of $\triangle$.
Let $$X_1=\bigcup_{\triangle:\text{simplex of dimension $m$ in X}} \triangle_1;$$ $$X_2=\bigcup_{\triangle:\text{simplex of dimension $m$ in X}} \triangle_2.$$ Notice that:
1. $X_1$ is strongly Lipschitz homotopy equivalent to $$\{c(\triangle): \text{$\triangle$ is $m$-dimensional simplex in X}\};$$
2. $X_2$ is strongly Lipschitz homotopy equivalent to $X^{(m-1)}$;\
3. $X^{(m)}=X_1 \cup X_2$ and $X_1 \cap X_2$ is the disjoint union of the boundaries of all $m$-dimensional $\triangle_1$ in $X^{(m)}$.
\(1) and (2) together with the strongly Lipschitz homotopy invariance of the dual $L^p$-$K$-homology and the induction hypothesis, imply that the theorem holds for $X_1$ and $X_2$.
By the strongly Lipschitz homotopy invariance of $K_*(B^{p,\ast}_L(X))$, $K_*(C^*_L(X))$ and the cutting and pasting exact sequence, we also know that our lemma holds for $X_1\cap X_2$.
Thus we can complete our induction process by using the cutting and pasting exact sequence and the five lemma.
Take the similar argument for $\phi$, we have the following theorem.
\[Thm:pIndenpendentStarLocalization\] Let $X$ be a finite dimensional simplicial complex endowed with the $\ell^1$-metric, then for any $p\in (1,\infty)$, $\phi$ induces an isomorphism between $K_*(B^{p,\ast}_L(X))$ and $K_*(B^p_L(X))$.
By the Theorem \[Thm:pIndependentLocalization\] and \[Thm:pIndenpendentStarLocalization\], we obtain that the $K$-theory for $L^p$ localization algebra is independent of $p$.
Let $X$ be a finite dimensional simplicial complex endowed with the $\ell^1$-metric, then for any $p\in (1,\infty)$, $K_*(B^p_L(X))$ does not depend on $p$.
Further more, we have the following $p$-indenpendency of $K$-theory for $L^p$ Roe algebra.
Let $X$ be a separable proper metric space, assume that there exists an anti-Čech system $\{C_k\}_k$ for $X$ such that $N_{C_k}$ is a finite dimensional simplicial complex for all $k$. Then if for any $p\in (1,\infty)$, the $L^p$ coarse Baum-Connes conjecture is true for $X$, we have that $K_*(B^p(X))$ does not depend on $p$.
By the Theorem \[Thm:LpBCCFAD\], we have the following theorem.
\[Thm:MainThmPIndpendency\] Let $X$ be a separable proper metric space. If $X$ has finite asymptotic dimension, then $K_*(B^p(X))$ does not depend on $p$ for $p\in (1,\infty)$.
Open problems
=============
In this last section, we list several interesting open problems.
There are several versions of $L^p$ $K$-homology. Are they all the same?
Similar to the $L^2$-version of $K$-homology, there are many different versions of $K$-homology
1. Kasparov’s $K$-homology [@KasparovKhomology]
2. $K$-theory of dual algebra by Paschke [@PaschkeDuality]
3. $K$-theory of localization algebra of Guoliang Yu [@YuLocalization][@QiaoRoe]
4. Localization $K$-homology by Xiaoman Chen and Qin Wang [@ChenWangLocalization]
5. $E$-theory by Connes and Higson [@ConnesHigson]
In the $L^2$ case, all above concepts are the same. In this paper, we have seen that the $L^p$ counterpart of all above notions are equivalent for finite dimensional simplicial complex, since they all agree for zero dimensional complexes, and preserve under strong Lipschitz homotopy. But we are not very optimistic that they are equivalent for general topological spaces. To prove the equivalence, we need some deep theorems, say the Voiculescu Theorem [@Voiculescu] and the Kasparov Technical Lemma [@Kasparov88], for $L^p$ spaces.
Is it possible to prove that the $K$-theory of $L^p$ Roe algebras are independent of $p$ without using the coarse Baum-Connes conjecture?
Up to now, all the results about the $p$ indenpendency of the $K$-theory of the group algebras, crossed products and Roe algebras rely on the Baum-Connes conjecture or the coarse Baum-Connes conjecture, since the $K$-homology sides are easier to maneuver. A more direct approach without using the (coarse) Baum-Connes conjecture would shed some light on a Banach algebra approach to the (coarse) Baum-Connes conjecture. For example, if we know certain groups admitting proper isometric actions on $L^p$-spaces and their $K$-theory of the $L^p$ group algebras does not depend on $p$, by the result of Kasparov and Yu [@KasparovYuPBC], we can verify the Baum-Connes conjecture for these groups.
Can we develop an $L^p$ version of Dirac-dual-Dirac method for the $L^p$ Baum-Connes conjecture for amenable groupoids?
In [@TuATmenable], Tu showed the Baum-Connes conjecture is true for amenable groupoids, or more generally, a-T-menable groupoids. For a space with finite asymptotic dimension, or more generally finite decomposition complexity, the coarse groupoid is amenable. For a dynamical system with finite dynamical complexity, the corresponding transformation groupoid is also amenable [@GuentnerWillettYuFDC]. It would be great if we can modify Tu’s method to deal with $L^p$ groupoid algebra and give a unified proof for the Chung’s result on $L^p$ crossed products and the results in this paper on $L^p$ Roe algebras.
Are there any topological and geometric implication of the $L^p$ (coarse) Baum-Connes conjecture?
For example, does it imply the Gromov conjecture [@GromovConjecture] that uniformly contractible manifolds with bounded geometry admit no uniform positive scalar curvature?
Are there any counter-examples for the injectivity of the $L^p$ coarse Baum-Connes conjecture?
In [@HigsonLafforgueSkandalis][@WillettYuExpander], Higson-Lafforgue-Skandalis and Willett-Yu showed that Magulis type expanders and expanders with large girth are counter-examples for the surjectivity of the coarse Baum-Connes conjecture. In [@ChungNowakPCBC], Chung and Nowak showed that Magulis type expanders are still counter-example for the $L^p$ coarse Baum Connes conjecture. However, the existence of injectivity counterexample of the $L^p$ coarse Baum Connes conjecture is still open. In [@YuFAD], Guoliang Yu gave a counterexample of the injectivity of coarse Baum-Connes conjecture. The proof relies on a positive scalar curvature argument. Is Yu’s counter-example still a counter-example for the $L^p$ version?
|
---
abstract: 'We use a random matrix model to study chiral symmetry breaking in QCD at finite chemical potential $\mu$. We solve the model and compute the eigenvalue density of the Dirac matrix on a complex plane. A naive “replica trick” fails for $\mu\neq0$: we find that quenched QCD is not a simple $n\to0$ limit of QCD with $n$ quarks. It is the limit of a theory with $2n$ quarks: $n$ quarks with original action and $n$ quarks with conjugate action. The results agree with earlier studies of lattice QCD at $\mu\neq0$ and provide a simple analytical explanation of a long-standing puzzle.'
address: |
Department of Physics, University of Illinois at Urbana-Champaign,\
1110 West Green Street, Urbana, IL 61801-3080, USA
author:
- 'M. A. Stephanov[^1]'
date: March 1996
title: Random matrix model of QCD at finite density and the nature of the quenched limit
---
Introduction
============
The spontaneous breaking of chiral symmetry is one of the most important dynamical properties of QCD which shapes the hadronic spectrum. A great deal of understanding of this nonperturbative phenomenon at zero and finite temperature has been achieved by various methods [@Sh95]. In particular, we expect that the chiral symmetry is restored above a certain critical temperature. The study of this new chirally symmetric phase of hot QCD is one of the primary objectives of heavy ion colliders. In contrast, the behavior of QCD at large baryon density (conditions which can arise in the heavy ion colliders or in neutron stars) is not well understood. The main puzzle has for a long time been a contradiction between a straightforward physical expectation and numerical results from quenched lattice QCD [@Ba86; @KoLo95]. Simulations with dynamical quarks, on the other hand, are very inefficient at finite $\mu$ — the fermion determinant is complex.
The puzzle concerns the dependence of the order parameter (the chiral condensate $\langle\bar\psi\psi\rangle$) on the baryon chemical potential. A non-analytical change in the value of $\langle\bar\psi\psi\rangle$ should occur when $\mu > \mu_c \approx
m_B/3$, where $m_B$ is the mass of the lightest baryon. At this point the production of baryons becomes energetically favorable. For smaller $\mu$ the value of $\langle\bar\psi\psi\rangle$ is nonzero. In contrast, lattice simulations of quenched QCD indicate that $\mu_c=0$ (at zero bare quark mass), i.e., the chiral condensate vanishes if $\mu\neq0$ [@Ba86; @KoLo95]. A number of possible explanations has been suggested [@explain]. However, the answer to this puzzle remains unclear.
This work was motivated by a desire to shed some light on this question using the random matrix approach which received considerable interest recently [@ShVe93; @VeZa93; @Ve94; @JaVe95; @WeSc96; @St96; @JaSe96; @JuNo96]. It is based on the idea that, for the purpose of studying chiral symmetry breaking, fluctuations of the Dirac operator in the background of the gauge fields can be approximated by purely random fluctuations of its matrix elements in a suitable basis. For example, in the instanton liquid model this basis can be formed from the Dirac zero modes for individual (anti)instantons, which due to overlaps form a band of small eigenvalues responsible for the chiral symmetry breaking [@DiPe86]. A similar random matrix approach is fruitful in the studies of spectra of systems with a high level of disorder, such as spectra of heavy nuclei [@Po65]. Introduction of chemical potential into such a model of chiral symmetry breaking is straightforward. The resulting Dirac matrix (times $i$) is [*non-hermitian*]{}. Thus the eigenvalues lie in the complex plane rather than on a line. Such random matrix models have not received much attention previously and this study is a step in an unexplored direction.
In this Letter we show how to solve such a model in the thermodynamic limit and discuss the implications.
The resolvent
=============
In order to study chiral symmetry breaking we shall calculate the resolvent of the Dirac operator $D$: $$\label{G}
G = \left\langle \mbox{ tr }(z - D)^{-1} \right\rangle,$$ as a function of the bare quark mass $z$ which we take to be a complex variable $z=x+iy$. The average is over fluctuations of the random matrix elements of $D$. It should be obvious that $G$ is the same as $\langle\bar\psi\psi\rangle$. The resolvent can be expressed through the average eigenvalue density $\rho$: $$\label{Grho}
G(x,y)=\int\,dx'\,dy'\,\rho(x',y'){1\over z - z'}.$$ A vector $\vec G =(\mbox{Re}G, -\mbox{Im}G)$ is the electric field created by the charge distribution $\rho$. This makes the inversion of (\[Grho\]) obvious: $$\label{rhoG}
\rho={1\over2\pi}\vec\nabla\vec G
= {1\over\pi}{ \partial \over \partial z^* } G,$$ where $\partial/\partial z^*
\equiv (\partial/\partial x +i \partial/\partial y)/2$.
Analytical properties of $G$ are very closely related to the chiral symmetry breaking. From (\[rhoG\]) we see that $\rho$ vanishes if the function $G$ is holomorphic. A discontinuity of $G$ along a cut going through $z=0$ is the signature of the spontaneous chiral symmetry breaking: $\langle\bar\psi\psi\rangle(+0)\neq\langle\bar\psi\psi\rangle(-0)$. This observation together with (\[rhoG\]) leads to the Banks-Casher relation [@BaCa80]: $\langle\bar\psi\psi\rangle = \pi
\rho(0)$, where $\rho(0)$ is the density per [*length*]{} on the cut at $z=0$. However, (\[rhoG\]) is more general and can be applied to a case when the non-analyticity is not in the form of a cut but occupies a 2-dimensional patch, which is the case in our model.
The matrix model and naive replica trick
========================================
The matrix $D$ has the form: $$D=\left(
\begin{array}{cc}
0 & iX\\
iX^\dagger & 0
\end{array}
\right) + \left(
\begin{array}{cc}
0 & \mu \\
\mu & 0
\end{array}
\right)$$ where we added the chemical potential term $\mu\gamma_0$ to the Dirac matrix [@ShVe93]. The $N\times N$ matrix elements of $X$ are independently distributed complex Gaussian random variables: $P(X) =
\mbox{const}\times\exp\{-N\mbox{ Tr }XX^\dagger\}$. The unit of mass in the model is set by $n_4/\langle\bar\psi\psi\rangle_0 \sim
200\mbox{ MeV}$, where $n_4\sim1\mbox{ fm}^{-4}$ is the number of small eigenvalues in a unit volume (instanton density [@Sh88]) and $\langle\bar\psi\psi\rangle_0\sim(200\mbox{ MeV})^3$ is the chiral condensate at $T=0$, $\mu=0$.
In order to find the resolvent (\[G\]) we introduce ${n}$ quark fields (replicas) and calculate: $$\label{Vn}
V_{{n}}=-{1\over{n}}\ln \left\langle {\det}^{{n}} ( z - D ) \right\rangle.$$ This quantity continued to ${n}\to0$ (quenched limit) becomes: $$V = -\left\langle \ln \det ( z - D ) \right\rangle,$$ from which we find $G$: $$\label{GV}
G = -{\partial\over\partial z} V.$$ The trace in (\[G\]) is normalized as $\mbox{ tr }1=\mbox{Tr }1/(2N)$. Following the electrostatic analogy of the previous section one can view Re$V$ as the scalar potential for $\vec G$.
Using Hubbard-Stratonovitch transformation we obtain: $$\label{expV}
\exp\{-{n} V_{{n}}\}
= \int {\cal D} a\, {\det}^N \left(
\begin{array}{cc}
z+a & \mu \\
\mu & z+a^\dagger
\end{array} \right)
\exp\{-N\mbox{ Tr }aa^\dagger\},$$ where $a$ is an auxiliary complex ${n}\times {n}$ matrix field. For large $N$ the calculation of the integral amounts to finding its saddle point. If we assume that the replica symmetry is not broken (i.e., $a$ is proportional to a unit matrix) we arrive at the saddle point equation: $$\label{a}
(z+a)=a[(z+a)^2-\mu^2],$$ The complex value of $a$ in (\[a\]) is the analytical continuation of the real part of the diagonal matrix elements of $a$ in (8). The imaginary part is zero in the saddle point. The solution of this cubic equation is straightforward. It is the same as in a similar model [@St96] with $\omega\to i\mu$. Finally, it is easy to find using (\[GV\],\[expV\]) that $G=a$ where $a$ is the saddle point given by (\[a\]).
The $V_{n}$ does not depend on ${n}$ and the limit ${n}\to0$ seems obvious. However, in the next section we shall compare this expectation to numerical data and see that the limit $n\to0$ is in fact very different! Now let us summarize the properties of this model for $n>0$.
We see that $G(z)$ is a holomorphic function. It has 3 Riemann sheets and we select the one where $G\to1/z$ for $z\to\infty$ – which follows from $\int dxdy\,\rho=1$ and the Gauss theorem. The only singularities on the physical sheet are the pole at $z=\infty$ and 2 (for $\mu^2\le1/8$) or 4 (for $\mu^2>1/8$) branch points connected by cuts. The brunch points are where 2 of the 3 solutions of (\[a\]) coincide. The trajectory of a cut is determined by a condition that $a$ is the deepest minimum (out of 3) of: $\mbox{ Re } [a^2 -
\ln((z+a)^2-\mu^2)]$.
At $\mu=0$ the cut along imaginary axis connects two singularities at $z=\pm2i$. For nonzero $\mu$ the singularities start moving towards each other along the imaginary axis. At $\mu^2=1/8$ each of the branch points bifurcates in two ones which move off the $y$ axis into the complex plane. The cut goes through the origin (along the $y$ axis) until $\mu^2=0.278...$. At this point it splits into two cuts connecting complex conjugate points. This means that $\mu_c^2=0.278...$ in such a model.
Numerical results and the solution of the model
===============================================
For $n=0$ one can easily determine the density of eigenvalues numerically by calculating the eigenvalues of the random matrix $D$ and plotting them on a complex plane. The density of points on such a scatter plot is proportional to $\rho$. The results for different values of $\mu^2$ are shown in Fig. 1. They contradict naive expectations from the previous section. At $\mu=0$ all eigenvalues are distributed between points $z=\pm2i$ on the $y$ axis. However, already at very small nonzero $\mu^2\ll1/8$ the eigenvalue density is nonzero in a “blob” of finite width in $x$ direction which grows with $\mu$. The same behavior is seen in quenched lattice QCD [@Ba86] and gives rise to the paradox described in the Introduction: there is no discontinuity in the value of $\langle\bar\psi\psi\rangle$ at any $\mu>0$. The matrix model has an advantage: it is amenable to exact treatment which clarifies the nature of the problem.
The failure of the naive replica approach can be understood if we look at the expression (\[Vn\]): it does not contain $z^*$! On the other hand, eq. (\[rhoG\]) tells us that $\rho\neq0$ if $G$ depends on $z^*$, i.e., if it is not holomorphic. In fact, the correct replica trick for a non-hermitian matrix should start from the quantity: $$\label{Vn2}
V_{{n,n}} = -{1\over {n}}\ln\left\langle
{\det}^{{n}} (z - D) (z^* - D^\dagger) \right\rangle,$$ which is now real due to introduction of the quarks with conjugate Dirac matrix. Naively, in the limit ${n}\to0$ the conjugate quarks decouple but, as we shall see, this is not always the case! In mathematics an analogous construction is called a V-transform [@Gi88] and allows one to study spectra of non-hermitian matrices. In the present context this formal construction has a clear and simple physical meaning.
We can calculate (\[Vn2\]) using the same method as for (\[Vn\]). Now, however, we have to introduce 4 auxiliary complex ${n}\times {n}$ fields, and we arrive at: $$\begin{aligned}
\exp(- {n} V_{{n,n}})
&=& \int {\cal D}a\,{\cal D}b\,{\cal D}c\,{\cal D}d\,\,
{\det}^{N} \left(
\begin{array}{cccc}
z+a & \mu & 0 & id \\
\mu & z+a^\dagger & ic & 0 \\
0 & id^\dagger & z^*+b^\dagger & \mu \\
ic^\dagger & 0 & \mu & z^*+b
\end{array}
\right)
\nonumber \\&&
\times\exp\left\{-N(|a|^2 + |b|^2 + |c|^2 + |d|^2)\right\}.\end{aligned}$$
The set of solutions of the saddle point equation is richer in this case. There is a solution with $c=d=0$. In this case the conjugate quarks do decouple and we obtain the same holomorphic function $G$ as before. However, there is another solution in which the condensates $c$ and $d$ are not zero! Then the function $G$ is not holomorphic and therefore $\rho\neq0$. This saddle point dominates the integral at small $z$ for $0<\mu<1$.
The condensates $c$ and $d$ are bilinears of the type $\langle\bar\psi\chi\rangle$, mixing original $\psi$ and conjugate $\chi$ quarks. These condensates do not break the original chiral symmetry but a spurious (replica type) symmetry involving both original and conjugate quarks. Similar condensates carrying baryon number were discussed in the $SU(2)$ model of QCD [*with*]{} quarks [@Da86-7]. In the quenched theory, as in [@Da86-7], the original chiral symmetry is always restored at $\mu>0$. The spurious symmetry is spontaneously broken for $\mu<1$ and is restored for $\mu>1$.
The boundary of the $\rho\neq0$ region is given by: $$\label{yx}
y^2 = (\mu^2 - x^2)^{-2}
[4 \mu^4 (1 - \mu^2) - (1 + 4\mu^2 - 8\mu^4) x^2 - 4 \mu^2 x^4].$$ It is plotted on Fig. 1 for comparison with numerical data. The baryonic condensates $c$ and $d$ inside of the “blob” are given by: $$|c|^2=|d|^2=
{\mu^2\over\mu^2-x^2} - \mu^2 - {x^2\over4(\mu^2-x^2)^2}
- {y^2\over4}.$$ On the boundary (\[yx\]) they vanish and the two solutions (holomorphic and non-holomorphic) match. In the outer region: $c=d=0$ and $G=a$ is the solution of the cubic equation (\[a\]). Inside of the “blob” the resolvent is given by: $$G = a = {1\over2} {x\over \mu^2 - x^2} - x - {iy\over 2},$$ and the density of the eigenvalues (\[rhoG\]) is: $$\rho = {1\over4\pi}\left( {{x^2+\mu^2\over(\mu^2-x^2)^2} - 1}\right).$$
To appreciate non-triviality of this result one should notice that expression (\[G\]) which defines the resolvent appears to depend only on $z$! The limit $n\to0$ must be taken with great care, as is well-known in the replica approach [@MePa87].
Conclusions
===========
The fermion determinant in QCD is complex at nonzero chemical potential. Lattice simulations of such a theory are extremely inefficient. Therefore all reliable data from lattice QCD so far have been obtained for a quenched theory. We learn from the random matrix model that the quenched theory at finite $\mu$ behaves qualitatively different from the QCD with dynamical quarks. Rather, the quenched approximation describes a theory where each of the quarks has a conjugate partner, so that the fermion determinant is non-negative. We see that for such a theory the result $\mu_c=0$ is natural. Similar arguments have been given by several authors in different settings and using less realistic models [@explain]. Here it can be demonstrated in a very clean and explicit way.
Simulations with dynamical quarks at strong coupling are possible in $SU(2)$ and $SU(4)$ QCD [@Da86-7] and also agree with our results. The $\mu_c$ is finite in the $SU(4)$ theory. On the other hand, in the $SU(2)$ theory, where the quarks are self-conjugate, $\mu_c=0$ due to the baryonic condensates.
The matrix model describes many features of the chiral symmetry breaking in QCD very well [@ShVe93; @VeZa93; @Ve94; @JaVe95; @WeSc96; @St96; @JaSe96; @JuNo96]. One of the apparent limitations, however, is that it is static — there are no kinetic terms and we cannot study spectrum of masses. In the quenched QCD $\mu_c$ appears to coincide with half of the mass of the so-called baryonic pion [@explain] — a bound state of a quark and a conjugate antiquark. It is degenerate with the $\pi$-meson but carries a nonzero baryon number. From the exact solution (\[yx\]) we find $\mu_c\approx\sqrt{m/2}$, for small quark mass $m\ll1$. If we had $f_\pi$ in our model we could relate $\mu_c$ to the mass of the pion. The model also does not account for the confinement of quarks. It remains to be seen if the confinement plays a role in the case under consideration.
The author is grateful to E. Fradkin, A. Kocic, J. Kogut, and M. Tsypin for discussions and suggestions some of which were crucial for this work. The work was supported by the National Science Foundation, NSF-PHY92-00148.
For a review see, e.g., E. V. Shuryak, hep-ph/9503427, to be published in [*Quark-Gluon Plasma*]{}, ed. R. Hwa. I. Barbour [*et al*]{}, Nucl. Phys. B [**275**]{} \[FS17\] (1986) 296. J. B. Kogut, M. P. Lombardo and D. K. Sinclair, Phys. Rev. D [**51**]{} (1995) 1282; Nucl. Phys. B, Proc. Suppl. [**42**]{} (1995) 514. P. E. Gibbs, Phys. Lett. B [**182**]{} (1986) 369; A. Gocksch, Phys. Rev. D [**37**]{} (1988) 1014; C. T. H. Davies and E. G. Klepfish, Phys. Lett. B [**256**]{} (1991) 68.
E. V. Shuryak and J. J. M. Verbaarschot, Nucl. Phys. A [**560**]{} (1993) 306. J. J. M. Verbaarschot and I. Zahed, Phys. Rev. Lett. [**70**]{} (1993) 3852. J. J. M. Verbaarschot, Phys. Rev. Lett. [**72**]{} (1994) 2531. A. D. Jackson and J. J. M. Verbaarschot, hep-ph/9509324. T. Wettig, A. Schafer, and H. A. Weidenmuller, Phys. Lett. B [**367**]{} (1996) 28. M. A. Stephanov, hep-lat/9601001, to be published in Phys. Lett. B. A. D. Jackson, M. K. Sener, and J. J. M. Verbaarschot, hep-ph/9602225. J. Jurkiewicz, M. A. Nowak, I. Zahed, hep-ph/9603308. D. Diakonov and V. Petrov, Nucl. Phys. B [**272**]{} (1986) 457. C. E. Porter, [*Statistical theories of spectra: fluctuations*]{}, (Academic Press, New York, 1965). T. Banks and A. Casher, Nucl. Phys. B [**169**]{} (1980) 103. E. Shuryak, Nucl. Phys. B [**302**]{} (1988) 559. V. L. Girko, [*Spektralnaia teoriia sluchainykh matrits*]{}, (Nauka, Moscow, 1988). E. Dagotto, F. Karsch, and A. Moreo, Phys. Lett. B [**169**]{} (1986) 421; E. Dagotto, A. Moreo, and U. Wolff, Phys. Rev. Lett. [**57**]{} (1986) 1292; Phys. Lett. B [**186**]{} (1987) 395. M. Mezard, G. Parisi, and M. A. Virasoro, [*Spin glass theory and beyond*]{}, (World Scientific, Philadelphia, 1987).
\
2em\
2em
[^1]: Electronic mail address: *misha@uiuc.edu.*
|
[**$W$ boson polarization as a measure of gauge-Higgs anomalous couplings at the LHC**]{}
.3cm
Kumar Rao\
.1cm [*Physics Department, Indian Institute of Technology Bombay,\
Powai, Mumbai 400076, India*]{}\
.1cm Saurabh D. Rindani\
.1cm [*Theoretical Physics Division, Physical Research Laboratory,\
Navrangpura, Ahmedabad 380009, India*]{}\
.3cm
[We show how the $W$ boson polarization in the process of associated $W^{\pm}H$ production at the Large Hadron Collider (LHC) can be used to constrain anomalous $WWH$ couplings. We first calculate the spin density matrix for the $W$ to linear order in the anomalous couplings, which are assumed to be small. We then evaluate angular asymmetries in the decay distributions of leptons produced in the decay of the $W$ and show how they can be used to measure the individual elements of the polarization tensor. We estimate the limits that can be placed on the anomalous $WWH$ couplings at a future run of the LHC. ]{}
[**1. Introduction**]{}\
After the discovery of the Higgs boson with a mass of around 125 GeV, several measurements at the Large Hadron Collider (LHC) indicate that its couplings are consistent with those predicted by the standard model (SM). However, a complete confirmation that the Higgs boson $H$ discovered at the LHC is indeed the Higgs boson of the SM will require precise determination of all the couplings of $H$, including Higgs self-couplings. A simplistic analysis, usually adopted in the interpretation of Higgs data, attempts to measure the ratio $\kappa$ of the coupling to that in the standard model. In this procedure, the so-called $\kappa$ framework, the forms of the interactions assumed are the same as in the SM at tree level. An attempt to introduce more general tensor forms of couplings is not permitted by the present accuracy of the experiments. However, in future experiments at higher luminosities, it is hoped that such general forms of couplings will be constrained. This could include measurement of differential cross sections, which would be highly data intensive. Alternatively, one could measure partial cross sections, or angular or energy asymmetries of final state particles.
An interesting additional variable which we consider in this work is the polarization of the $W^{\pm}$ produced in association with the Higgs. Measurement of polarization of a heavy particle requires the observation of decay distributions of the particle. Again one can construct appropriate asymmetries from the kinematical distributions of the decay particles. In particular, charged lepton distributions in the decay of the $W$ would enable the measurement of $W$ polarization parameters, which in turn would constrain the strengths of the tensor structures of the $WWH$ interactions.
$W$ polarization has been discussed recently in the context of polarized top decays and diboson resonances at the LHC [@aguilar], and earlier in the context of various single, pair and associated $W$ production processes [@stirling]. For details of the formalism in the context of LEP experiments, see [@bailey]. $Z$ polarization has been studied in the context of new physics at $e^+e^-$ colliders [@aguilar2; @Rahaman:2016pqj].
$W$ helicity fractions, which measure the degree of longitudinal or transverse polarizations, have been measured in top decay $t\to bW$ at the LHC from the polar-angle distributions, integrated over the azimuthal angle [@helfrac]. These correspond to the diagonal elements of the $W$ production spin-density matrix. In what follows, we also consider measurement of the off-diagonal density-matrix elements [@aguilar3; @belyaev; @Velusamy:2018ksp; @Rahaman:2017qql; @Boudjema:2009fz] through angular asymmetries of the leptons produced in $W$ decay.
The asymmetries we consider are defined in the rest frame of the decaying $W$. Measurement of these asymmetries would therefore involve transforming laboratory-frame kinematic variables to the $W$ rest frame. This in turn needs the knowledge of the $W$ four-momentum. This is a potential problem because the $W$ decays into a neutrino, which is not detected. While the transverse momentum of the neutrino can be reconstructed with good accuracy using momentum conservation, the longitudinal momentum cannot be measured directly. The usual procedure [@helfrac] is to constrain the invariant mass of the $W$ decay products to be equal to the $W$ mass. Moreover, the construction of the polarization asymmetries, which are related to the elements of the $W$ density matrix requires the $W$ to be on-shell [@Rahaman:2017qql; @Boudjema:2009fz]. Since the on-shell constraint gives rise to a quadratic equation, there is a two-fold ambiguity in the determination of the neutrino longitudinal momentum. Various procedures have been considered to choose one of the two solutions allowed. One procedure followed in a recent study of $WH$ production by ATLAS is to take the smaller of the two solutions [@Aaboud:2017cxo]. Another suggestion [@Godbole:2014cfa] is to compare the longitudinal boosts $\beta_z^W$ and $\beta_z^H$ of the reconstructed $W$ and the $H$, and choose the solution which gives the lower value for $\vert \beta_z^W - \beta_z^H
\vert $, which was found in simulations to give the true neutrino momentum in 65% of the cases.
$W$ and $Z$ polarization in associated Higgs production has been studied recently in [@Nakamura:2017ihk], with which our work has considerable overlap. While [@Nakamura:2017ihk] contains expressions for $W$ spin density matrices which we obtained independently, their analysis deals with hadronic decay of the vector bosons, whereas we concentrate on leptonic decay of the $W$. While the hadronic branching ratios are larger, it is not possible to determine the charge of the jets. On the other hand, though the branching ratio of $W$ into leptons is smaller, greater precision is possible, as well as charge discrimination is available.
The $WWH$ vertex for a process $W^{+*} \to W^+H$ may be written in a model-independent way as \[WWH\] \_ = g m\_W, where $q$ is the incoming $W^*$ momentum and $k$ is the outgoing $W$ momentum, and $\nu$, $\mu$ are their respective polarization indices. $g$ is the weak coupling constant, and $a_W =1$ in the SM at tree level. $b_W$ and $\tilde b_W$ which are vanishing in the SM at tree level, are anomalous couplings, taken to be complex form factors. An analogous vertex for the process $W^{-*} \to W^-H$ may also be written. While the first two terms would arise from terms in an effective Lagrangian and are invariant under CP, the $\tilde b_W$ term would correspond to a CP-violating term in the Lagrangian. The anomalous couplings could arise at one or more loops in the SM, or in extensions of the SM, with heavy particles (the top quark, $W$, $Z$ and $H$ in the SM, or other additional particles in SM extensions) occurring in the loops, and coupling to the Higgs boson. However, we will not be concerned here with predictions of any specific model.
[**2. Helicity amplitudes and density matrix**]{}\
We consider the process $pp \to W^{\pm}H X$ at the LHC, which at the partonic level proceeds via the process $q\bar q' \to W^* \to W^{\pm}H$, where $q$ and $q'$ are quarks. After calculating the helicity amplitudes for the process in the presence of anomalous $WWH$ couplings, we evaluate the production density matrix elements for the spin of the $W$ at the partonic level and consequently for a hadronic initial state, to linear order in the anomalous couplings. We further examine how each of these polarization tensor elements may be measured from various angular asymmetries of charged leptons produced in the decay of the $W$, and also estimate the sensitivity of these measurements for an assumed integrated luminosity of the experiment.
To calculate the helicity amplitudes for the production process in the quark-antiquark c.m. (centre-of-mass) frame, \[prodwplus\] u(p\_1)+|d(p\_2) W\^+(k) + H, where $u$ and $d$ are respectively up-type and down-type quarks of any generation, we make use of the following representation for the polarization vectors of the $W$: \[epspm\] \^(k,) ( 0,,-,), \[eps0\] \^(k,0) (,, 0,) where $E_W$ is the energy of the $W$ and $\vec k_W$ its momentum, with polar angle $\theta$ with respect to the direction of the $u$ quark taken as the $z$ axis.
The nonzero helicity amplitudes in the limit of massless quarks are given by \[helamps1\] M(-,+,-) = -g\^2V\_[qq’]{}m\_W \[helamps2\] M(-,+,0) = -g\^2V\_[qq’]{}E\_W \[helamps3\] M(-,+,+) = -g\^2V\_[qq’]{}m\_W where $\sqrt{\hat s}$ is the total energy in the parton c.m. frame, $\beta_W=|\vec k_W|/E_W$, and $V_{qq'}$ is the appropriate element of the Cabibbo-Kobayashi-Maskawa matrix, and the first two entries in $M$ correspond to helicities $-1/2$ and $+1/2$ of the quark and anti-quark, respectively, and the third entry is the $W$ helicity.
The helicity amplitudes for the $W^-$ production process \[prodwminus\] d(p\_1)+|u(p\_2) W\^-(k) + H are also given by eqns. (\[helamps1\])-(\[helamps3\]), with the first two entries in $M$ denoting the helicities of the $d$ and $\bar u$, and $\theta$ representing the angle between $W^-$ and $d$. Here it is assumed that the same couplings $a_W$, $b_W$ and $\tilde b_W$ occur in the process $W^{-*} \to W^-H$ as in $W^{+*} \to W^+H$, as in an effective field theory approach [@Nakamura:2017ihk].
In terms of the helicity amplitudes, the spin-density matrix for $W$ production is defined as \[defrho\] (i,j) = M(h\_q,h\_[|q]{},i)M(h\_q,h\_[|q]{},j)\^\*, the sum and average being over initial helicities $h_q$, $h_{\bar q}$ of the quark and anti-quark, respectively, and also over initial colour states, not shown explicitly. The diagonal elements for $i=j$ would correspond to production probabilities with definite $W$ polarization labelled by $i=j$ as applicable, for example, in the study of helicity fractions. However, in the description of $W$ production followed by decay, where measurement is made on the decay products, the full density matrix description, which includes off-diagonal elements, is needed. This is because a full description requires multiplying the helicity amplitudes for production with the helicity amplitudes for decay in a coherent fashion (see, for example, [@Leader:2001gr]).
The density matrix elements derived from the helicity amplitudes (\[helamps1\])-(\[helamps3\]), to linear order in the couplings $b_W$ and $\tilde b_W$, setting $a_W=1$ are as follows. \[rhopmpm\] (,) = |V\_[qq’]{}|\^2 (1 )\^2 \[rho00\] (0,0) = |V\_[qq’]{}|\^2 \^2$$\begin{aligned}
\label{rhomp0}
\rho(\mp,0)\!\! &\!\!=\!\!& \frac{g^4}{12}\frac{\hat s m_W
E_W}{2\sqrt{2}(\hat s - m_W^2)^2} |V_{qq'}|^2\sin\theta(1\pm \cos\theta)
\\
&\hskip -1.7cm \times & \hskip -1cm \left[1
-{\rm Re}b_W \sqrt{\hat s}\frac{(E_W^2+m_W^2)}{E_Wm_W^2}
-i{\rm Im}b_W \sqrt{\hat s}
\frac{\beta_W ^2 E_W}{m_W^2}
\mp i\beta_W \tilde b_W \frac{\sqrt{\hat s}E_W}{m_W^2}
\right]\end{aligned}$$ \[rhomppm\] (,) = |V\_[qq’]{}|\^2 \^2
We have used the analytical manipulation software FORM [@form] to check these expressions.
Defining an integral of this density matrix over an appropriate kinematic range as $\sigma(i,j)$, the latter can be parametrized in terms of the linear polarization $\vec P$ and the tensor polarization $T$ as follows. [@Leader:2001gr] \[vectensorpol\] (i,j) (
[ccc]{} + + & + &\
+ & - & -\
& - & - +
) where $\sigma(i,j)$ is the integral of $\rho(i,j)$, and $\sigma$ is the production cross section, = (+,+) + (-,-) + (0,0).
The vector and tensor polarizations then can be obtained by inverting eqn. (\[vectensorpol\]): $$\begin{aligned}
P_x& =& \frac{1}{(\sqrt{2}\sigma)}[\sigma(+,0)+ \sigma(0,+)+\sigma(-,0)+\sigma(0,-)]\\
P_y& =& \frac{i}{(\sqrt{2}\sigma)}[\sigma(+,0)- \sigma(0,+)-\sigma(-,0)+\sigma(0,-)]\\
P_z& =& \frac{1}{\sigma}[\sigma(+,+)- \sigma(-,-)]\\
T_{xy}& =& \frac{i\sqrt{6}}{(4\sigma)}[\sigma(+,-)- \sigma(-,+)]\\
T_{xz}& =& \frac{\sqrt{3}}{(4\sigma)}[\sigma(+,0)+ \sigma(0,+)-\sigma(-,0)-\sigma(0,-)]\\
T_{yz}& =& \frac{i\sqrt{3}}{(4\sigma)}[\sigma(+,0)- \sigma(0,+)+\sigma(-,0)-\sigma(0,-)]\\
T_{xx}-T_{yy}& =& \frac{\sqrt{6}}{(2\sigma)}[\sigma(+,-)+ \sigma(-,+)]\\
T_{zz}& =& \frac{\sqrt{6}}{(6\sigma)}[\sigma(+,+)+ \sigma(-,-)-2 \sigma(0,0)],\end{aligned}$$
[**3. Leptonic asymmetries**]{}\
Obtaining spin information of the $W$ requires measurements to be made on the decay products of the $W$. Using leptonic decays is more convenient than using hadronic decays because charge identification is difficult, if not impossible, for that latter case. Expressions may be obtained for the decay-lepton distribution in the $W$ production process by combining the relevant production-level density matrix elements with appropriate decay density matrix elements and integrating over the appropriate phase space. As mentioned before, a full measurement of the lepton distribution would require a very large number of events. It is more economical to use integrated angular asymmetries, which utilize all relevant events. We therefore adopt this approach and define different angular asymmetries of the charged lepton.
Following [@Rahaman:2016pqj], we define angular asymmetries of the lepton arising from $W$ decay, evaluated in the rest frame of the $W$, which isolate various elements of the polarization tensor: $$\begin{aligned}
A_x &=& \frac
{\sigma(\cos\phi^* > 0) - \sigma(\cos\phi^*<0)}
{\sigma(\cos\phi^* > 0) + \sigma(\cos\phi^*<0)}, \\
A_y &=& \frac
{\sigma(\sin\phi^* > 0) - \sigma(\sin\phi^*<0)}
{\sigma(\sin\phi^* > 0) + \sigma(\sin\phi^*<0)}, \\
A_z &=& \frac
{\sigma(\cos\theta^* > 0) - \sigma(\cos\theta^*<0)}
{\sigma(\cos\theta^* > 0) + \sigma(\cos\theta^*<0)}, \\
A_{xy} &=& \frac
{\sigma(\sin2\phi^* > 0) - \sigma(\sin2\phi^*<0)}
{\sigma(\sin2\phi^* > 0) + \sigma(\sin2\phi^*<0)}, \\
A_{xz} &=& \frac
{\sigma(\cos\theta^*\cos\phi^* < 0) - \sigma(\cos\theta^*\cos\phi^*>0)}
{\sigma(\cos\theta^*\cos\phi^* > 0) + \sigma(\cos\theta^*\cos\phi^*<0)}, \\
A_{yz} &=& \frac
{\sigma(\cos\theta^*\sin\phi^* > 0) - \sigma(\cos\theta^*\sin\phi^*<0)}
{\sigma(\cos\theta^*\sin\phi^* > 0) + \sigma(\cos\theta^*\sin\phi^*<0)}, \\
A_{x^2-y^2} &=& \frac
{\sigma(\cos2\phi^* > 0) - \sigma(\cos2\phi^*<0)}
{\sigma(\cos2\phi^* > 0) + \sigma(\cos2\phi^*<0)}, \\
A_{zz} &=& \frac
{\sigma(\sin3\theta^* > 0) - \sigma(\sin3\theta^*<0)}
{\sigma(\sin3\theta^* > 0) + \sigma(\sin3\theta^*<0)}. \end{aligned}$$ The direction of the quark momentum is defined as the $z$ axis, and the $x$ axis chosen so that the $W$ lies in the $xz$ plane. Using these axes, the angles $\theta^*$ and $\phi^*$ are the polar and azimuthal angles of the decay lepton, defined in the rest frame of the $W$, with respect to the boost direction of the $W$.
It may be observed that since the sign of the triple vector product of the beam direction, the $W$ momentum direction and the lepton momentum direction determines the sign of $\sin\phi^*$, the asymmetries $A_y$, $A_{xy}$, $A_{yz}$ which are linear in $\sin\phi^*$ are measures of this triple vector product. These asymmetries are therefore odd under naive time reversal operation T$_{\rm N}$, which is simply reversal of all momentum and spin directions. Hence these asymmetries would be either proportional to the T-odd parameter $\tilde b_W$, or proportional to the T-even coupling $b_W$, but to satisfy unitarity and the CPT theorem, proportional only to its imaginary part. This will be seen in the numerical expressions or asymmetries which follow later on.
The above results assume that the quark and antiquark directions can be identified unambiguously. This is not true in the case of the LHC, where the quark could arise from either proton, and the choice of the $z$ axis is not unique. Taking into account the two possibilities when the quark (and antiquark) arise from the two oppositely directed proton beams, we find that the density matrix elements $\sigma(\pm,0)$ and $\sigma(0,\pm)$ vanish, as also the polarizations $P_x$, $P_y$, $P_{xz}$, $P_{yz}$ and the corresponding asymmetries $A_x$, $A_y$, $A_{xz}$, $A_{yz}$.
In what follows we will take the $z$ axis to be defined by the direction of the reconstructed momentum of the combination $WH$. In this case, the density matrix elements, polarizations and asymmetries which were vanishing when the $z$ was chosen to be the beam direction now turn out to be nonzero.
[**4. Numerical results**]{}\
To start with, we have evaluated the production spin density matrix elements after integrating over the parton distribution functions as well as the final-state phase space. We do not restrict ourselves to any particular decay mode of the Higgs, but assume that full identification is possible. In practice, one would have to apply kinematic cuts for lepton identification, elimination of backgrounds, etc., as also take into account the Higgs detection efficiency, which will require a more refined analysis.
We use the MMHT2014 parton distributions [@MMHT] with factorization scale chosen as the square root of the partonic c.m. energy. For the two cases of $W^+$ and $W^-$ production, though the partonic level cross sections and density matrices have the same expressions, the parton densities corresponding to the initial states are different. Hence the numerical results are different.
As mentioned before, we choose as $z$ axis the direction of the combined momenta of $W$ and $H$.
The results for the density matrices for $W^+$ production and $W^-$ production are shown respectively in Table \[sigmaijplus\] and Table \[sigmaijminus\].
--------------------------------------------------------------------------------------
SM Re $b_W$ Im $b_W$ Re $\tilde b_W$ Im $\tilde b_W$
------------------- ------- ---------- ----------- ----------------- -----------------
$\sigma(\pm,\pm)$ 165.8 $-1757$ 0 0 $\mp 1273$
$\sigma(0,0)$ 388.7 $-1757$ 0 0 0
$\sigma(\pm,\mp)$ 82.91 $-878.6$ 0 $\pm i 636.8$ 0
$\sigma(\pm,0)$ 95.96 $-872.8$ $-i431.7$ $\pm i 518.9$ $\mp
518.9$
$\sigma(0,\pm)$ 95.96 $-872.8$ $i431.7$ $\mp i 518.9$ $\mp
518.9$
--------------------------------------------------------------------------------------
: \[sigmaijplus\]Production spin density matrix elements for the $W^+$ (in units of fb) for the SM and the coefficients of various couplings in each matrix element
SM Re $b_W$ Im $b_W$ Re $\tilde b_W$ Im $\tilde b_W$
------------------- ------- ---------- ----------- ----------------- -----------------
$\sigma(\pm,\pm)$ 110.2 $-1140$ 0 0 $\mp 817.1$
$\sigma(0,0)$ 251.5 $-1140$ 0 0 0
$\sigma(\pm,\mp)$ 55.10 $-570.0$ 0 $\pm i 408.5$ 0
$\sigma(\pm,0)$ 49.86 $-439.6$ $-i209.9$ $\pm i 255.0$ $\mp 255.0$
$\sigma(0,\pm)$ 49.86 $-439.6$ $i209.9$ $\mp i 255.0$ $\mp 255.0$
: \[sigmaijminus\]Production spin density matrix elements for the $W^-$ (in units of fb) for the SM and the coefficients of various couplings in each matrix element
The total cross section for $W^+$ production has the expression \[numcsplus\] = (720.2 - 5271 [Re]{} b\_W) [fb]{}. and that for $W^-$ production the expression \[numcsminus\] = (471.8 - 3420 [Re]{} b\_W) [fb]{}. The total cross section for $W^+$ production could put a limit on Re $b_W$ of $2.28\times 10^{-4}$ with an integrated luminosity $L=500\,{\rm fb}^{-1}$, and of $1.61\times 10^{-4}$ with $L=1000\,{\rm
fb}^{-1}$. The corresponding limits using cross section for $W^-$ production are $2.84\times 10^{-4}$ and $2.01\times 10^{-4}$. Measurement of the cross section using only electron and muon decay modes of the $W^+$ assuming branching ratios of 10.71% and 10.63% respectively, we can therefore set a limit of $4.93\times 10^{-4}$ on the coupling Re $b_W$ for $L=500$ fb$^{-1}$, and $3.49\times 10^{-4}$ for $L=1000$ fb$^{-1}$. The corresponding numbers for $W^-$ are respectively $6.15\times
10^{-4}$ and $4.35\times 10^{-4}$.
The leptonic asymmetries corresponding to the different polarizations in $W^+$ production and decay, in an obvious notation, are given by A\_x = -0.282 + 0.502 [Re]{}b\_W A\_y = 1.52 [Re]{}b\_W A\_z = 2.60 [Im]{}b\_W A\_[xy]{} = -0.563 [Re]{}b\_W A\_[xz]{} = 0.649[Im]{}b\_W A\_[yz]{} = 0.540[Im]{}b\_W A\_[x\^2 -y\^2]{} = 0.0733 - 0.240 [Re]{}b\_W A\_[zz]{} = -0.116 - 0.849 [Re]{}b\_W The corresponding asymmetries in $W^-$ production and decay are A\_x = -0.224 + 0.351 [Re]{}b\_W A\_y = 1.15 [Re]{}b\_W A\_z = 2.65 [Im]{}b\_W A\_[xy]{} = -0.551 [Re]{}b\_W A\_[xz]{} = 0.487[Im]{}b\_W A\_[yz]{} = 0.401[Im]{}b\_W A\_[x\^2 -y\^2]{} = 0.0744 - 0.230 [Re]{}b\_W A\_[zz]{} = -0.112 - 0.814 [Re]{}b\_W
As remarked earlier, the reconstruction of the $W$ rest frame in which the above asymmetries are defined usually requires constraining the $\ell\nu$ invariant mass to be equal to the $W$ mass. We have checked that if we do not use this restriction and allow an off-shell $W$ to produce the $\ell\nu$ pair, the asymmetries do not change by more than a few per cent in most cases. Thus, the usual algorithms for constructing the $W$ rest frame would work with good accuracy.
In order to evaluate the 1-$\sigma$ limit $C_{\rm limit}$ on a coupling $C$ which can be obtained from the asymmetries, assuming one coupling to be nonzero at a time, and an integrated luminosity $L$, we use the expression \[sens\] C\_[limit]{} = , where $A$ is the asymmetry for unit value of the coupling $C$. For $W^+$ production, for integrated luminosities of 500 fb$^{-1}$ and 1000 fb$^{-1}$, we obtain the limits shown in Table \[limitswplus\].
--------------- ----------------- ------------------------- --------------------------
Asymmetry Coupling Limit (in $10^{-3}$) Limit (in $10^{-3}$)
($L=500~{\rm fb}^{-1}$) ($L=1000~{\rm fb}^{-1}$)
$A_x$ Re $b_W$ 6.9 4.9
$A_y$ Re $\tilde b_W$ 2.4 1.7
$A_z$ Im $\tilde b_W$ 1.4 0.96
$A_{xy}$ Re $\tilde b_W$ 6.4 4.5
$A_{xz}$ Im $\tilde b_W$ 5.6 3.9
$A_{yz}$ Im $b_W$ 6.7 4.7
$A_{x^2-y^2}$ Re $b_W$ 15 11
$A_{zz}$ Re $b_W$ 4.2 3.0
--------------- ----------------- ------------------------- --------------------------
: \[limitswplus\]1-$\sigma$ limits which could be obtained from various leptonic asymmetries in $W^+$ production and decay, with integrated luminosities of 500 and 1000 fb$^{-1}$.
The corresponding limits from $W^-$ production and decay are shown in Table \[limitswminus\].
--------------- ----------------- ------------------------- --------------------------
Asymmetry Coupling Limit (in $10^{-3}$) Limit (in $10^{-3}$)
($L=500~{\rm fb}^{-1}$) ($L=1000~{\rm fb}^{-1}$)
$A_x$ Re $b_W$ 12 8.7
$A_y$ Re $\tilde b_W$ 3.9 2.7
$A_z$ Im $\tilde b_W$ 1.7 1.2
$A_{xy}$ Re $\tilde b_W$ 8.1 5.7
$A_{xz}$ Im $\tilde b_W$ 9.2 6.5
$A_{yz}$ Im $b_W$ 11 7.9
$A_{x^2-y^2}$ Re $b_W$ 19 14
$A_{zz}$ Re $b_W$ 5.4 3.9
--------------- ----------------- ------------------------- --------------------------
: \[limitswminus\]1-$\sigma$ limits which could be obtained from various leptonic asymmetries in $W^-$ production and decay, with integrated luminosities of 500 and 1000 fb$^{-1}$.
The cross sections give the best limits on Re $b_W$. The results on the limits from leptonic asymmetries show that the asymmetries which are the most sensitive ones are $A_{zz}$ for Re $b_W$, $A_{yz}$ (the only one) for Im $b_W$, $A_y$ for Re $\tilde b_W$ and $A_z$ for Im $\tilde b_W$. The limits from $W^+H$ production are better than those from $W^-H$ production in all cases. However, it would be advantageous to combine results from both final states to improve the results.
[**5. Conclusions**]{}\
It is important to obtain complete information about the Higgs boson discovered at the LHC, including the tensor form of the couplings. A proposal to measure form and magnitude of the coupling of the Higgs boson to a pair of $W$ bosons through the polarization data of the $W$ is investigated here. The polarization density matrix elements of the $W$ can be measured through certain angular asymmetries of the charged lepton produced in $W$ decay, and we have studied the sensitivity of these asymmetries to the anomalous couplings $b_W$ and $\tilde b_W$ defined in eqn. (\[WWH\]). Our results for $W^+$ and $W^-$ are shown in tables \[limitswplus\] and \[limitswminus\].
We see that a high degree of accuracy could be obtained in the measurement of the $WWH$ anomalous couplings from the measurement of the $W$ polarization parameters through suitable angular asymmetries of leptons assuming an integrated luminosity of 500 fb$^{-1}$. There is considerable improvement, as expected, if the luminosity is increased to 1000 fb$^{-1}$. The 1-$\sigma$ limits in most cases are of the order of a few times $10^{-3}$.
As mentioned earlier, the angular asymmetries we discuss are defined in the rest frame of the $W$. The reconstruction of the $W$ rest frame in the presence of the undetected neutrino has its drawbacks, and would entail some loss in efficiency. We have also not taken into account acceptance and isolation cuts on leptons. We also assume 100% efficiency for the detection of the Higgs. To get some idea of the effect of cuts, we did evaluate the angular asymmetries and the sensitivities in the presence of generic LHC acceptance cuts on the transverse momentum and the rapidity of the leptons. We found that the asymmetries do not change much. A full-scale analysis using an event generator coupled with all appropriate cuts relevant to the decay channels of the Higgs would be able to refine the actual sensitivities that we have obtained. It would also be profitable to combine the results from $W^+$ and $W^-$ production processes, which would improve the accuracy.
.2cm [**Acknowledgement**]{}: We thank Pankaj Sharma for collaboration in the initial stages of the work. KR acknowledges support from IIT Bombay, grant no. 12 IRCCSG032. SDR acknowledges support from the Department of Science and Technology, India, under the J.C. Bose National Fellowship programme, Grant No. SR/SB/JCB-42/2009. We thank Rohini Godbole for discussions. We thank the referee for improvements and the suggestion for the choice of $z$ axis.
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|
---
author:
- 'J.Štěpán'
bibliography:
- 'bibs.bib'
title: |
Polarization diagnostics of proton beams\
in solar flares
---
Introduction
============
There is an observational evidence for the fast electron beams () and also for the fast proton beams ($\sim 10$ MeV) bombarding the solar chromosphere during the impulsive and gradual phase of solar flares [@korchak67; @orrall76]. This bombardment is consistent with the “standard model” of solar flare which assumes an injection of the high energetic particles from a coronal reconnection site to the chromosphere. It results in heating and nonthermal excitation of the chromospheric gas. The standard semiempirical models of the flaring chromosphere [@machado80] are based on the series of continuum and lines observations and do not take into account a nonthermal excitation; they rather overestimates the chromospheric temperature to explain an increased radiative emission.
Also the low energy proton beams (below 1 MeV) could play a significant role in the flare physics but their presence in the chromospheric layers is still uncertain due to their negligible bremsstrahlung radiation. There are however different ways how to diagnose them. For instance:
1. Ly$\alpha$ red wing emission due to charge exchange effect [@cc85; @zhao98].
2. Modification of the lines intensity profiles due to nonthermal excitation [@henoux93; @xu05b].
3. Detection of a linear polarization of the lines due to an anisotropic collisional excitation (i.e. impact atomic polarization).
Moreover, several unanswered questions still remain about the electron beams. Especially the physics of the return currents creation and their influence on lines formation.
Measurement of linear polarization in solar flares is a complicated task due to the high spatial and time gradients. Lot of observations report the polarization degree of H$\alpha$ of the order of few percent [@henoux90b; @vogt99; @hanaoka03; @xu05a]. Direction of this polarization is usually found to be either radial or tangential with respect to the limb and its degree is up to 5% or even exceeds 10% in the few cases. The standard interpretation of these measurements is anisotropic excitation by proton and/or electron beams. By contrast, no measurable linear polarization above the noise level at 0.1% was found in the set of many different flares by another authors [@bianda03; @bianda05].
Our objective is to find out if the measurements of a linear polarization degree of the hydrogen H$\alpha$ line are a promising tool for the diagnostics of the low-energy proton beams. Moreover, we present a first quantitative estimation of the polarizing effect of the return electric currents taking properly into account depolarizing collisions and radiative transfer.
We start with the standard semiempirical model of the flaring atmosphere F1 [@machado80] with a fixed temperature structure. We include the nonthermal collisional rates of excitation and ionization to obtain the new line profiles. We use a stationary modeling of the polarized radiation transfer in the chromospheric hydrogen lines using a multilevel NLTE polarized transfer code.
H$\alpha$ impact polarization
=============================
Let us assume a beam of the unidirectional charged colliders and let their energy be high enough to excite the $n=3$ level (i.e. higher than 12.1 eV). In fact, the $n=3$ level is composed of five quasi-degenerated fine structure levels. Some of them can be polarized, hence the photons emitted in the H$\alpha$ (and Ly$\beta$) line are in general polarized. An anisotropical excitation of these levels by either protons or electrons is one of the processes which can lead to polarization of these levels. The orientation of electric vector of this radiation can be either parallel or perpendicular to the velocity of the beam (cf. Figure \[fig:turnover\]). A beam is expected to follow the direction of magnetic field which is assumed to be vertical in the atmosphere; hence the linear polarization is either radial or tangential with respect to the solar limb.
![Linear polarization of the photons emitted in the H$\alpha$ line perpendicularly to the collisional trajectory (i.e. in the direction of a maximum polarization degree). For both the electrons and protons impacts, low energies of collisions lead to linear polarization of the emitted photons in the direction parallel to the collisional trajectory. A polarization direction above the so-called turnover energy is perpendicular to it. []{data-label="fig:turnover"}](Stepanf1){width="\columnwidth"}
The assumptions above should lead to an upper theoretical limit for the H$\alpha$ linear polarization. The coordinate system definition is given in Figure \[fig:refframe\]. In such a reference frame, only the Stokes parameters $I$ and $Q$ are in general nonzero.
![We use a right-handed coordinate system with $Y$-axis in the collisional plane and $Z$-axis in the line of sight direction. Due to cylindrical symmetry of the model, only Stokes parameters $I$ and $Q$ do not generally vanish in this coordinate system.[]{data-label="fig:refframe"}](Stepanf2){width="\columnwidth"}
We are interested especially in a modeling of the radial polarization. It follows from Figure \[fig:turnover\] that this orientation is associated with the low-energy electrons or the low-energy protons moving vertically in the atmosphere.[^1]
Formalism and methods
=====================
A vertical magnetic field is assumed to be of the strength of few hundreds gauss [@vogt97]. It can be shown that in this regime and in the plasma conditions under consideration (see below) fine structure splitting of the hydrogen levels up to $n=3$ is a good approximation [@sahal96]. The hyperfine splitting of the levels can be neglected because it does not significantly affect a linear polarization degree [@bommier86a]. In addition, a lifetime of the fine structure levels is reduced by collisions with the background protons and electrons and thus the hyperfine levels of the excited states completely overlap. To describe an atomic state we adopt the formalism of the atomic density matrix in the basis of irreducible tensorial operators $\rho^K_Q(nlj)$ [@fano57], with the traditional meaning of the symbols. We neglect all the quantum coherences between different fine structure levels due to their large separation in comparison to their width and due to the selection rules for the optical transitions. A magnetic field is able to destroy all the quantum coherences between the Zeeman sublevels of any level. In the formalism of irreducible tensors it means that all the multipole components of $\rho$ with $Q\neq 0$ are identically zero, $\rho^K_Q(nlj)=\delta_{Q0}\rho^K_Q(nlj)$. On the other hand, the strength of a magnetic field is not so high to induce a Zeeman splitting large enough to lead to the complicated level-crossing effects. Finally, because of the assumption of absence of any circularly polarized radiation, the odd ranks $K$ of the density matrix are identically zero. All these assumptions lead to the simplifications of the formalism.
The results of our modeling are strongly dependent on the collisional cross-sections of the different excitation and ionization transitions. We consider the following collisional processes and cross-sections data:
- Excitation and charge exchange for a proton beam using the close-coupling cross-sections data of @balanca98.
- The dipolar transitions between the fine structure levels $nlj\to n l\pm 1 j'$ for interaction with the background electrons, protons, and the beam are calculated using the semiclassical theory of @sahal96.
- Excitation of the levels populations by the background electrons (the cross-sections data are taken from the AMDIS database, http://www-amdis.iaea.org).
For the purposes of impact polarization studies it is necessary to calculate all the collisional transition rates $$C^{K\to K'}_{nlj\to n'l'j'}=N_{\rm P}\int{\rm d}^3\vec{v}f(\vec{v})v
\sigma^{K\to K'}_{nlj\to n'l'j'}(\vec{v})\;,$$ for transitions $\rho^K_0(nlj)\to\rho^{K'}_0(n'l'j')$. These rates enter the equations of statistical equilibrium together with the radiative rates $R^{K\to K'}_{nlj\to n'l'j'}$ in the so-called impact approximation [@landi84; @bommier91]. The equations of statistical equilibrium have the form[^2] $$\sum_{nljK} (C^{K\to K'}_{nlj\to n'l'j'}+R^{K\to K'}_{nlj\to n'l'j'})
\rho^K_0(nlj)=0\;.
\label{eq:ese}$$
To take properly into account a coupling of the atomic states at the different atmospheric points, it is necessary to solve a system of equations (\[eq:ese\]) coupled by the radiative transfer equation for Stokes vector $\vec{I}=(I,Q)^{\rm T}$. Its evolution along the path $s$ is given by $$\frac{{\rm d}\vec{I}}{{\rm d}s}
=\vec{J}-\vec{K}\,\vec{I}\;.
\label{eq:rte}$$ In this equation, $\vec J$ is the emissivity vector and $\vec K$ is the so-called propagation matrix. We use our multigrid code [@stepan06spw] to solve this problem in the plan-parallel geometry, out of the LTE approximation (the case of the chromospheric H$\alpha$ line).
Results for the proton beams
============================
The temperature structure of the atmosphere is fixed and given by the F1 model. A new ionization degree and the atomic states along the chromosphere are obtained by solution of the system of the Eqs. (\[eq:ese\]) and (\[eq:rte\]).
![Energy distribution of a beam flux (in arbitrary units), $\delta=4$, $E_{\rm c}=150~{\rm keV}$. The initial sharp-peaked power-law energy distribution (upper curve) becomes flatter in the region where the slowest protons are stopped (lower curve). []{data-label="fig:distrib"}](Stepanf3){width="\columnwidth"}
The initial energy distribution of the proton beam at the top of the chromosphere is (as usually) expected to be given by the power-law $$F(E)\sim E^{-\delta}\;,\qquad E>E_{\rm c}\;,$$ with the lower energy cut-off $E_{\rm c}$ and the spectral index $\delta$. After crossing some column depth, the distribution is modified by collisions with various atmospheric species (see Figure \[fig:distrib\]), [@emslie78; @cc85]. The protons most efficient in creation of the $n=3$ level polarization have a small energy about $5~{\rm keV}$ [@balanca98]. A number of such protons in the region of the H$\alpha$ line core formation is very small compared to the number of protons of an energy about $E_{\rm c}$ at the injection site [cf. @vogt97; @vogt01]. As a result, there is only a small number of low-energy protons even for the high initial beam fluxes.
The electron (proton) densities of the chromosphere obtained as a result of the NLTE calculations are of the order of $10^{12}~{\rm cm^{-3}}$ in the regions of the H$\alpha$ centre and near wings formation (Figure \[fig:fioniz\]). These densities were computed using the code of @kasparova02 adapted for the proton beams studies.
![Electron density for the model F1. The plot corresponds to the case of $\delta=4$, $E_{\rm c}=150\;{\rm keV}$. Thermal case is plotted by a solid line, the non-thermal beam fluxes are $\mathcal{E}_0=10^8$ (dot), $10^9$ (dash), $10^{10}$ (dash-dot), and $10^{11}\,{\rm erg\,cm^{-2}\,s^{-1}}$ (dash-dots). []{data-label="fig:fioniz"}](Stepanf4){width="\columnwidth"}
It was shown by @bommier86b that depolarization of the hydrogen levels by collisions with the background electrons and protons becomes important even at the densities of the order of $10^{10}~{\rm cm^{-3}}$. It is therefore expected that the depolarization effect will play a crucial role in the chromospheric conditions.
![H$\alpha$ line intensity profiles for the same set of models as in the Figure \[fig:fioniz\]. []{data-label="fig:ihalpha"}](Stepanf5){width="\columnwidth"}
![Emergent fractional linear polarization $Q/I$ profiles (%) computed close to the limb ($\mu=0.11$) for the same set of models as in the Figure \[fig:fioniz\]. A positive sign of $Q/I$ means the tangential direction of polarization, while a negative sign is the radial one. []{data-label="fig:qihalpha"}](Stepanf6){width="\columnwidth"}
The theoretical line profiles calculated for a set of the proton beam energy fluxes are plotted in the Figures \[fig:ihalpha\] and \[fig:qihalpha\]. As it can be seen in Figure \[fig:qihalpha\], a polarization degree of the emergent H$\alpha$ line close to the limb is extremely small. Moreover, its orientation is tangential and not radial as it was expected for the low-energy proton impacts. This polarization is mainly due to resonance scattering of the radiation, while the impact polarization effect is not seen. The degree of 0.02% is well below any measurable value to day.
Effect of electronic return currents
====================================
It seems that linear polarization of the H$\alpha$ line due to slow proton beams is not a good diagnostics tool even if very restrictive conditions on the beam anisotropy are postulated. Another phenomena which could explain the observed radial polarization are the return currents (RC) associated with the electron beams. The detailed studies of the collective plasma processes show that the return currents which neutralize a huge electric current of the beam can be formed if several plasma conditions in the atmosphere are fulfilled [@norman78]. The atomic excitations caused by the return current were shown to be more important than the effect of the beam itself in some depths [@karlicky04].
@karlicky02 propose the anisotropical impacts of the slow (few ) RC as an explanation for the observed linear polarization of the H$\alpha$ line. However, all these calculations neglected the effects of depolarizing collisions and polarized radiative transfer. We have studied a simple model of monoenergetic electron beam with the initial energy flux of $1.2\times 10^{12}~{\rm erg\;cm^{-2}\;s^{-1}}$ composed of the electrons with the energy 10 keV penetrating the F1 atmosphere. The electron-hydrogen cross-sections for the dipolar transitions have been calculated by the semiclassical method with momentum transfer [@bommier05]. These calculations show that return current is locally able to produce a significant atomic polarization, but it is decreased by the depolarizing collisions and high intensity of radiation. If all these effects are taken into account together with radiation transfer, the emergent linear polarization degree in the centre of H$\alpha$ is only about 0.25% in the radial direction. That is still one order of magnitude below the measured values.
Conclusions
===========
A net impact polarization of an electron beam + RC seems to be a more promising explanation of the observed linear polarization than the proton beam impacts. However, collisional depolarization by background electrons and protons together with the increased radiation intensity can destroy most of the impact atomic polarization.
Several things have been simplified to make our models traceable:
1. The semiempirical model F1 overestimates the atmosphereric temperature. As a result, depolarization and lines intensities are also overestimated.
2. No time dependence was considered. Detailed time-dependent modeling could possibly explain some aspects of polarization generation.
3. The plane-parallel model has a limited applicability and more complex geometry could be considered to interpret the observations.
4. If a RC is carried by a substantial number of background electrons, they do not contribute to the collisional depolarization in the same way as the thermal electrons. That was not taken fully into account in our models.
5. The beams are never purely directional and vertical and their scattering may lead to further decrease of the impact polarization. The monoenergetic electron beam + RC is not a satisfactory physical model.
A quantitative estimation of all these effects should be a subject of the future studies.
[^1]: The case of horizontal motions at high energies is unlikely for the physical reasons.
[^2]: Additional terms have to be added to take into account the bound-free transitions. They are not expressed in the Eq. (\[eq:ese\]) for the reasons of simplicity.
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abstract: |
The solution of the continuous time filtering problem can be represented as a ratio of two expectations of certain functionals of the signal process that are parametrized by the observation path. We introduce a class of discretization schemes of these functionals of arbitrary order. The result generalizes the classical work of Picard, who introduced first order discretizations to the filtering functionals. For a given time interval partition, we construct discretization schemes with convergence rates that are proportional with the $m$-power of the mesh of the partition for arbitrary $m\in\mathbb{N}$. The result paves the way for constructing high order numerical approximation for the solution of the filtering problem.
**MSC 2010**: 60G35, 60F05, 60F25, 60H35, 60H07, 93E11.
**Key words**: Non-linear filtering, Kallianpur-Striebel’s formula, high order time discretization.
author:
- 'D. Crisan[^1] [^2] and S. Ortiz-Latorre[^3] [^4]'
title: 'A high order time discretization of the solution of the non-linear filtering problem'
---
Introduction
============
Partially observed dynamical systems are ubiquitous in a multitude of real-life phenomena. The dynamical system is typically modelled by a continuous time stochastic process called the signal process $X$. The signal process cannot be measured directly, but only via a related process $Y$, called the observation process. The filtering problem is that of estimating the current state of the dynamical system at the current time given the observation data accumulated up to that time. Mathematically the problem entails computing the conditional distribution of the signal process $X_{t}$, denoted by $\pi_{t}$, given $\mathcal{Y}_{t},$ the $\sigma$-algebra generated by $Y$. In a few special cases, $\pi_{t}$ can be expressed in closed form as a functional of the observation path. For example, the celebrated Kalman-Bucy filter does this in the linear case. In general, an explicit formula for $\pi_{t}$ is not available and inferences can only be made by numerical approximations of $\pi_{t}$. As expected the problem has attracted a lot of attention in the last fifty years (see Chapter 8 of [@BaCr08] for a survey of existing numerical methods for approximating $\pi_{t}$.
The basis of this class of numerical methods is the representation of $\pi_{t}$ given by the KallianpurStriebel formula (see (\[eq: Kallianpur-Striebel\]) below). In the case when the signal process is modelled by the solution of a stochastic differential equation (SDE) and the observation process is a function of the signal perturbed by white noise (see Section \[sec: Main Result\] below for further details), the formula entails the computation of expectations of functionals of the solution of the signal SDE that are parametrized by the observation path. The numerical approximation of $\pi_{t}$ requires three procedures:
$\bullet$ the discretization of the functionals (corresponding to a partition of the interval $\left[0,t\right]$).
$\bullet$ the approximation of the law of the signal with a discrete measure.
$\bullet$ the control of the computational effort.
The first step is typically achieved by the discretization scheme introduced by Picard in [@Pi84]. This offers a first order approximation for the functionals appearing in formula (\[eq: Kallianpur-Striebel\]). More precisely, the $L^{1}$-rate of convergence of the approximation is proportional with the mesh of the partition of the time interval $\left[0,t\right]$ (see Theorem 21.5 in [@Cris11]). The second and the third step are achieved by a combination of an Euler approximation of the solution of the SDE, a Monte Carlo step that gives a sample from the law of the Euler approximation and a re-sampling step that acts as a variance reduction method and keeps the computational effort in control. There are a variety of algorithms that follow this template. Further details can be found, for instance, in Part VII of [@CrRo11]. It is worth pointing out that once the functional discretization and the Euler approximation have been applied, the problem can be reduced to one where the signal evolves and is observed in discrete time. The discrete version of the filtering problem is popular both with practitioners and with theoreticians. The majority of the existing theoretical results and the numerical algorithms are constructed and analyzed in the discrete framework. For more details, the interested reader can consult the comprehensive theoretical monograph [@Delm04] and the reference therein and the equally comprehensive methodological volume [@DFG01] and the references therein with some updates in Part VII of [@CrRo11].
The first order discretization introduced by Picard creates a bottleneck: There exist higher order schemes for approximating the law of the signal that can be used, but which won’t bring any substantial improvements because of this. For example, in the recent paper [@CrOr2013], the authors employ high order cubature methods to approximate the law of the signal with only minimal improvements due to the low order discretization of the required functionals. The aim of this paper is to address this issue. More precisely, we introduce a class of high order discretizations of the functionals. As we shall see, we prove that the $L^{1}$-rate of convergence of the approximations is proportional with the $m$-power of the mesh of the partition of the time interval $\left[0,t\right]$. For details, see Theorem \[thm: Main Filtering\_2\] below. In a work in progress, this discretization procedure is employed to produce a second order particle filter. It is hoped that this discretization will be used in conjunction with other high order approximations of the law of the signal, in particular with cubature methods. We are not aware of any other similar high order discretization schemes.
The paper is organized as follows: In Section \[sec: Main Result\] we introduce some basic definitions and state the main result of the paper, Theorem \[thm: Main Filtering\_2\]. Section \[sec: ProofMainResult\] is devoted to prove our main result. We start by proving several auxiliary results on iterated stochastic integrals and on the integrability of the likelihood function and its discretizations. These lead to the two main results of the section, Proposition \[prop: Main1\] and Proposition \[prop: Main2\], from which we will deduce our main result. In Section \[sec:Technical-Lemmas\] we address the most technical aspects of the paper. We first introduce some technical tools on Malliavin calculus (subsection \[subsec:Malliavin-calculus\]), the Stroock-Taylor formula (subsection \[subsec:MartingaleRep\]) and backward martingales (subsection \[subsec:BackwardMart\]). Then, with the aid of the these tools, we prove in subsection \[subsec:CondExpEst\] the estimates on the conditional expectation with respect to $\mathcal{Y}_{t}$ that are essential in proving Proposition \[prop: Main1\].
Basic framework and statement of the main result\[sec: Main Result\]
====================================================================
Let $(\Omega,\mathcal{F},P)$ be a probability space together with a filtration $(\mathcal{F}_{t})_{t\geq0}$ which satisfies the usual conditions. On $(\Omega,\mathcal{F},P)$ we consider a $d_{X}\times d_{Y}$-dimensional partially observed system $(X,Y)$ satisfying $$\begin{aligned}
X_{t} & =X_{0}+\int_{0}^{t}f(X_{s})ds+\int_{0}^{t}\sigma(X_{s})dV_{s},\\
Y_{t} & =\int_{0}^{t}h(X_{s})ds+W_{t},\end{aligned}$$ where $V$ is a standard $\mathcal{F}_{t}$-adapted $d_{V}$-dimensional Brownian motion and and $W$ is a a standard $\mathcal{F}_{t}$-adapted $d_{Y}$-dimensional Brownian motion, independent of each other. We also assume that $X_{0}$ is a random variable independent of $V$ and $W$ and denote by $\pi_{0}$ its law. We assume that $f=(f_{i})_{i=1,...,d_{X}}:\mathbb{R}^{d_{X}}\rightarrow\mathbb{R}^{d_{X}}$ and $\sigma=(\sigma_{i,j})_{i=1,...d_{X},j=1,...,d_{V}}:\mathbb{R}^{d_{X}}\rightarrow\mathbb{R}^{d_{X}\times d_{V}}$ are globally Lipschitz continuous. In addition, we assume that $h=\left(h_{i}\right)_{i=1,...,d_{Y}}:\mathbb{R}^{d_{X}}\rightarrow\mathbb{R}^{d_{Y}}$ is measurable and has linear growth.
Let $\mathbb{Y}=\{\mathcal{Y}_{t}\}_{t\geq0}$ be the usual augmentation of the filtration generated by the process $Y,$ that is, $\mathcal{Y}_{t}=\sigma\left(Y_{s},s\in\left[0,t\right]\right)\vee\mathcal{N},$ where $\mathcal{N}$ are all the $P$-null sets of $(\Omega,\mathcal{F},P)$. We are interested in determining $\pi_{t},$ the conditional law of the signal $X$ at time $t$ given the information accumulated from observing $Y$ in the interval $[0,t].$ More precisely, for any Borel measurable and bounded function $\varphi,$ we want to compute $\pi_{t}\left(\varphi\right)=\mathbb{E}[\varphi\left(X_{t}\right)|\mathcal{Y}_{t}].$ By an application of Girsanov’s theorem (see, for example, Chapter 3 in [@BaCr08]) one can construct a new probability measure $\tilde{P}$, absolutely continuous with respect to $P$, under which $Y$ becomes a Brownian motion independent of $X$ and the law of $X$ remains unchanged. The Radon-Nikodym derivative of $\tilde{P}$ with respect to $P$ is given by the process $Z(X,Y)=\left(Z_{t}(X,Y)\right)_{t\geq0}$ given by $$Z_{t}(X,Y)=\exp\left(\sum_{i=1}^{d_{Y}}\int_{0}^{t}h_{i}(X_{s})dY_{s}^{i}-\frac{1}{2}\sum_{i=1}^{d_{Y}}\int_{0}^{t}h_{i}^{2}\left(X_{s}\right)ds\right),\quad t\geq0,\label{eq: Likelihood}$$ which is an $\mathcal{F}_{t}$-adapted martingale under $\tilde{P}$ under the assumptions introduced above. We will denote by $\mathbb{\tilde{E}}$ to be the expectation with respect to $\tilde{P}$. In the following we will make use of the measure valued process $\rho=\left(\rho_{t}\right)_{t\geq0},$ defined by the formula $\rho_{t}\left(\varphi\right)=\mathbb{\tilde{E}}\left[\varphi\left(X_{t}\right)Z_{t}|\mathcal{Y}_{t}\right],$ for any bounded Borel measurable function $\varphi$. The processes $\pi$ and $\rho$ are connected through the Kallianpur-Striebel’s formula: $$\begin{aligned}
\pi_{t}\left(\varphi\right)=\frac{\rho_{t}\left(\varphi\right)}{\rho_{t}\left(\boldsymbol{1}\right)} & =\frac{\mathbb{\tilde{E}}\left[\varphi\left(X_{t}\right)\exp\left.\left(\sum_{i=1}^{d_{Y}}\int_{0}^{t}h_{i}(X_{s})dY_{s}^{i}-\frac{1}{2}\sum_{i=1}^{d_{Y}}\int_{0}^{t}h_{i}^{2}\left(X_{s}\right)ds\right)\right\vert \mathcal{Y}_{t}\right]}{\mathbb{\tilde{E}}\left[\exp\left.\left(\sum_{i=1}^{d_{Y}}\int_{0}^{t}h_{i}(X_{s})dY_{s}^{i}-\frac{1}{2}\sum_{i=1}^{d_{Y}}\int_{0}^{t}h_{i}^{2}\left(X_{s}\right)ds\right)\right\vert \mathcal{Y}_{t}\right]},\label{eq: Kallianpur-Striebel}\end{aligned}$$ $P$-a.s., where $\boldsymbol{1}$ is the constant function. As a result, $\rho$ is called the unnormalized conditional distribution of the signal. For further details on the filtering framework, see [@BaCr08].
It follows from (\[eq: Kallianpur-Striebel\]) that $\pi_{t}\left(\varphi\right)$ is a ratio of two conditional expectations of functionals of the signal that depend on the stochastic integrals with respect to the process $Y.$ In the following we will introduce a class of time discretization schemes for these conditional expectations which, in turn, will generate time discretisation schemes $\pi_{t}$ (of any order). This is the main result of the paper and is stated Theorem \[thm: Main Filtering\_2\] below.
We first introduce some useful notation and definitions. We denote by:
- $\mathcal{B}_{b}$ the space of bounded Borel-measurable functions.
- $\mathcal{B}_{P}$ the space of Borel-measurable functions with polynomial growth.
- $C_{b}^{k}$ the space of continuously differentiable functions up to order $k\in\mathbb{Z}_{+}$ with bounded derivatives of order greater or equal to one.
- $C_{P}^{k}$ the space of continuously differentiable functions up to order $k\in\mathbb{Z}_{+}$ such that the function and its derivatives have at most polynomial growth.
- $L^{p}(\Omega,\mathcal{F},\tilde{P})$ the space of $p$-integrable random variables (with respect to $\tilde{P}$) and denote by $\left\vert \left\vert \cdot\right\vert \right\vert _{p}$ the corresponding norm on $L^{p}(\Omega,\mathcal{F},\tilde{P})$, i.e., for $\xi\in L^{p}(\Omega,\mathcal{F},\tilde{P})$, $\left\vert \left\vert \xi\right\vert \right\vert _{p}\triangleq\mathbb{\tilde{E}}[\left\vert \xi\right\vert ^{p}]^{1/p}$.
In the following, we will use the notation introduced in Section 5.4 in Kloeden and Platen [@KlPl92]. More precisely, let $S$ be a subset of $\mathbb{Z}_{+}$ and denote by $\mathcal{M}^{\ast}(S)$ the set of all multi-indices with values in $S.$ In addition, define $\mathcal{M}(S)\triangleq\mathcal{M}^{\ast}(S)\cup\{v\},$ where $v$ denotes the multi-index of lenght zero . For $\alpha=(\alpha_{1},...,\alpha_{k})\in\mathcal{M}(S)$ define the following operations $$\begin{aligned}
\left\vert \alpha\right\vert & \triangleq k,\\
\left\vert \alpha\right\vert _{0} & \triangleq\#\{j:\alpha_{j}=0,j=1...,k\},\\
\alpha- & \triangleq(\alpha_{1},...,\alpha_{k-1}),\\
-\alpha & \triangleq(\alpha_{2},...,\alpha_{k}),\end{aligned}$$ where $\left|v\right|=0,-v=v-=v$. Given two multi-indices $\alpha,\beta\in\mathcal{M}(S)$ we denote its concatenation by $\alpha\ast\beta$ . Itô-Taylor expansions are usually done with a particular subsets of multi-indices, the so called hierarchical sets. We call a subset $\mathcal{A}\subset\mathcal{M\left(S\right)}$ a hierarchical set if $\mathcal{A}$ is nonempty, $\sup_{\alpha\in\mathcal{A}}\left|\alpha\right|<\infty,$ and $$-\alpha\in\mathcal{A\mbox{ \quad if}\quad}\alpha\in\mathcal{A}\setminus\{v\}.$$ For any given hierarchichal set $\mathcal{A}$ we define the remainder set $\mathcal{R\left(A\right)}$ of $\mathcal{A}$ by $$\mathcal{R\left(A\right)}\triangleq\left\{ \alpha\in\mathcal{M\left(S\right)}\setminus\mathcal{A}:-\alpha\in\mathcal{A}\right\} .$$ We will consider the hierarchical set $\mathcal{M}_{m}(S)$ and its associated remainder set $\mathcal{R}\left(\mathcal{M}_{m}(S)\right),$ that is, $$\mathcal{M}_{m}(S)\triangleq\{\alpha\in\mathcal{M}(S):\left\vert \alpha\right\vert \leq m\},$$ and $$\mathcal{R}\left(\mathcal{M}_{m}(S)\right)\triangleq\{\alpha\in\mathcal{M}(S):\left\vert \alpha\right\vert =m+1\}.$$ Observe that $\mathcal{R}\left(\mathcal{M}_{m}(S)\right)=\mathcal{M}_{m+1}(S)\setminus\mathcal{M}_{m}(S)$. We shall use the sets of multi-indices with values in the sets $S_{0}=\{0,1,...,d_{V}\}$ and $S_{1}=\{1,...,d_{V}\}$. Note also that the set $\mathcal{R}\left(\mathcal{M}_{m}(S_{0})\right)$ can be partioned in the following way $$\mathcal{R}\left(\mathcal{M}_{m}(S_{0})\right)={\displaystyle \biguplus\limits _{k=0}^{m+1}}\mathcal{R}\left(\mathcal{M}_{m}(S_{0})\right)_{k},$$ where $\mathcal{R}\left(\mathcal{M}_{m}(S_{0})\right)_{k}=\{\alpha\in\mathcal{R}\left(\mathcal{M}_{m}(S_{0})\right):\left\vert \alpha\right\vert _{0}=k\},k=0,...,m+1,$ that is, $\mathcal{R}\left(\mathcal{M}_{m}(S_{0})\right)_{k}$ is the set of multi-indices of lenght $m+1$ with values in $S_{0}$ which contains $k$ zeros.
To simplify the notation, it is convenient to add an additional component to the Brownian motion $V.$ Let $V_{s}^{0}=s,$ for all $s\geq0$ and consider the $(d_{V}+1)$-dimensional process $V=(V^{i})_{i=0}^{d_{V}}.$ We will consider the filtration $\mathbb{F}^{0,V}=\{\mathcal{F}_{s}^{0,V}\}_{s\geq0}$ defined to be the usual augmentation of the filtration generated by the process $V$ and initially enlarged with the random variable $X_{0}.$ Moreover, for fixed $t$, we will also consider the filtration $\mathbb{H}^{t}=\{\mathcal{H}_{s}^{t}\triangleq\mathcal{F}_{s}^{0,V}\lor\mathcal{Y}_{t}\}_{s\geq0}$. For $\alpha\in\mathcal{M}(S_{0}),$ denote by $I_{\alpha}(h_{\cdot})_{s,t}$ the following Itô iterated integral $$I_{\alpha}(h_{\cdot})_{s,t}=\left\{ \begin{array}{ccc}
h_{t} & \text{if} & \alpha=v\\
\int_{s}^{t}I_{\alpha-}(h_{\cdot})_{s,u}dV_{u}^{\alpha_{|\alpha|}} & \text{if} & \left\vert \alpha\right\vert \geq1
\end{array}\right.,$$ where $h=\{h_{s}\}_{s\geq0}$ is an $\mathbb{H}^{t}$-adapted process (satisfying appropriate integrability conditions). We introduce the differential operators $L^{0},L^{r},r=1,...,d_{V}$ defined by $$\begin{aligned}
L^{0}g(x) & \triangleq\sum_{k=1}^{d_{X}}f^{k}(x)\frac{\partial g}{\partial x^{k}}(x)+\frac{1}{2}\sum_{k,l=1}^{d_{X}}\sum_{r=1}^{d_{V}}\sigma_{k,r}(x)\sigma_{l,r}(x)\frac{\partial^{2}g}{\partial x^{k}\partial x^{l}}(x).\\
L^{r}g(x) & \triangleq\sum_{k=1}^{d_{X}}\sigma_{k,r}(x)\frac{\partial g}{\partial x^{k}}(x),\quad r=1,...,d_{V},\end{aligned}$$ where $g:\mathbb{R}^{d_{X}}\rightarrow\mathbb{R}$ belongs to $C_{P}^{2}\left(\mathbb{R}^{d_{X}};\mathbb{R}\right).$ For $\alpha\in\mathcal{M}(S_{0}),$ with $\alpha=(\alpha_{1},...,\alpha_{k}),$ and the differential operator $L^{\alpha}$ is defined by $$\begin{aligned}
L^{\alpha}g & =L^{\alpha_{1}}\circ L^{\alpha_{2}}\circ\cdots\circ L^{\alpha_{k}}g,\end{aligned}$$ and, by convention $L^{v}g=g$. Finally, let $\tau\triangleq\{0=t_{0}<\cdots<t_{i}<\cdots<t_{n}=t\}$ be a partition of $[0,t].$ Associated to $\tau$ we define the following elements $$\begin{aligned}
\delta_{i} & \triangleq t_{i}-t_{i-1},\quad i=1,...,n,\\
\delta & \triangleq\max_{i=1,...,n}\delta_{i},\\
\delta_{\mbox{min}} & \triangleq\min_{i=1,...,n}\delta_{i},\\
\tau(s) & \triangleq t_{i-1},\quad s\in[t_{i-1},t_{i}),i=1,...,n,\\
\eta(s) & \triangleq t_{i},\quad s\in[t_{i-1},t_{i}),i=1,...,n.\end{aligned}$$ We will only consider partitions satisfying the following condition $$\delta\leq C\delta_{\mbox{min}},\label{eq: UnifPartition}$$ for some finite constant $C\geq1$. We denote by $\Pi(t)$ the set of all partitions of $[0,t]$ satisfying $\left(\ref{eq: UnifPartition}\right)$ and such that $\delta$ converges to zero when $n$ tends to infinity. We denote by $\Pi(t,\delta_{0})$ the set of all partitions of $[0,t]$ satisfying $\left(\ref{eq: UnifPartition}\right)$, such that $\delta$ converges to zero when $n$ tends to infinity and $\delta<\delta_{0}$.
\[rem: Uniform\]Under the assumption $\left(\ref{eq: UnifPartition}\right)$ one has that $$n\leq t\delta_{\mbox{min}}^{-1}\leq Ct\delta^{-1}.\label{eq: UnifPartition2}$$
To simplify the notation, we will add an additional component to the Brownian motion $Y.$ Let $Y^{0}$ be the process $Y_{s}^{0}=s,$ for all $s\geq0$ and consider the $(d_{Y}+1)$-dimensional process $Y=(Y^{i})_{i=0}^{d_{Y}}.$ Then the martingale $Z=\left(Z_{t}\right)_{t\geq0}$ defined in (\[eq: Likelihood\]) can be written as $Z_{t}=\exp\left(\xi_{t}\right),t\geq0,$ where $$\xi_{t}=\sum_{i=0}^{d_{Y}}\int_{0}^{t}h_{i}(X_{s})dY_{s}^{i},\quad t\geq0,$$ and $h_{0}=-\frac{1}{2}\sum_{i=1}^{d_{Y}}h_{i}^{2}.$ For $\tau\in\Pi(t)$ and $m\in\mathbb{N}$ we consider the processes $$\begin{aligned}
\xi_{t}^{\tau,m} & \triangleq\sum_{j=0}^{n-1}\xi_{t}^{\tau,m}\left(j\right)\triangleq\sum_{j=0}^{n-1}\sum_{i=0}^{d_{Y}}\sum_{\alpha\in\mathcal{M}_{m-1}(S_{0})}L^{\alpha}h_{i}(X_{t_{j}})\int_{t_{j}}^{t_{j+1}}I_{\alpha}(\boldsymbol{1})_{t_{j},s}dY_{s}^{i}\\
& =\sum_{i=0}^{d_{Y}}\int_{0}^{t}\left\{ \sum_{\alpha\in\mathcal{M}_{m-1}(S_{0})}L^{\alpha}h_{i}(X_{\tau(s)})I_{\alpha}(\boldsymbol{1})_{\tau(s),s}\right\} dY_{s}^{i}.\end{aligned}$$ For $m>2$, we can write
$$\xi_{t}^{\tau,m}=\xi_{t}^{\tau,2}+\sum_{j=0}^{n-1}\mu^{\tau,m}\left(j\right),$$ where $$\mu^{\tau,m}\left(j\right)\triangleq\sum_{i=0}^{d_{Y}}\sum_{\alpha\in\mathcal{M}_{1,m-1}(S_{0})}L^{\alpha}h_{i}(X_{t_{j}})\int_{t_{j}}^{t_{j+1}}I_{\alpha}(\boldsymbol{1})_{t_{j},s}dY_{s}^{i},$$ and $$\begin{aligned}
\mathcal{M}_{1,m-1}(S_{0}) & \triangleq\mathcal{M}_{m-1}(S_{0})\setminus\mathcal{M}_{1}(S_{0})\\
& =\left\{ \alpha:\left\vert \alpha\right\vert \in\left[2,m-1\right],\alpha_{k}\in\{0,...,d_{V}\},k=1,...,\left\vert \alpha\right\vert \right\} .\end{aligned}$$ The processes $\xi^{\tau,m}$ are obtained by replacing $h_{i}(X_{s})$ in the formula for the process $\xi$ with the truncation of degree $(m-1)$ of the corresponding stochastic Taylor expansion of $h_{i}(X_{s})$. They are used to produce discretization schemes of order 1 and 2 for $\pi_{t}(\varphi)$. They *cannot* be used to produce discretization schemes of order $m>2$ as they don’t have finite exponential moments (required to define the discretization schemes). More precisely, the quantities $\mu^{\tau,m}\left(j\right)$ do not have finite exponential moments because of the high order iterated integral involved. For this, we need to introduce a truncation of $\mu^{\tau,m}\left(j\right)$ resulting in a (partial) taming procedure to the stochastic Taylor expansion of $h_{i}(X)$. We define the processes $$\bar{\xi}_{t}^{\tau,m}\triangleq\sum_{j=0}^{n-1}\bar{\xi}_{t}^{\tau,m}\left(j\right),$$ where $$\bar{\xi}_{t}^{\tau,i}\left(j\right)=\left\{ \begin{array}{ll}
\xi_{t}^{\tau,i}\left(j\right) & \mathrm{if}\ \ i=1,2\\
\xi_{t}^{\tau,2}\left(j\right)+\Gamma_{m-\frac{1}{2},\delta_{j}}\left(\mu^{\tau,m}\left(j\right)\right) & \mathrm{if}\ \ i>2
\end{array}\right.,~~j=0,...,n-1$$ with the truncation function $\Gamma$ being defined as $$\Gamma_{q,\delta}\left(z\right)=\frac{z}{1+\left(z/\delta\right)^{2q}},\qquad z\in\mathbb{R}$$ for some $\delta>0$ and $q\in\mathbb{N}$. Finally, for $\tau\in\Pi(t)$ and $m\in\mathbb{N}$ consider the processes $Z^{\tau,m}=$$\left(Z_{t}^{\tau,m}\right)_{t\geq0}$ given by $$\begin{aligned}
Z_{t}^{\tau,m} & =\exp\left(\bar{\xi}_{t}^{\tau,m}\right).\label{eq: Discrete Z}\end{aligned}$$ For any Borel measurable function $\varphi$ such that $\varphi\left(X_{t}\right)Z_{t}^{\tau,m}\in L^{1}(\Omega,\mathcal{F},\tilde{P})$ define the $m$-th order discretizations $$\begin{aligned}
\rho_{t}^{\tau,m}\left(\varphi\right) & \triangleq\mathbb{\tilde{E}}\left[\varphi\left(X_{t}\right)Z_{t}^{\tau,m}|\mathcal{Y}_{t}\right],\end{aligned}$$ and $$\pi_{t}^{\tau,m}\left(\varphi\right)\triangleq\rho_{t}^{\tau,2}\left(\varphi\right)/\rho_{t}^{\tau,m}\left(\boldsymbol{1}\right),$$ of $\rho_{t}$ and $\pi_{t}$, respectively.
Let $m\in\mathbb{N}$, our main assumption is the following:
We have that:\
$\bullet$ $f=\left(f_{i}\right){}_{i=1,...,d_{X}}:\mathbb{R}^{d_{X}}\rightarrow\mathbb{R}^{d_{X}}\in\mathcal{B}_{b}\cap C_{b}^{2\vee(2m-1)},$\
$\bullet$ $\sigma=\left(\sigma_{i,j}\right){}_{i=1,...d_{X},j=1,...,d_{V}}:\mathbb{R}^{d_{X}}\rightarrow\mathbb{R}^{d_{X}\times d_{V}}\in\mathcal{B}_{b}\cap C_{b}^{2m},$\
$\bullet$ $h=\left(h_{i}\right){}_{i=0,...,d_{Y}}:\mathbb{R}^{d_{X}}\rightarrow\mathbb{R}^{d_{Y}+1}\in\mathcal{B}_{b}\cap C_{b}^{2m+1},$\
$\bullet$ $X_{0}$ has moments of all orders.
Note that if assumption **H**$(m)$ holds for some $m\in\mathbb{N},$ then it also holds for any $n\leq m$.
\[thm: Main Filtering\_2\]Let assumption **$\mathbf{H}\left(m\right)$** be satisfied. Then, there exists constants $\delta_{0},C>0$ not depending on the choice of the partition $\tau\in\Pi(t,\delta_{0}),$ such that $$\left\Vert \rho_{t}\left(\varphi\right)-\rho_{t}^{\tau,m}\left(\varphi\right)\right\Vert _{2}\leq C\delta^{m},$$ for $\varphi\in C_{P}^{m+1}$. Moreover, if $\sup_{\tau\in\Pi(t,\delta_{0})}\left\Vert \pi_{t}^{\tau,m}(\varphi)\right\Vert _{2+\varepsilon}<\infty,$ for some $\varepsilon>0,$ then $$\mathbb{E}\left[\left\vert \pi_{t}\left(\varphi\right)-\pi_{t}^{\tau,m}\left(\varphi\right)\right\vert \right]\leq\bar{C}\delta^{m},$$ where $\bar{C}$ is another constant independent of $\tau\in\Pi(t,\delta_{0})$.
The assumption $\sup_{\tau\in\Pi(t,\delta_{0})}\mathbb{\tilde{E}}\left[\left\vert \pi_{t}^{\tau,m}(\varphi)\right\vert ^{2+\varepsilon}\right]<\infty$ for some $\varepsilon>0$ is satisfied if $\varphi$ is bounded. If $\varphi$ is unbounded, note that by using Jensen’s inequality one has $$\begin{aligned}
\mathbb{\tilde{E}}\left[\left\vert \pi_{t}^{\tau,m}(\varphi)\right\vert ^{2+\varepsilon}\right] & =\mathbb{\tilde{E}}\left[\left\vert \mathbb{\tilde{E}}\left[\frac{\varphi(X_{t})Z_{t}^{\tau,m}}{\mathbb{\tilde{E}}\left[Z_{t}^{\tau,m}|\mathcal{Y}_{t}\right]}|\mathcal{Y}_{t}\right]\right\vert ^{2+\varepsilon}\right]\\
& \leq\mathbb{\tilde{E}}\left[|\varphi(X_{t})|^{2+\varepsilon}\mathbf{\exp}((2+\varepsilon)(\bar{\xi}_{t}^{\tau,m}-\mathbb{\tilde{E}}[\bar{\xi}_{t}^{\tau,m}|\mathcal{Y}_{t}]))\right].\end{aligned}$$ Hence, one can reason as in Lemma \[lem: Xi\_Tau2\^p\_Integrability\] to justify that $\sup_{\tau\in\Pi(t,\delta_{0})}\mathbb{\tilde{E}}\left[\left\vert \pi_{t}^{\tau,m}(\varphi)\right\vert ^{2+\varepsilon}\right]<\infty.$
\[rem: High Order\] $\left.\right.$\
i. In the case $m=1$ we can consider any partition $\tau\in\Pi(t)$. For $m\geq2,$ we must consider partitions $\tau$ with mesh $\delta$ smaller than $$\delta_{0}=\frac{1}{2\left\Vert Lh\right\Vert _{\infty}\sqrt{d_{Y}d_{V}}},\label{eq: Delta_0}$$ where $$\left\Vert Lh\right\Vert _{\infty}\triangleq\max_{i=1,...d_{Y}\ r=1,...,d_{V}}\left\Vert L^{r}h^{i}\right\Vert _{\infty}.$$
ii\. The functional discretization given in (\[eq: Discrete Z\]) is recursive. More precisely, if $\tau^{\prime}\in\Pi(t+s)$ is a partition that includes $t$ as an intermediate point, for example $\tau^{\prime}\triangleq\{0=t_{0}<\cdots<t_{k}=t<t_{k+1}\cdots<t_{n}=t+s\}$ with $0<k<n$, then $$\begin{aligned}
Z_{t+s}^{\tau^{\prime},m} & =Z_{t}^{\tau,m}\prod_{j=k}^{n-1}\exp\left(\bar{\xi}_{t}^{\tau,m}\left(j\right)\right).\end{aligned}$$ This property is essential for implementation purposes as at every discretization time we only need to use the previous functional discretization and the term corresponding to the next interval to obtain the new functional discretization.
iii\. The discretization introduced by Picard in [@Pi84] corresponds to the case $m=1$. In this case, $\rho_{t}^{\tau,m}$ can be explicitly written as $$\rho_{t}^{\tau,m}\left(\varphi\right)\triangleq\mathbb{\tilde{E}}\left[\varphi\left(X_{t}\right)\left.\exp\left(\sum_{j=0}^{n-1}\sum_{i=0}^{d_{Y}}h_{i}(X_{t_{j}})\left(Y_{t_{j+1}}^{i}-Y_{t_{j}}^{i}\right)\right)\right\vert \mathcal{Y}_{t}\right],\label{PicardFilter}$$ This discretization scheme leads to a wealth of numerical methods that can be used to approximate $\pi_{t}$. Among them, particle methods[^5] are algorithms which approximate $\pi_{t}$ with discrete random measures of the form $\sum_{i}a_{i}(t)\delta_{v_{i}(t)},$ in other words with empirical distributions associated with sets of randomly located particles of stochastic masses $a_{1}(t)$,$a_{2}(t)$, …, which have stochastic positions $v_{1}(t),v_{2}(t),\ldots$ . These methods are currently among the most successful and versatile for numerically solving the filtering problem. Based on , the “garden variety‘" particle filter uses particles that evolve according to the signal equation (or, rather, the Euler approximation of the signal) and carry exponential weights. These weights are proportional with $$\exp\left(\sum_{i=0}^{d_{Y}}h_{i}(v_{t_{n}}^{j})\left(Y_{t_{n+1}}^{i}-Y_{t_{n}}^{i}\right)\right),$$ where $v^{j}$ is the process modelling the trajectory of the particle and $t_{n}$ is the update time. The method also involves a variance reduction procedure (for further details, see for example Chapter 9 in [@BaCr08]). Alternatively one can use a cubature method to approximate the law of the signal, see [@CrOr2013]. In both cases, higher order approximations of the signal can be used, but this would not improve the rate of convergence of the method as Picard’s discretisation has an error of order 1. The remedy is to exploit the result in this paper and use a higher order discretisation. The second author is working on a particle filter that uses the second order discretisation presented in this paper.
Proof of the main result \[sec: ProofMainResult\]
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We start by recalling and introducing some basic results on iterated integrals and martingale representations. Throughout the rest of the paper we will be assuming that **H**$(m)$ holds, without recalling it in each result statement. Moreover, $C$ will denote a constant that usually depends on $d_{V},$ $d_{X},$ $d_{Y},$ $f,$ $\sigma,$ $h$ and possibly other parameters but NOT on the partition $\tau.$ As we are interested in showing a rate of convergence for our approximations, the particular form of dependence of $C$ with respect to these parameters is not relevant and, hence, omitted. Of course, the choice of the constant $C$ may change from line to line.
\[rem: M\] Some immediate consequences of assumption **H**$(m)$ are the following:
1. The signal process $X$ has moments of all orders and for any $p\geq1,$ we have $$\mathbb{\tilde{E}}\left[\sup_{s\in[0,t]}\left|X_{s}^{i}\right|{}^{p}\right]<\infty,$$ for all $i\in\{1,...,d_{X}\}$.
2. If $\Upsilon:\mathbb{R}^{d_{X}}\rightarrow\mathbb{R}$ is a function with polynomial growth we have $$\mathbb{\tilde{E}}\left[\sup_{s\in[0,t]}\left|\Upsilon(X_{s})\right|{}^{p}\right]<\infty,$$ in particular, $$\mathbb{\tilde{E}}\left[\sup_{s\in[0,t]}\left|L^{\alpha}h_{i}(X_{s})\right|^{p}\right]<\infty,$$ for $i=0,...,d_{Y}$ and $\alpha\in\mathcal{M}_{m}\left(S_{0}\right)=\mathcal{M}_{m-1}\left(S_{0}\right)\biguplus\mathcal{R}\left(\mathcal{M}_{m-1}\left(S_{0}\right)\right).$
3. The processes $\xi_{t}$ and $\xi_{t}^{\tau,m},m\in\mathbb{N},$ as defined above have finite moments of all orders.
\[rem:TruncProc\]Consider the truncation function $$\Gamma_{q,\delta}(z)=\frac{z}{1+(z/\delta)^{2q}}.$$ defined as above corresponding to the real parameters $q\geq1$ and $\delta>0$.
1. For any $z\in\mathbb{R}$, $\left\vert \Gamma_{q,\delta}(z)\right\vert \le\delta$. To check this observe that if $\left\vert z\right\vert \leq\delta$ we have that $1+(z/\delta)^{2q}\geq1$ and then $$\left\vert \Gamma_{q,\delta}(z)\right\vert =\frac{\left\vert z\right\vert }{1+(z/\delta)^{2q}}\leq\left\vert z\right\vert \leq\delta.$$ On the other hand, if $\left\vert z\right\vert >\delta$ we have that $\left\vert z/\delta\right\vert ^{-1}+\left\vert z/\delta\right\vert ^{2q-1}>1$ and then $$\left\vert \Gamma_{q,\delta}(z)\right\vert =\frac{\left\vert z\right\vert }{1+\left\vert z\right\vert ^{2q}\delta^{-2q}}=\frac{1}{\left\vert z\right\vert ^{-1}+\left\vert z\right\vert ^{2q-1}\delta^{-2q}}=\frac{\delta}{\left\vert z/\delta\right\vert ^{-1}+\left\vert z/\delta\right\vert ^{2q-1}}\leq\delta.$$
2. Moreover, if we define $$\mathcal{E}_{q,\delta}(z)\triangleq\Gamma_{q,\delta}(z)-z,$$ we get that $$\begin{aligned}
\left|\mathcal{E}_{q,\delta}(z)\right| & =\left\vert \Gamma_{q,\delta}(z)-z\right\vert =\left\vert \frac{z}{1+(z/\delta)^{2q}}-z\right\vert =\frac{\left\vert z\right\vert ^{2q+1}\delta^{-2q}}{1+(z/\delta)^{2q}}=\frac{\left\vert z\right\vert ^{2q+1}}{\delta^{2q}+z^{2q}},\nonumber \\
& =\delta^{-2q}\frac{\left\vert z\right\vert ^{2q+1}}{1+\left(z/\delta\right)^{2q}}\leq\delta^{-2q}\left\vert z\right\vert ^{2q+1},\quad\forall z\in\mathbb{R}.\label{eq:EpsilonInequality}\end{aligned}$$
3. Finally, note that for $m$$\geq3$ we can write $$\begin{aligned}
\bar{\xi}_{t}^{\tau,m} & \triangleq\xi_{t}^{\tau,2}+\sum_{j=0}^{n-1}\Gamma_{m-\frac{1}{2},\delta_{j}}\left(\mu^{\tau,m}\left(j\right)\right)\nonumber \\
& =\xi_{t}^{\tau,m}+\sum_{j=0}^{n-1}\mathcal{E}_{m-\frac{1}{2},\delta_{j}}\left(\mu^{\tau,m}\left(j\right)\right)\label{eq:BigEpsilon}\end{aligned}$$
\[subsec:IteratedIntegrals\]Iterated integrals
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The following two results are well known and can be found in Kloeden and Platen [@KlPl92], Theorem 5.5.1 and Lemma 5.7.5, respectively.
\[thm: Ito Taylor Expansion\]Let $\rho_{1}$ and $\rho_{2}$ be two stopping times with $0\leq\rho_{1}\leq\rho_{2}\leq t,$ a.s., let $\mathcal{A}\subset\mathcal{M}(S_{0})$ be a hierarchical set and $g:\mathbb{R}^{d_{X}}\rightarrow\mathbb{R}.$ Then, the Itô-Taylor expansion $$g(X_{\rho_{2}})=\sum_{\alpha\in\mathcal{A}}L^{\alpha}g(X_{\rho_{1}})I_{\alpha}(\boldsymbol{1})_{\rho_{1},\rho_{2}}+\sum_{\alpha\in\mathcal{R}\left(\mathcal{A}\right)}I_{\alpha}(L^{\alpha}g(X_{\cdot}))_{\rho_{1},\rho_{2}},\label{eq: Ito Taylor Expansion}$$ holds, provided all of the derivatives of $g,f$ and $\sigma$ and all of the iterated Itô integrals appearing in $\left(\ref{eq: Ito Taylor Expansion}\right)$ exist.
\[lem: Moments Iterated Integral\] Let $\alpha\in\mathcal{M}(S_{0}),$ let $\theta=\{\theta_{s}\}_{s\in[0,t]}$ be an $\mathbb{H}^{t}$-adapted process, let $p\geq1$ and let $\rho_{1}$ and $\rho_{2}$ be two stopping times with $0\leq\rho_{1}\leq\rho_{2}\leq t$ and $\rho_{2}$ being $\mathcal{H}_{\rho_{1}}^{t}$-measurable. Then, $$\mathbb{\tilde{E}}\left[\left\vert I_{\alpha}(\theta_{\cdot})_{\rho_{1},\rho_{2}}\right\vert ^{2p}|\mathcal{H}_{\rho_{1}}^{t}\right]\leq CR(\theta)\left(\rho_{2}-\rho_{1}\right)^{p\{|\alpha|+|\alpha|_{0}\}},$$ where $$R(\theta)=\mathbb{\tilde{E}}\left[\sup_{\rho_{1}\leq s\leq\rho_{2}}\left\vert \theta_{s}\right\vert ^{2p}|\mathcal{H}_{\rho_{1}}^{t}\right].$$
The following lemma gives a basic estimate on the difference between the log likelihood functional $\xi_{t}$ and its $m$-th order discretization. Its proof relies on Theorem \[thm: Ito Taylor Expansion\] and Lemma \[lem: Moments Iterated Integral\].
\[lem: Difference Moment Estimate\]We have that $$\xi_{t}-\xi_{t}^{\tau,m}=\sum_{i=0}^{d_{Y}}\int_{0}^{t}\left\{ \sum_{\alpha\in\mathcal{R}\left(\mathcal{M}_{m-1}(S_{0})\right)}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}\right\} dY_{s}^{i},\label{eq: Difference Taylor Expansion}$$ and $$\mathbb{\tilde{E}}\left[\left\vert \xi_{t}-\xi_{t}^{\tau,m}\right\vert ^{2p}\right]\leq C\delta^{pm}.$$
By Remark \[rem: M\], we can apply Theorem \[thm: Ito Taylor Expansion\] and get equation \[eq: Difference Taylor Expansion\]. Applying the Itô isometry and Jensen’s inequality (or Jensen’s inequality directly if $i=0$), we obtain the following bound $$\begin{aligned}
\mathbb{\tilde{E}}\left[\left|\xi_{t}-\xi_{t}^{\tau,m}\right|^{2p}\right] & =\mathbb{\tilde{E}}\left[\left\vert \sum_{i=0}^{d_{Y}}\int_{0}^{t}\left\{ \sum_{\alpha\in\mathcal{R}\left(\mathcal{M}_{m-1}(S_{0})\right)}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}\right\} dY_{s}^{i}\right\vert ^{2p}\right]\\
& \leq C\sum_{\alpha\in\mathcal{R}\left(\mathcal{M}_{m-1}(S_{0})\right)}\int_{0}^{t}\mathbb{\tilde{E}}\left[\left\vert I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}\right\vert ^{2p}\right]ds.\end{aligned}$$ Let $\alpha\in\mathcal{R}\left(\mathcal{M}_{m-1}(S_{0})\right),$ by Lemma \[lem: Moments Iterated Integral\] and Remark \[rem: M\] we get that $$\begin{aligned}
\mathbb{\tilde{E}}\left[\left\vert I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}\right\vert ^{2p}\right] & \leq C\mathbb{\tilde{E}}\left[\sup_{\tau(s)\leq u\leq s}\left\vert L^{\alpha}h_{i}(X_{u})\right\vert ^{2p}\right]\left(s-\tau(s)\right)^{p\{|\alpha|+|\alpha|_{0}\}}\leq C\delta^{pm},\end{aligned}$$ where in the last inequality we have used that $|\alpha|+|\alpha|_{0}\geq m$ for $\alpha\in\mathcal{R}\left(\mathcal{M}_{m-1}(S_{0})\right).$ From the previous inequality the result follows easily.
\[lem: SumEpsilon\]Let $p,q\geq1$ and $m\geq3.$ Then, we have that $$\mathbb{\tilde{E}}\left[\left\vert \sum_{j=0}^{n-1}\mathcal{E}_{q,\delta_{j}}\left(\mu^{\tau,m}\left(j\right)\right)\right\vert ^{2p}\right]\leq C\left(t,d_{Y},p,m\right)\delta^{p\left(2q+1\right)}.$$
We have that $$\begin{aligned}
\mathbb{\tilde{E}}\left[\left\vert \sum_{j=0}^{n-1}\mathcal{E}_{q,\delta_{j}}\left(\mu^{\tau,m}\left(j\right)\right)\right\vert ^{2p}\right] & \leq C\left(p\right)n^{2p-1}\sum_{j=0}^{n-1}\mathbb{\tilde{E}}\left[\left|\mathcal{E}_{q,\delta_{j}}\left(\mu^{\tau,m}\left(j\right)\right)\right|^{2p}\right].\end{aligned}$$ Moreover, using similar arguments as in Lemma \[lem: Difference Moment Estimate\], for any $r\geq1$, we have that $$\begin{aligned}
\mathbb{\tilde{E}}\left[\left|\mu^{\tau,m}\left(j\right)\right|^{r}\right] & \leq C\left(d_{Y},m\right)\delta^{\frac{r}{2}(1+\left|\alpha\right|+\left|\alpha\right|_{0})}\\
& \leq C\left(d_{Y},m\right)\delta^{\frac{3}{2}r},\end{aligned}$$ because as $\alpha\in\mathcal{M}_{1,m-1}(S_{0}),m\geq3$ then $\left|\alpha\right|\in[2,m-1]$ and $\left|\alpha\right|_{0}\in[0,m-1]$. Then, using equation $\left(\ref{eq:EpsilonInequality}\right)$ and Remark \[rem: Uniform\] we get that $$\begin{aligned}
n^{2p-1}\sum_{j=0}^{n-1}\mathbb{\tilde{E}}\left[\left|\mathcal{E}_{q,\delta_{j}}\left(\mu^{\tau,m}\left(j\right)\right)\right|^{2p}\right] & \leq n^{2p-1}\sum_{j=0}^{n-1}\delta^{-4pq}\mathbb{\tilde{E}}\left[\left|\mu^{\tau,m}\left(j\right)\right|^{2p\left(2q+1\right)}\right]\\
& \leq C\left(t,d_{Y},m\right)n\delta^{-\left(2p-1\right)}\delta^{-4pq}\delta^{3p(2q+1)}\\
& \leq C\left(t,d_{Y},m\right)\delta^{p(2q+1)}.\end{aligned}$$
\[lem: CondExpect\]Let $\theta=\{\theta_{s}\}_{s\in[0,t]}$ and $\Psi=\{\Psi_{s}\}_{s\in[0,t]}$ be two $\mathbb{H}^{t}$-adapted process. Then:
1. \[lem: CE1\]For $\alpha\in\mathcal{M}(S_{0})$ and $0\leq s_{1}\leq s_{2}\leq s_{3}\leq t,$ we have that $$\mathbb{\tilde{E}}\left[I_{\alpha}(\theta)_{s_{2},s_{3}}|\mathcal{H}_{s_{1}}^{t}\right]=\boldsymbol{1}_{\{\left\vert \alpha\right\vert =\left\vert \alpha\right\vert _{0}\}}I_{\alpha}\left(\mathbb{\tilde{E}}[\theta|\mathcal{H}_{s_{1}}^{t}]\right)_{s_{2},s_{3}}.$$
2. \[lem: CE2\]For $\alpha\in\mathcal{M}(S_{0})$ with $\left\vert \alpha\right\vert \neq\left\vert \alpha\right\vert _{0},r\in\{1,...,d_{V}\},0\leq s_{1}\leq s_{2}\leq t$ and $0\leq s_{3}\leq s_{4}\leq t$ we have that $$\begin{aligned}
\mathbb{\tilde{E}}\left[\left(\int_{s_{3}}^{s_{4}}\Psi_{s}dV_{s}^{r}\right)I_{\alpha}(\theta)_{s_{1},s_{2}}|\mathcal{Y}_{t}\right] & =\boldsymbol{1}_{\{\alpha_{\left\vert \alpha\right\vert }=0\}}\int_{s_{1}}^{s_{2}}\mathbb{\tilde{E}}\left[\left(\int_{s_{3}}^{s_{4}}\Psi_{s}dV_{s}^{r}\right)I_{\alpha-}(\theta)_{s_{1},u}|\mathcal{Y}_{t}\right]du\\
& \quad+\boldsymbol{1}_{\{\alpha_{\left\vert \alpha\right\vert }=r\}}\int_{s_{1}\vee s_{3}}^{s_{2}\wedge s_{4}}\mathbb{\tilde{E}}\left[\Psi_{u}I_{\alpha-}(\theta)_{s_{1},u}|\mathcal{Y}_{t}\right]du.\end{aligned}$$
3. \[lem: CE3\]For $\alpha\in\mathcal{M}(S_{0})$ with $\left\vert \alpha\right\vert \geq2,\alpha_{\left|\alpha\right|}\neq0,\alpha_{\left|\alpha\right|-1}\neq0,r_{1},r_{2}\in\{1,...,d_{V}\},0\leq s_{1}\leq s_{2}\leq t$ and $0\leq s_{3}\leq s_{4}\leq s_{5}\leq s_{6}\leq t$ we have that $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\left(\int_{s_{3}}^{s_{6}}\int_{s_{3}}^{s_{5}}\Psi_{s_{4}}dV_{s_{4}}^{r_{1}}dV_{s_{5}}^{r_{2}}\right)I_{\alpha}(\theta)_{s_{1},s_{2}}|\mathcal{Y}_{t}\right]\\
& =\boldsymbol{1}_{\{\alpha_{\left\vert \alpha\right\vert }=r_{2}\}}\int_{s_{1}\vee s_{3}}^{s_{2}\wedge s_{6}}\mathbb{\tilde{E}}\left[\left(\int_{s_{3}}^{s_{5}}\Psi_{s_{4}}dV_{s_{4}}^{r_{1}}\right)I_{\alpha-}(\theta)_{s_{1},u}|\mathcal{Y}_{t}\right]du\\
& =\boldsymbol{1}_{\{\alpha_{\left\vert \alpha\right\vert }=r_{2},\alpha_{\left\vert \alpha\right\vert -1}=r_{1}\}}\int_{s_{1}\vee s_{3}}^{s_{2}\wedge s_{6}}\int_{s_{1}\vee s_{3}}^{u}\mathbb{\tilde{E}}\left[\Psi_{v}I_{\left(\alpha-\right)-}(\theta)_{s_{1},v}|\mathcal{Y}_{t}\right]dvdu\end{aligned}$$
$\left.\right.$\
\[lem: CE1\]. If $\left\vert \alpha\right\vert \neq\left\vert \alpha\right\vert _{0},$ then the iterated integral $I_{\alpha}(\theta)_{s_{2},s_{3}}$ contains a Brownian differential $dV^{r}$ and it vanishes when we take the conditional expectation with respect to $\mathcal{H}_{s_{1}}^{t}$. If $\left\vert \alpha\right\vert =\left\vert \alpha\right\vert _{0},$ all the differentials in the iterated integral $I_{\alpha}(\theta)_{s_{2},s_{3}}$ are Lebesgue differentials and we can write the conditional expectation inside the inner integral.\
\[lem: CE2\]. Note that if $\alpha_{\left|\alpha\right|}=0$ we can write $$\mathbb{\tilde{E}}\left[\left(\int_{s_{3}}^{s_{4}}\Psi_{s}dV_{s}^{r}\right)\int_{s_{2}}^{s_{3}}I_{\alpha-}\left(\theta\right)_{s_{1},u}du|\mathcal{Y}_{t}\right]=\mathbb{\tilde{E}}\left[\int_{s_{2}}^{s_{3}}\left(\int_{s_{3}}^{s_{4}}\Psi_{s}dV_{s}^{r}\right)I_{\alpha-}\left(\theta\right)_{s_{2},u}du|\mathcal{Y}_{t}\right],$$ because we can push the random variable $\left(\int_{s_{3}}^{s_{4}}\Psi_{s}dV_{s}^{r}\right)$ inside the Lebesgue integral. If $\alpha_{\left|\alpha\right|}\neq0$ we can write $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\left(\int_{s_{3}}^{s_{4}}\Psi_{s}dV_{s}^{r}\right)\int_{s_{1}}^{s_{2}}I_{\alpha-}(\theta)_{s_{1},u}dV_{u}^{\alpha_{\left|\alpha\right|}}|\mathcal{Y}_{t}\right]\\
& =\mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[\left(\int_{s_{3}}^{s_{4}}\Psi_{s}dV_{s}^{r}\right)\int_{s_{1}}^{s_{2}}I_{\alpha-}(\theta)_{s_{1},u}dV_{u}^{\alpha_{\left|\alpha\right|}}|\mathcal{H}_{0}^{t}\right]|\mathcal{Y}_{t}\right],\\
& =\boldsymbol{1}_{\{\alpha_{\left\vert \alpha\right\vert }=r\}}\int_{s_{1}\vee s_{3}}^{s_{2}\wedge s_{4}}\mathbb{\tilde{E}}\left[\Psi_{u}I_{\alpha-}(\theta)_{s_{1},u}|\mathcal{Y}_{t}\right]du\end{aligned}$$ where we have just applied the $\mathcal{H}_{s}^{t}$-semimartingale covariation formula.\
\[lem: CE3\]. Same reasoning as for statement \[lem: CE2\].
\[subsec:IntegrabilityLikelihood\]Integrability of the likelihood functional and its discretizations
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In this section we state some integrability results for the likelihood functional and its discretizations. The first result is on the integrability of the likelihood functional. It follows from the basic fact that any Gaussian distribution has exponential moments of all orders.
\[lem: Z\_t\^p\_Integrability\]Assume that $\mathbf{H}\left(1\right)$ holds. Let $p\geq1$ and $\tau$ be any partition. Then, one has that $$\mathbb{\tilde{E}}\left[\left|Z_{t}\right|^{p}\right]=\mathbb{\tilde{E}}\left[\exp(p\xi_{t})\right]\leq\mathbb{\tilde{E}}\left[\exp(p\left|\xi_{t}\right|)\right]<\infty,$$ and $$\mathbb{\tilde{E}}\left[\left|Z_{t}^{\tau,1}\right|^{p}\right]=\mathbb{\tilde{E}}\left[\exp(p\xi_{t}^{\tau,1})\right]<\infty.$$
We have that $$\begin{aligned}
\mathbb{\tilde{E}}\left[\exp(p\left|\xi_{t}\right|)\right] & =\mathbb{\tilde{E}}\left[\exp\left(\left|p\sum_{i=1}^{d_{Y}}\int_{0}^{t}h^{i}(X_{s})dY_{s}^{i}-\frac{p}{2}\sum_{i=1}^{d_{Y}}\int_{0}^{t}h^{i}\left(X_{s}\right)^{2}ds\right|\right)\right]\\
& \leq\exp\left(\frac{p}{2}d_{Y}\left\Vert h\right\Vert _{\infty}^{2}t\right)\mathbb{\tilde{E}}\left[\exp\left(p\sum_{i=1}^{d_{Y}}\left|\int_{0}^{t}h^{i}(X_{s})dY_{s}^{i}\right|\right)\right].\end{aligned}$$ Recall that if $Z\thicksim\mathcal{N}\left(0,\sigma^{2}\right)$ under $\tilde{P}$, then $$\mathbb{\tilde{E}}\left[e^{p\left|Z\right|}\right]=2e^{\frac{p^{2}\sigma^{2}}{2}}\Phi\left(p\sigma\right),$$ where $\Phi$ is the cumulative distribution function of a standard normal random variable. As $Y$ is a Brownian motion independent of $X$ under $\tilde{P}$, we have that $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\exp\left(p\sum_{i=1}^{d_{Y}}\left|\int_{0}^{t}h^{i}(X_{s})dY_{s}^{i}\right|\right)\right]\\
& =\mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[\exp\left(p\sum_{i=1}^{d_{Y}}\left|\int_{0}^{t}h^{i}(X_{s})dY_{s}^{i}\right|\right)|\mathcal{F}_{t}^{0,V}\right]\right]\\
& =\mathbb{\tilde{E}}\left[2\exp\left(\frac{p^{2}}{2}\sum_{i=1}^{d_{Y}}\int_{0}^{t}h^{i}(X_{s})^{2}ds\right)\Phi\left(p\left(\sum_{i=1}^{d_{Y}}\int_{0}^{t}h^{i}(X_{s})^{2}ds\right)^{1/2}\right)\right]\\
& \leq2\exp\left(\frac{p^{2}}{2}d_{Y}\left\Vert h\right\Vert _{\infty}^{2}t\right),\end{aligned}$$ and we can conclude that $\mathbb{\tilde{E}}\left[\exp(p\left|\xi_{t}\right|)\right]<\infty.$ The proof that $\mathbb{\tilde{E}}\left[\left|Z_{t}^{\tau,1}\right|^{p}\right]<\infty$ follows by similar arguments.
The following lemma ensures the $L^{p}\left(\text{\ensuremath{\Omega}}\right)$ integrability of the second order discretization of the likelihood function, provided the discretization is done on a sufficiently fine partition. We give a bound on the mesh of the partition in terms of $p$, the uniform bounds on the sensor function $h$ and its derivatives and the dimensions of the noise driving the signal and the observation process. The proof is based on the fact that the square of a centered Gaussian random variable has finite exponential moment of order sufficiently small.
\[lem: Xi\_Tau2\^p\_Integrability\]Assume that $\mathbf{H}\left(2\right)$ holds. Let $p\geq1$ and $\tau$ be a partition with mesh size $$\delta<\left(p\left\Vert Lh\right\Vert _{\infty}\sqrt{d_{Y}d_{V}}\right)^{-1},$$ where $$\left\Vert Lh\right\Vert _{\infty}\triangleq\max_{\substack{i=1,...d_{Y}\\
r=1,...,d_{V}
}
}\left\Vert L^{r}h^{i}\right\Vert _{\infty}.$$ Then, one has that $$\mathbb{\tilde{E}}\left[\left|Z_{t}^{\tau,2}\right|^{p}\right]=\mathbb{\tilde{E}}\left[\exp(p\xi^{\tau,2})\right]<\infty.$$
We can write $\exp\left(p\xi^{\tau,2}\right)\triangleq{\displaystyle \prod\limits _{i=1}^{4}}\left(K_{t}^{\tau,2,i}\right)^{p},$ where $$\begin{aligned}
K_{t}^{\tau,2,1} & \triangleq\exp\left(\sum_{i=1}^{d_{Y}}\sum_{r=1}^{d_{V}}\int_{0}^{t}L^{r}h^{i}(X_{\tau(s)})(V_{s}^{r}-V_{\tau(s)}^{r})dY_{s}^{i}\right),\\
K_{t}^{\tau,2,2} & \triangleq\exp\left(\sum_{i=1}^{d_{Y}}\int_{0}^{t}\{h^{i}(X_{\tau(s)})+L^{0}h^{i}(X_{\tau(s)})(s-\tau(s))\}dY_{s}^{i}\right),\\
K_{t}^{\tau,2,3} & \triangleq\exp\left(-\frac{1}{2}\sum_{i=1}^{d_{Y}}\int_{0}^{t}\{(h^{i})^{2}\left(X_{\tau(s)}\right)+L^{0}((h^{i})^{2})(X_{\tau(s)})(s-\tau(s))\}ds\right),\\
K_{t}^{\tau,2,4} & \triangleq\exp\left(-\frac{1}{2}\sum_{i=1}^{d_{Y}}\sum_{r=1}^{d_{V}}\int_{0}^{t}L^{r}((h^{i})^{2})(X_{\tau(s)})(V_{s}^{r}-V_{\tau(s)}^{r})\}ds\right).\end{aligned}$$ Let $\varepsilon>0,$ then, by Hölder inequality, we have $$\mathbb{\tilde{E}}\left[\exp\left(p\xi^{\tau,2}\right)\right]\leq\mathbb{\tilde{E}}\left[\left\vert K_{t}^{\tau,2,1}\right\vert ^{p(1+\varepsilon)}\right]^{\frac{1}{1+\varepsilon}}\mathbb{\tilde{E}}\left[{\displaystyle \prod\limits _{i=2}^{4}}\left\vert K_{t}^{\tau,2,i}\right\vert ^{p\frac{(1+\varepsilon)}{\varepsilon}}\right]^{\frac{\varepsilon}{1+\varepsilon}}.$$ Hence, the result follows by showing that $K_{t}^{\tau,2,1}$ has finite $p(1+\varepsilon)$-moment and $$\mathbb{\tilde{E}}\left[{\displaystyle \prod\limits _{i=2}^{4}}\left\vert K_{t}^{\tau,2,i}\right\vert ^{p\frac{(1+\varepsilon)}{\varepsilon}}\right]<\infty.\label{eq: ProductK}$$ Applying Hölder inequality twice, condition $\left(\ref{eq: ProductK}\right)$ follows by showing that $K_{t}^{\tau,2,i},i=2,...,4$ have finite moments of all orders. In what follows, let $q\geq1$ be a fixed real constant. We start by the easiest term, $K_{t}^{\tau,2,3}.$ We have that $$\mathbb{\tilde{E}}\left[\left\vert K_{t}^{\tau,2,3}\right\vert ^{q}\right]\leq\exp\left(\frac{qd_{Y}}{2}t(\left\Vert h\right\Vert _{\infty}^{2}+\delta\left\Vert L^{0}h^{2}\right\Vert _{\infty}\right)<\infty,$$ because $\left\Vert h\right\Vert _{\infty}^{2}$ and $\left\Vert L^{0}h^{2}\right\Vert _{\infty}=\max_{i=1,...,d_{Y}}\left\Vert L^{0}(h_{i}^{2})\right\Vert _{\infty}$ are finite due to the assumptions on $f,\sigma$ and $h.$ For the term $K_{t}^{\tau,2,4},$ we can write $$\begin{aligned}
\mathbb{\tilde{E}}\left[\left\vert K_{t}^{\tau,2,4}\right\vert ^{q}\right] & \leq\mathbb{\tilde{E}}\left[\exp\left(\frac{qd_{Y}d_{V}}{2}\left\Vert L((h)^{2})\right\Vert _{\infty}\int_{0}^{t}\left\vert V_{s}^{1}-V_{\tau(s)}^{1}\right\vert ds\right)\right]\\
& =\mathbb{\tilde{E}}\left[\exp\left(\frac{qd_{Y}d_{V}}{2}\left\Vert L((h)^{2})\right\Vert _{\infty}\left(\int_{0}^{t}(s-\tau(s))ds\right)\left\vert V_{1}^{1}\right\vert \right)\right]\\
& \leq\mathbb{\tilde{E}}\left[\exp\left(\frac{qd_{Y}d_{V}}{2}\left\Vert L((h)^{2})\right\Vert _{\infty}t\sqrt{\delta}\left\vert V_{1}^{1}\right\vert \right)\right]<\infty,\end{aligned}$$ because $\left\Vert L((h)^{2})\right\Vert _{\infty}=\max_{\substack{i=1,...d_{Y}\\
r=1,...,d_{V}
}
}\left\Vert L^{r}(h_{i}^{2})\right\Vert _{\infty}$ is finite, the law of $V_{s}^{1}-V_{\tau(s)}^{1}$ coincides with the law of $(s-\tau(s))^{1/2}V_{1}^{1}$ by the scaling properties of the Brownian motion and $\left\vert V_{1}^{1}\right\vert $ has exponential moments of any order.
For the term $K_{t}^{\tau,2,2},\ $we first condition with respect to $\mathcal{F}_{t}^{V}=\sigma(V_{s},0\leq s\leq t)$ and use the fact that, conditionally to $\mathcal{F}_{t}^{V}$, the stochastic integrals with respect to $Y$ are Gaussian. We get $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\left\vert K_{t}^{\tau,2,2}\right\vert ^{q}\right]=\mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[\exp\left(q\sum_{i=1}^{d_{Y}}\int_{0}^{t}\{h^{i}(X_{\tau(s)})+L^{0}h^{i}(X_{\tau(s)})(s-\tau(s))\}dY_{s}^{i}\right)|\mathcal{F}_{t}^{V}\right]\right]\\
& =\mathbb{\tilde{E}}\left[\exp\left(\frac{q^{2}}{2}\sum_{i=1}^{d_{Y}}\int_{0}^{t}\left\{ h^{i}(X_{\tau(s)})+L^{0}h^{i}(X_{\tau(s)})(s-\tau(s))\right\} ^{2}ds\right)\right]\\
& =\exp(q^{2}d_{Y}t\{\left\Vert h\right\Vert _{\infty}^{2}+\left\Vert L^{0}h\right\Vert _{\infty}^{2}\})<\infty.\end{aligned}$$
Finally, the term $K_{t}^{\tau,2,1}$ is more delicate because, in order to show that has finite $(p+\varepsilon)$-moment, a relationship between the mesh of the partition $\delta$ and $p+\varepsilon$ is needed. Proceeding as with the term $K_{t}^{\tau,2,2},$ we obtain $$\begin{aligned}
\mathbb{\tilde{E}}\left[\left\vert K_{t}^{\tau,2,1}\right\vert ^{p(1+\varepsilon)}\right] & =\mathbb{\tilde{E}}\left[\exp\left(p(1+\varepsilon)\sum_{i=1}^{d_{Y}}\sum_{r=1}^{d_{V}}\int_{0}^{t}L^{r}h^{i}(X_{\tau(s)})(V_{s}^{r}-V_{\tau(s)}^{r})dY_{s}^{i}\right)\right]\\
& =\mathbb{\tilde{E}}\left[\prod_{i=1}^{d_{Y}}\mathbb{\tilde{E}}\left[\exp\left(\int_{0}^{t}p(1+\varepsilon)\sum_{r=1}^{d_{V}}L^{r}h^{i}(X_{\tau(s)})(V_{s}^{r}-V_{\tau(s)}^{r})dY_{s}^{i}\right)|\mathcal{F}_{t}^{V}\right]\right].\end{aligned}$$ Now, conditionally to $\mathcal{F}_{t}^{V},$ the terms in the exponential are centered Gaussian random variables and we get that $$\begin{aligned}
\mathbb{\tilde{E}}\left[\left\vert K_{t}^{\tau,2,1}\right\vert ^{p(1+\varepsilon)}\right] & =\mathbb{\tilde{E}}\left[\prod_{i=1}^{d_{Y}}\exp\left(\frac{p^{2}(1+\varepsilon)^{2}}{2}\int_{0}^{t}\left(\sum_{r=1}^{d_{V}}L^{r}h^{i}(X_{\tau(s)})(V_{s}^{r}-V_{\tau(s)}^{r})\right)^{2}ds\right)\right]\\
& \leq\mathbb{\tilde{E}}\left[\prod_{i=1}^{d_{Y}}\exp\left(\frac{p^{2}(1+\varepsilon)^{2}d_{V}}{2}\int_{0}^{t}\left(\sum_{r=1}^{d_{V}}|L^{r}h^{i}(X_{\tau(s)})|^{2}(V_{s}^{r}-V_{\tau(s)}^{r})^{2}\right)ds\right)\right]\\
& =\mathbb{\tilde{E}}\left[\exp\left(\frac{p^{2}(1+\varepsilon)^{2}d_{Y}d_{V}\left\Vert Lh\right\Vert _{\infty}^{2}}{2}\sum_{r=1}^{d_{V}}\int_{0}^{t}(V_{s}^{r}-V_{\tau(s)}^{r})^{2}ds\right)\right]\\
& =\mathbb{\tilde{E}}\left[\exp\left(\frac{p^{2}(1+\varepsilon)^{2}d_{Y}d_{V}\left\Vert Lh\right\Vert _{\infty}^{2}}{2}\int_{0}^{t}(V_{s}^{1}-V_{\tau(s)}^{1})^{2}ds\right)\right]^{d_{V}}.\end{aligned}$$ So we need to find conditions on $\beta>0,$ such that $\mathbb{\tilde{E}}\left[\exp\left(\beta\int_{0}^{t}(V_{s}^{1}-V_{\tau(s)}^{1})^{2}ds\right)\right]<\infty.$ We can write $$\begin{aligned}
\mathbb{\tilde{E}}\left[\exp\left(\beta\int_{0}^{t}(V_{s}^{1}-V_{\tau(s)}^{1})^{2}ds\right)\right] & =\mathbb{\tilde{E}}\left[\exp\left(\beta\sum_{j=1}^{n}\int_{t_{j-1}}^{t_{j}}(V_{s}^{1}-V_{t_{j-1}}^{1})^{2}ds\right)\right]\\
& =\prod_{j=1}^{n}\mathbb{\tilde{E}}\left[\exp\left(\beta\int_{t_{j-1}}^{t_{j}}(V_{s}^{1}-V_{t_{j-1}}^{1})^{2}ds\right)\right]\\
& \triangleq\prod_{j=1}^{n}\Theta\left(\beta,\delta_{j}\right).\end{aligned}$$ Denote by $M_{t}\triangleq\sup_{0\leq s\leq t}V_{s}^{1}$ and recall that the density of $M_{t}$ is given by $$\begin{aligned}
f_{M_{t}}\left(x\right) & =\frac{2}{\sqrt{2\pi t}}e^{-\frac{x^{2}}{2t}}\boldsymbol{1}_{(0,\infty)},\end{aligned}$$ see Karatzas and Shreve [@KS91], page 96. Moreover, note that for any $A>0,$ $$\frac{2}{\sqrt{2\pi\sigma^{2}}}\int_{0}^{\infty}\exp\left\{ -A\frac{x^{2}}{2\sigma^{2}}\right\} dx=A^{-1/2}.$$ Then, we have that $$\begin{aligned}
\Theta\left(\beta,\delta_{j}\right) & \leq\mathbb{\tilde{E}}[\exp(\beta\delta_{j}M_{\delta}^{2}]=\int_{0}^{\infty}\frac{2}{\sqrt{2\pi\delta_{j}}}\exp\left\{ \beta\delta_{j}x^{2}-\frac{x^{2}}{2\delta_{j}}\right\} \\
& =\int_{0}^{\infty}\frac{2}{\sqrt{2\pi\delta_{j}}}\exp\left\{ -\left(1-2\beta\delta_{j}^{2}\right)\frac{x^{2}}{2\delta_{j}}\right\} =\left(1-2\beta\delta_{j}^{2}\right)^{-1/2}<\infty,\end{aligned}$$ as long as $1-2\beta\delta_{j}^{2}>0.$ On the other hand, $$\begin{aligned}
\left(1-2\beta\delta_{j}^{2}\right)^{-1} & =\sum_{k=0}^{\infty}\left(2\beta\delta_{j}^{2}\right)^{k}=1+2\beta\delta_{j}^{2}\left(\sum_{k=0}^{\infty}\left(2\beta\delta_{j}^{2}\right)^{k}\right)\\
& \leq1+2\beta\delta_{j}^{2}\left(\sum_{k=0}^{\infty}\left(2\beta\delta^{2}\right)^{k}\right)=1+\frac{2\beta\delta_{j}^{2}}{1-2\beta\delta^{2}}\\
& \leq\exp\left(\frac{2\beta\delta_{j}^{2}}{1-2\beta\delta^{2}}\right),\end{aligned}$$ and, therefore, $$\begin{aligned}
\prod_{j=1}^{n}\Theta\left(\beta,\delta_{j}\right) & \leq\prod_{j=1}^{n}\exp\left(\frac{\beta\delta_{j}^{2}}{1-2\beta\delta^{2}}\right)\leq\exp\left(\frac{\beta\sum_{j=1}^{n}\delta_{j}^{2}}{1-2\beta\delta^{2}}\right)\\
& \leq\exp\left(\frac{\beta\delta t}{1-2\beta\delta^{2}}\right)<\infty.\end{aligned}$$
As $\beta=\frac{p^{2}(1+\varepsilon)^{2}d_{Y}d_{V}\left\Vert Lh\right\Vert _{\infty}^{2}}{2}$ and $\varepsilon>0$ can be made arbitrary small we get the following condition for the partition mesh $\delta<\left(p\left\Vert Lh\right\Vert _{\infty}\sqrt{d_{Y}d_{V}}\right)^{-1}.$
We complete the section with an application of the previous two lemmas. Note that, in order to control the high order discretizations of the likelihood function, we reduce the problem to the control of the second order discretization via the truncation procedure as described in Remark \[rem:TruncProc\].
\[cor: BoundsFi\]Let $\varphi\in\mathcal{B}_{P}$. One has that:
1. \[enu: condL\]If $\mathbf{H}\left(1\right)$ holds, then there exists $\varepsilon>0$ such that $$\mathbb{\tilde{E}}\left[\left\vert \varphi(X_{t})e^{\xi_{t}}\right\vert ^{2+\varepsilon}\right]<\infty,\label{eq: CondL}$$ and $$\sup_{\tau\in\Pi(t)}\mathbb{\tilde{E}}\left[\left\vert \varphi(X_{t})e^{\xi_{t}^{\tau,1}}\right\vert ^{2+\varepsilon}\right]<\infty.\label{eq: CondL1Pi}$$
2. \[enu: condL2\]If $\mathbf{H}\left(2\right)$ holds, then there exists $\varepsilon>0$ and $\delta_{0}=\delta_{0}\left(h,f,\sigma,\right)>0$ such that $$\sup_{\tau\in\Pi(t,\delta_{0})}\mathbb{\tilde{E}}\left[\left\vert \varphi(X_{t})e^{\xi_{t}^{\tau,2}}\right\vert ^{2+\varepsilon}\right]<\infty.\label{eq: CondL2Pi}$$
3. \[enu: condLm\]If $\mathbf{H}\left(m\right)$ with $m\geq3$ holds, then there exists $\varepsilon>0$ and $\delta_{0}>0$ such that $$\sup_{\tau\in\Pi(t,\delta_{0})}\mathbb{\tilde{E}}\left[\left\vert \varphi(X_{t})e^{\bar{\xi}_{t}^{\tau,m}}\right\vert ^{2+\varepsilon}\right]<\infty.\label{eq: CondLmPi}$$
Combining Lemmas \[lem: Z\_t\^p\_Integrability\] and \[lem: Xi\_Tau2\^p\_Integrability\] with Hölder inequality and Remark \[rem: M\] we obtain $\left(\ref{eq: CondL}\right)$, $\left(\ref{eq: CondL1Pi}\right)$ and $\left(\ref{eq: CondL2Pi}\right)$. Moreover, for $m\geq3$, note that $$\begin{aligned}
\bar{\xi}_{t}^{\tau,m} & =\xi_{t}^{\tau,2}+\sum_{j=0}^{n-1}\Gamma_{m-\frac{1}{2},\delta_{j}}\left(\mu^{\tau,m}\left(j\right)\right)\\
& \leq\xi_{t}^{\tau,2}+\sum_{j=0}^{n-1}\left|\Gamma_{m-\frac{1}{2},\delta_{j}}\left(\mu^{\tau,m}\left(j\right)\right)\right|\\
& \leq\xi_{t}^{\tau,2}+\sum_{j=0}^{n-1}\delta_{j}=\xi_{t}^{\tau,2}+t,\end{aligned}$$ and (\[eq: CondLmPi\]) follows from (\[eq: CondL2Pi\]).
Proof of the Theorem \[thm: Main Filtering\_2\]\[sec: Proof Main Theorem\]
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In this section we prove the main theorem of the paper. We start by stating and proving two main propositions.
\[prop: Main1\]Let $m\in\mathbb{N}$ and assume that condition **H**$(m)$ holds and $\varphi\in C_{P}^{m+1}$. Then, there exists a constant $C$ independent of the partition $\pi\in\Pi$ such that $$\mathbb{\tilde{E}}\left[\left\vert \mathbb{\tilde{E}}\left[(\xi_{t}-\xi_{t}^{\tau,m})\varphi(X_{t})e^{\xi_{t}}|\mathcal{Y}_{t}\right]\right\vert ^{2}\right]\leq C\delta^{2m}.$$
By Lemma \[lem: Difference Moment Estimate\] we can write $$(\xi_{t}-\xi_{t}^{\tau,m})\varphi(X_{t})e^{\xi_{t}}=\varphi(X_{t})e^{\xi_{t}}\left(\sum_{i=0}^{d_{Y}}\int_{0}^{t}\left\{ \sum_{\alpha\in\mathcal{R}(\mathcal{M}_{m-1}(S_{0}))}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}\right\} dY_{s}^{i}\right).$$ For $i=0$, the result follows from Lemmas \[lem: Alpha0=00003D00003Dm\] and \[lem: ds\]. Recall that $$\mathcal{R}\left(\mathcal{M}_{m-1}(S_{0})\right)_{k}=\{\alpha\in\mathcal{R}\left(\mathcal{M}_{m-1}(S_{0})\right):\left\vert \alpha\right\vert _{0}=k\},k=0,..,m,$$ that is, $\mathcal{R}\left(\mathcal{M}_{m-1}(S_{0})\right)_{k}$ is the set of multi-indices in $\mathcal{R}\left(\mathcal{M}_{m-1}(S_{0})\right)$ that contain $k$ zeros. This collection of sets are obviously a disjoint partition of $\mathcal{R}\left(\mathcal{M}_{m-1}(S_{0})\right),$ that is, $\mathcal{R}\left(\mathcal{M}_{m-1}(S_{0})\right)={\displaystyle \biguplus\limits _{k=0}^{m}}\mathcal{R}\left(\mathcal{M}_{m-1}(S_{0})\right)_{k}.$ For $i\neq0$, we will divide the proof of the theorem in cases, depending on $\alpha$ belonging to one of these subsets. The cases for $m\in\left\{ 1,2\right\} $ are:\
$\bullet$ $m=1,\alpha\in\mathcal{R}\left(\mathcal{M}_{0}(S_{0})\right)_{0}$: Lemma \[lem: dY\_m=00003D00003D1\].\
$\bullet$ $m=1,\alpha\in\mathcal{R}\left(\mathcal{M}_{0}(S_{0})\right)_{1}$: Lemma \[lem: Alpha0=00003D00003Dm\].\
$\bullet$ $m=2,\alpha\in\mathcal{R}\left(\mathcal{M}_{1}(S_{0})\right)_{0}$: Lemma \[lem: dY\_m=00003D00003D2\_alpha=00003D00003D1\].\
$\bullet$ $m=2,\alpha\in\mathcal{R}\left(\mathcal{M}_{1}(S_{0})\right)_{1}$: Lemma \[lem: dY\_m=00003D00003D2\_alpha=00003D00003D2\].\
$\bullet$ $m=2,\alpha\in\mathcal{R}\left(\mathcal{M}_{1}(S_{0})\right)_{2}$: Lemma \[lem: Alpha0=00003D00003Dm\].
For arbitrary $m$$>2$, the proof follows the same ideas as for $m\in\{1,2\}.$ In the case that $\alpha\in\mathcal{R}\left(\mathcal{M}_{m-1}(S_{0})\right)_{m}$ the result follows from applying Lemma \[lem: Alpha0=00003D00003Dm\]. For $\mathcal{R}\left(\mathcal{M}_{m-1}(S_{0})\right)_{k}$ with $k\in\{0,m-1\}$, first one needs to use the truncated Stroock-Taylor formula of order $k$ to express $\varphi(X_{t})e^{\xi_{t}}$ as a sum of iterated integrals with respect to the Brownian motion. The goal is to use the covariance between the iterated integrals in the Stroock-Taylor expansion of $\varphi(X_{t})e^{\xi_{t}}$ and $I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}$ in order to generate the right order of convergence in $\delta$. However, this is not straightforward due to the presence of the stochastic integral with respect to Y. The process Y as an integrator makes impossible to use directly an integration by parts formula because the two iterated integrals are semimartingales with respect to different filtrations. To overcome this difficulty, the idea is to compute this covariance along a partition. We use an integration by parts formula, in each subinterval and only to the integral with respect to Y, to obtain $$\int_{t_{j}}^{t_{j+1}}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dY_{s}^{i}=\int_{t_{j}}^{t_{j+1}}\left(Y_{t_{j+1}}^{i}-Y_{s}^{i}\right)I_{\alpha-}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dV^{\alpha_{|\alpha|}}.$$ The term on the right hand side in the last expression is an $\mathbb{H}^{t}$-semimartingale and we can compute its covariation with the terms in the Stroock-Taylor expansion of $\varphi(X_{t})e^{\xi_{t}}$, see Lemmas \[lem: dY\_m=00003D00003D1\], \[lem: dY\_m=00003D00003D2\_alpha=00003D00003D1\] and \[lem: dY\_m=00003D00003D2\_alpha=00003D00003D2\].
\[prop: Main2\]Let $m\in\mathbb{N}$ and assume that condition **H**$(m)$ holds and $\varphi\in\mathcal{B}_{P}$. Then, there exist $\delta_{0}>0$ and constant $C$ independent of any partition $\pi\in\Pi\left(t,\delta_{0}\right)$ such that $$\mathbb{\tilde{E}}\left[\left\vert \left\vert \varphi(X_{t})\right\vert \left(e^{\xi_{t}}+e^{\bar{\xi}_{t}^{\tau,m}}\right)(\xi_{t}-\bar{\xi}_{t}^{\tau,m})^{2}\right\vert ^{2}\right]\leq C\delta^{2m}.$$
As **H**$(m)$ holds, let $\varepsilon>0$ such as in Corollary \[cor: BoundsFi\]. By Hölder inequality we have that $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\left\vert \left\vert \varphi(X_{t})\right\vert \left(e^{\xi_{t}}+e^{\bar{\xi}_{t}^{\tau,m}}\right)(\xi_{t}-\bar{\xi}_{t}^{\tau,m})^{2}\right\vert ^{2}\right]\\
& \leq\mathbb{\tilde{E}}\left[\left\vert \left\vert \varphi(X_{t})\right\vert \left(e^{\xi_{t}}+e^{\bar{\xi}_{t}^{\tau,m}}\right)\right\vert ^{2+\varepsilon}\right]^{\frac{2}{2+\varepsilon}}\mathbb{\tilde{E}}\left[\left\vert \xi_{t}-\bar{\xi}_{t}^{\tau,m}\right\vert ^{4\frac{2+\varepsilon}{\varepsilon}}\right]^{\frac{\varepsilon}{2+\varepsilon}},\end{aligned}$$ Corollary \[cor: BoundsFi\] yields that there exists $\delta_{0}>0$ such that $$\sup_{\tau\in\Pi\left(t,\delta_{0}\right)}\mathbb{\tilde{E}}\left[\left\vert \left\vert \varphi(X_{t})\right\vert \left(e^{\xi_{t}}+e^{\bar{\xi}_{t}^{\tau,m}}\right)\right\vert ^{2+\varepsilon}\right]<\infty.\label{eq: BoundAuxiliary}$$ On the other hand, by equation $\left(\ref{eq:BigEpsilon}\right)$, for any $p\geq1$ we get that $$\begin{aligned}
\left\vert \xi_{t}-\bar{\xi}_{t}^{\tau,m}\right\vert ^{2p} & \leq C\left\{ \left|\xi_{t}-\xi_{t}^{\tau,m}\right|^{2p}+\left|\sum_{j=0}^{n-1}\mathcal{E}_{m-\frac{1}{2}}\left(\mu^{\tau,m}\left(j\right)\right)\right|^{2p}\right\} .\end{aligned}$$ By Lemma \[lem: Difference Moment Estimate\], we obtain $$\mathbb{\tilde{E}}\left[\left|\xi_{t}-\xi_{t}^{\tau,m}\right|^{2p}\right]\leq C\delta^{pm},$$ and by Lemma \[lem: SumEpsilon\] with $q=m-\frac{1}{2}$ we have that $$\mathbb{\tilde{E}}\left[\left\vert \sum_{j=0}^{n-1}\mathcal{E}_{m-\frac{1}{2},\delta_{j}}\left(\mu^{\tau,m}\left(j\right)\right)\right\vert ^{2p}\right]\leq C\left(t,d_{Y},p,m\right)\delta^{2pm}$$ Hence, setting $p=2\left(2+\varepsilon\right)/\varepsilon$, we obtain $$\mathbb{\tilde{E}}\left[\left\vert \xi_{t}-\bar{\xi}_{t}^{\tau,m}\right\vert ^{4\left(2+\varepsilon\right)/\varepsilon}\right]^{\varepsilon/\left(2+\varepsilon\right)}\leq C\delta^{2m}.$$
We are finally ready to put everything together and deduce Theorem \[thm: Main Filtering\_2\].
To get the desired rate of convergence for the unnormalised conditional distribution $\rho_{t}^{\tau,m},$ we can write $$\begin{aligned}
\rho_{t}\left(\varphi\right)-\rho_{t}^{\tau,m}\left(\varphi\right) & =\mathbb{\tilde{E}}[\varphi(X_{t})(\xi_{t}-\bar{\xi}_{t}^{\tau,m})e^{\xi_{t}}|\mathcal{Y}_{t}]\\
& +\mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}-\varphi(X_{t})e^{\bar{\xi}_{t}^{\tau,m}}-\varphi(X_{t})(\xi_{t}-\bar{\xi}_{t}^{\tau,m})e^{\xi_{t}}|\mathcal{Y}_{t}\right].\end{aligned}$$ Using the inequality $$\left\vert e^{x}-e^{y}-(x-y)e^{x}\right\vert \leq\frac{e^{x}+e^{y}}{2}(x-y)^{2},$$ we get that $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\left\vert \rho_{t}\left(\varphi\right)-\rho_{t}^{\tau,m}\left(\varphi\right)\right\vert ^{2}\right]\\
& \leq C\left\{ \mathbb{\tilde{E}}\left[\left\vert \mathbb{\tilde{E}}\left[(\xi_{t}-\bar{\xi}_{t}^{\tau,m})\varphi(X_{t})e^{\xi_{t}}|\mathcal{Y}_{t}\right]\right\vert ^{2}\right]+\mathbb{\tilde{E}}\left[\left\vert \left\vert \varphi(X_{t})\right\vert \frac{e^{\xi_{t}}+e^{\bar{\xi}_{t}^{\tau,m}}}{2}(\xi_{t}-\bar{\xi}_{t}^{\tau,m})^{2}\right\vert ^{2}\right]\right\} .\end{aligned}$$ Now, Propositions \[prop: Main1\] and \[prop: Main2\] yield $$\mathbb{\tilde{E}}\left[\left\vert \rho_{t}\left(\varphi\right)-\rho_{t}^{\tau,m}\left(\varphi\right)\right\vert ^{2}\right]\leq C\delta^{2m}.$$
To prove the rate for the normalised conditional distribution observe that we can write $$\pi_{t}^{\tau,m}\left(\varphi\right)-\pi_{t}\left(\varphi\right)=\frac{1}{\rho_{t}\left(\boldsymbol{1}\right)}\frac{\rho_{t}^{\tau,m}\left(\varphi\right)}{\rho_{t}^{\tau,m}\left(\boldsymbol{1}\right)}\left(\rho_{t}\left(\boldsymbol{1}\right)-\rho_{t}^{\tau,m}\left(\boldsymbol{1}\right)\right)+\frac{1}{\rho_{t}\left(\boldsymbol{1}\right)}\left(\rho_{t}^{\tau,m}\left(\varphi\right)-\rho_{t}\left(\varphi\right)\right).$$ Hence, $$\begin{aligned}
& \mathbb{E}\left[\left\vert \pi_{t}\left(\varphi\right)-\pi_{t}^{\tau,m}\left(\varphi\right)\right\vert \right]\\
& \leq C\mathbb{\tilde{E}}\left[\frac{Z_{t}}{\left\vert \rho_{t}\left(\boldsymbol{1}\right)\right\vert }\left\{ \left\vert \pi_{t}^{\tau,m}\left(\varphi\right)\right\vert \left\vert \rho_{t}\left(\boldsymbol{1}\right)-\rho_{t}^{\tau,m}\left(\boldsymbol{1}\right)\right\vert +\left\vert \rho_{t}^{\tau,m}\left(\varphi\right)-\rho_{t}\left(\varphi\right)\right\vert \right\} \right]\\
& =C\mathbb{\tilde{E}}\left[\frac{\mathbb{\tilde{E}}[Z_{t}|\mathcal{Y}_{t}]}{\left\vert \rho_{t}\left(\boldsymbol{1}\right)\right\vert }\left\{ \left\vert \pi_{t}^{\tau,m}\left(\varphi\right)\right\vert \left\vert \rho_{t}\left(\boldsymbol{1}\right)-\rho_{t}^{\tau,m}\left(\boldsymbol{1}\right)\right\vert +\left\vert \rho_{t}^{\tau,m}\left(\varphi\right)-\rho_{t}\left(\varphi\right)\right\vert \right\} \right]\\
& \leq C\left\{ \mathbb{\tilde{E}}\left[\left\vert \pi_{t}^{\tau,m}\left(\varphi\right)\right\vert \left\vert \rho_{t}\left(\boldsymbol{1}\right)-\rho_{t}^{\tau,m}\left(\boldsymbol{1}\right)\right\vert \right]+\mathbb{\tilde{E}}\left[\left\vert \rho_{t}^{\tau,m}\left(\varphi\right)-\rho_{t}\left(\varphi\right)\right\vert \right]\right\} .\\
& \leq C\left\{ \mathbb{\tilde{E}}\left[\left\vert \pi_{t}^{\tau,m}\left(\varphi\right)\right\vert ^{2}\right]^{1/2}\mathbb{\tilde{E}}\left[\left\vert \rho_{t}\left(\boldsymbol{1}\right)-\rho_{t}^{\tau,m}\left(\boldsymbol{1}\right)\right\vert ^{2}\right]^{1/2}+\mathbb{\tilde{E}}\left[\left\vert \rho_{t}^{\tau,m}\left(\varphi\right)-\rho_{t}\left(\varphi\right)\right\vert ^{2}\right]^{1/2}\right\} ,\end{aligned}$$ where in the last inequality we have applied Hölder inequality. Combining the bounds for the unnormalised distribution and the hypothesis on $\pi_{t}^{\tau,m}\left(\varphi\right)$ we can conclude.
\[sec:Technical-Lemmas\]Technical Lemmas
========================================
We collate in this section the technical lemmas required to prove the main results. We begin with some limited background material on Malliavin Calculus (and partial Maliavin Calculus) with a view to deduce the necessary properties of the functionals to be discretised.
\[subsec:Malliavin-calculus\]Malliavin calculus
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Let $B$=$\left\{ B_{t}\right\} _{t\in\left[0,T\right]}$ be a $d$-dimensional standard Brownian motion defined on a complete probability space $\left(\Omega,\mathcal{F},P\right).$ Let $\mathcal{S}$ denote the class of smooth random variables such that a random variable $F$$\in\mathcal{S}$ has the form $$F=f\left(B_{t_{1}};...;B_{t_{n}}\right),$$ where the function $f\left(x^{11},...,x^{d1};...;x^{1n},...,x^{dn}\right)$ belongs to $C_{b}^{\infty}\left(\mathbb{R}^{dn}\right)$ and $t_{1},...,t_{n}\in\left[0,T\right]$. The Malliavin derivative of a smooth functional $F$ can be defined as the $d$-dimensional stochastic processes given by $$\left(DF\right)_{t}^{j}=\sum_{i=1}^{n}\frac{\partial f}{\partial x^{ji}}\left(B_{t_{1}};...;B_{t_{n}}\right)\mathbf{1}_{\left[0,t_{i}\right]}\left(t\right),$$ for $t\in\left[0,T\right]$ and $j=1,...,d$. The derivative $DF$ can be regarded as a random variable taking values in the Hilbert space $H=L^{2}\left(\left[0,T\right];\mathbb{R}^{d}\right)$. Noting the isometry $L^{2}\left(\Omega\times\left[0,T\right];\mathbb{R}^{d}\right)\simeq L^{2}\left(\Omega;H\right)$ we can identify $\left(DF\right)_{t}^{j}$ as the value at time $t$ of the $j$th component of and $\mathbb{R}$$^{d}-$valued stochastic process. We will also the notation $D_{t}^{j}F$ for $\left(DF\right)_{t}^{j}$. One can see that the operator $D$ is closable from $L^{p}\left(\Omega\right)$ to $L$$^{p}\left(\Omega;H\right)$, $p\geq1$ and we will denote the domain of $D$ in $L$$^{p}\left(\Omega\right)$ by $\mathbb{D}^{1,p}.$ That is, meaning that $\mathbb{D}^{1,p}$ is the closure of smooth random variables $\mathcal{S}$ with respect to the norm $$\left\Vert F\right\Vert _{\mathbb{D}^{1,p}}=\left(\mathbb{E}\left[\left|F\right|^{p}\right]+\mathbb{E}\left[\left\Vert DF\right\Vert _{H}^{p}\right]\right)^{1/p}.$$ We define the $k$-th derivative of $F$, $D$$^{k}F$, as the $H$$^{\otimes k}$-valued random variable $$\left(D^{k}F\right)_{s_{1},...,s_{k}}^{j_{1},...,j_{k}}=\sum_{i_{1},...,i_{k}=1}^{n}\frac{\partial^{k}f}{\partial x^{j_{1}i_{1}}\cdots\partial x^{j_{k}i_{k}}}\left(B_{t_{1}};...;B_{t_{n}}\right)\mathbf{1}_{\left[0,t_{i_{1}}\right]}\left(s_{1}\right)\cdots\mathbf{1}_{\left[0,t_{i_{k}}\right]}\left(s_{k}\right),$$ where $s_{1},...,s_{k}$$\in\left[0,T\right]$ and $j_{1},...,j_{k}=1,...,d$. We will also write $D_{s_{1},...,s_{k}}^{j_{1},...,j_{k}}F$ for $\left(D^{k}F\right)_{s_{1},...,s_{k}}^{j_{1},...,j_{k}}$ and notice that it coincides with the iterated derivative $D_{t_{1}}^{j_{1}}\cdots D_{t_{k}}^{j_{k}}F$. For any integer $k$$\geq1$ and any real number $p>1$ we introduce the norm on $\mathcal{S}$ given by $$\left\Vert F\right\Vert _{k,p}=\left(\mathbb{E}\left[\left|F\right|^{p}\right]+\sum_{j=1}^{k}\mathbb{E}\left[\left\Vert D^{j}F\right\Vert _{H^{\otimes j}}^{p}\right]\right)^{1/p},$$ where $$\left\Vert D^{k}F\right\Vert _{H^{\otimes k}}=\left(\sum_{j_{1}\cdots j_{k}=1}^{d}\int_{\left[0,T\right]^{k}}\left|D_{s_{1},...,s_{k}}^{j_{1},...,j_{k}}F\right|^{2}ds_{1}\cdots ds_{k}\right)^{1/2}.$$ We will denote by $\mathbb{D}^{k,p}$ the completion of the family of random variables $\mathcal{S}$ with respect to the norm $\left\Vert \cdot\right\Vert _{k,p}$. We also define the space $\mathbb{D}^{k,\infty}=\bigcap_{p\geq1}\mathbb{D}^{k,p}$. We have the following chain rule formula for the Malliavin derivative.
\[prop: ChainRule\]Let $\varphi:\mathbb{R}^{m}\rightarrow\mathbb{R}$ be of class $C_{P}^{1}$$\left(\mathbb{R}^{m}\right)$. Suppose that $F=\left(F^{1},...,F^{m}\right)$ is a random vector whose components belong to $\mathbb{D}^{1,\infty}$. Then, $\varphi\left(F\right)$$\in\mathbb{D}^{1,\infty}$ and $$D_{t}^{j}\varphi\left(F\right)=\sum_{i=1}^{m}\frac{\partial\varphi}{\partial x^{i}}\left(F\right)D_{t}^{j}F^{i},$$ where $t\in\left[0,T\right]$ and $\dot{j}=1,...,d.$
The proof follows the same ideas as the proof of Proposition 1.2.3 in Nualart [@Nu06], where is proved for $\varphi\in C_{b}^{1}\left(\mathbb{R}^{m}\right)$ and $F$$\in\mathbb{D}^{1,p}$. One can extend the result to $\varphi\in C_{P}^{1}\left(\mathbb{R}^{m}\right)$ by requiring $F\in\mathbb{D}^{1,\infty}$ and using Hölder inequality.
As a corollary of Proposition \[prop: ChainRule\] one obtains that the product rule and the binomial formula holds for the Malliavin derivative of products of random variables in $\mathbb{D}^{1,\infty}$. However, Proposition \[prop: ChainRule\] does not apply to the exponential function. In order to show that the likelihood functional $e^{\xi_{t}}$ is smooth in the Malliavin sense we need the following lemma.
\[lem: MallDerExponential\]Let $F$$\in\mathbb{D}^{1,\infty}$ and such that $$\mathbb{E}\left[\exp\left(p\left|F\right|\right)\right]<\infty,\label{eq:ExpAbsMoments}$$ for all $p\geq1.$ Then $G=e^{F}$$\in\mathbb{D}^{1,\infty}$ and $$D_{t}^{j}G=GD_{t}^{j}F,\label{eq:DerivExpo}$$ where $t\in\left[0,T\right]$ and $\dot{j}=1,...,d.$
Define $G_{n}=\sum_{k=0}^{n}\frac{F^{k}}{k!}.$ As $F$$\in\mathbb{D}^{1,\infty},$ Proposition \[prop: ChainRule\] yields that $G_{n}\in\mathbb{D}^{1,\infty}$ and $$DG_{n}=\sum_{k=1}^{n}\frac{kF^{k-1}}{k!}DF=G_{n-1}DF.$$ In order to prove that $G\in\mathbb{D}^{1,\infty}$ and that the identity $\left(\ref{eq:DerivExpo}\right)$ is satisfied, it suffices to show that for all $p\geq1$ one has that $G_{n}$ converges to $G$ in $L^{p}\left(\Omega\right)$ and $$\mathbb{E}\left[\left\Vert DG-DG_{n}\right\Vert _{H}^{p}\right]\longrightarrow0,$$ when $n$ tends to infinity. Note that $$\begin{aligned}
\mathbb{E}\left[\left\Vert DG-DG_{n}\right\Vert _{H}^{p}\right] & =\mathbb{E}\left[\left|G-G_{n-1}\right|^{p}\left|\int_{0}^{T}\left|D_{t}F\right|^{2}dt\right|^{p/2}\right]\\
& \leq\mathbb{E}\left[\left|G-G_{n-1}\right|^{2p}\right]^{1/2}\mathbb{E}\left[\left\Vert DF\right\Vert _{L^{2}\left(\left[0,T\right]\right)}^{2p}\right]^{1/2}.\end{aligned}$$ Hence, the problem is reduced to show that $G_{n}$ converges to $G$ in $L$$^{p}$ for all $p$$\geq1$. Equivalently, defining $$G_{n}^{c}:=G-G_{n}=\sum_{k=n+1}^{\infty}\frac{F^{k}}{k!},$$ it suffices to prove that $G_{n}^{c}$ converges to $0$ in $L$$^{p}$ for all $p$$\geq1$. Clearly, $G_{n}^{c}$ converges to $0$ almost surely and, thanks to assumption $\left(\ref{eq:ExpAbsMoments}\right)$, the dominated convergence theorem yields that $G_{n}^{c}$ also converges to 0 in $L^{p}\left(\Omega\right)$ for all $p$$\geq1$.
We also have the following relationship between the conditional expectation and the Malliavin derivative.
\[lem:CondExpMallDer\]Let $F$$\in\mathbb{D}^{1,2}$ and $\mathbb{F}=\left\{ \mathcal{F}_{t}\right\} _{t\in\left[0,T\right]}$ be the P-augmented natural filtration generated by $B$. Then $\mathbb{E}\left[F|\mathcal{F}_{t}\right]\in\mathbb{D}^{1,2}$ and $D_{s}^{j}\mathbb{E}\left[F|\mathcal{F}_{t}\right]=\mathbb{E}\left[D_{s}^{j}F|\mathcal{F}_{t}\right]\mathbf{1}_{\left[0,t\right]}\left(s\right)$$,j=1,...,d.$
The lemma is a particular case of Proposition 1.2.8 in Nualart [@Nu06].
The following is an important result regarding the Malliavin differentiability of the solution of a stochastic differential equation.
\[lem: MomentMalliavinDerivative\]If $X_{t}\in\mathbb{R}^{n}$ is the solution to $$X_{t}=x+\int_{0}^{t}V_{0}\left(X_{s}\right)ds+\int_{0}^{t}V\left(X_{s}\right)dB_{s},$$ where the components of V$_{0}$ and $V$ are $m$-times continuously differentiable with bounded derivatives of order greater or equal than one and $B_{t}=(B_{t}^{1},...,B_{t}^{d})$ is a $d$-dimensional Brownian motion. Then, $X_{t}^{i}\in\mathbb{D}^{m,\infty},t\in[0,T],i=1,...,n.$ Furthermore, for any $p\geq1$ one has that $$\sup_{r_{1},r_{2},...,r_{k}\in[0,T]}\mathbb{E}\left[\sup_{r_{1}\vee r_{2}\vee\cdots\vee r_{k}\leq t\leq T}\left\vert D_{r_{1},r_{2},...,r_{k}}^{j_{1},j_{2},...,j_{k}}X_{t}^{i}\right\vert ^{p}\right]<\infty,$$ for all $p\geq1,$$i=1,...,n,$ $j_{k}\in\left\{ 1,...,d\right\} $ and $1\leq k\leq m$.
See Nualart [@Nu06], Theorem 2.2.1. and 2.2.2.
\[rem:CommuteDandY\]We will be using a variation of the classical Malliavin calculus known as *partial Malliavin calculus*. This calculus was introduced in Kusuoka and Stroock [@KuStr84] and Nualart and Zakai [@NuZa89] with a view towards its application to the stochastic filtering problem, see also Tanaka [@Ta14]. The idea is to consider only the Malliavin derivative operator with respect some of the components of the Brownian motion $B$. In our setting $B$=$\left(V,Y\right)$ is a $d_{V}+d_{Y}$-dimensional Brownian motion under $\tilde{P}$ and the Malliavin differentiation will be only with respect to the Brownian motion $V$. The main consequence of this approach is that the Malliavin derivative with respect to $V$ commutes with the stochastic integral with respect to $Y$.
\[lem: BoundedIteratedMD1\]Let $m\in\mathbb{N}$ and assume that **H**$(m)$ holds and $\varphi\in C_{P}^{m+1}$. Then, the random variable $\varphi(X_{t})e^{\xi_{t}}$ belongs to $\mathbb{D}^{m+1,\infty}.$ Moreover, $$\sup_{r_{1},...,r_{\left\vert \alpha\right\vert }\in[0,t]}\mathbb{\tilde{E}}\left[\left\vert D_{r_{1},...,r_{_{\left\vert \alpha\right\vert }}}^{\alpha_{1},...,\alpha_{\left\vert \alpha\right\vert }}(\varphi(X_{t})e^{\xi_{t}})\right\vert ^{p}\right]<\infty,$$ for all $p\geq1$ and $\alpha\in\mathcal{M}_{m+1}(S_{1}).$
To ease the notation we are only going to give the proof for $d_{V}=d_{Y}=d_{X}=1.$ We will also use the notation $D_{r_{1},...,r_{k}}^{k}F=D_{r_{1},...,r_{k}}^{1,\overset{k}{\overbrace{...}},1}F$. Lemma \[lem: MomentMalliavinDerivative\] yields that $X_{t}\in\mathbb{D}^{1,\infty}$. Applying iteratively Proposition \[prop: ChainRule\] we obtain that $\varphi(X_{t})\in\mathbb{D}^{m+1,\infty}$ and $h$$(X_{t})\in\mathbb{D}^{m+1,\infty}$. Taking into account Remark \[rem:CommuteDandY\], we have that $\xi_{t}\in\mathbb{D}^{m+1,\infty}$. Moreover, thanks to Lemma \[lem: Z\_t\^p\_Integrability\], we can apply iteratively Lemma \[lem:CondExpMallDer\] and conclude that $e^{\xi_{t}}\in\mathbb{D}^{m+1,\infty}$. For any $\alpha\in\mathcal{M}_{m+1}(S_{1}),$ by Leibniz’s rule, we can write $$\begin{aligned}
D_{r_{1},...,r_{_{\left\vert \alpha\right\vert }}}^{\alpha_{1},...,\alpha_{\left\vert \alpha\right\vert }}\left(\varphi(X_{t})e^{\xi_{t}}\right) & =D_{r_{1},...,r_{_{\left\vert \alpha\right\vert }}}^{\left\vert \alpha\right\vert }\left(\varphi(X_{t})e^{\xi_{t}}\right)\\
& =\sum_{k=0}^{\left\vert \alpha\right\vert }\binom{\left\vert \alpha\right\vert }{k}\left(D_{r_{1},...,r_{k}}^{k}\varphi(X_{t})\right)(D_{r_{1},...,r_{\left\vert \alpha\right\vert -k}}^{\left\vert \alpha\right\vert -k}e^{\xi_{t}}),\end{aligned}$$ and applying Schwartz’s inequality one has that $$\begin{aligned}
\mathbb{\tilde{E}}\left[\left\vert D_{r_{1},...,r_{_{\left\vert \alpha\right\vert }}}^{\left\vert \alpha\right\vert }\left(\varphi(X_{t})e^{\xi_{t}}\right)\right\vert ^{p}\right] & \leq C\sum_{k=0}^{\left\vert \alpha\right\vert }\binom{\left\vert \alpha\right\vert }{k}\mathbb{\tilde{E}}\left[\left\vert \left(D_{r_{1},...,r_{k}}^{k}\varphi(X_{t})\right)(D_{r_{1},...,r_{\left\vert \alpha\right\vert -k}}^{\left\vert \alpha\right\vert -k}e^{\xi_{t}})\right\vert ^{p}\right]\\
& \leq C\sum_{k=0}^{\left\vert \alpha\right\vert }\binom{\left\vert \alpha\right\vert }{k}\mathbb{\tilde{E}}\left[\left\vert D_{r_{1},...,r_{k}}^{k}\varphi(X_{t})\right\vert ^{2p}\right]^{1/2}\mathbb{\tilde{E}}\left[\left\vert D_{r_{1},...,r_{\left\vert \alpha\right\vert -k}}^{\left\vert \alpha\right\vert -k}e^{\xi_{t}}\right\vert ^{2p}\right]^{1/2}.\end{aligned}$$ Hence, the result follows if we show that $$\begin{aligned}
\sup_{r_{1},...,r_{k}\in[0,t]}\mathbb{\tilde{E}}\left[|D_{r_{1},...,r_{k}}^{k}\varphi(X_{t})|^{p}\right] & <\infty,\quad0\leq k\leq\left\vert \alpha\right\vert ,\label{eq: MDF}\\
\sup_{r_{1},...,r_{k}\in[0,t]}\mathbb{\tilde{E}}[|D_{r_{1},...,r_{k}}^{k}e^{\xi_{t}}|^{p}] & <\infty,\quad0\leq k\leq\left\vert \alpha\right\vert ,\label{eq: MDG}\end{aligned}$$ for any $p\geq1$.
*Proof of (\[eq: MDF\])*$:$
If $k=0,$ using that $\mathbf{H}\left(m\right)$ holds and $\varphi\in C_{P}^{m+1}$, we have that $\mathbb{\tilde{E}}[|\varphi(X_{t})|^{p}]<\infty$, by Remark \[rem: M\]. If $1\leq k\leq\left\vert \alpha\right\vert ,$ we use Faà di Bruno’s formula to obtain an expression for $D_{r_{1},...,r_{k}}^{k}\varphi(X_{t})$ in terms of the so called partial Bell polynomials, which are given by $$B_{k,a}(x_{1},...,x_{k})=\sum_{(j_{1},...,j_{k})\in\Lambda(k,a)}\frac{k!}{j_{1}!\left(1!\right)^{j_{1}}j_{2}!\left(2!\right)^{j_{2}}\cdots j_{k}!(k!)^{j_{k}}}x_{1}^{j_{1}}x_{2}^{j_{2}}\cdots x_{k}^{j_{k}},$$ where $1\leq a\leq k$ and $$\Lambda(k,a)=\{(j_{1},...,j_{k})\in\mathbb{Z}_{+}^{k}:j_{1}+2j_{2}+\cdots+kj_{k}=k,j_{1}+j_{2}+\cdots+j_{k}=a\}.$$ In particular, we have that $$D_{r_{1},...,r_{k}}^{k}\varphi(X_{t})=\sum_{a=1}^{k}\varphi^{(a)}(X_{t})B_{k,a}(D_{r_{1}}^{1}X_{t},D_{r_{1},r_{2}}^{2}X_{t},...,D_{r_{1},...,r_{k}}^{k}X_{t}).$$ Hence, for any $p\geq1,$ applying Cauchy-Schwarz inequality we get $$\begin{aligned}
& \mathbb{\tilde{E}}[|D_{r_{1},...,r_{k}}^{k}\varphi(X_{t})|^{p}]\\
& \leq C\sum_{a=1}^{k}\mathbb{\tilde{E}}[|\varphi^{(a)}(X_{t})B_{k,a}(D_{r_{1}}^{1}X_{t},D_{r_{1},r_{2}}^{2}X_{t},...,D_{r_{1},...,r_{k}}^{k}X_{t})|^{p}]\\
& \leq C\sum_{a=1}^{k}\mathbb{\tilde{E}}[|\varphi^{(a)}(X_{t})|^{2p}]^{1/2}\mathbb{\tilde{E}}[|B_{k,a}(D_{r_{1}}^{1}X_{t},D_{r_{1},r_{2}}^{2}X_{t},...,D_{r_{1},...,r_{k}}^{k}X_{t})|^{2p}]^{1/2}.\end{aligned}$$ The terms $\mathbb{\tilde{E}}[\left\vert \varphi^{(a)}(X_{t})\right\vert ^{2p}]<\infty,a=1,...,k,$ due to Remark \[rem: M\] combined with that $\mathbf{H}\left(m\right)$ holds and $\varphi\in C_{P}^{m+1}.$ On the other hand, using the generalized version of Hölder’s inequality, Lemma \[lem: Generalized Holder\], we can bound $$\mathbb{\tilde{E}}[|B_{k,a}(D_{r_{1}}^{1}X_{t},D_{r_{1},r_{2}}^{2}X_{t},...,D_{r_{1},...,r_{k}}^{k}X_{t})|^{2p}],\quad1\leq a\leq k,$$ by a sum of products of expectations of powers of Malliavin derivatives of $X$ of different orders. Combining this bound with Lemma \[lem: MomentMalliavinDerivative\] we get that the integrability condition $\left(\ref{eq: MDF}\right)$ is satisfied.
*Proof of (\[eq: MDG\])*$:$
If $k=0,$ we have that $\mathbb{\tilde{E}}\left[|e^{\xi_{t}}|^{p}\right]<\infty$ due to Lemma \[lem: Z\_t\^p\_Integrability\]. If $1\leq k\leq\left\vert \alpha\right\vert ,$ using again Faà di Bruno’s formula we get $$\begin{aligned}
D_{r_{1},...,r_{k}}^{k}e^{\xi_{t}} & =\sum_{a=1}^{k}\left.\frac{d^{a}}{dx^{a}}e^{x}\right\vert _{x=\xi_{t}}B_{k,a}(D_{r_{1}}^{1}\xi_{t},D_{r_{1},r_{2}}^{2}\xi_{t},...,D_{r_{1},...,r_{k}}^{k}\xi_{t})\\
& =\sum_{a=1}^{k}\exp(\xi_{t})B_{k,a}(D_{r_{1}}^{1}\xi_{t},D_{r_{1},r_{2}}^{2}\xi_{t},...,D_{r_{1},...,r_{k}}^{k}\xi_{t}).\end{aligned}$$ We can repeat exactly the same arguments as in the proof of $\left(\ref{eq: MDF}\right),$ due to the fact that by Lemma \[lem: Z\_t\^p\_Integrability\] $e^{\xi_{t}}$ has moment of all orders, provided we can show that $$\sup_{r_{1},...,r_{a}\in[0,t]}\mathbb{\tilde{E}}[|D_{r_{1},...,r_{a}}^{a}\xi_{t}|^{p}]<\infty,\quad1\leq a\leq k,\label{eq: MDXi}$$ for any $p\geq1.$ As noted in Remark \[rem:CommuteDandY\], the Malliavin derivative commute with the stochastic integral with respect to $Y$ and we can write $$\begin{aligned}
D_{r_{1},...,r_{a}}^{a}\xi_{t} & =D_{r_{1},...,r_{a}}^{a}\left(\int_{0}^{t}h(X_{s})dY_{s}-\frac{1}{2}\int_{0}^{t}h^{2}(X_{s})ds\right)\\
& =\int_{0}^{t}D_{r_{1},...,r_{a}}^{a}h(X_{s})dY_{s}-\frac{1}{2}\int_{0}^{t}D_{r_{1},...,r_{a}}^{a}\left(h^{2}(X_{s})\right)ds.\end{aligned}$$
Hence, by Burkholder-Davis-Gundy inequality and Jensen’s inequality, we get for any $p\geq1$ that $$\begin{aligned}
& \mathbb{\tilde{E}}[|D_{r_{1},...,r_{a}}^{a}\xi_{t}|^{2p}]\\
& \leq C\left\{ \mathbb{\tilde{E}}\left[\left\vert \int_{0}^{t}D_{r_{1},...,r_{a}}^{a}h(X_{s})dY_{s}\right\vert ^{2p}\right]+\mathbb{\tilde{E}}\left[\left\vert \int_{0}^{t}D_{r_{1},...,r_{a}}^{a}\left(h^{2}(X_{s})\right)ds\right\vert ^{2p}\right]\right\} \\
& \leq C\left\{ \mathbb{\tilde{E}}\left[\int_{0}^{t}\left\vert D_{r_{1},...,r_{a}}^{a}h(X_{s})\right\vert ^{2p}ds\right]+\mathbb{\tilde{E}}\left[\int_{0}^{t}\left\vert D_{r_{1},...,r_{a}}^{a}\left(h^{2}(X_{s})\right)\right\vert ^{2p}ds\right]\right\} \\
& \leq C\left\{ A_{1}+A_{2}\right\} .\end{aligned}$$ Applying Faà di Bruno formula we can write $$\begin{aligned}
A_{1} & \leq C\sum_{l=1}^{a}\int_{0}^{t}\mathbb{\tilde{E}}[\left\vert h^{(l)}(X_{s})B_{a,l}(D_{r_{1}}^{1}X_{s},D_{r_{1},r_{2}}^{2}X_{s},...,D_{r_{1},...,r_{a}}^{a}X_{s})\right\vert ^{2p}]ds\\
& \leq C\left\Vert h\right\Vert _{\infty,a}^{q}\sum_{l=1}^{a}\int_{0}^{t}\mathbb{\tilde{E}}[\left\vert B_{a,l}(D_{r_{1}}^{1}X_{s},D_{r_{1},r_{2}}^{2}X_{s},...,D_{r_{1},...,r_{a}}^{a}X_{s})\right\vert ^{2p}]ds,\end{aligned}$$ where $$\left\Vert h\right\Vert _{\infty,a}\triangleq\sum_{i=0}^{d_{Y}}\sum_{l=0}^{a}\left\Vert h_{i}^{(l)}\right\Vert _{\infty}<\infty,$$ because $\mathbf{H}\left(m\right)$ holds. Therefore, using the generalized version of Hölder inequality, Lemma \[lem: Generalized Holder\], and Lemma \[lem: MomentMalliavinDerivative\] we get $A_{1}<\infty$. We can repeat the same argument for $A_{2}$ and obtain $\left(\ref{eq: MDXi}\right).$
\[subsec:MartingaleRep\]Martingale representations and Clark-Ocone formula
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In this section we recall the Clark-Ocone formula. This formula relates the kernels in the Itô martingale representation of Malliavin differentiable functionals with the Malliavin derivatives of such functionals. We present a truncated version of the well known Stroock-Taylor formula, see Stroock [@Stro87], that can be seen as an extension of the Clark-Ocone formula and it will be essential in deducing several conditional expectation estimates (see Section \[subsec:CondExpEst\]). We also show that, if the coefficients $f$ and $\sigma$ of of the SDE modeling the signal, the sensor function $h$ and the test function $\varphi$ are regular enough with bounded derivatives then, the kernels in the truncated Stroock-Taylor formula for $\varphi\left(X_{t}\right)e^{\xi_{t}}$ satisfy a uniform integrability property. Finally, we show that those kernels also satisfy a Hölder continuity property.
\[thm: Integral Representation\]Let $F\in L^{2}(\Omega,\mathcal{H}_{t}^{t},\tilde{P})$ **.** Then, $F$ admits the following martingale representation $$F=\mathbb{\tilde{E}}[F|\mathcal{H}_{0}^{t}]+\sum_{r=1}^{d_{V}}\int_{0}^{t}J_{s}^{r}dV_{s}^{r},$$ where $J^{r}=\{J_{s}^{r},s\in\left[0,t\right]\},r=1,...,d_{V}$ are $\mathcal{H}_{s}^{t}$-progressively measurable processes such that $$\mathbb{\tilde{E}}\left[\int_{0}^{t}\left\vert J_{s}^{r}\right\vert ^{2}ds_{1}\right]<\infty,\quad r=1,...,d_{V}.$$ Moreover, if $F$$\in\mathbb{D}^{1,2}$ then $$J_{s}^{r}=\mathbb{\tilde{E}}[D_{s}^{r}F|\mathcal{H}_{s}^{t}],\quad s\in\left[0,t\right],$$ which is known as the Clark-Ocone formula.
The proof is similar to that of Lemma 17 in Crisan [@Cris11] and the proof of the Clark-Ocone formula can be found in Nualart [@Nu06], Proposition 1.3.14.
By applying Theorem \[thm: Integral Representation\] to the kernels $J^{r},r=1,...,d_{V}$ one can get the following result.
\[thm: ST formula\]Assume that $F\in L^{2}(\Omega,\mathcal{H}_{t}^{t},\tilde{P}).$ Then, for $m\in\mathbb{N}$ we can write $$F=\sum_{\beta\in\mathcal{M}_{m-1}(S_{1})}I_{\beta}\left(\mathbb{\tilde{E}}\left[J_{s_{1},...,s_{\left\vert \beta\right\vert }}^{\beta}|\mathcal{H}_{0}^{t}\right]\right){}_{0,t}+\sum_{\beta\in\mathcal{R}\left(\mathcal{M}_{m-1}(S_{1})\right)}I_{\beta}\left(J_{s_{1},...,s_{\left\vert \beta\right\vert }}^{\beta}\right){}_{0,t},$$ where the kernels $J_{s_{1},...,s_{\left\vert \beta\right\vert }}^{\beta}$ for $\beta\in\mathcal{M}_{m}(S_{1})$ are obtained from the martingale representation of $J_{s_{2},...,s_{\left\vert \beta\right\vert }}^{-\beta}$, that is, they satisfy the following relationship
$$\begin{aligned}
J^{v} & \triangleq F,\\
J_{s_{2},...,s_{\left\vert \beta\right\vert }}^{-\beta} & =\mathbb{\tilde{E}}\left[J_{s_{2},...,s_{\left\vert \beta\right\vert }}^{-\beta}|\mathcal{H}_{0}^{t}\right]+\sum_{\beta_{1}=1}^{d_{V}}\int_{0}^{s_{2}}J_{s_{1},...,s_{\left\vert \beta\right\vert }}^{\beta_{1}*\left(-\beta\right)}dV_{s_{1}}^{\beta_{1}}.\end{aligned}$$
Moreover, if $\varphi(X_{t})e^{\xi_{t}}$$\in\mathbb{D}^{m,2}$ then $$J_{s_{1},...,s_{\left\vert \beta\right\vert }}^{\beta}=\mathbb{\tilde{E}}\left[D_{s_{1},...,s_{\left\vert \beta\right\vert }}^{\beta}F|\mathcal{H}_{s_{1}}^{t}\right],\quad\beta\in\mathcal{M}_{m}(S_{1}).$$
We prove the result by induction. For $m=1$, the result is precisely Theorem \[thm: Integral Representation\]. We assume that the result holds for $m-1\ge0$ and prove that this implies that it also holds for $m$. By the induction hypothesis we have that $$F=\sum_{\beta\in\mathcal{M}_{m-2}(S_{1})}I_{\beta}\left(\mathbb{\tilde{E}}\left[J_{s_{1},...,s_{\left\vert \beta\right\vert }}^{\beta}|\mathcal{H}_{0}^{t}\right]\right){}_{0,t}+\sum_{\beta\in\mathcal{R}\left(\mathcal{M}_{m-2}(S_{1})\right)}I_{\beta}\left(J_{s_{1},...,s_{\left\vert \beta\right\vert }}^{\beta}\right){}_{0,t}.$$ Applying Theorem \[thm: Integral Representation\] to $J_{s_{1},...,s_{\left\vert \beta\right\vert }}^{\beta},\beta\in\mathcal{R}\left(\mathcal{M}_{m-2}(S_{1})\right)$ we get $$\begin{aligned}
F & =\sum_{\beta\in\mathcal{M}_{m-2}(S_{1})}I_{\beta}\left(\mathbb{\tilde{E}}\left[J_{s_{1},...,s_{\left\vert \beta\right\vert }}^{\beta}|\mathcal{H}_{0}^{t}\right]\right){}_{0,t}+\sum_{\beta\in\mathcal{R}\left(\mathcal{M}_{m-2}(S_{1})\right)}I_{\beta}\left(\mathbb{\tilde{E}}\left[J_{s_{1},...,s_{\left\vert \beta\right\vert }}^{\beta}|\mathcal{H}_{0}^{t}\right]\right){}_{0,t}\\
& +\sum_{\beta\in\mathcal{R}\left(\mathcal{M}_{m-2}(S_{1})\right)}\sum_{r=1}^{d_{V}}I_{r*\beta}\left(J_{s,s_{1},...,s_{\left\vert \beta\right\vert }}^{r*\beta}\right){}_{0,t}\\
& =\sum_{\beta\in\mathcal{M}_{m-1}(S_{1})}I_{\beta}\left(\mathbb{\tilde{E}}\left[J_{s_{1},...,s_{\left\vert \beta\right\vert }}^{\beta}|\mathcal{H}_{0}^{t}\right]\right){}_{0,t}+\sum_{\beta\in\mathcal{R}\left(\mathcal{M}_{m-1}(S_{1})\right)}I_{\beta}\left(J_{s_{1},...,s_{\left\vert \beta\right\vert }}^{\beta}\right){}_{0,t},\end{aligned}$$ where in the last equality we have used that $$\mathcal{M}_{m-1}(S_{1})=\mathcal{M}_{m-2}(S_{1})\biguplus\mathcal{R}\left(\mathcal{M}_{m-2}(S_{1})\right),$$ the definitions of $\mathcal{R}\left(\mathcal{M}_{m-1}(S_{1})\right)$ and the concatenation of multi-indices.
The Clark-Ocone representation of the kernels also follows from a straightforward induction.
\[prop: UnifBoundKernels\]Let $m\in\mathbb{N}$ and assume that** H**$(m)$ holds and $\varphi\in C_{P}^{m+1}$. Then, the kernels $J_{s_{1},...,s_{\left\vert \beta\right\vert }}^{\beta},$ $\beta\in\mathcal{M}_{m+1}(S_{1})$ appearing in the Stroock-Taylor formula of order $m+1$ for $\varphi(X_{t})e^{\xi_{t}}$ satisfy $$\sup_{0\leq s_{1}<\cdots<s_{\left|\beta\right|}\leq t}\mathbb{\tilde{E}}\left[\left\vert J_{s_{1},...,s_{\left|\beta\right|}}^{\beta}\right\vert ^{p}\right]<\infty,$$ for $p\geq1$
It is a straightforward combination of Theorem \[thm: ST formula\], Jensen’s inequality for conditional expectations and Lemma \[lem: BoundedIteratedMD1\].
\[lem: RegularityKernels\]Assume that **H**$(1)$ holds and $\varphi\in C_{P}^{2}$. Then, the kernels $J^{r}=\{J_{s}^{r},s\in\left[0,t\right]\},r=1,...,d_{V}$ in the martingale representation of $\varphi(X_{t})e^{\xi_{t}}$ satisfy the following Hölder continuity property: $$\mathbb{\tilde{E}}\left[\left\vert J_{s}^{r}-J_{u}^{r}\right\vert ^{2p}\right]\leq C\left(s-u\right)^{p},\qquad0\leq u\leq s\leq t,$$ for $p\geq1.$
The idea is to use the Clark-Ocone formula, Theorem \[thm: Integral Representation\]. That is, one has the following representation $$J_{s}^{r}=\mathbb{\tilde{E}}\left[D_{s}^{r}\left\{ \varphi\left(X_{t}\right)e^{\xi_{t}}\right\} |\mathcal{H}_{s}^{t}\right],\qquad0\leq s\leq t,$$ where $D_{s}^{r}$ denotes the Malliavin derivative with respect to $V^{r}.$ Hence, we can write $$\begin{aligned}
J_{s}^{r}-J_{u}^{r} & =\mathbb{\tilde{E}}\left[D_{s}^{r}\left\{ \varphi\left(X_{t}\right)e^{\xi_{t}}\right\} -D_{u}^{r}\left\{ \varphi\left(X_{t}\right)e^{\xi_{t}}\right\} |\mathcal{H}_{s}^{t}\right]\\
& \quad+\mathbb{\tilde{E}}\left[D_{u}^{r}\left\{ \varphi\left(X_{t}\right)e^{\xi_{t}}\right\} |\mathcal{H}_{s}^{t}\right]-\mathbb{\tilde{E}}\left[D_{u}^{r}\left\{ \varphi\left(X_{t}\right)e^{\xi_{t}}\right\} |\mathcal{H}_{u}^{t}\right]\\
& \triangleq A_{1}+A_{2}.\end{aligned}$$ For the term $A_{1},$ note that we can write $$\begin{aligned}
& D_{s}^{r}\left\{ \varphi\left(X_{t}\right)e^{\xi_{t}}\right\} -D_{u}^{r}\left\{ \varphi\left(X_{t}\right)e^{\xi_{t}}\right\} \\
& =e^{\xi_{t}}\left(D_{s}^{r}\varphi\left(X_{t}\right)-D_{u}^{r}\varphi\left(X_{t}\right)\right)+\varphi\left(X_{t}\right)\left(D_{s}^{r}e^{\xi_{t}}-D_{u}^{r}e^{\xi_{t}}\right)\\
& =e^{\xi_{t}}\sum_{j=1}^{d_{X}}\partial_{j}\varphi\left(X_{t}\right)\left(D_{s}^{r}X_{t}^{j}-D_{u}^{r}X_{t}^{j}\right)+e^{\xi_{t}}\varphi\left(X_{t}\right)\left(D_{s}^{r}\xi_{t}-D_{u}^{r}\xi_{t}\right),\end{aligned}$$ and $$\begin{aligned}
D_{s}^{r}\xi_{t}-D_{u}^{r}\xi_{t} & =\sum_{i=1}^{d_{Y}}\sum_{k=1}^{d_{X}}\int_{0}^{t}\partial_{k}h^{i}\left(X_{v}\right)\left(D_{s}^{r}X_{v}^{j}-D_{u}^{r}X_{v}^{j}\right)dY_{v}^{i}\\
& \quad+\frac{1}{2}\sum_{i=1}^{d_{Y}}\sum_{k=1}^{d_{X}}\int_{0}^{t}\partial_{k}\{h^{i}\left(X_{v}\right)^{2}\}\left(D_{s}^{r}X_{v}^{j}-D_{u}^{r}X_{v}^{j}\right)dv.\end{aligned}$$ Hence, the result follows from the fact that $$\mathbb{\tilde{E}}\left[\left\vert D_{s}^{r}X_{v}^{j}-D_{u}^{r}X_{v}^{j}\right\vert ^{2p}\right]\leq C\left(s-u\right)^{p},$$ $D_{s}^{r}X_{t}^{j}$ satisfies an evolution equation drive by a Brownian motion, see Section 2.2.2 in Nualart [@Nu06]. For the term $A_{2}$ the result follows from the martingale representation theorem, Theorem \[thm: Integral Representation\], applied to the random variable $D_{u}\left\{ \varphi\left(X_{t}\right)e^{\xi_{t}}\right\} \in L^{2}\left(\Omega,\mathcal{H}_{t}^{t},\tilde{P}\right)$ which yields $$D_{u}^{r}\left\{ \varphi\left(X_{t}\right)e^{\xi_{t}}\right\} =\mathbb{\tilde{E}}\left[\text{\ensuremath{D_{u}^{r}\left\{ \varphi\left(X_{t}\right)e^{\xi_{t}}\right\} }\ensuremath{|}}\mathcal{H}_{0}^{t}\right]+\sum_{r_{1}=1}^{d_{V}}\int_{0}^{t}G_{v}^{r_{1}}dV_{v}^{r_{1}},$$ and, hence, $A_{2}=\sum_{r_{1}=1}^{d_{V}}\int_{u}^{s}G_{v}^{r_{1}}dV_{v}^{r_{1}}$ and $$\begin{aligned}
\mathbb{\tilde{E}}\left[\left|A_{2}\right|^{2p}\right] & \leq C\sum_{r_{1}=1}^{d_{V}}\mathbb{\tilde{E}}\left[\left|\int_{u}^{s}\left|G_{v}^{r_{1}}\right|^{2}dv\right|^{p}\right]\\
& \leq C\left(s-u\right)^{p-1}\sum_{r_{1}=1}^{d_{V}}\int_{u}^{s}\mathbb{\tilde{E}}\left[\left|G_{v}^{r_{1}}\right|^{2p}\right]dv\\
& \leq C\left(s-u\right)^{p}\sum_{r_{1}=1}^{d_{V}}\sup_{0\leq v\leq t}\mathbb{\tilde{E}}\left[\left|G_{v}^{r_{1}}\right|^{2p}\right]\\
& \leq C\left(s-u\right)^{p},\end{aligned}$$ where in the last inequality we have used Proposition \[prop: UnifBoundKernels\].
\[rem: HighRegularityKernel\]If **H**$(m)$ holds and $\varphi\in C_{P}^{m+1}$, using the same reasonings as in Lemma \[lem: RegularityKernels\], one can show that the kernels $J^{\beta},\beta\in\mathcal{M}_{m}\left(S_{1}\right)$ in the Stroock-Taylor formula for $\varphi(X_{t})e^{\xi_{t}}$ satisfy the following Hölder continuity property: $$\mathbb{\tilde{E}}\left[\left\vert J_{s_{1},...,s,...s_{\left|\beta\right|}}^{\beta}-J_{s_{1},...,s_{i-1},u,s_{i+1},...s_{\left|\beta\right|}}^{\beta}\right\vert ^{2p}\right]\leq C\left|s-u\right|^{p},$$ for $p\geq1,$ $s_{i-1}\leq u\leq s\leq s_{i+1}$, $i=2,...,m-1$.
\[subsec:BackwardMart\]Backward martingales estimates
-----------------------------------------------------
In this section we start reviewing some basic concepts of backward Itô integration that can be found, for instance, in Pardoux and Protter [@ParPro87], Bensoussan [@Ben92] and Applebaum [@App09]. Then we compute some technical estimates related to products of backward Itô integrals and backward stochastic exponentials that will be useful in the next section.
We know that under $\tilde{P}$ the observation process $Y$ is a Brownian motion with respect to the filtration $\mathcal{\mathbb{Y}}$. For fixed $t\geq0,$ we can consider the process $\overleftarrow{Y}=\{\overleftarrow{Y}_{s}\triangleq Y_{s}-Y_{t}\}_{0\leq s\leq t}$ which is a Brownian motion with respect to the backward filtration $$\mathcal{\mathbb{Y}}^{t}=\left\{ \mathcal{Y}_{s}^{t}\triangleq\sigma\left(\overleftarrow{Y}_{u},s\leq u\leq t\right)\vee\mathcal{N}\right\} _{0\leq s\leq t},$$ where $\mathcal{N}$ are all the $P$-null sets of $(\Omega,\mathcal{F},P)$. We can also consider the filtration $\mathbb{Y}^{0,V}=\left\{ \mathcal{Y}_{s}^{0,V}\triangleq\mathcal{F}_{t}^{0,V}\vee\mathcal{Y}_{s}\right\} _{0\leq s\leq t}$ and the backward filtration $\mathbb{Y}^{0,V,t}=\{\mathcal{Y}_{s}^{0,V,t}\triangleq\mathcal{F}_{t}^{0,V}\vee\mathcal{Y}_{s}^{t}\}$$_{0\leq s\leq t}$. As $X$$_{0}$ and $V$ are independent of $Y$ under $\tilde{P}$, we also have that $Y$ is a $\mathbb{Y}^{0,V}$-Brownian motion and $\overleftarrow{Y}$ is $\mathbb{Y}^{0,V,t}$-Brownian motion.
If $\eta=\left\{ \eta_{s}^{1},...,\eta_{s}^{d_{Y}}\right\} _{0\leq s\leq t}$ is a square integrable measurable process adapted to $\mathcal{\mathbb{Y}}^{0,V,t}$ we can define the backward Itô integral of $\eta$ with respect to $\overleftarrow{Y}$ by $$\begin{aligned}
\int_{s}^{t}\eta_{u}d\overleftarrow{Y}_{u} & \triangleq\sum_{i=1}^{d_{Y}}\int_{s}^{t}\eta_{u}^{i}d\overleftarrow{Y^{i}}_{u}\\
& \triangleq L^{2}(\tilde{P})-\lim_{\tau\in\Pi\left(t\right),\left|\tau\right|\rightarrow0}\sum_{i=1}^{d_{Y}}\sum_{j=0}^{n-1}\eta_{t_{j+1}}\left(\overleftarrow{Y^{i}}_{t_{j+1}\lor s}-\overleftarrow{Y^{i}}_{t_{j}\lor s}\right),\\
& =L^{2}(\tilde{P})-\lim_{\tau\in\Pi\left(t\right),\left|\tau\right|\rightarrow0}\sum_{i=1}^{d_{Y}}\sum_{j=0}^{n-1}\eta_{t_{j+1}}\left(Y_{t_{j+1}\lor s}^{i}-Y_{t_{j}\lor s}^{i}\right).\end{aligned}$$
\[rem: Backward Ito integral\]If a square integrable process $\theta=\left\{ \theta_{u}\right\} _{0\leq u\leq t}$ is simultaneously adapted to $\mathbb{Y}^{0,V}$and $\mathcal{\mathbb{Y}}^{0,V,t}$ both, the Itô and the backward Itô integrals, can be defined over the same interval but, in general, they will be different. However, if $\theta_{u}$ is measurable with respect to respect to $\mathcal{Y}_{0}^{0,V}=\mathcal{Y}_{t}^{0,V,t}=\mathcal{F}_{t}^{0,V}$ for all $0\leq u\leq t$, then both integrals coincide. In fact, they coincide with the Stratonovich integral, see Pardoux and Protter [@ParPro87]. This means that in the statement of Lemma \[lem: Main Backward\] we can change all backward Itô integrals by Itô integrals and the estimates will hold true. However, in the proof of Lemma \[lem: Main Backward\] we use the properties of the backward integral and for that reason we keep the notation of backward integration.
The backward Itô integral is analogous to the Itô integral. In particular, the backward Itô integral has zero expectation and it is a backward martingale with respect to $\mathcal{\mathbb{Y}}^{0,V,t}$, that is $$\mathbb{\tilde{E}}\left[\int_{s_{1}}^{t}\eta_{u}d\overleftarrow{Y}_{u}|\mathcal{Y}_{s_{2}}^{0,V,t}\right]=\int_{s_{2}}^{t}\eta_{u}d\overleftarrow{Y}_{u},\quad0\leq s_{1}<s_{2}\leq t.$$ A backward Itô process is a process of the following form $$Z_{s}=Z_{t}+\int_{s}^{t}\nu_{u}du+\int_{s}^{t}\eta_{u}d\overleftarrow{Y}_{u},\quad0\leq s\leq t,$$ where $\nu$ and $\eta$ are two square integrable, measurable and $\mathcal{\mathbb{Y}}^{0,V,t}$-adapted processes of the appropriate dimensions. For backward Itô processes and $f$$\in C^{1,2}\left(\left(0,t\right)\times\mathbb{R}^{d_{Z}};\mathbb{R}\right)$ we have the following Itô formula, see Bensoussan [@Ben92], $$\begin{aligned}
f\left(s,Z_{s}\right) & =f\left(t,Z_{t}\right)+\int_{s}^{t}\left\{ -\partial_{t}f\left(u,Z_{u}\right)+\nu_{u}Df\left(u,Z_{u}\right)+\frac{1}{2}\text{tr}\left(D^{2}f\left(u,Z_{u}\right)\eta_{u}\eta_{u}^{T}\right)\right\} du\\
& \quad+\int_{s}^{t}Df\left(u,Z_{u}\right)\eta_{u}d\overleftarrow{Y}_{u},\end{aligned}$$ where $D$ and $D^{2}$ stand for the gradient and the Hessian, respectively, with respect to the space variables. As a corollary, one gets the integration by parts formula $$\begin{aligned}
\left(\int_{s}^{t}\eta_{u}^{i}d\overleftarrow{Y^{i}}_{u}\right)\left(\int_{s}^{t}\eta_{u}^{j}d\overleftarrow{Y^{j}}_{u}\right) & =\int_{s}^{t}\left(\int_{u}^{t}\eta_{v}^{i}d\overleftarrow{Y^{i}}_{v}\right)\eta_{u}^{j}d\overleftarrow{Y^{j}}_{u}\label{eq: Backward_IBP}\\
& \quad+\int_{s}^{t}\left(\int_{u}^{t}\eta_{v}^{j}d\overleftarrow{Y^{j}}_{v}\right)\eta_{u}^{i}d\overleftarrow{Y^{i}}_{u}\nonumber \\
& \quad+\mathbf{1}_{\{i=j\}}\int_{s}^{t}\eta_{u}^{i}\eta_{u}^{j}du.\nonumber \end{aligned}$$ and $$\tilde{\mathbb{E}}\left[\left(\int_{s}^{t}\eta_{u}^{i}d\overleftarrow{Y^{i}}_{u}\right)\left(\int_{s}^{t}\eta_{u}^{j}d\overleftarrow{Y^{j}}_{u}\right)\right]=\mathbf{1}_{\{i=j\}}\tilde{\mathbb{E}}\left[\int_{s}^{t}\eta_{u}^{i}\eta_{u}^{j}du\right].\label{eq: EquBackMartVariation}$$ Let $\phi\in\mathcal{B}_{b}\left(\mathbb{R}^{d_{X}};\mathbb{R}^{d_{Y}}\right)$ be a bounded measurable function and let $M^{t}\left(\phi\right)=\left\{ M_{s}^{t}\left(\phi\right)\right\} _{0\leq s\leq t}$ be the process $$M_{s}^{t}\left(\phi\right)=\exp\left(\sum_{i=1}^{d_{Y}}\int_{s}^{t}\phi_{i}(X_{u})d\overleftarrow{Y_{u}^{i}}-\frac{1}{2}\sum_{i=1}^{d_{Y}}\int_{s}^{t}\phi_{i}^{2}\left(X_{u}\right)du\right).$$ It is easy to show that $M^{t}$$\left(\phi\right)$ is a backward martingale with respect to $\mathcal{\mathbb{Y}}^{0,V,t}$ and applying the backward Itô formula one finds the following formula for the increments of $M^{t}$$\left(\phi\right)$ $$M_{s_{1}}^{t}\left(\phi\right)=M_{s_{2}}^{t}\left(\phi\right)+\sum_{i=1}^{d_{Y}}\int_{s_{1}}^{s_{2}}M_{u}^{t}\left(\phi\right)\phi_{i}\left(X_{u}\right)d\overleftarrow{Y_{u}^{i}}.\label{eq: EquBackMartIncrem}$$ Moreover, by the same reasoning as in Lemma \[lem: Z\_t\^p\_Integrability\], we have for all $p\geq1$ $$\mathbb{\tilde{E}}\left[\left|M_{s}^{t}\left(\phi\right)\right|^{p}\right]<\infty.\label{eq:MomentsMts}$$
\[lem: Backward Martingale\]Let $0\leq s_{1}\leq s_{2}\leq s_{3}\leq t$, $\Psi$ be $\mathcal{Y}_{s_{3}}^{0,V,t}$-measurable random variable and $\theta=\left\{ \theta_{u}\right\} _{0\leq u\leq t}$ a square integrable and measurable process such that $\theta_{u}$ is measurable with respect to $\mathcal{Y}_{s_{3}}^{0,V,t}$ for all $s_{2}\leq u\leq s_{3}$. Then, $$\mathbb{\tilde{E}}\left[\Psi M_{s_{1}}^{t}\left(\phi\right)\int_{s_{2}}^{s_{3}}\theta_{u}d\overleftarrow{Y_{u}^{i}}\right]=\mathbb{\tilde{E}}\left[\Psi M_{s_{3}}^{t}\left(\phi\right)\int_{s_{2}}^{s_{3}}\phi_{i}\left(X_{u}\right)\theta_{u}du\right].$$
By the backward martingale properties of $M_{s}^{t}\left(\phi\right)$ and equation $\left(\ref{eq: EquBackMartIncrem}\right)$ we can write $$\begin{aligned}
\mathbb{\tilde{E}}\left[\Psi M_{s_{1}}^{t}\left(\phi\right)\int_{s_{2}}^{s_{3}}\theta_{u}d\overleftarrow{Y_{u}^{i}}\right] & =\mathbb{\tilde{E}}\left[\Psi\mathbb{\tilde{E}}\left[M_{s_{1}}^{t}\left(\phi\right)|\mathcal{Y}_{s_{2}}^{0,V,t}\right]\int_{s_{2}}^{s_{3}}\theta_{u}d\overleftarrow{Y_{u}^{i}}\right]\\
& =\mathbb{\tilde{E}}\left[\Psi M_{s_{2}}^{t}\left(\phi\right)\int_{s_{2}}^{s_{3}}\theta_{u}d\overleftarrow{Y_{u}^{i}}\right]\\
& =\mathbb{\tilde{E}}\left[\Psi M_{s_{3}}^{t}\left(\phi\right)\int_{s_{2}}^{s_{3}}\theta_{u}d\overleftarrow{Y_{u}^{i}}\right]\\
& \quad+\sum_{i_{1}=1}^{d_{Y}}\mathbb{\tilde{E}}\left[\Psi\left(\int_{s_{2}}^{s_{3}}M_{u}^{t}\left(\phi\right)\phi_{i_{1}}\left(X_{u}\right)d\overleftarrow{Y_{u}^{i_{1}}}\right)\left(\int_{s_{2}}^{s_{3}}\theta_{u}d\overleftarrow{Y_{u}^{i}}\right)\right]\end{aligned}$$ Next, note that $$\mathbb{\tilde{E}}\left[\Psi M_{s_{3}}^{t}\left(\phi\right)\int_{s_{2}}^{s_{3}}\theta_{u}d\overleftarrow{Y_{u}^{i}}\right]=\mathbb{\tilde{E}}\left[\Psi M_{s_{3}}^{t}\left(\phi\right)\mathbb{\tilde{E}}\left[\int_{s_{2}}^{s_{3}}\theta_{u}d\overleftarrow{Y_{u}^{i}}|\mathcal{Y}_{s_{3}}^{0,V,t}\right]\right]=0,$$ and using equation $\left(\ref{eq: EquBackMartVariation}\right)$ we have $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\Psi\left(\int_{s_{2}}^{s_{3}}M_{u}^{t}\left(\phi\right)\phi_{i_{1}}\left(X_{u}\right)d\overleftarrow{Y_{u}^{i_{1}}}\right)\left(\int_{s_{2}}^{s_{3}}\theta_{u}d\overleftarrow{Y_{u}^{i}}\right)\right]\\
& =\mathbf{1}_{\left\{ i_{1}=i\right\} }\mathbb{\tilde{E}}\left[\Psi\left(\int_{s_{2}}^{s_{3}}M_{u}^{t}\left(\phi\right)\phi_{i_{1}}\left(X_{u}\right)\theta_{u}du\right)\right]\\
& =\mathbf{1}_{\left\{ i_{1}=i\right\} }\mathbb{\tilde{E}}\left[\Psi\left(\int_{s_{2}}^{s_{3}}\mathbb{\tilde{E}}\left[M_{u}^{t}\left(\phi\right)|\mathcal{Y}_{s_{3}}^{0,V,t}\right]\phi_{i_{1}}\left(X_{u}\right)\theta_{u}du\right)\right]\\
& =\mathbf{1}_{\left\{ i_{1}=i\right\} }\mathbb{\tilde{E}}\left[\Psi M_{s_{3}}^{t}\left(\phi\right)\left(\int_{s_{2}}^{s_{3}}\phi_{i_{1}}\left(X_{u}\right)\theta_{u}du\right)\right].\end{aligned}$$ Hence, the result follows.
\[lem: Backward Martingale II\]Let $0\leq s_{1}\leq s_{2}\leq s_{3}\leq t$, $\Psi$ be $\mathcal{Y}_{s_{3}}^{0,V,t}$-measurable random variable and $\theta^{1}=\left\{ \theta_{u}^{1}\right\} _{0\leq u\leq t}$ and $\theta^{1}=\left\{ \theta_{u}^{1}\right\} _{0\leq u\leq t}$ be two square integrable measurable processes such that $\theta_{u}^{1}$ and $\theta_{u}^{2}$ are also measurable with respect to $\mathcal{Y}_{s_{3}}^{0,V,t}$ for all $s_{2}\leq u\leq s_{3}$. Then, $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\Psi M_{s_{1}}^{t}\left(\phi\right)\left(\int_{s_{2}}^{s_{3}}\theta_{u}^{1}d\overleftarrow{Y_{u}^{i_{1}}}\right)\left(\int_{s_{2}}^{s_{3}}\theta_{u}^{2}d\overleftarrow{Y_{u}^{i_{2}}}\right)\right]\\
& =\mathbb{\tilde{E}}\left[\Psi M_{s_{3}}^{t}\left(\phi\right)\int_{s_{2}}^{s_{3}}\phi_{i_{1}}\left(X_{u}\right)\theta_{u}^{1}\left(\int_{u}^{s_{3}}\phi_{i_{2}}\left(X_{v}\right)\theta_{v}^{2}dv\right)du\right]\\
& \quad+\mathbb{\tilde{E}}\left[\Psi M_{s_{3}}^{t}\left(\phi\right)\int_{s_{2}}^{s_{3}}\phi_{i_{2}}\left(X_{u}\right)\theta_{u}^{2}\left(\int_{u}^{s_{3}}\phi_{i_{1}}\left(X_{v}\right)\theta_{v}^{1}dv\right)du\right]\\
& \quad+\mathbf{1}_{\left\{ i_{1}=i_{2}\right\} }\mathbb{\tilde{E}}\left[\Psi M_{s_{3}}^{t}\left(\phi\right)\int_{s_{2}}^{s_{3}}\theta_{u}^{1}\theta_{u}^{2}du\right].\end{aligned}$$
Using the integration by parts formula $\left(\ref{eq: Backward_IBP}\right)$ we can write $$\begin{aligned}
\mathbb{\tilde{E}}\left[\Psi M_{s_{1}}^{t}\left(\phi\right)\left(\int_{s_{2}}^{s_{3}}\theta_{u}^{1}d\overleftarrow{Y_{u}^{i_{1}}}\right)\left(\int_{s_{2}}^{s_{3}}\theta_{u}^{2}d\overleftarrow{Y_{u}^{i_{2}}}\right)\right] & =\mathbb{\tilde{E}}\left[\Psi M_{s_{1}}^{t}\left(\phi\right)\int_{s_{2}}^{s_{3}}\theta_{u}^{1}\left(\int_{u}^{s_{3}}\theta_{v}^{2}d\overleftarrow{Y_{v}^{i_{2}}}\right)d\overleftarrow{Y_{u}^{i_{1}}}\right]\\
& \quad+\mathbb{\tilde{E}}\left[\Psi M_{s_{1}}^{t}\left(\phi\right)\int_{s_{2}}^{s_{3}}\theta_{u}^{2}\left(\int_{u}^{s_{3}}\theta_{v}^{1}d\overleftarrow{Y_{v}^{i_{1}}}\right)d\overleftarrow{Y_{u}^{i_{2}}}\right]\\
& \quad+\mathbf{1}_{\left\{ i_{1}=i_{2}\right\} }\mathbb{\tilde{E}}\left[\Psi M_{s_{1}}^{t}\left(\phi\right)\int_{s_{2}}^{s_{3}}\theta_{u}^{1}\theta_{u}^{2}du\right].\end{aligned}$$ Then, using the same reasonings as in Lemma \[lem: Backward Martingale\] we get that $$\begin{aligned}
\mathbb{\tilde{E}}\left[\Psi M_{s_{1}}^{t}\left(\phi\right)\int_{s_{2}}^{s_{3}}\theta_{u}^{1}\left(\int_{u}^{s_{3}}\theta_{v}^{2}d\overleftarrow{Y_{v}^{i_{2}}}\right)d\overleftarrow{Y_{u}^{i_{1}}}\right] & =\mathbb{\tilde{E}}\left[\Psi\int_{s_{2}}^{s_{3}}M_{u}^{t}\left(\phi\right)\phi_{i_{1}}\left(X_{u}\right)\theta_{u}^{1}\left(\int_{u}^{s_{3}}\theta_{v}^{2}d\overleftarrow{Y_{v}^{i_{2}}}\right)du\right]\end{aligned}$$ Next, by Fubini’s theorem, Lemma \[lem: Backward Martingale\] and Fubini’s theorem again we obtain $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\Psi\int_{s_{2}}^{s_{3}}M_{u}^{t}\left(\phi\right)\phi_{i_{1}}\left(X_{u}\right)\theta_{u}^{1}\left(\int_{u}^{s_{3}}\theta_{v}^{2}d\overleftarrow{Y_{v}^{i_{2}}}\right)du\right]\\
& =\int_{s_{2}}^{s_{3}}\mathbb{\tilde{E}}\left[\Psi\phi_{i_{1}}\left(X_{u}\right)\theta_{u}^{1}M_{u}^{t}\left(\phi\right)\left(\int_{u}^{s_{3}}\theta_{v}^{2}d\overleftarrow{Y_{v}^{i_{2}}}\right)\right]du\\
& =\int_{s_{2}}^{s_{3}}\mathbb{\tilde{E}}\left[\Psi\phi_{i_{1}}\left(X_{u}\right)\theta_{u}^{1}M_{s_{3}}^{t}\left(\phi\right)\left(\int_{u}^{s_{3}}\phi_{i_{2}}\left(X_{u}\right)\theta_{v}^{2}dv\right)\right]du\\
& =\mathbb{\tilde{E}}\left[\Psi M_{s_{3}}^{t}\left(\phi\right)\int_{s_{2}}^{s_{3}}\phi_{i_{1}}\left(X_{u}\right)\theta_{u}^{1}\left(\int_{u}^{s_{3}}\phi_{i_{2}}\left(X_{u}\right)\theta_{v}^{2}dv\right)\right]du\end{aligned}$$ By symmetry we get an analogous expression for the term $\int_{s_{2}}^{s_{3}}\theta_{u}^{2}\left(\int_{u}^{s_{3}}\theta_{v}^{1}d\overleftarrow{Y_{v}^{i_{1}}}\right)d\overleftarrow{Y_{u}^{i_{2}}}$ . Finally, for the last term we only need to take conditional expectation with respect to $\mathcal{Y}_{s_{3}}^{0,V,t}$ and use that $M_{s}^{t}\left(\phi\right)$ is a $\mathbb{Y}^{0,V,t}$-martingale.
The next lemma is a well known generalization of Hölder’s inequality.
\[lem: Generalized Holder\]Let $p_{i}>1,i=1,...,m$ such that $\sum_{i=1}^{m}\frac{1}{p_{i}}=1,$ and $X_{i}\in L^{p_{i}}(\Omega,\mathcal{F},\tilde{P}),i=1,...,m$. Then, $$\tilde{\mathbb{E}}\left[\left|\prod_{i=1}^{m}X_{i}\right|\right]\leq\prod_{i=1}^{m}\tilde{\mathbb{E}}\left[\left|X_{i}\right|^{p_{i}}\right]^{1/p_{i}}<\infty.$$
\[lem: Main Backward\]Let $\tau=\left\{ 0=t_{0}<t_{1}<\cdots<t_{n}=t\right\} $ be a partition of $\left[0,t\right]$, $\phi\in\mathcal{B}_{b}\left(\mathbb{R}^{d_{X}};\mathbb{R}^{d_{Y}}\right)$, $\Upsilon$$\in L^{p}(\Omega,\mathcal{Y}_{t}^{0,V,t},\tilde{P})$ for any $p\geq1$, $\beta_{s},$ be a deterministic processes satisfying $$\left|\beta_{s}\mathbf{1}_{\left[t_{j},t_{j+1}\right]}\left(s\right)\right|\leq\delta^{m}\label{eq: BoundBeta}$$ for some $m\in\mathbb{N}$ and $\theta^{1},\theta^{2},\kappa^{1},\kappa^{2},$ be stochastic processes measurable with respect to $\mathcal{Y}_{t}^{0,V,t}$, such that $$\begin{aligned}
\sup_{0\leq s\leq t}\mathbb{\tilde{E}}\left[\left|\theta_{s}^{j}\right|^{p}\right] & <\infty,\quad j=1,2,\label{eq: MomentsTheta}\\
\sup_{0\leq s\leq t}\mathbb{\tilde{E}}\left[\left|\kappa_{s}^{j}\right|^{p}\right] & <\infty,\quad j=1,2,\nonumber \end{aligned}$$ for any $p\geq1$. Then:
1. For $i\in\left\{ 1,...,d_{Y}\right\} $, we have that $$\left|\mathbb{\tilde{E}}\left[\Upsilon M_{0}^{t}\left(\phi\right)\left(\int_{t_{j}}^{t_{j+1}}\beta_{s}d\overleftarrow{Y_{s}^{i_{1}}}\right)\left(\int_{t_{k}}^{t_{k+1}}\beta_{s}d\overleftarrow{Y_{s}^{i_{1}}}\right)\right]\right|\leq C\left\{ \mathbf{1}_{\left\{ j\neq k\right\} }\delta^{2m+2}+\mathbf{1}_{\left\{ j=k\right\} }\delta^{2m+1}\right\} .$$
2. For $i,i_{1}\in\left\{ 1,...,d_{Y}\right\} $, we have that $$\begin{aligned}
& \left|\mathbb{\tilde{E}}\left[\Upsilon M_{0}^{t}\left(\phi\right)\left(\int_{t_{k+1}}^{t}\theta_{s}^{1}d\overleftarrow{Y_{s}^{i_{1}}}\right)\left(\int_{t_{j}}^{t_{j+1}}\beta_{s}d\overleftarrow{Y_{s}^{i}}\right)\left(\int_{t_{k}}^{t_{k+1}}\beta_{s}d\overleftarrow{Y_{s}^{i}}\right)\right]\right|\\
& \leq C\left\{ \mathbf{1}_{\left\{ j\neq k\right\} }\delta^{2m+2}+\mathbf{1}_{\left\{ j=k\right\} }\delta^{2m+1}\right\} .\end{aligned}$$
3. For $i,i_{1},a_{1}\in\left\{ 1,...,d_{Y}\right\} $, we have that $$\begin{aligned}
& \left|\mathbb{\tilde{E}}\left[\Upsilon M_{0}^{t}\left(\phi\right)\left(\int_{t_{j+1}}^{t}\theta_{s}^{1}d\overleftarrow{Y_{s}^{i_{1}}}\right)\left(\int_{t_{k+1}}^{t}\kappa_{s}^{1}d\overleftarrow{Y_{s}^{a_{1}}}\right)\left(\int_{t_{j}}^{t_{j+1}}\beta_{s}d\overleftarrow{Y_{s}^{i}}\right)\left(\int_{t_{k}}^{t_{k+1}}\beta_{s}d\overleftarrow{Y_{s}^{i}}\right)\right]\right|\\
& \leq C\left\{ \mathbf{1}_{\left\{ j\neq k\right\} }\delta^{2m+2}+\mathbf{1}_{\left\{ j=k\right\} }\delta^{2m+1}\right\} .\end{aligned}$$
4. For $i,i_{1},i_{2}a_{1},a_{2}\in\left\{ 1,...,d_{Y}\right\} $, we have that $$\begin{aligned}
& \left|\mathbb{\tilde{E}}\left[\Upsilon M_{0}^{t}\left(\phi\right)\prod_{l=1}^{2}\left\{ \left(\int_{t_{j+1}}^{t}\theta_{s}^{l}d\overleftarrow{Y_{s}^{i_{l}}}\right)\left(\int_{t_{k+1}}^{t}\kappa_{s}^{l}d\overleftarrow{Y_{s}^{a_{l}}}\right)\right\} \left(\int_{t_{j}}^{t_{j+1}}\beta_{s}d\overleftarrow{Y_{s}^{i}}\right)\left(\int_{t_{k}}^{t_{k+1}}\beta_{s}d\overleftarrow{Y_{s}^{i}}\right)\right]\right|\\
& \leq C\left\{ \mathbf{1}_{\left\{ j\neq k\right\} }\delta^{2m+2}+\mathbf{1}_{\left\{ j=k\right\} }\delta^{2m+1}\right\} .\end{aligned}$$
The full proof of the lemma is lengthy and depends on applying Lemmas \[lem: Backward Martingale\] and \[lem: Backward Martingale II\] repeatedly. We do not present it in full, but only write in detail the proof of the statement $\left(4\right)$, the others being similar and easier. Note that by the assumptions on $\Upsilon$,$\beta^{j}$ and $\theta^{j}$ and $\phi$ the expectations in the statement of the lemma are finite. We start with some preliminary estimations. In what follows $C\left(t\right)$,$C$$\left(\phi\right)$and $C\left(t,\phi\right)$ will denote constants that only depends on $t$, on $\phi$ and on $t$ and $\phi$, respectively. For any $p$$\geq1$ we have
- Let $0\leq s\leq t$, then $$\begin{aligned}
\mathbb{\tilde{E}}\left[\left|M_{s}^{t}\left(\phi\right)\right|^{p}\right] & =\mathbb{\tilde{E}}\left[M_{s}^{t}\left(p\phi\right)\exp\left(\frac{p^{2}-p}{2}\sum_{i=1}^{d_{Y}}\int_{s}^{t}\phi_{i}^{2}\left(X_{u}\right)du\right)\right]\\
& \leq\exp\left(d_{Y}\left\Vert \phi\right\Vert _{\infty}^{2}\frac{p^{2}-p}{2}\left(t-s\right)\right),\end{aligned}$$ where we have used that for $\phi\in\mathcal{B}_{b}\left(\mathbb{R}^{d_{X}};\mathbb{R}^{d_{Y}}\right)$ one has that $M_{s}^{t}\left(\phi\right)$ is a backward martingale with expectation equal to one. The previous estimate yields that $$\sup_{0\leq s\leq t}\mathbb{\tilde{E}}\left[\left|M_{s}^{t}\left(\phi\right)\right|^{p}\right]\leq\exp\left(d_{Y}\left\Vert \phi\right\Vert _{\infty}^{2}\frac{p^{2}-p}{2}t\right)\leq C\left(t,\phi\right).\label{eq:BE_M}$$
- Let $0\leq s_{1}\leq s_{2}\leq t$, then $$\begin{aligned}
\mathbb{\tilde{E}}\left[\left|\int_{s_{1}}^{s_{2}}\theta_{u}\overleftarrow{Y_{u}^{i}}\right|^{p}\right] & \leq\mathbb{\tilde{E}}\left[\left(\int_{s_{1}}^{s_{2}}\left|\theta_{u}\right|^{2}du\right)^{p/2}\right]\nonumber \\
& \leq\mathbb{\tilde{E}}\left[\left(s_{2}-s_{1}\right)^{p/2-1}\left(\int_{s_{1}}^{s_{2}}\left|\theta_{u}\right|^{p}du\right)\right]\nonumber \\
& \leq\sup_{0\leq s\leq t}\mathbb{\tilde{E}}\left[\left|\theta_{s}\right|^{p}\right]\left(s_{2}-s_{1}\right)^{p/2}\nonumber \\
& \leq C\left(s_{2}-s_{1}\right)^{p/2},\label{eq: BE_theta_delta}\end{aligned}$$ where we have used the Burkholder-Davis-Gundy inequality for backward martingales, Jensen’s inequality and Fubini’s theorem. The previous estimate yields that $$\sup_{0\leq s_{1}\leq s_{2}\leq t}\mathbb{\tilde{E}}\left[\left|\int_{s_{1}}^{s_{2}}\theta_{u}\overleftarrow{Y_{u}^{i}}\right|^{p}\right]\leq\sup_{0\leq s\leq t}\mathbb{\tilde{E}}\left[\left|\theta_{s}\right|^{p}\right]t^{p/2}\leq C\left(t\right).\label{eq: BE_theta}$$ Moreover, using Jensen’s inequality and Fubini’s theorem we have that $$\begin{aligned}
\mathbb{\tilde{E}}\left[\left|\int_{s_{1}}^{s_{2}}\phi_{i}\left(X_{s}\right)\theta du\right|^{p}\right] & \leq\left\Vert \phi\right\Vert _{\infty}^{p}\mathbb{\tilde{E}}\left[\left|\int_{s_{1}}^{s_{2}}\left|\theta\right|du\right|^{p}\right]\\
& \leq\left\Vert \phi\right\Vert _{\infty}^{p}\mathbb{\tilde{E}}\left[\left(s_{2}-s_{1}\right)^{p-1}\int_{s_{1}}^{s_{2}}\left|\theta\right|du\right]\\
& \leq\left\Vert \phi\right\Vert _{\infty}^{p}\sup_{0\leq s\leq t}\mathbb{\tilde{E}}\left[\left|\theta_{s}\right|^{p}\right]\left(s_{2}-s_{1}\right)^{p}.\end{aligned}$$ The previous estimate yields that $$\mathbb{\tilde{E}}\left[\left|\int_{s_{1}}^{s_{2}}\phi_{i}\left(X_{s}\right)\theta du\right|^{p}\right]\leq\left\Vert \phi\right\Vert _{\infty}^{p}\sup_{0\leq s\leq t}\mathbb{\tilde{E}}\left[\left|\theta_{s}\right|^{p}\right]t^{p/2}\leq C\left(t,\phi\right).\label{eq: E_FiTheta}$$
- Let $0\leq s_{1}\leq s_{2}\leq t$, then similar reasonings as in the previous point and hypothesis $\left(\ref{eq: BoundBeta}\right)$ give $$\begin{aligned}
\mathbb{\tilde{E}}\left[\left|\int_{s_{1}}^{s_{2}}\beta_{u}\overleftarrow{Y_{u}^{i}}\right|^{p}\right] & \leq\mathbb{\tilde{E}}\left[\left(\int_{s_{1}}^{s_{2}}\left|\beta_{u}\right|^{2}du\right)^{p/2}\right]\nonumber \\
& \leq\left(s_{2}-s_{1}\right)^{p/2-1}\int_{s_{1}}^{s_{2}}\left|\beta_{u}\right|^{p}du\nonumber \\
& \leq t^{p/2-1}\sum_{j=1}^{n}\int_{t_{j-1}}^{t_{j}}\left|\beta_{u}\right|^{p}du\nonumber \\
& \leq t^{p/2}\delta^{mp}\leq C\left(t\right)\label{eq: EB_Beta}\end{aligned}$$ Moreover, if $s_{1}=t_{j}$ and $s_{2}=t_{j+1}$for some $j\in\{0,...,n-1\}$ we can conclude using hypothesis $\left(\ref{eq: BoundBeta}\right)$ that $$\mathbb{\tilde{E}}\left[\left|\int_{s_{1}}^{s_{2}}\beta_{u}\overleftarrow{Y_{u}^{i}}\right|^{p}\right]\leq\delta^{p/2-1}\int_{t_{j}}^{t_{j+1}}\delta^{mp}du\leq\delta^{\left(m+\frac{1}{2}\right)p}.\label{eq: EB_Beta_delta}$$ Finally, $$\mathbb{E}\left[\left|\int_{t_{j}}^{t_{j+1}}\phi_{i}\left(X_{s}\right)\beta_{s}ds\right|^{p}\right]\leq\left\Vert \phi\right\Vert _{\infty}^{p}\delta^{\left(m+1\right)p}=C\left(\phi\right)\delta^{\left(m+1\right)p}\label{eq: E_Beta_Fi}$$
**Case $j$$=k$:**
Using Cauchy-Schwarz inequality, Lemma \[lem: Generalized Holder\] and inequalities $\left(\ref{eq:BE_M}\right)$, $\left(\ref{eq: BE_theta}\right)$ and $\left(\ref{eq: EB_Beta_delta}\right)$ we can write $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\Upsilon M_{0}^{t}\left(\phi\right)\prod_{l=1}^{2}\left\{ \left(\int_{t_{j+1}}^{t}\theta_{s}^{l}d\overleftarrow{Y_{s}^{i_{l}}}\right)\left(\int_{t_{j+1}}^{t}\kappa_{s}^{l}d\overleftarrow{Y_{s}^{a_{l}}}\right)\right\} \left(\int_{t_{j}}^{t_{j+1}}\beta_{s}d\overleftarrow{Y_{s}^{i}}\right)^{2}\right]\\
& \leq\mathbb{\tilde{E}}\left[\left|\Upsilon M_{0}^{t}\left(\phi\right)\prod_{l=1}^{2}\left\{ \left(\int_{t_{j+1}}^{t}\theta_{s}^{l}d\overleftarrow{Y_{s}^{i_{l}}}\right)\left(\int_{t_{j+1}}^{t}\kappa_{s}^{l}d\overleftarrow{Y_{s}^{a_{l}}}\right)\right\} \right|^{2}\right]^{1/2}\mathbb{\tilde{E}}\left[\left(\int_{t_{j}}^{t_{j+1}}\beta_{s}d\overleftarrow{Y_{s}^{i_{1}}}\right)^{4}\right]^{1/2}\\
& \leq\left(\mathbb{\tilde{E}}\left[\left|\Upsilon\right|^{12}\right]\mathbb{\tilde{E}}\left[\left|M_{0}^{t}\left(\phi\right)\right|^{12}\right]\prod_{l=1}^{2}\mathbb{\tilde{E}}\left[\left|\int_{t_{j+1}}^{t}\theta_{s}^{l}d\overleftarrow{Y_{s}^{i_{l}}}\right|^{12}\right]\mathbb{\tilde{E}}\left[\left|\int_{t_{j+1}}^{t}\kappa_{s}^{l}d\overleftarrow{Y_{s}^{a_{l}}}\right|^{12}\right]\right)^{1/12}\delta^{\frac{1}{2}\left(m+\frac{1}{2}\right)4}\\
& \leq C\left(t\right)\delta^{2m+1}\end{aligned}$$ **Case $j$$<k$:**
Using Lemma \[lem: Backward Martingale\], we can write $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\Upsilon M_{0}^{t}\left(\phi\right)\prod_{l=1}^{2}\left\{ \left(\int_{t_{j+1}}^{t}\theta_{s}^{l}d\overleftarrow{Y_{s}^{i_{l}}}\right)\left(\int_{t_{k+1}}^{t}\kappa_{s}^{l}d\overleftarrow{Y_{s}^{a_{l}}}\right)\right\} \left(\int_{t_{j}}^{t_{j+1}}\beta_{s}d\overleftarrow{Y_{s}^{i}}\right)\left(\int_{t_{k}}^{t_{k+1}}\beta_{s}d\overleftarrow{Y_{s}^{i}}\right)\right]\\
& =\mathbb{\tilde{E}}\left[\Upsilon\left(\int_{t_{k}}^{t_{k+1}}\beta_{s}d\overleftarrow{Y_{s}^{i}}\right)\prod_{l=1}^{2}\left\{ \left(\int_{t_{j+1}}^{t}\theta_{s}^{l}d\overleftarrow{Y_{s}^{i_{l}}}\right)\left(\int_{t_{k+1}}^{t}\kappa_{s}^{l}d\overleftarrow{Y_{s}^{a_{l}}}\right)\right\} \right.\\
& \qquad\left.\times M_{t_{j+1}}^{t}\left(\phi\right)\left(\int_{t_{j}}^{t_{j+1}}\phi_{i}\left(X_{s}\right)\beta_{s}ds\right)\right]\\
& =\mathbb{\tilde{E}}\left[\Upsilon_{1}M_{t_{j+1}}^{t}\left(\phi\right)\prod_{l=1}^{2}\left\{ \left(\int_{t_{j+1}}^{t}\theta_{s}^{l}d\overleftarrow{Y_{s}^{i_{l}}}\right)\right\} \right]\\
& \triangleq\sum_{i=1}^{9}A_{i},\end{aligned}$$ Where $$\Upsilon_{1}=\Upsilon\left(\int_{t_{k}}^{t_{k+1}}\beta_{s}d\overleftarrow{Y_{s}^{i}}\right)\left(\int_{t_{j}}^{t_{j+1}}\phi_{i}\left(X_{s}\right)\beta_{s}ds\right)\prod_{l=1}^{2}\left(\int_{t_{k+1}}^{t}\kappa_{s}^{l}d\overleftarrow{Y_{s}^{a_{l}}}\right),$$ and $$\begin{aligned}
A_{1} & =\mathbb{\tilde{E}}\left[\Upsilon_{1}M_{t_{j+1}}^{t}\left(\phi\right)\left(\int_{t_{j+1}}^{t_{k}}\theta_{s}^{1}d\overleftarrow{Y_{s}^{i_{1}}}\right)\left(\int_{t_{j+1}}^{t_{k}}\theta_{s}^{2}d\overleftarrow{Y_{s}^{i_{2}}}\right)\right]\\
A_{2} & =\mathbb{\tilde{E}}\left[\Upsilon_{1}M_{t_{j+1}}^{t}\left(\phi\right)\left(\int_{t_{j+1}}^{t_{k}}\theta_{s}^{1}d\overleftarrow{Y_{s}^{i_{1}}}\right)\left(\int_{t_{k}}^{t_{k+1}}\theta_{s}^{2}d\overleftarrow{Y_{s}^{i_{2}}}\right)\right]\\
A_{3} & =\mathbb{\tilde{E}}\left[\Upsilon_{1}M_{t_{j+1}}^{t}\left(\phi\right)\left(\int_{t_{j+1}}^{t_{k}}\theta_{s}^{1}d\overleftarrow{Y_{s}^{i_{1}}}\right)\left(\int_{t_{k+1}}^{t}\theta_{s}^{2}d\overleftarrow{Y_{s}^{i_{2}}}\right)\right]\\
A_{4} & =\mathbb{\tilde{E}}\left[\Upsilon_{1}M_{t_{j+1}}^{t}\left(\phi\right)\left(\int_{t_{k}}^{t_{k+1}}\theta_{s}^{1}d\overleftarrow{Y_{s}^{i_{1}}}\right)\left(\int_{t_{j+1}}^{t_{k}}\theta_{s}^{2}d\overleftarrow{Y_{s}^{i_{2}}}\right)\right]\\
A_{5} & =\mathbb{\tilde{E}}\left[\Upsilon_{1}M_{t_{j+1}}^{t}\left(\phi\right)\left(\int_{t_{k}}^{t_{k+1}}\theta_{s}^{1}d\overleftarrow{Y_{s}^{i_{1}}}\right)\left(\int_{t_{k}}^{t_{k+1}}\theta_{s}^{2}d\overleftarrow{Y_{s}^{i_{2}}}\right)\right]\\
A_{6} & =\mathbb{\tilde{E}}\left[\Upsilon_{1}M_{t_{j+1}}^{t}\left(\phi\right)\left(\int_{t_{k}}^{t_{k+1}}\theta_{s}^{1}d\overleftarrow{Y_{s}^{i_{1}}}\right)\left(\int_{t_{k+1}}^{t}\theta_{s}^{2}d\overleftarrow{Y_{s}^{i_{2}}}\right)\right]\\
A_{7} & =\mathbb{\tilde{E}}\left[\Upsilon_{1}M_{t_{j+1}}^{t}\left(\phi\right)\left(\int_{t_{k+1}}^{t}\theta_{s}^{1}d\overleftarrow{Y_{s}^{i_{1}}}\right)\left(\int_{t_{j+1}}^{t_{k}}\theta_{s}^{2}d\overleftarrow{Y_{s}^{i_{2}}}\right)\right]\\
A_{8} & =\mathbb{\tilde{E}}\left[\Upsilon_{1}M_{t_{j+1}}^{t}\left(\phi\right)\left(\int_{t_{k+1}}^{t}\theta_{s}^{1}d\overleftarrow{Y_{s}^{i_{1}}}\right)\left(\int_{t_{k}}^{t_{k+1}}\theta_{s}^{2}d\overleftarrow{Y_{s}^{i_{2}}}\right)\right]\\
A_{9} & =\mathbb{\tilde{E}}\left[\Upsilon_{1}M_{t_{j+1}}^{t}\left(\phi\right)\left(\int_{t_{k+1}}^{t}\theta_{s}^{1}d\overleftarrow{Y_{s}^{i_{1}}}\right)\left(\int_{t_{k+1}}^{t}\theta_{s}^{2}d\overleftarrow{Y_{s}^{i_{2}}}\right)\right]\end{aligned}$$ The treatment of some of the terms is completely analogous. We distinguish four subcases:
**Subcase** $1$: This subcase covers term $A_{9}.$ We apply Lemma \[lem: Backward Martingale II\] to write $$\begin{aligned}
A_{9} & =\mathbb{\tilde{E}}\left[\Upsilon_{1}M_{t_{j+1}}^{t}\left(\phi\right)\left(\int_{t_{k+1}}^{t}\theta_{s}^{1}d\overleftarrow{Y_{s}^{i_{1}}}\right)\left(\int_{t_{k+1}}^{t}\theta_{s}^{2}d\overleftarrow{Y_{s}^{i_{2}}}\right)\right]\\
& =\mathbb{\tilde{E}}\left[\Gamma_{1}M_{t_{k+1}}^{t}\left(\phi\right)\left(\int_{t_{j}}^{t_{j+1}}\phi_{i}\left(X_{s}\right)\beta_{s}ds\right)\left(\int_{t_{k}}^{t_{k+1}}\phi_{i}\left(X_{s}\right)\beta_{s}ds\right)\right],\end{aligned}$$ where $$\Gamma_{1}\triangleq\Upsilon\prod_{l=1}^{2}\left(\int_{t_{k+1}}^{t}\kappa_{s}^{l}d\overleftarrow{Y_{s}^{a_{l}}}\right)\left(\int_{t_{k+1}}^{t}\theta_{s}^{l}d\overleftarrow{Y_{s}^{i_{l}}}\right).$$ Hence, using Lemma \[lem: Generalized Holder\] and inequality $\left(\ref{eq: BE_theta}\right)$we have, for $p\geq1,$ that $$\mathbb{E}\left[\left|\Gamma_{1}\right|^{p}\right]\leq C\left(t\right),$$ and using Lemma \[lem: Generalized Holder\] and inequalities $\left(\ref{eq:BE_M}\right)$ and $\left(\ref{eq: E_Beta_Fi}\right)$ we obtain $$\begin{aligned}
\left|A_{9}\right| & \leq\mathbb{\tilde{E}}\left[\left|\Gamma_{1}\right|^{4}\right]^{1/4}\mathbb{\tilde{E}}\left[\left|M_{t_{k+1}}^{t}\left(\phi\right)\right|^{4}\right]^{1/4}\mathbb{E}\left[\left|\int_{t_{j}}^{t_{j+1}}\phi_{i}\left(X_{s}\right)\beta_{s}ds\right|^{4}\right]^{1/4}\\
& \qquad\times\mathbb{E}\left[\left|\int_{t_{k}}^{t_{k+1}}\phi_{i}\left(X_{s}\right)\beta_{s}ds\right|^{4}\right]^{1/4}\\
& \leq C\left(t,\phi\right)\delta^{2m+2}.\end{aligned}$$
**Subcase** $2$: The terms $A_{2},A_{4},A_{5},A_{6}$ and $A_{8}$ are treated analogously. We will write the proof for $A_{2}.$ By Lemma \[lem: Backward Martingale II\], we can write $$\begin{aligned}
A_{2} & =\mathbb{\tilde{E}}\left[\Upsilon_{1}M_{t_{j+1}}^{t}\left(\phi\right)\left(\int_{t_{j+1}}^{t_{k}}\theta_{s}^{1}d\overleftarrow{Y_{s}^{i_{1}}}\right)\left(\int_{t_{k}}^{t_{k+1}}\theta_{s}^{2}d\overleftarrow{Y_{s}^{i_{2}}}\right)\right]\\
& =\mathbb{\tilde{E}}\left[\Gamma_{2}M_{t_{j+1}}^{t}\left(\phi\right)\left(\int_{t_{j}}^{t_{j+1}}\phi_{i}\left(X_{s}\right)\beta_{s}ds\right)\left(\int_{t_{k}}^{t_{k+1}}\beta_{s}d\overleftarrow{Y_{s}^{i}}\right)\left(\int_{t_{k}}^{t_{k+1}}\theta_{s}^{2}d\overleftarrow{Y_{s}^{i_{2}}}\right)\right],\end{aligned}$$ where $$\Gamma_{2}\triangleq\Upsilon\left(\int_{t_{j+1}}^{t_{k}}\theta_{s}^{1}d\overleftarrow{Y_{s}^{i_{1}}}\right)\left(\int_{t_{k}}^{t_{k+1}}\theta_{s}^{2}d\overleftarrow{Y_{s}^{i_{2}}}\right)\prod_{l=1}^{2}\left(\int_{t_{k+1}}^{t}\kappa_{s}^{l}d\overleftarrow{Y_{s}^{a_{l}}}\right).$$ Hence, using Lemma \[lem: Generalized Holder\] and inequality $\left(\ref{eq: BE_theta}\right)$we have, for $p\geq1,$ that $$\mathbb{E}\left[\left|\Gamma_{2}\right|^{p}\right]\leq C\left(t\right),$$ and using Lemma \[lem: Generalized Holder\] and inequalities $\left(\ref{eq:BE_M}\right)$,$\left(\ref{eq: E_Beta_Fi}\right)$, $\left(\ref{eq: EB_Beta_delta}\right)$ and $\left(\ref{eq: BE_theta_delta}\right)$ we obtain $$\begin{aligned}
\left|A_{2}\right| & \leq\mathbb{E}\left[\left|\Gamma_{2}\right|^{5}\right]^{1/5}\mathbb{E}\left[\left|M_{t_{j+1}}^{t}\left(\phi\right)\right|^{5}\right]^{1/5}\mathbb{E}\left[\left|\int_{t_{j}}^{t_{j+1}}\phi_{i}\left(X_{s}\right)\beta_{s}ds\right|^{5}\right]^{1/5}\\
& \qquad\times\mathbb{E}\left[\left|\int_{t_{k}}^{t_{k+1}}\beta_{s}d\overleftarrow{Y_{s}^{i}}\right|^{5}\right]^{1/5}\mathbb{E}\left[\left|\int_{t_{k}}^{t_{k+1}}\theta_{s}^{2}d\overleftarrow{Y_{s}^{i_{2}}}\right|^{5}\right]^{1/5}\\
& \leq C\left(t,\phi\right)\delta^{\left(m+1\right)}\delta^{\left(m+\frac{1}{2}\right)}\delta^{1/2}=C\left(t,\phi\right)\delta^{2m+2}.\end{aligned}$$
**Subcase** 3: The terms $A_{3}$and $A_{7}$ are treated analogously. We will write the proof for $A_{3}$. We apply Lemma \[lem: Backward Martingale II\] twice to write $$\begin{aligned}
A_{3} & =\mathbb{\tilde{E}}\left[\Upsilon_{1}M_{t_{j+1}}^{t}\left(\phi\right)\left(\int_{t_{j+1}}^{t_{k}}\theta_{s}^{1}d\overleftarrow{Y_{s}^{i_{1}}}\right)\left(\int_{t_{k+1}}^{t}\theta_{s}^{2}d\overleftarrow{Y_{s}^{i_{2}}}\right)\right]\\
& =\mathbb{\tilde{E}}\left[\Gamma_{3}\left(\int_{t_{k}}^{t_{k+1}}\beta_{s}d\overleftarrow{Y_{s}^{i}}\right)M_{t_{k}}^{t}\left(\phi\right)\left(\int_{t_{j+1}}^{t_{k}}\phi_{i_{1}}\left(X_{s}\right)\theta_{s}^{1}ds\right)\left(\int_{t_{j}}^{t_{j+1}}\phi_{i}\left(X_{s}\right)\beta_{s}ds\right)\right]\\
& =\mathbb{\tilde{E}}\left[\Gamma_{3}\left(\int_{t_{j+1}}^{t_{k}}\phi_{i_{1}}\left(X_{s}\right)\theta_{s}^{1}ds\right)M_{t_{k+1}}^{t}\left(\phi\right)\left(\int_{t_{k}}^{t_{k+1}}\phi_{i}\left(X_{s}\right)\beta_{s}ds\right)\left(\int_{t_{j}}^{t_{j+1}}\phi_{i}\left(X_{s}\right)\beta_{s}ds\right)\right]\end{aligned}$$ where $$\Gamma_{3}=\Upsilon\left(\int_{t_{k+1}}^{t}\theta_{s}^{2}d\overleftarrow{Y_{s}^{i_{2}}}\right)\prod_{l=1}^{2}\left(\int_{t_{k+1}}^{t}\kappa_{s}^{l}d\overleftarrow{Y_{s}^{a_{l}}}\right).$$ Using Lemma \[lem: Generalized Holder\] and inequalities $\left(\ref{eq: BE_theta}\right)$ and $\left(\ref{eq: E_FiTheta}\right)$ we have, for $p\geq1,$ that $$\mathbb{E}\left[\left|\Gamma_{3}\left(\int_{t_{j+1}}^{t_{k}}\phi_{i_{1}}\left(X_{s}\right)\theta_{s}^{1}ds\right)\right|^{p}\right]\leq C\left(t,\phi\right),$$ and using Lemma \[lem: Generalized Holder\] and inequalities $\left(\ref{eq:BE_M}\right)$,$\left(\ref{eq: E_Beta_Fi}\right)$, $\left(\ref{eq: EB_Beta_delta}\right)$ and $\left(\ref{eq: BE_theta_delta}\right)$ $$\begin{aligned}
\left|A_{3}\right| & \leq\mathbb{\tilde{E}}\left[\left|\Gamma_{3}\left(\int_{t_{j+1}}^{t_{k}}\phi_{i_{1}}\left(X_{s}\right)\theta_{s}^{1}ds\right)\right|^{4}\right]^{1/4}\mathbb{\tilde{E}}\left[\left|M_{t_{k+1}}^{t}\left(\phi\right)\right|^{5}\right]^{1/5}\\
& \qquad\times\mathbb{\tilde{E}}\left[\left|\int_{t_{k}}^{t_{k+1}}\phi_{i}\left(X_{s}\right)\beta_{s}ds\right|^{4}\right]^{1/4}\mathbb{\tilde{E}}\left[\left|\int_{t_{j}}^{t_{j+1}}\phi_{i}\left(X_{s}\right)\beta_{s}ds\right|^{4}\right]^{1/4}\\
& \leq C\left(t,\phi\right)\delta^{m+1}\delta^{m+1}=C\delta^{2m+1}.\end{aligned}$$
**Subcase** 4: This subcase corresponds to the term $A_{1}.$ Applying Lemma \[lem: Backward Martingale II\] and Lemma \[lem: Backward Martingale\] we can write $$\begin{aligned}
A_{1} & =\mathbb{\tilde{E}}\left[\Upsilon_{1}M_{t_{j+1}}^{t}\left(\phi\right)\left(\int_{t_{j+1}}^{t_{k}}\theta_{s}^{1}d\overleftarrow{Y_{s}^{i_{1}}}\right)\left(\int_{t_{j+1}}^{t_{k}}\theta_{s}^{2}d\overleftarrow{Y_{s}^{i_{2}}}\right)\right]\\
& =\mathbb{\tilde{E}}\left[\Gamma_{4}\Gamma_{5}M_{t_{k}}^{t}\left(\phi\right)\left(\int_{t_{k}}^{t_{k+1}}\beta_{s}d\overleftarrow{Y_{s}^{i}}\right)\left(\int_{t_{j}}^{t_{j+1}}\phi_{i}\left(X_{s}\right)\beta_{s}ds\right)\right]\\
& =\mathbb{\tilde{E}}\left[\Gamma_{4}\Gamma_{5}M_{t_{k+1}}^{t}\left(\phi\right)\left(\int_{t_{k}}^{t_{k+1}}\phi_{i}\left(X_{s}\right)\beta_{s}ds\right)\left(\int_{t_{j}}^{t_{j+1}}\phi_{i}\left(X_{s}\right)\beta_{s}ds\right)\right]\end{aligned}$$ where $$\begin{aligned}
\Gamma_{4} & =\Upsilon\prod_{l=1}^{2}\left(\int_{t_{k+1}}^{t}\kappa_{s}^{l}d\overleftarrow{Y_{s}^{a_{l}}}\right)\\
\Gamma_{5} & =\mathbf{1}_{\left\{ i_{1}=i_{2}\right\} }\int_{t_{j+1}}^{t_{k}}\theta_{s}^{1}\theta_{s}^{2}du\\
& \qquad+\int_{t_{j+1}}^{t_{k}}\phi_{i_{1}}\left(X_{u}\right)\theta_{u}^{1}\left(\int_{u}^{t_{k}}\phi_{i_{2}}\left(X_{v}\right)\theta_{v}^{2}dv\right)du\\
& \qquad+\int_{t_{j+1}}^{t_{k}}\phi_{i_{2}}\left(X_{u}\right)\theta_{u}^{2}\left(\int_{u}^{t_{k}}\phi_{i_{1}}\left(X_{v}\right)\theta_{v}^{1}dv\right)du\end{aligned}$$ To finish the proof one follows the same reasonings in the previous subcase taking into account that $\Gamma_{4}$ and $\Gamma_{5}$ have moments of all orders that only depend on $t$ and $\phi$.
**Case $j$$>k$:**
This is completely symmetric to the previous case by swapping the role of the $\theta'$s by the $\kappa'$s.
\[rem: BackwardEstimate\]Under the assumptions of Lemma \[lem: Main Backward\] one can prove that for any $q\in\{1,...,m\}$ and $i,i_{1},...,i_{q}a_{1},...,a_{q}\in\left\{ 1,...,d_{Y}\right\} $, we have that $$\begin{aligned}
& \left|\mathbb{\tilde{E}}\left[\Upsilon M_{0}^{t}\left(\phi\right)\prod_{l=1}^{q}\left\{ \left(\int_{t_{j+1}}^{t}\theta_{s}^{l}d\overleftarrow{Y_{s}^{i_{l}}}\right)\left(\int_{t_{k+1}}^{t}\kappa_{s}^{l}d\overleftarrow{Y_{s}^{a_{l}}}\right)\right\} \left(\int_{t_{j}}^{t_{j+1}}\beta_{s}d\overleftarrow{Y_{s}^{i}}\right)\left(\int_{t_{k}}^{t_{k+1}}\beta_{s}d\overleftarrow{Y_{s}^{i}}\right)\right]\right|\\
& \leq C\left\{ \mathbf{1}_{\left\{ j\neq k\right\} }\delta^{2m+2}+\mathbf{1}_{\left\{ j=k\right\} }\delta^{2m+1}\right\} .\end{aligned}$$
\[subsec:CondExpEst\]Conditional expectation estimates
------------------------------------------------------
In this subsection we will show the main estimates for conditional expectations with respect to the observation filtration that will allow the proof of our result. Throughout this section we assume that $\varphi\in\mathcal{B}_{P}$, which ensures that Corollary \[cor: BoundsFi\] holds, and $m\in\mathbb{N}$.
\[lem: Alpha0=00003D00003Dm\]Assume that **H**$(m)$ holds. For $\alpha\in\mathcal{R}\left(\mathcal{M}_{m-1}(S_{0})\right)$ with $\left\vert \alpha\right\vert _{0}=m$ and $i\in\left\{ 0,1,...,d_{Y}\right\} $ we have $$\mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}\int_{0}^{t}I_{\alpha}(L^{\alpha}h_{0}(X_{\cdot}))_{\tau(s),s}dY_{s}^{i}|\mathcal{Y}_{t}\right]^{2}\right]\leq C\delta^{2m}.$$
Using Jensen inequality, Hölder inequality and Burkholder-Davis-Gundy inequality, if $i\neq0$, or Jensen inequality, if $i=0$, we get $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\left\vert \mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}\int_{0}^{t}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dY_{s}^{i}|\mathcal{Y}_{t}\right]\right\vert ^{2}\right]\\
& \leq\mathbb{\tilde{E}}\left[\left\vert \varphi(X_{t})e^{\xi_{t}}\int_{0}^{t}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dY_{s}^{i}\right\vert ^{2}\right]\\
& \leq\mathbb{\tilde{E}}\left[\left\vert \varphi(X_{t})e^{\xi_{t}}\right\vert ^{2+\varepsilon}\right]^{\frac{2}{2+\varepsilon}}\mathbb{\tilde{E}}\left[\left\vert \int_{0}^{t}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dY_{s}^{i}\right\vert ^{2\frac{2+\varepsilon}{\varepsilon}}\right]^{\frac{\varepsilon}{2+\varepsilon}}\\
& \leq C\left\Vert \varphi(X_{t})e^{\xi_{t}}\right\Vert _{2+\varepsilon}^{2}\mathbb{\tilde{E}}\left[\left\vert \int_{0}^{t}\left\vert I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}\right\vert ^{2}ds\right\vert ^{\frac{2+\varepsilon}{\varepsilon}}\right]^{\frac{\varepsilon}{2+\varepsilon}}.\end{aligned}$$ Next, using Jensen inequality again, Fubini’s Theorem, Assumption **H**$(m)$, Lemma \[lem: Moments Iterated Integral\] and that $|\alpha|+|\alpha|_{0}=2m$ we get $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\left\vert \int_{0}^{t}\left\vert I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}\right\vert ^{2}ds\right\vert ^{\frac{2+\varepsilon}{\varepsilon}}\right]\\
& \leq C\mathbb{\tilde{E}}\left[\left\vert I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}\right\vert ^{2\frac{2+\varepsilon}{\varepsilon}}\right]ds\\
& \leq C\int_{0}^{t}\mathbb{\tilde{E}}[\sup_{\tau(s)\leq u\leq s}\left\vert L^{\alpha}h_{i}(X_{u})\right\vert ^{2\frac{2+\varepsilon}{\varepsilon}}]\left(s-\tau(s)\right)^{\frac{2+\varepsilon}{\varepsilon}\{|\alpha|+|\alpha|_{0}\}}ds\\
& \leq C\delta^{2m\frac{2+\varepsilon}{\varepsilon}},\end{aligned}$$ from which follows the result.
\[lem: ds\]Assume that **H**$(m)$ holds. For $\alpha\in\mathcal{R}\left(\mathcal{M}_{m-1}(S_{0})\right)$ with $\left\vert \alpha\right\vert _{0}\neq m$ $$\mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}\int_{0}^{t}I_{\alpha}(L^{\alpha}h_{0}(X_{\cdot}))_{\tau(s),s}ds|\mathcal{Y}_{t}\right]^{2}\right]\leq C\delta^{2m}$$
We will give only the proof for the case $m\in\{1,2\}.$ The proof for $m>2$ follows the same ideas but it is tedious to write down and we leave it to the reader. We split the proof depending on $\left\vert \alpha\right\vert _{0}$, the number of zeros in $\alpha$. If $m=1,\left\vert \alpha\right\vert _{0}\in\{0\}$ and if $m=2,\left\vert \alpha\right\vert _{0}\in\{0,1\}.$ We group the three cases into two: $\left\vert \alpha\right\vert _{0}=m-1$ and $\left\vert \alpha\right\vert _{0}=0,$ (of course the two overlap when $m=1$).
Assume that $\left\vert \alpha\right\vert _{0}=m-1.$ Then, using Theorem \[thm: Integral Representation\] we can write $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}\int_{0}^{t}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}ds|\mathcal{Y}_{t}\right]\\
& =\int_{0}^{t}\mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}|\mathcal{Y}_{t}\right]ds\\
& =\int_{0}^{t}\mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}|\mathcal{H}_{0}^{t}\right]I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}|\mathcal{Y}_{t}\right]ds\\
& \quad+\sum_{r=1}^{d_{V}}\int_{0}^{t}\mathbb{\tilde{E}}\left[\left(\int_{0}^{t}J_{u}^{r}dV_{u}^{r}\right)I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}|\mathcal{Y}_{t}\right]ds\\
& =\int_{0}^{t}\mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}|\mathcal{H}_{0}^{t}\right]\mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}|\mathcal{H}_{0}^{t}\right]|\mathcal{Y}_{t}\right]ds\\
& \quad+\sum_{r=1}^{d_{V}}\int_{0}^{t}\mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[\left(\int_{0}^{t}J_{u}^{r}dV_{u}^{r}\right)I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}|\mathcal{H}_{0}^{t}\right]|\mathcal{Y}_{t}\right]ds.\end{aligned}$$ Moreover, by Lemma \[lem: CondExpect\] (\[lem: CE1\]), we get $\mathbb{\tilde{E}}\left[I_{\alpha}(L^{\alpha}h_{0}(X_{\cdot}))_{\tau(s),s}|\mathcal{H}_{0}^{t}\right]=0$ and, by Lemma \[lem: CondExpect\] (\[lem: CE2\]), for $r=1,...,d_{V}$ we have $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\left(\int_{0}^{t}J_{u}^{r}dV_{u}^{r}\right)I_{\alpha}(L^{\alpha}h_{0}(X_{\cdot}))_{\tau(s),s}|\mathcal{Y}_{t}\right]\\
& =\boldsymbol{1}_{\{\alpha_{m}=0\}}\int_{\tau(s)}^{s}\mathbb{\tilde{E}}\left[\left(\int_{\tau(s)}^{s}J_{v}^{r}dV_{v}^{r}\right)I_{\alpha-}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),u}|\mathcal{Y}_{t}\right]du\\
& \quad+\boldsymbol{1}_{\{\alpha_{m}=r\}}\int_{\tau(s)}^{s}\mathbb{\tilde{E}}\left[J_{u}^{r}I_{\alpha-}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),u}|\mathcal{Y}_{t}\right]du.\end{aligned}$$ Next, using Jensen’s inequality, Cauchy-Schwartz inequality, Itô isometry, Lemma \[lem: Moments Iterated Integral\] and Remark \[rem: M\] we get $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\left\vert \int_{\tau(s)}^{s}\mathbb{\tilde{E}}\left[\left(\int_{\tau(s)}^{s}J_{v}^{r}dV_{v}^{r}\right)I_{\alpha-}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),u}|\mathcal{Y}_{t}\right]du\right\vert ^{2}\right]\\
& \leq C\delta\int_{\tau(s)}^{s}\mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[\left(\int_{\tau(s)}^{s}J_{v}^{r}dV_{v}^{r}\right)I_{\alpha-}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),u}|\mathcal{Y}_{t}\right]^{2}\right]du\\
& \leq C\delta\int_{\tau(s)}^{s}\mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[\left\vert \int_{\tau(s)}^{s}J_{v}^{r}dV_{v}^{r}\right\vert ^{2}|\mathcal{Y}_{t}\right]\mathbb{\tilde{E}}\left[\left\vert I_{\alpha-}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),u}\right\vert ^{2}|\mathcal{Y}_{t}\right]\right]du\\
& \leq C\delta\int_{\tau(s)}^{s}\mathbb{\tilde{E}}\left[\left\vert \int_{\tau(s)}^{s}J_{v}^{r}dV_{v}^{r}\right\vert ^{2}\right]\mathbb{\tilde{E}}\left[\left\vert I_{\alpha-}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),u}\right\vert ^{2}\right]du\\
& \leq C\delta\int_{\tau(s)}^{s}\int_{\tau(s)}^{s}\mathbb{\tilde{E}}\left[\left\vert J_{v}^{r}\right\vert ^{2}\right]dv(u-\tau(s))^{\left\vert \alpha-\right\vert +\left\vert \alpha-\right\vert _{0}}\mathbb{\tilde{E}}\left[\sup_{0\leq v\leq t}\left\vert L^{\alpha}h_{i}(X_{v})\right\vert ^{2}\right]du\\
& \leq C\delta^{\left\vert \alpha-\right\vert +\left\vert \alpha-\right\vert _{0}+3}=C\delta^{m-1+m-2+3}=C\delta^{2m},\end{aligned}$$ and using similar reasonings we get $$\begin{aligned}
\mathbb{\tilde{E}}\left[\left\vert \int_{\tau(s)}^{s}\mathbb{\tilde{E}}\left[J_{u}^{r}I_{\alpha-}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),u}|\mathcal{Y}_{t}\right]du\right\vert ^{2}\right] & \leq C\delta^{\left\vert \alpha-\right\vert +\left\vert \alpha-\right\vert _{0}+2}\\
& =C\delta^{m-1+m-1+2}=C\delta^{2m},\end{aligned}$$ and the result for the case $\left\vert \alpha\right\vert _{0}=m-1$ follows.
The last case is $\left\vert \alpha\right\vert _{0}=0$ and $m=2.$ Applying Theorem \[thm: Integral Representation\] we can write $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}\int_{0}^{t}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}ds|\mathcal{Y}_{t}\right]\\
& =\int_{0}^{t}\mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}|\mathcal{H}_{0}^{t}\right]I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}|\mathcal{Y}_{t}\right]ds\\
& \quad+\sum_{r=1}^{d_{V}}\int_{0}^{t}\mathbb{\tilde{E}}\left[\left(\int_{0}^{t}\mathbb{\tilde{E}}\left[J_{u}^{r}|\mathcal{H}_{0}^{t}\right]dV_{u}^{r}\right)I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}|\mathcal{Y}_{t}\right]ds\\
& \quad+\sum_{r_{1},r_{2}=1}^{d_{V}}\int_{0}^{t}\mathbb{\tilde{E}}\left[\left(\int_{0}^{t}\int_{0}^{u_{2}}J_{u_{1},u_{2}}^{r_{1},r_{2}}dV_{u_{1}}^{r_{1}}dV_{u_{2}}^{r_{2}}\right)I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}|\mathcal{Y}_{t}\right]ds.\\
& \triangleq A_{1}+\sum_{r=1}^{d_{V}}A_{2}(r)+\sum_{r_{1},r_{2}=1}^{d_{V}}A_{3}\left(r_{1},r_{2}\right).\end{aligned}$$ Applying Lemma \[lem: CondExpect\] (\[lem: CE1\]), we see that the term $A_{1}$ vanishes. Applying Lemma \[lem: CondExpect\] (\[lem: CE2\]) and, then, Lemma \[lem: CondExpect\] (\[lem: CE1\]), for $r=1,...,d_{V}$, we can write $$\begin{aligned}
A_{2}\left(r\right) & =\mathbf{1}_{\left\{ \alpha_{2}=r\right\} }\int_{0}^{t}\int_{\tau(s)}^{s}\mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[J_{u}^{r}|\mathcal{H}_{0}^{t}\right]I_{\alpha_{1}}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),u}|\mathcal{Y}_{t}\right]duds\\
& =\mathbf{1}_{\left\{ \alpha_{2}=r\right\} }\int_{0}^{t}\int_{\tau(s)}^{s}\mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[J_{u}^{r}|\mathcal{H}_{0}^{t}\right]\mathbb{\tilde{E}}\left[I_{\alpha_{1}}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),u}|\mathcal{H}_{0}^{t}\right]|\mathcal{Y}_{t}\right]duds\\
& =\mathbf{1}_{\left\{ \alpha_{2}=r,\left|\alpha_{1}\right|=\left|\alpha_{1}\right|_{0}\right\} }\int_{0}^{t}\int_{\tau(s)}^{s}\mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[J_{u}^{r}|\mathcal{H}_{0}^{t}\right]\right.\\
& \quad\times\left.I_{\alpha_{1}}\left(\mathbb{\tilde{E}}\left[L^{\alpha}h_{i}(X_{\cdot})|\mathcal{H}_{0}^{t}\right]\right){}_{\tau(s),u}|\mathcal{Y}_{t}\right]duds\end{aligned}$$ which is equal to zero because $1=\left|\alpha_{1}\right|\neq\left|\alpha_{1}\right|_{0}=0.$ Applying Lemma \[lem: CondExpect\] (\[lem: CE3\]), for $r_{1},r_{2}=1,...,d_{V}$ we can write $$\begin{aligned}
A_{3}\left(r_{1},r_{2}\right) & =\boldsymbol{1}_{\{\alpha_{2}=r_{2},\alpha_{1}=r_{1}\}}\int_{0}^{t}\int_{\tau(s)}^{s}\int_{\tau(s)}^{u_{2}\wedge u}\mathbb{\tilde{E}}\left[J_{v,u}^{r_{1},r_{2}}I_{\left(\alpha-\right)-}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),v}|\mathcal{Y}_{t}\right]dvduds\\
& =\boldsymbol{1}_{\{\alpha_{2}=r_{2},\alpha_{1}=r_{1}\}}\int_{0}^{t}\int_{\tau(s)}^{s}\int_{\tau(s)}^{u}\mathbb{\tilde{E}}\left[J_{v,u}^{r_{1},r_{2}}L^{\alpha}h_{i}(X_{v})|\mathcal{Y}_{t}\right]dvduds\\
& \leq\boldsymbol{1}_{\{\alpha_{2}=r_{2},\alpha_{1}=r_{1}\}}\int_{0}^{t}\int_{\tau(s)}^{s}\int_{\tau(s)}^{u}\mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[\left|J_{v,u}^{r_{1},r_{2}}\right|^{2}|\mathcal{H}_{0}^{t}\right]^{1/2}\right.\\
& \quad\times\left.\mathbb{\tilde{E}}\left[\left|L^{\alpha}h_{i}(X_{v})\right|^{2}|\mathcal{H}_{0}^{t}\right]^{1/2}|\mathcal{Y}_{t}\right]dvduds,\end{aligned}$$ Hence, using Jensen’s inequality, Cauchy-Schwartz inequality and Remark \[rem: M\] we have $$\begin{aligned}
\mathbb{\tilde{E}}\left[\left|A_{3}\left(r_{1},r_{2}\right)\right|^{2}\right] & \leq C\boldsymbol{1}_{\{\alpha_{2}=r_{2},\alpha_{1}=r_{1}\}}\delta^{2}\int_{0}^{t}\int_{\tau(s)}^{s}\int_{\tau(s)}^{u}\mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[\left|J_{v,u}^{r_{1},r_{2}}\right|^{2}|\mathcal{H}_{0}^{t}\right]\right.\\
& \quad\times\left.\mathbb{\tilde{E}}\left[\left|L^{\alpha}h_{i}(X_{v})\right|^{2}|\mathcal{H}_{0}^{t}\right]\right]dvduds\\
& \leq C\boldsymbol{1}_{\{\alpha_{2}=r_{2},\alpha_{1}=r_{1}\}}\delta^{2}\int_{0}^{t}\int_{\tau(s)}^{s}\int_{\tau(s)}^{u}\mathbb{\tilde{E}}\left[\left|J_{v,u}^{r_{1},r_{2}}\right|^{2}\right]dvduds\\
& \leq C\boldsymbol{1}_{\{\alpha_{2}=r_{2},\alpha_{1}=r_{1}\}}\delta^{4}=C\boldsymbol{1}_{\{\alpha_{2}=r_{2},\alpha_{1}=r_{1}\}}\delta^{2m},\end{aligned}$$ and we can conclude.
\[lem: Convenient\]Let $m\in\{1,2\}$ and assume that **H**$(m)$ holds. For $\alpha\in\mathcal{R}\left(\mathcal{M}_{m-1}(S_{0})\right)$, $\left\vert \alpha\right\vert _{0}\neq\left|\alpha\right|$ and $i\neq0$, we can write
$$\begin{aligned}
& \mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}\int_{0}^{t}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dY_{s}^{i}|\mathcal{Y}_{t}\right]\nonumber \\
& =\sum_{r=1}^{d_{V}}\sum_{j=0}^{n-1}\mathbb{\tilde{E}}\left[\left(\int_{t_{j}}^{t_{j+1}}J_{s}^{r}dV_{s}^{r}\right)\left(\int_{t_{j}}^{t_{j+1}}\left(Y_{t_{j+1}}^{i}-Y_{t_{j}}^{i}\right)I_{\alpha-}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dV_{s}^{\alpha_{|\alpha|}}\right)|\mathcal{Y}_{t}\right],\label{eq: Convenient_1}\end{aligned}$$
and $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}\int_{0}^{t}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dY_{s}^{i}|\mathcal{Y}_{t}\right]\nonumber \\
& =\sum_{r_{1}=1}^{d_{V}}\sum_{j=0}^{n-1}\mathbb{\tilde{E}}\left[\left(\int_{t_{j}}^{t_{j+1}}\mathbb{\tilde{E}}\left[J_{s}^{r_{1}}|\mathcal{H}_{0}^{t}\right]dV_{s}^{r_{1}}\right)\right.\label{eq: Convenient_2}\\
& \quad\times\left.\left(\int_{t_{j}}^{t_{j+1}}\left(Y_{t_{j+1}}^{i}-Y_{t_{j}}^{i}\right)I_{\alpha-}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dV_{s}^{\alpha_{|\alpha|}}\right)|\mathcal{Y}_{t}\right]\nonumber \\
& \quad+\sum_{r_{1},r_{2}=1}^{d_{V}}\sum_{j=0}^{n-1}\mathbb{\tilde{E}}\left[\left(\int_{t_{j}}^{t_{j+1}}\int_{0}^{s_{2}}J_{s_{1},s_{2}}^{r_{1},r_{2}}dV_{s_{1}}^{r_{1}}dV_{s_{2}}^{r_{2}}\right)\right.\nonumber \\
& \quad\times\left.\left(\int_{t_{j}}^{t_{j+1}}\left(Y_{t_{j+1}}^{i}-Y_{t_{j}}^{i}\right)I_{\alpha-}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dV_{s}^{\alpha_{|\alpha|}}\right)|\mathcal{Y}_{t}\right].\nonumber \end{aligned}$$
Note that, as $\left\vert \alpha\right\vert _{0}\neq\left|\alpha\right|,$ by Lemma \[lem: CondExpect\] $\left(\ref{lem: CE1}\right)$ we have that if $0\leq u\leq v\leq w\leq t$ then $$\mathbb{\tilde{E}}\left[I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{v,w}|\mathcal{H}_{u}^{t}\right]=0.\label{eq: NullCondExp}$$ Using Theorem \[thm: Integral Representation\] we can write $$\begin{aligned}
& \varphi(X_{t})e^{\xi_{t}}\int_{0}^{t}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dY_{s}^{i}\\
& =\varphi(X_{t})e^{\xi_{t}}\sum_{j_{1}=0}^{n-1}\int_{t_{j_{1}}}^{t_{j_{1}+1}}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{t_{j_{1}},s}dY_{s}^{i}\\
& =\mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}|\mathcal{H}_{0}^{t}\right]\sum_{j_{1}=0}^{n-1}\int_{t_{j_{1}}}^{t_{j_{1}+1}}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{t_{j_{1}},s}dY_{s}^{i}\\
& \quad+\sum_{r=1}^{d_{V}}\sum_{j_{1},j_{2}=0}^{n-1}\left(\int_{t_{j_{2}}}^{t_{j_{2}+1}}J_{s}^{r}dV_{s}^{r}\right)\left(\int_{t_{j_{1}}}^{t_{j_{1}+1}}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{t_{j_{1}},s}dY_{s}^{i}\right).\end{aligned}$$ Next, for $j_{1}\geq0$ we get, using equation $\left(\ref{eq: NullCondExp}\right)$ that $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}|\mathcal{H}_{0}^{t}\right]\int_{t_{j_{1}}}^{t_{j_{1}+1}}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{t_{j_{1}},s}dY_{s}^{i}|\mathcal{H}_{0}^{t}\right]\\
& =\mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}|\mathcal{H}_{0}^{t}\right]\int_{t_{j_{1}}}^{t_{j_{1}+1}}\mathbb{\tilde{E}}\left[I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{t_{j_{1}},s}|\mathcal{H}_{0}^{t}\right]dY_{s}^{i}=0.\end{aligned}$$ Moreover, for $r=1,...,d_{V},$ if $j_{2}>j_{1}$ we get that $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\left(\int_{t_{j_{2}}}^{t_{j_{2}+1}}J_{s}^{r}dV_{s}^{r}\right)\left(\int_{t_{j_{1}}}^{t_{j_{1}+1}}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{t_{j_{1}},s}dY_{s}^{i}\right)|\mathcal{H}_{t_{j_{2}}}^{t}\right]\\
& =\left(\int_{t_{j_{1}}}^{t_{j_{1}+1}}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{t_{j_{1}},s}dY_{s}^{i}\right)\mathbb{\tilde{E}}\left[\int_{t_{j_{2}}}^{t_{j_{2}+1}}J_{s}^{r}dV_{s}^{r}|\mathcal{H}_{t_{j_{2}}}^{t}\right]=0,\end{aligned}$$ and if $j_{2}<j_{1}$ we get that $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\left(\int_{t_{j_{2}}}^{t_{j_{2}+1}}J_{s}^{r}dV_{s}^{r}\right)\left(\int_{t_{j_{1}}}^{t_{j_{1}+1}}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{t_{j_{1}},s}dY_{s}^{i}\right)|\mathcal{H}_{t_{j_{1}}}^{t}\right]\\
& =\left(\int_{t_{j_{2}}}^{t_{j_{2}+1}}J_{s}^{r}dV_{s}^{r}\right)\int_{t_{j_{1}}}^{t_{j_{1}+1}}\mathbb{\tilde{E}}\left[I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{t_{j_{1}},s}|\mathcal{H}_{t_{j_{1}}}^{t}\right]dY_{s}^{i}=0.\end{aligned}$$ Hence, using the tower property of the conditional expectation we can write $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}\int_{0}^{t}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dY_{s}^{i}|\mathcal{Y}_{t}\right]\nonumber \\
& =\sum_{r=1}^{d_{V}}\mathbb{\tilde{E}}\left[\sum_{j=0}^{n-1}\left(\int_{t_{j}}^{t_{j+1}}J_{s}^{r}dV_{s}^{r}\right)\left(\int_{t_{j}}^{t_{j+1}}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dY_{s}^{i}\right)|\mathcal{Y}_{t}\right].\label{eq: Convenient Intermediate}\end{aligned}$$ By integration by parts formula for $\mathcal{F}_{s}^{V,0}\vee\mathcal{Y}_{s}$-semimartingales we have $$\begin{aligned}
\int_{t_{j}}^{t_{j+1}}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dY_{s}^{i} & =\left(Y_{t_{j+1}}^{i}-Y_{t_{j}}^{i}\right)I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{t_{j},t_{j+1}}\\
& \quad-\int_{t_{j}}^{t_{j+1}}\left(Y_{s}^{i}-Y_{t_{j}}^{i}\right)I_{\alpha-}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dV_{s}^{\alpha_{1}}.\end{aligned}$$ Moreover, we can rewrite the right hand side of the previous equality as a well defined $\mathcal{H}_{s}^{t}$-iterated integral and obtain $$\begin{aligned}
\int_{t_{j}}^{t_{j+1}}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dY_{s}^{i} & =\int_{t_{j}}^{t_{j+1}}\left(Y_{t_{j+1}}^{i}-Y_{s}^{i}\right)I_{\alpha-}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dV^{\alpha_{|\alpha|}},\end{aligned}$$ which combined with equation $\left(\ref{eq: Convenient Intermediate}\right)$ gives equation $\left(\ref{eq: Convenient_1}\right)$. Finally, using Theorem \[thm: ST formula\] with $k$=1 and repeating the same reasonings as before we get equation $\left(\ref{eq: Convenient_2}\right)$.
\[lem: dY\_m=00003D00003D1\]Assume that **H**$(1)$ holds and $\varphi\in C_{P}^{2}$. For $\alpha\in\mathcal{R}\left(\mathcal{M}_{0}(S_{0})\right)$ with $\left\vert \alpha\right\vert _{0}\neq1$ and $i\neq0$ we have that $$\begin{aligned}
\mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}\int_{0}^{t}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dY_{s}^{i}|\mathcal{Y}_{t}\right]^{2}\right] & \leq C\delta^{2}.\end{aligned}$$
We divide the proof into several steps.
**Step 1**. First we will find a more convenient expression for $$\mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}\int_{0}^{t}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dY_{s}^{i}|\mathcal{Y}_{t}\right].$$ Recall that $\alpha\in\mathcal{R}\left(\mathcal{M}_{0}(S_{0})\right)$ with $\left\vert \alpha\right\vert _{0}\neq1$ concides with the set of multiindices $\alpha=\left(\alpha_{1}\right)$ with $\alpha_{1}\in\left\{ 1,...,d_{V}\right\} $. Using Lemma \[lem: Convenient\], equation $\left(\ref{eq: Convenient_1}\right)$, and taking into account that $I_{\alpha-}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}=L^{\alpha_{1}}h_{i}(X_{s}),$ we can write $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}\int_{0}^{t}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dY_{s}^{i}|\mathcal{Y}_{t}\right]\\
& =\sum_{r=1}^{d_{V}}\sum_{j=0}^{n-1}\mathbb{\tilde{E}}\left[\left(\int_{t_{j}}^{t_{j+1}}J_{s}^{r}dV_{s}^{r}\right)\left(\int_{t_{j}}^{t_{j+1}}\left(Y_{t_{j+1}}^{i}-Y_{t_{j}}^{i}\right)L^{\alpha_{1}}h_{i}(X_{s})dV_{s}^{\alpha_{1}}\right)|\mathcal{Y}_{t}\right]\end{aligned}$$ Next, by Lemma \[lem: CondExpect\] (\[lem: CE2\]) we get that $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}\int_{0}^{t}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dY_{s}^{i}|\mathcal{Y}_{t}\right]\\
& =\sum_{r=1}^{d_{V}}\boldsymbol{1}_{\left\{ \alpha_{1}=r\right\} }\mathbb{\tilde{E}}\left[\sum_{j=0}^{n-1}\int_{t_{j}}^{t_{j+1}}J_{s}^{r}\left(Y_{t_{j+1}}^{i}-Y_{s}^{i}\right)L^{\alpha_{1}}h_{i}(X_{s})ds|\mathcal{Y}_{t}\right]\\
& =\sum_{r=1}^{d_{V}}\boldsymbol{1}_{\left\{ \alpha_{1}=r\right\} }\mathbb{\tilde{E}}\left[\int_{0}^{t}J_{s}^{r}\left(Y_{\eta(s)}^{i}-Y_{s}^{i}\right)L^{\alpha_{1}}h_{i}(X_{s})ds|\mathcal{Y}_{t}\right]\\
& =\sum_{r=1}^{d_{V}}\boldsymbol{1}_{\left\{ \alpha_{1}=r\right\} }\left(B_{1}\left(r\right)+B_{2}\left(r\right)+B_{3}\left(r\right)\right),\end{aligned}$$ where $$\begin{aligned}
B_{1}\left(r\right) & \triangleq\mathbb{\tilde{E}}\left[\int_{0}^{t}\left(J_{s}^{r}-J_{\eta(s)}^{r}\right)\left(Y_{\eta(s)}^{i}-Y_{s}^{i}\right)L^{r}h_{i}(X_{s})ds|\mathcal{Y}_{t}\right],\\
B_{2}\left(r\right) & \triangleq\mathbb{\tilde{E}}\left[\int_{0}^{t}J_{\eta(s)}^{r}\left(Y_{\eta(s)}^{i}-Y_{s}^{i}\right)\left(L^{r}h_{i}(X_{s})-L^{r}h_{i}(X_{\tau(s)})\right)ds|\mathcal{Y}_{t}\right],\\
B_{3}\left(r\right) & \triangleq\mathbb{\tilde{E}}\left[\int_{0}^{t}J_{\eta(s)}^{r}L^{r}h_{i}(X_{\tau(s)})\left(Y_{\eta(s)}^{i}-Y_{s}^{i}\right)ds|\mathcal{Y}_{t}\right].\end{aligned}$$
**Step 2**. Next, we prove the result for $B_{1}\left(r\right).$ Applying Jensen inequality, Cauchy-Schwarz inequality, Hôlder inequality, Remark \[rem: M\], that $Y$$^{i}$ is a Brownian motion under $\tilde{P}$ and Lemma \[lem: RegularityKernels\] we have that $$\begin{aligned}
\mathbb{\tilde{E}}\left[\left|B_{1}\left(r\right)\right|^{2}\right] & \leq C(t)\int_{0}^{t}\mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[\left(J_{s}^{r}-J_{\eta(s)}^{r}\right)^{2}\left(Y_{\eta(s)}^{i}-Y_{s}^{i}\right)^{2}|\mathcal{Y}_{t}\right]\mathbb{\tilde{E}}\left[\left|L^{r}h_{i}(X_{s})\right|^{2}|\mathcal{Y}_{t}\right]\right]ds\\
& \leq C(t)\int_{0}^{t}\mathbb{\tilde{E}}\left[\left(J_{s}^{r}-J_{\eta(s)}^{r}\right)^{2}\left(Y_{\eta(s)}^{i}-Y_{s}^{i}\right)^{2}\right]ds\\
& \leq C(t)\int_{0}^{t}\mathbb{\tilde{E}}\left[\left(J_{s}^{r}-J_{\eta(s)}^{r}\right)^{2+\varepsilon}\right]^{2/(2+\varepsilon)}\mathbb{\tilde{E}}\left[\left(Y_{\eta(s)}^{i}-Y_{s}^{i}\right)^{2\frac{2+\varepsilon}{\varepsilon}}\right]^{\varepsilon/(2+\varepsilon)}ds\\
& \leq C(t)\delta\int_{0}^{t}\mathbb{\tilde{E}}\left[\left(J_{s}^{r}-J_{\eta(s)}^{r}\right)^{2+\varepsilon}\right]^{2/(2+\varepsilon)}ds\\
& \leq C\left(t\right)t\delta^{2}.\end{aligned}$$
**Step 3**. Here, we prove the result for $B_{2}\left(r\right).$ Applying Jensen inequality and Cauchy-Schwarz inequality we get $$\begin{aligned}
\mathbb{\tilde{E}}\left[\left|B_{2}\left(r\right)\right|^{2}\right] & \leq C(t)\int_{0}^{t}\mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[\left|J_{\eta(s)}^{r}\left(Y_{\eta(s)}^{i}-Y_{s}^{i}\right)\right|^{2}|\mathcal{Y}_{t}\right]\mathbb{\tilde{E}}\left[\left|L^{r}h_{i}(X_{s})-L^{r}h_{i}(X_{\tau(s)})\right|^{2}|\mathcal{Y}_{t}\right]\right]ds\\
& \leq C(t)\int_{0}^{t}\mathbb{\tilde{E}}\left[\left|J_{\eta(s)}^{r}\left(Y_{\eta(s)}^{i}-Y_{s}^{i}\right)\right|^{2}\right]\mathbb{\tilde{E}}\left[\left|L^{r}h_{i}(X_{s})-L^{r}h_{i}(X_{\tau(s)})\right|^{2}\right]ds.\end{aligned}$$ Applying Hölder inequality and Proposition \[prop: UnifBoundKernels\] we can conclude that $$\begin{aligned}
\mathbb{\tilde{E}}\left[\left|J_{\eta(s)}^{r}\left(Y_{\eta(s)}^{i}-Y_{s}^{i}\right)\right|^{2}\right] & \leq\mathbb{\tilde{E}}\left[\left|J_{\eta(s)}^{r}\right|^{2+\varepsilon}\right]^{2/(2+\varepsilon)}\mathbb{\tilde{E}}\left[\left|\left(Y_{\eta(s)}^{i}-Y_{s}^{i}\right)\right|^{2\frac{2+\varepsilon}{\varepsilon}}\right]^{\varepsilon/\left(2+\varepsilon\right)}\\
& \leq\delta\sup_{0\leq s\leq t}\mathbb{\tilde{E}}\left[\left|J_{s}^{r}\right|^{2+\varepsilon}\right]^{2/(2+\varepsilon)}\leq C\delta.\end{aligned}$$ On the other hand, we can write $$L^{r}h_{i}(X_{s})-L^{r}h_{i}(X_{\tau(s)})=\int_{\tau\left(s\right)}^{s}L^{(0,r)}h_{i}(X_{u})du+\sum_{r_{1}=1}^{d_{V}}\int_{\tau\left(s\right)}^{s}L^{(r_{1},r)}h_{i}(X_{u})dV_{u}^{r_{1}}.$$ As the worst rate is achieved by the terms with the stochastic integral, it suffices to show that $$\mathbb{\tilde{E}}\left[\left|\int_{\tau\left(s\right)}^{s}L^{(r_{1},r)}h_{i}(X_{u})dV_{u}^{r_{1}}\right|^{2}\right]\leq C\delta,$$ which easily follows by Itô isometry and Remark \[rem: M\].
**Step 4**. Finally, we prove the result for $B_{3}\left(r\right).$ We can write $$\begin{aligned}
B_{3}\left(r\right) & =\sum_{j=0}^{n-1}\mathbb{\tilde{E}}\left[\int_{t_{j}}^{t_{j+1}}J_{t_{j+1}}^{r}L^{r}h_{i}(X_{t_{j}})\left(Y_{t_{j+1}}^{i}-Y_{s}^{i}\right)ds|\mathcal{Y}_{t}\right]\\
& =\sum_{j=0}^{n-1}\mathbb{\tilde{E}}\left[J_{t_{j+1}}^{r}L^{r}h_{i}(X_{t_{j}})\int_{t_{j}}^{t_{j+1}}\left(Y_{t_{j+1}}^{i}-Y_{s}^{i}\right)ds|\mathcal{Y}_{t}\right]\\
& =\sum_{j=0}^{n-1}\mathbb{\tilde{E}}\left[J_{t_{j+1}}^{r}L^{r}h_{i}(X_{t_{j}})\int_{t_{j}}^{t_{j+1}}\left(s-t_{j}\right)dY_{s}^{i}|\mathcal{Y}_{t}\right]\\
& \triangleq\sum_{j=0}^{n-1}\mathbb{\tilde{E}}\left[J_{t_{j+1}}^{r}L^{r}h_{i}(X_{t_{j}})\int_{t_{j}}^{t_{j+1}}\beta_{s}^{j}dY_{s}^{i}|\mathcal{Y}_{t}\right]\end{aligned}$$ Moreover, $$J_{t_{j+1}}^{r}=\mathbb{\tilde{E}}\left[D_{t_{j+1}}^{r}\left[\varphi\left(X_{t}\right)e^{\xi_{t}}\right]|\mathcal{H}_{t_{j+1}}^{t}\right],$$ by the Clark-Ocone formula. Using the for the Malliavin derivative, we get $$\begin{aligned}
D_{t_{j+1}}^{r}\left[\varphi\left(X_{t}\right)e^{\xi_{t}}\right] & =e^{\xi_{t}}D_{t_{j+1}}^{r}\varphi\left(X_{t}\right)+\varphi\left(X_{t}\right)D_{t_{j+1}}^{r}e^{\xi_{t}}\end{aligned}$$ Therefore, using the tower property of the conditional expectation and the previous expression for the Malliavin derivative, we have $$\begin{aligned}
& \mathbb{\tilde{E}}\left[J_{t_{j+1}}^{r}L^{r}h_{i}(X_{t_{j}})\int_{t_{j}}^{t_{j+1}}\beta_{s}^{j}dY_{s}^{i}|\mathcal{Y}_{t}\right]\\
& =\mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[D_{t_{j+1}}^{r}\left[\varphi\left(X_{t}\right)e^{\xi_{t}}\right]|\mathcal{H}_{t_{j+1}}^{t}\right]L^{r}h_{i}(X_{t_{j}})\int_{t_{j}}^{t_{j+1}}\beta_{s}^{j}dY_{s}^{i}|\mathcal{Y}_{t}\right]\\
& =\mathbb{\tilde{E}}\left[e^{\xi_{t}}D_{t_{j+1}}^{r}\varphi\left(X_{t}\right)L^{r}h_{i}(X_{t_{j}})\int_{t_{j}}^{t_{j+1}}\beta_{s}^{j}dY_{s}^{i}|\mathcal{Y}_{t}\right]\\
& \quad+\mathbb{\tilde{E}}\left[\varphi\left(X_{t}\right)D_{t_{j+1}}^{r}e^{\xi_{t}}L^{r}h_{i}(X_{t_{j}})\int_{t_{j}}^{t_{j+1}}\beta_{s}^{j}dY_{s}^{i}|\mathcal{Y}_{t}\right].\end{aligned}$$ Then, $$\begin{aligned}
\mathbb{\tilde{E}}\left[\left|B_{3}\left(r\right)\right|^{2}\right] & \leq\mathbb{\tilde{E}}\left[\left|\sum_{j=0}^{n-1}\mathbb{\tilde{E}}\left[J_{t_{j+1}}^{r}L^{r}h_{i}(X_{t_{j}})\int_{t_{j}}^{t_{j+1}}\beta_{s}^{j}dY_{s}^{i}|\mathcal{Y}_{t}\right]\right|^{2}\right]\\
& \leq2\mathbb{\tilde{E}}\left[\left|\sum_{j=0}^{n-1}e^{\xi_{t}}D_{t_{j+1}}^{r}\varphi\left(X_{t}\right)L^{r}h_{i}(X_{t_{j}})\int_{t_{j}}^{t_{j+1}}\beta_{s}^{j}dY_{s}^{i}\right|^{2}\right]\\
& \quad+2\mathbb{\tilde{E}}\left[\left|\sum_{j=0}^{n-1}\varphi\left(X_{t}\right)D_{t_{j+1}}^{r}e^{\xi_{t}}L^{r}h_{i}(X_{t_{j}})\int_{t_{j}}^{t_{j+1}}\beta_{s}^{j}dY_{s}^{i}\right|^{2}\right]\\
& \triangleq2A_{1}\left(r\right)+2A_{2}\left(r\right).\end{aligned}$$ Next, note that $$\begin{aligned}
D_{t_{j+1}}^{r}e^{\xi_{t}} & =e^{\xi_{t}}\left\{ \sum_{k=1}^{d_{Y}}\int_{0}^{t}D_{t_{j+1}}^{r}h_{k}(X_{s})dY_{s}^{k}-\frac{1}{2}\sum_{k=1}^{d_{Y}}\int_{0}^{t}D_{t_{j+1}}^{r}\left[h_{k}(X_{s})^{2}\right]ds\right\} \\
& =e^{\xi_{t}}\left\{ \sum_{k=1}^{d_{Y}}\int_{t_{j+1}}^{t}D_{t_{j+1}}^{r}h_{k}(X_{s})dY_{s}^{k}-\frac{1}{2}\sum_{k=1}^{d_{Y}}\int_{t_{j+1}}^{t}D_{t_{j+1}}^{r}\left[h_{k}(X_{s})^{2}\right]ds\right\} \\
& \triangleq e^{\xi_{t}}\left\{ \sum_{k=1}^{d_{Y}}\int_{t_{j+1}}^{t}\alpha_{s}^{j,k,1}dY_{s}^{k}-\frac{1}{2}\sum_{k=1}^{d_{Y}}\int_{t_{j+1}}^{t}\alpha_{s}^{j,k,2}ds\right\} ,\end{aligned}$$ where we have used that $D_{u}^{r}h_{k}(X_{s})=0,s<u<t$. In addition, note that $$e^{2\xi_{t}}=M_{0}^{t}\left(2h\right)\exp\left(\sum_{i=1}^{d_{Y}}\int_{0}^{t}h_{i}^{2}\left(X_{u}\right)du\right),$$ where $$M_{s}^{t}\left(h\right)=\exp\left(\sum_{i=1}^{d_{Y}}\int_{s}^{t}h_{i}(X_{u})dY_{u}^{i}-\frac{1}{2}\sum_{i=1}^{d_{Y}}\int_{s}^{t}h_{i}^{2}\left(X_{u}\right)du\right),$$ is an exponential martingale. Defining $$\Gamma(j_{1},j_{2})\triangleq D_{t_{j_{1}+1}}^{r}\varphi\left(X_{t}\right)D_{t_{j_{2}+1}}^{r}\varphi\left(X_{t}\right)L^{r}h_{i}(X_{t_{j_{1}}})L^{r}h_{i}(X_{t_{j_{2}}})\exp\left(\sum_{i=1}^{d_{Y}}\int_{0}^{t}h_{i}^{2}\left(X_{u}\right)du\right),$$ and $$\varLambda(j_{1},j_{2})\triangleq\varphi\left(X_{t}\right)^{2}L^{r}h_{i}(X_{t_{j_{1}}})L^{r}h_{i}(X_{t_{j_{2}}})\exp\left(\sum_{i=1}^{d_{Y}}\int_{0}^{t}h_{i}^{2}\left(X_{u}\right)du\right),$$ we can write $$A_{1}\left(r\right)=\sum_{j_{1},j_{2}=0}^{n-1}\mathbb{\tilde{E}}\left[\Gamma(j_{1},j_{2})M_{0}^{t}\left(2h\right)\int_{t_{j_{1}}}^{t_{j_{1}+1}}\beta_{s}^{j_{1}}dY_{s}^{i}\int_{t_{j_{2}}}^{t_{j_{2}+1}}\beta_{s}^{j_{2}}dY_{s}^{i}\right],$$ and $$A_{2}\left(r\right)=\sum_{k_{1},k_{2}=1}^{d_{Y}}A_{2,1}\left(r,k_{1},k_{2}\right)-\frac{1}{2}A_{2,2}\left(r,k_{1},k_{2}\right)-\frac{1}{2}A_{2,3}\left(r,k_{1},k_{2}\right)+\frac{1}{4}A_{2,4}\left(r,k_{1},k_{2}\right),$$ where $$\begin{aligned}
A_{2,1}\left(r,k_{1},k_{2}\right) & \triangleq\sum_{j_{1},j_{2}=0}^{n-1}\mathbb{\tilde{E}}\left[\varLambda(j_{1},j_{2})M_{0}^{t}\left(2h\right)\int_{t_{j_{1}+1}}^{t}\alpha_{s}^{j_{1},k_{1},1}dY_{s}^{k_{1}}\int_{t_{j_{2}+1}}^{t}\alpha_{s}^{j_{2},k_{2},1}dY_{s}^{k_{2}}\right.\\
& \quad\times\left.\int_{t_{j_{1}}}^{t_{j_{1}+1}}\beta_{s}^{j_{1}}dY_{s}^{i}\int_{t_{j_{2}}}^{t_{j_{2}+1}}\beta_{s}^{j_{2}}dY_{s}^{i}\right],\\
A_{2,2}\left(r,k_{1},k_{2}\right) & \triangleq\sum_{j_{1},j_{2}=0}^{n-1}\mathbb{\tilde{E}}\left[\varLambda(j_{1},j_{2})M_{0}^{t}\left(2h\right)\int_{t_{j_{1}+1}}^{t}\alpha_{s}^{j_{1},k_{1},1}dY_{s}^{k_{1}}\int_{t_{j_{2}+1}}^{t}\alpha_{s}^{j_{2},k_{2},2}ds\right.\\
& \quad\times\left.\int_{t_{j_{1}}}^{t_{j_{1}+1}}\beta_{s}^{j_{1}}dY_{s}^{i}\int_{t_{j_{2}}}^{t_{j_{2}+1}}\beta_{s}^{j_{2}}dY_{s}^{i}\right],\\
A_{2,3}\left(r,k_{1},k_{2}\right) & \triangleq\sum_{j_{1},j_{2}=0}^{n-1}\mathbb{\tilde{E}}\left[\varLambda(j_{1},j_{2})M_{0}^{t}\left(2h\right)\int_{t_{j_{1}+1}}^{t}\alpha_{s}^{j_{1},k_{1},2}ds\int_{t_{j_{2}+1}}^{t}\alpha_{s}^{j_{2},k_{2},1}dY_{s}^{k_{2}}\right.\\
& \quad\times\left.\int_{t_{j_{1}}}^{t_{j_{1}+1}}\beta_{s}^{j_{1}}dY_{s}^{i}\int_{t_{j_{2}}}^{t_{j_{2}+1}}\beta_{s}^{j_{2}}dY_{s}^{i}\right],\\
A_{2,4}\left(r,k_{1},k_{2}\right) & \triangleq\sum_{j_{1},j_{2}=0}^{n-1}\mathbb{\tilde{E}}\left[\varLambda(j_{1},j_{2})M_{0}^{t}\left(2h\right)\int_{t_{j_{1}+1}}^{t}\alpha_{s}^{j_{1},k_{1},2}ds\int_{t_{j_{2}+1}}^{t}\alpha_{s}^{j_{2},k_{2},2}ds\right.\\
& \quad\times\left.\int_{t_{j_{1}}}^{t_{j_{1}+1}}\beta_{s}^{j_{1}}dY_{s}^{i}\int_{t_{j_{2}}}^{t_{j_{2}+1}}\beta_{s}^{j_{2}}dY_{s}^{i}\right].\end{aligned}$$ The result follows by applying Lemma \[lem: Main Backward\], taking into account Remark \[rem: Backward Ito integral\], to the terms $A_{1},$$A_{2,1},A_{2,2},$$A_{2,3}$ and $A_{2,4}$.
\[lem: dY\_m=00003D00003D2\_alpha=00003D00003D1\]Assume that **H**$(2)$ holds and $\varphi\in C_{P}^{3}$. For $\alpha\in\mathcal{R}\left(\mathcal{M}_{1}(S_{0})\right)$ with $\left\vert \alpha\right\vert _{0}=1$ and $i\neq0$ we have that $$\begin{aligned}
\mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}\int_{0}^{t}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dY_{s}^{i}|\mathcal{Y}_{t}\right]^{2}\right] & \leq C\delta^{4}.\end{aligned}$$
The proof of this lemma is analogous to the proof of Lemma \[lem: dY\_m=00003D00003D1\]. Using Lemma \[lem: Convenient\], we can write $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}\int_{0}^{t}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dY_{s}^{i}|\mathcal{Y}_{t}\right]\\
& =\sum_{r=1}^{d_{V}}\sum_{j=0}^{n-1}\mathbb{\tilde{E}}\left[\left(\int_{t_{j}}^{t_{j+1}}J_{s}^{r}dV_{s}^{r}\right)\left(\int_{t_{j}}^{t_{j+1}}\left(Y_{t_{j+1}}^{i}-Y_{t_{j}}^{i}\right)I_{\alpha-}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dV_{s}^{\alpha_{|\alpha|}}\right)|\mathcal{Y}_{t}\right]\\
& \triangleq\sum_{r=1}^{d_{V}}A\left(r\right).\end{aligned}$$ Therefore, by Lemma \[lem: CondExpect\] $\left(\ref{lem: CE2}\right)$, we have that $$\begin{aligned}
& A\left(r\right)\\
& =\sum_{j=0}^{n-1}\mathbb{\tilde{E}}\left[\left(\int_{t_{j}}^{t_{j+1}}J_{s}^{r}dV_{s}^{r}\right)\left(\int_{t_{j}}^{t_{j+1}}\left(Y_{t_{j+1}}^{i}-Y_{s}^{i}\right)\left(\int_{t_{j}}^{s}L^{\alpha}h_{i}\left(X_{u}\right)dV_{u}^{\alpha_{1}}\right)dV_{s}^{\alpha_{2}}\right)|\mathcal{Y}_{t}\right]\\
& =\mathbf{1}_{\left\{ \alpha_{1}=0,\alpha_{2}=r\right\} }\mathbb{\tilde{E}}\left[\left(\int_{0}^{t}\left(Y_{\eta(s)}^{i}-Y_{s}^{i}\right)J_{s}^{r}\left(\int_{\tau(s)}^{s}L^{\alpha}h_{i}\left(X_{u}\right)du\right)ds\right)|\mathcal{Y}_{t}\right]\\
& \quad+\mathbf{1}_{\left\{ \alpha_{1}=r,\alpha_{2}=0\right\} }\mathbb{\tilde{E}}\left[\left(\int_{0}^{t}\left(Y_{\eta(s)}^{i}-Y_{s}^{i}\right)\left(\int_{\tau(s)}^{s}J_{u}^{r}L^{\alpha}h_{i}\left(X_{u}\right)du\right)ds\right)|\mathcal{Y}_{t}\right]\\
& \triangleq\mathbf{1}_{\left\{ \alpha_{1}=0,\alpha_{2}=r\right\} }A_{1}\left(r\right)+\mathbf{1}_{\left\{ \alpha_{1}=r,\alpha_{2}=0\right\} }A_{2}\left(r\right).\end{aligned}$$ Next, the proof follows by similar reasonings as in Lemma \[lem: dY\_m=00003D00003D1\].
\[lem: dY\_m=00003D00003D2\_alpha=00003D00003D2\]Assume that **H**$(2)$ holds and $\varphi\in C_{P}^{3}$. For $\alpha\in\mathcal{R}\left(\mathcal{M}_{1}(S_{0})\right)$ with $\left\vert \alpha\right\vert _{0}=0$ and $i\neq0$ we have that $$\begin{aligned}
\mathbb{\tilde{E}}\left[\mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}\int_{0}^{t}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dY_{s}^{i}|\mathcal{Y}_{t}\right]^{2}\right] & \leq C\delta^{4}.\end{aligned}$$
We divide the proof into several steps.
**Step 1**. Using Lemma \[lem: Convenient\], we can write $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}\int_{0}^{t}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dY_{s}^{i}|\mathcal{Y}_{t}\right]\\
& =\sum_{r_{1}=1}^{d_{V}}\sum_{j=0}^{n-1}\mathbb{\tilde{E}}\left[\left(\int_{t_{j}}^{t_{j+1}}\mathbb{\tilde{E}}\left[J_{s}^{r_{1}}|\mathcal{H}_{0}^{t}\right]dV_{s}^{r_{1}}\right)\right.\\
& \quad\times\left.\left(\int_{t_{j}}^{t_{j+1}}\left(Y_{t_{j+1}}^{i}-Y_{t_{j}}^{i}\right)I_{\alpha-}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dV_{s}^{\alpha_{|\alpha|}}\right)|\mathcal{Y}_{t}\right]\\
& \quad+\sum_{r_{1},r_{2}=1}^{d_{V}}\sum_{j=0}^{n-1}\mathbb{\tilde{E}}\left[\left(\int_{t_{j}}^{t_{j+1}}\int_{0}^{s_{2}}J_{s_{1},s_{2}}^{r_{1},r_{2}}dV_{s_{1}}^{r_{1}}dV_{s_{2}}^{r_{2}}\right)\right.\\
& \quad\times\left.\left(\int_{t_{j}}^{t_{j+1}}\left(Y_{t_{j+1}}^{i}-Y_{t_{j}}^{i}\right)I_{\alpha-}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dV_{s}^{\alpha_{|\alpha|}}\right)|\mathcal{Y}_{t}\right]\\
& \triangleq\sum_{r_{1}=1}^{d_{V}}\sum_{j=0}^{n-1}A\left(r_{1},j\right)+\sum_{r_{1,}r_{2}=1}^{d_{V}}\sum_{j=0}^{n-1}A\left(r_{1},r_{2},j\right).\end{aligned}$$ Therefore, by Lemma \[lem: CondExpect\] $\left(\ref{lem: CE2}\right)$, we have that $$\begin{aligned}
& A\left(r_{1},j\right)\\
& =\mathbb{\tilde{E}}\left[\left(\int_{t_{j}}^{t_{j+1}}\mathbb{\tilde{E}}\left[J_{s}^{r_{1}}|\mathcal{H}_{0}^{t}\right]dV_{s}^{r_{1}}\right)\left(\int_{t_{j}}^{t_{j+1}}\left(Y_{t_{j+1}}^{i}-Y_{s}^{i}\right)\left(\int_{t_{j}}^{s}L^{\alpha}h_{i}\left(X_{u}\right)dV_{u}^{\alpha_{1}}\right)dV_{s}^{\alpha_{2}}\right)|\mathcal{Y}_{t}\right]\\
& =\mathbf{1}_{\left\{ \alpha_{2}=r_{1}\right\} }\mathbb{\tilde{E}}\left[\left(\int_{t_{j}}^{t_{j+1}}\mathbb{\tilde{E}}\left[J_{s}^{r_{1}}|\mathcal{H}_{0}^{t}\right]\left(Y_{t_{j+1}}^{i}-Y_{s}^{i}\right)\left(\int_{t_{j}}^{s}L^{\alpha}h_{i}\left(X_{u}\right)dV_{u}^{\alpha_{1}}\right)ds\right)|\mathcal{Y}_{t}\right]\\
& =\mathbf{1}_{\left\{ \alpha_{2}=r_{1}\right\} }\mathbb{\tilde{E}}\left[\left(\int_{t_{j}}^{t_{j+1}}\mathbb{\tilde{E}}\left[J_{s}^{r_{1}}|\mathcal{H}_{0}^{t}\right]\left(Y_{t_{j+1}}^{i}-Y_{s}^{i}\right)\mathbb{\tilde{E}}\left[\left(\int_{t_{j}}^{s}L^{\alpha}h_{i}\left(X_{u}\right)dV_{u}^{\alpha_{1}}\right)|\mathcal{H}_{t_{j}}^{t}\right]ds\right)|\mathcal{Y}_{t}\right]\\
& =0.\end{aligned}$$ and, by Lemma \[lem: CondExpect\] $\left(\ref{lem: CE3}\right)$ and Lemma \[lem: CondExpect\] $\left(\ref{lem: CE2}\right)$, we obtain $$\begin{aligned}
& A\left(r_{1},r_{2},j\right)\\
& =\mathbb{\tilde{E}}\left[\left(\int_{t_{j}}^{t_{j+1}}\int_{0}^{s_{2}}J_{s_{1},s_{2}}^{r_{1},r_{2}}dV_{s_{1}}^{r_{1}}dV_{s_{2}}^{r_{2}}\right)\left(\int_{t_{j}}^{t_{j+1}}\left(Y_{t_{j+1}}^{i}-Y_{s}^{i}\right)\left(\int_{t_{j}}^{s}L^{\alpha}h_{i}\left(X_{u}\right)dV_{u}^{\alpha_{1}}\right)dV_{s}^{\alpha_{2}}\right)|\mathcal{Y}_{t}\right]\\
& =\mathbf{1}_{\left\{ \alpha_{2}=r_{2}\right\} }\mathbb{\tilde{E}}\left[\left(\int_{t_{j}}^{t_{j+1}}\left(Y_{t_{j+1}}^{i}-Y_{s}^{i}\right)\left(\int_{0}^{s}J_{s_{1},s}^{r_{1},r_{2}}dV_{s_{1}}^{r_{1}}\right)\left(\int_{t_{j}}^{s}L^{\alpha}h_{i}\left(X_{u}\right)dV_{u}^{\alpha_{1}}\right)ds\right)|\mathcal{Y}_{t}\right]\\
& =\mathbf{1}_{\left\{ \alpha_{2}=r_{2},\alpha_{1}=r_{1}\right\} }\mathbb{\tilde{E}}\left[\left(\int_{t_{j}}^{t_{j+1}}\left(Y_{t_{j+1}}^{i}-Y_{s}^{i}\right)\left(\int_{t_{j}}^{s}J_{u,s}^{r_{1},r_{2}}L^{\alpha}h_{i}\left(X_{u}\right)du\right)ds\right)|\mathcal{Y}_{t}\right].\end{aligned}$$ Hence, we can write $$\begin{aligned}
& \mathbb{\tilde{E}}\left[\varphi(X_{t})e^{\xi_{t}}\int_{0}^{t}I_{\alpha}(L^{\alpha}h_{i}(X_{\cdot}))_{\tau(s),s}dY_{s}^{i}|\mathcal{Y}_{t}\right]\\
& =\sum_{r_{1,}r_{2}=1}^{d_{V}}\mathbf{1}_{\left\{ \alpha_{2}=r_{2},\alpha_{1}=r_{1}\right\} }\mathbb{\tilde{E}}\left[\left(\int_{0}^{t}\left(Y_{\eta(s)}^{i}-Y_{s}^{i}\right)\left(\int_{\tau(s)}^{s}J_{u,s}^{r_{1},r_{2}}L^{\alpha}h_{i}\left(X_{u}\right)du\right)ds\right)|\mathcal{Y}_{t}\right]\\
& =\sum_{r_{1,}r_{2}=1}^{d_{V}}\mathbf{1}_{\left\{ \alpha_{2}=r_{2},\alpha_{1}=r_{1}\right\} }\left(B_{1}\left(r_{1},r_{2}\right)+B_{2}\left(r_{1},r_{2}\right)+B_{3}\left(r_{1},r_{2}\right)+B_{4}\left(r_{1},r_{2}\right)\right),\end{aligned}$$ where $$\begin{aligned}
B_{1}\left(r_{1},r_{2}\right) & =\mathbb{\tilde{E}}\left[\left(\int_{0}^{t}\left(Y_{\eta(s)}^{i}-Y_{s}^{i}\right)\left(\int_{\tau(s)}^{s}\left(J_{u,s}^{r_{1},r_{2}}-J_{s,s}^{r_{1},r_{2}}\right)L^{\alpha}h_{i}\left(X_{u}\right)du\right)ds\right)|\mathcal{Y}_{t}\right],\\
B_{2}\left(r_{1},r_{2}\right) & =\mathbb{\tilde{E}}\left[\left(\int_{0}^{t}\left(Y_{\eta(s)}^{i}-Y_{s}^{i}\right)J_{s,s}^{r_{1},r_{2}}\left(\int_{\tau(s)}^{s}\left(L^{\alpha}h_{i}\left(X_{u}\right)-L^{\alpha}h_{i}\left(X_{\tau(s)}\right)\right)du\right)ds\right)|\mathcal{Y}_{t}\right]\\
B_{3}\left(r_{1},r_{2}\right) & =\mathbb{\tilde{E}}\left[\left(\int_{0}^{t}\left(Y_{\eta(s)}^{i}-Y_{s}^{i}\right)\left(J_{s,s}^{r_{1},r_{2}}-J_{\eta\left(s\right),\eta\left(s\right)}^{r_{1},r_{2}}\right)L^{\alpha}h_{i}\left(X_{\tau(s)}\right)\left(\int_{\tau(s)}^{s}du\right)ds\right)|\mathcal{Y}_{t}\right]\\
B_{4}\left(r_{1},r_{2}\right) & =\mathbb{\tilde{E}}\left[\left(\int_{0}^{t}\left(Y_{\eta(s)}^{i}-Y_{s}^{i}\right)J_{\eta\left(s\right),\eta\left(s\right)}^{r_{1},r_{2}}L^{\alpha}h_{i}\left(X_{\tau(s)}\right)\left(\int_{\tau(s)}^{s}du\right)ds\right)|\mathcal{Y}_{t}\right]\end{aligned}$$
**Step 2**. That the terms $B_{1}\left(r_{1},r_{2}\right),B_{2}\left(r_{1},r_{2}\right)$ and $B_{3}\left(r_{1},r_{2}\right)$ have the right order is deduced analogously to the **Steps 2** and **3** in Lemma \[lem: dY\_m=00003D00003D1\].
**Step 3**. Finally, we prove the result for $B_{4}\left(r_{1},r_{2}\right).$ We can write $$\begin{aligned}
B_{4}\left(r_{1},r_{2}\right) & =\sum_{j=0}^{n-1}\mathbb{\tilde{E}}\left[\int_{t_{j}}^{t_{j+1}}J_{t_{j+1},t_{j+1}}^{r_{1},r_{2}}L^{\left(r_{1},r_{2}\right)}h_{i}\left(X_{t_{j}}\right)\left(\int_{t_{j}}^{s}du\right)\left(Y_{t_{j+1}}^{i}-Y_{s}^{i}\right)ds|\mathcal{Y}_{t}\right]\\
& =\sum_{j=0}^{n-1}\mathbb{\tilde{E}}\left[J_{t_{j+1},t_{j+1}}^{r_{1},r_{2}}L^{\left(r_{1},r_{2}\right)}h_{i}\left(X_{t_{j}}\right)\int_{t_{j}}^{t_{j+1}}\left(\int_{t_{j}}^{s}du\right)\left(Y_{t_{j+1}}^{i}-Y_{s}^{i}\right)ds|\mathcal{Y}_{t}\right]\\
& =\sum_{j=0}^{n-1}\mathbb{\tilde{E}}\left[J_{t_{j+1},t_{j+1}}^{r_{1},r_{2}}L^{\left(r_{1},r_{2}\right)}h_{i}(X_{t_{j}})\int_{t_{j}}^{t_{j+1}}\frac{\left(s-t_{j}\right)^{2}}{2}dY_{s}^{i}|\mathcal{Y}_{t}\right]\\
& \triangleq\sum_{j=0}^{n-1}\mathbb{\tilde{E}}\left[J_{t_{j+1},t_{j+1}}^{r_{1},r_{2}}L^{\left(r_{1},r_{2}\right)}h_{i}(X_{t_{j}})\int_{t_{j}}^{t_{j+1}}\beta_{s}^{j}dY_{s}^{i}|\mathcal{Y}_{t}\right],\end{aligned}$$ where $\left|\beta_{s}^{j}\right|\leq\delta^{2}.$ Moreover, $$J_{t_{j+1},t_{j+1}}^{r_{1},r_{2}}=\mathbb{\tilde{E}}\left[D_{t_{j+1},t_{j+1}}^{r_{1},r_{2}}\left\{ \varphi\left(X_{t}\right)e^{\xi_{t}}\right\} |\mathcal{H}_{t_{j}}^{t}\right],$$ by the Clark-Ocone formula. Using the , we get $$\begin{aligned}
D_{t_{j+1},t_{j+1}}^{r_{1},r_{2}}\left\{ \varphi\left(X_{t}\right)e^{\xi_{t}}\right\} & =D_{t_{j+1}}^{r_{1}}\left(D_{t_{j+1}}^{r_{2}}\left\{ \varphi\left(X_{t}\right)e^{\xi_{t}}\right\} \right)\\
& =D_{t_{j+1}}^{r_{1}}\left(e^{\xi_{t}}D_{t_{j+1}}^{r_{2}}\varphi\left(X_{t}\right)+\varphi\left(X_{t}\right)D_{t_{j+1}}^{r_{2}}e^{\xi_{t}}\right)\\
& =D_{t_{j+1}}^{r_{1}}e^{\xi_{t}}D_{t_{j+1}}^{r_{2}}\varphi\left(X_{t}\right)+e^{\xi_{t}}D_{t_{j+1},t_{j+1}}^{r_{1},r_{2}}\varphi\left(X_{t}\right)\\
& +D_{t_{j+1}}^{r_{1}}\varphi\left(X_{t}\right)D_{t_{j+1}}^{r_{2}}e^{\xi_{t}}+\varphi\left(X_{t}\right)D_{t_{j+1},t_{j+1}}^{r_{1},r_{2}}e^{\xi_{t}}\end{aligned}$$ Reasoning as in Step 4 of Lemma \[lem: dY\_m=00003D00003D1\], we get that $$\begin{aligned}
\mathbb{\tilde{E}}\left[\left|B_{4}\left(r_{1},r_{2}\right)\right|^{2}\right] & \leq\mathbb{\tilde{E}}\left[\left|\sum_{j=0}^{n-1}\mathbb{\tilde{E}}\left[J_{t_{j+1},t_{j+1}}^{r_{1},r_{2}}L^{\left(r_{1},r_{2}\right)}h_{i}(X_{t_{j}})\int_{t_{j}}^{t_{j+1}}\beta_{s}^{j}dY_{s}^{i}|\mathcal{Y}_{t}\right]\right|^{2}\right]\\
& \leq C\mathbb{\tilde{E}}\left[\left|\sum_{j=0}^{n-1}D_{t_{j+1}}^{r_{1}}e^{\xi_{t}}D_{t_{j+1}}^{r_{2}}\varphi\left(X_{t}\right)L^{\left(r_{1},r_{2}\right)}h_{i}(X_{t_{j}})\int_{t_{j}}^{t_{j+1}}\beta_{s}^{j}dY_{s}^{i}\right|^{2}\right]\\
& \quad+C\mathbb{\tilde{E}}\left[\left|\sum_{j=0}^{n-1}e^{\xi_{t}}D_{t_{j+1},t_{j+1}}^{r_{1},r_{2}}\varphi\left(X_{t}\right)L^{\left(r_{1},r_{2}\right)}h_{i}(X_{t_{j}})\int_{t_{j}}^{t_{j+1}}\beta_{s}^{j}dY_{s}^{i}\right|^{2}\right]\\
& \quad+C\mathbb{\tilde{E}}\left[\left|\sum_{j=0}^{n-1}D_{t_{j+1}}^{r_{1}}\varphi\left(X_{t}\right)D_{t_{j+1}}^{r_{2}}e^{\xi_{t}}L^{\left(r_{1},r_{2}\right)}h_{i}(X_{t_{j}})\int_{t_{j}}^{t_{j+1}}\beta_{s}^{j}dY_{s}^{i}\right|^{2}\right]\\
& \quad+C\mathbb{\tilde{E}}\left[\left|\sum_{j=0}^{n-1}\varphi\left(X_{t}\right)D_{t_{j+1},t_{j+1}}^{r_{1},r_{2}}e^{\xi_{t}}L^{\left(r_{1},r_{2}\right)}h_{i}(X_{t_{j}})\int_{t_{j}}^{t_{j+1}}\beta_{s}^{j}dY_{s}^{i}\right|^{2}\right]\\
& \triangleq C\left\{ F_{1}\left(r_{1},r_{2}\right)+F_{2}\left(r_{1},r_{2}\right)+F_{3}\left(r_{1},r_{2}\right)+F_{4}\left(r_{1},r_{2}\right)\right\} .\end{aligned}$$ The term $F_{2}\left(r_{1},r_{2}\right)$ is analogous to the term $A$$_{1}\left(r\right)$ in Lemma \[lem: dY\_m=00003D00003D1\] and the terms $F_{2}\left(r_{1},r_{2}\right)$ and $F_{3}\left(r_{1},r_{2}\right)$ are analogous to the term $A_{2}\left(r\right)$ in Lemma \[lem: dY\_m=00003D00003D1\]. For the term $F_{4}\left(r_{1},r_{2}\right)$ we have that $$\begin{aligned}
D_{t_{j+1},t_{j+1}}^{r_{1},r_{2}}e^{\xi_{t}} & =D_{t_{j+1}}^{r_{1}}\left\{ e^{\xi_{t}}\left\{ \sum_{k=1}^{d_{Y}}\int_{t_{j+1}}^{t}D_{t_{j+1}}^{r_{2}}h_{k}(X_{s})dY_{s}^{k}-\frac{1}{2}\sum_{k=1}^{d_{Y}}\int_{t_{j+1}}^{t}D_{t_{j+1}}^{r_{2}}\left[h_{k}(X_{s})^{2}\right]ds\right\} \right\} \\
& =e^{\xi_{t}}\left\{ \sum_{k=1}^{d_{Y}}\int_{t_{j+1}}^{t}D_{t_{j+1}}^{r_{1}}h_{k}(X_{s})dY_{s}^{k}-\frac{1}{2}\sum_{k=1}^{d_{Y}}\int_{t_{j+1}}^{t}D_{t_{j+1}}^{r_{1}}\left[h_{k}(X_{s})^{2}\right]ds\right\} \\
& \qquad\times\left\{ \sum_{k=1}^{d_{Y}}\int_{t_{j+1}}^{t}D_{t_{j+1}}^{r_{2}}h_{k}(X_{s})dY_{s}^{k}-\frac{1}{2}\sum_{k=1}^{d_{Y}}\int_{t_{j+1}}^{t}D_{t_{j+1}}^{r_{2}}\left[h_{k}(X_{s})^{2}\right]ds\right\} \\
& +e^{\xi_{t}}\left\{ \sum_{k=1}^{d_{Y}}\int_{t_{j+1}}^{t}D_{t_{j+1},t_{j+1}}^{r_{1},r_{2}}h_{k}(X_{s})dY_{s}^{k}-\frac{1}{2}\sum_{k=1}^{d_{Y}}\int_{t_{j+1}}^{t}D_{t_{j+1},t_{j+1}}^{r_{1},r_{2}}\left[h_{k}(X_{s})^{2}\right]ds\right\} \end{aligned}$$ All the terms obtained in the previous expression can be dealt analogously to the terms in Lemma \[lem: dY\_m=00003D00003D1\] except the terms $$G\left(j,r_{1},r_{2}\right)\triangleq e^{\xi_{t}}\sum_{k_{1},k_{2}=1}^{d_{Y}}\left(\int_{t_{j+1}}^{t}D_{t_{j+1}}^{r_{1}}h_{k_{1}}(X_{s})dY_{s}^{k_{1}}\right)\left(\int_{t_{j+1}}^{t}D_{t_{j+1}}^{r_{2}}h_{k_{2}}(X_{s})dY_{s}^{k_{2}}\right).$$ Let $$H\triangleq\mathbb{\tilde{E}}\left[\left|\sum_{j=0}^{n-1}\varphi\left(X_{t}\right)G\left(j,r_{1},r_{2}\right)L^{\left(r_{1},r_{2}\right)}h_{i}(X_{t_{j}})\int_{t_{j}}^{t_{j+1}}\beta_{s}^{j}dY_{s}^{i}\right|^{2}\right].$$ Defining $$\varLambda(j_{1},j_{2})\triangleq\varphi\left(X_{t}\right)^{2}L^{\left(r_{1},r_{2}\right)}h_{i}(X_{t_{j_{1}}})L^{\left(r_{1},r_{2}\right)}h_{i}(X_{t_{j_{2}}})\exp\left(\sum_{i=1}^{d_{Y}}\int_{0}^{t}h_{i}^{2}\left(X_{u}\right)du\right),$$ we can write $$\begin{aligned}
H & =\sum_{k_{1},...,k_{4}=1}^{d_{Y}}\sum_{j_{1},j_{2}=0}^{n-1}\mathbb{\tilde{E}}\left[\varLambda(j_{1},j_{2})M_{0}^{t}\left(2h\right)\right.\\
& \qquad\times\left(\int_{t_{j_{1}+1}}^{t}D_{t_{j_{1}+1}}^{r_{1}}h_{k_{1}}(X_{s})dY_{s}^{k_{1}}\right)\left(\int_{t_{j_{1}+1}}^{t}D_{t_{j_{1}+1}}^{r_{2}}h_{k_{2}}(X_{s})dY_{s}^{k_{2}}\right)\\
& \qquad\times\left(\int_{t_{j_{2}+1}}^{t}D_{t_{j_{2}+1}}^{r_{1}}h_{k_{3}}(X_{s})dY_{s}^{k_{3}}\right)\left(\int_{t_{j_{2}+1}}^{t}D_{t_{j_{2}+1}}^{r_{2}}h_{k_{4}}(X_{s})dY_{s}^{k_{4}}\right)\\
& \qquad\times\left.\left(\int_{t_{j_{1}}}^{t_{j_{1}+1}}\beta_{s}^{j_{1}}dY_{s}^{i}\right)\left(\int_{t_{j_{2}}}^{t_{j_{2}+1}}\beta_{s}^{j_{2}}dY_{s}^{i}\right)\right],\end{aligned}$$ and the result follows from Lemma \[lem: Main Backward\] and Remark \[rem: Backward Ito integral\].
Following Remarks \[rem: HighRegularityKernel\] and \[rem: BackwardEstimate\], the results in Lemmas \[lem: dY\_m=00003D00003D2\_alpha=00003D00003D1\] and \[lem: dY\_m=00003D00003D2\_alpha=00003D00003D2\] can be extended analogously to $m>2$ and $\alpha\in\mathcal{R}\left(\mathcal{M}_{m-1}(S_{0})\right)$ with $\left\vert \alpha\right\vert _{0}\in\{0,...,m-1\}$ without any additional difficulties.
[10]{} Applebaum, D. (2009). *Lévy Processes and Stochastic Calculus*. *Second Edition*. Cambridge Studies in Advanced Mathematics, vol. **116**. Cambridge University Press, Cambridge.
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Crisan, D. (2011). Discretizing the Continuous Time Filtering Problem. Order of Convergence. In *The Oxford Handbook of Nonlinear Filtering*. Oxford University Press, Oxford.
Crisan, D. and Rozovsky, B. (2011). *The Oxford handbook of nonlinear filtering*. Oxford University Press, Oxford.
Crisan, D. and Ortiz-Latorre, S. (2013). A Kusuoka-Lyons-Victoir particle filter. *Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.* **469**, no. 2156.
Del Moral, P. (2004). *Feynman-Kac formulae. Genealogical and interacting particle systems with applications*. Probab. Appl. . Springer-Verlag, New York.
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Karatzas, I. and Shreve, S.E. (1991). *Brownian Motion and Stochastic Calculus*. Graduate Texts in Mathematics, vol. **113**, Second Edition. Springer-Verlag, New York.
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Kusuoka, S. and Stroock, D. (1984). The partial Malliavin Calculus and its application to non-linear filtering. *Stochastics*, **12**, 83-142.
Nualart, D. (2006). *The Malliavin Calculus and Related Topics*. Springer Verlag, New York.
Nualart, D. and Zakai, M. (1989). The partial Malliavin calculus. Séminaire de Probabilités XXIII. Lecture Notes in Math. **1372**, 362-381, Springer Verlag, Berlin.
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[^1]: The work of D. C. was partially supported by the EPSRC Grant EP/H0005500/1.
[^2]: Department of Mathematics, Imperial College London, Huxley’s Building,180 Queen’s Gate, London SW7 2AZ, United Kingdom, E-mail: dcrisan@imperial.ac.uk
[^3]: The work of S. O-L. was supported by the BP-DGR 2009 grant and the project *Energy Markets: Modeling, Optimization and Simulation (EMMOS)*, funded by the Norwegian Research Council under grant Evita/205328.
[^4]: Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N-0316 Oslo, Norway, E-mail: salvadoo@math.uio.no
[^5]: Also known as *particle filters* or *sequential Monte Carlo methods*.
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abstract: 'In framework of eigen-functional bosonization method, we introduce an imaginary phase field to uniquely represent electron correlation, and demonstrate that the Landau Fermi liquid theory and the Tomonaga-Luttinger liquid theory can be unified. It is very clear in this framework that the Tomonaga-Luttinger liquid behavior of one-dimensional interacting electron gases originates from their Fermi structure, and the non-Landau-Fermi liquid behavior of 2D interacting electron gases is induced by the long-range electron interaction, while 3D interacting electron gases generally show the Landau Fermi liquid behavior.'
address: 'Center for Advanced Study, Tsinghua University, Beijing 100084, People’s Republic of China'
author:
- 'Yu-Liang Liu'
title: Unified theory of strongly correlated electron systems
---
Since the discovery of high Tc cuprate superconductors[@1], the strongly correlated electron systems has been extensively studied theoretically[@2; @3; @4; @4a; @4b; @4c; @5; @6; @7]. Now a common consensus is reached that the low energy physics properties of cuprate superconductors, such as anomalous normal state behavior and high superconducting transition temperature, are determined by the strong electron correlation in their copper-oxide plane(s). However, up to now there is not a microscopic or phenomenal theory to successfully explain the normal and superconducting physical properties of the cuprate superconductors, because one cannot exactly and effectively treat the strong electron correlation of the systems.
It is well-known that usual metals can be described by the Landau Fermi liquid theory[@8; @9], in which there is weak correlation among electrons, and near the Fermi surface there exist well-defined quasi-particles (holes), thus the fundamental assumption of the Landau Fermi liquid theory is satisfied, i.e., the states of an interacting electron gas can be put into a one-to-one correspondence via adiabatic continuation with those of the free electron gas. However, for strong electron correlation, this fundamental assumption of the Landau Fermi liquid theory may fail, and one does not have well-defined quasi-particles (holes) near the Fermi surface. A one-dimensional interacting electron gas is a good example, in which there is strong electron correlation even though for small electron interaction, thus one does not have well-defined quasi-particles (holes) near the Fermi levels $\pm k_{F}$ (its Fermi surface is composed of two points, $\pm k_{F}$, defined by the Fermi momentum $k_{F}$). It is described by the Tomonaga-Luttinger liquid theory[@10; @11; @12; @13; @14], where in low energy regime electron Green’s function and other correlation functions present power-law behavior, and the correlation exponents are not universal, and depend upon the electron interaction strength. Therefore, the one-dimensional interacting electron gas is a strongly correlated electron system even for weak electron interaction.
Generally, the low energy behavior of the one-dimensional (1D) interacting electron gas is qualitatively different from that of three-dimensional (3D) interacting electron gas, the former is represented by the Tomonaga-Luttinger liquid theory, while the latter is represented by the Landau Fermi liquid theory (see below). This difference derives from their different Fermi surface strctures. For 1D electron gas, its Fermi surface is composed of two points $\pm k_{F}$, defined by the Fermi momentum $k_{F}$, and its Hilbert space is drastically suppressed. There are only two kinds of elementary excitation modes, one is the excitation modes near these two Fermi levels $\pm k_{F}$, respectively, where the energy spectrum is approximately linear; and another one is the excitation modes with large momentum ($2nk_{F}$, $n=1,2,...$) transfer between these Fermi levels $\pm k_{F}$, which usually make the system become an insulator ( for a 1D electron gas this kind of excitation is absent). This drastically suppressed Hilbert space will induce the strong electron correlation even for weak electron interaction, thus it is true that for a 1D interacting electron gas the strong electron correlation originates from its special Fermi surface structure, and the Tomonaga-Luttinger liquid theory is an universal theory of 1D electron gases. In contrast with the 1D electron gas, the Fermi surface of the 3D electron gas is a sphere with a radius $k_{F}$, and its Hilbert space is enlarged comparing with that of the 1D electron gas. The low energy elementary excitation modes are quasi-particles and quasi-holes. It is well-known that even for long-range Coulomb interaction, the 3D electron gas still can be described by the Landau Fermi liquid theory, where strong electron interaction does not mean that there is the strong electron correlation, and it only produces weak and short-range electron correlation. Thus the Landau Fermi liquid theory is an universal theory of 3D electron gases.
For 2D electron gas, the situation is different. Its Fermi surface is a circle with a radius $k_{F}$, and for long-range Coulomb interaction, it shows non-Landau-Fermi liquid behavior in the low energy regime[@15; @16]. We shall demonstrate that this anomalous behavior of the 2D interacting electron gas derives from the two aspects, one is its Fermi structure, and another one is the long-range electron interaction. These both effects induce the strong electron correlation, thus the non-Landau-Fermi liquid behavior of the 2D strongly correlated electron gas cannot be completely represented by the Tomonaga-Luttinger liquid theory. It shows not only some characters of the Landau Fermi liquid, but also some characters of the Tomonaga-Luttinger liquid.
It is very desirable to find an unified theory to represent not only 1D interacting electron gases, but also 2D and 3D interacting electron gases. In the framework of the eigen-functional bosonization method[@17; @18], we try to give such the unified theory of strongly correlated electron gases, which not only reduces to the Tomonaga-Luttinger liquid theory for the 1D interacting electron gas and to the Landau Fermi liquid thoery for the 3D interacting electron gas, but also can represent the low energy behavior of the 2D interacting electron gas. We clearly demonstrate that an imaginary phase field which naturally appears in the eigen-functional bosonization, is a key parameter field hiden in the strongly correlated electron systems and represents the electron correlation, and the eigen-functionals can be used to define the states of the interacting electrons. By calculation the overlap of these eigen-functionals with the eigen-functions of the free electrons, we can judge whether or not there exist well-defined quasi-particles (holes) near the Fermi surface. The imaginary phase field is a very important quantity in our representation, and it determines the low energy behavior of the system. Moreover, as $d\geq 2$ it does not contribute to the action of the system.
In general, we consider the Hamiltonian (omitting spin label $\sigma$ of the electron operators $c_{k}$) $$H=\sum_{k}\epsilon(k)c^{\dagger}_{k}c_{k}+\frac{1}{2L^{d}}\sum_{q}
v(q)\rho(q)\rho(-q)
\label{1}$$ where $\epsilon(k)=k^{2}/(2m)-\mu$, $\mu$ is the chemical potential, $d$ is the dimension number, $\rho(q)=(1/L^{d})\sum_{k}c^{\dagger}_{k}
c_{k+q}$ is the electron density operator, and $v(q)$ is the Coulomb potential. The last term represents usual electron’s Coulomb interaction (four-fermion interaction). Usually, one can introduce the Hubbard-Stratonovich (HS) field $\phi(x,t)$ to descouple this four-fermion interaction term. With the standard path integral method[@19], we have the action of the system, $$\begin{aligned}
S &=& \displaystyle{ \int dt\int d^{d}x\left\{\Psi^{\dagger}(x,t)\left[
i\frac{\partial}{\partial t}+\mu+\frac{\nabla^{2}}{2m}-\phi(x,t)\right]
\Psi(x,t)\right\}} \nonumber \\
&+& \displaystyle{\frac{1}{2TL^{d}}\sum_{q,\Omega}\frac{1}{v(q)}
\phi(-q,-\Omega)\phi(q,\Omega)}
\label{2}\end{aligned}$$ where $\Psi(x,t)$ is the electron field. The system now reduces into that the electrons move in the HS field $\phi(x,t)$, and we have the eigen-functional equation, $$\left[i\frac{\partial}{\partial t}+\mu+\frac{\nabla^{2}}{2m}-\phi(x,t)
\right]\Psi_{k\omega}(x,t,[\phi])=E_{k\omega}[\phi]\Psi_{k\omega}(x,t,[\phi])
\label{3}$$ where the eigen-value $E_{k\omega}[\phi]=\omega-\epsilon(k)-\Sigma_{k}[\phi]$, and the self-energy $\Sigma_{k}[\phi]=\int^{1}_{0}d\xi\int dt d^{d}x \phi(x,t)
\Psi^{\dagger}_{k\omega}(x,t,[\xi\phi])\Psi_{k\omega}(x,t,[\xi\phi])$ is a regular function, and independent of $\omega$. The eigen-functionals $\Psi_{k\omega}(x,t,[\phi])$ can be used to define the elementary low energy excitation modes, i.e., quasi-particles (holes) and/or collective excitation modes, and have the formal solutions, $$\Psi_{k\omega}(x,t,[\phi])=A_{k}\left(\frac{1}{TL^{d}}\right)^{1/2}
e^{Q_{k}(x,t)}e^{i{\bf k}\cdot{\bf x}-i(\omega-\Sigma_{k}[\phi])t}
\label{4}$$ where $|A_{k}|\sim 1$ is the normalization constant, and $T\rightarrow\infty$ is the time length of the system. These eigen-functionals are composed of two parts, one presents the free electrons, and another one presents the correlation of the electrons produced by the electron interaction. Thus we can formally write $$\begin{aligned}
\Psi_{k\omega}(x,t,[\phi])=\psi_{k\omega}(x,t)e^{Q_{k}(x,t)}
\nonumber\end{aligned}$$ The phase fields $Q_{k}(x,t)$ determine the low energy behavior of the system, and satisfy the usual Eikonal equation with the condition $Q_{k}(x,t)=0$ as $\phi(x,t)=0$, $$\left[i\frac{\partial}{\partial t}+i\frac{{\bf k}\cdot{\bf \nabla}}{m}+
\frac{\nabla^{2}}{2m}\right]Q_{k}(x,t)-\phi(x,t)=-\frac{1}{2m}
\left({\bf \nabla}Q_{k}(x,t)\right)^{2}
\label{5}$$ which can be exactly solved by a series expansion of the HS field $\phi(x,t)$ and/or by computer calculations[@20; @21].
With the solution of the eigen-functionals $\Psi_{k\omega}(x,t,[\phi])$ (\[4\]), the action of the system reads (omitting constant terms), $$\begin{aligned}
S &=& \displaystyle{
\frac{1}{2TL^{d}}\sum_{q,\Omega}\frac{\phi(-q,-\Omega)\phi(q,\Omega)}
{v(q)}} \nonumber \\
&-& \displaystyle{\frac{1}{2}\int dt d^{d}x \phi(x,t)\left[ F_{1}(x,t,\delta)
+F_{2}(x,t)\right]_{{\bf \delta}\rightarrow 0}}
\label{6}\end{aligned}$$ where $F_{1}(x,t,\delta)=-(1/(2\pi)^{d})\int d^{d}k\theta(-\epsilon(k))
\sin({\bf k}\cdot{\bf \delta}){\bf \delta}\cdot{\bf \nabla}Q^{I}_{k}(x,t)$, $F_{2}(x,t)=(2/(2\pi^{d}))\int d^{d}k\theta(-\epsilon(k))Q^{R}_{k}(x,t)$, and $\theta(x)$ is a step function, i.e., $\theta(x)=1$ for $x>0$ and $\theta(x)=0$ for $x<0$. We have written the phase field as $Q_{k}(x,t)=Q^{R}_{k}(x,t)+iQ^{I}_{k}(x,t)$, a real part $Q^{R}_{k}(x,t)$ and an imaginary part $Q^{I}_{k}(x,t)$. We shall demonstrate that the phase field $Q^{I}_{k}(x,t)$ completely determine the low energy behavior of the system. It is also noted that the action of the system is composed of two parts, one is from the real phase field $Q^{R}_{k}(x,t)$, and another one is from the imaginary phase field $Q^{I}_{k}(x,t)$. Due to the Fermi surface structure of the system, the momentum integral in $F_{1}(x,t,\delta)$ can be written as[@22], $\int d^{d}k=S_{d-1}\int d|k| |k|^{d-1}\int^{\pi}_{0}d\theta
(\sin\theta)^{d-2}$, where $S_{d}=2\pi^{d/2}/\Gamma(d/2)$. As $d\geq 2$, the integration of $\sin({\bf k}\cdot{\bf \delta})$ is regular, thus as ${\bf
\delta}\rightarrow 0$ the function $F_{1}(x,t,\delta)=0$, the imaginary phase filed $Q^{I}_{k}(x,t)$ has no contribution to the action of the system. Only at $d=1$, it has contribution to the action, where the real phase field $Q^{R}_{k}(x,t)$ is zero (see below). This property is independent of the electron interaction, it is completely determined by the Fermi surface structure of the system.
We now consider the 1D interacting electron gas, in which the Fermi surface is composed of two Fermi levels, $\pm k_{F}$, and the electron energy spectrum can be written as, $\epsilon(k)=\pm v_{F}k$, where $v_{F}$ is the electron Fermi velocity. The branch $\epsilon(k)=v_{F}k$ represents the right-moving electrons, and the branch $\epsilon(k)=-v_{F}k$ represents the left-moving electrons. In general, the electron interaction term reads, $(1/L)\sum_{q}V(q)\rho_{R}(q)\rho_{L}(-q)$, where $\rho_{R(L)}(q)$ are the right- and left-moving electron densities, respectively, and $V(q)\sim V$, a constant. To decouple this four-fermion interaction, we introduce two HS fields $\phi_{R}(x,t)$ and $\phi_{L}(x,t)$, and have two sets of eigen-functionals representing the right- and left-moving electrons, respectively, $$\begin{aligned}
\Psi_{Rk\omega}(x,t,[\phi]) &=& \displaystyle{ \left(\frac{1}{TL}\right)^{1/2}
e^{Q_{R}(x,t)}e^{ikx-i(\omega-\Sigma_{R}[\phi])t}} \nonumber \\
\Psi_{Lk\omega}(x,t,[\phi]) &=& \displaystyle{ \left(\frac{1}{TL}\right)^{1/2}
e^{Q_{L}(x,t)}e^{ikx-i(\omega-\Sigma_{L}[\phi])t}}
\label{7}\end{aligned}$$ where $\Sigma_{R(L)}[\phi]$ is a regular quantity, and independent of $k$ and $\omega$. The phase fields $Q_{R}(x,t)$ and $Q_{L}(x,t)$ are independent of $k$, and satisfy the simplified Eikonal equation[@17; @23], $$\begin{aligned}
\displaystyle{
\left(i\frac{\partial}{\partial t}+iv_{F}\frac{\partial}{\partial x}\right)
Q_{R}(x,t)-\phi_{R}(x,t)} &=& 0 \nonumber \\
\displaystyle{
\left(i\frac{\partial}{\partial t}-iv_{F}\frac{\partial}{\partial x}\right)
Q_{L}(x,t)-\phi_{L}(x,t)} &=& 0
\label{8}\end{aligned}$$ These linear differential equations can be easily solved, and the phase fields $Q_{R}(x,t)$ and $Q_{L}(x,t)$ are imaginary because the HS fields $\phi_{R(L)}(x,t)$ are real. It is worthy noted that the imaginary phase fields $Q_{R(L)}(x,t)$ not only determine the electron correlation, but also contribute to the action of the system. This is qualitatively different from that in 2D and 3D electron gases, where the imaginary part of the phase field $Q_{k}(x,t)$ does not contribute to the action due to their Fermi surface structures.
It is very simple to prove that the 1D interacting electron gas is a strongly correlated system even for very weak electron interaction $V\sim 0$. With the action (\[6\]), by the simple calculation we can obtain the relations, $$\begin{aligned}
<\psi_{R(L)k\omega}(x,t)\Psi^{\dagger}_{R(L)k\omega}(x',t,[\phi])>_{\phi}
&\sim & \displaystyle{\left(\frac{1}{L}\right)^{\alpha}e^{ik(x-x')}}
\label{9} \\
<e^{Q_{R(L)}(x,t)}e^{-Q_{R(L)}(x',t)}>_{\phi} &\sim &
\displaystyle{ \left(\frac{1}{|x-x'|}\right)^{2\alpha}, \;\;\; |x-x'|
\rightarrow\infty}
\nonumber\end{aligned}$$ where $\alpha\sim (1-V/(2\pi\hbar v_{F}))/2$ for $V\sim 0$, is the dimensionless coupling strength parameter, the eigen-functions $\psi_{R(L)k\omega}(x,t)$ present the right(left)-moving free electrons, and $<...>_{\phi}$ means taking functional average over the HS field $\phi_{R(L)}(x,t)$. The first equation presents the zero overlap between the eigen-functionals $\Psi_{R(L)k\omega}(x,t,[\phi])$ of the interaction electrons and the eigen-functions of the free electrons ($V=0$) as $L\rightarrow\infty$, thus the states of the interacting electron gas does not have one-to-one correspondence via adiabatic continuation with those of the free electron gas, and there are not well-defined quasi-particles (holes) near its two Fermi levels $\pm k_{F}$. The second equation presents the strong electron correlation, in which the electron correlation length is infinity even for weak electron interaction, this can be easily seen by re-writing the right side of the last equation in (\[9\]) as, $\exp\{-2\alpha\ln|x-x'|\}$. Due to this strong electron correlation, the low energy excitation modes of the 1D interacting electron gas are those collective excitation modes, such as the charge and spin density waves, and there are not well-defined quasi-particles (holes) near the two Fermi levels $\pm k_{F}$. Thus the Tomonaga-Luttinger liquid theory is a strongly correlated theory, and is universal for 1D interacting electron gases.
As $d\geq 2$, the real phase field $Q^{R}_{k}(x,t)$ is finite, and the imaginary phase field $Q^{I}_{k}(x,t)$ does not contribute to the action of the system. For simplicity, we can solve the Eikonal equation by neglecting the quadratic term $({\bf \nabla}Q_{k})^{2}$. This approximation is reasonable because for the long-range Coulomb interaction, only the states near the Fermi surface with momentum $q<q_{c}$ ($q_{c}\ll k_{F}$) are important in the low energy regime, and for the smooth function $Q_{k}(x,t)$ this quadratic term is proportional to $(q_{c}/k_{F})^{2}\sim 0$. With this approximation, we can obtain the simple effective action, $$S_{eff.}=\frac{1}{2TL^{d}}\sum_{q,\Omega}\left(\frac{1}{v(q)}-\chi(q,\Omega)
\right)|\phi(q,\Omega)|^{2}
\label{11}$$ where $\chi(q,\Omega)$ is usual Lindhard function[@19]. In fact, the above approximation is equivalent to usual random-phase approximation (RPA). However, in present framework, it gives more useful and important informations than usual RPA perturbation method, because we have not only the real phase field, but also the imaginary phase field. This imaginary phase field does not contribute to the action of the system, but it determines the electron correlation. For a 3D electron gas with long-range Coulomb interaction $v(q)=e^{2}/(4\pi q^{2})$, by simple calculation, we have the relations, $$\begin{aligned}
\displaystyle{
<\psi_{k\omega}(x,t)\Psi^{\dagger}_{k\omega}(x',t,[\phi])>_{\phi}} &\sim &
\displaystyle{Z_{k}e^{i{\bf k}\cdot({\bf x}-{\bf x'})}}
\nonumber \\
\displaystyle{
<e^{iQ^{I}_{k}(x,t)}e^{-iQ^{I}_{k}(x',t)}>_{\phi}} &\sim &
\displaystyle{ e^{z^{I}_{k}(x-x')}}
\label{12}\end{aligned}$$ where $Z_{k}$ is finite, and $z^{I}_{k}(x)$ is a smooth function. As $|{\bf x}|
\rightarrow\infty$, the function $z^{I}_{k}(x)$ goes to zero. The first equation presents that the eigen-functionals $\Psi_{k\omega}(x,t,[\phi])$ have strong overlap with the eigen-functions $\psi_{k\omega}(x,t)$, and the second equation presents that there is weak electron correlation even for long-range Coulomb interaction. Thus the fundamental assumption of the Landau Fermi liquid theory is satisfied, thus there are well-defined quasi-articles (holes) near the Fermi surface. The Landau Fermi liquid theory is universal for 3D interacting electron gases. Therefore, we can generally conclude that there does not exist non-Landau-Fermi liquid behavior in the 3D electron gases as $q_{c}\ll k_{F}$. For a 2D electron gas with the long-range Coulomb interaction $v(q)=e^{2}/
(4\pi q^{2})$, the situation is different from that in the 3D electron gas. By simple calculation, we can obtain the relations, $$\begin{aligned}
\displaystyle{ <\psi_{k\omega}(x,t)\Psi^{\dagger}_{k\omega}(x',t,[\phi])
>_{\phi}} &\sim & \displaystyle{ \left(\frac{1}{q_{c}L}\right)^{\beta}
e^{i{\bf k}\cdot({\bf x}-{\bf x'})}} \label{13} \\
\displaystyle{ <e^{iQ^{I}_{k}(x,t)}e^{-iQ^{I}_{k}(x',t)}>_{\phi}} &\sim &
\displaystyle{e^{-2\beta\ln(q_{c}|{\bf x}-{\bf x'}|)}, \;\;\;
|{\bf x}-{\bf x'}|\rightarrow\infty}
\nonumber\end{aligned}$$ where $\beta=e^{2}/(2(4\pi)^{2}\omega_{p})$ is the dimensionless coupling strength parameter, where $\omega_{p}$ is the plasma frequency. In fact, $q_{c}L\rightarrow\infty$, the first equation presents that the eigen-functionals $\Psi_{k\omega}(x,t,[\phi])$ has no (or infinitesimal) overlap with the eigen-functions $\psi_{k\omega}(x,t)$, and the second equation presents the strong electron correlation. In this case, the fundamental assumption of the Landau Fermi liquid theory fails, and the system shows the non-Landau-Fermi liquid behavior. Even though the asymptotic behavior of the single electron Green’s function is similar to that of the 1D interacting electron gas (see, (\[9\]) and (\[13\])), while they originate from different physics. The former is producted by the plasmon excitation of the system, while the latter is induced by the gapless charge (spin) density wave(s). Thus this non-Landau-Fermi liquid behavior is different from the usual Tomonaga-Luttinger liquid behavior, because the former is induced by the strong long-range electron interaction, while the latter is induced by the Fermi structure of the 1D electron gases. However, this difference can also be clearly seen from their effective actions of the density field. For 1D interacting electron gases, this action can be easily obtained by usual bosonization method[@13; @14], where the density field has well-defined propagator (density wave). For the 2D electron gas with long-range Coulomb interaction, the effective action (\[11\]) can be written as with the density field, $$S_{eff.}[\rho]=\frac{1}{2TL^{2}}\sum_{q,\Omega}\left(\frac{1}{\chi(q,
\Omega)}-v(q)\right)|\rho(q,\Omega)|^{2}
\label{14}$$ where the density field does not have well-defined propagator. It was shown in Ref.[@16] that as $e^{2}\gg 4\pi$ the system can be described as a Landau Fermi liquid formed by chargeless quasi-particles which has vanishing wavefunction overlap with the bare electrons in the system. In the short range $\xi\sim 1/q_{c}$, one can also define local quasi-particles (holes). For short range and/or weak electron interaction, the 2D interacting electron gases would show the Landau Fermi liquid behavior in the low energy region.
In summary, in the simple framework of the eigen-functional bosonization method, we have demonstrated that the imaginary phase field $Q^{I}_{k}(x,t)$ can completely represent the electron correlation, and the Landau Fermi liquid theory and the Tomonaga-Luttinger liquid theory can be unified. It is very clear in this framework that: a). the Tomonaga-Luttinger liquid behavior of 1D interacting electron gases originates from their Fermi structure which is composed of two Fermi levels $\pm k_{F}$; the phase field is imaginary, and contributes to the action of the systems. b). the non-Landau-Fermi liquid behavior of 2D interacting electron gas is induced by the long-range Coulomb interaction. The imaginary phase field $Q^{I}_{k}(x,t)$ does not contribute to the action of the system. However, for other anomalous gauge interaction, the system can also show the non-Landau-Fermi liquid behavior if the imaginary phase field $Q^{I}_{k}(x,t)$ satisfies the relations in (\[13\]). Generally, 2D interacting electron gases may show the Landau Fermi liquid behavior or non-Landau-Fermi liquid behavior in the low energy region, which completely depends upon the electron interaction. c). 3D interacting electron gases generally show the Landau Fermi liquid behavior. The electron interaction cannot alter this property as $q_{c}\ll k_{F}$, and it only modifies the weak electron correlation length and the overlap weight with the bare electron wavefunctions.
This method can also be used to treat a 2D electron gas with transverse gauge interaction, and gives the same results as that in Ref.[@27; @28]. However, this method can clearly show that the non-Fermi liquid behavior of this system originates from the imaginary phase field which is a function of transverse gauge fields.
We thank T. K. Ng for helpful discussions.
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ABSTRACT. Measuring a strength of dependence of random variables is an important problem in statistical practice. In this paper, we propose a new function valued measure of dependence of two random variables. It allows one to study and visualize explicit dependence structure, both in some theoretical models and empirically, without prior model structure. This provides a comprehensive view of association structure and makes possible much detailed inference than based on standard numeric measures of association. We present theoretical properties of the new measure of dependence and discuss in detail estimation and application of copula-based variant of it. Some artificial and real data examples illustrate the behavior and practical utility of the measure and its estimator.\
[KEY WORDS:]{} Copula; Dependence measures; Graphical method; Local correlation; Nonparametric association; Rank tests.
Teresa Ledwina, Institute of Mathematics, Polish Academy of Sciences, ul. Kopernika 18, 51-617 Wroc[ł]{}aw, Poland ( E-mail: [*ledwina@impan.pan.wroc.pl*]{}). Research was supported by the Grant N N201 608440 from the National Science Center, Poland. The author thanks Dr. Grzegorz Wy[ł]{}upek for his help in preparing figures and tables of this article and for useful remarks.\
[**1. Introduction**]{}\
Measuring a strength of dependence of two random variables has long history and wide applications. For brief overview see Jogdeo (1982) and Lancaster (1982). More detailed information can be found in Drouet Mari and Kotz (2001) as well as Balakrishnan and Lai (2009), for example. Most of measures of dependence, introduced in vast literature on the subject, are scalar ones. Such indices are called global measures of dependence. However, nowadays there is strong evidence that an attempt to represent complex dependence structure via a single number can be misleading. To overcome this drawback, several local indices have been proposed; see Section 6.3 of Drouet Mari and Kotz (2001) for details. Many of these indices were introduced in the context of regression models or survival analysis. Some local dependence functions have been introduced as well. In particular, Kowalczyk and Pleszczyńska (1977) invented function valued measure of monotonic dependence, based on some conditional expectations and adjusted to detect dependence weaker than the quadrant one. Next, Bjerve and Doksum (1993), Bairamov et al. (2003) and Li et al. (2014), among others, introduced local dependence measures based on regression concepts. See the last mentioned paper for more information. Holland and Wang (1987) defined the local dependence function, which mimics cross-product ratios for bivariate densities and treats the two variables in a symmetrical way. This function valued measure has several appealing properties and received considerable attention in the literature; cf. Jones and Koch (2003) for discussion and references. However, on the other hand, this measure has some limitations: it is not normalized, requires existence of densities of the bivariate distribution, and is intimately linked to strong form of dependence, the likelihood ratio dependence. Recently, Tj[ø]{}stheim and Hufthammer (2013) extensively discussed the role and history of local dependence measures in finance and econometrics. They also proposed the new local dependence measure, the local correlation function, based on approximating of bivariate density locally by a family of Gaussian densities. Similarly as the measure of Holland and Wang (1987), this measure treats both variables on the same basis. Though the idea behind the construction of this measure is intuitive one its computation and estimation is a difficult and complex problem. The asymptotic theory developed in Tj[ø]{}stheim and Hufthammer (2013) treats in detail the problems mentioned above, in a scope that covers some time series models. In Berentsen et al. (2013) this theory is applied to describe dependence structure of different copula models.
In this paper, we propose the new function valued measure of dependence of two random variables $X$ and $Y$ and present its properties. The measure has simple form and its definition exploits cumulative distribution functions (cdf’s), only. In particular, we do not assume existence of a density of the observed vector. The measure takes values in \[-1,1\] and treats both variables in a symmetrical way. The measure preserves the correlation order, or equivalently the concordance order, which is the quadrant order restricted to the class of distributions with fixed marginals. In particular, it is non-negative (non-positive) if and only if $X$ and $Y$ are positively (negatively) quadrant dependent. Quadrant dependence is relatively weak, intuitive and useful dependence notion, widely used in insurance and economics; see Dhaene et al. (2009) for an evidence and further references. The new measure obeys several properties formulated in the literature as useful or desirable. We introduce two variants of the measure. In Section 2 we consider general case, assuming that the vector $(X,Y)$ has joint cdf $H$ and marginals $F$ and $G$, respectively. In Section 3 we discuss its copula-based counterpart which corresponds to some cdf $C$ on $[0,1]^2$ with uniform marginals. Both variants allows for readable visualization of departures from independence. We focus our presentation on the copula-based variant. Simple and natural estimator of the copula-based measure in the i.i.d. case is proposed and its appealing properties are discussed. The estimator can be effectively exploited to assess graphically underlying bivariate dependence structure and to build some formal local and global tests. Some illustrative examples are given in Section 4 to support utility of new solution. Section 6 concludes.\
[**2. General case**]{}\
Consider a pair of random variables $X$ and $Y$ with cdf’s $F(x)=P(X\leq x)$ and $G(y)=P(Y\leq y)$, respectively and a joint cdf $H(x,y)=P(X\leq x,Y\leq y)$. Set $\mathbbm{D} =\{(x,y): 0<F(x)<1,0<G(y)<1\}$ and define $$q(x,y)=q_H(x,y)=\frac{H(x,y)-F(x)G(y)}{\sqrt{F(x)G(y)[1-F(x)][1-G(y)]}}\;\;\; \mbox{for}\;\;(x,y)\in \mathbbm{D}.
\eqno(1)$$ From (1) it is seen that $q$ treats both variables $X$ and $Y$ symmetrically and a knowledge of $q$ and the marginal distributions allows one to recover $H$. The measure $q$ fulfills the following properties, motivated by the axioms formulated in Schweitzer and Wolff (1981) and updated in Embrechts et al. (2002).\
[**Proposition 1.**]{}\
1. $q$ is defined for any $X$ and $Y$.\
2. $-1 \leq q \leq 1$ for all $(x,y) \in \mathbbm{D}$.\
3. If the variables $X$ and $Y$ are exchangeable then $q(x,y)=q(y,x)$ for $(x,y) \in \mathbbm{D}$.\
4. $q(x,y)\equiv 0$ if and only if $X$ and $Y$ are independent.\
5. $q$ is non-negative (non-positive) if and only if $(X,Y)$ are positively (negatively) quadrant dependent.\
6. $q$ is maximal (minimal) if and only if $Y=f(X)$ and $f$ is non-decreasing (non-increasing) a.s. on the range of $X$.\
7. $q$ respects concordance ordering, i.e. for cdf’s $H_1$ and $H_2$ with the same marginals, $H_1(x,y) \leq H_2(x,y)$ for all $(x,y) \in \mathbbm{R}^2$ implies $q_{H_1}(x,y) \leq q_{H_2}(x,y)$ for all $(x,y) \in \mathbbm{D}$.\
[**Proof**]{}. Most of the above mentioned properties are obvious. The property 6 is an immediate consequence of Fréchet-Hoeffding bounds and their properties. To justify 2 it is enough to show that $q_H(x,y)$ is the correlation coefficient of some random variables. For this purpose, for $(x,y) \in \mathbbm{D}$ and $(s,t) \in \mathbbm{R}^2$ set $$\phi_x(s)=-\sqrt{\frac{1-F(x)}{F(x)}}\mathbbm{1}_{(-\infty ,x]}(s) + \sqrt{\frac{F(x)}{1-F(x)}}\mathbbm{1}_{(x, +\infty)}(s),$$ $$\psi_y(t)=-\sqrt{\frac{1-G(y)}{G(y)}}\mathbbm{1}_{(-\infty ,y]}(t) + \sqrt{\frac{G(y)}{1-G(y)}}\mathbbm{1}_{(y, +\infty)}(t).$$ Then, by an elementary argument one gets $$q(x,y)=q_H(x,y)=E_H\phi_x(X)\psi_y(Y)=Cov_H\phi_x(X)\psi_y(Y)=Corr_H\phi_x(X)\psi_y(Y).
\eqno(2)$$ $\square$\
[**Remark 1.**]{} The last expression in (2) shows that the function $q$ is based on aggregated local correlations. Moreover, note that $\int_{\mathbbm{R}}\phi_x(s)dF(s)=\int_{\mathbbm{R}}\psi_y(t)dG(t)=0 $ and $\int_{\mathbbm{R}}\phi_x^2(s)dF(s)=\int_{\mathbbm{R}}\psi_y^2(t)dG(t)=1 $, for all $(x,y) \in \mathbbm{D}$. Therefore, the value $q(x,y)$ can be interpreted as the Fourier coefficient of the cdf $H(s,t)$ pertaining to the quasi-monotone function $\phi_x(s) \psi_y(t),\;(s,t) \in \mathbbm{R}^2$.\
Given random sample $(X_1,Y_1),\ldots,(X_n,Y_n)$ from cdf $H$ with marginals $F$ and $G$, set $H_n$, $F_n$, and $G_n$ for respective empirical cdf’s. A natural estimator $\hat {q}_H$ of $q_H(x,y)$ results by plugging these empirical cdf’s into (1). This, given $(x,y)$, yields rank statistics. Note that the values $\chi_{ni}$ given by $\hat {q}_H(X_i,Y_i),\; i=1,\ldots,n$, have been already introduced in Fisher and Switzer (1985), as one of the two components of the so-called chi-plots, designed to investigate possible patterns of association of two random variables.
We shall not study the estimator $\hat {q}_H$ in this paper. In the next section we comment in more detail on special case of (1), and the related problems, in the case when the role of $H$ is played by the pertaining copula.\
[**3. Copula-based measure of dependence**]{}\
In this section, to avoid technicalities and to concentrate on the main idea, we restrict attention to cdf’s $H$ with continuous marginals $F$ and $G$. Under such a restriction there exists a unique copula $C$ such that $H(x,y)=C(F(x),G(y))$. In other words, $C$ is the restriction to the unit square of the joint cdf of $(F(X),G(Y))$. The copula captures the dependence structure among $X$ and $Y$, irrespective of their marginal cdf’s. This is important in many applications. For the related discussion see Póczos et al. (2012).\
[*3.1. The form and further properties of q*]{}\
We have $$q(u,v)=q_{C}(u,v)=\frac{C(u,v)-uv}{\sqrt{uv(1-u)(1-v)}} = w(u,v)[C(u,v)-uv], \;\;\;\;(u,v) \in (0,1)^2,
\eqno(3)$$ where $$w(u,v)=\frac{1}{\sqrt{uv(1-u)(1-v)}}.
\eqno(4)$$ The interpretation of $q$ in terms of correlations, given in (2), is still valid with some obvious adjustment. Namely, now for $u \in (0,1)$ and $s\in (0,1)$ we consider $$\phi_u(s)=\psi_u(s)=-\sqrt{\frac{1-u}{u}}\mathbbm{1}_{[0,u]}(s)+\sqrt{\frac{u}{1-u}}\mathbbm{1}_{(u,1]}(s).$$
[**Proposition 2.**]{} The copula based measure of dependence $q$, given by (3), additionally to 1-7, has the following properties.\
8. $q$ is invariant to strictly increasing a.s. on ranges of $X$ and $Y$, respectively, transformations.\
9. If $X$ and $Y$ are transformed by strictly decreasing a.s. functions then $q(u,v)$ transforms to $q(1-u,1-v)$.\
10. If $f$ and $g$ are strictly decreasing a.s. on ranges of $X$ and $Y$, respectively, then $q$’s for the pairs $(f(x),Y)$ and $(X,g(Y))$ take the forms $-q(1-u,v)$ and $-q(u,1-v)$, accordingly.\
11. The equation $q(u,v)\equiv c$, $c$ a constant, can hold true if and only if $c=0$.\
12. If $(X,Y)$ and $(X_n,Y_n),\;n=1,2,\ldots,$ are pairs of random variables with joint cdf’s $H$ and $H_n$, and the pertaining copulas $C$ and $C_n$, respectively, then weak convergence of $\{H_n\}$ to $H$ implies $q_{C_n}(u,v) \to q_C(u,v)$ for each $(u,v) \in (0,1)^2$.\
[**Proof.**]{} Properties 8-10 follow from Theorem 3 in Schweizer and Wolff (1981). The convergence in 12 is due to continuity of $C$. To justify 11 observe that the equation is equivalent to $C(u,v)=C_c(u,v)=uv+c\sqrt{uv(1-u)(1-v)}$. Since $C$ is quasi-monotone, then $C_c(u,v)$ should also possess such a property. Since $C_c(u,v)$ is absolutely continuous then quasi-monotonicity is equivalent to $\frac{\partial^2}{\partial u \partial v} C_c(u,v) \geq 0$ for almost all $(u,v) \in [0,1]$ (in the Lebesgue measure); cf. Cambanis et al. (1976). However, $\frac{\partial^2}{\partial u \partial v} C_c(u,v) = 1 + c[u-1/2][v-1/2]w(u,v)$ and for $c\not= 0$ this expression can be negative on the set of positive Lebesgue measure.$\square$\
[**Remark 2.**]{} The properties 4 and 8-10 provide some compromise to too demanding postulates P4 and P5 discussed in Embrechts et al. (2002).\
[**Remark 3.**]{} The property 11 is very different from respective property of the local dependence function of Holland and Wang (1987) which is constant for the bivariate normal distribution and some other models; cf. Jones (1998) for details.\
[**Remark 4.**]{} By Fréchet-Hoeffding bounds for copulas, the property 2 saying that $q(u,v) \in [-1,1]$ can be further sharpened to $$B_*(u,v) \leq q_C(u,v) \leq B^*(u,v),\;\;\;(u,v) \in (0,1)^2$$ where $B_*(u,v)=w(u,v)[\max\{u+v-1,0\} -uv]$ and $B^*(u,v)=w(u,v)[\min\{u,v\} -uv]$. Note that $B_*(u,1-u) = -1$ for $u \in (0,1)$ while $-1 < B_*(u,v) \leq 0$ otherwise. Similarly, $B^*(u,u)=+1$ for $u \in (0,1)$ and $0 \leq B^*(u,v)<1$ in the remaining cases. In Figure 1 we show these bounds on the the regular grid $$\mathbbm{G}_{16} = \{(u,v): u=i/16, v=j/16,\; i,j=1,\ldots,15\}.$$
[*Fig. 1.*]{} Left panel: lower bound $B_{*}(u, v)$ of $q_C(u, v)$ on the grid $\mathbbm {G}_{16}$; right panel: upper bound $B^{*}(u, v)$ of $q_C(u, v)$ on the grid $\mathbbm {G}_{16}$.\
[**Remark 5.**]{} Similarly as $B_*$ and $B^*$, the measure $q_C$ can be displayed in a graphical form. This is useful feature helping to visualize a structure of departures from independence. It is also worth emphasizing that though we focused of cdf’s with continuous marginals we still do not assume that $H$ or $C$ posses a density. In Figures 2 and 3 we show plots of $q_C$ pertaining to classical Marshall-Olkin and recently introduced Mai-Scherer (2011) extreme value copulas. The displays are accompanied by scatter plots of simulated data and pertaining heat maps. Therefore, we introduce first a natural estimator of $q_C$ and present all illustrative examples in Section 4.\
[*3.2. Estimates of $q_C$*]{}\
Let $(X_1,Y_1),\ldots,(X_n,Y_n)$ be a random sample from cdf $H$. Furthermore, let $R_i$ be the rank of $X_i,\;i=1,\ldots,n$, in the sample $X_1,\ldots,X_n$ and $S_i$ the rank of $Y_i,\;i=1,\ldots,n,$ within $Y_1,\ldots,Y_n$. Simple estimate of $q_C$ has the form $$\hat q_C (u,v)=w(u,v)[{D_n(u,v)-uv}], \;\;\;\;(u,v) \in (0,1)^2,$$ where $D_n$ is rank-based empirical copula estimator of $C$, i.e. $$D_n(u,v)=\frac{1}{n}\sum_{i=1}^n \mathbbm{1}\Bigl(\frac{R_i}{n}\leq u,\frac{S_i}{n}\leq v\Bigr), \quad
(u,v) \in [0,1]^2.
\eqno(5)$$ The paper of Swanepoel and Allison (2013) provides exact mean and variance of $D_n(u,v)$. Ledwina and Wy[ł]{}upek (2014) have shown that $D_n(u,v)$ preserves the quadrant order. More precisely, if a copula $C_1$ has larger quadrant dependence than a copula $C_{2}$ then, under any fixed $(u,v) \in [0,1]^2$, any $c \in \mathbb{R}$ and any $n$, it holds that $$P_{C_1}\Bigl(D_n(u,v) \geq c\Bigr) \geq P_{C_{2}}\Bigl(D_n(u,v)\geq c\Bigr).
\eqno(6)$$ An obvious consequence of (6) is analogous order preserving property of $\hat q_C (u,v)$.
Asymptotic properties of the empirical copula process $Z_n(u,v)=\{\sqrt n [D_n(u,v)-uv],\;u,v \in [0,1]\}$ have been studied by many authors; cf. Fermanian et al. (2004) for the related results and references. Since $uv(1-u)(1-v)$ is the asymptotic variance of $Z_n(u,v)$ under independence, $\sqrt n \hat q_C (u,v)$ coincides with natural weighted empirical copula process while Theorem 3 of Fermanian et al. (2004) implies that, under independence, $\sqrt n \hat q_C (u,v)$ is asymptotically $N(0,1)$ for each $u,v \in (0,1)$. The estimate $D_n(u,v),$ in a series of papers originated by Deheuvels (1979), has been called empirical dependence function.
In application oriented papers, more popular variant of rank-based estimator of $C$ is $$C_n(u,v)=\frac{1}{n}\sum_{i=1}^n \mathbbm{1}\Bigl(\frac{R_i}{n+1}\leq u,\frac{S_i}{n+1}\leq v\Bigr), \quad
(u,v) \in [0,1]^2.
\eqno(7)$$ The variables $(R_i/(n+1),S_i/(n+1)),\; i=1,\ldots,n$ are called pseudo-observations in the literature. Obviously, the finite sample and basic asymptotic properties of $C_n(u,v)$ are inherited after $D_n(u,v)$. Therefore, we shall consider the following estimator of $q_C$ $$Q_n(u,v)=w(u,v)[{C_n(u,v)-uv}] = \frac{C_n(u,v)-uv}{\sqrt{uv(1-u)(1-v)}}, \;\;\;\;(u,v) \in (0,1)^2,
\eqno(8)$$ which, similarly as $\hat q_C (u,v)$, is the rank statistic. Moreover, we set $$L_n(u,v)=\sqrt n Q_n(u,v)
\eqno(9)$$ for the standardized version of this estimate. Simple algebra yields that for any $(u,v) \in (0,1)^2$ it holds $$L_n(u,v)=\frac{1}{\sqrt n} \sum_{i=1}^n\phi_u(\frac{R_i}{n+1})\phi_v(\frac{S_i}{n+1}) + O(\frac{1}{\sqrt n}).
\eqno(10)$$ So, up to deterministic term of the order $O(1/\sqrt n)$, the standardized estimator $L_n(u,v)$ is linear rank statistic with the quasi-monotone score generating function $\phi_u \times \phi_v$. Moreover, the definition of $L_n$ and (6) yield that $$P_{C_1}\Bigl(L_n(u,v) \geq c\Bigr) \geq P_{C_{2}}\Bigl(L_n(u,v)\geq c\Bigr)
\eqno(11)$$ for any $(u,v) \in (0,1)^2$, any $c$, any $n$, and any two copulas $C_1$ and $C_2$ such that $C_1$ has larger quadrant dependence than $C_2$. Summarizing the above mentioned results, let us note that under independence $L_n(u,v)$ is distribution free. So, given $n$, under independence, the significance of the obtained values of this statistic can be easily assessed on a basis of simple simulation experiment. For large $n$ one can rely on asymptotic normality of $L_n(u,v)$. Due to (11), similar conclusions follow if one likes to verify hypothesis asserting that $q_C(u,v) \geq 0.$ Moreover, (11) implies that different levels of strength of quadrant dependence of the underlying $H$’s shall be adequately quantified by order preserving $L_n(u,v)$’s. These results make the values of $L_n(u,v), \;(u,v) \in (0,1)^2,$ a useful diagnostic tool. For example, since quadrant dependence is relatively weak notion, significantly negative values of $L_n(u,v)$ for some $(u,v)$’s make questionable positive quadrant dependence, and many other forms of positive dependence of the data at hand, as well.
To close, note that, given $u$ and $u$, the score generating function $\phi_u \times \phi_v$, appearing in (10), is not smooth one and takes on at most four possible values, only. This causes that, under independence, the convergence of $L_n(u,v)$ to the limiting $N(0,1)$ law is not very fast. Moreover, the rate of convergence is expected to depend on $u$ and $v$, with the least favorable situation when $(u,v)$ is close to the vertices of the unit square. We illustrate these aspects in Table 1, where simulated critical values of the test rejecting independence for large values of $|L_n(u,v)|$ are given under some $(u,v)$’s, five different sample sizes, and two selected significance levels $\alpha$. In cases when finite sample distribution of $L_n$ is far from continuous one, the problem of uniqueness of sample quantiles arises. Through, to calculate sample quantiles we apply Gumbel’s approach, described in Hyndman and Fan (1996) by Definition 7.\
Table 1. Simulated critical values of the test rejecting independence for large values of $|L_n(u,v)|$ for selected $(u,v)$, versus $n$ and $\alpha$.\
------------------------------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
$(u,v)$
200 300 400 500 600 200 300 400 500 600
$(\frac{1}{2},\frac{1}{2})$ 2.546 2.540 2.600 2.504 2.613 1.980 1.848 2.000 1.968 1.960
$(\frac{1}{12},\frac{1}{12})$ 2.520 2.960 2.800 2.716 2.583 1.594 1.575 1.782 1.968 2.049
$(\frac{1}{16},\frac{1}{16})$ 2.753 2.879 2.933 2.349 2.591 1.546 1.894 2.080 1.586 1.894
$(\frac{1}{20},\frac{1}{20})$ 2.233 2.735 3.158 2.589 3.008 2.233 1.519 2.105 1.648 2.149
------------------------------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
\
The sample sizes in Table 1 are close to that we shall consider in Section 4. The results exhibit that, for this range of sample sizes, the simulated quantiles of $|L_n(1/2,1/2)|$ are reasonably close to the limiting ones. We studied some selection of regular grids $\{(i/g,j/g),\;i,j=1,\ldots,g-1\}$, for some $g$’s, the related behavior of $L_n$’s and reported in Table 1 the results on $L_n(1/g,1/g)$ for $g=12, 16, 20$. They illustrate the conclusion that too dense grid shall imply too much inaccuracy in the simulated quantiles while too conservative choice not necessarily provides much progress. In view of these observations, we decided to apply the regular grid $\mathbbm{G}_{16}$ through. With this choice, for the sample sizes under consideration, the significance of observed values of $|L_n(i/g,j/g)|$ can be easily approximately evaluated looking at the heat maps, that we provide in each case. For more precise evaluation extra simulations are needed.
Note also that there are available some smooth nonparametric estimators of copulas. See Janssen et al. (2012) and Omelka et al. (2009) for recent contributions and extensive overview. However, we do prefer to insert the estimator $C_n$, since it is naturally linked to local correlations defined in the paper and obeys the property (11), which is crucial in finite sample inference on dependence. Counterparts of (11) for more general rank statistics are not available, according to the best our knowledge. For some related discussion see Ledwina and Wy[ł]{}upek (2014), Section 3.\
[**4. Illustration**]{}\
[*4.1. Example 1: Extreme value copulas*]{}\
We start with two simulated data sets of size $n=500$ from Marshall-Olkin and Mai-Scherer (2001) copulas given by $C(u,v)=C_{a,b}^{MO}(u,v)=\min\{u^{1-a}v,uv^{1-b}\},\;a=0.50, b=0.75$, and $C(u,v)=C_{a,b}^{MS}(u,v)=\min\{u^a,v^b\}\min\{u^{1-a},v^{1-b}\},\;a=0.9, b=0.5$; cf. Nelsen (2006), p. 53 and Mai-Scherer (2001), p. 313, respectively. Both copulas possess a singular part. In Figures 2 and 3 we show dependence functions $q_C(u,v)$ for these models. The functions are accompanied by scatter plots of pseudo-observations $(R_i/(n+1),S_i/(n+1)), i=1,\ldots,500$, from the simulated samples. The scatter plots nicely exhibit the singularities and show some tendencies in the data. Right panels in these figures display respective heat maps of standardized correlations $L_n(u,v)$’s calculated on the grid $\mathbbm {G}_{16}$. Each square of size $0.0625\times0.0625$ represents the respective value of $L_n$ in its upper-right corner. To simplify reading, each heat map is accompanied with two numbers $$L_* = \min_{1\leq i,j \leq 15} L_n(i/16,j/16) \;\;\;\mbox{and}\;\;\; L^*=\max_{1\leq i,j \leq 15} L_n(i/16,j/16).
\eqno(12)$$ Both copulas represent positively quadrant dependent distributions. Under such dependence large values of $U$ tend to associate to large values of $V$ and similar pattern applies to small values. This tendency is nicely seen in the figures. Intuitively, the tendency is stronger for $C_{0.9,0.5}^{MS}$ than for $C_{0.50,0.75}^{MO}$ and this is indeed well reflected by the heat maps. The points of the grid $\mathbbm{G}_{16}$ in which the estimated correlations $Q_n$ are significant on the levels 0.05 and 0.01 can be easily identified; cf. Table 1. Some possibility of testing for positive local and/or global dependence is sketched in Section 4.2.\
Marshall-Olkin copula, $\alpha = 1/2$, $\beta = 3/4$\
[*Fig. 2.*]{} Left panel: dependence function $q_C(u, v)$ for the Marshall-Olkin copula; middle panel: scatter plot of $(R_i/(n+1),S_i/(n+1))$, $i = 1,\ldots,n$, $n = 500$, of simulated observations from the copula; right panel: standardized estimator $L_n(u,v)$ of $q_C(u, v)$ on the grid $\mathbbm {G}_{16}$. $L_{*} = -0.2$, $L^{*} = 12.2$.\
Mai-Scherer copula, $a = 0.9$, $b = 0.5$\
[*Fig. 3.*]{} Left panel: dependence function $q_C(u, v)$ for the Mai-Scherer copula; middle panel: scatter plot of $(R_i/(n+1),S_i/(n+1))$, $i = 1,\ldots,n$, $n = 500$, of simulated observations from the copula; right panel: standardized estimator $L_n(u,v)$ of $q_C(u, v)$ on the grid $\mathbbm {G}_{16}$. $L_{*} = 1.5$, $L^{*} = 16.1$.\
Next examples follow similar pattern. They concern three real data sets considered earlier by Jones and Koch (2003). In each example of our paper we use the same scale of intensity of colors in the heat maps. This allows one to compare how different degrees of association are reflected by our estimators.\
[*4.2. Example 2: Automobile data*]{}\
We shall consider two data sets of size $n=392$ available through [www/http://lib.stat.\
cmu.edu/datasets/cars]{}. This is 1983 [*ASA Data Exposition*]{} data set, collected by Ernesto Ramos and David Donoho.
The first sample collects observations of engine power (variable $X$), measured in horsepower, and fuel consumption (variable $Y$). This example was already investigated by Hawkins (1994), who fitted to the original data points decreasing regression function. Strong negative association is also clearly manifested by the scatter plot, which is based of transformed observations.\
, $n = 392$\
[*Fig. 4.*]{} Left panel: scatter plot of $(R_i/(n+1),S_i/(n+1))$, $i = 1,\ldots,n$, $n = 392$; middle panel: estimator $Q_n(u,v)$ of $q_C(u, v)$ on the grid $\mathbbm {G}_{16}$; right panel: standardized estimator $L_n(u,v) = \sqrt{n}\, Q_n(u,v)$ on the grid $\mathbbm {G}_{16}$. $L_{*} = -16.0$, $L^{*} = -1.2$.\
The heat map indicates visible negative trend and strong negative dependence in most of the points of the grid $\mathbbm {G}_{16}$. Only for some points close to the edges (0,0) and (1,1) the correlations are small in absolute value. To allow for some immediate quantitative analysis we give in Table 2 simulated quantiles of $L_n(u,v)$ for $n=392$ and two choices of $(u,v)$’s.\
Table 2. Simulated $\alpha$-quantiles of $L_n(u,v)$ for two selected $(u,v)$ and $n=392$.\
------------------------------- -------- -------- -------- ------- ------- -------
$(u,v)$ 0.01 0.05 0.10 0.90 0.95 0.99
$(\frac{1}{2},\frac{1}{2})$ -2.424 -1.616 -1.212 1.212 1.616 2.222
$(\frac{1}{16},\frac{1}{16})$ -1.266 -1.266 -1.266 1.320 2.182 3.044
------------------------------- -------- -------- -------- ------- ------- -------
\
The local correlations can be used to test independence and to verify local negative and positive dependence, as well. In particular, observe that except five values close to the vertices (0,0) and (1,1), where the standardized empirical local correlations are in \[-1.212,-2.000), the remaining ones are strictly less than -2.000. This, along with the rough information contained in Table 2, allow one to expect that all local correlations (in the grid points $\mathbbm {G}_{16}$) shall be accepted to be non positive on the standard level $\alpha$=0.05.
Obviously, the local correlations can be also used to form a new test statistic on global negative dependence. A reasonable candidate is the test rejecting such hypothesis for large values of $L^*$, see (12). By Theorem 1 of Ledwina and Wy[ł]{}upek (2014) such statistic preserves the correlation order. For the data under consideration, simulated in 10 000 MC runs, $p$-value of this test is equal to 1.
Our conclusion is that the heat map displayed in Figure 4 of this paper supports more simple picture of the overall dependence structure than this one presented in Figure 4 of Jones and Koch (2003) and obtained via kernel methods applied to the original data.\
The second sample of automobile data consists of observations of acceleration time ($X$) and fuel consumption ($Y$). Both, the original data display in Jones and Koch (2003) and the scatter plot provided in Figure 5 of our paper, suggest some not very strong positive dependence. The plot of $Q_n$ supports this suggestion. The heat map visualizes standardized correlations and gives better insight into the strength of this dependence. The strongest local correlations are observed close to the lower tails of both (transformed) variables and the strength of the dependence is getting weaker towards the upper tails. Again our look at the data reveals a simpler structure of the dependence that this one provided in Figure 5 of Jones and Koch (2003). In particular, we do not notice zero local dependence between moderately large values of both variables. Test rejecting positive local dependence for small values of $L_n(i/16,j/16), i,j=1,\ldots,15,$ can be applied in each grid point while global positive dependence can be verified by the test rejecting it for small values of $L_*$, see (12) and (11). For the given data, the simulated, on the basis of 10 000 observations, $p$-value of such global test on positive quadrant dependence is equal to 1.\
, $n = 392$\
[*Fig. 5.*]{} Left panel: scatter plot of $(R_i/(n+1),S_i/(n+1))$, $i = 1,\ldots,n$, $n = 392$; middle panel: estimator $Q_n(u,v)$ of $q_C(u, v)$ on the grid $\mathbbm {G}_{16}$; right panel: standardized estimator $L_n(u,v) = \sqrt{n}\, Q_n(u,v)$ on the grid $\mathbbm {G}_{16}$. $L_{*} = 0.8$, $L^{*} = 10.2$.\
[*4.3. Example 3: Aircraft data*]{}\
Consider $n=230$ aircraft span and speed data, on log scales, from years 1956-1984, reported and analyzed in Bowman and Azzalini (1997). We summarize the data in Figure 6. Since in this example both negative and positive correlations appear, we added respective signs to the colors in the heat maps. The figure exhibits that small and moderately large values of log speed are positively correlated with log span, while for the remaining cases the relation is reversed. Two, approximately symmetrically located, regions of relatively strong dependence are seen. In general, the strength of dependence is weaker than in previous cases. Similarly as in the previous example, also here our approach provides simpler and more regular picture of the dependence structure than this one presented in Figure 2 of Jones and Koch (2003).\
, $n = 230$\
[*Fig. 6.*]{} Left panel: scatter plot of $(R_i/(n+1),S_i/(n+1))$, $i = 1,\ldots,n$, $n = 230$; middle panel: estimator $Q_n(u,v)$ of $q_C(u, v)$ on the grid $\mathbbm {G}_{16}$; right panel: standardized estimator $L_n(u,v) = \sqrt{n}\, Q_n(u,v)$ on the grid $\mathbbm {G}_{16}$. $L_{*} = -6.5$, $L^{*} = 4.6$.\
Bowman and Azzalini (1997) used these data to discuss some drawbacks of standard correlation measures. Indeed, for these data classical Pearson’s, and Spearman’s and Blomqvist’s rank statistics for assessing an association yield simulated $p$-values 0.81, 0.74, and 0.79, respectively. Kendall’s rank correlation gives simulated $p$-value 0.31, which also seems to be too high, when one is looking at the magnitude of standardized local correlations in Figure 6. Combining the local correlation into global statistic $L^o =\max_{1 \leq i,j \leq 15} |L_n(i/16,j/16)|$, with large values being significant, basing on simulation of size 10 000, we get $p$-value 0.00 for such global independence test. This shows that local correlations are more informative than each of the above single classical global indices of association.\
[**5. Discussion**]{}\
We have introduced the novel function valued measure of dependence of two random variables. Its definition, based on Studentized difference of two cdf’s, is general, simple, and natural. In our considerations, we mainly focus attention on copula-based variant of the measure. It allows for simple estimation and guarantees appealing finite sample properties of the resulting estimate. The estimate is tightly linked to the popular scatter plot and helps to extract explicit dependence structure from it. Both, the measure and the estimate, allow for comparison and visualization of different association structures. The value of the measure in a fixed point has useful interpretation as correlation coefficient of some specific increasing functions of the marginals. Also, the proposed estimate features simple interpretation and easy implementation. Its performance in real data analysis yields relatively simple, in comparison to alternative method, dependence structure. We believe that the proposed approach will be useful in practice.
Also, simple and reliable tests for local and global association, based on estimated dependencies have been proposed. It is worth noticing that, in particular, statistic like $L^*$, cf. (12), can be considered as a usable approximation of empirical isotonic canonical correlation coefficient, introduced in Schriever (1987). Similarly, $L^o$, given in (13), can be serve as an easy to implement approximate exemplification of Rényi’s idea to calculate maximal correlation over large class of functions. In Ledwina and Wy[ł]{}upek (2014) empirical correlations close to $L_n(u,v)$’s were successfully applied to construct highly sensitive test for detection of positive quadrant dependence. Recently, there is much of interest in detecting dependencies in some conditional copulas; see Veraverbeke et al. (2011), and Li et al. (2014) for discussion and further references. It seems that some graphical presentation of dependence structure of two random variables, conditionally upon some fixed values of a covariate, and formal application of pertaining counterparts of $L^o, L^*$, and $L_*$ could be useful in such considerations, as well. It is also worthy noting that the definition of $q$ can be naturally extended to higher dimensions and applied to construct tests for positive orthant dependence, for example. These questions are however beyond the scope of this initial article.\
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|
---
abstract: |
The paper presents new simple sharp bounds for transition density functions for time-homogeneous diffusions processes. The bounds are obtained under mild conditions on the drift and diffusion coefficients, extending and substantially improving previous results in the literature which were limited to drifts satisfying a linear growth condition. They lead to an asymptotic expression for the time $t$ transition density as $t \rightarrow 0$. While the focus is on the one-dimensional case, an extension to multiple dimensions is discussed. Results are illustrated by numerical examples.\
\
*Keywords:* diffusion process; bounds for transition density\
*2000 Mathematics Subject Classification:* Primary 60J60; Secondary 60J35.
author:
- 'Andrew N. Downes[^1]'
bibliography:
- 'BoundaryCrossing.bib'
- 'TransDensBounds.bib'
- 'GeneralBooks.bib'
title: 'Bounds for the Transition Density of Time-Homogeneous Diffusion Processes'
---
Introduction
============
Let $(U_t)$ be a one-dimensional time-homogeneous diffusion process satisfying the stochastic differential equation $$dU_t = \nu(U_t) dt + {\sigma}(U_t) dW_t, \qquad U_0 = u_0,$$ where $(W_t)$ denotes a Brownian motion. The aim of this note is to bound, from above and below, the transition probability density function for $(U_t)$, $p_U(t, u_0, w) := \frac{d}{dw} P_{U}(t,u_0,w)$, where $P_{U}(t,u_0,w) := {\mathbb{P}}(U_t \leq w | U_0 = u_0)$. While the focus is on the one-dimensional case, the results are easily extended to some special cases in $\mathbb{R}^n$, $n \geq 2$ (see remark at the end of this section). Some simple bounds for the distribution function are also considered.
Except for a few special cases, the transition functions are unknown for general diffusion processes, so finding approximations to them is an important alternative approach. We use Girsanov’s theorem and then a transformation of the Radon-Nikodym density of the type suggested in [@Baldi_etal_0802] to relate probabilities for a general diffusion $(U_t)$ to those of a ‘reference diffusion’. Using a reference diffusion with known transition functions, we are able to derive various bounds for the transition functions under mild conditions on the original process. The results have a simple form and are readily evaluated.
As an aside, the generator of the diffusion $(U_t)$ is given by $$Af(x) = \nu(x) {\frac{\partial{f}}{\partial{x}}} + \frac{1}{2} {\sigma}^2(x) {\frac{\partial^{2} f}{\partial x^{2} }},$$ and the transition probability density function is the minimal fundamental solution to the parabolic equation $$\left(A - {\frac{\partial{}}{\partial{t}}}\right)u(t,x) = 0.$$ Thus the results presented here also bound solutions to certain types of parabolic partial differential equations.
Several papers on this topic are available in the literature, especially for bounding the transition probability density. Most recently, [@Qian_etal_0504] proposed upper and lower bounds for diffusions whose drift satisfied a linear growth constraint. This appears to be the first such paper to relax the assumption of a bounded drift term. The results in [@Qian_etal_0504] will be compared with those obtained in the current paper, although the former can not be used for processes not satisfying the linear growth constraint. To the best of our knowledge, the bounds presented in the current paper are the only ones to relax this constraint, and also appear to generally offer a tightening of the bounds previously available. For further background on diffusions with bounded drift, see e.g. [@Qian_etal_1103] and references therein.
In addition, the same ideas allow us to obtain bounds for other functions related to the diffusions. This is not the focus of this note and is not discussed in great detail here, but as an example at the end of Section \[sec:main\_result\] we consider the density of the process and its first crossing time. This has application in many areas, such as the pricing of financial barrier options. Bounds for other probabilities may be derived in the same manner.
Consider a one-dimensional time-homogeneous non-explosive diffusion $(U_t)$ governed by the stochastic differential equation (SDE) $$\begin{aligned}
\label{eq:original_diff}
dU_t = \nu (U_t) dt + {\sigma}(U_t) dW_t,
\end{aligned}$$ where $(W_t)$ is a Brownian motion and ${\sigma}(y)$ is differentiable and non-zero inside the diffusion interval (that is, the the smallest interval $I \subseteq \mathbb{R}$ such that $U_t \in I$ a.s.). As is well-known, one can transform the process to one with unit diffusion coefficient by letting $$\begin{aligned}
\label{eq:fn_transform}
F(y) := \int_{y_0}^y \frac{1}{{\sigma}(u)} du
\end{aligned}$$ for some $y_0$ from the diffusion interval of $(U_t)$ and then considering $X_t := F(U_t)$ (see e.g. [@Rogers_xx85], p.161). By [Itô]{}’s formula, $(X_t)$ will have unit diffusion coefficient and a drift coefficient $\mu(y)$ given by the composition $$\mu(y) := \left( \frac{\nu}{{\sigma}} - \frac{1}{2} {\sigma}'\right) \circ F^{-1}(y).$$ From here on we work with the transformed diffusion process $(X_t)$ governed by the SDE $$\begin{aligned}
dX_t = \mu(X_t) dt + dW_t, \qquad X_0 = F(U_0) =: x.
\end{aligned}$$ Conditions mentioned throughout refer to the transformed process $(X_t)$ and its drift coefficient $\mu$.
We will consider the following two cases only (the results extend to diffusions with other diffusion intervals with one finite endpoint by employing appropriate transforms):
1. The diffusion interval of $(X_t)$ is the whole real line $\mathbb{R}$.
2. The diffusion interval of $(X_t)$ is $(0, \infty)$.
The results extend to diffusions with other diffusion intervals with one finite endpoint by employing appropriate transforms.
For the diffusion $(X_t)$ we will need a reference diffusion $(Y_t)$ with certain characteristics. The reference diffusion must have the same diffusion interval as $(X_t)$ and a unit diffusion coefficient, so that Girsanov’s theorem may be applied to $(X_t)$. To be of any practical use, the reference process must also have known transition functions. In case \[A\], we use the Brownian motion as the reference process, while in case \[B\] we use the Bessel process of an arbitrary dimension $d \geq 3$.
Recall the definition of the Bessel process $(R_t)$ of dimension $d = 3, 4, \ldots,$ starting at a point $x>0$. This process gives the Euclidean norm of the $d$-dimensional Brownian motion originating at $(x,0, \ldots, 0)$, that is, $$R_t = \sqrt{\bigl(x +W_t^{(1)}\bigr)^2 + \cdots + \bigl(W_t^{(d)}\bigr)^2},$$ where the $\bigl(W_t^{(i)}\bigr)$ are independent standard Brownian motions, $i = 1, \ldots, d$. As is well known (see e.g. [@Revuz_etal_xx99], p.445), $(R_t)$ satisfies the SDE $$\begin{aligned}
dR_t = \frac{d-1}{2} \frac{1}{R_t}dt + dW_t.
\end{aligned}$$ Note that for non-integer values of $d$ the Bessel process of ‘dimension’ $d$ is defined using the above SDE. The process has the transition density function $$\begin{aligned}
p_R(t,y,z) = z \left(\frac{z}{y}\right)^{\eta} t^{-1} e^{-(y^2 + z^2)/2t} {{{\mathcal I}}}_{\eta} \left(\frac{yz}{t} \right),
\end{aligned}$$ where $\eta = d/2 -1$ and ${{{\mathcal I}}}_{\eta}(z)$ is the modified Bessel function of the first kind. For further information, see Chapter XI in [@Revuz_etal_xx99].
We denote by ${\mathbb{P}}_x$ and ${\mathbb{E}}_x$ probabilities and expectations conditional on the process in question ($(X_t)$ or some other process, which will be obvious from the context) starting at $x$. We work with the natural filtration ${{\mathcal F}}_s := {\sigma}\left(X_u: u \leq s\right)$.
Finally, note that the present work can be easily extended to a class of $n$-dimensional diffusions for $n \geq 2$. If $(X_t)$ is an $n$-dimensional diffusion satisfying the SDE $$dX_t = \mu(X_t) dt + dW_t,$$ $(W_t)$ being an $n$-dimensional Brownian motion, then the majority of results can be extended assuming $\mu(\cdot)$ is curl-free. The extension is straight-forward, and in this note we shall only concern ourselves with the one-dimensional case.
Main Results {#sec:main_result}
============
This section states and proves a result relating probabilities for the diffusion $(X_t)$ to expectations under an appropriate reference measure. In the case \[A\], the result may be known, and we state it here for completeness. The extension to case \[B\] is straight-forward. We then apply this proposition to obtain bounds for transition densities and distributions.
**Relation to the Reference Process**
We define the functions $G(y)$ and $N(t)$ as follows, according to the diffusion interval of $(X_t)$:
1. If the diffusion interval of $(X_t)$ is $\mathbb{R}$, then we define, for some fixed $y_0 \in \mathbb{R}$, $$\begin{array}{rl}
G(y) \!\!\!&:= \displaystyle\int_{y_0}^{y} \mu(z) dz,\\
\vphantom{.}\\
N(t) \!\!\!&:= \displaystyle\int_0^t \left(\mu'(X_u) + \mu^2(X_u)\right) du.
\end{array}
\label{eq:bm_G}$$
2. If the diffusion interval of $(X_t)$ is $(0, \infty)$, then we define, for some fixed $d \geq 3$ (the dimension of the reference Bessel process) and $y_0 > 0$, $$\begin{aligned}
G(y) &:= \int_{y_0}^{y} \left(\mu(z) - \frac{d-1}{2z} \right) dz,\\
N(t) &:= \int_0^t \left( \mu'(X_u) - \frac{(d-1)(d-3)}{4X_u^2} + \mu^2(X_u) \right) du.
\end{aligned}$$
For diffusions on $(0, \infty)$, the choice of $d$ is arbitrary subject to $d \geq 3$. Therefore this choice can be used to optimise any bounds presented in the next subsection.
\[th:gen\_int\] Assume the the drift coefficient $\mu$ of $(X_t)$ is absolutely continuous. Then, for any $A \in \mathcal{F}_t$, $${\mathbb{P}}_x(A) = {\hat{{\mathbb{E}}}_x}\left[ e^{G(X_t) - G(x)} e^{-(1/2) N(t)}{\mathbbm{1}}_A \right],$$ where ${\hat{{\mathbb{E}}}_x}$ denotes expectation with respect to the law of the reference process.
In terms of the original process $(U_t)$ defined in , the condition of absolute continuity of $\mu(y)$ requires $\nu(z)$ and ${\sigma}'(z)$ to be absolutely continuous.
The proof is a straight-forward application of Girsanov’s theorem and its idea is similar to the one used in [@Baldi_etal_0802]. We present the proof for case \[A\], the proof for case \[B\] is completed similarly (see [@Downes_etal_xx08] for the general approach).
Define ${\mathbb{Q}}_x$ to be the reference measure such that under ${\mathbb{Q}}_x$, $X_0 = x$ and $$dX_s = d\widetilde{W}_s,$$ for a ${\mathbb{Q}}_x$ Brownian motion $(\widetilde{W}_s)$. Set $$\begin{aligned}
{\zeta}_s &:= \frac{d{\mathbb{P}}_x}{d{\mathbb{Q}}_x} = \exp \left\{ \int_0^s \mu (X_u) d\widetilde{W}_u - \frac{1}{2} \int_0^s \mu^2(X_u) du \right\},
\end{aligned}$$ so by Girsanov’s theorem under ${\mathbb{P}}_x$ we regain the original process $(X_s)$ satisfying $$dX_s = \mu(X_s) ds + dW_s,$$ for a ${\mathbb{P}}_x$ Brownian motion $(W_s)$. The regularity conditions allowing this application of Girsanov’s theorem are satisfied (see e.g. Theorem 7.19 in [@Liptser_etal_xx01]), since under both ${\mathbb{P}}_x$ and ${\mathbb{Q}}_x$ the process $(X_s)$ is non-explosive and $\mu(y)$ is locally bounded so we have, for any $t>0$, $${\mathbb{P}}_x \left( \int_0^t \mu^2(X_s) ds < \infty \right) = {\mathbb{Q}}_x \left( \int_0^t \mu^2(X_s) ds < \infty \right) = 1.$$ We then have, under ${\mathbb{Q}}_x$, using [Itô]{}’s formula and , $$\begin{aligned}
\label{eq:dG}
d G (X_s) &= \mu(X_s) dX_s + \frac{1}{2} \mu'(X_s) (dX_s)^2\notag\\
&= \mu(X_s) d\widetilde{W}_s + \frac{1}{2} \mu'(X_s) ds.
\end{aligned}$$ Note that in order to apply [Itô]{}’s formula, we only require $\mu$ to be absolutely continuous with Radon-Nikodym derivative $\mu'$ (see e.g. Theorem 19.5 in [@Kallenberg_xx97]). This also implies the above is defined uniquely only up to a set of Lebesgue measure zero, and we are free to assign an arbitrary value to $\mu'$ at points of discontinuity.
Rearranging gives $$\int_0^s \mu (X_u) d\widetilde{W}_u = G(X_s) - G(X_0) - \frac{1}{2} \int_0^s \mu'(X_u) du.$$ Hence $${\zeta}_s = \exp \left\{ G(X_s) - G(X_0) - \frac{1}{2} \int_0^s \left( \mu'(X_u) + \mu^2 (X_u) \right) du \right\},$$ which together with $$\begin{aligned}
{\mathbb{P}}_x(A) = {\mathbb{E}}_x[{\mathbbm{1}}_A] = \int {\mathbbm{1}}_A d{\mathbb{P}}_x = \int {\mathbbm{1}}_A {\zeta}_t d{\mathbb{Q}}_x = {\hat{{\mathbb{E}}}_x}[ {\zeta}_t {\mathbbm{1}}_A ],
\end{aligned}$$ completes the proof of the proposition.
**Bounds for Transition Densities and Distributions**
Define $L$ and $M$ as follows, according to the diffusion interval of $(X_t)$:
1. If the diffusion interval of $(X_t)$ is $\mathbb{R}$, then $$\begin{aligned}
L &:= \displaystyle \mbox{ess sup}\left(\mu'(y) + \mu^2(y)\right),\\
M &:= \displaystyle \mbox{ess inf}\left(\mu'(y) + \mu^2(y)\right),
\end{aligned}$$ where the essential supremum/infimum is taken over $\mathbb{R}$.
2. If the diffusion interval of $(X_t)$ is $(0, \infty)$, then, for some fixed $d \geq 3$ (the dimension of the reference Bessel process), we put $$\begin{aligned}
L &:= \displaystyle \mbox{ess sup} \left( \mu'(y) - \frac{(d-1)(d-3)}{4y^2} + \mu^2(y) \right),\\
M &:= \displaystyle \mbox{ess inf} \left( \mu'(y) - \frac{(d-1)(d-3)}{4y^2} + \mu^2(y) \right),
\end{aligned}$$ where the essential supremum/infimum is taken over $(0, \infty)$.
Note that in what follows, in the case \[B\], the dimension of the reference Bessel process may be chosen so as to optimise the particular bound. Recall also that $(Y_t)$ denotes the reference process (the Weiner process in case \[A\], the $d$-dimensional Bessel process in case \[B\]).
\[cor:trans\_dens\] The transition density of the diffusion $(X_t)$ is bounded according to $$\begin{aligned}
\label{eq:trans_dens}
e^{-tL/2} \leq \frac{p_X(t,x,w)}{e^{G(w) - G(x)} p_Y(t,x,w)} \leq e^{-tM/2}.
\end{aligned}$$
The bound is sharp: for a constant drift coefficient $\mu$, equalities hold in .
Recall (see the proof of Proposition \[th:gen\_int\]) we only required $\mu$ to be absolutely continuous, and its value on a set of Lebesgue measure zero is irrelevant. Hence $L$ (respectively $M$) gives an upper (lower) bound for the integrand in $N(t)$ for all paths. Applying Proposition \[th:gen\_int\] with $A = \{X_t \in [w, w+h)\}$, $h>0$, gives $$\begin{aligned}
\inf_{w \leq y \leq w+h} e^{G(y) - G(x)} e^{-tL/2} {\mathbb{P}}_x(Y_t \in [w, &w+h)) \leq {\mathbb{P}}_x(X_t \in [w, w+h))\\
&\leq \sup_{w \leq y \leq w+h} e^{G(y) - G(x)} e^{-tM/2} {\mathbb{P}}_x(Y_t \in [w, w+h)).
\end{aligned}$$ Taking the limits as $h \rightarrow 0$ gives the required result.
In the case of bounded $L$ and $M$ this immediately gives an asymptotic expression for the density $p_X(t,x,w)$ as $t \rightarrow 0$.
If $-\infty < L, M < \infty$, then, as $t \rightarrow 0$, $$p_X(t,x,w) \sim e^{G(w) - G(x)} p_Y(t,x,w),$$ uniformly in $x$, $w$.
While the tightest bounds for the transition distribution are obtained by integrating the bounds for the density given above, this does not, in general, yield a simple closed form expression. We mention other, less tight bounds that are simple and are obtained by a further application of Proposition \[th:gen\_int\].
\[cor:trans\_dist\] The transition distribution function of the diffusion $(X_t)$ admits the following bound: for any $w \in \mathbb{R}$, $$\begin{aligned}
\inf_{ \ell \leq y \leq w} e^{G(y) - G(x)} e^{-tL/2} P_Y(t,x,w) \leq P_X(t,x,w)
\leq \sup_{\ell \leq y \leq w} e^{G(y) - G(x)} e^{-tM/2} P_Y(t,x,w),
\end{aligned}$$ where $\ell$ is the lower bound of the diffusion interval.
The assertion of the corollary immediately follows from that of Proposition \[th:gen\_int\] with $A = \{X_t \leq w \}$.
By considering other events (e.g. $A = \{ X_t > w \}$), other similar bounds can be derived.
**Further Probabilities**
While the focus of this note is on bounds for the transition functions, Proposition \[th:gen\_int\] can be used to obtain other useful results. For example, consider $$\eta_X(t, x, y, w) := \frac{d}{dw} {\mathbb{P}}_x \left( \sup_{0 \leq s \leq t} X_s \geq y, X_t \leq w \right).$$ Such a function has applications in many areas, for example the pricing of barrier options in financial markets. Using ideas similar to the proof of Corollary \[cor:trans\_dens\] immediately gives
\[cor:other\_probs\] For the diffusion $(X_t)$, $$e^{ -tL/2} \leq \frac{\eta_X(t,x,y,w)}{e^{G(w) - G(x)} \eta_Y(t,x,y,w)} \leq e^{ -tM/2}.$$
Note that for such probabilities the bounds may be improved, if desired, by replacing $L$ and $M$ with appropriate constants on a case-by-case basis. For example, if we are considering the probability our diffusion stays between two constant boundaries at the levels $c_1 < c_2$, then the supremum (for $L$) and infimum (for $M$) need only be taken over the range $c_1 \leq y \leq c_2$.
Other probabilities may be considered in a similar way.
Numerical Results {#sec:num_results}
=================
Here we illustrate the precision of the results from the previous section for transition densities. Bounds from Corollary \[cor:trans\_dens\] are compared with known transition density functions and previously available bounds for the Ornstein-Uhlenbeck process in the case \[A\]. For the case \[B\], we only compare the bounds obtained in the current paper with exact results, since there appears to be no other known bounds in the literature. We also construct a ‘truncated Ornstein-Uhlenbeck’ process in order to compare our results with other bounds available in the literature. For the Ornstein-Uhlenbeck process we also consider an example to illustrate Corollary \[cor:other\_probs\].
**The Ornstein-Uhlenbeck Process**
We consider an Ornstein-Uhlenbeck process $(S_t)$, which satisfies the SDE $$dS_t = -S_t dt + dW_t.$$ This process has the transition density $$p_S(t,x,w) = \frac{ e^{t}}{\sqrt{\pi (e^{2 t}-1)}} \exp\left(\frac{\left(w e^{t} - x\right)^2}{1-e^{2 t}} \right),$$ see e.g. (1.0.6) in [@Borodin_etal_xx02], p.522, and we begin by comparing this with the bound obtained in Corollary \[cor:trans\_dens\]. Since $\mu(z) = -z$, we have $$M = -1, \qquad G(w) - G(x) = \frac{1}{2}(x^2 - w^2),$$ giving the estimate $$p_S(t,x,w) \leq e^{\frac{1}{2}(x^2 - w^2 + t)} p_W(t,x,w).$$ Clearly in this case the bound is tighter for smaller values of $|x|$ and $t$. Figure \[fig:OU\_dens\_cent\] displays a plot of the right-hand side of this bound together with the exact density for $x=0$ and $t=1,2$.
To compare our results with other known bounds for transition functions, we look at the bound given by (3.3) in [@Qian_etal_0504] (which, to the best of the author’s knowledge, is the only bound available for such a process). Figure \[fig:OU\_dens\_comp\] compares this bound with that obtained in Corollary \[cor:trans\_dens\] and the exact transition density. The values $x=0$ and $t=1$ are used (for the bound in [@Qian_etal_0504], $q=1.2$ seemed to give the best result, see [@Qian_etal_0504] for further information on notation). Note that [@Qian_etal_0504] gives a sharper bound for $w$ close to zero, but quickly grows to very large values as $|w|$ increases, and in general the bounds presented in this note offer a large improvement. This is typical for all values of $t$, with the effect becoming more pronounced as $t$ decreases. A meaningful lower bound for this process is unavailable by the methods of the present paper, since $L = -\infty$.
For this example, we briefly look at the bound obtained in Corollary \[cor:other\_probs\]. We have, see e.g. (1.1.8) in [@Borodin_etal_xx02], p. 522, $$\eta_S(t,x,0,z) = \frac{1}{\sqrt{\pi (1 - e^{-2 t})}} \exp \left( -\frac{(|z| - x e^{-t})^2}{1 - e^{-2 t}} \right).$$ Figure \[fig:ou\_eta\] compares this as a function of $t \in [0,1]$ with the bound obtained in Corollary \[cor:other\_probs\], $$\begin{aligned}
\eta_S(t,x,0,z) &\leq \exp \left\{\frac{1}{2} (x^2 - z^2 + t) \right\} \eta_W(t,x,0,z)\\
&= \exp \left\{ \frac{1}{2} (x^2 - z^2 + t) \right\} \frac{1}{\sqrt{2 \pi t}} \exp \left\{ - \frac{1}{2t} (|z| - x)^2 \right\},
\end{aligned}$$ where $\eta_W(t,x,0,z)$ is given by (1.1.8) on p. 154 of [@Borodin_etal_xx02].
**The Truncated Ornstein-Uhlenbeck Process**
Other density bounds available in the literature hold only for processes which have bounded drift. For completeness we compare one such bound with the results of this paper. We use the bound in [@Qian_etal_1103], which is the most recent for bounded drift and seems to give the best results over a large domain. To use these results, however, we need a process with bounded drift. As such, we have chosen the ‘truncated Ornstein-Uhlenbeck’ process, which we define as a process $(\overline{S}_t)$ satisfying the SDE $$d\overline{S}_t = \mu(\overline{S}_t) dt + dW_t,$$ where, for a fixed $c > 0$, $$\mu(z) =
\begin{cases}
c, & \qquad z < -c,\\
-z, & \qquad |z| \leq c,\\
-c, & \qquad z > c.
\end{cases}$$ For this process we again have $M=-1$ and, assuming $|x| \leq c$, $$G(w) - G(x) =
\begin{cases}
\frac{1}{2}(c^2 + x^2) + cw, & \qquad w < -c,\\
\frac{1}{2}(x^2 - w^2), & \qquad |w| \leq c,\\
\frac{1}{2}(c^2 + x^2) - cw, & \qquad w > c.
\end{cases}$$ Figure \[fig:trunc\_ou\] displays the bounds from Corollary \[cor:trans\_dens\] together with those in [@Qian_etal_1103] for different values of $c$ with $x=0$ and $t=1$. Smaller values of $c$ move the bounds closer together, however for the given choice of $x$ and $t$ they do not touch until we use the (rather severe) truncation $c \approx 0.45$. In general the method outlined in this note provides a dramatic improvement. We have also plotted an estimate for the transition density using simulation. The simulation was performed using the predictor-corrector method (see e.g. [@Kloeden_etal_xx94] p.198), with $10^5$ simulations and $100$ time-steps.
**A Diffusion on $(0, \infty)$**
Finally we consider a process from the case \[B\]. The author believes this is the first paper to present a bound on transition densities without the linear growth constraint. The process $(V_t)$ satisfying the SDE $$\begin{aligned}
\label{eq:feller_sde}
dV_t = (p V_t + q)dt + \sqrt{2 r V_t} dW_t
\end{aligned}$$ with $p$, $q \in \mathbb{R}$ and $r>0$, has a known transition density (see (26) in [@Giorno_etal_0686]). After applying the transform $Z_t = F(V_t)$, with $F(y) = \sqrt{\frac{2}{r}y}$ by , we obtain the process $$dZ_t = \mu(Z_t) dt + dW_t,$$ where $$\mu(y) = \frac{p}{2} y + \frac{1}{y}\left( \frac{q}{r} - \frac{1}{2} \right).$$ For $q > r$ this dominates the drift of a Bessel process of order $2q/r > 2$ so is clearly a diffusion on $(0, \infty)$.
We take the values $q=2.5$, $r=1$ and $p=1$. Using these values, we have $$M = \inf_{0 \leq y \leq \infty} \left[ \frac{y^2}{4} + 2.5 + \frac{1}{y^2} \left(2 - \frac{(d-1)(d-3)}{4}\right) \right],$$ and $$\begin{aligned}
G(y) - G(x) = \frac{1}{4}(y^2 - x^2) + c \log \left( \frac{y}{x} \right),
\end{aligned}$$ where $d$ is the order of the reference Bessel process and $c = 2 - (d-1)/2$.
It remains to choose the order of the reference Bessel process. It is not clear how to define the ‘best’ order of the reference process for a range of $z$ values, as for fixed $t$ and $x$ the upper bound for $p_Z(t,x,w)$ is minimised for different values of $d$ depending on the value of $z$. In Figure \[fig:rplus\_dens\] we have taken $t=x=0.5$ and used $d=4.7$, however depending on the relevant criterion improvements can be made. Again, a meaningful lower bound for this process is unavailable by the methods of this paper, since $L= -\infty$.
[**Acknowledgements:**]{} This research was supported by the ARC Centre of Excellence for Mathematics and Statistics of Complex Systems. The author is grateful for many useful discussions with K. Borovkov which lead to improvements in the paper.
[^1]: Department of Mathematics and Statistics, University of Melbourne, a.downes@ms.unimelb.edu.au
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abstract: 'Semantics-based knowledge representations such as ontologies are found to be very useful in automatically generating meaningful factual questions. Determining the difficulty-level of these system generated questions is helpful to effectively utilize them in various educational and professional applications. The existing approaches for finding the difficulty-level of factual questions are very simple and are limited to a few basic principles. We propose a new methodology for this problem by considering an educational theory called Item Response Theory (IRT). In the IRT, knowledge proficiency of end users (learners) are considered for assigning difficulty-levels, because of the assumptions that a given question is perceived differently by learners of various proficiencies. We have done a detailed study on the features/factors of a question statement which could possibly determine its difficulty-level for three learner categories (experts, intermediates, and beginners). We formulate ontology-based metrics for the same. We then train three logistic regression models to predict the difficulty-level corresponding to the three learner categories. The output of these models is interpreted using the IRT to find the question’s overall difficulty-level. The performance of the models based on cross-validation is found to be satisfactory and, the predicted difficulty-levels of questions (chosen from four domains) were found to be close to their actual difficulty-levels determined by domain experts. Comparison with the state-of-the-art method shows an improvement of 8.5% in correctly predicating the difficulty-levels of benchmark questions.'
address:
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Department of Computer Science and Engineering, Indian Institute of Technology Madras, Chennai, India\
E-mail: {vinuev,psk}@cse.iitm.ac.in
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title: 'Difficulty-level Modeling of Ontology-based Factual Questions'
---
and
Difficulty-level estimation,Item response theory,Question generation
ß
Introduction
============
A considerable amount of effort has been invested into the creation of a semantics-based knowledge representations such as ontologies where information is formalized into machine-interpretable formats. Among these are SNOMED CT[^2], BioPortal[^3], Disease ontology[^4], to name a few, which capture domain-specific knowledge. Given these knowledge repositories, the opportunity for creating automated systems which utilize the underlying knowledge is enormous. Making use of the semantics of the information, such systems could perform various intelligently challenging operations. For example, a challenging task which often required in an e-Learning system is to generate questions about a given topic which match the end users’ (learners’) educational need and their proficiency level.
The problem of generating question items from ontologies has recently gained much attention in the computer science community [@asma; @ontogen; @mining; @owled; @zito; @swrl]. This is mainly due to the utility of the generated questions in various educational and professional activities, such as learner assessments in e-Learning systems, quality control in human computational tasks and, fraud detection in crowd-sourcing platforms [@Seyler3], to name a few.
Traditionally, question generation (QG) approaches have largely focused on retrieving questions from raw text, databases and other non-semantics based data sources. However, since these sources do not capture the semantics of the domain of discourse, the generated questions cannot be machine-processed, making them less employable in many of the real-world applications. For example, questions that are generated from raw text are suitable only for language learning tasks [@tahaniphd]. Using semantics-based knowledge sources in QG has various advantages, such as (1) in ontologies, we model the semantic relationships between domain entities, which help in generating meaningful and machine-processable questions (2) ontologies enable standard reasoning and querying services over the knowledge, providing a framework for generating questions more easily.
Many efforts in the ontology-based QG are accompanied by methods for automating the task of difficulty-level estimation. In the E-ATG system [@EV2016SWJ], a state-of-the-art QG system, we have proposed an interesting method for predicting difficulty-level of the system generated factual questions. To recall, in that method, we assign a relatively high difficulty score to a question, if the concepts and roles in the question form a rare combination/pattern. For example, considering movie domain, if a question contains the roles: *is based on* and *won oscar*, which rarely appear together, the question is likely to be more difficult than those questions which are formed using a common role combination, say, *is directed by* and *is produced by*. Even though this method can correctly predict the difficulty-levels to a large extent, there are cases where this method fails. This is because there are other factors which influence the difficulty-level of a question.
An early effort to identify factors that could potentially predict the difficulty-level was by Seyler et. al [@Seyler1; @Seyler2]. They have introduced a method to classify a question as *easy* or *hard* by finding the features of the similar question entities in the Linked Open Data (LOD). Feature values for the classification task are obtained based on the connectivity of the question entities in the LOD. We observed that, rather than mapping to LOD – which is not always possible in the case of highly specific domains/domain-entities – incorporating domain knowledge in the form of terminological axioms and following an educational theory called Item Response Theory (IRT), the prediction can be made more accurate.
The contributions of this paper can be listed as follows.
- We reformulate some of the existing factors/features and propose new factors which influence the difficulty-level of a question, by taking into account the learners’ knowledge level (or learners’ category).
- We introduce ontology-based metrics for finding the feature values.
- With the help of standard feature selection methods in machine learning and by using a test dataset, we study the influence of these factors in predicting hardness of a question for three standard learner categories.
- We then propose three learner-specific regression models trained only with the respective influential features and, the output of the models is interpreted using the IRT to find the overall difficulty-level of a question.
This paper is organized as follows. Section 2 contains the preliminaries required for understanding the paper. Section 3 discusses the outline of the proposed method. In Section \[rw\], we give an account of the related works. Section \[factor\] proposes the set of features of a question which determines its difficulty-level. In Section \[ml\], we explain the machine learning methods that we have adopted to develop the Difficulty-level Model (DLM). Further, we discuss the performance of DLM in Section \[regx\]. A comparison with the state-of-the-art method is given in Section \[compx\]. Conclusions and future line of research are detailed at the end.
Preliminaries
=============
We assume the reader to be familiar with Description Logics[@Baader] (DLs). DLs are decidable fragments of first-order logic with the following building blocks: unary predicates (called *concepts*), binary predicates (called *roles*), instances of concepts (called *individuals*) and values in role assertions (called *literals*). A DL ontology is thought of as a body of knowledge describing some domain using a finite set of DL axioms. The concept assertions and role assertions form the assertion component (or ABox) of the ontology. The concept inclusion, concept equality, role hierarchy etc. (the type of axioms depend on the expressivity of the DL) form the terminological component (or TBox) of the ontology.
Question generation using patterns\[patternx\]
----------------------------------------------
For a detailed study of difficulty-level estimation, we use the *pattern-based* method, employed in the E-ATG system, for generating factual questions from the ABox of the given ontologies.
In the pattern-based question generation, a question can be considered as a set of *conditions* that asks for a solution which is explicitly present in the ontology. The set of conditions is formed using different combinations of concepts and roles assertions associated with an individual in the ontology. Example-\[eg1\] is on such question, framed from the following assertions that are associated with the (*key*) individual `birdman`.
`Movie(birdman)`
`isDirectedBy(birdman,alejandro)`
`hasReleaseDate(birdman,Aug 27 2014)`
\[eg1\]-0.5em Name the *Movie* that *is directed by Alejandro* and *has release date [Aug 27, 2014]{}*.
For generating a question of the above type, we may need to use a (generic) SPARQL query template as shown below. The resultant tuples are then associated with a question pattern (E.g., Name the \[?C\], that is \[?R1\] \[?o1\] and \[?R2\] \[?o2\]. (key: ?s)) to frame the questions.
SELECT ?s ?C ? R1 ?o1 ?R2 ?o2 WHERE
{
?s a ?C . ?s ?R1 ?o1 . ?s ?R2 ?o2 .
?R1 a owl:ObjectProperty .
?R2 a owl:DatatypeProperty .
}
In [@EV2016SWJ], the authors have studied all the possible generic question patterns that are useful in generating common factual questions. They have also proposed methods for selecting *domain-relevant* resultant tuple55s/questions for conducting domain related assessments. A resultant tuple of the above query (for example, `?s = birdman, ?C = Movie, ?R1 = isDirectedBy, ?o1 = alejandro, ?R2 = hasReleaseDate, ?o2 = Aug 27 2014`) can be represented in the form of a set of triples (`{(birdman, a, Movie), (birdman, isDirectedBy, alejandro), (birdman, hasReleaseDate, Aug 27 2014)}`). These triples, without the key, give rise to concept expressions that represent the conditions in the question. For example, the concept expression of “`(___, a, Movie)`” is the concept `Movie` itself. Similarly, the concept expression of “`(___, isDirectedBy, alejandro)`” is `\existsisdirectedBy.{alejandro}`. The conditions for the question given in Example-1 are:
- [**Conditions:** ]{}`Movie, \existsisdirectedBy.{alejandro}, ~~~~~~~~~~\existshasReleaseDate.{Aug 27 2014}`
It should be noted that, `\existsdirectedBy.{alejandro}` does not imply that the movie is directed *only* by Alejandro, but it is mandatory that he should be a director of the movie.
For the ease of understanding, all examples presented in this paper are from the Movie domain.
Item Response Theory\[irt\]
---------------------------
Item Response Theory (IRT) [@sage] models relationship between the ability or trait of a person and his responses to the *items* in an experiment. The term *item* denotes an entry, statement or a question used in the experiment. The item response can be *dichotomous* (yes or no; correct or incorrect; true or false) or *polytomous* (more than two options such as rating of a product). The quality measured by the item may be knowledge proficiency, aptitude, belief or even attitude. This theory was first proposed in the field of psychometrics, later, the theory was employed widely in educational research to calibrate and evaluate questions items in the world-wide examinations such as the Scholastic Aptitude Test (SAT) and Graduate Record Examination (GRE) [@IRT2].
In our experiments, we use the simplest IRT model often called *Rasch model* or the *one-parameter logistic model* (1PL) [@IRT]. According to this model, a learner’s response to a question item is determined by her knowledge proficiency level (a.k.a. *trait level*) and the difficulty of the item. 1PL is expressed in terms of the probability that a learner with a particular trait level will correctly answer a question that has a particular difficulty-level. [@sage] represents this model as: $$\label{hdns}
\begin{split}
P(R_{li} = 1|\theta_{l},\alpha_{i})=\frac{e^{(\theta_{l} - \alpha_{i})}}{1 + e^{(\theta_{l} - \alpha_{i})}}
\end{split}$$
In the equation, $R_{li}$ refers to the response ($R$) made by the learner $l$ for the question item $i$ (where $R_{li}=1$ refers to a correct response), $\theta_{l}$ denotes the trait level of the learner $l$, $\alpha_{i}$ represents the difficulty score of item $i$. $\theta_{l}$ and $\alpha_{i}$ values are normalized to be in the range \[-1.5 to 1.5\]. $P(R_{li} = 1 | \theta_{l},\alpha_{i})$ denotes the conditional probability that a learner $l$ will respond to item $i$ correctly. For example, the probability that a below-average trait level (say, $\theta_{l} = -1.4$) learner will correctly answer a question that has a relatively high hardness (say, $\alpha = 1.3$) is: $$\label{egdd}
\begin{split}
P=\frac{e^{(-1.4 - 1.3)}}{1 + e^{(-1.4 - 1.3)}}=\frac{e^{(-2.7)}}{1 + e^{(-2.7)}}=0.063
\nonumber
\end{split}$$ In the paper, we intend to find the $\alpha_{i}$ of the factual questions which are meant for learners, whose trait levels are known to be either high, medium or low. We find the trait levels of the learners by gathering (and normalizing) their grades or marks obtained for a standard test of subject matter conducted in their enrolled institutions. The corresponding $P$ values are obtained by finding the ratio of the number of learners (in the trait level under consideration) who have correctly answered the item, to the total number of learners at that trait level. On getting the values for $\theta_{l}$ and $P$, the value for $\alpha_{i}$ was calculated using the Equation-\[vv\]. $$\label{vv}
\begin{split}
\alpha_{i}=\theta_{l}- log_{e}(\frac{P}{1-P})
\end{split}$$
In the equation, $\alpha_{i} = \theta_{l}$, when $P$ is 0.50. That is, a question’s difficulty is defined as the trait level required for a learner to have 50 percent probability of answering the question item correctly. Therefore, for a trait level of $\theta_{l}=1.5$, if $\alpha_{i}\approx 1.5$, we can consider that the question as having a high difficulty-level. Similarly, for a trait level of $\theta_{l}=0$, if $\alpha_{i}\approx 0$, the question has a medium difficulty-level. In the same sense, for a trait level of $\theta_{l}=-1.5$, if $\alpha_{i}\approx -1.5$, then question has a low difficulty-level.
Outline of the proposed method\[bg\]
====================================
In this paper, based on the insights obtained by the study of the questions that are generated from the ATG[@flairs15] and E-ATG systems, we propose features/factors that can positively or negatively influence the difficulty-level of a question. Albeit there are existing methods which utilize some of these factors for predicting difficulty-level, studying the psychometric aspects of these factors by considering learners’ perspective about the question, has given us further insight into the problem.
As we saw in Section \[irt\], IRT is an item oriented theory which could be used to find the difficulty-level of a question by knowing the question’s hardness (difficult or not difficult) with respect to various learner categories. Therefore, on finding the hardness of a given question based each on learner category, we can effectively use the IRT model for interpreting its overall difficulty-level.
According to IRT, a question is assigned a *high* difficulty-level if it is difficult for an expert learner to answer it correctly. A question is said to be difficult for an expert if the probability of a group of expert learners answering the question correctly is $\le 0.5$. [Similarly, a question can be assigned a *medium* and *low* difficulty-level if the probability with which the question is answered by a group of intermediate learners is $\le 0.5$ and a group of beginner level learners is $\le 0.5,$ respectively.]{} Table \[dla\] shows the difficulty-level assignment of three questions: $Q_{1}, Q_{2}$ and $Q_{3},$ based on whether they are difficult (denoted as $d$) or not difficult (represented as $nd$) for three learner categories.
--------- -------- ----------- ---------- ------------
Qn. Expert Intermed. Beginner Difficulty
-level
$Q_{1}$ $d$ $d$ $d$ $high$
$Q_{2}$ $nd$ $d$ $d$ $medium$
$Q_{3}$ $nd$ $nd$ $d$ $low$
--------- -------- ----------- ---------- ------------
: Assigning one of the three difficulty-levels: *high, medium* and *low*, by considering whether the question is difficult ($d$) or not-difficult ($nd$) for three learner categories.\[dla\]
![ Block diagram of the proposed model for predicting a question’s difficulty-level\[classifiers\]](idm.png){width="42.00000%"}
We consider three standard categories of learners: *beginners, intermediates* and *experts*, and model three classifiers for predicting the difficulty corresponding to the three learner categories, as shown in Fig. \[classifiers\]. Since the hardness ($d/nd$) corresponding to the three categories of learners should be predicted first from the feature values, machine learning models/classifiers which can learn from available training data is an obvious choice. We consider only those factors which are influential for a given learner category for training the models. The output of the three classifiers is matched with the content of Table \[dla\] to find the question’s overall difficulty-level.
Related Work: Difficulty-level Estimation\[rw\]
===============================================
A simple notion to find the difficulty-level of an ontology-generated multiple choice questions (MCQs) was first introduced by Cubric and Tosic[@Cubric2010]. Later, in [@simieee], Alsubait et al. extended the idea and proposed a similarity-based theory for controlling the difficulty of ontology-generated MCQs. In [@mining], they have applied the theory on analogy type MCQs. In [@lesson], the authors have experimentally verified their approach in a student-course setup. The practical solution which they have suggested to find out the difficulty-level of an MCQ is with respect to the degree of similarity of the distractors to the key. If the distractors are very similar to the key, students may find it very difficult to answer the question, and hence it can be concluded that the MCQ is difficult.
In many a case, the question statement in an MCQ is also a deciding factor for the difficulty of an MCQ. For instance, the predicate combination or the concepts used in a question can be chosen such that they can make the MCQ difficult or easy to answer. This is the reason why in this paper we focus on finding difficulty-level of questions having no choices (i.e., non-MCQs). An initial investigation of this aspect was done in [@EV2016SWJ]. Concurrently, there was another relevant work by Seyler et. al[@Seyler1; @Seyler2], focusing on QG from knowledge graphs (KGs) such as DBpedia. For judging the difficulty-level of such questions, they have designed a classifier trained on Jeopardy! data. The classifier features were based on statistics computed from the KGs (Linked Open Data) and Wikipedia. However, they have not considered the learner’s knowledge level, as followed in the IRT, while formulating the feature metrics. This makes their measures less employable in sensitive applications such as in an e-Learning system. While considering ontology-based questions, one of the main limitation of their approach is that the feature values were determined based on the connectivity of question entities in the KG, whereas in the context of DL ontologies, the terminological axioms can be also incorporated to derive more meaningful feature metrics. In addition, the influence of the proposed factors in determining the difficulty using feature selection methods was not studied.
Proposed Factors to determine Difficulty-level of Questions\[factor\]
======================================================================
In this section, we look at a set of factors which can possibly influence the difficulty-level of a question and propose ontology-based metrics to calculate them. The intuitions for choosing those factors are also detailed.
To recall, a given question can be thought of as a set of conditions. For example, consider the following questions (where the underlined portions denote the equivalent ontology concepts/roles used).
- [**Qn-1:** ]{}*Name the that was Clint Eastwood*.
- [**Qn-2:** ]{}*Name the that was Clint Eastwood*.
The equivalent set of conditions of the two questions can be written as:
- [**Conditions in Qn-1:** ]{}`Movie,`\
`~~~~~~~~~~~~~~~~~\existsdirectedBy.{clint_eastwood}`
- [**Conditions in Qn-2:** ]{}`Oscar_movie,`\
`~~~~~~~~~~~~~~~~~\existsdirectedBy.{clint_eastwood}`
Popularity
----------
Popularity is considered as a factor because of the intuition that the greater the popularity of the entities that form the question, more likely that a learner answers the question correctly. (We observe that this notion is applicable for learners of all categories.) Therefore, the question becomes easier to answer if the popularity of the concepts and roles that are present in the question is high. For example, out of the following two questions, Qn-3 is likely to be easy to answer than Qn-4, since `Oscar_movie` is a popular concept than `Thriller_movie`.
- [**Qn-3:** ]{}*Name an* .
- [**Qn-4:** ]{}*Name a* .
Our approach for measuring popularity is based on the observation that, (similar to what we see in Wikipedia data) if more articles talk about a certain entity, the more important, or popular, this entity is. In Wikipedia, when an article mentions a related entity, it is usually denoted by a link to the corresponding Wikipedia page. These links form a graph which is exploited for measuring the importance of an entity within Wikipedia. Keeping this in mind, we can define the popularity of an entity (individual) in an ontology as the number of object properties which are linked to it from other individuals. For obtaining a measure in the interval \[0,1\], we divide the number of in-links by the total amount of individuals in the ontology.
To find the popularity of a concept $C$ in ontology $\mathcal{O},$ we find the mean of the popularities of all the individuals which satisfy $C$ in $\mathcal{O}$. If the condition in a question is a role restriction, then the [concept expression]{} of it will be considered, and popularity is calculated. The overall popularity of the question is determined by taking the mean of the popularities of all the concepts and role restrictions present in it.
Selectivity\[sel\]
------------------
Selectivity of the conditions in a question helps in measuring the quality of the hints that are present in it [@Seyler1]. Given a condition, selectivity refers to the number of individuals that satisfy it. When the selectivity is high, a question tends to be easy to answer. For example, among the following questions, clearly, Qn-5 is easier to answer than Qn-6. This is because finding an actor who has acted in at least a movie is easy to answer than finding an actor who has acted in a particular movie; finding the latter requires more specific knowledge.
- [**Qn-5:** ]{}*Name an [actor]{} who acted in a movie*.
- [**Qn-6:** ]{}*Name an actor who acted in [Argo]{}*.
To formalize such a notion, we can look at the *answer space* corresponding to each of the conditions in the questions. Answer space simply denotes the *count of individuals* satisfying a given condition. We will represent answer space of a condition $c$ as $ASpace(c).$
The conditions in the above questions are:
- [**Conditions in Qn-5:** ]{}`Actor, \existsactedIn.Movie`
- [**Conditions in Qn-6:** ]{}`Actor, \existsactedIn.{argo}`
Since [$ASpace($`\existsactedIn.{argo}`)]{} is very much lesser than [$ASpace($`\existsactedIn.Movie`),]{} we can say that Qn-6 is difficult to answer than Qn-5. (Actors who acted only in dramas are not possible answers to Qn-5.)
As a question can have more than one conditions present in it, answer spaces of all the condition have to be taken into account while calculating the overall difficulty score of the question. It is debatable that including a specific condition in the question can always make the question difficult to answer – sometimes a specific condition can give a better hint to a (proficient) learner.
For example, the following question is more difficult to answer than Qn-5 and Qn-6 for a non-expert, since [$ASpace$(`American_actor`) $<<$ $ASpace$(`Actor`)]{}. -2 em
- [**Qn-7:** ]{}*Name an American actor who acted in [Argo]{}*.
However, for an expert, given that the actor is an American is an additional hint, making the question sometimes easier than Qn-5 and 6. Therefore, we can roughly assume the relation between difficulty-level and answer space as follows, where $D_{expert}$ and $D_{beginner}$ correspond to the difficulty-level for an expert learner and difficulty-level for a beginner respectively. We will closely look at these relations in the following subsections. -2 em $$\label{DRequ1}
D_{expert}\propto {ASpace}
\nonumber$$-2 em $$\label{DRequ2}
D_{beginner}\propto \frac{1}{ASpace}
\nonumber$$
When a question contains multiple conditions, we do an aggregation of their normalized (or relative) answer spaces (denoted as $RASpace$) to find the overall answer space (addressed as $ASpaceOverall$) of the question. We find the $RASpace$ of a concept by dividing the count of individuals satisfying the concept by the total count of individuals in the apex concept (Thing class) of the ontology. For instance, $RASpace$(`{argo}`) = $ASpace$(`{argo}`)/$ASpace$(`owl:Thing`). Similarly, if the condition is a role related restriction, corresponding domain concept of the role will be used to find the relative answer space. For [`\existsactedIn.{argo}`, $RASpace$]{} is calculated as: [$ASpace($`\existsactedIn.{argo}`)/$ASpace($Domain(`actedIn`))]{}. The overall answer space can be found by taking the average of all the relative answer spaces of the conditions in the question, where $C_{S}=\{t_{1}, t_{2},..., t_{n}\}$ is the set of conditions in the question $S$, and $|C_{S}|=n.$-.7 em $$\label{Dxyz}\small
ASpaceOverall(C_{S}) = \frac{\sum_{i=1}^{n} RASpace(t_{i})}{n}$$-.3 em
In the following paragraphs, we discuss how the selectivity feature would affect the difficulty-level of an item. We discuss the cases of expert, intermediate and beginner learners separately. In the process, we define two selectivity based features and specify how to compute them using the knowledge base and the domain ontology.
#### **Expert learner**
![ Relation between selectivity and answer space for experts\[test1\]](sel1.png){width=".55\textwidth"}
An expert learner is assumed to have a well developed structured knowledge about the domain of discourse. She is supposed to clearly distinguish the terminologies of the domain and is capable of doing reasoning over them. Therefore, in general, selectivity can be assumed to be directly proportional to the difficulty-level; that is, when the *ASpaceOverall* increases, the underlying hints becomes poor and the question is likely to become difficult for her. However, intuitively, below and beyond particular *ASpaceOverall* values, a question’s difficulty does not necessarily follow this proportionality. As pointed out in [@EV2016SWJ; @flairs15] when a question pattern becomes rare, it becomes difficult to answer the question correctly. Therefore, in Fig. \[test1\], towards the left of the point A, the question tends to become difficult, since the answer space becomes too small. Similarly, towards the right of the point B, the question tends to become more generic and its difficulty diminishes. To accurately predict whether a question is difficult or not, it is necessary to statistically determine the positions of the points A and B. Based on the initial analysis of the empirical data obtained from [@EV2016SWJ], we processed with an assumption that the question tends to become too generic when the $ASpaceOverall$ $\ge 50\%$ of the total number of individuals in the ontology. Similarly, the question starts to become difficult when the $ASpaceOverall$ $\le 10\%$ of the total number of individuals. The selectivity corresponding to an expert is expressed as *Selectivity$_{Ex}$*. Knowing the overall answer space of a question, selectivity is computed directly from the graph in Fig. \[test1\] – in the graph, Max, A(10%) and B(50%) are known points.
#### **Beginner learner\[DMB\]**
A beginner is assumed to have a less developed internal knowledge structure. She can be assumed to be familiar with the generic (sometimes popular) information about the domain and is less aware about the detailed specifics.
![ Relation between selectivity and answer space for beginners\[test2\]](sel2.png){width=".38\textwidth"}
We assume that the *selectivity* factor behaves proportionally to the *ASpaceOverall*, unlike what we saw in the experts’ case. The intuition behind this assumption is that, when the overall answer space increases, as in the case of an expert the so-called hints in the question cannot be expected to become poor; this is because, a person with poorly developed domain knowledge may not be able to differentiate the quality or property of the hint, making it rather a factor for generalizing the question (thereby making the question easily answerable). Therefore, we can follow a linear proportionality relation as shown in Fig. \[test2\], to find the difficulty for a beginner, and we can denote this new selectivity as *Selectivity*$_{Bg}$.
#### **Intermediate learner\[DMI\]**
An intermediate learner can be assumed to have partially both the perspective of an expect as well that of a beginner. Therefore, we can assume her selectivity value as combination of *Selectivity*$_{Ex}$ and *Selectivity*$_{Bg}$ – considering them as two factors.
Coherence
---------
In the current context, coherence captures the semantic relatedness of entities (individuals and concepts) in a question. It can be best compared to measuring the co-occurrences of individuals and concepts in the text. While considering coherence as a factor, we assume that higher the coherence between individuals/concepts in a question, lower is its difficulty-level and vice versa, because intuitively, the facts about highly coherent entities are likely to be recalled easier than the facts about less coherent entities. It is observed that this notion is applicable for learners of all categories.
- [**Qn-8:** ]{}*Name the [hollywood-movie]{} [starring]{} Anil Kapoor and Tom Cruise.*
- [**Qn-9:** ]{}*Name the hollywood-movie starring Tom Cruise and Tim Robbins*.
Considering the above two questions, coherence between the concept `HollywoodMovie` and the individuals: `anil_kapoor, tom_cruise`, is lesser (since there is only one movie they both have acted together) than the coherence between `HollywoodMovie, tom_cruise` and `tim_robbins`, making the former question difficult to answer than the latter.
Given an ontology, we measure the coherence between two of its individuals as the sum of the ratio between the size of the set of entities that point to both individuals and the size of the union of the sets of entities that point to either one of the individuals, and the ratio between the size of the set of entities that are pointed by both individuals and the size of the union of the sets of entities that are pointed by either one of the individuals. Formally, the coherence between two individuals $p$ and $q$ can be represented as in Eq. \[dd1\], where $I_{i}$ is the set of entities from which the individual $i$ is having incoming relations and $O_{i}$ is the set of entities to which $i$ is having outgoing relations. $$\label{dd1}
Coherence(p,q) = \frac{|I_{p}\cap I_{q}|}{|I_{p}\cup I_{q}|}+\frac{|O_{p}\cap O_{q}|}{|O_{p}\cup O_{q}|}$$ Each portion of the measure is known as the Jaccard similarity coefficient, which is a statistical method to compare the similarity of sets.
Specificity
-----------
Specificity refers to how specific a question is. For example, among the following questions, Qn-2 is more specific question than Qn-10 and requires more knowledge proficiency to answer it correctly. We consider Qn-2 as more difficult to answer than Qn-10.
- [**Qn-2:** ]{}*Name an that was Clint Eastwood*.
- [**Qn-10:** ]{}*Name the that Clint Eastwood*.
For a learner, the difficulty-level depends on how detailed the question is. Intuitively, if a question contains domain specific conditions, the probability of a learner for correctly answering the question will reduce. (This notion is observed to be applicable for all categories of learners.) To capture this notion, we utilize the concept and role hierarchies in the domain ontology. We relate the depths of the concepts and roles that are used in the question to the concept and role hierarchies of the ontology, to determine the question difficulty. To achieve this, we introduce *depthRatio* for each predicate $p$ in an ontology. *depthRatio* is defined as: $$\label{DRequ}\small
\emph{depthRatio}_\mathcal{O}(p)= \frac{
\splitfrac{
\text{~~~~~~Depth (or length) of $p$}
}{\text{from the root of the hierarchy~~}}
}{
\splitfrac{\text{~~~Maximum length of }}
{ \text{the path containing $p$~~~~~~}}
}
$$
For a question $S$, generated from an ontology $\mathcal{O}$, with $x$ as key and $P$ as the set of concepts/roles in $S$, let $\mathcal{C}$ denote the set of concepts satisfied by $x$, and let $\mathcal{R}$ represents the set of roles such that either $x$ is present at their domain (subject) or range (object) position (i.e., $R \in \mathcal{R} \implies \mathcal{O}\models R(x,i) \lor R(i,x)$, where $i$ is an arbitrary instance in $\mathcal{O}$). For each $p\in P$, we find the largest subset in $\mathcal{C}$ (if $p$ is a concept) or we find the largest subset in $\mathcal{R}$ (if $p$ is a role), such that the elements in the subset can be related using the relation $\sqsubseteq$, and $p$ is an element in that subset. The cardinality of such a subset forms the denominator of Eq. \[DRequ\], and the numerator is the position of the predicate $p$ from the right (right represents the top concept or top role) when the elements in the subset are arranged using the relation $\sqsubseteq$.
A stem can have more than one predicate present in it. In that case, we assume that the predicate with a highest depthRatio (associated with the reference individual) could potentially make the stem more specific. Therefore, we define the overall depthRatio of a stem (called the *specificity*) as the product of the average depthRatio with the maximum of all the depthRatios.
Difficulty-level Modeling of Questions\[ml\]
=============================================
In the previous section, we have proposed a set of features which possibly influence the difficulty-level of a question. In this section, we do a feature selection study using three widely used filter models to find out the amount of influence of the proposed factors in predicting question difficulty. We then train three logistic regression models ($RM_{e}, RM_{i}, RM_{b}$) for each learner category (experts, intermediates and beginners, respectively) using the selected prominent features. Their predictions for a given question are taken to find the overall difficulty-level. Ten-fold cross validation is used to find the performance of the three models.
#### **Training data**
The training data consisted of a set of 520 questions that were generated from four ontologies (DSA, MAHA, GEO and PD ontologies – see our project website[^5] for details) available online. These questions were classified as *difficult* or *not-difficult* for each of the three learner categories (we denote the training data for experts, intermediate and beginners respectively as $TD_{e}, TD_{i}$ and $TD_{b}$). The classification is done by either of the two ways,
Item identifier: dsa_1
Popularity: 0.231
Selectivity_Ex: 0.320
Selectivity_Bg: 0.113
Coherence: 0.520
Specificity: 0.440
Difficulty: d
\(1) in a classroom setting by using IRT or (2) with the help of subject matter experts. In the former case, we find the probability by which a particular question is answered correctly by a learner of specific knowledge proficiency level and assign it as difficulty ($d$) or not ($nd$). In the latter case, more than 5 domain experts were asked to do the ratings and their majority ratings were considered for assigning $d$ or $nd$. All the question that were used for training had been previously used as benchmark sets in [@Vinu2015; @EV2016SWJ; @flairs15]. In the training data, the question identifiers are accompanied by five feature values tabulated from the respective ontologies along with their difficulty assignment. The feature values are normalized to values between 0 and 1. An instance of the training data is given in Fig. \[data\].
#### **Feature Selection**
In order to find out the amount of influence of each of the proposed factors, we did an attribute evaluation study using three popular feature selection approaches: [Information Gain[@IGFR] (IG), ReliefF[@RFilter] (RF) and Correlation-based[@CBFS] (CB)]{} methods. These feature selection approaches select a subset of features that minimize redundancy and maximize relevance to the target such as the class labels in classification. The ranking scores/weights obtained for the features are given in Table \[rank\].
[@lXXX@[ ]{}XXX@[ ]{}XXX@]{}& & &\
&$TD_{e}$&$TD_{i}$&$TD_{b}$&$TD_{e}$&$TD_{i}$&$TD_{b}$&$TD_{e}$&$TD_{i}$&$TD_{b}$\
Popularity & 0.7613 & 0.6344 & 0.7925 & 0.831 & 0.378 & 0.178 & 0.766 & 0.621 & 0.562\
Selectivity$_{Ex}$ & 0.8802 & [0.6913]{} & & 0.738 & & & 0.699 & &\
Selectivity$_{Bg}$ & & 0.6553 & 0.9996 & & [0.668]{} & 0.177 & & 0.442 & 0.249\
Coherence & 0.5638 & 0.3251 & 0.8112 & 0.761 & 0.487 & 0.211 & 0.731 & 0.538 & 0.315\
Specificity & 0.6577 & 0.5436 & 0.5751 & 0.651 & 0.521 & 0.459 & 0.657 & 0.761 & 0.602\
In Table \[rank\], we can see that, the least prominent feature for finding the difficulty for experts is the Selectivity$_{Bg}$, since all the three filter models ranked it as the least influential one – see the fields shaded in blue in the three $TD_{e}$ columns. In the case of predicting difficulty for intermediates, the ranking scores of Selectivity$_{Ex}$ is less than that of Selectivity$_{Bg}$ when the models used are RF and CB – see the fields shaded in red. When it comes to beginner learners, the factor Selectivity$_{Ex}$ is found to have the least influence – see the fields shaded in gray. While developing the IDM, we have ignored the least influential features for training the regression models.
#### **Observations**
Consistent to what we have postulated in Section \[sel\], Selectivity$_{Ex}$ is found to be a more influential factor than Selectivity$_{Bg}$, for deciding the difficulty of a question for an expert learner. Similarly, for a beginner, Selectivity$_{Bg}$ is found to be more influential than Selectivity$_{Ex}$.
Performance of regression models\[regx\]
----------------------------------------
The performances of three learner-specific regression models: $RM_{e}, RM_{i}, RM_{b}$, considering all the 5 features are 76.73%, 78.6% and 84.23% respectively. These percentage values indicate the ratio of the number of instances classified correctly to the total number of instances given for classification, under 10-fold cross-validation setting. After removing the least influential features, the performance of the classifiers became 76.9%, 79.8%, and 85% respectively. The difference in the performance before and after feature selection is roughly the same because the model can theoretically assign minimum or zero weight to non-influential features. However, we did the feature selection and ranking to evaluate our hypothesis about what features are influential in which case.
When the overall system was run on the available dataset of factual questions from different domains (520 questions), it is observed that DLM correctly classifies about 77% of them. These questions are relevant to the domain selected using the heuristics given in [@EV2016SWJ].
Non-classifiable Questions
--------------------------
Following from what we have seen in Section \[bg\], the DLM could not assign a difficulty-level to a given question if the outcomes of the three regression models
![Classifiable Vs Non-classifiable questions\[PNP\]](chart3.png){width=".5\textwidth"}
do not agree with the three possible assignments (see Fig. \[dla\]). We call such questions as *non-classifiable* ones and the others as *classifiable* questions. We investigated the percentage of such non-classifiable cases by analyzing questions generated from five ontologies available online (available in our project website). We used questions that were generated in [@flairs15] for our study, the statistics of the non-classifiable cases are given in Fig. \[PNP\] (Note that the X-axis begins with 50%). On an average, 10% of all the questions generated from an ontology are found to be non-classifiable.
An analysis of the non-classifiable questions taken from the DSA ontology shows that incompleteness of the ontology and the way the domain is modeled as an ontology are the main reasons for this discrepancy. For example, *Name a doubly linked list* is a question which is assigned to be difficult for an expert (since the Doubly Linked List concept has only one individual and the incompleteness of data makes it a less popular concept), and not difficult for a beginner (the depthRatio is high for the concept `Doubly_Linked_List` because, it forms a specific concept in the ontology. However, this issue would not have appeared, if the concept `Doubly_Linked_List` had been modeled as an individual.
Comparison with existing method\[compx\]
========================================
In this section, we compare the predictions of difficult-levels by the proposed model and the method given in [@EV2016SWJ]. We call the latter as *E-ATG method*. We do not report a comparison with the model proposed in [@Seyler1; @Seyler2] because their difficulty-level model is not a domain ontology-based model and prediction is possible only if the question components can be mapped to Linked Open Data entities. In addition, they could predict the question difficulty either as *easy* or *hard*, whereas our model classifies the question into three standard difficulty-levels: *high, medium* and *low*.
In [@EV2016SWJ], effectiveness of E-ATG method is established by comparing the predicated difficulty-levels with their actual difficulty-levels determined in a classroom setting. DSA ontology was used for the study. Twenty four representative questions (given in Appendix A), selected from 128213 generated questions, were utilized for the comparison. (More details about the selection process can be found at [@EV2016SWJ]). A correlation of 67% between the predicted and actual difficulty-levels was observed[^6]. We tested the approach proposed in this paper on the above set of questions.
While comparing the proposed difficulty-levels with the actual difficult-levels, we found that 21 out of 24 are matching (87.5% correlation), and one benchmark question is identified as non-classifiable.
Discussion
----------
The E-ATG method mainly considered only one feature, the *triviality score* (which denotes how rare the property combination in the stem are), for doing the predication. Our results (8.5% improvement) show that the proposed set of new features could improve the correctness of the prediction. The current model is trained only using 520 training samples. We expect the system to perform even better after training with more data as and when they are available, and by identifying other implicit features. Due to unavailability of large training data, unsupervised feature learning methods cannot be effectively applied in this context.
Conclusions and Future Work
===========================
Establishing mechanisms to control and predict the difficulty of assessment questions is clearly a big gap in existing question generation literature. Our contributions have covered the deeper aspects of the problem, and proposed strategies, that exploit ontologies and associated measures, to provide a better difficulty-level predicting model, that can address this gap. We developed the difficulty-level model (DLM) by introducing three learner-specific logistic regression models for predicting the difficulty of a given question for three categories of learners. The output of these three models was then interpreted using the Item Response Theory to assign *high, medium* or *low* difficulty-level. The overall performance of the DLM and the individual performance of the three regression models based on cross-validation were reported and they are found to be satisfactory. Comparison with the state-of-the-art method shows an improvement of 8.5% in correctly predicating the difficulty-levels of benchmark questions.
The model proposed in this paper for predicting the difficulty-level of questions is limited to ABox-based factual questions. It would be interesting to extend this model to questions that are generated using the TBox-based approaches. However, the challenges to be addressed would be much more, since, in the TBox-based methods, we have to deal with many complex restriction types (unlike in the case of ABox-based methods) and their influence on the difficulty-level of the question framed out of them needs a detailed investigation.
For establishing the propositions and techniques stated in this paper, we have implemented a system which demonstrates the feasibility of the methods on medium sized ontologies. It would be interesting to investigate the working of the system on large ontologies.
Acknowledgements {#acknowledgements .unnumbered}
================
This project is funded by Ministry of Human Resource Development, Gov. of India. We express our fullest gratitude to the participants of our evaluation process: [Dr. S.Gnanasambadan]{} (Director of Plant Protection, Quarantine $\&$ Storage), Ministry of Agriculture, Gov. of India; [Mr. J. Delince]{} and [Mr. J. M. Samraj]{}, Department of Social Sciences AC $\&$ RI, Killikulam, Tamil Nadu, India; [Ms. Deepthi.S]{} (Deputy Manager), Vegetable and Fruit Promotion Council Keralam (VFPCK), Kerala, India; [Dr. K.Sreekumar]{} (Professor) and students, College of Agriculture, Vellayani, Trivandrum, Kerala, India. We also thank all the undergraduate and post-graduate students of Indian Institute of Technology, Madras, who have participated in the empirical study.
Appendix A {#app:A .unnumbered}
==========
Tables \[egmcqs1\]-\[egmcqs3\] contain benchmark stems that are generated from the Data Structures and Algorithms (DSA) ontology. In those tables, the stems 1 to 6, 7 to 16 and 17 to 24 correspond to high, medium and low (actual) difficulty-levels respectively. These stems were employed in the experiment mentioned in Section \[compx\]. Stems- 7, 8 and 20 are the uncorrelated ones. Stem-20 is identified as non-classifiable questions by our approach.\
\
Item No. Stems of MCQs
---------- -----------------------------------------------------------------------------------------------------------------------
1 Name a polynomial time problem with application in computing canonical form of the difference between bound matrices.
2. Name an NP-complete problem with application in pattern matching and is related to frequent subtree mining problem.
3. Name an all pair shortest path algorithm that is faster than Floyd-Warshall Algorithm.
4. Name an application of an NP-complete problem that is also known as Rucksack problem.
5. Name a string matching algorithm that is faster than Robin-Karp algorithm.
6. Name a polynomial time problem that is also known as maximum capacity path problem.
: []{data-label="egmcqs1"}
Item No. Stems of MCQs
---------- ----------------------------------------------------------------------------------------------------
7. Name an NP-hard problem with application in logistics.
8. Name the one whose worst time complexity is n exp 2 and with Avg time complexity n exp 2.
9. Name the one which operates on output restricted dequeue and operates on input restricted dequeue.
10. Name the operation of a queue that operates on a priority queue.
11. Name a queue operation that operates on double ended queue and operates on a circular queue.
12. Name the ADT that has handling process “LIFO”.
13. Name an internal sorting algorithm with worse time complexity m plus n.
14. Name a minimum spanning tree algorithm with design technique greedy method.
15. Name an internal sorting algorithm with time complexity n log n.
16. Name an Internal Sorting Algorithm with worse time complexity n exp 2.
: Questions generated from the DSA ontology that are having *medium* actual difficulty-levels.\[egmcqs2\]
Item No. Stems of MCQs
---------- -----------------------------------------
17. Name a file operation.
18. Name a heap operation.
19. Name a tree search algorithm.
20. Name a queue with operation dequeue.
21. Name a stack operation.
22. Name a single shortest path algorithm.
23. Name a matrix multiplication algorithm.
24. Name an external sorting algorithm.
: Questions generated from the DSA ontology that are having *low* actual difficulty-levels.\[egmcqs3\]
[^1]: Corresponding author. E-mail: vinuev@cse.iitm.ac.in, mvsquare1729@gmail.com
[^2]: http://www.snomed.org/
[^3]: http://bioportal.bioontology.org/
[^4]: http://www.berkeleybop.org/ontologies/doid.owl
[^5]: Project website: https://sites.google.com/site/ontoassess/
[^6]: To get more accurate result, the calculations were redone with $\theta$ values between: \[-1.5, 1.5\], and 1.25, 0 and -1.25 as the medians of the $\alpha$ values for experts, beginners and intermediates, respectively with $\pm$.25 standard deviation
|
---
abstract: 'Radiative corrections to parity violating deep inelastic electron scattering (PVDIS) are reviewed including a discussion of the renormalization group evolution (RGE) of the weak mixing angle. Recently obtained results on hypothetical $Z^\prime$ bosons — for which parity violating observables play an important rôle — are also presented.'
author:
- Jens Erler
date: 'Received: date / Accepted: date'
title: 'Radiative Corrections and $Z^\prime$'
---
[example.eps]{} gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore
Effective electroweak interactions {#intro}
==================================
The first two terms of the Lagrangian, ${\cal L} = {\cal L}_{\rm fermion} + {\cal L}_{\rm Yukawa} + {\cal L}_{\rm gauge} + {\cal L}_{\rm Higgs}$, of the electroweak Standard Model (SM) contain the free fermionic part and the interactions, $${\cal L}_A + {\cal L}_{\rm W} + {\cal L}_{\rm Z} = - {g\over 2} \left(2 \sin^2\theta_W J^\mu_A A_\mu +
J^\mu_W W^-_\mu + J^{\mu\dagger}_W W^+_\mu + {1\over \cos^2\theta_W} J^\mu_Z Z_\mu \right),$$ in terms of the electromagnetic current, $J^\mu_A = \sum\limits_{i = 1}^3
\left( {2\over 3} \bar{u}^i \gamma^\mu u^i - {1\over 3} \bar{d}^i \gamma^\mu d^i - \bar{e}^i \gamma^\mu e^i \right)$,\
the weak charged current (CC), $J^{\mu \dagger}_W = \sqrt{2} \sum\limits_{i = 1}^3
\left( \bar{u}^{i0} \gamma^\mu P_L d^{i0} + \bar{\nu}^{i0} \gamma^\mu P_L e^{i0} \right)$,\
and the weak neutral current (NC), $J^\mu_Z \equiv \sum\limits_{i=1}^{N_\psi} \bar\psi^i \gamma^\mu [g_V^i - g_A^i \gamma^5] \psi^i =
- 2 \sin^2\theta_W J^\mu_A + \sum\limits_{i = 1}^3 \left( \bar{u}^i \gamma^\mu P_L u^i - \bar{d}^i \gamma^\mu P_L d^i +
\bar{\nu}^i \gamma^\mu P_L\nu^i - \bar{e}^i \gamma^\mu P_L e^i \right)$, where $P_L \equiv {1 - \gamma^5\over 2}$. At the tree-level, the NC couplings, $g_V^i = {1\over 2} \tau_3^{ii} - 2 Q^i \sin^2\theta_W$ and $g_A^i = {1\over 2} \tau_3^{ii}$, with $Q^i$ ($\tau_3$) denoting the electric charge (third Pauli matrix), give rise to the effective 4-Fermi Hamiltonian, $${\cal H}_{\rm eff} = {1\over 2} \left( {g\over 2 \cos^2\theta_W M_Z} \right)^2 J^\mu_Z J_{\mu Z} =
{G_F\over \sqrt{2}} J^\mu_Z J_{\mu Z} =
{G_F\over \sqrt{2}} \sum\limits_{MNij} h_{MN}^{ij} \bar{\psi}^i \Gamma^M \psi^i \bar{\psi}^j \Gamma^N \psi^j,$$ where $\Gamma^V = \gamma^\mu$, $\Gamma^A = \gamma^\mu \gamma^5$, and $h_{MN}^{ij} = g_M^i g_N^j$. Unfortunately, there is no generally accepted notation, normalization, and sign convention for the $h_{MN}^{ij}$ in the literature. For parity violating $eq$ interactions one defines $C_{1q} \equiv 2 h_{AV}^{eq}$ and $C_{2q} \equiv 2 h_{VA}^{eq}$. Parity violation in heavy atoms [@Budker] is basically driven by the $C_{1q}$, while PVDIS [@Souder] determines approximately the combination, $\omega_{\rm PVDIS} \equiv(2\ C_{1u} - C_{1d})+0.84\ (2\ C_{2u} - C_{2d})$.
Radiative corrections
=====================
Including one-loop electroweak radiative corrections one obtains the expressions [@Marciano:1982mm], $$2\ C_{1u} - C_{1d} = - {3\over 2} \left[ \rho_{\rm NC} - {\alpha\over 2\pi} \right] \left[ 1 - {20\over 9}
\left( \sin^2\hat\theta_W(0) -{ 2\alpha\over 9\pi} \right) \right] + \Box_{WW} + \Box_{ZZ} + \Box_{\gamma Z}$$ $$\label{eq:c1}
+ {5\hat\alpha\over9\pi} [1 - 4 \sin^2\hat\theta_W(M_Z)] \left[ \ln {M_Z\over m_e} + {1\over 12} \right],$$ $$2\ C_{2u} - C_{2d} = - {3\over 2} \left[ \rho_{\rm NC} - {\alpha\over 6\pi} \right] \left[ 1 - 4
\left( \sin^2\hat\theta_W(0) - {2\alpha\over 9\pi} \right) \right] +\Box_{WW} + \Box_{ZZ} + \Box_{\gamma Z}$$ $$\label{eq:c2}
+ {5\hat\alpha\over 9\pi} [1 - {12\over 5} \sin^2\hat\theta_W(M_Z)] \left[ \ln {M_Z\over m_q} + {1\over 12} \right]
- {8\hat\alpha\over 9\pi} \left[ \ln {M_W\over m_q} + {1\over 12} \right],$$ where $ \rho_{\rm NC} \approx 1.0007$ collects various propagator and vertex corrections relative to $\mu$-decay, and the second lines are from the $e$ and $q$ charge radii. With $\hat{s}^2 \equiv \sin^2\hat\theta_W(M_Z)$, $$\Box_{WW} = - {9 \hat\alpha\over 8\pi \hat{s}^2} \left[ 1 - {\hat\alpha_s(M_W)\over 3\pi} \right],
\Box_{\gamma Z} = - {3 \hat\alpha\over 4\pi} [ 1 - 4\hat{s}^2] \left[ \ln {M_Z\over M_\rho} + {3\over 4} \right] ,
\Box_{ZZ} \ll \Box_{WW}$$ are the box contributions except that for $2\ C_{2u} - C_{2d}$ the $\alpha_s$ correction to the $WW$-box [@Erler:2003yk], $\Box_{WW}$, is not yet known and $\Box_{\gamma Z}$ is obtained from above by replacing $4 \hat{s}^2$ by $28 \hat{s}^2/9$ and the constant 3/4 by 5/12. The numerical results are summarized in Table \[tab:PVDIS\].
[llll]{} & $2\ C_{1u} - C_{1d}$ &$2\ C_{2u} - C_{2d}$ & $\omega_{\rm PVDIS}$\
tree + QED & $-0.7060$ & $-0.0715$ & $-0.7660$\
charge radii & +0.0013 & $-0.0110$ & $-0.0079$\
$\Box_{WW}$ & $-0.0120$ & $-0.0120$ & $-0.0220$\
$\Box_{\gamma Z}$ & $-0.0008$ & $-0.0027$ & $-0.0031$\
other & $-0.0009$ & $-0.0011$ & $-0.0018$\
TOTAL & $-0.7184$ & $-0.0983$ & $-0.8010$\
Eqs. (\[eq:c1\]) and (\[eq:c2\]) were originally obtained for atomic parity violation. For PVDIS, the one-loop expressions with the full kinematic dependance (in analogy with Ref. [@Czarnecki:1995fw] for polarized Møller scattering) need to be computed, plus the $\hat\alpha_s$ corrections to $\Box_{WW}$ and $\Box_{ZZ}$. In practice, one would want to define new $C_{2q}$ at these kinematics since these would supersede the ones at very low $Q^2$ with their large hadronic uncertainties.
The $\overline{\rm MS}$ scheme (marked by a caret) weak mixing angle enters Eqs. (\[eq:c1\]) and (\[eq:c2\]) evaluated at the renormalization scale $\mu = 0$. Introducing the quantity $\hat{X} \equiv \sum_i N_C^i \gamma^i \hat g_V^i Q^i$, where $N_C = 3$ (1) for quarks (leptons) and $\gamma^i = 4$ (22) for chiral fermions (gauge bosons), one can show that $d\hat{X}/X = d\hat{\alpha}/\alpha$, [*i.e.*]{}, the RGE for $\hat\alpha(\mu)$ implies that for $\sin^2\hat\theta_W(\mu)$ (see Fig. \[fig:running\_s2w\]) including experimental constraints from $e^+ e^-$ annihilation and $\tau$ decays that enter the dispersion integral for the non-perturbative regime, [*provided*]{} that any one of the following conditions is satisfied: (i) no mass threshold is crossed; (ii) perturbation theory applies ($W^\pm$, leptons, $b$ and $c$ quarks); (iii) equal coefficients (like for $d$ [*vs.*]{} $s$ quarks); or (iv) symmetries like $SU(2)_I$ or $SU(3)_F$ may be applied.
![Renormalization scale dependance of the weak mixing angle. Various measurements are shown at their nominal energy scales, [*i.e.*]{}, not necessarily at their typical momentum transfers.[]{data-label="fig:running_s2w"}](s2w_2008_12.eps){width="100.00000%"}
This leaves as the only problem area the treatment of the $u$ [*vs.*]{} the ($d$, $s$) quark thresholds, or —considering that $m_s \neq m_d \approx m_u$ — the separation of the $s$ quarks from the ($u$, $d$) doublet. Our strategy [@Erler:2004in] is to define threshold masses (absorbing QCD matching effects), $\bar{m}_q = \xi_q M_{1S}/2$, in terms of $1S$ resonance masses. The $\xi_q$ are between 0 (chiral limit) and 1 (infinitely heavy quarks). One expects $\xi_b >\xi_c >\xi_s >\xi_d >\xi_u$ and we explicitly verified $\xi_b >\xi_c$ in perturbative QCD. Now, $\xi_s = \xi_c$ defines the heavy quark limit for the $s$ quark, implying $\bar{m}_s < 387$ MeV. On the other hand, $\xi_s = \xi_d \approx \xi_u$ together with the dispersion result for the three-flavor RGE for $\hat\alpha$ below $\mu = \bar{m}_c$, $\Delta\hat\alpha^{(3)} (\bar{m}_c)$, yields an upper limit on the $s$ quark contribution and $\bar{m}_s > 240$ MeV. Besides parametric uncertainties from the input values of $\hat{m}_b$, $\bar{m}_c$, and $\hat\alpha_s$, this procedure introduces an experimental error through $\Delta\hat\alpha^{(3)} (\bar{m}_c)$ ($\pm 3\times 10^{-5}$), $SU(3)_F$ breaking masses, $\bar{m}_u = \bar{m}_d \neq \bar{m}_s$ ($\pm 5\times 10^{-5}$), and $SU(2)_I$ breaking masses, $\bar{m}_u \neq \bar{m}_d$ ($\pm 8\times 10^{-6}$). Starting at three-loop order there is also the (OZI rule violating) singlet (QCD annihilation) contribution to the RGE for $\hat\alpha$ (but by virtue of $Q_u + Q_d + Q_s = \tau^{uu}_3 + \tau^{dd}_3 = 0$ not present in $\hat{X}$) introducing another $\pm 3\times 10^{-5}$ error.
$Z^\prime$ physics: the search for a fifth force
================================================
Extra $Z^\prime$ bosons are predicted in virtually all scenarios for TeV scale physics beyond the SM, including grand unified theories, left-right models, superstrings, technicolor, large extra dimensions and little Higgs theories and in all these cases one expects $M_{Z^\prime} = {\cal O}({\rm TeV})$ and 100 (1,000) fb$^{-1}$ of LHC data will explore $M_{Z^\prime}$ values up to 5 (6) TeV [@Godfrey:2002tna]. Angular distributions of leptons may help to discriminate spin-1 ($Z^\prime$) against spin-0 (sneutrino) and spin-2 (Kaluza-Klein graviton) resonances [@Osland:2009tn]. The LHC will also have some diagnostic tools to narrow down the underlying $Z'$ model by studying, [*e.g.,*]{} leptonic forward-backward asymmetries and heavy quark final states [@Barger:2006hm; @Godfrey:2008vf].
$Z^\prime$ models based on the gauge group $E_6$ without kinetic mixing correspond to extending the SM by a $U(1)^\prime = \cos\beta\ U(1)_\chi + \sin\beta\ U(1)_\psi$ ($-90^\circ < \beta \leq 90^\circ$). Particular values for $\beta$ give $Z^\prime$ models of special interest, namely (i) $\beta = 0^\circ \Longrightarrow Z_\chi$ and is defined by the breaking of $SO(10) \to SU(5) \times U(1)_\chi$; (ii) $\beta = 90^\circ \Longrightarrow Z_\psi$ defined by the breaking of $E_6 \to SO(10) \times U(1)_\psi$; (iii) $\beta \approx - 52.2^\circ \Longrightarrow Z_\eta$ and appears in a class of heterotic string models compactified on Calabi-Yau manifolds; (iv) $\beta \approx 37.8^\circ \Longrightarrow Z_I \perp Z_\eta$ and is hadrophobic in that it doesn’t couple to up-type quarks; (v) $\beta \approx 23.3^\circ \Longrightarrow Z_S$ and gives rise to the so-called secluded $U(1)^\prime$ breaking model addressing both the little hierarchy problem ($M_Z \ll M_{Z^\prime}$) [@Erler:2002pr] and electroweak baryogenesis [@Kang:2004pp]; and (vi) $\beta \approx 75.5^\circ \Longrightarrow Z_N$ with no couplings to right-handed neutrinos and therefore allowing the (ordinary) see-saw mechanism. Adding kinetic mixing is equivalent to considering the more general combination, $Z^\prime = \cos\alpha \cos\beta\ Z_\chi + \sin\alpha \cos\beta\ Z_Y + \sin\beta\ Z_\psi.$ Then the values (vii) $(\alpha,\beta) \approx (50.8^\circ, 0^\circ) \Longrightarrow Z_R$ defined by the breaking of $SU(2)_R \to U(1)_R$; (viii) in left-right symmetric models appears the $Z_{LR} \propto 1.53\ Z_R - 0.33\ Z_{B-L}$, where $(\alpha,\beta) \approx (-39.2^\circ, 0^\circ) \Longrightarrow Z_{B-L} \perp Z_R$; while (ix) $(\alpha,\beta) \approx (28.6^\circ, -48.6^\circ) \Longrightarrow Z_{\not{L}}$ with no couplings to charged leptons and left-handed neutrinos. Finally, (x) the sequential $Z_{SM}$ couples like and could be an excited state of the ordinary $Z$ boson.
$Z^\prime$ bosons can have various effects on precision observables. The $Z$-$Z^\prime$ mixing angle, $\theta_{ZZ^\prime}$, is strongly constrained by the $M_W$-$M_Z$ interdependence (even for the $Z_{\not{L}}$) and by the $Z$-pole (because $\theta_{ZZ^\prime}$ affects the very precisely measured $Z$ couplings to fermions). Conversely, if $\theta_{ZZ^\prime} = 0$ the $Z$ pole observables are rather blind to $Z^\prime$ physics because the $Z$ and $Z^\prime$ amplitudes are almost completely out of phase and one needs to go off-peak, [*i.e.*]{}, to LEP 2 and low energies. There are also loop effects which are small but not necessarily negligible. [*E.g.*]{}, the $M_W$-$G_F$ relation, parametrized by $\Delta \hat{r}_W$, is shifted, $$\label{deltar}
\delta (\Delta \hat{r}_W) = - {5\over 2} {\alpha\over \pi \cos^2\theta_W} \lambda \epsilon_L^e \epsilon_L^\mu
{M_W^2\over M_{Z^\prime}^2 - M_W^2} \ln {M_{Z^\prime}^2\over M_W^2},$$ where the $\epsilon_L^f$ denote $U(1)^\prime$ charges and $\lambda$ is a model dependent parameter of ${\cal O}(1)$. $Z^\prime$ bosons would also yield an apparent violation of first row CKM unitarity, $\delta( V_{ud}^2 + V_{us}^2 + V_{ub}^2)$, given by the r.h.s. of Eq. (\[deltar\]) upon replacing $\epsilon_L^\mu$ by $-2 (\epsilon_L^\mu - \epsilon_L^d)$. Finally, the muon anomalous magnetic moment [@Hertzog] would receive a (usually tiny) correction, $\delta a_\mu = 5 /36\ \alpha /\pi \cos^2\theta_W\ \lambda (V_\mu^2 - 5 A_\mu^2)\ m_\mu^2/M_{Z^\prime}^2$, with some interest for the $Z_\psi$ which is insensitive to most other precision data (since it does not possess any vector couplings $V_f$) while the axial coupling $A_\mu$ comes enhanced in $\delta a_\mu$.
Results from a global analysis [@Erler:2009jh] are shown in Table \[tab:limits\]. Some $Z'$ models give a fairly low minimum $\chi^2$, especially the $Z_\psi$ and $Z_R$. Technically, there is a 90% C.L. [*upper*]{} bound on the $Z_R$ mass of about 29 TeV. Of course, at present there is little significance to this observation since there are two additional fit parameters ($M_Z'$ and $\theta_{ZZ'}$) and various adjustable charges (like the angles $\alpha$ and $\beta$). Still this surprises given that the SM fit is quite good with $\chi^2_{\rm min} = 48.0/45$ (with $M_H$ unconstrained). It is interesting that the improvement, $\Delta\chi^2_{\rm min} = - 2.9$, is mainly from PAVI observables, namely from polarized Møller [@Anthony:2005pm] ($-1.7$) and $e^-$-hadron scattering [@Young:2007zs] ($-0.9$). The best fit with $M_{Z'} = 667$ GeV implies shifts in the so-called weak charges, $\delta |Q_W(e,p)| = - 0.0073$, corresponding to $6.6\sigma$ and $2.5\sigma$, respectively, for the proposed MOLLER [@Kumar] and Qweak [@Page] experiments at JLab. Similarly, expect $\delta |\omega_{\rm PVDIS}|= - 0.0200$ ($4.2\sigma$).
[lrrrrrccc]{} $Z'$ & electroweak & CDF & LEP 2 & $\theta_{ZZ'}^{\rm min}$ & $\theta_{ZZ'}^{\rm max}$ & $\chi^2_{\rm min}$\
$Z_\chi$ & 1,141 & 892 & 673 & $-0.0016$ & 0.0006 & 47.3\
$Z_\psi$ & 147 & 878 & 481 & $-0.0018$ & 0.0009 & 46.5\
$Z_\eta$ & 427 & 982 & 434 & $-0.0047$ & 0.0021 & 47.7\
$Z_I$ & 1,204 & 789 & & $-0.0005$ & 0.0012 & 47.4\
$Z_S$ & 1,257 & 821 & & $-0.0013$ & 0.0005 & 47.3\
$Z_N$ & 623 & 861 & & $-0.0015$ & 0.0007 & 47.4\
$Z_R$ & 442 & & & $-0.0015$ & 0.0009 & 46.1\
$Z_{LR}$ & 998 & 630 & 804 & $-0.0013$ & 0.0006 & 47.3\
$Z_{\not{L}}$ & (803) & (740) & & $-0.0094$ & 0.0081 & 47.7\
$Z_{SM}$ & 1,403 & 1,030 & 1,787 & $-0.0026$ & 0.0006 & 47.2\
It is a pleasure to thank Paul Langacker, Shoaib Munir and Eduardo Rojas for collaboration. This work is supported by CONACyT project 82291–F.
[99]{}
D. Budker, these proceedings
SOLID Collaboration: P. Souder [*et al.*]{}, these proceedings
W.J. Marciano and A. Sirlin, Phys. Rev. D [**27**]{}, 552 (1983) and [*ibid.*]{} [**29**]{}, 75 (1984) J. Erler, A. Kurylov and M.J. Ramsey-Musolf, Phys. Rev. D [**68**]{}, 016006 (2003) A. Czarnecki and W.J. Marciano, Phys. Rev. D [**53**]{}, 1066 (1996) J. Erler and M.J. Ramsey-Musolf, Phys. Rev. D [**72**]{}, 073003 (2005) S. Godfrey, arXiv:hep-ph/0201093; proceedings of Snowmass 2001 P. Osland, A.A. Pankov, A.V. Tsytrinov and N. Paver, Phys. Rev. D [**79**]{}, 115021 (2009) V. Barger, T. Han and D.G.E. Walker, Phys. Rev. Lett. [**100**]{}, 031801 (2008) S. Godfrey and T.A.W. Martin, Phys. Rev. Lett. [**101**]{}, 151803 (2008) J. Erler, P. Langacker and T. Li, Phys. Rev. D [**66**]{}, 015002 (2002) J. Kang, P. Langacker, T. Li and T. Liu, Phys. Rev. Lett. [**94**]{}, 061801 (2005) New ($g-2$) Collaboration: D. Hertzog [*et al.*]{}, these proceedings
J. Erler, P. Langacker, S. Munir and E. Rojas, JHEP [**0908**]{}, 017 (2009) SLAC E158 Collaboration: P.L. Anthony [*et al.*]{}, Phys. Rev. Lett. [**95**]{}, 081601 (2005) R.D. Young, R.D. Carlini, A.W. Thomas and J. Roche, Phys. Rev. Lett. [**99**]{}, 122003 (2007) MOLLER Collaboration: K. Kumar [*et al.*]{}, these proceedings
Qweak Collaboration: S. Page [*et al.*]{}, these proceedings
|
---
abstract: 'In this paper, we introduce a new class of distributions which is obtained by compounding the extended Weibull and power series distributions. The compounding procedure follows the same set-up carried out by Adamidis and Loukas (1998) and defines at least new 68 sub-models. This class includes some well-known mixing distributions, such as the Weibull power series (Morais and Barreto-Souza, 2010) and exponential power series (Chahkandi and Ganjali, 2009) distributions. Some mathematical properties of the new class are studied including moments and generating function. We provide the density function of the order statistics and obtain their moments. The method of maximum likelihood is used for estimating the model parameters and an EM algorithm is proposed for computing the estimates. Special distributions are investigated in some detail. An application to a real data set is given to show the flexibility and potentiality of the new class of distributions.'
address: |
Universidade Federal de Pernambuco\
Departamento de Estatística, Cidade Universitária, 50740-540 Recife, PE, Brazil
author:
- 'Rodrigo B. Silva'
- 'Marcelo B. Pereira'
- 'Cícero R. B. Dias'
- 'Gauss M. Cordeiro'
title: The compound class of extended Weibull power series distributions
---
EM algorithm ,Extended Weibull distribution ,Extended Weibull power series distribution ,Order statistic ,Power series distribution.
Introduction
============
The modeling and analysis of lifetimes is an important aspect of statistical work in a wide variety of scientific and technological fields. Several distributions have been proposed in the literature to model lifetime data by compounding some useful lifetime distributions. Adamidis and Loukas (1998) introduced a two-parameter exponential-geometric (EG) distribution by compounding an exponential distribution with a geometric distribution. In the same way, the exponential Poisson (EP) and exponential logarithmic (EL) distributions were introduced and studied by Kus (2007) and Tahmasbi and Rezaei (2008), respectively. Recently, Chahkandi and Ganjali (2009) proposed the exponential power series (EPS) family of distributions, which contains as special cases these distributions. Barreto-Souza [*et al.*]{} (2010) and Lu and Shi (2011) introduced the Weibull-geometric (WG) and Weibull-Poisson (WP) distributions which naturally extend the EG and EP distributions, respectively. In a very recent paper, Morais and Barreto-Souza (2011) defined the Weibull power series (WPS) class of distributions which contains the EPS distributions as sub-models. The WPS distributions can have an increasing, decreasing and upside down bathtub failure rate function.
Now, consider the class of extended Weibull (EW) distributions, as proposed by Gurvich [*et al.*]{} (1997), having the cumulative distribution function (cdf) $$\label{extweibull}
G(x;\, \alpha, \boldsymbol{\xi}) = 1 - \mathrm{e}^{-\alpha\, H(x;\, \boldsymbol{\xi})}, \quad x>0, \,\,\, \alpha>0,$$ where $H(x;\, \boldsymbol{\xi})$ is a non-negative monotonically increasing function which depends on a parameter vector $\boldsymbol{\xi}$. The corresponding probability density function (pdf) is given by $$\label{pdfextweibull}
g(x;\, \alpha, \boldsymbol{\xi}) = \alpha\, h(x;\, \boldsymbol{\xi}) \,\mathrm{e}^{-\alpha \,H(x;\,\boldsymbol{\xi})}, \quad x > 0, \,\,\, \alpha>0,$$ where $h(x;\, \boldsymbol{\xi})$ is the derivative of $H(x; \,\boldsymbol{\xi})$.
Note that many well-known models are special cases of equation (\[extweibull\]) such as:\
([*i*]{}) $H(x; \boldsymbol{\xi}) = x$ gives the exponential distribution;\
([*ii*]{}) $H(x; \boldsymbol{\xi}) = x^2$ yields the Rayleigh distribution (Burr type-X distribution);\
([*iii*]{}) $H(x; \boldsymbol{\xi}) = \log(x/k)$ leads to the Pareto distribution;\
([*iv*]{}) $H(x; \boldsymbol{\xi}) = \beta^{-1}[\exp(\beta x)-1]$ gives the Gompertz distribution.\
In this article, we define the extended Weibull power series (EWPS) class of univariate distributions obtained by compounding the extended Weibull and power series distributions. The compounding procedure follows the key idea of Adamidis and Loukas (1998) or, more generally, by Chahkandi and Ganjali (2009) and Morais and Barreto-Souza [*et al.*]{} (2011). The new class of distributions contains as special models the WPS distributions, which in turn extends the EPS distributions and defines at least new 68 (17 $\times$ 4) sub-models as special cases. The hazard function of our class can be decreasing, increasing, bathtub and upside down bathtub.
We are motivated to introduce the EWPS distributions because of the wide usage of the general class of Weibull distributions and the fact that the current generalization provides means of its continuous extension to still more complex situations.
This paper is organized as follows. In Section 2, we define the EWPS class of distributions and demonstrate that there are many existing models which can be deduced as special cases of the proposed unified model. In Section 3, we provide the density, survival and hazard rate functions and derive some useful expansions. In Section 4, we obtain its quantiles, ordinary and incomplete moments. Further, the order statistics are discussed and their moments are determined. Section 5 deals with reliability and average lifetime. Estimation of the parameters by maximum likelihood using an EM algorithm and large sample inference are investigated in Section 6. In Section 7, we present suitable constraints leading to the maximum entropy characterization of the new class. Three special cases of the proposed class are studied in Section 8. In Section 9, we provide an application to a real data set. The paper is concluded in Section 10.
The new class
=============
Our class can be derived as follows. Given $N$, let $X_1, \ldots, X_N$ be independent and identically distributed (iid) random variables following (\[extweibull\]). Here, $N$ is a discrete random variable following a power series distribution (truncated at zero) with probability mass function $$\label{powerseries}
p_n = P(N=n)=\frac{a_n \, \theta^n}{C(\theta)}, n=1,2,\ldots,\\$$ where $a_n$ depends only on $n$, $C(\theta) = \sum_{n=1}^{\infty}a_n \,\theta^n$ and $\theta>0$ is such that $C(\theta)$ is finite. Table \[table1\] summarizes some power series distributions (truncated at zero) defined according to (\[powerseries\]) such as the Poisson, logarithmic, geometric and binomial distributions. Let $X_{(1)} = \mbox{min}\left\{X_i\right\}^{N}_{i=1}$. The conditional cumulative distribution of $X_{(1)}|N = n$ is given by $$G_{X_{(1)}|N=n}(x) = 1-\mathrm{e}^{-n\alpha H(x; \boldsymbol{\xi})},$$ i.e., $X_{(1)}|N = n$ follows a general class of distributions with parameters $n\alpha$ and $\boldsymbol{\xi}$ based on the same $H(x; \boldsymbol{\xi})$ function. Hence, we obtain $$P(X_{(1)} \leq x, N=n) = \frac{a_n\, \theta^n}{C(\theta)}\left[1-\mathrm{e}^{-n\alpha H(x; \boldsymbol{\xi})}\right], \quad x>0,
\quad n \geq 1.$$
The EWPS class of distributions can then be defined by the marginal cdf of $X_{(1)}$:
$$\label{cdf}
F(x;\theta,\alpha, \boldsymbol{\xi}) = 1-\frac{C(\theta \,
\mathrm{e}^{-\alpha H(x; \boldsymbol{\xi})})}{C(\theta)}, \quad x>0.$$
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Distribution $a_n$ $C(\theta)$ $C'(\theta)$ $C''(\theta)$ $C(\theta)^{-1}$ $\Theta$
-------------- -- ---------------- -- ------------------------- -- --------------------- -- ----------------------------------- -- -------------------------- --------------------------
Poisson $n!^{-1}$ $\mathrm{e}^{\theta}-1$ $e^{\theta}$ $e^{\theta}$ $\log(\theta+1)$ $\theta \in (0, \infty)$
Logarithmic $n^{-1}$ $-\log(1-\theta)$ $(1-\theta)^{-1}$ $(1-\theta)^{-2}$ $1-\mathrm{e}^{-\theta}$ $\theta \in (0,1)$
Geometric 1 $\theta(1-\theta)^{-1}$ $(1-\theta)^{-2}$ $2(1-\theta)^{-3}$ $\theta(\theta+1)^{-1}$ $\theta \in (0,1)$
Binomial $\binom{m}{n}$ $(\theta+1)^{m} - $m(\theta+1)^{m-1}$ $\frac{m(m-1)}{(\theta+1)^{2-m}}$ $(\theta-1)^{1/m}-1$ $\theta \in (0, 1)$
1$
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Useful quantities for some power series distributions.[]{data-label="table1"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Distribution $H(x;\boldsymbol{\xi})$ $h(x;\boldsymbol{\xi})$ $\alpha$ $\boldsymbol{\xi}$ References
------------------------------------------ --------------------------------------------------- ----------------------------------------------------------------------------------------------- ---------- --------------------------------------------- -----------------------------
Exponential ($x\geq0$) $x$ 1 $\alpha$ $\emptyset$ Johnson *et al*. (1994)
Pareto ($x\geq k$) $\log(x/k)$ $1/x$ $\alpha$ $k$ Johnson *et al*. (1994)
Rayleigh ($x\geq0$) $x^2$ $2x$ $\alpha$ $\emptyset$ Rayleigh (1880)
Weibull ($x\geq0$) $x^{\gamma}$ $\gamma x^{\gamma-1}$ $\alpha$ $\gamma$ Johnson *et al*. (1994)
Modified Weibull ($x\geq0$) $x^{\gamma}\exp(\lambda x)$ $x^{\gamma - 1}\exp(\lambda $\alpha$ $[\gamma,\, \lambda]$ Lai *et al*. (2003)
x)(\gamma+\lambda x)$
Weibull extension ($x\geq0$) $\lambda[\exp(x/\lambda)^{\beta}-1]$ $\beta\exp(x/\lambda)^{\beta}(x/\lambda)^{\beta-1}$ $\alpha$ $[\gamma,\, \lambda,\,\beta]$ Xie *et al*. (2002)
Log-Weibull ($-\infty<x<\infty$) $\exp[(x-\mu)/\sigma]$ $(1/\sigma)\exp[(x-\mu)/\sigma]$ 1 $[\mu,\, \sigma]$ White (1969)
Phani ($0<\mu < x<\sigma<\infty$) $[(x-\mu)/(\sigma-x)]^{\beta}$ $\beta[(x-\mu)/(\sigma-x)]^{\beta-1}[(\sigma-\mu)/(\sigma-t)^2]$ $\alpha$ $[\mu,\, \sigma,\,\beta]$ Phani (1987)
Weibull Kies ($0<\mu < x<\sigma<\infty$) $(x-\mu)^{\beta_1}/(\sigma-x)^{\beta_2}$ $(x-\mu)^{\beta_1 -1}(\sigma-x)^{-\beta_2 - 1}[\beta_1(\sigma-x)+\beta_2(x-\mu)]$ $\alpha$ $[\mu,\,\sigma,\,\beta_1,\,\beta_2]$ Kies (1958)
Additive Weibull ($x\geq0$) $(x/\beta_1)^{\alpha_1} + (x/\beta_2)^{\alpha_2}$ $(\alpha_1/\beta_1)(x/\beta_1)^{\alpha_1 - 1} + (\alpha_2/\beta_2)(x/\beta_2)^{\alpha_2 - 1}$ 1 $[\alpha_1,\,\alpha_2,\,\beta_1,\,\beta_2]$ Xie and Lai (1995)
Traditional Weibull ($x\geq0$) $x^{b}[\exp(cx^{d}-1)]$ $b x^{b - 1}[\exp(cx^{d})-1] + cdx^{b + d -1}\exp(cx^d)$ $\alpha$ $[b,\, Nadarajah and Kotz (2005)
c,\,d]$
Gen. power Weibull ($x\geq0$) $[1+(x/\beta)^{\alpha_1}]^{\theta} - 1$ $(\theta\alpha_1/\beta)[1+(x/\beta)^{\alpha_1}]^{\theta-1}(x/\beta)^{\alpha_1}$ 1 $[\alpha_1,\, \beta,\,\theta]$ Nikulin and Haghighi (2006)
Flexible Weibull extension($x\geq0$) $\exp(\alpha_1x-\beta/x)$ $\exp(\alpha_1x-\beta/x)(\alpha_1+\beta/x^2) 1 $[\alpha_1,\,\beta]$ Bebbington *et al*. (2007)
$
Gompertz ($x\geq0$) $\beta^{-1}[\exp(\beta x) - 1]$ $\exp(\beta x)$ $\alpha$ $\beta$ Gompertz (1825)
Exponential power ($x\geq0$) $\exp[(\lambda x)^{\beta}]-1$ $\beta\lambda\exp[(\lambda x)^{\beta}](\lambda x)^{\beta-1}$ 1 $[\lambda,\,\beta]$ Smith and Bain (1975)
Chen ($x\geq0$) $\exp(x^{b})-1$ $b x^{b-1}\exp(x^b)$ $\alpha$ $b$ Chen (2000)
Pham ($x\geq0$) $ (a^{x})^{\beta} - 1$ $\beta(a^{x})^{\beta}\log(a)$ 1 $[a,\,\beta]$ Pham (2002)
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The random variable $X$ following (\[cdf\]) with parameters $\theta$ and $\alpha$ and the vector $\boldsymbol{\xi}$ of parameters is denoted by $X\sim \mbox{EWPS}(\theta,\alpha, \boldsymbol{\xi})$. Equation (\[cdf\]) extends several distributions which have been studied in the literature. The EG distribution (Adamidis and Loukas, 1998) is obtained by taking $H(x;\,\boldsymbol{\xi})=x$ and $C(\theta) = \theta(1-\theta)^{-1}$ with $\theta \in (0,1)$. Further, for $H(x;\,\boldsymbol{\xi})=x$, we obtain the EP (Kus, 2007) and EL (Tahmasbi and Rezaei, 2008) distributions by taking $C(\theta) = \mathrm{e}^{\theta}-1, \theta >0$, and $C(\theta) = -\log(1-\theta), \theta \in (0,1)$, respectively. In the same way, for $H(x;\,\boldsymbol{\xi})=x^\gamma$, we obtain the WG (Barreto-Souza [*et al*]{}., 2009) and WP (Lu and Shi, 2011) distributions. The EPS distributions are obtained from (\[cdf\]) by mixing $H(x;\,\boldsymbol{\xi})=x$ with any $C(\theta)$ listed in Table \[table1\] (see Chahkandi and Ganjali, 2009). Finally, we obtain the WPS distributions from (\[cdf\]) by compounding $H(x;\,\boldsymbol{\xi})=x^\gamma$ with any $C(\theta)$ in Table \[table1\] (see Morais and Barreto-Souza, 2011). Table \[table2\] displays some useful quantities and respective parameter vectors for each particular distribution.
Density, survival and hazard functions
======================================
The density function associated to (\[cdf\]) is given by $$\label{pdf}
f(x;\theta,\alpha, \boldsymbol{\xi}) = \theta \, \alpha \,
h(x;\boldsymbol{\xi})\, \mathrm{e}^{-\alpha H(x;\,
\boldsymbol{\xi})}\, \frac{C'(\theta \, \mathrm{e}^{-\alpha H(x;\,
\boldsymbol{\xi})})}{C(\theta)}, \quad x>0.$$
\[prop1\] The EW class of distributions with parameters $c\alpha$ and $\boldsymbol{\xi}$ is a limiting special case of the EWPS class of distributions when $\theta \rightarrow 0^+$, where $c= \min\left\{n \in \mathbb{N}: a_n >0\right\}$.
This proof uses a similar argument to that found in Morais and Barreto-Souza (2011). Define $c= \min\left\{n \in \mathbb{N}: a_n >0\right\}$. We have $$\begin{aligned}
\lim_{\theta \rightarrow 0^+} F(x) &= 1 - \lim_{\theta \rightarrow 0^+} \frac{\displaystyle{\sum_{n=c}^\infty a_n\left(\theta \, \mathrm{e}^{-\alpha H(x; \boldsymbol{\xi})}\right)^n}}{\displaystyle{\sum_{n=c}^\infty a_n\, \theta^n}}\\
&= 1 - \lim_{\theta \rightarrow 0^+} \frac{\displaystyle{\mathrm{e}^{-c \alpha H(x; \boldsymbol{\xi})} + a_c^{-1} \sum_{n=c+1}^\infty a_n \,\theta^{n-c}\mathrm{e}^{-n\alpha H(x; \boldsymbol{\xi})}}}{\displaystyle{1 + a_c^{-1}\sum_{n=c+1}^\infty a_n\, \theta^{n-c}}}\\
&= 1-\mathrm{e}^{-c\alpha H(x; \boldsymbol{\xi})},\end{aligned}$$ for $x>0$.
We now provide an interesting expansion for ($\ref{pdf}$). We have $C'(\theta) = \sum_{n=1}^{\infty}n\, a_n\, \theta^{n-1}$. By using this result in (\[pdf\]), it follows that $$\label{linearcomb}
f(x;\theta,\alpha, \boldsymbol{\xi}) = \sum_{n=1}^{\infty}p_n\,
g(x;\, n\alpha, \xi),$$ where $g(x;\, n\alpha, \boldsymbol{\xi})$ is given by (\[pdfextweibull\]). Based on equation , we obtain $$F(x;\theta,\alpha, \boldsymbol{\xi}) = 1 - \sum_{n=1}^\infty p_n\,
\mathrm{e}^{-n\alpha H(x;\, \boldsymbol{\xi})}.$$
Hence, the EWPS density function is an infinite mixture of EW densities. So, some mathematical quantities (such as ordinary and incomplete moments, generating function and mean deviations) of the EWPS distributions can be obtained by knowing those quantities for the baseline density function $g(x;\, n\alpha, \boldsymbol{\xi})$.
The EWPS survival function is given by $$\label{survewps}
S(x; \theta, \alpha, \boldsymbol{\xi})= \frac{C(\theta \,\mathrm{e}^{-\alpha H(x;\,\boldsymbol{\xi})})}{C(\theta)}$$ and the corresponding hazard rate function becomes $$\tau(x; \theta, \alpha, \boldsymbol{\xi}) = \theta \alpha\, h(x;\,
\boldsymbol{\xi}) \, \mathrm{e}^{-n\alpha H(x; \boldsymbol{\xi})}
\,\frac{C'(\theta \, \mathrm{e}^{-\alpha H(x;\,
\boldsymbol{\xi})})}{C(\theta \, \mathrm{e}^{-\alpha H(x;\,
\boldsymbol{\xi})})}.$$
Quantiles, moments and order statistics
=======================================
The EWPS distributions are easily simulated from (\[cdf\]) as follows: if $U$ has a uniform $U(0,1)$ distribution, then the solution of the nonlinear equation $$\begin{aligned}
X = H^{-1}\left\{-\frac{1}{\alpha}\log\left[\frac{C^{-1}(C(\theta)(1-U))}{\theta}\right]\right\}\end{aligned}$$ has the EWPS$(\theta,\alpha, \boldsymbol{\xi})$ distribution, where $H^{-1}(\cdot)$ and $C^{-1}(\cdot)$ are the inverse functions of $H(\cdot)$ and $C(\cdot)$, respectively. To simulate data from this nonlinear equation, we can use the matrix programming language Ox through [*SolveNLE*]{} subroutine (see Doornik, 2007).
We now derive a general expression for the $r$th raw moment of $X$, which may be determined by using (\[linearcomb\]) and the monotone convergence theorem. So, for $r \in \mathbb{N}$, we obtain $$\operatorname{E}(X^r) = \sum_{n=1}^{\infty}p_n\, \operatorname{E}(Z^r),\\$$ where $Z$ is a random variable with pdf $g(z; n\alpha,
\boldsymbol{\xi})$.
The incomplete moments and moment generating function (mgf) follow by using (\[linearcomb\]) and the monotone convergence theorem: $$\begin{aligned}
I_X(y) &= \int^{y}_{0}x^r\, f(x)dx = \sum_{n=1}^{\infty}p_n\, I_{Z}(y)\\
\intertext{and} M_{X}(t) &=
\sum_{n=1}^{\infty}p_n\,\operatorname{E}\left(\mathrm{e}^{tZ}\right).\end{aligned}$$ where $Z$ is defined as before.
Order statistics are among the most fundamental tools in non-parametric statistics and inference. They enter in the problems of estimation and hypothesis tests in a variety of ways. Therefore, we now discuss some properties of the order statistics for the proposed class of distributions. The pdf $f_{i:m}(x)$ of the $i$th order statistic for a random sample $X_1, \ldots, X_m$ from the EWPS distribution is given by $$\label{osewps}
f_{i:m}(x) = \frac{m!}{(i-1)!(m-i)!}f(x;\theta,\alpha, \boldsymbol{\xi})\left[1-\frac{C(\theta\, \mathrm{e}^{-\alpha H(x;\,
\boldsymbol{\xi})})}{C(\theta)}\right]^{i-1}\left[\frac{C(\theta
\mathrm{e}^{-\alpha H(x;\,
\boldsymbol{\xi})})}{C(\theta)}\right]^{m-i}, \quad x> 0,$$ where $f (x; \theta, \alpha, \boldsymbol{\xi})$ is the pdf given by (\[pdf\]). By using the binomial expansion, we can write (\[osewps\]) as $$\label{osewpsexpansion}
f_{i:m}(x) = \frac{m!}{(i-1)!(m-i)!}f(x;\theta,\alpha, \boldsymbol{\xi})\sum_{j=0}^{i-1} (-1)^j \,\binom{i-1}{j}\, S(x;\theta,\alpha,
\boldsymbol{\xi})^{m+j-i},$$ where $S(x; \theta, \alpha, \boldsymbol{\xi})$ is given by . The corresponding cumulative function is $$F_{i:m}(x) = \sum_{j=0}^{\infty}\sum_{k=i}^{m}(-1)^j\,\binom{k}{j}\,\binom{m}{k}\,S(x;\theta,\alpha,
\boldsymbol{\xi})^{m+j-k}.$$
An alternative form for can be obtained from as $$\label{altosewps}
f_{i:m}(x) = \frac{m!}{(i-1)! (m-i)!} \sum_{n=1}^\infty \sum_{j=0}^{i-1} \omega_j\, p_n\, g(x; n\alpha,
\boldsymbol{\xi})S(x;\theta,\alpha, \boldsymbol{\xi})^{m+j-1},$$ where $\omega_j = (-1)^j \binom{i-1}{j}$. So, the $s$th raw moment $X_{i:m}$ comes immediately from the above equation $$\label{eosewps}
\operatorname{E}\left(X_{i:m}^s\right) = \frac{m!}{(i-1)! (m-i)!}
\sum_{n=1}^\infty \sum_{j=0}^{i-1} \omega_j\, p_n\,
\operatorname{E}\left[Z^s S(Z)^{m+j-i}\right],$$ where $Z\sim \mbox{EW}(n\alpha, \boldsymbol{\xi})$ is defined before.
Reliability and average lifetime
================================
In the context of reliability, the stress-strength model describes the life of a component which has a random strength $X$ subjected to a random stress $Y$. The component fails at the instant that the stress applied to it exceeds the strength, and the component will function satisfactorily whenever $X > Y$. Hence, $R = \operatorname{P}(X > Y)$ is a measure of component reliability. It has many applications, especially in engineering concepts. The algebraic form for R has been worked out for the majority of the well-known distributions. Here, we obtain the form for the reliability $R$ when $X$ and $Y$ are independent random variables having the same EWPS distribution.
The quantity $R$ can be expressed as $$\label{eqR}
R = \int_0^{\infty}f(x; \theta, \alpha, \boldsymbol{\xi})F(x;\theta,\alpha, \boldsymbol{\xi}) dx.$$
Substituting (\[cdf\]) and (\[pdf\]) into equation (\[eqR\]), we obtain $$\begin{aligned}
\label{eqR2}
R &=& \int_{0}^{\infty}\theta \, \alpha \,
h(x;\boldsymbol{\xi})\, \mathrm{e}^{-\alpha H(x;\,
\boldsymbol{\xi})}\, \frac{C'(\theta \mathrm{e}^{-\alpha H(x;\,
\boldsymbol{\xi})})}{C(\theta)}\left[1-\frac{C(\theta
\mathrm{e}^{-\alpha H(x; \boldsymbol{\xi})})}{C(\theta)}\right] dx\\
&=&1 - \sum_{n=1}^{\infty}p_n\int_{0}^{\infty}
g(x;n\alpha,\boldsymbol{\xi})S(x;\theta,\alpha, \boldsymbol{\xi})dx,\end{aligned}$$ where the integral can be calculated from the baseline EW distribution.
The average lifetime is given by $$t_m =
\sum_{n=1}^{\infty}p_n\int\limits_{0}^{\infty}\operatorname{e}^{-n\alpha
H(x;\, \boldsymbol{\xi})}dx.$$
Given that there was no failure prior to $x_0$, the residual life is the period from time $x_0$ until the time of failure. The mean residual lifetime can be expressed as $$\begin{aligned}
m(x_0;\theta,\alpha,\boldsymbol{\xi}) &=& \left[\operatorname{Pr}(X>x_0)\right]^{-1}\int\limits_{0}^{\infty}y\,f(x_0+y;\theta,\alpha,\boldsymbol{\xi}) dy\\
&=&[S(x_0)]^{-1}\sum_{n=1}^{\infty}p_n\int\limits_{0}^{\infty}y\,g(x_0+y;n\alpha,\boldsymbol{\xi}) dy.\end{aligned}$$
The last integral can be computed from the baseline EW distribution. Furthermore, $m(x_0;\theta,\alpha,\boldsymbol{\xi}) \rightarrow \operatorname{E}(X)$ as $x_0 \rightarrow 0$.
Maximum likelihood estimation
=============================
Preliminaries
-------------
Here, we determine the maximum likelihood estimates (MLEs) of the parameters of the EWPS class of distributions from complete samples only. Let $X_1, \ldots, X_n$ be a random sample with observed values $x_1, \ldots, x_n$ from an EWPS distribution with parameters $\theta, \alpha$ and $\boldsymbol{\xi}$. Let $\Theta= (\theta,\alpha, \boldsymbol{\xi})^\top$ be the $p \times 1$ parameter vector. The total log-likelihood function is given by $$\begin{aligned}
\label{loglik} \nonumber
\ell_n &=& \ell_{n}(x; \Theta) = n\left[\log\theta + \log\alpha - \log C(\theta)\right] - \alpha\sum_{i=1}^{n}H(x_i;\, \boldsymbol{\xi}) +
\sum_{i=1}^{n}\log h(x_i;\, \boldsymbol{\xi})\\
&+& \sum_{i=1}^{n}\log C'(\theta\, \mathrm{e}^{-\alpha H(x_i;\, \boldsymbol{\xi})}).\end{aligned}$$
The log-likelihood can be maximized either directly by using the SAS (PROC NLMIXED) or the Ox program (sub-routine MaxBFGS) (see Doornik, 2007) or by solving the nonlinear likelihood equations obtained by differentiating (\[loglik\]). The components of the score function $U_n(\Theta) = \left(\partial \ell_n/\partial \theta, \partial \ell_n/\partial \alpha, \partial \ell_n/\partial \xi\right)^\top$ are $$\begin{aligned}
\frac{\partial \ell_n}{\partial \alpha} &= \frac{n}{\alpha} - \sum_{i=1}^{n} H(x_i;\, \boldsymbol{\xi}) - \theta\sum_{i=1}^n H(x_i;\, \boldsymbol{\xi}) \mathrm{e}^{-\alpha H(x_i;\, \boldsymbol{\xi})}\,\frac{C''(\theta\, \mathrm{e}^{-\alpha H(x_i;\, \boldsymbol{\xi})})}{C'(\theta\, \mathrm{e}^{-\alpha H(x_i;\, \boldsymbol{\xi})})},\\
\frac{\partial \ell_n}{\partial \theta} &= \frac{n}{\theta} - n\frac{C'(\theta)}{C(\theta)} + \sum_{i=1}^n \mathrm{e}^{-\alpha H(x_i;\, \boldsymbol{\xi})}\,\frac{C''(\theta\, \mathrm{e}^{-\alpha H(x_i;\, \boldsymbol{\xi})})}{C'(\theta\, \mathrm{e}^{-\alpha H(x_i;\, \boldsymbol{\xi})})}\\
\intertext{and} \frac{\partial \ell_n}{\partial \boldsymbol{\xi}_k} &=
\sum_{i=1}^{n} \frac{\partial \log h(x_i;\, \boldsymbol{\xi})}{\partial \boldsymbol{\xi}_k} - \alpha \sum_{i=1}^{n} \frac{\partial H(x_i;\, \xi)}{\partial
\boldsymbol{\xi}_k}\left[1 + \theta \mathrm{e}^{-\alpha H(x_i; \, \boldsymbol{\xi})} \frac{C''(\theta\, \mathrm{e}^{-\alpha H(x_i;\,\boldsymbol{\xi})})}{C'(\theta\, \mathrm{e}^{-\alpha H(x_i;\,
\boldsymbol{\xi})})}\right].\end{aligned}$$
For interval estimation on the model parameters, we require the observed information matrix $$J_n(\Theta)=-\left( \begin{array}{cccc}
U_{\theta\theta} & U_{\theta\alpha} & |&U_{\theta\boldsymbol{\xi}}^\top \\
U_{\alpha\theta} & U_{\alpha\alpha} & |&U_{\alpha\boldsymbol{\xi}}^\top \\
-- & --& --& --\\
U_{\theta\boldsymbol{\xi}} & U_{\alpha\boldsymbol{\xi}} & | & U_{\boldsymbol{\xi}\boldsymbol{\xi}} \end{array} \right),$$ whose elements are listed in \[apA\]. Let $\widehat{\Theta}$ be the MLE of $\Theta$. Under standard regular conditions stated in Cox and Hinkley (1974) that are fulfilled for our model whenever the parameters are in the interior of the parameter space, we have that the asymptotic distribution of $\sqrt{n}\left(\widehat{\Theta} - \Theta\right)$ is multivariate normal $N_p(0, K(\Theta)^{-1})$, where $K(\Theta) = \lim_{n \rightarrow \infty}J_n(\Theta)$ is the unit information matrix and $p$ is the number of parameters of the compounded distribution.
The EM algorithm
----------------
Here, we propose an EM algorithm (Dempster [*et al.*]{}, 1977) to estimate $\Theta$. The EM algorithm is a recurrent method such that each step consists of an estimate of the expected value of a hypothetical random variable and then maximizes the log-likelihood for the complete data. Let the complete-data be $X_1, \ldots, X_n$ with observed values $x_1, \ldots, x_n$ and the hypothetical random variables $Z_1, \ldots, Z_n$. The joint probability function is such that the marginal density of $X_1, \ldots, X_n$ is the likelihood of interest. Then, we define a hypothetical complete-data distribution for each $(X_i, Z_i)^\top, i=1, \ldots, n$, with a joint probability function in the form $$g(x, z; \Theta) = \frac{\alpha\, z\, a_z\, \theta^z}{C(\theta)}\,h(x;\, \xi)\, \mathrm{e}^{-\alpha z H(x;\, \xi)},$$ where $\theta$ and $\alpha$ are positive, $x>0$ and $z \in \mathbb{N}$. Under this formulation, the E-step of an EM cycle requires the expectation of $Z|X$; $\Theta^{(r)} = (\theta^{(r)}, \alpha^{(r)},
\boldsymbol{\xi}^{(r)})^\top$ as the current estimate (in the rth iteration) of $\Theta$. The probability function of $Z$ given $X$, say $g(z|x)$, is given by $$g(z|x) = \frac{z\, a_z\, \theta^{\theta-1}}{C'(\theta e^{-\alpha H(x_i;\, \boldsymbol{\xi})})}\,\mathrm{e}^{-\alpha(z-1) H(x_i;\, \boldsymbol{\xi})}$$ and its expected value is $$E(Z|X) = 1 + \theta e^{-\alpha H(x;\, \boldsymbol{\xi})}\,\frac{C''(\theta\, \mathrm{e}^{-\alpha H(x;\, \boldsymbol{\xi})})}{C'(\theta\, \mathrm{e}^{-\alpha H(x;\, \boldsymbol{\xi})})}.$$
The EM cycle is completed with the M-step by using the maximum likelihood estimation over $\Theta$, where the missing $Z's$ are replaced by their conditional expectations given before. The log-likelihood for the complete-data is $$\begin{aligned}
\textstyle
\ell_n^*(x_1, \ldots, x_n; \, z_1, \ldots, z_n; \,\alpha, \,\theta, \,\boldsymbol{\xi}) &\propto n\log\alpha + \log \theta \sum_{i=1}^n z_i + \sum_{i=1}^n \log h(x_i;\, \boldsymbol{\xi})\\
&- \alpha \sum_{i=1}^n z_i H(x_i;\, \boldsymbol{\xi}) - n\log C(\theta).\end{aligned}$$
So, the components of the score function $U^*_n(\Theta) = \left(\partial l^*_n/\partial \theta, \partial l^*_n/\partial \alpha, \partial l^*_n/\partial \boldsymbol{\xi}\right)^\top$ are $$\begin{aligned}
\frac{\partial l^*_n}{\partial \theta} &= \frac{n}{\theta} -
\sum_{i=1}^{n} z_i - n\frac{C'(\theta)}{C(\theta)}, \quad \quad
\frac{\partial l^*_n}{\partial \alpha} = \frac{n}{\alpha} -
\sum_{i=1}^{n} z_i H(x_i;\, \boldsymbol{\xi}) \quad \intertext{and}
\frac{\partial l^*_n}{\partial \boldsymbol{\xi}_k} &= \sum_{i=1}^{n}
\frac{\partial \log h(x_i;\, \boldsymbol{\xi})}{\partial \xi_k} - \alpha \sum_{i=1}^{n} z_i \frac{\partial H(x_i;\,\boldsymbol{\xi})}{\partial \boldsymbol{\xi}_k}.\end{aligned}$$
From a nonlinear system of equations $U^*_n(\widehat{\Theta}) = 0$, we obtain the iterative procedure of the EM algorithm $$\begin{aligned}
&\hat{\alpha}^{(t+1)} = \frac{n}{\sum_{i=1}^{n} z_i^{(t)} H(x_i;\,\boldsymbol{\xi}^{(t)})}, \quad \quad \hat{\theta}^{(t+1)} =
\frac{C(\hat{\theta}^{(t+1)})}{C'(\hat{\theta}^{(t+1)})}\frac{1}{n}\sum_{i=1}^{n}
z_i^{(t)} \intertext{and} &\sum_{i=1}^{n} \frac{\partial \log h(x_i;\,\hat{\boldsymbol{\xi}}^{(t+1)})}{\partial \boldsymbol{\xi}_k} -
\hat{\alpha}^{(t)}\sum_{i=1}^{n} z_i^{(t)}\frac{\partial H(x_i;\,\hat{\boldsymbol{\xi}}^{(t+1)})}{\partial \boldsymbol{\xi}_k} = 0,\end{aligned}$$ where $\hat{\theta}^{(t+1)}, \hat{\alpha}^{(t+1)}$ and $\hat{\xi}^{(t+1)}$ are obtained numerically. Here, for $i=1,
\ldots, n$, we have $$z_i^{(t)} = 1 +\hat{\theta}^{(t)} \mathrm{e}^{-\hat{\alpha}^{(t)} H(x_i;\,\hat{\boldsymbol{\xi}}^{(t)})} \frac{C''(\hat{\theta}^{(t)}
\mathrm{e}^{-\hat{\alpha}^{(t)} H(x_i;\,\hat{\boldsymbol{\xi}}^{(t)})})}{C'(\hat{\theta}^{(t)}
\mathrm{e}^{-\hat{\alpha}^{(t)} H(x_i;\,\hat{\boldsymbol{\xi}}^{(t)})})}.$$
Note that, in each step, $\theta, \alpha$ and $\boldsymbol{\xi}$ are estimated independently. The EWPS distributions can be very useful in modeling lifetime data and practitioners may be interested in fitting one of our models.
Maximum entropy identification
==============================
Shannon (1948) introduced the probabilistic definition of entropy which is closely connected with the definition of entropy in statistical mechanics. Let $X$ be a random variable of a continuous distribution with density $f$. Then, the Shannon entropy of $X$ is defined by $$\label{shannon}
\mathbb{H}_{Sh}(f) = - \int_{\mathbb{R}}f(x;\theta,\alpha,
\boldsymbol{\xi})\log\left[f(x;\theta,\alpha, \boldsymbol{\xi})\right] dx.$$
Jaynes (1957) introduced one of the most powerful techniques employed in the field of probability and statistics called the maximum entropy method. This method is closely related to the Shannon entropy and considers a class of density functions $$\label{jaynes}
\mathbb{F} = \left\{f(x;\theta,\alpha, \boldsymbol{\xi}):
\operatorname{E}_{f}(T_i(X)) = \alpha_i,\, i = 0, \ldots, m\right\},$$ where $T_i (X), i = 1, \ldots, m$, are absolutely integrable functions with respect to $f$, and $T_0(X) = a_0 = 1$. In the continuous case, the maximum entropy principle suggests deriving the unknown density function of the random variable $X$ by the model that maximizes the Shannon entropy in , subject to the information constraints defined in the class $\mathbb{F}$. Shore and Johnson (1980) treated axiomatically the maximum entropy method. This method has been successfully applied in a wide variety of fields and has also been used for the characterization of several standard probability distributions; see, for example, Kapur (1989), Soofi (2000) and Zografos and Balakrishnan (2009).
The maximum entropy distribution is the density of the class F, denoted by $f^{ME}$, which is obtained as the solution of the optimization problem $$f^{ME}(x;\theta,\alpha, \boldsymbol{\xi}) = \arg \max_{f \in \mathbb{F}} \mathbb{H}_{Sh}.$$
Jaynes (1957, p. 623) states that the maximum entropy distribution $f^{ME}$, obtained by the constrained maximization problem described above, “is the only unbiased assignment we can make; to use any other would amount to arbitrary assumption of information which by hypothesis we do not have". It is the distribution which should not incorporate additional exterior information other than which is specified by the constraints.
We now derive suitable constraints in order to provide a maximum entropy characterization for our class of distributions defined by . For this purpose, the next result plays an important role.
\[constraints\] Let X be a random variable with pdf given by . Then,
C1.
: $\operatorname{E}\left[\log( C'(\theta \, \mathrm{e}^{-\alpha H(X;\, \boldsymbol{\xi})}))\right] = \dfrac{\theta}{C(\theta)}\operatorname{E}\left[C'(\theta \, \mathrm{e}^{-\alpha H(Y;\, \boldsymbol{\xi})})\log( C'(\theta \, \mathrm{e}^{-\alpha H(Y;\, \boldsymbol{\xi})}))\right];$
C2.
: $\operatorname{E}\left[\log( h(X; \,\boldsymbol{\xi}))\right] = \dfrac{\theta}{C(\theta)}\operatorname{E}\left[C'(\theta \, \mathrm{e}^{-\alpha H(Y;\, \boldsymbol{\xi})})\log( h(Y;\,\boldsymbol{\xi}) )\right];$
C3.
: $\operatorname{E}\left[H(X; \,\boldsymbol{\xi})\right] = \dfrac{\theta}{C(\theta)}\operatorname{E}\left[C'(\theta \, \mathrm{e}^{-\alpha H(Y;\, \boldsymbol{\xi})})H(Y;\,\boldsymbol{\xi} )\right],$
where Y follows the EW distribution with density function .
The constraints C1, C2 and C3 are easily obtained and therefore their demonstrations are omitted.
The next proposition reveals that the EWPS distribution has maximum entropy in the class of all probability distributions specified by the constraints stated in the previous proposition.
The pdf f of a random variable X, given by , is the unique solution of the optimization problem $$f(x;\theta,\alpha, \boldsymbol{\xi}) = \arg \max_{h}
\mathbb{H}_{Sh},$$ under the constraints $\rm{C1}$, $\rm{C2}$ and $\rm{C3}$ presented in the Proposition \[constraints\].
Let $\tau$ be a pdf which satisfies the constraints C1, C2 and C3. The Kullback-Leibler divergence between $\tau$ and $f$ is $$D(\tau, f) = \int_{\mathbb{R}} \tau(x;\theta,\alpha, \boldsymbol{\xi}) \log\left(\frac{\tau(x;\theta,\alpha, \boldsymbol{\xi})}{f(x;\theta,\alpha, \boldsymbol{\xi})}\right) dx.$$
Following Cover and Thomas (1991), we obtain $$\begin{aligned}
0 \leq D(\tau, f) &=& \int_{\mathbb{R}} \tau(x;\theta,\alpha, \boldsymbol{\xi}) \log\left[ \tau(x;\theta,\alpha, \boldsymbol{\xi})\right] dx - \int_{\mathbb{R}} \tau(x;\theta,\alpha, \boldsymbol{\xi}) \log\left[f(x;\theta,\alpha, \boldsymbol{\xi})\right] dx\\
&=& -\mathbb{H}_{Sh}(\tau;\theta,\alpha, \boldsymbol{\xi}) - \int_{\mathbb{R}} \tau(x;\theta,\alpha, \boldsymbol{\xi}) \log\left[
f(x;\theta,\alpha, \boldsymbol{\xi})\right]dx.\end{aligned}$$ From the definition of $f$ and based on the constraints C1, C2 and C3, it follows that $$\begin{aligned}
\hspace{-2cm}\int_{\mathbb{R}} \tau(x) \log\left[f(x)\right] dx &=& \log(\theta \alpha) + \frac{\theta}{C(\theta)}\operatorname{E}\left\{C'(\theta \, \mathrm{e}^{-\alpha H(Y;\, \boldsymbol{\xi})})\log\left[ h(Y;\,\boldsymbol{\xi}) \right]\right\} - \log\left[ C(\theta)\right]\\
&-&\alpha \frac{\theta}{C(\theta)}\operatorname{E}\left[C'(\theta \, \mathrm{e}^{-\alpha H(Y;\, \boldsymbol{\xi})})H(Y;\,\boldsymbol{\xi} )\right] \\
&+& \frac{\theta}{C(\theta)}\operatorname{E}\left\{\log\left[ C'(\theta \, \mathrm{e}^{-\alpha H(Y;\, \boldsymbol{\xi})})\right] C'(\theta \, \mathrm{e}^{-\alpha H(Y;\, \boldsymbol{\xi})})\right\}\\
&=& \int_{\mathbb{R}} f(x;\theta,\alpha, \boldsymbol{\xi}) \log\left[
f(x;\theta,\alpha, \boldsymbol{\xi})\right] dx = - \mathbb{H}_{Sh}(f),\end{aligned}$$ where $Y$ is defined as before. So, we have $\mathbb{H}_{Sh}(\tau) \leq \mathbb{H}_{Sh}(f)$ with equality if and only if $\tau(x;\theta,\alpha, \boldsymbol{\xi}) = f
(x;\theta,\alpha, \boldsymbol{\xi})$ for all $x$, except for a set of measure 0, thus proving the uniqueness.
The intermediate steps in the above proof in fact provide the following explicit expression for the Shannon entropy of the EWPS distribution $$\begin{aligned}
\nonumber
&&\mathbb{H}_{Sh}(f) = -\log(\theta \alpha) - \frac{\theta}{C(\theta)}\operatorname{E}\left\{C'(\theta \, \mathrm{e}^{-\alpha H(Y;\, \boldsymbol{\xi})})\log\left[h(Y;\,\boldsymbol{\xi}) \right]\right\}+ \log\left[C(\theta)\right]\\
&&+\alpha \frac{\theta}{C(\theta)}\operatorname{E}\left[C'(\theta \, \mathrm{e}^{-\alpha H(Y;\, \boldsymbol{\xi})})H(Y;\,\boldsymbol{\xi} )\right] - \frac{\theta}{C(\theta)}\operatorname{E}\left\{C'(\theta \, \mathrm{e}^{-\alpha H(Y;\, \boldsymbol{\xi})})\log\left[C'(\theta \, \mathrm{e}^{-\alpha H(Y;\, \boldsymbol{\xi})})\right]
\right\}.\end{aligned}$$
For some EWPS distributions, the above results can only be obtained numerically.
Special models
==============
In this section, we investigate some special cases of the EWPS class of distributions. We offer some expressions for moments and moments of the order statistics. To illustrate the flexibility of these distributions, we provide plots of the density and hazard rate functions for selected parameter values.
Modified Weibull geometric distribution
---------------------------------------
The modified Weibull geometric (MWG) distribution is defined by the cdf (\[cdf\]) with $H(x;\, \boldsymbol{\xi}) = x^\gamma$ and $C(\theta) = \theta(1-\theta)^{-1}$ leading to $$F(x;\theta,\alpha, \gamma,\lambda) = 1 -
\frac{(1-\theta)\exp\left(-\alpha x^\gamma \mathrm{e}^{\lambda
x}\right)}{1 - \theta \exp\left(-\alpha x^\gamma \mathrm{e}^{\lambda
x}\right)}, \quad x > 0,$$ where $\theta \in (0,1)$. The associated pdf and hazard rate function are $$\begin{aligned}
f(x;\theta,\alpha, \gamma,\lambda) &= \alpha (1-\theta) (\gamma +
\lambda x)\,x^{\gamma-1}\frac{\exp\left(\lambda x - \alpha x^\gamma
\mathrm{e}^{\lambda x}\right)}{\left[1-\theta \exp\left(-\alpha
x^\gamma \mathrm{e}^{\lambda x}\right)\right]^2} \intertext{and}
\tau(x;\theta,\alpha, \gamma,\lambda) &= \alpha(\gamma+\lambda
x)\,x^{\gamma-1}\frac{\exp\left(\lambda x\right)}{1 - \theta
\exp\left(-\alpha x^\gamma \mathrm{e}^{\lambda x}\right)}\end{aligned}$$ for $x > 0$, respectively. The MWG distribution contains the WG distribution (Barreto-Souza *et al*. (2010)) as the particular choice $\lambda = 0$. Further, for $\lambda = 0$ and $\alpha=1$, we obtain the EG distribution (Adamidis and Loukas (1998)). Figures $\ref{fig:densityfigewps}$ and $\ref{fig:hazardfigewps}$ display the density and hazard functions of the MWG distribution for selected parameter values.
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The $r$th raw moment of the random variable $X$ having the MWG distribution has closed-form. It is calculated from (\[linearcomb\]) as $$\label{momentmwg}
E(X^r) = \sum_{n=1}^{\infty} p_n\, \mu_r(n),$$ where $\mu_r(n) = \int_{0}^{\infty}x^r g(x;\, n\alpha, \gamma, \lambda)dx$ denotes the $r$th raw moment of the MW distribution with parameters $n\alpha, \gamma$ and $\lambda$. Here $p_n$ corresponds to the probability function of the geometric distribution. Carrasco [*et al.*]{} (2008) determined an infinite representation for the $r$th raw moment of the MW distribution with these parameters expressed as $$\label{carrascoetal}
\mu_r(n) = \sum_{i_1, \ldots, i_r=1}^{\infty} \frac{A_{i_1, \ldots, i_r}\,\Gamma(s_r/\gamma + 1)}{(n\alpha)^{s_r/\gamma}},$$ where $$A_{i_1, \ldots, i_r} = a_{i_1}, \ldots, a_{i_r} \,\,\,\,\, \mbox{and} \,\,\,\,\, s_r = i_1, \ldots, i_r,$$ and $$a_i = \frac{(-1)^{i+1}i^{i-2}}{(i-1)!}\left(\frac{\lambda}{\gamma}\right)^{i-1}.$$ Hence, the moments of the MWG distribution can be obtained directly from equations (\[momentmwg\]) and (\[carrascoetal\]).
-- -- --
-- -- --
The density of the $i$th order statistic $X_{i:m}$ in a random sample of size $m$ from the MWG distribution is given by (for $i = 1, \ldots, m$) $$f_{i:m}(x) = \frac{m!}{(i-1)! (m-i)!} \sum_{n=1}^\infty
\sum_{j=0}^{i-1} \omega_j\, p_n\,
\left[\frac{(1-\theta)\exp\left(-\alpha x^\gamma \mathrm{e}^{\lambda
x}\right)}{1 - \theta \exp\left(-\alpha x^\gamma \mathrm{e}^{\lambda
x}\right)}\right]^{m+j-i} g(x; n\alpha, \gamma, \lambda),$$ where $g(x; n\alpha, \gamma, \lambda)$ denotes the MW density function with parameters $n\alpha, \gamma$ and $\lambda$. From , we obtain $$\operatorname{E}\left(X_{i:m}^s\right) = \frac{m!}{(i-1)! (m-i)!}\sum_{n=1}^\infty \sum_{j=0}^{i-1} \omega_j\, p_n\,
\operatorname{E}\left\{X^s \left[\frac{(1-\theta)\exp\left(-\alpha
X^\gamma \mathrm{e}^{\lambda X}\right)}{1 - \theta \exp\left(-\alpha
X^\gamma \mathrm{e}^{\lambda X}\right)}\right]^{m+j-i}\right\}.$$
Pareto Poisson distribution
---------------------------
The Pareto Poisson (PP) distribution is defined by taking $H(x;\, \boldsymbol{\xi}) = \log(x/k)$ and $C(\theta) = \mathrm{e}^{\theta}-1$ in , which yields $$F(x; \theta,\alpha, k) = 1 - \frac{\exp\left[\theta \left(k/x\right)^{\alpha}\right]-1}{\mathrm{e}^{\theta}-1}, \quad x\geq k.$$
The pdf and hazard functions of the PP distribution are $$f(x; \theta,\alpha, k) = \frac{\theta\,\alpha\, k^{\alpha} \exp\left[\theta \left(k/x\right)^{\alpha}\right]}{(\mathrm{e}^{\theta}-1)\,x^{\alpha+1}}$$ and $$\tau(x; \theta,\alpha, k) = \frac{\theta\,\alpha \,k^{\alpha}\exp\left[\theta\left(k/x\right)^{\alpha}\right]}{x^{\alpha+1}\left\{\exp\left[\theta\left(k/x\right)^{\alpha}\right]-1\right\}}.$$
We obtain the Pareto distribution as a sub-model when $\theta \rightarrow 0$. The $r$th moment of the random variable $X$ following the PP distribution becomes $$\label{paretomoments}
E(X^r) = \frac{\alpha k^r}{(\mathrm{e}^{\theta}-1)}\sum_{n=1}^{\infty}\frac{\theta^n}{(n-1)!\,(n\alpha-r)},
\quad n\alpha > r.$$
In particular, setting $r = 1$ in , the mean of $X$ reduces to $$\mu = \frac{\alpha k}{\mathrm{e}^{\theta}-1}\sum_{n=1}^{\infty}\frac{\theta^n}{(n-1)!\,(n\alpha-1)}, \quad n\alpha > 1.$$
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-- -- --
-- -- --
-- -- --
From equation (\[eosewps\]), the $s$th moment of the $i$th order statistic, for $i = 1, \ldots, m,$ is given by $$\operatorname{E}\left(X_{i:m}^s\right) = \frac{m!}{(i-1)! (m-i)!}
\sum_{n=1}^\infty \sum_{j=0}^{i-1} \omega_j\, p_n\,
\operatorname{E}\left[X^s \left(\frac{\exp(\theta \left(k/X\right)^{\alpha})-1}{\mathrm{e}^{\theta}-1}\right)^{m+j-i}\right],$$ where $p_n$ denotes the Poisson probability function. Furthermore, after some algebra, the Shannon entropy for the PP distribution reduces to $$\mathbb{H}_{Sh}(f) = \log\left(\frac{\mathrm{e}^\theta-1}{\theta \alpha}\right) - \frac{\theta}{\mathrm{e}^\theta-1}\left(\mu_1 - \alpha \mu_2 + \mu_3\right),$$ where $$\begin{aligned}
\mu_1 &= \operatorname{E}\left[\exp\left\{\theta \left(\frac{k}{X}\right)^\alpha\right\}\log \left(\frac{1}{X}\right)\right] = \frac{1}{2(\mathrm{e}^\theta-1)}\left\{\frac{\mathrm{Chi}(2\theta)-\log(2\theta)+\mathrm{Shi}(2\theta)-\gamma}{\alpha} - (\mathrm{e}^{2\theta}-1)\log k\right\},\\
\mu_2 &= \operatorname{E}\left[\exp\left\{\theta \left(\frac{k}{X}\right)^\alpha\right\}\log \left(\frac{X}{k}\right)\right] = \frac{\mathrm{Chi}(2\theta)-\log(2\theta)+\mathrm{Shi}(2\theta)-\gamma}{2\alpha(\mathrm{e}^\theta-1)}
\intertext{and}
\mu_3 &= \operatorname{E}\left[\theta \exp\left\{\theta \left(\frac{k}{X}\right)^\alpha\right\}\left(\frac{k}{X}\right)^\alpha\right] = \frac{\alpha\,\theta\, k^{2\alpha}}{4(\mathrm{e}^{\theta}-1)}\left\{1- (2\theta+1)\mathrm{e}^{2\theta}\right\},\end{aligned}$$ where $$\operatorname{Chi}(z) = \gamma + \log z + \int_{0}^{z}\frac{\operatorname{cosh}(t)-1}{t}dt$$ is the hyperbolic cosine integral, $$\operatorname{Shi}(z) = \int_{0}^{z}\frac{\operatorname{sinh}(t)-1}{t}dt$$ is the hyperbolic sine integral and $\gamma \approx 0.577216$ is the Euler-Mascheroni constant.
Chen logarithmic distribution
-----------------------------
The Chen logarithmic (CL) distribution is defined by the cdf (\[cdf\]) with $H(x;\, \boldsymbol{\xi}) = \exp(x^\beta)-1$ and $C(\theta) = -\log(1-\theta)$, leading to $$F(x) = 1 - \frac{\log\left\{1-\theta \exp\left[-\alpha (\exp(x^\beta)-1)\right]\right\}}{\log(1-\theta)} , \quad x > 0,$$ where $\theta \in (0,1)$. The associated pdf and hazard rate function (for $x>0$) are $$f(x) = \frac{\theta \alpha b x^{b-1} \exp\left\{x^b - \alpha\left[\exp(x^b)-1\right]\right\}}{\log(1-\theta)\left\{\theta \exp\left[- \alpha(\exp(x^b)-1)\right]-1\right\}}$$ and $$\tau(x) = \frac{\theta \alpha b x^{b-1}\exp\left[x^b - \alpha (\exp(x^b)-1)\right]}{\left\{\theta \exp\left[- \alpha (\exp(x^b)-1)\right]-1\right\}\log\left\{1-\theta \exp\left[- \alpha (\exp(x^b)-1)\right]\right\}},$$ respectively.
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-- -- --
As expected by proposition \[prop1\], we obtain the Chen distribution as a limiting special case when $\theta \rightarrow 0^+$.
-- -- --
-- -- --
The density of the $i$th order statistic $X_{i:m}$ in a random sample of size $m$ from the CL distribution is given by (for $i = 1, \ldots, m$) $$f_{i:m}(x) = \frac{m!}{(i-1)! (m-i)!} \sum_{n=1}^\infty \sum_{j=0}^{i-1} \omega_j^*\, p_n\, g(x; n\alpha, b) \left\{\log\left[1-\theta\exp(\alpha-\alpha\operatorname{e}^{x^b})\right]\right\}^{m+j-1},$$ where $g(x; n\alpha, b)$ is the pdf of the Chen distribution with parameters $n\alpha$ and $b$ and $p_n$ denotes the logarithmic probability mass function and $$\omega_j^* = (-1)^j \binom{i-1}{j}\left[\frac{1}{\log(1-\theta)}\right]^{m+j-1}.$$ In the same way, the $s$th raw moment of $X_{i:m}$ is obtained directly from $$\operatorname{E}\left(X_{i:m}^s\right) = \frac{m!}{(i-1)! (m-i)!}
\sum_{n=1}^\infty \sum_{j=0}^{i-1} \omega_j\, p_n\,
\operatorname{E}\left\{Z^s \exp\left[n\alpha(m+j-1)(1-\exp(Z^b))\right]\right\},$$ where $Z\sim \mbox{Chen}(n\alpha, b)$.
Application
===========
Fonseca and França (2007) studied the soil fertility influence and the characterization of the biologic fixation of $\mathrm{N}_2$ for the *Dimorphandra wilsonii rizz growth*. For 128 plants, they made measures of the phosphorus concentration in the leaves. The data are listed in Table \[tab1aplic\]. We fit the MWG, Gompertz Poisson (GP), PP, Chen Poisson (CP) and CL models to these data. We also fit the three-parameter WG distribution introduced by Barreto-Souza *et al*. (2010). The required numerical evaluations are implemented using the SAS (PROCNLMIXED) and R softwares.
------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
0.22 0.17 0.11 0.10 0.15 0.06 0.05 0.07 0.12 0.09 0.23 0.25 0.23
0.24 0.20 0.08 0.11 0.12 0.10 0.06 0.20 0.17 0.20 0.11 0.16 0.09
0.10 0.12 0.12 0.10 0.09 0.17 0.19 0.21 0.18 0.26 0.19 0.17 0.18
0.20 0.24 0.19 0.21 0.22 0.17 0.08 0.08 0.06 0.09 0.22 0.23 0.22
0.19 0.27 0.16 0.28 0.11 0.10 0.20 0.12 0.15 0.08 0.12 0.09 0.14
0.07 0.09 0.05 0.06 0.11 0.16 0.20 0.25 0.16 0.13 0.11 0.11 0.11
0.08 0.22 0.11 0.13 0.12 0.15 0.12 0.11 0.11 0.15 0.10 0.15 0.17
0.14 0.12 0.18 0.14 0.18 0.13 0.12 0.14 0.09 0.10 0.13 0.09 0.11
0.11 0.14 0.07 0.07 0.19 0.17 0.18 0.16 0.19 0.15 0.07 0.09 0.17
0.10 0.08 0.15 0.21 0.16 0.08 0.10 0.06 0.08 0.12 0.13
------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
: Phosphorus concentration in leaves data set.[]{data-label="tab1aplic"}
-------- --------- --------- -------- --------- -------- --------
Min. $Q_{1}$ $Q_{2}$ Mean $Q_{3}$ Max. Var.
0.0500 0.1000 0.1300 0.1408 0.1800 0.2800 0.0030
-------- --------- --------- -------- --------- -------- --------
: Descriptive statistics. []{data-label="tab2aplic"}
Tables \[tab2aplic\] and \[tab3aplic\] display some descriptive statistics and the MLEs (with corresponding standard errors in parentheses) of the model parameters. Since the values of the Akaike information criterion (AIC), Bayesian information criterion (BIC) and consistent Akaike information criterion (CAIC) are smaller for the CL distribution compared with those values of the other models, this new distribution seems to be a very competitive model for these data.
------- ----------- ----------- ---------- ----------- -- ---------- ---------- ----------
Model $\theta$ $\alpha$ $\gamma$ $\lambda$ AIC BIC AICC
MWG 0.7200 409.07 3.6545 $-$0.5727 $-$385.6 $-$374.2 $-$385.3
(0.2418) (1174.76) (0.821) (6.6673)
WG 0.9995 2.4471 4.2041 $-$ $-$378.5 $-$370.0 $-$378.3
(0.0017) (8.7059) (0.3022) $-$
$\theta$ $\alpha$ $\beta$
GP 2.9478 0.3169 19.7047 $-$368.7 $-$360.2 $-$368.5
(1.2627) (0.1473) (1.6135)
$\theta$ $\alpha$ $k$
PP 80.0903 0.0131 0.0500 $-$271.4 $-$265.7 $-$271.3
(69.7770) (0.0115)
$\theta$ $\alpha$ $b$
CP 15.4386 14.7817 2.9212 $-$383.7 $-$375.2 $-$383.5
(22.8318) (28.1576) (0.2634)
CL 0.9999 52232 7.5882 $-$395.8 $-$387.2 $-$395.6
(0.0001) (0.0000) (0.2039)
------- ----------- ----------- ---------- ----------- -- ---------- ---------- ----------
: MLEs of the model parameters, the corresponding SEs (given in parentheses) and the statistics AIC, BIC and AICC.[]{data-label="tab3aplic"}
Plots of the estimated pdf and cdf of the MWG, WG, GP, PP, CP and CL models fitted to these data are displayed in Figure \[fig:cdfplot\]. They indicate that the CL distribution is superior to the other distributions in terms of model fitting.
-- --
-- --
Table \[tab4aplic\] lists the values of the Kolmogorov-Smirnov (K-S) statistic and the values of $-2\ell(\widehat{\Theta})$. From these figures, we conclude that the CL distribution provides a better fit to these data than the MWG, WG, GP, PP and CP models.
Model K–S $-2\ell(\widehat{\Theta})$
------------------ -- -- -- -------- -- -- -- ---------------------------- -- -- -- --
MWG 0.0943 $-$393.6
WG 0.0873 $-$384.5
Gompertz Poisson 0.1201 $-$374.7
Pareto Poisson 0.3017 $-$374.7
Chen Poisson 0.1159 $-$389.7
Chen Logarithmic 0.0678 $-$401.8
: K-S statistics and $-2\ell(\widehat{\Theta})$ for the exceedances of phosphorus concentration in leaves data set.[]{data-label="tab4aplic"}
Concluding remarks
==================
We define a new lifetime class of distributions, called the extended Weibull power series (EWPS), which generalizes the Weibull power series class of distributions proposed by Morais and Barreto-Souza (2011), which in turn extends the exponential power series class of distributions (Chahkandi and Ganjali, 2009). We provide a mathematical treatment of the new distribution including expansions for the density function, moments, generating function and incomplete moments. Further, explicit expressions for the order statistics and Shannon entropy are derived. The EWPS density function can be expressed as a mixture of EW density functions. This property is important to obtain several other results. Our formulas related with the EWPS model are manageable, and with the use of modern computer resources with analytic and numerical capabilities, they may turn into adequate tools comprising the arsenal of applied statisticians. The estimation of the model parameters is approached by the method of maximum likelihood using the EM algorithm. The observed information matrix is derived. Further, maximum entropy identification for the EWPS distributions was discussed and some special models are studied in some detail. Finally, we fit the EWPS model to a real data set to show the usefulness of the proposed class. We hope that this generalization may attract wider applications in the literature of the fatigue life distributions.
Acknowledgements {#acknowledgements .unnumbered}
================
We also gratefully acknowledge financial support from CAPES and CNPq.
\[apA\] The elements of the $p \times p$ information matrix $J_n(\Theta)$ are $$\begin{aligned}
J_{\theta \theta} &= - \frac{n}{\theta^2} - n\left[\frac{C''(\theta)}{C(\theta)} - \left(\frac{C'(\theta)}{C(\theta)}\right)^2\right] + \theta \sum_{i=1}^n \left(\frac{z_{2i}}{z_{1i}}\right)^2 H(x_i;\, \xi) \mathrm{e}^{-2\alpha H(x_i;\, \xi)}\\
&- \theta \sum_{i=1}^n \frac{z_{3i}}{z_{1i}} H(x_i;\, \boldsymbol{\xi}) \mathrm{e}^{-2\alpha H(x_i;\, \boldsymbol{\xi})}\\
J_{\alpha \alpha} &= - \frac{n}{\alpha^2} + \theta \sum_{i=1}^n \frac{z_{2i}}{z_{1i}} H^2(x_i; \boldsymbol{\xi}) \mathrm{e}^{-\alpha H(x_i;\, \xi)} + \theta^2 \sum_{i=1}^n \frac{(z_{3i}-z_{2i}^2)}{z_{1i}}H^2(x_i; \xi) \mathrm{e}^{-2\alpha H(x_i;\, \xi)}\\
J_{\alpha \theta} &= \theta \sum_{i=1}^n \left[\left(\frac{z_{2i}}{z_{1i}}\right)^2 - \frac{z_{3i}}{z_{1i}}\right] H^2(x_i; \xi) \mathrm{e}^{-2\alpha H(x_i;\, \xi)} - \sum_{i=1}^n \frac{z_{2i}}{z_{1i}} H^2(x_i; \xi) \mathrm{e}^{-\alpha H(x_i;\, \xi)} \\
J_{\alpha \boldsymbol{\xi}_k} &= - \sum_{i=1}^n \frac{\partial H(x_i;\, \boldsymbol{\xi})}{\partial \xi_k} - \theta \sum_{i=1}^n \frac{z_{2i}}{z_{1i}}\frac{\partial H(x_i;\, \xi)}{\partial \xi_k} \mathrm{e}^{-\alpha H(x_i;\, \boldsymbol{\xi})} \left[1 - \alpha H(x_i;\, \boldsymbol{\xi})\right] \\
&+ \alpha \theta^2 \sum_{i=1}^n \left[\frac{z_{3i}}{z_{1i}}-\left(\frac{z_{2i}}{z_{1i}}\right)^2 \right] \frac{\partial H(x_i;\, \xi)}{\partial \boldsymbol{\xi}_k}H(x_i;\, \xi)\mathrm{e}^{-2\alpha H(x_i;\, \boldsymbol{\xi})}\\
J_{\theta \boldsymbol{\xi}_k} &= \theta \alpha \sum_{i=1}^n \left[\left(\frac{z_{2i}}{z_{1i}}\right)^2 - \frac{z_{3i}}{z_{1i}}\right] \frac{\partial H(x_i;\, \boldsymbol{\xi})}{\partial \boldsymbol{\xi}_k} \mathrm{e}^{-2\alpha H(x_i;\, \xi)} - \alpha \sum_{i=1}^n \frac{z_{2i}}{z_{1i}} \frac{\partial H(x_i;\, \boldsymbol{\xi})}{\partial \boldsymbol{\xi}_k} \mathrm{e}^{-\alpha H(x_i;\, \boldsymbol{\xi})}\\
J_{\xi_k \boldsymbol{\xi}_l} &= -\alpha \sum_{i=1}^n \frac{\partial^2 H(x_i;\, \boldsymbol{\xi})}{\partial \boldsymbol{\xi}_k \partial \boldsymbol{\xi}_l} - \sum_{i=1}^n \frac{1}{H(x_i;\, \boldsymbol{\xi})^2}\frac{\partial H(x_i;\, \boldsymbol{\xi})}{\partial \boldsymbol{\xi}_k}\frac{\partial H(x_i;\, \boldsymbol{\xi})}{\partial \boldsymbol{\xi}_l} + \sum_{i=1}^n \frac{1}{H(x_i;\, \boldsymbol{\xi})}\frac{\partial^2 H(x_i;\, \boldsymbol{\xi})}{\partial \boldsymbol{\xi}_k \partial \boldsymbol{\xi}_l}\\
&+ (\alpha \theta)^2 \sum_{i=1}^n \left[\left(\frac{z_{2i}}{z_{1i}}\right)^2 + \frac{z_{3i}}{z_{1i}}\right] \frac{\partial H(x_i;\, \boldsymbol{\xi})}{\partial \xi_k} \frac{\partial H(x_i;\, \boldsymbol{\xi})}{\partial \boldsymbol{\xi}_l} \mathrm{e}^{-2\alpha H(x_i;\, \boldsymbol{\xi})} \\
&- \alpha \theta \sum_{i=1}^n \frac{z_{2i}}{z_{1i}} \frac{\partial^2
H(x_i;\, \xi)}{\partial \boldsymbol{\xi}_k \partial
\boldsymbol{\xi}_l} \mathrm{e}^{-\alpha H(x_i;\, \boldsymbol{\xi})} + \alpha^2
\theta \sum_{i=1}^n \frac{z_{2i}}{z_{1i}} \frac{\partial H(x_i;
\xi)}{\partial \boldsymbol{\xi}_k} \frac{\partial H(x_i;
\boldsymbol{\xi})}{\partial \boldsymbol{\xi}_l}\mathrm{e}^{-\alpha H(x_i;
\boldsymbol{\xi})}\end{aligned}$$ where $z_{1i} = C'(\theta e^{-\alpha H(x_i;\, \boldsymbol{\xi})}),
z_{2i} = C''(\theta \mathrm{e}^{-\alpha H(x_i;\, \boldsymbol{\xi})})$ and $z_{3i} = C'''(\theta \mathrm{e}^{-\alpha H(x_i;\, \boldsymbol{\xi})})$, for $i
= 1, \ldots, n$.
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|
---
abstract: |
The following multi-determinantal algebraic variety plays a central role in algebra, algebraic geometry and computational complexity theory: ${{\rm SING}}_{n,m}$, consisting of all $m$-tuples of $n\times n$ complex matrices which span [*only*]{} singular matrices. In particular, an efficient deterministic algorithm testing membership in ${{\rm SING}}_{n,m}$ will imply super-polynomial circuit lower bounds, a holy grail of the theory of computation.
A sequence of recent works suggests such efficient algorithms for memberships in a general class of algebraic varieties, namely the [*null cones*]{} of linear group actions. Can this be used for the problem above? Our main result is negative: ${{\rm SING}}_{n,m}$ is [*not*]{} the null cone of any (reductive) group action! This stands in stark contrast to a non-commutative analog of this variety, and points to an inherent structural difficulty of ${{\rm SING}}_{n,m}$.
To prove this result we identify precisely the group of symmetries of ${{\rm SING}}_{n,m}$. We find this characterization, and the tools we introduce to prove it, of independent interest. Our work significantly generalizes a result of Frobenius for the special case $m=1$, and suggests a general method for determining the symmetries of algebraic varieties.
address:
- 'School of Mathematics, Institute for Advanced Study, Princeton'
- 'School of Mathematics, Institute for Advanced Study, Princeton'
author:
- Visu Makam
- Avi Wigderson
bibliography:
- 'refs.bib'
title: |
Singular tuples of matrices is not a null cone\
(and, the symmetries of algebraic varieties)
---
Introduction {#sec:intro}
============
We start the introduction with a general discussion of the main problems and their motivations. Next we turn to describe our main object of study - singular spaces of matrices. We end by formally stating our main results. While a few technical terms here may be unfamiliar to some readers, we will have a simple running example to demonstrate all essential notions. Throughout, the underlying field is the complex numbers ${{\mathbb C}}$.
Motivation and main problems
----------------------------
Consider a (reductive[^1]) group $G$ acting (algebraically) on a vector space $V$ by linear transformations. Understanding this very general setting is the purview of [*invariant theory*]{}. As a simple, and very relevant running example, consider the following.
\[Running example\] \[E-running\] Consider $G= {\operatorname{SL}}_n$ acting on $n\times n$ matrices (namely $V={{\mathbb C}}^{n^2}$) by left multiplication, i.e, the action of $P \in {\operatorname{SL}}_n$ sends the matrix $X$ to $PX$.
A group action partitions $V$ into [*orbits*]{}: the orbit of $v$ is the set of all points in $V$ it can be moved to by an element $g\in G$. An even more natural object in our setting is the [*orbit closure*]{}: all limit points of an orbit[^2].
The [*null cone*]{} of a group action is the set of points $v \in V$ whose orbit closure contains the origin, namely the point $0$. Null cones of group actions are central to invariant theory, and are interesting algebraic objects to study in mathematics and physics. More recently, connections to fundamental problems in computational complexity have surfaced. Diverse problems (see [@GGOW16; @BGFOWW]) such as bipartite matching, equivalence of non-commutative rational expressions, tensor scaling and quantum distillation, can each be formulated (for specific choices of $G,V$ and an action) as a [*null cone membership problem*]{} – given a point $v \in V$, decide if it is in the null cone. Note that in our running example, i.e., Example \[E-running\], the null cone is precisely the set of singular matrices.
A closely related problem is the [*orbit closure intersection problem*]{} – given $v,w \in V$, decide if the orbit closure of $v$ and $w$ intersect. The orbit closure intersection problem is a generalization of the null cone membership problem, and this too has many connections with arithmetic complexity. For example, the graph isomorphism problem can be phrased as an orbit closure intersection problem! We refer to [@GCTV] for more details on the aforementioned problems and their relevance in the Geometric Complexity Theory (GCT) program, which is an algebro-geometric approach to the VP vs VNP problem (an algebraic analog of P vs NP). Note that in Example \[E-running\], the orbit closure of two matrices $X$ and $Y$ intersect precisely when $\det(X)=\det(Y)$.
In an exciting series of recent works, efficient algorithms for the null cone membership and orbit closure intersection problems in various cases have been discovered, and moreover techniques have developed that may allow significant generalization of their applicability [@GGOW16; @IQS2; @FS; @DM; @DMOC; @GGOW18; @AGLOW18; @BGFOWW; @BGFOWW2; @CF; @DM-arbchar; @DM-siq]. Curiously, Geometric Complexity Theory (morally) predicts efficient algorithms for null cone membership problems in great generality (see [@GCTV] for precise formulations), although establishing this remains an elusive goal.
What is remarkable is the possibility that such efficient algorithms, through the work of [@KI], may enable proving non-trivial [*lower bounds*]{} on computation, the major challenge of computational complexity. Specifically, what is needed is a deterministic polynomial time algorithm for a problem called Symbolic Determinant Identity Testing (SDIT)[^3] that is central to this work, and will be defined soon. SDIT happens to be a membership problem in an [*algebraic subvariety*]{}, a context generalizing null cones.
A subset $S \subseteq V$ is called an algebraic subvariety[^4] (or simply a [*subvariety*]{}) if it is the zero locus of a collection of polynomial functions on $V$. Many algorithmic problems can be phrased as “membership in a subvariety”, and is non-trivial when the underlying set of polynomials is given implicitly or are difficult to compute. It is a fundamental result of invariant theory that [*every null cone is an algebraic subvariety*]{}, a connection which goes through [*invariant polynomials*]{} of group actions. A polynomial function $f$ on $V$ is called invariant if it is constant along orbits, i.e., $f(gv) = f(v)$ for all $g \in G, v \in V$. Invariant polynomials form a graded subring of ${{\mathbb C}}[V]$, the ring of polynomial functions on $V$. Mumford proved that the orbit closures of any two points $v,w \in V$ intersect, if and only if $f(v) = f(w)$ for all invariant polynomials[^5], see [@Mumford]. As a consequence, the null cone can also be described as the zero locus of all (non-constant) homogenous invariant polynomials. Indeed, this analytic-algebraic connection provides the path to structural and algorithmic understanding of the null cone membership and orbit closure intersection problems via invariant theory.
Summarizing, if a subvariety $S \subseteq V$ happens to be a null cone for some group action, then the aforementioned algorithms can be used to decide “membership in $S$”, with the exciting possibility that they could very well be efficient. Of course, not every subvariety is a null cone, which leads to the following interesting problem:
\[prob.nullcone\] Given a subvariety $S \subseteq V$, is it the null cone for the (algebraic) action of a (reductive) group $G$ on $V$?
We specifically refer to $S$ as a [*subvariety*]{} of $V$ rather than just call it a [*variety*]{} for the following reason. In the above problem, it is important that we view $S$ as a subset of $V$. As an abstract variety, a different embedding of $S$ into another vector space could very well make it a null cone.[^6] Our setting of a given embedding makes the problem well-defined.
We now make an important observation. If $S$ is to be the null cone for the action of a group $G$, then the group must “preserve” $S$, i.e., for all $g \in G$, we must have $gS = S$. We define the [*group of symmetries*]{} to be the (largest) subgroup of ${\operatorname{GL}}(V)$ consisting of all linear transformations that preserve $S$. With reference to Example \[E-running\], one might ask which is the largest group of symmetries in ${\operatorname{GL}}_{n^2}$ which preserves the set $n\times n$ the singular matrices (which is defined by the zeros of the single determinant polynomial). This question was resolved by Frobenius [@Frob] as we will later see, and is a very special case of our main technical result.
So, the (hypothetical) acting group $G$ must be a subgroup[^7] of the group of symmetries of $S$. Roughly speaking, this provides an important “upper bound” to the groups that one must consider while resolving Problem \[prob.nullcone\].
\[prob.gos\] Given a subvariety $S \subseteq V$, compute its group of symmetries.
Needless to say, the important role of symmetries in mathematics in present just about every branch, and exploiting symmetries is an immensely powerful tool. Specifically, the fact that the determinant and permanent polynomials are [*defined*]{} by their symmetries form the starting point to the GCT of Mulmuley and Sohoni [@MS01; @MS02] program mentioned earlier towards the VP $\neq$ VNP conjecture. Computing the group of symmetries of an algebraic variety is an extremely natural problem (even in the absence of Problem \[prob.nullcone\]!), and may be useful for other purposes. We now elaborate informally on the path we take to solve Problem \[prob.gos\], and another natural problem it raises.
The group of symmetries of an algebraic subvariety $S \subseteq V$ is always an algebraic subgroup of ${\operatorname{GL}}(V)$ (and hence a Lie subgroup). Suppose that $H$ is an algebraic group that acts linearly on a vector space $V$. It is a fact that the null cone for the action of its identity component[^8] (denoted $H^\circ$) is the same as the null cone for the action of $H$. Thus, for Problem \[prob.nullcone\], one might as well study the [*connected group of symmetries*]{}, i.e., the identity component of the group of symmetries. Indeed, if $S$ is the null cone for the action of a reductive group $G$, then it is the null cone for the action of its identity component $G^\circ$, which must be a subgroup of the connected group of symmetries. Thus, we are led to problem below.
\[prob.cgos\] Given a subvariety $S \subseteq V$, compute its connected group of symmetries.
To understand that Problem \[prob.cgos\] really is much easier than Problem \[prob.gos\], one needs to realize that connected group of symmetries is a connected algebraic subgroup of ${\operatorname{GL}}(V)$, and so in particular is determined by its Lie algebra (which is a Lie subalgebra of the Lie algebra of ${\operatorname{GL}}(V)$). Roughly speaking, we will use this to “linearize” the problem.
Algebraic subvarieties are defined as the zero locus of a collection of polynomials. Suppose we have a collection of homogeneous polynomials $\{f_i:i \in I\}$, and let $S$ be its zero locus. If the ring of invariants for the action of some group $G$ is precisely ${{\mathbb C}}[f_i : i \in I]$, then $S$ would be the null cone (recall that the null cone can be seen as the zero locus of non-constant homogeneous invariant polynomials). This brings us to another interesting problem, which can be seen as a scheme-theoretic version of Problem \[prob.nullcone\]
\[prob.invring\] Given a collection of polynomials $\{f_i: i \in I\}$ on $V$, is there a group $G$ acting on $V$ by linear transformations such that the ring of invariants is ${{\mathbb C}}[\{f_i: i \in I\}]$.
Curiously, the above problem is in some sense is an inverse problem to the classical one in invariant theory: there, given a group action on $V$, we seek its invariant polynomials, whereas here we are given the polynomials, and seek the group which makes them all invariant.
Both Problem \[prob.gos\] and Problem \[prob.invring\] belong to a general class of problems called [*linear preserver problems*]{}. We refer the reader to the survey [@LP-survey] which contains in particular some general techniques for approaching linear preserver problems. These techniques do not seem to be sufficient for us.
Finally, let us mention that all the aforementioned problems are very natural, interesting in their own right, and could potentially use tools from invariant theory, representation theory, Lie theory, algebraic geometry, commutative algebra and computational complexity.
The algebraic variety SING and the computational problem SDIT
-------------------------------------------------------------
Having introduced the problems of interest, let us introduce the subvariety which we will be the main focus of this paper. Let ${\operatorname{Mat}}_{n}$ denote $n \times n$ matrices with entries in ${{\mathbb C}}$. Let $t_1,\dots,t_m$ be indeterminates, and let ${{\mathbb C}}(t_1,\dots,t_m)$ denote the function field in $m$ indeterminates. Define $$\begin{aligned}
{{\rm SING}}_{n,m} \triangleq \left\{X = (X_1,\dots,X_m)\in {\operatorname{Mat}}_{n}^m\ |\ \sum_{i=1}^m t_i X_i \text{ singular (over ${{\mathbb C}}(t_1,\dots,t_m)$)}\right\}.\end{aligned}$$
Note that ${{\rm SING}}_{n,m} \subseteq V={{\mathbb C}}^{mn^2}$, given by the zero locus of all polynomials $\{\det(c_1X_1 + c_2X_2+ \dots + c_mX_m)\,:\, c_i \in {{\mathbb C}}\}$. While this is an uncountable set, one can easily make it finite. Another important note is that the case $m=1$ is the null cone for our simple running example (Example \[E-running\]) of the previous subsection!
The subvariety ${{\rm SING}}_{n,m}$ is of central importance in computational complexity. The membership problem for ${{\rm SING}}_{n,m}$ (i.e., given $X \in {\operatorname{Mat}}_{n}^m$, decide if $X \in {{\rm SING}}_{n,m}$) is often called Symbolic Determinant Identity Testing (SDIT). This problem SDIT is also sometimes referred to as the [*Edmonds’ problem*]{}, as Edmond’ paper [@Edm67] first explicitly defined it and asked if it has a polynomial time algorithm. Note that any [*fixed*]{} tuple $X = (X_1,\dots,X_m) \in {{\rm SING}}_{n,m}$ if and only if the [*symbolic*]{} determinant $\det(t_1X_1 + t_2X_2+ \dots t_mX_m)$ vanishes identically when viewed now as a polynomial in the new variables $t_1,\dots,t_m$. This viewpoint immediately provides an efficient [*probabilistic*]{} algorithm for the SDIT [@Lovasz'79]: given $X$, simply pick (appropriately) at random values for the variables $t_i$ and evaluate the resulting [*numeric*]{} determinant.
The importance of determining the complexity of SDIT stems from several central results in arithmetic complexity and beyond. First, Valiant’s completeness theorem for VP [@Valiant] implies that SDIT captures the general problem of Polynomial Identity Testing (PIT) problem (see the survey [@SY], for background and status of this problem, and more generally on arithmetic complexity). An equivalent way of phrasing Valiant’s result is that SDIT [*is*]{} the [*word problem*]{} for ${{\mathbb C}}(t_1,\dots,t_m)$, namely testing if a rational expression in ${{\mathbb C}}(t_1,\dots,t_m)$ is identically zero. A second, and far more surprising result we already mentioned, of Kabanets and Impagliazzo (see [@KI]), shows that efficient [*deterministic*]{} algorithms for PIT would imply circuit lower bounds, a holy grail of complexity theory. SDIT also plays an important role in the GCT program, see [@GCTV]. Finally, the structural study of the variety ${{\rm SING}}_{n,m}$, namely of singular spaces of matrices is a rich subject in linear algebra and geometry (see e.g. [@FR; @EH; @RW; @RM1; @RM2; @RM3]).
It is illustrative to compare with the non-commutative version of the above story, and we will do so. Let $t_1,\dots,t_m$ be now [*non-commuting*]{} indeterminates, and let ${{\mathbb C}}{\,\,(\!\!\!\!<\!}t_1,\dots,t_m {\!>\!\!\!\!)\,\,}$ denote the free skew field[^9]. Consider $${{\rm NSING}}_{n,m} \triangleq \left\{X = (X_1,\dots,X_m) \in {\operatorname{Mat}}_{n}^m\ | \ \sum_i t_iX_i \text{ singular (over ${{\mathbb C}}{\,\,(\!\!\!\!<\!}t_1,\dots,t_m {\!>\!\!\!\!)\,\,}$)} \right\},$$ which is clearly a non-commutative analog of ${{\rm SING}}_{n,m}$. Moreover, membership in ${{\rm NSING}}_{n,m}$ captures the word problem over the free skew field ${{\mathbb C}}{\,\,(\!\!\!\!<\!}t_1,\dots,t_m {\!>\!\!\!\!)\,\,}$ (often called non-commutative rational identity testing (RIT)) in precisely the same manner as membership in ${{\rm SING}}_{n,m}$ captures the word problem over the function field ${{\mathbb C}}(t_1,\dots,t_m)$.
The surprising fact is that membership in ${{\rm NSING}}_{n,m}$ [*does*]{} have polynomial time deterministic algorithms, see [@GGOW16; @IQS2]. The main point to note is that the algorithms use crucially the fact that ${{\rm NSING}}_{n,m}$ is a null cone! Indeed, it is the null cone for the so called left-right action of ${\operatorname{SL}}_n \times {\operatorname{SL}}_n$ on ${\operatorname{Mat}}_{n}^m$ which is defined by: $$(P,Q) \cdot (X_1,\dots,X_m) = (PX_1Q^t,PX_2Q^t,\dots,PX_mQ^t),$$ where $Q^t$ denotes the transpose of the matrix $Q$. In view of this, it is only natural to ask whether a similar story can be used to give an efficient algorithm for membership in ${{\rm SING}}_{n,m}$. This provides the principal motivation for studying Problem \[prob.nullcone\].
For $S = {{\rm SING}}_{n,m}$, in this paper, we will answer Problem \[prob.nullcone\] and Problem \[prob.gos\] (and hence also Problem \[prob.cgos\]). Moreover, recall that ${{\rm SING}}_{n,m}$ is the zero locus of a natural collection of polynomials, namely $\{\det(\sum_i c_i X_i) : c_i \in {{\mathbb C}}\}$. We also give a negative answer to Problem \[prob.invring\] for this collection of polynomials. We will now proceed to give precise statements.
Main results
------------
We begin by stating the main result, i.e., a negative answer to Problem \[prob.nullcone\] for ${{\rm SING}}_{n,m}$.
\[theo:nullcone\] Let $n,m \geq 3$. Let $G$ be any reductive group acting algebraically on ${\operatorname{Mat}}_{n}^m$ by linear transformations. Then the null cone for the action of $G$ is not equal to ${{\rm SING}}_{n,m}$.
First, and foremost, let us observe that the condition $n,m \geq 3$ cannot be removed or even improved. Indeed, if $n \leq 2$ or $m \leq 2$, we have ${{\rm SING}}_{n,m} = {{\rm NSING}}_{n,m}$ and hence it is a null cone! Thus, the above theorem gives the strongest possible statement of this nature. The above theorem follows from the following one, which has no restrictions on $n$ and $m$.
\[theo:nullcone2\] Let $G$ be any reductive group acting algebraically on $V = {\operatorname{Mat}}_{n}^m$ by linear transformations which preserve ${{\rm SING}}_{n,m}$ (i.e., $g \cdot {{\rm SING}}_{n,m} = {{\rm SING}}_{n,m}$ for all $g \in G$). Let $\mathcal{N} = \mathcal{N}_G(V)$ denote the null cone for this action. If the null cone $\mathcal{N} \subseteq{{\rm SING}}_{n,m}$, then the null cone $\mathcal{N} \subseteq {{\rm NSING}}_{n,m}$.
Indeed, Theorem \[theo:nullcone\] follows from the above theorem as $n,m \geq 3$ is precisely the condition needed to ensure that ${{\rm NSING}}_{n,m}$ is a proper subset of ${{\rm SING}}_{n,m}$.
A crucial component in the proof of the above theorem is the computation of the group of symmetries for ${{\rm SING}}_{n,m}$. The importance of this computation is well beyond the context of this paper. For example, it should serve as the starting point for any approach to SDIT that aims at utilizing symmetry. Let us formally define the group of symmetries for a subvariety.
\[Group of symmetries\] For a subvariety $S \subseteq V$, we define its group of symmetries $$\mathcal{G}_S = \{g \in {\operatorname{GL}}(V) \ |\ gS = S\}.$$ The group of symmetries $\mathcal{G}_S$ is always an algebraic subgroup of ${\operatorname{GL}}(V)$. We call its identity component (denoted $\mathcal{G}_S^\circ$) the connected group of symmetries.
In order to compute the group of symmetries for ${{\rm SING}}_{n,m}$, we first compute the connected group of symmetries. Viewing ${\operatorname{Mat}}_{n}^m$ as ${{\mathbb C}}^m \otimes {{\mathbb C}}^n \otimes {{\mathbb C}}^n$ elucidates a natural linear action of ${\operatorname{GL}}_m \times {\operatorname{GL}}_n \times {\operatorname{GL}}_n$ on ${\operatorname{Mat}}_{n}^m$. Concretely, the action is given by the formula: $$(P,Q,R) \cdot (X_1,\dots,X_m) = \left(\sum_{j=1}^m p_{1j} QX_jR^{-1}, \sum_{j=1}^m p_{2j} QX_j R^{-1},\dots, \sum_{j=1}^m p_{nj} QX_j R^{-1}\right),$$ where $p_{ij}$ denotes the $(i,j)^{th}$ entry of $P$. A linear action is simply a representation, so we have a map ${\operatorname{GL}}_m \times {\operatorname{GL}}_n \times {\operatorname{GL}}_n \rightarrow {\operatorname{GL}}({\operatorname{Mat}}_{n}^m)$. We will call the image of this map $G_{n,m}$.
\[theo:cgos\] Let $S = {{\rm SING}}_{n,m} \subseteq V = {\operatorname{Mat}}_{n}^m$. Then the connected group of symmetries $\mathcal{G}_S^\circ$ is the subgroup $G_{n,m}$.
We will discuss in the subsequent section, the strategy of proof in more detail for the above theorem. However, it is worth mentioning that it is essentially a linear algebraic computation on the level of Lie algebras, and is applicable in more generality. At this juncture, we note a classical result of Frobenius that addresses the special case of $m =1$ (see [@Frob; @Dieudonne]), which deals with our simple running example earlier. This result is essential for our proof of the above theorem for any value of $m$. We will also give our own proof of this result as it allows us to illustrate our proof strategy in the simple case.
\[Frobenius\] \[theo:frob\] Let $S = {{\rm SING}}_{n,1} \subseteq V = {\operatorname{Mat}}_{n}$. The group of symmetries $\mathcal{G}_S$ consists of linear transformations of the form $X \mapsto PXQ$ or of the form $X \mapsto PX^tQ$ where $P,Q \in {\operatorname{SL}}_n$.
First, note that the above result computes the entire group of symmetries! In the general case, let us first note that apriori there could be an incredible number of groups whose identity component is $G_{n,m}$. However, it turns out that they are actually manageable, and with some fairly elementary results on semisimple Lie algebras, we can determine the entire group of symmetries for any $m$.
\[theo:gos\] Let $S = {{\rm SING}}_{n,m} \subseteq V = {\operatorname{Mat}}_{n}^m$. Let $\tau$ denote the linear transformation that sends $X = (X_1,\dots,X_m) \mapsto (X_1^t,\dots,X_m^t)$. Then the group of symmetries $\mathcal{G}_S = G_{n,m} \cup G_{n,m} \cdot \tau = G_{n,m} \rtimes {{\mathbb Z}}/2$.
The key idea here is that the entire group of symmetries must normalize the connected group of symmetries, i.e., $G_{n,m}$. So, we compute the normalizer of $G_{n,m}$. To do so, we utilize heavily that the group $G_{n,m}$ is reductive, and use ad-hoc arguments that are particularly suited to this special case. A slightly more abstract approach via automorphisms of Dynkin diagrams such as the one in [@Guralnick] would work in this case (see also [@Lbook17]). We do not quite know a general strategy to bridge the gap between the connected group of symmetries and the entire group of symmetries. We also note that the same strategy yields the group of symmetries for ${{\rm NSING}}_{n,m}$
\[theo:ngos\] Let $S = {{\rm NSING}}_{n,m} \subseteq {\operatorname{Mat}}_{n}$. Then the group of symmetries $\mathcal{G}_S = G_{n,m} \rtimes {{\mathbb Z}}/2$ (as defined in the above theorem).
Once we compute the group of symmetries, the rest of the argument relies on an understanding of the Hilbert–Mumford criterion (see Theorem \[theo:HM\]) which tells us that the null cone is a union of $G$-orbits of coordinate subspaces (linear subspaces that are defined by the vanishing of a subset of coordinates, see Definition \[D-coordsubspace\]). In particular, we will show that the union of all the coordinate subspaces contained in ${{\rm SING}}_{n,m}$ moved around by the action of its group of symmetries does not cover all of ${{\rm SING}}_{n,m}$, which will give the contradiction. We explain this idea in more detail in Section \[subs:proofidea\].
\[Positive characteristic\] Our choice in working with ${{\mathbb C}}$ as a ground field is essentially for simplicity of the exposition and proofs. All our results above (specifically Theorems \[theo:nullcone\], \[theo:nullcone2\], \[theo:cgos\], \[theo:frob\], \[theo:gos\] and \[theo:ngos\]) hold for every algebraically closed fields of every characteristic. In Appendix \[app.pos.char\], we discuss the issues that arise in positive characteristic and the appropriate modifications needed to deal with them.
The subvariety ${{\rm SING}}_{n,m}$ is the zero locus of some very structured polynomials. Observe that for any $c_i \in {{\mathbb C}}$, the polynomial $\det(\sum_i c_iX_i)$ vanishes on ${{\rm SING}}_{n,m}$. It is easy to see that the zero locus of the collection of all $\det(\sum_i c_i X_i)$ (for all choices of $c_i$) is precisely ${{\rm SING}}_{n,m}$[^10]. We prove a negative result for Problem \[prob.invring\] for this collection of polynomials.
\[theo:invring\] Suppose $n,m \geq 3$. Then the subring $R = {{\mathbb C}}[\{\det(\sum_i c_i X_i) : c_i \in {{\mathbb C}}\}] \subseteq {{\mathbb C}}[{\operatorname{Mat}}_{n}^m]$ is not the invariant ring for any linear action of any group $G$ on ${\operatorname{Mat}}_{n}^m$.
If we restrict to reductive groups, then the above theorem is a simple consequence of Theorem \[theo:nullcone\] and the alternate definition of null cone as the zero locus of non-constant homogenous invariants. However, we use a different argument that works for [*any*]{} group, irrespective of reductivity.
Organization
------------
In Section \[sec:invthry\], we recall the basic notions from invariant theory and null cones as well as the crucial Hilbert–Mumford criterion. It also contains a sketch of the proof strategy for proving Theorem \[theo:nullcone\]. In Section \[sec:gos\], we present the theoretical statements that we will use in the computation of the group of symmetries, and in particular, we describe the role of Lie algebras. This is followed by Section \[sec:explicit\], which contains an explicit description of the action of the Lie algebra on polynomials, which is vital for our computations. The ideal of polynomials vanishing on ${{\rm SING}}_{n,m}$ is discussed in Section \[sec:van\], with proofs pushed into the appendix. The group of symmetries for ${{\rm SING}}_{n,1}$ (i.e., the important special case of $m = 1$) is computed in Section \[sec:frob\]. While the statement was already known (due to Frobenius), we present a different proof that serves to illustrate our strategy in the general case. Section \[sec:multi\] is a discussion of a particular multi-grading of the set of matrix-tuples, needed for computations. Section \[sec:intermediate\] tackles an intermediate problem, for a simpler group action. The group of symmetries for ${{\rm SING}}_{n,m}$ and ${{\rm NSING}}_{n,m}$ are computed in Section \[sec:symsingnm\], proving Theorem \[theo:gos\] and Theorem \[theo:ngos\]. Section \[sec:notnullcone\] contains the proofs of Theorem \[theo:nullcone\], our main negative result, and Theorem \[theo:nullcone2\] which implies it. In Section \[sec:inv.conv\], we prove Theorem \[theo:invring\]. Finally, in Section \[sec:disc\], we discuss some open problems and directions for future research.
In Appendix \[App.gos\], we recall the necessary algebraic geometry and Lie theory to prove the results in Section \[sec:gos\]. The results stated in Section \[sec:van\] are proved in Appendix \[App.rep\] with the help of representation theory. Finally in Appendix \[app.pos.char\], we discuss the modifications needed to extend the results to positive characteristic.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We would like to especially thank J. M. Landsberg for suggesting that we compute the group of symmetries, and Gurbir Dhillon for helping us with the Lie theoretic statements needed for the computation. In addition, we also thank Ronno Das, Harm Derksen, Ankit Garg, Robert Guralnick, Alexander Kleschev, Thomas Lam, Daniel Litt, Rafael Oliveira, Gopal Prasad, Akash Sengupta, Rahul Singh, Yuval Wigderson, John Wiltshire-Gordon and Jakub Witaszek for helpful discussions.
Invariant theory and null cones {#sec:invthry}
===============================
We will now recall the basic notions in invariant theory that we need. In particular, we will need the notions of rational group actions (rational representations), their invariant polynomials, null cones and their basic properties. Most of this material is well known and can be found in a standard text such as [@DK]. We will try to remain as elementary as possible. We remind the reader again that our underlying field is ${{\mathbb C}}$.
A [*linear algebraic group*]{} $G$ is a subgroup of ${\operatorname{GL}}_n$ (for some $n$) that is also an algebraic subvariety[^11]. In this paper, we will drop the prefix linear and simply refer to these as algebraic groups for brevity. The connected component of an algebraic group $G$ containing the identity element is itself a connected algebraic group, and we call this the identity component of $G$, and denote it by $G^\circ$.
For a linear algebraic group $G \subseteq {\operatorname{GL}}_n$, an $m$-dimensional representation is simply a map $\rho:G \rightarrow {\operatorname{GL}}_m$. We want to consider “algebraic representations”, so we want the map $\rho$ to be a morphism of algebraic varieties. So, for $X = (x_{ij}) \in G \subseteq {\operatorname{GL}}_n$, each coordinate of the $m \times m$ matrix $\rho(X) \in {\operatorname{GL}}_m$ is given as a rational function (ratio of polynomials) in the $x_{ij}$’s.[^12] This is why such representations are called [*rational*]{} representations. The definition itself is of course quite straightforward.
\[Rational representation\] A rational representation $V$ of an algebraic group $G$ is a morphism of algebraic groups $G \rightarrow {\operatorname{GL}}(V)$ (where $V$ is a vector space over ${{\mathbb C}}$). By a morphism of algebraic groups, we simply mean a group homomorphism that is also a morphism of varieties.
A morphism $G \rightarrow {\operatorname{GL}}(V)$ can also be interpreted as a morphism $G \times V \rightarrow V$, and we will write $g \cdot v$ or simply $gv$ to denote the image of $(g,v)$ under this map. The orbit of a point $v \in V$ is $G \cdot v = \{gv\ |\ g \in G\}$. All representations considered in this paper will be rational. Subrepresentations, direct sums etc are defined in the standard way. A representation is called [*irreducible*]{} if it has no subrepresentations.
For $V$ to be a rational representation of an algebraic group $G$ simply means that $G$ acts algebraically on $V$ by linear transformations. This is precisely the premise under which we define a null cone, and hence precisely the hypothesis in the main results (for example in Theorem \[theo:nullcone\]).
For a vector space $V$, we denote by ${{\mathbb C}}[V]$ the ring of polynomial functions on $V$ (a.k.a. the coordinate ring of $V$). Concretely, if we have a basis $e_1,\dots,e_n$ for $V$, and $x_1,\dots,x_n$ denote the corresponding coordinate functions, then ${{\mathbb C}}[V] = {{\mathbb C}}[x_1,\dots,x_n]$ is the polynomial ring in $\dim V = n$ variables.
\[Invariant function\] For a representation $V$ of a group $G$, a function $f \in {{\mathbb C}}[V]$ is invariant (for the action of $G$) if it is constant along orbits, i.e., $f(gv) = f(v)$ for all $v \in V$ and $g \in G$.
Invariant functions form a subring of the coordinate ring, which we will call the [*invariant ring*]{} or [*ring of invariants*]{}.
\[Invariant ring\] For a representation $V$ of a group $G$, we denote by ${{\mathbb C}}[V]^G$, the ring of invariants, i.e., $${{\mathbb C}}[V]^G = \{f \in {{\mathbb C}}[V]\ |\ f(gv) = f(v)\ \forall g\in G,v\in V\}.$$
Invariant rings are graded subrings of the polynomial ring ${{\mathbb C}}[V]$, i.e., ${{\mathbb C}}[V]^G = \bigoplus_{d \in {{\mathbb N}}} {{\mathbb C}}[V]^G_d$.
There are several equivalent definitions of a reductive group, particularly in characteristic zero. We pick a definition that would resonate with anyone who has had experience with representations of finite groups. In particular, we want to point to the fundamental result called Maschke’s theorem, which says that for a finite group $G$ (if characteristic is zero or doesn’t divide $|G|$), any representation can be written as a direct sum of irreducible representations (a.k.a. complete reducibility). This property is very useful because in order to study any representation of $G$, one can often reduce it to the study of the irreducible representations. Algebraic groups with this property are called reductive groups.
\[Reductive group\] An algebraic group $G$ is called reductive if any rational representation $V$ of $G$ is completely reducible, i.e., it can be written as a direct sum of irreducible representations.
Examples of reductive groups include all finite groups, tori (i.e., $({{\mathbb C}}^*)^n)$, and all classical groups such as ${\operatorname{GL}}_n$, ${\operatorname{SL}}_n$, ${\rm SO}_n$, ${\rm Sp}_n$ etc.
\[Null cone\] \[def:nullcone\] Let $V$ be a rational representation of a reductive group $G$. Then the null cone $\mathcal{N}_G(V)$ (or simply $\mathcal{N}_G$ or even $\mathcal{N}$ when there is no confusion) is defined as the set of points in $V$ whose orbit closure[^13] contains zero, i.e., $$\mathcal{N} = \mathcal{N}_G(V) = \{v \in V\ |\ 0 \in \overline{G \cdot v}\},$$ where $\overline{G \cdot v}$ denotes the closure of $G \cdot v$, the orbit of $v$.
The above definition of the null cone is analytic in nature, and as defined seems to be a feature of the geometry of orbits and their closures. However, there is an equivalent algebraic description via invariant polynomials that we state below due to Mumford (and known already to Hilbert for $G = {\operatorname{SL}}_n$). This interplay between the analytic and algebraic viewpoints has already proved extremely valuable (see for e.g. [@GGOW16; @BGFOWW; @BGFOWW2]), and perhaps yet to be exploited to its full capacity.
\[Mumford\] \[theo:alg.nullcone\] Let $V$ be a rational representation of a reductive group $G$. Then, the null cone $$\mathcal{N} = \mathcal{N}_G(V) = \text{ zero locus of } \bigcup\limits_{d= 1}^{\infty} {{\mathbb C}}[V]^G_d.$$
For a proof of the above result, we refer the reader to [@DK Section 2.5].
Null cones for tori {#sec:inv.thry.tori}
-------------------
The group ${{\mathbb C}}^* = {\operatorname{GL}}_1({{\mathbb C}})$ is clearly an algebraic group, which is moreover abelian. A direct product $T = ({{\mathbb C}}^*)^n$ is called a (complex) torus. Any connected abelian reductive group must a torus! Needless to say (non-abelian) reductive groups can of course be far more complicated than tori. However, an understanding of the null cone for tori is key to understanding the null cones for more general reductive groups, and this is captured by the celebrated Hilbert–Mumford criterion that we will discuss in the next subsection. The null cone in the “easy” abelian case of the torus has a simple description as a union of linear subspaces of a specific form (this is related to the linear programming problem in complexity).
For this subsection, let $T = ({{\mathbb C}}^*)^n$ be a (complex) torus. Let $\mathcal{X}(T)$ denote all the characters of $T$, i.e., all algebraic group homomorphisms $T \rightarrow {{\mathbb C}}^*$. One can identify $\mathcal{X}(T) = {{\mathbb Z}}^n$ as follows. For $\lambda = (\lambda_1,\dots,\lambda_n) \in {{\mathbb Z}}^n$, we have the corresponding character (also denoted $\lambda$ by abuse of notation) $\lambda: T \rightarrow {{\mathbb C}}^*$ defined by $\lambda(t_1,\dots,t_n) = \prod_{i=1}^n t_i^{\lambda_i}$. It is a well known result that these are all the algebraic characters of $T$.
Suppose $V$ is a rational representation of $T$. Then there is a [*weight space decomposition*]{} $$V = \bigoplus_{\lambda \in \mathcal{X}(T)} V_\lambda,$$ where for any $\lambda \in \mathcal{X}(T)$, the weight space $V_{\lambda} = \{v \in V \ |\ t \cdot v = \lambda(t) v\}$. One should think of this as a simultaneous eigenspace decomposition for the action of $T$. Indeed, the weight space $V_\lambda$ consists of eigenvectors for the action of every $t \in T$, although each $t$ will act by a different eigenvalue, i.e., $\lambda(t)$. Elements of $V_{\lambda}$ are called weight vectors of weight $\lambda$. Let $e_1,\dots,e_m$ denote a basis of $V$ consisting of weight vectors (thus identifying $V$ with ${{\mathbb C}}^m$), and let the weight of $e_i$ be $w_i \in {{\mathbb Z}}^n$. Let $x_1,\dots,x_m$ denote the corresponding coordinates.
\[R-tori.invpol\] A monomial $\prod_{i=1}^m x_i^{a_i}$ is an invariant monomial if and only if $\sum_i a_i \cdot w_i = 0$ (note that $a_i \in {{\mathbb N}}$ and $w_i \in {{\mathbb Z}}^n$, so this is an equality in ${{\mathbb Z}}^n$). As a vector space over ${{\mathbb C}}$, the ring of invariants ${{\mathbb C}}[V]^T$ is spanned by such invariant monomials. In particular, the null cone is precisely the zero locus of such invariant monomials (excluding the trivial monomial $\prod_{i = 1}^m x_i^0$ which is the constant function $1$).
\[Coordinate subspace\] \[D-coordsubspace\] For a subset $I \subseteq [m]$, we define $L_I$ to be the linear subspace of ${{\mathbb C}}^m$ that is defined as the zero locus of $\{x_j: j \notin I\}$. In other words, $L_I$ consists of all the vectors in ${{\mathbb C}}^m$ whose support (i.e., the set of non-zero coordinates) is a subset of $I$. We will call any subspace of the form $L_I$ a [*coordinate subspace*]{}.
For a subset $I \subseteq [m]$, consider the set of points $W_I = \{w_i : i \in I\} \subseteq {{\mathbb Z}}^n \subseteq {{\mathbb Q}}^n \subseteq {{\mathbb R}}^n$. Let $\Delta_I$ denote the convex hull of $W_I$. The following description of the null cone is the main takeaway from this subsection. We provide a proof for completeness.
Let $V$ be an $m$-dimensional rational representation of the torus $T$. Identify $V = {{\mathbb C}}^m$ using a basis of weight vectors. Using the notation above, the null cone $$\mathcal{N}_T(V) = \bigcup\limits_{I \subseteq [m], 0 \notin \Delta_I} L_I$$
First, let us show that for each $I$ such that $0 \notin \Delta_I$, $L_I \subseteq \mathcal{N}_T(V)$. By Remark \[R-tori.invpol\] and Theorem \[theo:alg.nullcone\], it suffices to show that every (non-constant) invariant monomial vanishes on $L_I$. Take such an invariant monomial $m = \prod_{i=1}^m x_i^{a_i}$. If $a_j > 0$ for some $j \notin I$, then clearly $m$ vanishes on $L_I$. Otherwise $m = \prod_{i \in I} x_i^{a_i}$, so for $m$ to be invariant, $\sum_{i \in I} a_i w_i = 0$, but this means that $0 \in \Delta_I$, which is a contradiction. Thus every non-constant invariant monomial vanishes on $L_I$. Thus, we have shown $\supseteq$.
For the reverse direction, it suffices to show that $v \notin {\rm R.H.S.}$ implies $v \notin \mathcal{N}_T(V)$. To this end, let $v \notin {\rm R.H.S.}$. Let $J$ be the support of $v$ (i.e., the set of all non-zero coordinates). Clearly $0 \in \Delta_J$. Thus, we have $0 = \sum_{i \in J} a_i w_i = 0$ for some $a_i > 0$ and $\sum_i a_i = 1$. If the $a_i$’s were (non-negative) integers, then $\prod_{i \in J} x_i^{a_i}$ would be an invariant monomial that doesn’t vanish at $v$, and we would be done. Even if the $a_i$’s are rational numbers, by removing denominators, the argument would still go through. However, we only know that $a_i$’s are real numbers, and we will need a little bit of work to ensure that we can get a non-negative integer linear combination of the $w_i$’s to add to zero.
W.l.o.g., we can assume that $a_i > 0$ for all $i \in J$ (else, replace $J$ with $\{i \in J\ | a_i >0\}$ and proceed with the argument). Let $K = \{(p_i)_{i \in J}\ | \ \sum_{i \in J} p_i w_i = 0\} \subseteq {{\mathbb R}}^J$. Then $K$ is the kernel of an $n \times |J|$ matrix whose columns are $w_i : i \in J$. Since this matrix has rational entries, there is a basis of $K$ with rational entries, i.e., $b_1,\dots,b_r \in K \cap {{\mathbb Q}}^J$ that span $K$ (as an ${{\mathbb R}}$-vector space). Now, since $(a_i)_{i \in J} \in K$, we can write $(a_i)_{i \in J} = \sum_{t = 1}^r \lambda_t b_t$ for some $\lambda_t \in {{\mathbb R}}$. Since $\sum_t \lambda_t b_t = (a_i)_{i \in J} \in {{\mathbb R}}_{> 0}^J$, we deduce by continuity that there exists $\epsilon > 0$ such that $\sum_t \mu_t b_t \in {{\mathbb R}}_{>0}^J$ for all $\mu_i$ such that $|\mu_i - \lambda_i| < \epsilon$. Let $\mu_i$ be rational numbers such that $|\mu_i - \lambda_i| < \epsilon$. Then let $(c_i)_{i \in J} = \sum_i \mu_i b_i \in K \cap {{\mathbb R}}_{>0}^J$, but clearly $c_i$ are rational, so $(c_i)_{i \in J} \in K \cap {{\mathbb Q}}_{>0}^J$. Thus $\sum_i c_i w_i = 0$ and $c_i \in {{\mathbb Q}}$. For some $D \in {{\mathbb N}}$, we have $Dc_j \in {{\mathbb N}}$ for all $j$. Then $m = \prod_{i \in J} x_i^{Dc_j}$ is an invariant monomial that does not vanish on $v$, so $v \notin \mathcal{N}_T(V)$.
Null cones for reductive groups: Hilbert–Mumford criterion
----------------------------------------------------------
Let $G$ be a reductive group, and let $T$ be a maximal torus[^14], i.e., a subgroup of $G$ that is a torus, and not contained in a larger torus. The celebrated result called the Hilbert–Mumford criterion that we state below essentially tells us that elements in the null cone for the action of $G$ are precisely those which can be moved (by applying an element of $G$) into the null cone for the torus $T$. In particular, this is one way to see that the null cone for the group $G$ is the same as the null cone for its identity component $G^\circ$.
The following statement can be found in [@Mumford] (see also [@DK Theorem 2.5.3]).
\[Hilbert–Mumford criterion\] \[theo:HM\] Let $V$ be a rational representation of a reductive group $G$. Then $$\mathcal{N}_G(V) = G \cdot \mathcal{N}_T(V).$$
The Hilbert–Mumford criterion is sometimes stated in more general fashion, which says that $v \in V$ is in the null cone for $G$ if and only if there is a $1$-parameter subgroup of $G$ that drives it to zero. To see that this is equivalent to the version we state above, one needs to understand two things. The first is that any $1$-parameter subgroup is contained in some maximal torus, and all maximal tori are conjugate. The second is an understanding of the $1$-parameter subgroups of a torus, which will show that our description of the null cone for tori agrees with the criterion in terms of $1$-parameter subgroups (this is not hard).
Proof idea of Theorem \[theo:nullcone2\] {#subs:proofidea}
----------------------------------------
Let us briefly give the idea behind the proof of Theorem \[theo:nullcone2\] (from which Theorem \[theo:nullcone\] follows easily). Indeed, suppose there is a reductive group $G$ (with maximal torus $T$) acting on $V = {\operatorname{Mat}}_{n}^m$ preserving ${{\rm SING}}_{n,m}$ such that the null cone is contained in ${{\rm SING}}_{n,m}$. Then the null cone for the torus $N_T(V)$ is also a subset of ${{\rm SING}}_{n,m}$. We know from the above discussion that $\mathcal{N}_T(V)$ is a union of coordinate subspaces. We will show that any coordinate subspace contained in ${{\rm SING}}_{n,m}$ must already be contained in ${{\rm NSING}}_{n,m}$ – we will see this in Section \[sec:notnullcone\]. Thus, whatever $\mathcal{N}_T(V)$ may be, it must be contained in ${{\rm NSING}}_{n,m}$.
This is the point where an understanding the group of symmetries is really needed. To be precise, the crucial result that drives the following argument is that the group of symmetries for ${{\rm SING}}_{n,m}$ is the [*same*]{} as the group of symmetries for ${{\rm NSING}}_{n,m}$! So, in particular, since $G$ preserves ${{\rm SING}}_{n,m}$, it also preserves ${{\rm NSING}}_{n,m}$. Thus, we have $G \cdot \mathcal{N}_T(V) \subseteq {{\rm NSING}}_{n,m}$. By the Hilbert–Mumford criterion (Theorem \[theo:HM\] above), we get that $\mathcal{N}_G(V) = G \cdot \mathcal{N}_T(V) \subseteq {{\rm NSING}}_{n,m}$ which is the required conclusion for Theorem \[theo:nullcone2\].
Computing the group of symmetries via polynomials {#sec:gos}
=================================================
In this section, we will explain the important statements that go into the calculation of the group of symmetries. The proofs will be postponed to an appendix so as to not interrupt the flow of the paper. The main purpose of this section is however to highlight the fact one can determine the connected group of symmetries by a linear algebraic computation (by passing to Lie algebras), and this works in great generality. Later on, we discuss a technique to determine the entire group of symmetries, but this works only in a more limited setting (which of course includes ${{\rm SING}}_{n,m}$).
Given a subset $S \subseteq V$, we define $$I_S = \{f \in {{\mathbb C}}[V]\ |\ f(s) =0\ \forall s \in S\},$$ which is called the ideal of polynomials vanishing on $S$. about $S$ can almost always be reformulated in terms of questions on $I_S$. The first observation is that for a group $G$ acting (algebraically) on $V$, there is a (natural) induced action of $G$ on ${{\mathbb C}}[V]$. To understand this action , we need to describe for $g \in G$ and a polynomial function $f \in {{\mathbb C}}[V]$, what the resulting polynomial function $g \cdot f$ is. To describe a polynomial function, one can simply give its evaluation on all points of $V$. The polynomial function $g \cdot f$ is defined by $$(g \cdot f) (v) = f(g^{-1}v)$$
There are other ways to describe this action, one of them being that we identify ${{\mathbb C}}[V]$ with the symmetric algebra over the dual space $V^*$. This point of view is not needed here, but will be helpful in a later technical section. For now, we note some key features. The most important feature is that $\deg(f) = \deg(g \cdot f)$ for any $g \in G$ and any (homogenous) $f \in {{\mathbb C}}[V]$. Hence, the linear subspace ${{\mathbb C}}[V]_a$ consisting of homogenous polynomials of degree $a$ is a $G$-stable subspace of ${{\mathbb C}}[V]$.
When $S$ is a cone (i.e., $\lambda \in {{\mathbb C}}, s\in S \implies \lambda s \in S$), then $I_S$ is graded, i.e., $I_S = \oplus_{a \in {{\mathbb N}}} (I_S)_a$, where $(I_S)_a$ denotes the polynomials in $I$ that are homogenous of degree $a$.
\[Lgos-graded\] Suppose $S \subseteq V$ is a cone, and let $a \in {{\mathbb N}}$. Then $$\mathcal{G}_S \subseteq \{g \in {\operatorname{GL}}(V)\ |\ g (I_S)_a \subseteq (I_S)_a\}.$$ Further, if the zero locus of $(I_S)_a$ is equal to $S$, then we have equality.
The key reason behind restricting ourselves to polynomials of a certain degree is to work with finite dimensional vector spaces rather than infinite dimensional ones.
Algebraic sets (in particular cones), are often described as the zero locus of a collection of (homogenous) polynomials $\{f_i: i \in I\}$. While it is difficult to compute a set of generators for the ideal $I_S$, the degrees of $\{f_i: i \in I\}$ can help us find a suitable $a$ to apply the above lemma (indeed the least common multiple of degrees of $f_i$ will suffice, but in specific cases, one can probably do much better). It is however another task to compute [*all*]{} the homogenous polynomials of a certain degree that vanish on $S$. In any specific case, this may be manageable, but we do not know of any general strategy. For the case of ${{\rm SING}}_{n,m}$, we will manage this (in a later section) with the help of representation theory of ${\operatorname{GL}}_m \times {\operatorname{GL}}_n \times {\operatorname{GL}}_n$.
Our technique to compute the group of symmetries $\mathcal{G}_S$ has two parts to it. The first is to determine the connected group of symmetries $\mathcal{G}_S^\circ$, i.e., the identity component of the group of symmetries – this will be done by appealing to the theory of Lie algebras, which reduces the problem to linear algebra. The second is to determine the component group $\mathcal{G}_S/\mathcal{G}_S^\circ$, which is always a finite group.
Connected group of symmetries via Lie algebras
----------------------------------------------
In this section, we discuss the first part of our technique, i.e., how to determine the connected group of symmetries. The first observation (and easy to see) is that the group of symmetries of an algebraic subset $S \subseteq V$ is a Zariski closed subgroup of ${\operatorname{GL}}(V)$, and so is an algebraic subgroup of ${\operatorname{GL}}(V)$ (and hence a Lie group). Consequently, the connected group of symmetries is a connected algebraic group (and hence connected Lie subgroup). Connected Lie subgroups of ${\operatorname{GL}}(V)$ are in $1-1$ correspondence with Lie subalgebras of ${{\mathfrak{gl}}}(V)$, the Lie algebra of ${\operatorname{GL}}(V)$[^15]. Thus, to determine the connected group of symmetries $\mathcal{G}_S^\circ$, it suffices to determine its Lie algebra (denoted ${\mathfrak{g}}_S$), which we will call the [*Lie algebra of symmetries*]{}. We should point out here that in general neither $\{M \in {{\mathfrak{gl}}}(V)\ |\ M \cdot S = S\}$ nor $\{M \in {{\mathfrak{gl}}}(V)\ |\ M \cdot S \subseteq S\}$ is equal to ${\mathfrak{g}}_S$. However, we have the following result:
\[P-Liealg-all\] Let $S \subseteq V$ be a cone. Then for any $a \in {{\mathbb N}}$, the Lie algebra of symmetries $${\mathfrak{g}}_S \subseteq \{M \in {{\mathfrak{gl}}}(V)\ |\ M \cdot (I_S)_a \subseteq (I_S)_a\}.$$ Finally, if the zero locus of $(I_S)_a$ is precisely the cone $S$, then we have equality.
In the appendix, we give a gentle and quick introduction to Lie algebras, and prove the proposition. It is however imperative for the reader to understand the action of the Lie algebra ${{\mathfrak{gl}}}(V)$ on polynomials in ${{\mathbb C}}[V]_a$ to be able to use the above proposition as a computational tool. For this purpose, in the next section, we describe this action explicitly.
For algebraic groups $G,H,\dots$, we will denote their Lie algebras by ${{\rm Lie}}(G),{{\rm Lie}}(H),\dots$ or by the corresponding gothic letters ${\mathfrak{g}},{\mathfrak{h}},\dots$ (to avoid cumbersome notation).
Component group
---------------
We have already discussed above how to compute the connected group of symmetries $\mathcal{G}_S^\circ$ for an algebraic subset $S \subseteq V$. To compute the entire group of symmetries $\mathcal{G}_S$, we observe that $\mathcal{G}_S$ is an algebraic subgroup of ${\operatorname{GL}}(V)$ whose identity component is $\mathcal{G}_S^\circ$. One deduces that $\mathcal{G}_S$ must be a subgroup of the normalizer of $\mathcal{G}_S^\circ$. In the event that $\mathcal{G}_S^\circ$ is a reductive group and acts irreducibly on $V$, its normalizer will be a finite extension (see for e.g. [@Guralnick])[^16]. In particular its normalizer is also an algebraic group whose identity component is $\mathcal{G}_S^\circ$. So, $\mathcal{G}_S$ is a union of some of the components of the normalizer of $\mathcal{G}_S^\circ$, and we just have to identity which ones.
Explicit description of the Lie algebra action on polynomials {#sec:explicit}
=============================================================
For the technical aspects of the computations we do, it is absolutely essential to understand the action of the Lie algebras on polynomial functions. Let $V$ be a vector space with basis $e_1,\dots,e_n$, and let the corresponding coordinates functions be denoted $x_1,\dots,x_n$. Using the basis, we identify $V = {{\mathbb C}}^n$, ${{\mathbb C}}[V] = {{\mathbb C}}[x_1,\dots,x_n]$, ${\operatorname{GL}}(V) = {\operatorname{GL}}_n$ and ${{\mathfrak{gl}}}(V) = {\operatorname{Mat}}_{n}$.
Let $E_{ij}$ denote the elementary matrix with a $1$ in $(i,j)^{th}$ entry and $0$’s everywhere else. As a linear transformation, $E_{ij}$ maps $e_j$ to $e_i$ and kills $e_k$ for $k \neq j$. The matrices $\{E_{ij}\}_{1 \leq i,j \leq n}$ form a basis for ${\operatorname{Mat}}_{n}$. We will describe the action of $E_{ij}$’s, and then extend by linearity to understand the action of ${{\mathfrak{gl}}}(V)$. The matrix $E_{ij}$ acts as the derivation $-x_j \partial_i$, where $\partial_i$ denotes the partial derivative with respect to $x_i$. In other words, for any $f \in {{\mathbb C}}[V]$, we have $$E_{ij} \cdot f = -x_j\partial_i f$$
To write it all out explicitly, a matrix $M \in {{\mathfrak{gl}}}(V) = {\operatorname{Mat}}_{n}$ acts on a polynomial $f \in {{\mathbb C}}[V] = {{\mathbb C}}[x_1,\dots,x_n]$ by the following formula:
$$M \cdot f = \left(- \sum_{1\leq i,j \leq n} m_{ij} x_j \partial_i \right) f,$$
where $m_{ij}$ denotes the $(i,j)^{th}$ entry of the matrix $M$. Also note that we can write $M = \sum_{i,j} m_{ij} E_{ij}$, so another point of view is that $m_{ij}$ are the coordinates of the matrix $M$ with respect to the (standard) basis $\{E_{ij}\}$.
Twisting by Cartan involution
-----------------------------
The action of the Lie algebra on polynomial functions is annoying due to the negative signs and these will be cumbersome to keep track of it in computation. To make the computations less confusing and more intuitive, we twist the action. This is done with the help of the Cartan involution.
\[Cartan involution\] The Cartan involution $\Theta: {{\mathfrak{gl}}}_n \rightarrow {{\mathfrak{gl}}}_n$ is the composition of negation and transpose, i.e., $$\Theta(X) = - X^t.$$ The Cartan involution is an automorphism of Lie algebras.
The main thing to observe about the map $\Theta$ is that it is an involution, i.e., $\Theta \circ \Theta$ is the identity map.
\[Twisted action\] For $M \in {{\mathfrak{gl}}}_n$, we define an action of $M$ on ${{\mathbb C}}[V] = {{\mathbb C}}[x_1,\dots,x_n]$ by $$M {\star}f = \Theta(M) \cdot f.$$ In particular, we have $$E_{ij} {\star}f = x_i \partial_j f,$$ and hence $$M {\star}f = (\sum_{i,j} m_{ij} x_i \partial_j ) f$$
The result on computing the Lie algebra of symmetries (i.e., Proposition \[P-Liealg-all\]) can be reformulated in terms of the twisted action:
\[star-action\] Let $S \subseteq V$ be a cone. Then for any $a \in {{\mathbb N}}$, we have $${\mathfrak{g}}_S \subseteq \{\Theta(M)\ |\ M {\star}(I_S)_a \subseteq (I_S)_a \} = \Theta \{M\ |\ M {\star}(I_S)_a \subseteq (I_S)_a \}.$$ In the above, we have equality if the zero locus of $(I_S)_a$ is precisely $S$.
Vanishing ideal of singular tuples of matrices {#sec:van}
==============================================
In order to apply the ideas of Section \[sec:gos\] to computing the Lie algebra of symmetries for $S$, we need to understand the ideal $I_S$. We will focus on the case of $S = {{\rm SING}}_{n,m}$. For this, one needs the representation theory of the general linear group (highest weight vectors, Cauchy formulas, Schur functors) as well as an understanding of the invariants in the left-right action of ${\operatorname{SL}}_n \times {\operatorname{SL}}_n$. The proofs are ad-hoc, and suited precisely to the case of ${{\rm SING}}_{n,m}$. It is highly unlikely that these ideas can be generalized to give results for other choices of $S$. For all these reasons, we postpone the proofs to an appendix.
In the case of $S = {{\rm SING}}_{n,m}$, we do not know how to determine the entire ideal $I_S$. However, we can determine it upto degree $n$, which turns out to suffice for our purposes.
\[P-ideal\] Let $S = {{\rm SING}}_{n,m} \subseteq V = {\operatorname{Mat}}_{n}^m$, and let $I_S \subseteq {{\mathbb C}}[{\operatorname{Mat}}_{n}^m]$ be the ideal of polynomial functions that vanish on $S$. Then,
1. $I_S$ is graded;
2. $(I_S)_a$ is empty if $a < n$;
3. $(I_S)_n = {{\rm span}}(\det(\sum_i c_iX_i): c_i \in {{\mathbb C}}).$
The significance of the above result is demonstrated by the following statement:
\[C-loscom\] Let $S = {{\rm SING}}_{n,m} \subseteq V = {\operatorname{Mat}}_{n}^m$. Then $$\mathcal{G}_S = \{g \in {\operatorname{GL}}(V) \ |\ g \cdot (I_S)_n \subseteq (I_S)_n \},$$ and hence $${\mathfrak{g}}_S = \{M \in {{\mathfrak{gl}}}(V)\ | M \cdot (I_S)_n \subseteq (I_S)_n \}.$$
This follows from Lemma \[Lgos-graded\], Proposition \[P-Liealg-all\] and the above proposition since the zero locus of $(I_S)_n$ is precisely ${{\rm SING}}_{n,m}$
The latter part of the corollary is the one that is extremely useful because now the computation of ${\mathfrak{g}}_S$ is feasible. In principle, for a fixed $n$ and $m$, one could run an explicit computer algorithm to compute ${\mathfrak{g}}_S$. However, we will actually be able to compute ${\mathfrak{g}}_S$ for all $n,m$, and for this, we will need to do the linear algebra by hand. To do so, we will (repeatedly) exploit the numerous symmetries and multilinearity of the determinant polynomial.
Symmetries of singular matrices {#sec:frob}
===============================
In this section we give a proof of Frobenius’ result, i.e., Theorem \[theo:frob\] which is the important case of $m=1$ in Theorem \[theo:gos\]. The technique we use is different from the existing proofs in [@Frob; @Dieudonne]. First, we use the the previous sections to write the computation of the Lie algebra of symmetries as a linear algebraic computation. Next, we define Kronecker product of matrices, and then describe the twisted action of ${{\mathfrak{gl}}}({\operatorname{Mat}}_{n})$. Then, we recall a few facts on the Symmetric group, and finally give the explicit computations to determine the Lie algebra of symmetries, which suffices to determine the connected group of symmetries. Finally, we compute the entire group of symmetries.
For this section, let $S = {{\rm SING}}_{n,1} = \{X \in {\operatorname{Mat}}_{n}\ | \det(X) = 0 \} \subseteq V = {\operatorname{Mat}}_{n}$. For $(P,Q) \in {\operatorname{GL}}_n \times {\operatorname{GL}}_n$, consider the linear transformation $X \mapsto PXQ^t$. The set of all such linear transformations is the group $G_{n,1}$ (as defined in Theorem \[theo:gos\]). We will write ${\mathfrak{g}}_{n,1}$ for the Lie algebra of $G_{n,1}$. In this section, we will first prove:
\[P1\] Let $S = {{\rm SING}}_{n,1} \subseteq V = {\operatorname{Mat}}_{n}$. Then, we have $\mathcal{G}_S^\circ = G_{n,1}
$ and hence ${\mathfrak{g}}_S = {\mathfrak{g}}_{n,1}.
$
Let $E_{ij}$ denote the $n \times n$ matrix with a $1$ in its $(i,j)^{th}$ spot and $0$’s everywhere else. Then $\{E_{ij}\}_{1 \leq i,j \leq n}$ form a basis for $V = {\operatorname{Mat}}_{n}$. The Lie algebra ${{\mathfrak{gl}}}(V) = {\operatorname{Mat}}_{n^2}$ can be identified canonically with ${{\rm End}}(V)$, the space of linear transformations from $V$ to $V$. Let us give a description of ${\mathfrak{g}}_{n,1}$. We have $${\mathfrak{g}}_{n,1} = \{ X \mapsto MX + XN\ |\ M,N \in {\operatorname{Mat}}_{n} \} \subseteq {{\mathfrak{gl}}}(V) = {{\rm End}}(V)\}.$$
The Lie algebra of symmetries consists of precisely those elements of ${{\mathfrak{gl}}}(V)$ for whose action the determinant polynomial is an eigenvector.
\[L-los-frob\] The Lie algebra of symmetries $${\mathfrak{g}}_S = \Theta(\{M \in {{\mathfrak{gl}}}(V) \ |\ M {\star}\det = c \cdot \det \text{ for some } c\in {{\mathbb C}}\})$$
This follows from Proposition \[P-ideal\], Corollary \[C-loscom\] and Lemma \[star-action\].
To make the necessary computations, we need to understand the action (or rather the twisted action) of ${{\mathfrak{gl}}}({\operatorname{Mat}}_{n})$ on polynomials in ${{\mathbb C}}[{\operatorname{Mat}}_{n}]$ explicitly. Prior to that, we recall the notion of Kronecker product of matrices.
Kronecker product of matrices
-----------------------------
For $A = (a_{ij}) \in {\operatorname{Mat}}_{n}$ and $B = (b_{ij}) \in {\operatorname{Mat}}_{n}$, we define the Kronecker (or tensor) product $$A \otimes B =
\begin{pmatrix}
a_{11}B & a_{12}B & \dots & a_{1n}B \\
a_{21}B & \ddots & \ddots & \vdots \\
\vdots & \ddots & \ddots & \vdots \\
a_{n1}B & \dots & \dots & a_{nn} B
\end{pmatrix} \in {\operatorname{Mat}}_{n^2}.$$
Using the Kronecker product, we will identify ${\operatorname{Mat}}_{n^2} = {\operatorname{Mat}}_{n} \otimes {\operatorname{Mat}}_{n}$. To do so, we will first index the rows and columns by $[n]\times [n]$ in lexicographic order (rather than $[n^2]$). Thus a basis for ${\operatorname{Mat}}_{n^2}$ is given by $\{E_{ij,kl}\ |\ ij,kl \in [n] \times [n]\}$. Note that we have the intuitive equality $$E_{ij} \otimes E_{pq} = E_{ip,jq}.$$
Twisted action of ${{\mathfrak{gl}}}({\operatorname{Mat}}_{n})$ on ${{\mathbb C}}[{\operatorname{Mat}}_{n}]$
------------------------------------------------------------------------------------------------------------
Recall that the (standard) basis for $V = {\operatorname{Mat}}_{n}$ is $\{E_{ij}\}$, and let $\{x_{ij}\}$ denote the corresponding coordinate functions. Note that $V$ is $n^2$-dimensional. The Lie algebra ${{\mathfrak{gl}}}(V) = {{\mathfrak{gl}}}({\operatorname{Mat}}_{n})$ can be identified with ${\operatorname{Mat}}_{n^2}$, to be viewed as ${\operatorname{Mat}}_{n} \otimes {\operatorname{Mat}}_{n}$, as described above. Observe that we can think of an $n^2 \times n^2$ matrix as an $n \times n$ block matrix whose blocks are also of size $n \times n$. Thus for any $M \in {{\mathfrak{gl}}}(V)$, we have
$$M = \begin{pmatrix}
M_{11} & \dots & M_{1n} \\
\vdots & \ddots & \vdots \\
M_{n1} & \dots & M_{nn}
\end{pmatrix},$$ where each $M_{ij}$ is an $n\times n$ matrix. Note that writing $M = \sum_{i,j} E_{ij} \otimes M_{ij}$ also defines $M_{ij}$.
The matrix $E_{ij} \otimes E_{pq} = E_{ip,jq} \in {\operatorname{Mat}}_{n^2} = {{\mathfrak{gl}}}({\operatorname{Mat}}_{n})$ acts (twisted action) on polynomial functions via the derivation $x_{ip} \partial_{jq}$, i.e., for $f \in {{\mathbb C}}[{\operatorname{Mat}}_{n}]$, we have $$E_{ip,jq} {\star}f = x_{ip}\partial_{jq} f.$$
There is an ${{\mathbb N}}^n$-grading on ${{\mathbb C}}[V] = {{\mathbb C}}[x_{ij}]$ given by setting $\deg(x_{ij}) = \delta_i := (0,\dots,0,\underbrace{1}_i,0,\dots,0) \in {{\mathbb N}}^n$. We have the decomposition $${{\mathbb C}}[V] = \bigoplus_{d \in {{\mathbb N}}^n} {{\mathbb C}}[V]_d,$$ where ${{\mathbb C}}[V]_d$ denotes the (multi)-homogeneous polynomials of degree $d$. The polynomial $\det = \sum_{\sigma \in S_n} \prod_{i=1}^n x_{i \sigma(i)}$ has degree $(1,1,\dots,1)$. We make a crucial observation
Let us understand the action of the Lie algebra with respect to this multi-degree. The twisted action of the matrix $E_{ip,jq}$ is by the derivation $x_{ip}\partial_{jq}$. Thus, $E_{ip,jq} : {{\mathbb C}}[V]_d \rightarrow {{\mathbb C}}[V]_{d - \delta_j + \delta_i}$.
\[Grading on ${{\mathfrak{gl}}}({\operatorname{Mat}}_{n})$\] We give a grading on the Lie algebra ${{\mathfrak{gl}}}({\operatorname{Mat}}_{n})$ by setting $\deg(E_{ip,jq}) = \delta_i - \delta_j$. We have $${{\mathfrak{gl}}}({\operatorname{Mat}}_{n}) = {\operatorname{Mat}}_{n^2} = {{\mathfrak{gl}}}({\operatorname{Mat}}_{n})_0 \bigoplus_{i \neq j}{{\mathfrak{gl}}}({\operatorname{Mat}}_{n})_{\delta_i -\delta_j}.$$
For any $M \in {\operatorname{Mat}}_{n^2}$, we write $M = \sum_{i,j} E_{ij} \otimes M_{ij}$ for matrices $M_{ij} \in {\operatorname{Mat}}_{n}$ as above. Then the degree $0$ part is $\sum_i E_{ii} \otimes M_{ii}$, and for $i \neq j$, $E_{ij} \otimes M_{ij}$ is the degree $\delta_i - \delta_j$ part. Thus, the decomposition of $M$ into homogeneous components is $$M = (\sum_i E_{ii} \otimes M_{ii}) \bigoplus_{i \neq j} E_{ij} \otimes M_{ij}$$
The following lemma is immediate from the preceding discussion.
If $M \in {\operatorname{Mat}}_{n^2}$ is homogeneous of degree $d$, and $f \in {{\mathbb C}}[V]$ is homogeneous of degree $d'$, then $M {\star}f$ is homogeneous of degree $d+ d'$
The following easy corollary of the above lemma will be useful to us:
Suppose $M = \sum_{ij} E_{ij} \otimes M_{ij} \in {{\mathfrak{gl}}}(V)$. Then $M {\star}\det = c \cdot \det$ for some $c \in {{\mathbb C}}$ if and only if the following conditions hold:
1. For $i \neq j$, $(E_{ij} \otimes M_{ij}) {\star}\det = 0$;
2. $\left(\sum_{i=1}^n E_{ii} \otimes M_{ii}\right) {\star}\det = c \cdot \det$.
This is straightforward from the decomposition of $M$ into homogenous components (given above) and the above lemma.
Before we unravel the above condition to compute the Lie algebra of the symmetries, a few words on the symmetric group.
Symmetric group
---------------
We denote by $S_n$, the symmetric group on $n$ letters. In other words, $S_n$ consists of all bijective maps $\sigma: [n] \rightarrow [n]$. The group operation is composition of maps. The pair $(i,j)$ with $i < j$ is called an inversion for $\sigma \in S_n$ if $\sigma(i) > \sigma(j)$. For $\sigma \in S_n$, we define it sign $${{\rm sgn}}(\sigma) = (-1)^{\text{ number of inversions in $\sigma$ }}.$$
For $\sigma \in S_n$, we will define $\iota(\sigma) \in S_n$ by $\iota(\sigma)(1) = \sigma(2)$, $\iota(\sigma(2)) = \sigma(1)$ and $\iota(\sigma)(k) = \sigma(k)$ for all $k > 2$. Thus $$\iota:S_n \rightarrow S_n$$ is an involution (without any fixed points!). Moreover, ${{\rm sgn}}(\iota(\sigma)) = - {{\rm sgn}}(\sigma)$ for any $\sigma \in S_n$.
Computation of Lie algebra of symmetries
----------------------------------------
To understand ${\mathfrak{g}}_S$ (i.e., the elements of ${{\mathfrak{gl}}}(V)$ for which $\det$ is an eigenvector by Lemma \[L-los-frob\]) it suffices to understand the two conditions in the previous corollary. For the rest of this subsection, let $M = \sum_{ij} E_{ij} \otimes M_{ij} \in {\operatorname{Mat}}_{n^2} = {{\mathfrak{gl}}}(V)$.
\[L1-offd\] For $i \neq j$, $(E_{ij} \otimes M_{ij}) {\star}\det = 0$ if and only if $M_{ij} = \kappa {\rm I}_n$ for some $\kappa \in {{\mathbb C}}$, where ${\rm I}_n$ denotes the identity matrix of size $n \times n$.
Let us first prove the forward direction. Suppose $E_{ij} \otimes M_{ij} {\star}\det = 0$. We want to prove that $M_{ij} = \kappa {\rm I}_n$.
First, observe that without loss of generality, we can consider $i = 2$ and $j = 1$, and we will do so. Let us denote the $(p,q)^th$ entry of $M_{21}$ by $\alpha_{pq}$. Then $E_{21} \otimes M_{21}$ acts by the derivation $D = \sum_{p,q} \alpha_{p,q} x_{2p} \partial_{1q}$.
Recall that $$\det = \sum_{\sigma \in S_n} {{\rm sgn}}(\sigma) x_{1\sigma(1)}x_{2\sigma(2)}\dots x_{n \sigma(n)}.$$
Thus $$\begin{aligned}
(E_{21} \otimes M_{21}) {\star}\det & = D \cdot \det \\
& = \sum_{\begin{array}{c} p,q,\sigma \\ \sigma(1) = q\end{array}} \alpha_{pq} {{\rm sgn}}(\sigma) x_{2p}x_{2\sigma(2)}x_{3\sigma(3)}\dots x_{n\sigma(n)}.\end{aligned}$$
Consider the monomial $m = x_{22}x_{22}x_{33}\dots x_{nn}$. Let us compute the coefficient of $m$ in $D \cdot \det$. To do so, let us check the choices of $p,q$ and $\sigma$ in the above summation contribute to the coefficient of $m$. Surely, we need $2 = p = \sigma(2)$, and $\sigma(k) = k$ for $k > 2$. This means that $\sigma$ must be the identity permutation, $p = 2$ and $q = \sigma(1) = 1$. So, there is only one contributing term, and that contributes a coefficient of $\alpha_{21}$. Since $D \cdot \det = 0$, we must have that $\alpha_{21} = 0$.
For any choice of $p \neq q$, a similar argument will show that $\alpha_{pq} = 0$ for $p \neq q$ (indeed consider instead of $m$, a monomial $x_{2q} x_{2q} x_{3\pi(3)}\dots x_{n\pi(n)}$ for some $\pi \in S_n$ such that $\pi(1) = p$ and $\pi(2) = q$). This proves that the off-diagonal entries of $M_{ij}$ are zero.
Now, consider the monomial $n = x_{21}x_{22}x_{33}\dots x_{nn}$ and let us compute its coefficient in $D \cdot \det$. Again, let us check the choices of $p,q$ and $\sigma$. Clearly need that $\sigma(k) = k$ for $k \geq 2$. Moreover, we need either $p = 1$ and $\sigma(2) = 2$ or $p = 2$ and $\sigma(2) = 1$. In the former case, we will have $\sigma$ to be the identity permutation and $q = \sigma(1) = 1$, so this contributes $\alpha_{11}$. Similarly the latter case contributes $- \alpha_{22}$. Thus, the coefficient of the $n$ is $\alpha_{11} - \alpha_{22}$ which must be zero. Hence $\alpha_{11} = \alpha_{22}$.
Again, a similar argument proves that $\alpha_{ii} = \alpha_{jj}$ for all $i,j$ (indeed, consider instead of $n$, the monomial $x_{2i}x_{2j}x_{3\pi(3)}\dots x_{n\pi(n)}$ for some $\pi \in S_n$ such that $\pi(1) = i$ and $\pi(2) = j$). Thus, $M_{ij} = \kappa {\rm I}_n$, where we take $\kappa = \alpha_{11}$. This shows that if $E_{ij} \otimes M_{ij} {\star}\det = 0$, then $M_{ij} = \kappa {\rm I}_n$.
For the converse direction, if $M_{ij} = \kappa {\rm I}_n$, then $E_{ij} \otimes M_{ij}$ acts by $$D = \kappa \sum_{i=1}^n x_{2i} \partial_{1i}.$$ Consider the action of $D$ on $t = \prod_{i=1}^n x_{i\sigma(i)}$. Unless $i = \sigma(1)$, the term $x_{2i} \partial_{1i}$ kills it. Thus, we get: $$\begin{aligned}
D \cdot t & = \kappa x_{2\sigma(1)} \partial_{1\sigma(1)} \cdot t \\
&= x_{2\sigma(1)} x_{2\sigma(2)} x_{3\sigma(3)} \dots x_{n \sigma(n)}. \\\end{aligned}$$
Thus, $$D \cdot \det = \kappa \sum_{\sigma \in S_n} {{\rm sgn}}(\sigma) x_{2\sigma(1)} \partial_{1\sigma(1)} \cdot t = \kappa(\sum_{\sigma \in S_n} {{\rm sgn}}(\sigma) x_{2\sigma(1)} x_{2\sigma(2)} x_{3\sigma(3)} \dots x_{n \sigma(n)}).$$
We claim that the sum is zero. To see this, notice that the terms corresponding to $\sigma$ and $\iota(\sigma)$ cancel. Hence, the whole sum cancels out as required.
\[L1-diag\] Suppose $\left(\sum_{i=1}^n E_{ii} \otimes M_{ii}\right) {\star}\det = c \cdot \det$. Then for all $i,j$, we have $M_{ii} - M_{jj} = \mu_{i,j} {\rm I}_n$ for some scalar $\mu_{i,j} \in {{\mathbb C}}$.
Let the $(p,q)^{th}$ entry of $M_{ii}$ be $\alpha^{i}_{p,q}$. Without loss of generality, take $i = 1$ and $j=2$, i.e., we will prove $M_{11} - M_{22} = \mu_{1,2} {\rm I}_n$ for some $\mu_{1,2} \in {{\mathbb C}}$. Note that $\sum_i E_{ii} \otimes M_{ii}$ acts by the derivation $D = \sum_{i,p,q} \alpha^{i}_{p,q} x_{ip} \partial_{iq}$. So
$$\label{Stupidity}
D \cdot \det = \sum_{\begin{array}{c} i,p,q,\sigma \\ \sigma(i) = q \end{array}} \alpha_{p,q}^i \cdot {{\rm sgn}}(\sigma) \cdot x_{1\sigma(1)}\dots x_{ip} \dots x_{n\sigma(n)}.$$
Fix $\pi \in S_n$, and consider the monomial $m = x_{1\pi(1)} x_{2\pi(1)} x_{3\pi(3)} \dots x_{n\pi(n)}$. The coefficient of $m$ in $c \cdot \det$ is $0$. So, the coefficient of $m$ in $D \cdot \det$ is also zero. We leave it to the reader to check from the above expression that the coefficient of $m$ in $D \cdot \det$ is ${{\rm sgn}}(\pi) (\alpha_{\pi(1),\pi(2)}^2 - \alpha_{\pi(1),\pi(2)}^1)$. Thus, we have $\alpha_{\pi(1),\pi(2)}^2 = \alpha_{\pi(1),\pi(2)}^1$. Running over all choices of $\pi$, we get that $\alpha_{p,q}^1 = \alpha_{p,q}^2$ for all $p \neq q$. This means that the off-diagonal entries of $M_{11}$ and $M_{22}$ are the same.
Again fix $\pi \in S_n$. Consider the monomial $n = \prod_{i=1}^n x_{i\pi(i)}$. Its coefficient in $c \cdot \det$ is $c \cdot {{\rm sgn}}(pi)$. So, its coefficient in $D \cdot \det$ should also be $c \cdot {{\rm sgn}}(\pi)$. From Equation \[Stupidity\], one can check again that the coefficient of $n$ in $D \cdot \det$ is ${{\rm sgn}}(\sigma) (\sum_i \alpha_{\pi(i),\pi(i)}^i).$ Thus, we must have $$\sum_i \alpha^i_{\pi(i),\pi(i)} = c.$$ This holds for all permutations, in particular, if we replace $\pi$ with $\iota(\pi)$. Thus, we have
$$\sum_i \alpha^i_{\pi(i),\pi(i)} = \sum_i \alpha^i_{(\iota\pi)(i),(\iota\pi)(i)}.$$
Hence, this means
$$\alpha^1_{\pi(1),\pi(1)} + \alpha^2_{\pi(2),\pi(2)} = \alpha^1_{\pi(2),\pi(2)} + \alpha^2_{\pi(1),\pi(1)}.$$
Again, this holds for all $\pi$, so we have
$$\alpha^1_{p,p} - \alpha^2_{p,p} = \alpha^1_{q,q} - \alpha^2_{q,q}.$$ for all $p,q$. Thus, if we set $\mu_{1,2} = \alpha^1_{p,p} - \alpha^2_{p,p}$, then the diagonal entries of $M_{11}$ and $M_{22}$ differ precisely by $\mu_{1,2}$. Since the off-diagonal entries of $M_{11}$ and $M_{22}$ agree (shown above), we have that $M_{11} - M_{22} = \mu_{1,2} {\rm I}_n$.
Suppose $M \in {\operatorname{Mat}}_{n^2} = {{\mathfrak{gl}}}({\operatorname{Mat}}_{n})$. Then $M {\star}\det = c \cdot \det$ for some $c\in {{\mathbb C}}$ if and only if $M$ is of the form $A \otimes {\rm I}_n + {\rm I}_n \otimes B$.
This follows from the previous two lemmas. Suppose $M$ is such that $M {\star}\det = c \cdot \det$ for some $c \in {{\mathbb C}}$. Observe that $M_{ij} = \kappa_{ij} {\rm I}_n$ by Lemma \[L1-offd\] for some scalars $\kappa_{ij} \in {{\mathbb C}}$. Then, by Lemma \[L1-diag\], we know that $M_{jj} = M_{11} + \mu_{j,1} {\rm I}_n$ for all $j$ (where $\mu_{j,1}$ is as defined in Lemma \[L1-diag\]).
Now, set $A$ to be the $n \times n$ matrix whose $(i,j)^{th}$ entry is $\kappa_{ij}$ if $i \neq j$, and $(i,i)^{th}$ entry is $\mu_{j,1}$. (Note that $\mu_{1,1} = 0$). Also, set $B = M_{11}$. This just means that $M = A \otimes {\rm I}_n + {\rm I}_n \otimes B$.
The Lie algebra of symmetries ${\mathfrak{g}}_S = \Theta \{A \otimes {\rm I}_n + {\rm I}_n \otimes B\ | A,B \in {\operatorname{Mat}}_{n}\} = \{A \otimes {\rm I}_n + {\rm I}_n \otimes B\ | A,B \in {\operatorname{Mat}}_{n}\}$.
It follows from the above corollary and Lemma \[L-los-frob\] that ${\mathfrak{g}}_S = \Theta \{A \otimes {\rm I}_n + {\rm I}_n \otimes B\ | A,B \in {\operatorname{Mat}}_{n}\}$. Observe that $\Theta( A \otimes {\rm I}_n + {\rm I}_n \otimes B) = -(A^t \otimes {\rm I}_n + {\rm I}_n \otimes B^t)$. Thus $\Theta \{A \otimes {\rm I}_n + {\rm I}_n \otimes B\ | A,B \in {\operatorname{Mat}}_{n}\} = \{A \otimes {\rm I}_n + {\rm I}_n \otimes B\ | A,B \in {\operatorname{Mat}}_{n}\}$.
The lie algebra ${\mathfrak{g}}_{n,1} = \{ A \otimes {\rm I}_n + {\rm I}_n \otimes B\ | A,B \in {\operatorname{Mat}}_{n}\} \subseteq {\operatorname{Mat}}_{n^2} = {{\mathfrak{gl}}}({\operatorname{Mat}}_{n})$.
We have the map $\phi: {\operatorname{GL}}_n \times {\operatorname{GL}}_n \rightarrow {\operatorname{GL}}_{n^2}$ given by $(P,Q) \mapsto P \otimes Q$. The image of $\phi$ is $G_{n,1}$ (by definition). The derivative of $\phi$, i.e., $d\phi: {{\mathfrak{gl}}}_n \times {{\mathfrak{gl}}}_n \rightarrow {{\mathfrak{gl}}}_{n^2} = {{\mathfrak{gl}}}({\operatorname{Mat}}_{n})$ is given by the formula $d\phi(A,B) = A \otimes {\rm I}_n + {\rm I}_n \otimes (-B)$. The lemma follows from the fact that the image of $d\phi$ is the Lie algebra of the image of $\phi$ (see Appendix \[App.gos\]), i.e., ${{\rm Lie}}(G_{n,1}) = {\mathfrak{g}}_{n,1}$.
From the above lemma and the corollary preceding it, we deduce Proposition \[P1\].
\[Proof of Proposition \[P1\]\] It follows from the above lemma and the preceding corollary that ${\mathfrak{g}}_S = {\mathfrak{g}}_{n,1}$. Hence, it follows that $G_S^\circ = G_{n,1}$.
The entire group of symmetries
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Much of the work has gone into computing the connected group of symmetries $\mathcal{G}_S^\circ$. Now, we want to determine the entire group of symmetries $\mathcal{G}_S$. To do so, we will need some results from the theory of semisimple Lie algebras, and we will only recall those facts that we need.
Let $G$ be a linear algebraic group and let $G^\circ$ denote its identity component. Then $G$ normalizes $G^\circ$, i.e., for all $g \in G$, we have $g G^\circ g^{-1} = G^\circ$.
Consider the map $\phi:G^\circ \rightarrow G$ given by $h \mapsto ghg^{-1}$. The image is a connected because $G^\circ$ is connected, and contains the identity $e \in G$ because $\phi(e) = e$. So, we must have $g G^\circ g^{-1} \subseteq G^\circ$. Since this holds for any $g \in G$, we must have $g^{-1} G^\circ g \subseteq G^\circ$, which implies that $G^\circ \subseteq g G^\circ g^{-1}$. Thus, we have $g G^\circ g^{-1} = G^\circ$.
Let $H \subseteq G$ be a subgroup. The normalizer of $H$ in $G$ is defined as $$N_G(H) = \{g \in G \ |\ gHg^{-1} = H\}.$$
Let $F_n \subseteq {\operatorname{GL}}({\operatorname{Mat}}_{n}) = {\operatorname{GL}}_{n^2}$ denote the subgroup of all linear transformations of the form $X \mapsto PXQ$ and $X \mapsto PX^tQ$ for some $P,Q \in {\operatorname{GL}}_n$. The proof of the following lemma is from [@Dokovic-Li], but we recall it as we will need to generalize it.
\[L-normfrob\] The normalizer of $H = G_{n,1}$ in $G = {\operatorname{GL}}({\operatorname{Mat}}_{n})$ is $F_n$.
Let $\tau:{\operatorname{Mat}}_{n} \rightarrow {\operatorname{Mat}}_{n}$ denote the transpose, i.e., $\tau(A) = A^t$. Clearly $\tau \in N_G(H)$, so $F_n \subseteq N_G(H)$.
Let $g \in N_G(H)$. Then $g$ normalizes $H$, and hence normalizes its derived group $[H,H] = G_{n,1}$. Therefore, it normalizes its Lie algebra ${\mathfrak{g}}_{n,1} = {{\rm Lie}}(G_{n,1}) = \{A \otimes {\rm I}_n + {\rm I}_n \otimes B\ |\ {\rm Tr}(A) = {\rm Tr}(B) = 0\}$. Abstractly, ${\mathfrak{g}}_{n,1}$ is equal to ${{\mathfrak{sl}}}_n \oplus {{\mathfrak{sl}}}_n$. Let $L_1 = \{A \otimes {\rm I}_n \ |\ {\rm Tr}(A) = 0\}$ and let $L_2 = \{{\rm I}_n \otimes B\ |\ {\rm Tr}(B) = 0\}$. Then ${\mathfrak{g}}_{n,1} = L_1 \oplus L_2$ is a semisimple Lie algebra, and each $L_i$ is a simple Lie algebra isomorphic to ${{\mathfrak{sl}}}_n$. Thus if $g$ normalizes ${\mathfrak{g}}_{n,1}$, then conjugation by $g$ is an automorphism of the Lie algebra. $L_1$ and $L_2$ are the only simple ideals of ${\mathfrak{g}}_{n,1}$, so such an automorphism must either preserve each $L_i$ or switch the two. Also, observe that conjugation by $\tau$ switches $L_1$ and $L_2$. Thus, by composing with $\tau$ if necessary, we assume w.l.o.g that conjugation by $g$ preserves each $L_i$.
Now, write $g = \sum_{i=1}^r P_i \otimes Q_i$ with $\{P_i\}$ a linearly independent subset of ${\operatorname{Mat}}_{n}$ and $\{Q_i\}$ also a linearly independent subset of ${\operatorname{Mat}}_{n}$. Since $gL_1g^{-1} = L_1$ (equivalently $gL_1 = L_1 g$), we have that for any $A \in {\operatorname{Mat}}_{n}$ (with trace zero), there exists $\widetilde{A} \in {\operatorname{Mat}}_{n}$ (with trace zero) such that $$\sum_i P_i A \otimes Q_i = \sum_i \widetilde{A} P_i \otimes Q_i.$$
Since $Q_i$ are linearly independent, we deduce that $P_i A = \widetilde{A} P_i$ for all $i$. Mutliplying by an appropriate $U \otimes {\rm I}_n$ on the left and $V \otimes {\rm I}_n$ on the right (both of which are in $H$), we can assume that $P_1 = \begin{pmatrix} {\rm I}_k & 0 \\ 0 & 0 \end{pmatrix}$ for some $k \leq n$. We claim that $k = n$. Otherwise, take $A$ to be $\begin{pmatrix} 0 & E \\ 0 & 0 \end{pmatrix}$ for some non-zero $E$, and observe that there is no $\widetilde{A}$ which can satisfy $P_1A = \widetilde{A} P_1$. Thus $k = n$, i.e., $P_1 = {\rm I}_n$. Hence $A = P_1 A = \widetilde{A} P_1 = \widetilde{A}$, i.e., $\widetilde{A} = A$ for all $A$. Thus we must have $P_i A = A P_i$ for all $i \geq 2$. This means that $P_i$ are scalar matrices, but we chose $P_i$ to be linearly independent. So, we must have $i = 1$.
This means that $g = P_1 \otimes Q_1$, with $P_1$ invertible. A similar argument shows that $Q_1$ is also invertible. In other words, $g \in H$.
To summarize, we have that either $g$ or $g \tau$ is in $H$, i.e, $g \in F_n$. This shows that $N_G(H) \subseteq F_n$. Thus $N_G(H) = F_n$ as required.
\[Proof of Theorem \[theo:frob\]\] We know that $\mathcal{G}_S$ is an algebraic subgroup of ${\operatorname{GL}}({\operatorname{Mat}}_{n})$ whose identity component is $G_{n,1}$ by Proposition \[P1\]. Thus, we must have $\mathcal{G}_S \subseteq N_{{\operatorname{GL}}({\operatorname{Mat}}_{n})}(G_{n,1}) = F_n$ by the above discussion. On the other hand, it is clear that $F_n \subseteq \mathcal{G}_S$, so $\mathcal{G}_S = F_n$ as required.
A very short proof (that hides much of the details) is to observe that $F_n \subseteq \mathcal{G}_S \subsetneq {\operatorname{GL}}({\operatorname{Mat}}_{n})$. Next, $F_n$ is a maximal proper subgroup of ${\operatorname{GL}}({\operatorname{Mat}}_{n})$ (see for example [@Dynkin; @Dokovic-Li; @Guralnick]), so $\mathcal{G}_S = F_n$. However, in the general case, such an argument will not suffice, but the argument we give here will be generalized.
Multi-grading on ${{\mathbb C}}[{\operatorname{Mat}}_{n}^m]$ {#sec:multi}
============================================================
For any vector space $V$, the ring of polynomial functions ${{\mathbb C}}[V]$ has a natural grading given by (total) degree. In the case when $V = {\operatorname{Mat}}_{n}^m$, we have a finer multi-grading which we will now describe.
We will denote the coordinate functions on ${\operatorname{Mat}}_{n}^m$ by $x^{(i)}_{jk}$. More precisely, let $x^{(i)}_{jk}$ denote the $(j,k)^{th}$ coordinate of the $i^{th}$ matrix. Thus, $${{\mathbb C}}[{\operatorname{Mat}}_{n}^m] = {{\mathbb C}}[x^{(i)}_{jk} : 1 \leq i \leq m, 1\leq j,k \leq n].$$ We can define an ${{\mathbb N}}^m$-grading on ${{\mathbb C}}[{\operatorname{Mat}}_{n}]$ by setting $$\deg(x^{(i)}_{jk}) = \delta_i = (0,\dots,0,\underbrace{1}_i,0 \dots 0) \in {{\mathbb N}}^m.$$ So, we have $${{\mathbb C}}[{\operatorname{Mat}}_{n}^m] = \bigoplus_{e = (e_1,\dots,e_m) \in {{\mathbb N}}^m} {{\mathbb C}}[{\operatorname{Mat}}_{n}^m]_{e}.$$ where ${{\mathbb C}}[{\operatorname{Mat}}_{n}^m]_{e}$ is the linear span of all monomials $m = \prod (x^{(i)}_{jk})^{a^{(i)}_{jk}}$ such that $\deg(m) = \sum a^{(i)_{jk}} \delta_i = e$.
We will now give another description of $(I_S)_n$ for the case $S = {{\rm SING}}_{n,m}$ that incorporates this multi-grading. First, recall that $(I_S)_n = {{\rm span}}(\det(\sum_i c_i X_i): c_i \in {{\mathbb C}})$. Let $t_1,\dots,t_m$ denote indeterminates. For $e \in {{\mathbb N}}^m$, let $t^e := t_1^{e_1} t_2^{e_2} \dots t_m^{e_m}$. Consider $\det(\sum_i t_i X_i) \in {{\mathbb C}}[{\operatorname{Mat}}_{n}]$. Write $$\det(\sum_i t_i X_i) = \sum_{e \in {{\mathbb N}}^m, \sum_i e_i = n} t^e f_e.$$
\[L-multi-I\] Let $S = {{\rm SING}}_{n,m} \subseteq V = {\operatorname{Mat}}_{n}$. Then we have
1. $(I_S)_n = \bigoplus\limits_{e \in {{\mathbb N}}^m, \sum_i e_i = n} (I_S)_e$ ;
2. $(I_S)_e = {{\rm span}}(f_e)$, and hence $1$-dimensional.
A standard interpolation argument tells us that ${{\rm span}}(f_e: e \in {{\mathbb N}}^m \text{ such that } \sum_i e_i = n) = {{\rm span}}(\det(\sum_i c_i X_i) : c_i \in {{\mathbb C}})$. It is also easy to see that the multi-degree $\deg(f_e) = e$. The lemma now follows.
Let us describe more explicitly the polynomials $f_e$. We call $p = (p_1,\dots,p_n) \in [m]^n$ $e$-compatible if $|\{i \ |\ p_i = j\}| = e_j$ for $1 \leq j \leq m$.
We have $$f_e = \sum_{p \text{ is } e\text{-compatible}} \sum_{\sigma \in S_n} {{\rm sgn}}(\sigma) x^{(p_1)}_{1\sigma(1)} x^{(p_2)}_{2\sigma(2)}\dots x^{(p_n)}_{n\sigma(n)}.$$
An intermediate problem: action of $({\operatorname{GL}}_n \times {\operatorname{GL}}_n)^{\times m}$ {#sec:intermediate}
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In this section, we consider the action of $({\operatorname{GL}}_n \times {\operatorname{GL}}_n)^{\times m}$ on ${\operatorname{Mat}}_{n}^m$ given by $$((P_1,Q_1),\dots,(P_m,Q_m)) \cdot (X_1,\dots,X_m) = (P_1X_1Q_1^{t},\dots,P_mX_mQ_m^{t}).$$
The main goal of this section is to compute the subgroup of $({\operatorname{GL}}_n \times {\operatorname{GL}}_n)^{\times m}$ which fixes ${{\rm SING}}_{n,m}$. Our approach here will be slightly different because we do not resort to a Lie algebra computation. Consider the homomorphism ${\operatorname{GL}}_n \times {\operatorname{GL}}_n \times ({{\mathbb C}}^*)^m \rightarrow ({\operatorname{GL}}_n \times {\operatorname{GL}}_n)^{\times m}$ given by $$(P,Q,(\lambda_1,\dots,\lambda_m),(\mu_1,\dots,\mu_m)) \mapsto ((\lambda_1P,\mu_1Q), (\lambda_2 P, \mu_2Q),\dots, (\lambda_m P, \mu_mQ)).$$ Let the image of this homomorphism be denoted $H$.
\[P-inter\] The subgroup of $({\operatorname{GL}}_n \times {\operatorname{GL}}_n)^{\times m}$ that preserves $S = {{\rm SING}}_{n,m} \subseteq V = {\operatorname{Mat}}_{n}$ is $H$, i.e., $$\{g \in ({\operatorname{GL}}_n \times {\operatorname{GL}}_n)^{\times m} \ |\ g S = S\} = H$$
We will derive the above proposition from the following lemma.
\[L-inter\] Let $S = {{\rm SING}}_{n,m} \subseteq V = {\operatorname{Mat}}_{n}$. The subgroup $\{ g \in ({\operatorname{GL}}_n \times {\operatorname{GL}}_n)^{\times m} \ |\ g (I_S)_n \subseteq (I_S)_n \} = H$.
We will prove Proposition \[P-inter\] assuming Lemma \[L-inter\] above, and then we will prove Lemma \[L-inter\].
\[Proof of Proposition \[P-inter\]\] Let $\rho:({\operatorname{GL}}_n \times {\operatorname{GL}}_n)^{\times m} \rightarrow {\operatorname{GL}}({\operatorname{Mat}}_{n}^m)$ be the group homomorphism that defines the representation above. Then, observe that $\{g \in ({\operatorname{GL}}_n \times {\operatorname{GL}}_n)^{\times m} \ |\ g S = S\} = \rho^{-1} (\mathcal{G}_S)$.
Similarly, $\{ g \in ({\operatorname{GL}}_n \times {\operatorname{GL}}_n)^{\times m} \ |\ g (I_S)_n \subseteq (I_S)_n \} = \rho^{-1} (\mathcal{G}_S)$ follows from Lemma \[Lgos-graded\] because the zero locus of $(I_S)_n$ is precisely $S$.
Thus, we have an equality $$\{g \in ({\operatorname{GL}}_n \times {\operatorname{GL}}_n)^{\times m} \ |\ g {{\rm SING}}_{n,m} = {{\rm SING}}_{n,m}\} =\{ g \in ({\operatorname{GL}}_n \times {\operatorname{GL}}_n)^{\times m} \ |\ g (I_S)_n \subseteq (I_S)_n \} = \rho^{-1}(\mathcal{G}_S).$$
Hence, Lemma \[L-inter\] implies Proposition \[P-inter\]
Now, all that is left to prove is Lemma \[L-inter\].
\[Proof of Lemma \[L-inter\]\] Let $g \in ({\operatorname{GL}}_n \times {\operatorname{GL}}_n)^{\times m}$ be such that $g (I_S)_n \subseteq (I_S)_n$. Write $g = ((P_1,Q_1^{t}),\dots,(P_m,Q_m^{t}))$ (We put transposes on the $Q$’s for convenience). For any $f \in (I_S)_n$, we have $$g^{-1} \cdot f (X) = f( gX) = f( P_1X_1Q_1,\dots, P_m X_m Q_m) = c \cdot f(X_1, L_2X_2R_2,\dots, L_mX_mR_m),$$
where $L_i = P_1^{-1}P_i$ and $R_i = Q_i Q_1^{-1}$, and $c = \det(P_1^{-1}Q_1^{-1})$. The last equality follows because $(I_S)_n$ is spanned by $\det(\sum_i c_i X_i)$, and $ \det(\sum_i c_i P_1^{-1} X_i Q_1^{-1}) = \det(P_1^{-1}Q_1^{-1}) \det(\sum_i c_i X_i) $
Recall the multi-degree defined in the previous section. Observe that for any $g \in ({\operatorname{GL}}_n \times {\operatorname{GL}}_n)^{\times m}$ and any (multi)-homogenous polynomial $f$ of degree $e$, the polynomial $g^{-1} \cdot f$ is also (multi)-homogenous of degree $e$.
From now on, let $$f = f_{(n-1,1,0,\dots,0)} = \sum_{\sigma \in S_n} \sum_{1 \leq r \leq n} x^{(1)}_{1\sigma(1)} \dots x^{(1)}_{(r-1),\sigma(r-1)} x^{(2)}_{r \sigma(r)} x^{(1)}_{(r+1),\sigma(r+1)} \dots x^{(1)}_{n\sigma(n)}.$$
Then we have $$g^{-1} \cdot f (X) = f(gX) = c f(X) \text{ for some } c \in {{\mathbb C}}^*.$$ This is because $g^{-1} \cdot f$ is a non-zero polynomial that is homogeneous of degree $(n-1,1,0,\dots,0)$ and must be in $(I_S)_n$, and so is in $(I_S)_{(n-1,1,0,\dots,0)}$ which is spanned by $f = f_{(n-1,1,0,\dots,0)}$ by Lemma \[L-multi-I\].
The matrix $L_2$ is diagonal.
\[Proof of Claim\] Suppose $(L_2)_{ij} \neq 0$ for some $i \neq j$ (Here $(L_2)_{ij}$ denotes the $(i,j)^{th}$ entry of $L_2$). Let $p$ and $q$ be such that $(R_2)_{pq} \neq 0$. This means that $(L_2)_{ij} (R_2)_{pq} \neq 0$. We have $$\begin{aligned}
g^{-1}\cdot f &= f(gX) \\
& = f(X_1,L_2X_2R_2,\dots,L_mX_mR_m) \\
& = \sum_{\sigma \in S_n} {{\rm sgn}}(\sigma) \sum_r x^{(1)}_{1\sigma(1)} \dots x^{(1)}_{(r-1),\sigma(r-1)} \left(\sum_{a,b} (L_2)_{ra} \cdot x^{(2)}_{ab} \cdot (R_2)_{b\sigma(r)} \right) x^{(1)}_{(r+1),\sigma(r+1)} \dots x^{(1)}_{n\sigma(n)}. \\
& = \sum_{\sigma,r,a,b} {{\rm sgn}}(\sigma) \cdot (L_2)_{ra} \cdot (R_2)_{b\sigma(r)} \cdot x^{(1)}_{1\sigma(1)} \dots x^{(1)}_{(r-1),\sigma(r-1)} x^{(2)}_{ab} x^{(1)}_{(r+1),\sigma(r+1)} \dots x^{(1)}_{n\sigma(n)}. \end{aligned}$$
Let $\pi \in S_n$ be such that $\pi(i) = q$. Let us compute the coefficient of the monomial $m = x^{(1)}_{1\pi(1)} \dots x^{(1)}_{(i-1),\pi(i-1)} x^{(2)}_{jp} x^{(1)}_{(i+1),\pi(i+1)} \dots x^{(1)}_{n\pi(n)}$ in $g^{-1} \cdot f$. In the expansion of $g^{-1} \cdot f$ above, let us see for which choices of $\sigma,r,a,b$ do we get the monomial $m$. Indeed we must have $ r = i$, and so we must have $\sigma(k) = \pi(k)$ for all $k \neq i$, which forces $\sigma =\pi$. Further, we must have $a = j$ and $b = q$. Hence, we conclude that the monomial $m$ appears in the above expansion exactly once and with a coefficient of ${{\rm sgn}}(\pi) (L_2)_{ij} (R_2)_{pq}$ (which is nonzero as noted above). However, the coefficient of the monomial $m$ in $c \dot f_{(n-1,1,0,\dots,0)}$ is zero. This is a contradiction. Therefore $(L_2)_{ij} = 0$ for all $i \neq j$. This means that $(L_2)$ is a diagonal matrix.
By a similar argument, all the $(L_i)$’s and $(R_i)$’s are all diagonal matrices. Now, let us write out $g^{-1} \cdot f$ again. Since all the $L_i$’s and $R_i$’s are diagonal, we have $$g^{-1}\cdot f = \sum_{\sigma \in S_n} \sum_r x^{(1)}_{1\sigma(1)} \dots x^{(1)}_{(r-1),\sigma(r-1)} \left( (L_2)_{rr} x^{(2)}_{r \sigma(r)} (R_2)_{\sigma(r),\sigma(r)} \right) x^{(1)}_{(r+1),\sigma(r+1)} \dots x^{(1)}_{n\sigma(n)}.$$
The matrices $L_i$ and $R_i$ are scalar matrices.
\[Proof of Claim\] The coefficient of $n = x^{(1)}_{1\sigma(1)} \dots x^{(1)}_{(r-1),\sigma(r-1)} x^{(2)}_{r \sigma(r)} x^{(1)}_{(r+1),\sigma(r+1)} \dots x^{(1)}_{n\sigma(n)}$ in $g^{-1} \cdot f$ is ${{\rm sgn}}(\sigma)(L_2)_{rr} (R_2)_{\sigma(r),\sigma(r)}$. The coefficient of $n$ in $c \cdot f$ is ${{\rm sgn}}(\sigma) \cdot c$. Thus, if we are to have $g^{-1} \cdot f = c \cdot f$, then we must have $(L_2)_{rr} (R_2)_{\sigma(r),\sigma(r)} = c$. This must hold for all choices of $r$ and $\sigma$, so we have $(L_2)_{ii} (R_2)_{jj} = c$ for all $i,j$. This means that $(L_2)_{ii} = (L_2)_{kk}$ for all $i,k$, i.e., $L_2$ is a scalar matrix, and so is $R_2$. Similarly all the $L_i$ and $R_i$ are scalar matrices.
Since the $L_i$’s and $R_i$’s are scalar matrices, we can write $L_i = \lambda_i {\rm I}_n$ and $R_i = \mu_i {\rm I}_n$ for scalars $\lambda_i,\mu_i \in {{\mathbb C}}^*$. Thus we have $P_i = \lambda_i P_1$ and $Q_i = \mu_i Q_1$ for $i \geq 2$. Thus, we have $$g = ((P_1,Q_1),(\lambda_2P_1,\mu_2Q_2),\dots,(\lambda_mP_1,\mu_mQ_1)) \in H.$$
To summarize, we have shown that $\{ g \in ({\operatorname{GL}}_n \times {\operatorname{GL}}_n)^{\times m} \ |\ g (I_S)_n \subseteq (I_S)_n \} \subseteq H$. The other inclusion is clear.
Symmetries of singular tuples of matrices {#sec:symsingnm}
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In this section, we will compute the group of symmetries for $S = {{\rm SING}}_{n,m}$, i.e., Theorem \[theo:gos\]. While the high-level strategy resembles that of Section \[sec:frob\] (which deals with the $m =1 $ case), we need to work a little harder in the computations. Below, we will recall the setup again for the convenience of the reader. Then, we will describe the action of ${{\mathfrak{gl}}}({\operatorname{Mat}}_{n}^m)$ on polynomials explicitly. We then present the explicit computation of the Lie algebra of symmetries. The main features that we utilize in the computation are the multi-grading (as defined in Section \[sec:multi\]), the description of the ideal $I_S$ in Lemma \[L-multi-I\] and the intermediate case resolved in the previous section. This is followed by the computation of the entire group of symmetries which parallels the computation in Section \[sec:frob\]. Finally we indicate how the same arguments also compute the group of symmetries for ${{\rm NSING}}_{n,m}$.
First, let us recall the setup again. We have $S = {{\rm SING}}_{n,m} \subseteq V = {\operatorname{Mat}}_{n}^m$, and we want to compute $\mathcal{G}_S,\mathcal{G}_S^\circ$ and ${\mathfrak{g}}_S$. We will first focus on ${\mathfrak{g}}_S$. From Corollary \[C-loscom\], we have $${\mathfrak{g}}_S = \{M \in {{\mathfrak{gl}}}({\operatorname{Mat}}_{n}^m) \ |\ M \cdot (I_S)_n \subseteq (I_S)_n \}$$ Moreover, we have by Proposition \[P-ideal\] that $$(I_S)_n = {{\rm span}}(\det(\sum_i c_i X_i) : c_i \in {{\mathbb C}}).$$ From Lemma \[L-multi-I\], we know that $(I_S)_n$ is multi-graded, the explicit decomposition being $(I_S)_n = \bigoplus\limits_{e \in {{\mathbb N}}^m, \sum_i e_i = n} (I_S)_e$, where $(I_S)_e$ is $1$-dimensional and spanned by $f_e$ (as defined in Lemma \[L-multi-I\]). Moreover, we have an explicit formula for $f_e$, i.e., $$\label{fe}
f_e = \sum_{p \text{ is } e\text{-compatible}} \sum_{\sigma \in S_n} {{\rm sgn}}(\sigma) x^{(p_1)}_{1\sigma(1)}x^{(p_2)}_{2\sigma(2)} \dots x^{(p_n)}_{n\sigma(n)},$$ where we call $p = (p_1,\dots,p_n) \in [m]^n$ $e$-compatible if $|\{i \ |\ p_i = j\}| = e_j$ for $1 \leq j \leq m$. Of particular interest are the cases of $e = (n,0,\dots,0)$ and $e = (n-1,1,0,\dots,0)$. We have: $$f_{(n,0,\dots,0)} = \sum_{\sigma \in S_n} {{\rm sgn}}(\sigma) x^{(1)}_{1\sigma(1)}\dots x^{(1)}_{n\sigma(n)} = \det(X_1)$$ Similarly, we have: $$f_{(n-1,1,0,\dots,0)} = \sum_{j \in [n]} \sum_{\sigma \in S_n} {{\rm sgn}}(\sigma) x^{(1)}_{1\sigma(1)} \dots x^{(1)}_{j-1, \sigma(j-1)} x^{(2)}_{j\sigma(j)}x^{(1)}_{j+1,\sigma(j+1)} \cdots x^{(1)}_{n \sigma(n)}.$$
In the next subsection, we will write out the (twisted) action of ${{\mathfrak{gl}}}({\operatorname{Mat}}_{n}^m)$ explicitly, so that we can make the computations we need.
Action of ${{\mathfrak{gl}}}({\operatorname{Mat}}_{n}^m)$ on ${{\mathbb C}}[{\operatorname{Mat}}_{n}^m]$
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Let $V = {\operatorname{Mat}}_{n}^m$ be the space of $m$-tuples of $n \times n$ matrices. Let $E^{(i)}_{jk}$ denote the tuple of matrices with a $1$ in the $(j,k)^{th}$ entry of the $i^{th}$ matrix, and $0$’s everywhere else. Let $x^{(i)}_{jk}$ denote the coordinate function corresponding to the $(j,k)^{th}$ entry of the $i^{th}$ matrix. Then ${{\mathbb C}}[V] = {{\mathbb C}}[x^{(i)}_{jk} : 1\leq i \leq m, 1 \leq j,k \leq n]$.
Since $V$ is $mn^2$ dimensional, we can identify ${{\mathfrak{gl}}}(V) = {{\mathfrak{gl}}}({\operatorname{Mat}}_{n}^m)$ with ${\operatorname{Mat}}_{mn^2}$, but we will do so in a very specific way. We will think of ${{\mathfrak{gl}}}({\operatorname{Mat}}_{n}^m) = {{\mathfrak{gl}}}({{\mathbb C}}^m \otimes {\operatorname{Mat}}_{n}) = {\operatorname{Mat}}_{m} \otimes {\operatorname{Mat}}_{n^2}$. We have already been explicit in the earlier sections about how we view ${\operatorname{Mat}}_{n^2}$ as ${{\mathfrak{gl}}}({\operatorname{Mat}}_{n})$.
We index the rows and columns of $mn^2 \times mn^2$ matrices by $\{(i,j,k): 1 \leq i \leq m, 1 \leq j,k \leq n\}$ in lexicographic order. Thus, we can write $M \in {{\mathfrak{gl}}}({\operatorname{Mat}}_{n}^m) = {\operatorname{Mat}}_{mn^2}$ as
$$M = \begin{pmatrix} M_{11} & \dots & M_{1m} \\
\vdots & \ddots & \vdots \\
M_{m1} & \dots & M_{mm}
\end{pmatrix},$$ where each $M_{pq}$ is an $n^2 \times n^2$ matrix. Equivalently, we can write $M = \sum_{1 \leq p,q \leq m} E_{pq} \otimes M_{pq}$. With this indexing, we have the intuitive formula $$E_{pq} \otimes E_{ab,cd} = E_{pab, qcd}.$$
The twisted action of the matrix $E_{pab,qcd} \in {{\mathfrak{gl}}}({\operatorname{Mat}}_{n}^m)$ is via the derivation $x^{(p)}_{ab} \partial^{(q)}_{cd}$ where $\partial^{(q)}_{cd}$ denotes the partial derivative with respect to the coordinate $x^{(q)}_{cd}$.
Recall the ${{\mathbb N}}^m$-grading on ${{\mathbb C}}[{\operatorname{Mat}}_{n}^m]$. Observe that the twisted action of $E_{pab,qcd}$ maps ${{\mathbb C}}[V]_e$ to ${{\mathbb C}}[V]_{e + \delta_p - \delta_q}$, where $\delta_i = (0,\dots,0,\underbrace{1}_i,0\dots,0) \in {{\mathbb N}}^m$. Thus, we give an ${{\mathbb N}}^m$-grading on ${{\mathfrak{gl}}}({\operatorname{Mat}}_{n}) = {\operatorname{Mat}}_{mn^2}$.
\[Grading on ${{\mathfrak{gl}}}({\operatorname{Mat}}_{n}^m)$\] We give a grading on the Lie algebra ${{\mathfrak{gl}}}({\operatorname{Mat}}_{n}^m)$ by setting $\deg(E_{pab,qcd}) = \delta_p - \delta_q$. We have $${{\mathfrak{gl}}}({\operatorname{Mat}}_{n}^m) = {{\mathfrak{gl}}}({\operatorname{Mat}}_{n}^m)_0 \bigoplus_{p \neq q} {{\mathfrak{gl}}}({\operatorname{Mat}}_{n}^m)_{\delta_p - \delta_q}.$$ For any $M \in {\operatorname{Mat}}_{mn^2} = {{\mathfrak{gl}}}({\operatorname{Mat}}_{n}^m)$, we write $M = \sum_{1 \leq p,q \leq m} E_{pq} \otimes M_{pq}$, with $M_{pq} \in {\operatorname{Mat}}_{n^2}$. Then, the degree $0$ part is $\sum_{p = 1}^m E_{pp} \otimes M_{pp}$, and for $p \neq q$, $E_{pq} \otimes M_{pq}$ is the degree $\delta_p - \delta_q$ part. Thus the decomposition of $M$ into homogenous components is $$M = (\sum_{p=1}^m E_{pp} \otimes M_{pp} ) \bigoplus_{p \neq q} E_{pq} \otimes M_{pq}.$$
The following lemma is immediate from the preceding discussion:
\[L-grade-nm\] Let $M \in {\operatorname{Mat}}_{mn^2}$ be homogenous of degree $e$, and $f \in {{\mathbb C}}[{\operatorname{Mat}}_{n}^m]$ be homogenous of degree $e'$. Then $M {\star}f$ is homogenous of degree $e + e'$.
Computing the Lie algebra of symmetries
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For this subsection, let $M \in {{\mathfrak{gl}}}({\operatorname{Mat}}_{n}^m) = {\operatorname{Mat}}_{mn^2}$ be such that $M {\star}(I_S)_n \subseteq (I_S)_n$. Further, write $M = \sum_{p,q} E_{pq} \otimes M_{pq}$, where $M_{pq} \in {\operatorname{Mat}}_{n^2}$, i.e., $$M = \begin{pmatrix} M_{11} & \dots & M_{1m} \\
\vdots & \ddots & \vdots\\
M_{m1} & \dots & M_{mm}
\end{pmatrix}.$$
For $p \neq q$, we have $M_{pq} = \lambda_{pq} {\rm I}_{n^2}$ for some scalar $\lambda_{pq} \in {{\mathbb C}}$.
Without loss of generality, let us assume that $p = 2, q = 1$. We know that $M {\star}f_{(n,0,\dots,0)} \in (I_S)_n$. Consider the degree $(n-1,1,0,\dots,0)$ homogenous part of $M {\star}f_{(n,0,\dots,0)}$. By Lemma \[L-grade-nm\] and the description of the grading on ${{\mathfrak{gl}}}({\operatorname{Mat}}_{n}^m)$, we know that the degree $(n-1,1,0,\dots,0)$ homogenous part of $M {\star}f_{(n,0,\dots,0)}$ is $(E_{21} \otimes M_{21}) {\star}f_{(n,0,\dots,0)}$. Since, this must be in $(I_S)_{(n-1,1,0,\dots,0)}$ which is spanned by $f_{(n-1,1,0,\dots,0)}$, we must have $$(E_{21} \otimes M_{21}) {\star}f_{(n,0,\dots,0)} = c \cdot f_{(n-1,1,0,\dots,0)},$$ for some $c \in {{\mathbb C}}$. $M_{21} \in {\operatorname{Mat}}_{n^2}$ (and recall that we index the rows and columns of ${\operatorname{Mat}}_{n^2}$ by $[n] \times [n]$ in lexicographic order). Let the $(ab,cd)^{th}$ entry of $M_{21}$ be $\mu_{ab,cd}$.
Thus, $$\begin{aligned}
\label{M21}
(E_{21} \otimes M_{21}) {\star}f_{(n,0,\dots,0)} &= \left(\sum_{a,b,c,d} \mu_{ab,cd} \cdot x^{(2)}_{ab} \partial^{(1)}_{cd} \right) \left(\sum_{\sigma \in S_n} {{\rm sgn}}(\sigma) \cdot x^{(1)}_{1 \sigma(1)} \dots x^{(1)}_{n\sigma(n)} \right) \\
\label{M22}
& = \sum_{\begin{array}{c} a,b,c,d,\sigma \\ \sigma(c) = d \end{array}} \mu_{ab,cd} \cdot {{\rm sgn}}(\sigma) \cdot x^{(1)}_{1 \sigma(1)} \dots x^{(1)}_{(c-1),\sigma(c-1)} x^{(2)}_{ab} x^{(1)}_{(c+1)\sigma(c+1)} \dots x^{(1)}_{n\sigma(n)}.\end{aligned}$$
Since $(E_{21} \otimes M_{21}) {\star}f_{(n,0,\dots,0)} = c \cdot f_{(n-1,1,0,\dots,0)}$, we will match the coefficients of monomials on both sides to get conditions on the entries of $M_{21}$.
First, fix $\pi \in S_n$, $i \in [n]$ and let $\alpha \beta \neq i \pi(i)$. Now, consider the coefficient of the monomial $$m = x^{(1)}_{1 \pi(1)} \dots x^{(1)}_{(i-1),\sigma(i-1)} x^{(2)}_{\alpha,\beta} x^{(1)}_{(i+1)\sigma(i+1)} \dots x^{(1)}_{n\sigma(n)}.$$
In the expression Equation \[M22\], let us see what choices of $a,b,c,d,\sigma$ lead to this monomial. Clearly, we need $c = i$, $ d = \sigma(i)$. Moreover, we need $\sigma(k) = \pi(k)$ for all $k \neq i$, so $\sigma = \pi$ (and hence $\sigma(i) = \pi(i) = d$). Finally also observe that we also need $a = \alpha$ and $b = \beta$. Thus, there is precisely one choice for which can lead to the monomial $m$, and this means that the coefficient of the monomial $m$ is $ \mu_{\alpha\beta,i\pi(i)}\cdot {{\rm sgn}}(\pi)$. The coefficient of $m$ in $c \cdot f_{(n-1,1,0,\dots,0)}$ is zero, so we must have $\mu_{\alpha\beta,i\pi(i)} = 0$. Observe that as long as $i_1j_1 \neq i_2j_2$, we can choose $\alpha = i_1, \beta = j_1$, $i = i_2$ and $\pi$ such that $\pi(i) = j_2$ and satisfy the condition $\alpha\beta \neq i\pi(i)$. Thus, all the off-diagonal terms of $M_{21}$ are zero. In other words $M_{21}$ is a diagonal matrix.
Now that $M_{21}$ is a diagonal matrix, we have $$\begin{aligned}
(E_{21} \otimes M_{21}) {\star}f_{(n,0,\dots,0)} & =
\sum_{\begin{array}{c} a,b,\sigma \\ \sigma(a) = b \end{array}} \mu_{ab,ab} \cdot {{\rm sgn}}(\sigma) \cdot x^{(1)}_{1 \sigma(1)} \dots x^{(1)}_{(a-1),\sigma(a-1)} x^{(2)}_{ab} x^{(1)}_{(a+1)\sigma(a+1)} \dots x^{(1)}_{n\sigma(n)} \\
\label{M21-diag}
& = \sum_{a,\sigma} \mu_{a\sigma(a),a\sigma(a)} \cdot {{\rm sgn}}(\sigma) \cdot x^{(1)}_{1 \sigma(1)} \dots x^{(1)}_{(a-1),\sigma(a-1)} x^{(2)}_{a\sigma(a)} x^{(1)}_{(a+1)\sigma(a+1)} \dots x^{(1)}_{n\sigma(n)}.\end{aligned}$$
On the other hand $$\begin{aligned}
c \cdot f_{(n-1,1,0,\dots,0)} = \sum_{a,\sigma} c \cdot {{\rm sgn}}(\sigma) \cdot x^{(1)}_{1 \sigma(1)} \dots x^{(1)}_{(a-1),\sigma(a-1)} x^{(2)}_{a\sigma(a)} x^{(1)}_{(a+1)\sigma(a+1)} \dots x^{(1)}_{n\sigma(n)}.\end{aligned}$$
Thus, by matching coefficients of monomials, we get that $\mu_{a\sigma(a),a\sigma(a)} = c$. Since this is true for all choices of $a$ and $\sigma$, we have that $\mu_{ab,ab} = c$ for all $ab \in [n] \times [n]$. This means that $M_{21} = c {\rm I}_{n^2}$. Thus, with $\lambda_{21} = c$, we have $M_{21} = \lambda_{21} \cdot {\rm I}_{n^2}$ as required.
Recall the action of $({\operatorname{GL}}_n \times {\operatorname{GL}}_n)^{\times m}$ on ${\operatorname{Mat}}_{n}^m$ in Section \[sec:intermediate\]. This gives a homomorphism $\rho: ({\operatorname{GL}}_n \times {\operatorname{GL}}_n)^{\times m} \rightarrow {\operatorname{GL}}({\operatorname{Mat}}_{n}^m) = {\operatorname{GL}}_{mn^2}$. In coordinates the map is given explicitly by the formula $$((P_1,Q_1),(P_2,Q_2),\dots,(P_m,Q_m)) \mapsto \sum_i E_{ii} \otimes P_i \otimes Q_i.$$
Differentiating gives a Lie algebra homomorphism $d\rho: ({{\mathfrak{gl}}}_n \times {{\mathfrak{gl}}}_n)^{\times m} \rightarrow {{\mathfrak{gl}}}({\operatorname{Mat}}_{n}^m) = {\operatorname{Mat}}_{mn^2}$. Explicitly in coordinates, this is given by the formula $$((A_1,B_1),\dots,(A_m,B_m)) \mapsto \sum_i E_{ii} \otimes (A_i \otimes {\rm I}_n + {\rm I}_n \otimes B_i).$$
Recall the group $H$ defined in Section \[sec:intermediate\]. By Lemma \[L-inter\], we have $H = \{g \in ({\operatorname{GL}}_n \times {\operatorname{GL}}_n)^{\times m}\ |\ g (I_S)_n \subseteq (I_S)_n\} = \{g \in ({\operatorname{GL}}_n \times {\operatorname{GL}}_n)^{\times m}\ | \rho(g) \in \mathcal{G}_S\}$. It follows that $${{\rm Lie}}(H) = \{N \in ({{\mathfrak{gl}}}_n \times {{\mathfrak{gl}}}_n)^{\times m} \ |\ d\rho(N) \cdot (I_S)_n \subseteq (I_S)_n\} = \{N \in ({{\mathfrak{gl}}}_n \times {{\mathfrak{gl}}}_n)^{\times m} \ |\ d\rho(N) \in {\mathfrak{g}}_S\}.$$ The first equality essentially follows from the same argument in Proposition \[P-Liealg-all\], and the second equality is clear from Proposition \[P-Liealg-all\]. From the description of $H$ in Section \[sec:intermediate\], a straightforward computation gives $$d\rho ({{\rm Lie}}(H)) = \{{\rm I}_m \otimes A \otimes {\rm I}_n + {\rm I}_m \otimes {\rm I}_n \otimes B + D \otimes {\rm I}_{n} \otimes {\rm I}_n \ |\ A,B \in {\operatorname{Mat}}_{n}, D \in {\operatorname{Mat}}_{n} \text{ diagonal matrix}\}.$$
Consider the degree $0$ part of $M$, i.e., $M_0 = \sum_i E_{ii} \otimes M_{ii}$. Then $M_0 \in d\rho ({{\rm Lie}}(H))$
Let $M = \bigoplus_e M_e$ be its graded decomposition. We know that $M_{\delta_i - \delta_j} = E_{ij} \otimes M_{ij}$ for $i \neq j$, and $M_0 = \sum_i E_{ii} \otimes M_{ii}$. For all other $e$, $M_e = 0$. In particular, $M {\star}(I_S)_n \subseteq (I_S)_n$ implies that for all $e$ with $\sum_i e_i = n$, $M_0 {\star}f_e = \gamma_e f_e$ for some $\gamma_e \in {{\mathbb C}}$.
First, observe that $M_0 {\star}f_{(n,0,\dots,0)} = M {\star}\det(X_1) = \sum_i (E_{ii} \otimes M_{ii}) {\star}(\det(X_1)) = (E_{11} \otimes M_{11}) {\star}\det(X_1)$. This means that $(E_{11} \otimes M_{11}) {\star}\det(X_1) = \gamma_{(n,0,\dots,0)} \det(X_1)$. By Theorem \[theo:frob\], and Corollary \[C-loscom\], we get that $M_{11}$ is of the form $A_1 \otimes {\rm I}_n + {\rm I}_n \otimes B$ for some $A,B \in {\operatorname{Mat}}_{n}$. Similarly, each $M_{ii}$ is of the form $A_i \otimes {\rm I}_n + {\rm I}_n \otimes B_i$. Thus $M_0$ is in the image of $d\rho$, since $M_0 = \sum_i E_{ii} \otimes (A_i \otimes {\rm I}_n + {\rm I}_n \otimes B_i)$ (see the explicit formula for $d\rho$ above).
This means that $M_0$ is in the image of $d\rho$ such that $(M_0) {\star}(I_S)_n \subseteq (I_S)_n$. From the description of ${{\rm Lie}}(H)$ above, we get that $\Theta(M_0) \in d\rho ({{\rm Lie}}(H))$. But since $d\rho({{\rm Lie}}(H))$ is closed under $\Theta$, we get that $M_0 \in d \rho ({{\rm Lie}}(H))$.
Thus, putting the above two lemmas together, we get that $M$ is of the form $C \otimes {\rm I}_n \otimes {\rm I}_n + {\rm I}_m \otimes A \otimes {\rm I}_n + {\rm I}_m \otimes {\rm I}_n \otimes B$, i.e., $M \in {{\rm Lie}}(G_{n,m})$.
Thus, we conclude that $${\mathfrak{g}}_S = \Theta \{M \ |\ M {\star}(I_S)_n \subseteq (I_S)_n\} \subseteq \Theta({{\rm Lie}}(G_{n,m})) = {{\rm Lie}}(G_{n,m}).$$ since ${{\rm Lie}}(G_{n,m})$ is closed under $\Theta$.
The reverse inclusion is clear since $G_{n,m} \subseteq \mathcal{G}_S$ implies that ${{\rm Lie}}(G_{n,m}) \subseteq {{\rm Lie}}(\mathcal{G}_S) = {\mathfrak{g}}_S$. So, we conclude that $${\mathfrak{g}}_S = {{\rm Lie}}(G_{n,m}).$$ Further, this implies (by the Lie subgroups – Lie subalgebras correspondence) that $$\mathcal{G}_S^\circ = G_{n,m}.$$
Let us record this result.
Let $S = {{\rm SING}}_{n,m} \subseteq V = {\operatorname{Mat}}_{n}$. Then the connected group of symmetries $$\mathcal{G}_S^\circ = G_{n,m}.$$
In the next subsection, we will determine the entire group of symmetries. The argument is very similar to the one in Section \[sec:frob\]
The group of symmetries
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From the above discussion, we know that $G_{n,m}$ is the identity component of $\mathcal{G}_S$. The component group $\mathcal{G}_S / G_{n,m}$ is a finite group[^17]. In any case the fact that $G_{n,m}$ is the identity component of $\mathcal{G}_S$ means that $\mathcal{G}_S$ normalizes $G_{n,m}$. Thus $\mathcal{G}_S \subseteq N_{{\operatorname{GL}}({\operatorname{Mat}}_{n}^m)} (G_{n,m})$. Let us therefore compute this normalizer.
Consider the transpose map $\tau: {\operatorname{Mat}}_{n}^m \rightarrow {\operatorname{Mat}}_{n}^m$ given by $(X_1,\dots,X_m) \mapsto (X_1^t,\dots,X_m^t)$. Viewing ${\operatorname{Mat}}_{n}^m$ as ${{\mathbb C}}^m \otimes {{\mathbb C}}^n \otimes {{\mathbb C}}^n$, the $\tau$ is simply the linear map that switches the second and third tensor factors. When $m = n$, then all three tensor factors are ${{\mathbb C}}^n$, and there are linear transformations that permute them in any way. For a permutation $\sigma \in S_3$, let us denote by $\tau_{\sigma}$ the corresponding linear map. Precisely, we have $$\begin{aligned}
\tau_\sigma: {{\mathbb C}}^n \otimes {{\mathbb C}}^n \otimes {{\mathbb C}}^n &\longrightarrow {{\mathbb C}}^n \otimes {{\mathbb C}}^n \otimes {{\mathbb C}}^n \\
\sum_i v_{i,1} \otimes v_{i,2} \otimes v_{i,3} &\longmapsto \sum_i v_{i,\sigma(1)} \otimes v_{i,\sigma(2)} \otimes v_{i,\sigma(3)}.\end{aligned}$$
In particular, the transpose morphism $\tau = \tau_{\sigma}$ for $\sigma$ defined as $\sigma(1) = 1, \sigma(2) = 3, \sigma(3) = 2$.
Let us define $$\Sigma_{n,m} = \begin{cases} \{1,\tau\} & \text{ if $n \neq m$} \\
\{\tau_{\sigma}: \sigma \in S_3\} & \text{ if $n = m$}. \end{cases}$$
Observe that $\Sigma_{n,m}$ is a subset of linear transformations of ${{\mathbb C}}^m \otimes {{\mathbb C}}^n \otimes {{\mathbb C}}^n$. When $m = n$, $\Sigma_{n,m}$ consists of six linear transformations, and when $m \neq n$, it consists of two linear transformations.
The normalizer $$N_{{\operatorname{GL}}({\operatorname{Mat}}_{n}^m)}(G_{n,m}) = \{h_1h_2 \ | h_1 \in G_{n,m}, h_2 \in \Sigma_{n,m} \} = G_{n,m} \rtimes \Sigma_{n,m}.$$
The argument is very similar to the one for the $m=1$ case. Let $g \in {\operatorname{GL}}({\operatorname{Mat}}_{n}^m)$ be such that $g$ normalizes $G_{n,m}$. Thus, it normalizes its derived group which is isomorphic to ${\operatorname{SL}}_m \times {\operatorname{SL}}_n \times {\operatorname{SL}}_n$, and hence the Lie algebra of its derived group. This Lie algebra is ${{\mathfrak{sl}}}_m \times {{\mathfrak{sl}}}_n \times {{\mathfrak{sl}}}_n$ which embeds in ${{\mathfrak{gl}}}({\operatorname{Mat}}_{n}^m)$ as $$\{C \otimes {\rm I}_n \otimes {\rm I}_n + {\rm I}_m \otimes A \otimes {\rm I}_n + {\rm I}_m \otimes {\rm I}_n \otimes B\ | \ C \in {{\mathfrak{sl}}}_m, A,B, \in {{\mathfrak{sl}}}_n\}.$$ For simplicity we will continue to refer to this Lie subalgebra as ${{\mathfrak{sl}}}_m \times {{\mathfrak{sl}}}_n \times {{\mathfrak{sl}}}_n$. $g$ normalizes this Lie subalgebra. This Lie subalgebra has exactly $3$ simple ideals, so the (conjugation) action of $g$ has to permute them. There are two cases, when $m = n$, then there are 6 possible permutations, and when $m \neq n$, the ${{\mathfrak{sl}}}_m$ must remain fixed and the two ${{\mathfrak{sl}}}_n$’s can be permuted. Now, one observes (in both cases) that for some $h_2 \in \Sigma_{n,m}$, the action of $g' = gh_2$ (by conjugation) fixes the three simple ideals.
Thus, we have $g' \in {\operatorname{GL}}({\operatorname{Mat}}_{n}^m) = \subseteq {\operatorname{Mat}}_{mn^2} = {\operatorname{Mat}}_{m} \otimes {\operatorname{Mat}}_{n^2}$. Observe that in this decomposition, we identify ${\operatorname{Mat}}_{m}$ with linear transformations on ${{\mathbb C}}^m$ (the first tensor factor) and ${\operatorname{Mat}}_{n^2}$ with linear transformations on ${{\mathbb C}}^n \otimes {{\mathbb C}}^n$ (the second and third tensor factors). Note that $g'$ fixes ${{\mathfrak{sl}}}_m$, and write $g' = \sum_{i=1}^r P_i \otimes Q_i$ where $P_i \in {\operatorname{Mat}}_{m} = {{\mathfrak{gl}}}({{\mathbb C}}^m)$ and $Q_i \in {{\mathfrak{gl}}}({{\mathbb C}}^n \otimes {{\mathbb C}}^n) = {\operatorname{Mat}}_{n^2}$ such that $\{P_i\}$ is a linearly independent subset of ${\operatorname{Mat}}_{m}$ and $\{Q_i\}$ is a linearly independent subset of ${\operatorname{Mat}}_{n^2}$.
The same argument as in the proof of Lemma \[L-normfrob\] proves that $r = 1$ and $P_1 \in {\operatorname{GL}}_m$. Repeating the argument for the other tensor factors, we get that $g' = P_1 \otimes P_2 \otimes P_3$, where $P_1 \in {\operatorname{GL}}_m$, $P_2 \in {\operatorname{GL}}_n$ and $P_3 \in {\operatorname{GL}}_n$. In other words, $g' \in G_{n,m}$.
This proves that $g = g'h_2^{-1} \in G_{n,m} \rtimes \Sigma_{n,m}$. This proves that $N_{{\operatorname{GL}}({\operatorname{Mat}}_{n}^m)}(G_{n,m}) \subseteq G_{n,m} \rtimes \Sigma_{n,m}$. The reverse inclusion is clear.
\[Proof of Theorem \[theo:gos\]\] It is clear that $G_{n,m} \subseteq \mathcal{G}_S \subseteq G_{n,m} \rtimes \Sigma_{n,m}$. Any algebraic group sandwiched between $G_{n,m}$ and $G_{n,m} \times \Sigma_{n,m}$ must be a union of components, i.e., $\mathcal{G}_S = \cup_{h \in I} G_{n,m} \cdot h$ for some subgroup $I \subseteq \Sigma_{n,m}$. But this subgroup is easy to determine. Clearly the transpose morphism $\tau$ is in $I$, so $I = \Sigma_{n,m}$ when $m \neq n$.
Now, consider the case $m = n$. We still claim that $I = \{e,\tau\}$. Observe that $\{e,\tau\}$ is a proper maximal subgroup of $\Sigma_3$, and we have seen that $\{e,\tau\} \subseteq I$. Thus, it suffices to show that $I \subsetneq \Sigma_3$.
To see this, let $\sigma \in S_3$ be the permutation $\sigma(1) = 2$, $\sigma(2) = 1$ and $\sigma(3) = 3$. Let us take $X = ({\rm I}_n,0,\dots,0)$. Then observe that $\tau_\sigma (X) = (E_{11},E_{12},\dots,E_{1n})$. Observe that $X \notin {{\rm SING}}_{n,m}$ whereas $\tau_\sigma(X) \in {{\rm SING}}_{n,m}$. Thus $\tau_\sigma \notin I$. This forces $\{e,\tau\} \subseteq I \subsetneq \Sigma_{n,m}$. Thus $I = \{e,\tau\}$.
Thus irrespective of whether $m$ and $n$ are equal or not, we have $\mathcal{G}_S = G_{n,m} \rtimes {{\mathbb Z}}/2$ as required.
Symmetries of ${{\rm NSING}}_{n,m}$
-----------------------------------
All the work for computing the symmetries of ${{\rm NSING}}_{n,m}$ has already been done, and we just need to put it together.
\[Proof of Theorem \[theo:ngos\]\] Let us denote by $I \subseteq {{\mathbb C}}[{\operatorname{Mat}}_{n}^m]$ the vanishing ideal of ${{\rm SING}}_{n,m}$ and by $J\subseteq {{\mathbb C}}[{\operatorname{Mat}}_{n}^m]$ the vanishing ideal of ${{\rm NSING}}_{n,m}$. Our first claim is that $I_n = J_n$ (see Lemma \[L-N-ideal\]). Thus the lie algebra of symmetries for ${{\rm NSING}}_{n,m}$ is a subalgebra of $\{M \in {{\mathfrak{gl}}}({\operatorname{Mat}}_{n}^m)\ |\ M I_n \subseteq I_n\} = {\mathfrak{g}}_{n,m}$. Thus, the connected group of symmetries for ${{\rm NSING}}_{n,m}$ is a subgroup of $G_{n,m}$. But clearly $G_{n,m}$ preserves ${{\rm NSING}}_{n,m}$. Thus, the connected group of symmetries for ${{\rm NSING}}_{n,m}$ is also $G_{n,m}$. To determine the component group, the same analysis as in the previous subsection works. Thus the group of symmetries for ${{\rm NSING}}_{n,m}$ is exactly the same as the group of symmetries for ${{\rm SING}}_{n,m}$.
Singular tuples of matrices cannot be a null cone {#sec:notnullcone}
=================================================
In this section, we will prove our main theorem, i.e., Theorem \[theo:nullcone\] as well as Theorem \[theo:nullcone2\]. To do so, we need to understand the coordinate subspaces (see Definition \[D-coordsubspace\]) of ${{\rm NSING}}_{n,m}$ and ${{\rm SING}}_{n,m}$. The main point is that both ${{\rm NSING}}_{n,m}$ and ${{\rm SING}}_{n,m}$ have exactly the same coordinate subspaces. First a few definitions.
\[Support of a matrix\] For a matrix $M$, it support ${{\rm Supp}}(M) \subseteq [n] \times [n]$ is defined as the subset of positions with non-zero entries. In other words, $(j,k) \in {{\rm Supp}}(M)$ if and only if the $(j,k)^{th}$ entry of $M$ is non-zero.
\[Support and union support of a tuple of matrices\] For $X = (X_1,\dots,X_n) \in {\operatorname{Mat}}_{n}^m$, we define its support ${{\rm Supp}}(X) \subseteq [m] \times [n] \times [n]$ as the subset of positions with non-zero entries. More precisely ${{\rm Supp}}(X)$ consists of all $(i,j,k)$ such that the $(j,k)^{th}$ coordinate of $X_i$ is non-zero.
We also define its union support ${{\rm USupp}}(X) \subseteq [n] \times [n]$ to be $\cup_i {{\rm Supp}}(X_i)$. In other words, $(j,k) \in {{\rm USupp}}(X)$ if and only if the $(j,k)^{th}$ entry of some $X_i$ is non-zero.
Let us define a map $$\begin{aligned}
\pi_{2,3} : [m] \times [n] \times [n] & \longrightarrow [n] \times [n] \\
(i,j,k) & \longmapsto (j,k)\end{aligned}$$
For $X = (X_1,\dots,X_m) \in {\operatorname{Mat}}_{n}^m$, the union support ${{\rm USupp}}(X)$ can also be seen in the following equivalent ways
1. $\pi_{2,3} ({{\rm Supp}}(X))$;
2. ${{\rm Supp}}(\sum_i t_i X_i)$ for indeterminates $t_1,\dots, t_m$;
3. ${{\rm Supp}}(\sum_i c_i X_i)$ for generic $c_i \in {{\mathbb C}}$.
Recall that on $V = {\operatorname{Mat}}_{n}^m$, we denote by $x^{(i)}_{j,k}$ the $(j,k)^{th}$ coordinate of the $i^{th}$ matrix.
For $I \subseteq [m] \times [n] \times [n]$, we define the linear subspace of ${\operatorname{Mat}}_{n}^m$ $$L_I = \{X \in {\operatorname{Mat}}_{n}^m \ |\ {{\rm Supp}}(X) \subseteq I\}.$$ Equivalently, it can be seen as the zero locus of $\{x^{(i)}_{j,k}\ |\ (i,j,k) \notin I\}$.
Coordinate subspaces of ${{\rm NSING}}_{n,m}$ and ${{\rm SING}}_{n,m}$
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The following result will be derived easily from well known characterizations of ${{\rm NSING}}_{n,m}$. We say a subsset $J \subseteq [n] \times [n]$ contains a permutation $\sigma \in S_n$ if $\{(i,\sigma(i)) \ |\ 1 \leq i \leq n\} \subseteq J$. We say $J \subseteq [n] \times [n]$ is [*permutation free*]{} if it does not contain any permutation.
For $I \subseteq [m] \times [n] \times [n]$, $L_I \subseteq {{\rm NSING}}_{n,m}$ if and only if $\pi_{2,3}(I) \subseteq [n] \times [n]$ is permutation free.
Let us recall that ${{\rm NSING}}_{n,m}$ is the null cone for the action of ${\operatorname{SL}}_n \times {\operatorname{SL}}_n$. Let $T = {\operatorname{ST}}_n \times {\operatorname{ST}}_n$ denote the (standard) maximal torus, i.e., $T$ consists of pairs of diagonal matrices with determinant $1$. Recall that ${{\rm NSING}}_{n,m} = ({\operatorname{SL}}_n \times {\operatorname{SL}}_n) \cdot \mathcal{N}_T({\operatorname{Mat}}_{n}^m)$ by Theorem \[theo:HM\]. Further, from the description of the null cone for tori in Section \[sec:inv.thry.tori\], it can be deduced that $$\mathcal{N}_T({\operatorname{Mat}}_{n}^m) = \bigcup_{\pi_{2,3}(I) \text{ permutation free}} L_I.$$ Another simple way to see this is to understand that the invariant ring is generated by monomials of the form $\prod_{(i,j,k) \in J} x^{(i)}_{j,k}$ where $|J| = n$ and $\pi_{2,3}(J)$ is a permutation. Thus, we conclude that $L_I \subseteq {{\rm NSING}}_{n,m}$ if $\pi_{2,3}(I)$ is permutation free. Alternately, one can see from the description of ${\operatorname{SL}}_n \times {\operatorname{SL}}_n$ invariants (say for example [@DM Theorem 1.4]) that all non-constant homogenous invariants vanish on $L_I$.
Conversely, suppose $\pi_{2,3}(I)$ is not permutation free. So, $\pi_{2,3}(I)$ must contain some permutation, say $\sigma$. Thus for all $1 \leq i \leq n$, there exists $p_i \in [m]$, such that $(p_i,i,\sigma(i)) \in I$. Let $X = (X_1,\dots,X_m) \in {\operatorname{Mat}}_{n}^m$ be such that that $(i,\sigma(i))^{th}$ entry of $X_{p_i}$ is $1$ and all other entries are zero. Clearly $X \in L_I$ and further $\sum_i X_i$ is a permutation matrix (the one associated to $\sigma$), and hence non-singular. But this means that $X \notin {{\rm SING}}_{n,m}$. Hence $L_I \nsubseteq {{\rm SING}}_{n,m}$, and so $L_I \nsubseteq {{\rm NSING}}_{n,m}$ (because ${{\rm NSING}}_{n,m} \subseteq {{\rm SING}}_{n,m}$).
Indeed, observe that the proof of above also gives the following:
For $I \subseteq [m] \times [n] \times [n]$, $L_I \subseteq {{\rm SING}}_{n,m}$ if and only if $\pi_{2,3}(I) \subseteq [n] \times [n]$ is permutation free.
Thus we get the following corollary that is crucial for our purposes.
\[C-coordsubs\] For $I \subseteq [m] \times [n] \times [n]$, $L_I \subseteq {{\rm NSING}}_{n,m}$ if and only if $L_I \subseteq {{\rm SING}}_{n,m}$.
Proof of main result
--------------------
First, let us prove Theorem \[theo:nullcone2\].
Let $G$ be a reductive group acting on $V = {\operatorname{Mat}}_{n}^m$ preserving $S = {{\rm SING}}_{n,m}$ such that $\mathcal{N}_G(V) \subseteq S$. This action is given by a map $\rho: G \rightarrow {\operatorname{GL}}(V)$. The fact that $G$ preserves ${{\rm SING}}_{n,m}$ means that the image $\rho(G)$ is contained in the group of symmetries $\mathcal{G}_S = G_{n,m} \times {{\mathbb Z}}/2$. Now, consider a maximal torus $T$ of $G$. Then $\rho(T)$ is a subtorus of $\rho(G)$ and hence a subtorus of $G_{n,m}$. Thus, $\rho(T)$ is contained in a maximal torus of $G_{n,m}$ and all maximal tori are conjugate under the action of $G_{n,m}$. Thus, for some $g \in G_{n,m}$, we have that $g \rho(T) g^{-1}$ is a subtorus of the standard maximal torus $T_{n,m}$. The standard maximal torus $$T_{n,m} = \{D_1 \otimes D_2 \otimes D_3\ |\ D_1 \in T_m, D_2 \in T_n, D_3 \in T_n\},$$ where $T_k$ denotes the (standard) diagonal torus of ${\operatorname{GL}}_k$. Let $\widetilde{\rho}: G \rightarrow {\operatorname{GL}}(V)$ be defined by $\widetilde{\rho}(h) = g \rho(h) g^{-1}$. This is also an action that satisfies the hypothesis, in particular, $\mathcal{N}_{G,\widetilde{\rho}}(V) = g \cdot \mathcal{N}_{G,\rho}(V)$, and has the added feature that $\widetilde{\rho}(T) \subseteq T_{n,m}$. The point of the above discussion was to establish the fact that the standard basis $\{E_{ijk}\}$ of $V$ is a weight basis for the action defined by $\widetilde{\rho}$ (since it is a weight basis for $T_{n,m}$). Thus, the null cone for the torus $\mathcal{N}_{T,\widetilde{\rho}}(V)$ is a union of certain coordinate subspaces of ${{\rm SING}}_{n,m}$, and hence contained in ${{\rm NSING}}_{n,m}$ by Corollary \[C-coordsubs\]. Thus, the null cone $\mathcal{N}_{G,\widetilde{\rho}}(V) = \widetilde{\rho}(G) \cdot \mathcal{N}_{T,\widetilde{\rho}}(V) \subseteq {{\rm NSING}}_{n,m}$ because $\widetilde{\rho}(G) = g \rho(G) g^{-1} \subseteq G_{n,m} \rtimes {{\mathbb Z}}/2$, which is the group of symmetries of ${{\rm NSING}}_{n,m}$.
Now, we simply observe that $\mathcal{N}_{G,\rho}(V) = g^{-1} \cdot \mathcal{N}_{G,\widetilde{\rho}}(V) \subseteq g^{-1} {{\rm NSING}}_{n,m} = {{\rm NSING}}_{n,m}$, which is the required conclusion.
Before proving Theorem \[theo:nullcone\], let us quickly recollect the fact that ${{\rm NSING}}_{n,m}$ is a proper subset of ${{\rm SING}}_{n,m}$ precisely when $n,m \geq 3$. To begin, we refer the reader to [@GGOW16; @IQS] for many equivalent characterizations of the ${{\rm NSING}}_{n,m}$ (we will not recall them here). First, if $n= 1$ or $m = 1$, it is obvious. For $n = 2$, ${{\rm NSING}}_{2,m} = {{\rm SING}}_{2,m}$ follows from the fact that for $2 \times 2$ linear matrices, their commutative rank and non-commutative rank are the same (this follows from [@FR Remark 1] or [@DM Lemma 2.9]). For $m = 2$, it follows from the fact that polynomials of the form $\det(c_1 X_1 + c_2X_2)$ generates the invariant ring for the action of ${\operatorname{SL}}_n \times {\operatorname{SL}}_n$, see [@Happel1; @Happel2]. Thus ${{\rm NSING}}_{n,2}$ is the zero locus of $\{\det(c_1 X_1 + c_2 X_2) : c_i \in {{\mathbb C}}\}$, which is precisely ${{\rm SING}}_{n,2}$.
On the other hand, for $n = m = 3$, the $3$-tuple $$X = \left( \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 & 0 \\ 0 &0 & 1 \\ 0& -1 & 0 \end{pmatrix} \right) \in {\operatorname{Mat}}_{3}^3$$ is in ${{\rm SING}}_{3,3}$ but not in ${{\rm NSING}}_{3,3}$ (see for example [@FR] or [@DM-ncrk Example 1.1]). For larger $n$ and $m$, this example can be modified in straightforward ways to show that ${{\rm NSING}}_{n,m}$ is a proper subset of ${{\rm SING}}_{n,m}$.
\[Proof of Theorem \[theo:nullcone\]\] Since $n,m \geq 3$, we know that ${{\rm NSING}}_{n,m} \subsetneq {{\rm SING}}_{n,m}$ by the above discussion. Suppose there was a group $G$ acting on $V = {\operatorname{Mat}}_{n}^m$ such that the null cone is ${{\rm SING}}_{n,m}$. This means in particular that $G$ must preserve ${{\rm SING}}_{n,m}$. Thus, we can apply Theorem \[theo:nullcone2\] to deduce that the null cone is contained in ${{\rm NSING}}_{n,m}$, which is a contradiction.
Finally, let us reiterate that if $n$ or $m$ is less than $3$, then ${{\rm SING}}_{n,m} = {{\rm NSING}}_{n,m}$ is the null cone for the left-right action of ${\operatorname{SL}}_n \times {\operatorname{SL}}_n$.
The ring generated by determinantal polynomials is not invariant for [*any*]{} group action {#sec:inv.conv}
===========================================================================================
Let $D(n,m)$ denote the ring ${{\mathbb C}}[\{\det(\sum_i c_i X_i) : c_i \in {{\mathbb C}}\}] \subseteq {{\mathbb C}}[{\operatorname{Mat}}_{n}^m]$. We want to show that this is not the invariant ring for the linear action of [*any*]{} group on $V = {\operatorname{Mat}}_{n}$. First, it suffices to restrict ourselves to subgroups of $V$. Indeed, if there was such a group $G$ with an action, i.e., a map $\rho:G \rightarrow {\operatorname{GL}}(V)$, then the ring of invariants for the action of $G$ is the same as the ring of invariants for the action of $\rho(G)$ which is a subgroup of ${\operatorname{GL}}(V)$.
Let us look at the subgroup $G_D \subseteq {\operatorname{GL}}(V)$ consisting of all linear transformations that leave $\det(\sum_i c_i X_i)$ invariant for all choices of $c_i \in {{\mathbb C}}$, i.e., $$G_D = \{g \in {\operatorname{GL}}(V) = {\operatorname{GL}}_{mn^2} \ |\ g \cdot \det(\sum_i c_i X_i) = \det(\sum_i c_i X_i) \ \forall\ c_i \in C\}.$$
Let us also define $$G_{\rm det} = \{g \in {\operatorname{GL}}({\operatorname{Mat}}_{n}) = {\operatorname{GL}}_{n^2} \ |\ g \cdot \det = \det\}.$$
Recall from Frobenius that $G_{n,1} \rtimes {{\mathbb Z}}/2$ is the group of symmetries of ${{\rm SING}}_{n,1}$. If we define ${\operatorname{SL}}_{n,1} = \{A \otimes B\ |\ A,B \in {\operatorname{SL}}_n\} \subseteq G_{n,1}$, then it follows easily that $$G_{\rm det} = {\operatorname{SL}}_{n,1} \rtimes {{\mathbb Z}}/2.$$
We will prove the following proposition:
The group $G_D = \{ {\rm I}_m \otimes C \ | C \in G_{\rm det} \}$.
Let $g \in G \subseteq {\operatorname{GL}}_{mn^2}$. Then write $$g = \begin{pmatrix}
g_{11} & g_{12} & \dots & g_{1m} \\
\vdots & \ddots & \ddots & \vdots\\
g_{m1} & \dots & \dots & g_{mm}\\
\end{pmatrix},$$ where each block $g_{ij}$ is an $n^2 \times n^2$ matrix (which describes the map from the $j^{th}$ copy of ${\operatorname{Mat}}_{n}$ to the $i^{th}$ copy of ${\operatorname{Mat}}_{n}$). We will denote the action of $g$ by $\ast$ to avoid confusion with matrix multiplication.
Now, $g$ (and $g^{-1}$) fixes $\det(X_1)$, so we have $\det(X_1) = \det(g \ast X_1)$. Observe that $g \ast (X_1,0,\dots,0) = (g_{11} \ast X_1,\dots, g_{m1} \ast X_1)$. In particular, we must have $\det(X_1) = \det(g_{11} \ast X_1)$. Thus $g_{11} \in G_{\rm det}$.
In particular, let $$h = \begin{pmatrix}
g_{11}^{-1} & \dots & 0 \\
\vdots & \ddots & \vdots\\
0 & \dots & g_{11}^{-1}\\
\end{pmatrix},$$
Then since $g$ and $h$ preserve $\det(X_1)$, so does $gh$. Observe that we have $$(gh \ast X)_1 = X_1 + g_{12}g_{11}^{-1} \ast X_2 + \dots + g_{1m}g_{11}^{-1} \ast X_n.$$
Let us write $$L = g_{12}g_{11}^{-1} \ast X_2 + \dots + g_{1m}g_{11}^{-1} \ast X_n.$$
We now have $\det(X_1 + L) = \det(X_1)$, where $L$ is a matrix whose entries are linear functions in $(X_i)_{j,k}$ with $i \geq 2$. Suppose $L \neq 0$, then w.l.o.g, let us assume $L_{1,1} \neq 0$. When we expand out $\det(X_1 + L)$ with the definition as sum over all permutations, the term $L_{1,1}\cdot (X_1)_{2,2} \cdot (X_1)_{3,3}\cdot \dots \cdot (X_1)_{n,n}$ occurs, and cannot be cancelled. This is because in no other permutation can we get the subterm $(X_1)_{2,2} \cdot (X_1)_{3,3} \cdot \dots \cdot (X_1)_{n,n}$. But then, this means that $\det(X_1) \neq \det(X_1 + L)$, which is a contradiction. Hence, $L = 0$. Since $g_{11} \in {\operatorname{GL}}_{n^2}$ is invertible, this means that $g_{12}, \dots, g_{1m} = 0$.
The argument above generalizes in the following way: Consider the identification ${\operatorname{Mat}}_{n}^m = {\operatorname{Mat}}_{n} \otimes K^m$. Let the standard basis for this $K^m$ be $\{w_1,\dots,w_m\}$. Then the above argument simply says that $g$ preserves the “slice” ${\operatorname{Mat}}_{n} \otimes w_1$. Clearly, the same argument will show that $g$ preserves ${\operatorname{Mat}}_{n} \otimes w$ for all $w \in K^m$. Specializing to each $w_i$, we get that $g_{ij} = 0$ whenever $i \neq j$. So, we have
$$g = \begin{pmatrix}
g_{11} & \dots & 0 \\
\vdots & \ddots & \vdots\\
0 & \dots & g_{mm}\\
\end{pmatrix}.$$
Now, suppose $g_{11} \neq g_{22}$. Again, w.l.o.g, we can assume column 1 of $g_{11} \neq$ column 1 of $g_{22}$. Then, $g \ast (E_{1,1} \otimes (w_1 + w_2)) \notin {\operatorname{Mat}}_{n} \otimes (w_1 + w_2)$. Note that when viewing an $n^2 \times n^2$ matrix (say $N$) as linear transformations on ${\operatorname{Mat}}_{n}$, the first column tells us the image of $E_{1,1}$ under $N$. Thus $g_{11} = g_{22}$. By a similar argument $g_{ii} = g_{11}$ for all $i$. To summarize, we have $g = {\rm I}_m \otimes g_{11}$ with $g_{11} \in G_{\rm det}$. Thus $G_D \subseteq \{ {\rm I}_m \otimes C \ | C \in G_{\rm det} \}$. The other inclusion is obvious.
\[Proof of Theorem \[theo:invring\]\] Clearly, $D(n,m) \subseteq {{\mathbb C}}[V]^{G_D}$. Recall the left-right action of ${\operatorname{SL}}_n \times {\operatorname{SL}}_n$ on $V$ given by $(A,B) \cdot (X_1,\dots,X_m) = (AX_1B^t,\dots, AX_mB^t)$. This action is given by a map $\rho: {\operatorname{SL}}_n \times {\operatorname{SL}}_n \rightarrow {\operatorname{GL}}({\operatorname{Mat}}_{n})$. The image of a reductive group under a morphism of algebraic groups is reductive, so $\rho({\operatorname{SL}}_n \times {\operatorname{SL}}_n)$ is a connected reductive subgroup of ${\operatorname{GL}}(V)$. We observe that in fact this is precisely the identity component of $G_D$. Thus $G_D$ is a reductive group. Hence, the null cone for $G_D$ is the same as the null cone for its identity component $\rho({\operatorname{SL}}_n \times {\operatorname{SL}}_n)$, and hence equal to the null cone for ${\operatorname{SL}}_n \times {\operatorname{SL}}_n$, which we know is ${{\rm NSING}}_{n,m}$. Thus to summarize, the zero locus of all non-constant homogenous elements of ${{\mathbb C}}[V]^{G_D}$ is ${{\rm NSING}}_{n,m}$. On the other hand the zero locus of all non-constant homogenous elements of $D(n,m)$ is precisely ${{\rm SING}}_{n,m}$. Thus, this means that we have a proper inclusion $D(n,m) \subsetneq {{\mathbb C}}[V]^{G_D}$.
Now, suppose there was [*any*]{} group $G$ such that ${{\mathbb C}}[V]^G = D(n,m)$. Suppose the action is given by $\rho:G \rightarrow {\operatorname{GL}}(V)$. Then $\rho(G) \subseteq G_D$ by the previous proposition, so ${{\mathbb C}}[V]^G = {{\mathbb C}}[V]^{\rho(G)} \supseteq {{\mathbb C}}[V]^{G_D}$. But this is a contradiction because ${{\mathbb C}}[V]^G = D(n,m) \subsetneq {{\mathbb C}}[V]^{G_D} \subseteq {{\mathbb C}}[V]^G$. Thus, there is no such group.
Discussion and open questions {#sec:disc}
=============================
This paper demonstrates another collaboration of different fields in mathematics. Expanding on ongoing work cited in the introduction, here too fundamental problems in computational complexity have given rise to a new flavor of problems that are purely algebraic in nature, some of which arise from analyzing analytic (rather than symbolic) algorithms. We feel that it is important to introduce these problems to representation theorists, algebraic geometers and commutative algebraists. The results of this paper open the door for several further avenues of research, inviting a further collaboration between theoretical computer scientists and mathematicians to resolve them.
Let us begin with the stating that ${{\rm SING}}_{n,m}$ is a very important variety to study due to its connection to circuit lower bounds ([@KI]) that we mentioned earlier. Insights from any field of mathematics may be helpful! The major open problem is of course:
Is there a deterministic polynomial time algorithm for SDIT?
Various subclasses of SDIT (and PIT) have polynomial time algorithms. For example, we say an $m$-tuple of $n \times n$ matrices $X = (X_1,\dots,X_m)$ satisfies the property $(R1)$ if the linear subspace in ${\operatorname{Mat}}_{n}^m$ spanned by $X_1,\dots,X_m$ has a basis consisting of rank $1$ matrices. It turns out that if $X$ satisfies $(R1)$, then $X \in {{\rm SING}}_{n,m}$ if and only if $X \in {{\rm NSING}}_{n,m}$. Thus, SDIT restricted to tuples with the $(R1)$ property can be solved via a null cone membership algorithm! (this is implicit in [@Gur04]). One direction of future research is to consider the following natural generalization of the $(R1)$ property.
For fixed $k\in {{\mathbb Z}}_{\geq 1}$ We say $X = (X_1,\dots,X_m)$ satisfies the property $(Rk)$ if the linear subspace in ${\operatorname{Mat}}_{n}^m$ spanned by $X_1,\dots,X_m$ has a basis consisting of rank $\leq k$ matrices.
Is there a deterministic polynomial time algorithm for SDIT for tuples satisfying $(Rk)$? How about $(R2)$?
Next, we turn to the symmetry group of an algebraic subvariety.
What algorithms can one use to determine the group of symmetries of a subvariety? How efficient are these algorithms?
In this paper, we explicitly determined the group of symmetries of one family of variety. It is however very clear that most steps are algorithmic. Roughly speaking, if the generators for the ideal of polynomials vanishing on the subvariety are given as an input, then determining the Lie algebra of symmetries reduces to solving a system of linear equations. So, in terms of the input size of such generating polynomials given by their coefficients, this Lie algebra part is efficient. It is not clear to us how to obtain the group itself efficiently from the Lie algebra. Moreover, if we are given the generating polynomials that describes the subvariety (set-theoretically) in an implicit, concise way (as in SING) it seems that more work is needed even to define the computational task. It is possible that when the generators themselves have some symmetries, or rich relations (as in SING), one can do more.
Another general problem to be pursued is to get a better understanding of null cones (and orbit closure equivalence classes)
Can one classify null cones? What features must a subvariety satisfy in order to possibly be a null cone?
In this paper, we used mainly the fact that the null cone must be the translation (by a group element) of a union of coordinate subspaces (i.e., the Hilbert–Mumford criterion). It will be interesting to find other properties of null cones which distinguish them from arbitrary subvarieties.
A different direction to pursue is the following. The main result of this paper is that ${{\rm SING}}_{n,m}$ is not a null cone for any [*reductive*]{} group action. Natural as this condition is mathematically (and we use it and consequences of it here), it is not important algorithmically, and one can potentially implement and analyze null cone membership algorithms using non-reductive groups.[^18] So, what if we drop the reductivity assumption?
Can ${{\rm SING}}_{n,m}$ be the null cone for the action of a non-reductive group?
Now, we mention a few more problems which are a little bit more technical, and of interest to commutative algebraists and algebraic geometers.
Let $I$ be the ideal of polynomials vanishing on ${{\rm SING}}_{n,m}$. Determine the ideal generators of $I$. Do the determinantal polynomials $\det(\sum_i c_i X_i)$ generate the ideal?
Consider the ring ${{\mathbb C}}[\{\det(\sum_i c_i X_i) : c_i \in {{\mathbb C}}\}] \subseteq {{\mathbb C}}[{\operatorname{Mat}}_{n,n}^m]$. Is it Cohen–Macaulay? What is its regularity, etc?
Missing proofs for Section \[sec:gos\] {#App.gos}
======================================
In this appendix, we will give the complete details of the theoretical ideas that go into Lemma \[Lgos-graded\] and Proposition \[P-Liealg-all\]. First, we note a lemma that will find repeated use.
\[repeat\] Let $W$ be a (finite-dimensional) linear subspace and $U$ be a linear subspace of $W$. Then $$\{g \in {\operatorname{GL}}(W)\ |\ gU \subseteq U\} = \{g \in {\operatorname{GL}}(W)\ |\ gU = U\}.$$
The proof is straightforward and left to the reader.
Let $S \subseteq V$ be a subvariety, and let $I_S$ denote the ideal of all polynomials that vanish on $S$. Recall that the action of ${\operatorname{GL}}(V)$ on $V$ gives an induced action on the polynomial ring ${{\mathbb C}}[V]$. Further, this action preserves the degree of the polynomials, so ${{\mathbb C}}[V]_{\leq d}$ (subspace of polynomials of degree $\leq d$) and ${{\mathbb C}}[V]_a$ (subspace of homogeneous polynomials of degree $a$) are subrepresentations (for any non-negative integers $d$ and $a$). Recall that $\mathcal{G}_S = \{g \in {\operatorname{GL}}(V)\ |\ g S = S\}$ is the group of symmetries.
\[L-ode\] The group of symmetries $$\mathcal{G}_S = \{g \in {\operatorname{GL}}(V)\ | g \cdot I_S \subseteq I_S\} = \{g \in {\operatorname{GL}}(V)\ |\ g \cdot I_S = I_S\}.$$
Let us first prove the second equality. If we denote by $(I_S)_{\leq d}$ the subspace of $I_S$ consisting of polynomials of degree $\leq d$. Indeed, applying the above lemma to $U = (I_S)_{\leq d}$ and $W = {{\mathbb C}}[V]_{\leq d}$, it follows that $$\{g \in {\operatorname{GL}}(V)\ | g \cdot (I_S)_{\leq d} \subseteq (I_S)_{\leq d}\} = \{g \in {\operatorname{GL}}(V)\ |\ g \cdot(I_S)_{\leq d} = (I_S)_{\leq d}$$
Since $I_S = \cup_{d} (I_S)_{\leq d}$, the second equality follows. Note that we did not directly apply to $I_S \subseteq {{\mathbb C}}[V]$ because they are not finite-dimensional.
Now, let us prove the first equality. For $g \in \mathcal{G}_S$ and $f \in I_S$, we observe that for $s \in S$, $(g \cdot f) (s) = f(g^{-1} \cdot s) = 0$. Thus $g \cdot f \in I_S$. This shows that $g \cdot I_S \subseteq I_S$. This shows $\subseteq$.
For the reverse inclusion. Suppose $g \in {\operatorname{GL}}(V)$ is such that $g \cdot I_S \subseteq I_S$. Then, by the second equality, we know that $g \cdot I_S = I_S$ and hence $g^{-1} \cdot I_S = I_S$. Now, suppose $s \in S$. We want to show that $g \cdot s \in S$. For any $f \in I_S$, we have $f(g \cdot s) = (g^{-1} \cdot f) (s) = 0$ since $g^{-1} \cdot f \in I_S$. This means that $gS \subseteq S$. Moreover, suppose $v \notin S$. Then for some $f \in I_S$, we have $f(v) \neq 0$. Thus $(g\cdot f) (gv) = f(v) \neq 0$. Since $g \cdot f \in I_S$, we get that $gv \notin S$. Thus $gS^c \subseteq S^c$, where $S^c$ denotes the complement of $S$ in $V$.
Since $gS \subseteq S$ and $g S^c \subseteq S^c$, we have $gS = S$ (because $g$ is invertible). Thus $g \in \mathcal{G}_S$, and this concludes the proof.
The same proof gives the following statement.
\[asdfs\] Suppose the zero locus of $(I_S)_{\leq d}$ is precisely $S$. Then the group of symmetries $$\mathcal{G}_S = \{g \in {\operatorname{GL}}(V)\ | g \cdot (I_S)_{\leq d} \subseteq (I_S)_{\leq d}\}$$
Run the same argument as above, but even easier because the infinite dimensional issue doesn’t arise.
\[Proof of Lemma \[Lgos-graded\]\] Again run the same argument as above. The hypothesis of $S$ being a cone can be ignored. However, unless $S$ is a cone, the zero locus of $(I_S)_a$ cannot possibly equal $S$.
Let us note that to invoke Lemma \[asdfs\] (or Lemma \[Lgos-graded\]) in any explicit situation, one has to find an appropriate $d$ (or $a$), which is not always an easy task.
Next, we will carry over these results to the setting of Lie algebras, culminating in a proof of Proposition \[P-Liealg-all\]. First, a focused introduction to Lie theory.
### Lightning introduction to Lie theory
Let us start with an example.
Let $V$ be a complex vector space with basis $e_1,\dots,e_n$, and let the corresponding coordinate functions be denoted $x_1,\dots,x_n$. The group ${\operatorname{GL}}(V)$ consists of all invertible linear transformations from $V$ to itself, and can be identified with invertible $n \times n$ matrices using the chosen basis. Its Lie algebra ${{\mathfrak{gl}}}(V)$ consists of all linear transformations from $V$ to itself, and so can be identified with ${\operatorname{Mat}}_{n}$. There is an [*exponential map*]{} ${\rm exp}: {{\mathfrak{gl}}}(V) = {\operatorname{Mat}}_{n} \rightarrow {\operatorname{GL}}(V) = {\operatorname{GL}}_n$ that sends $M \mapsto {\rm exp}(M) = I + M + \frac{M^2}{2!} + \dots + \frac{M^n}{n!} + \dots$.
With this example in mind, let us give some definitions. A Lie group $G$ is a smooth manifold (over the real numbers ${{\mathbb R}}$) which is also a group such that the multiplication map and inverse map are smooth. To a Lie group $G$, one associates a Lie algebra denoted ${{\rm Lie}}(G)$ or ${\mathfrak{g}}$ (in general, we may use the corresponding fraktur letter to make notation less cumbersome). The Lie algebra ${{\rm Lie}}(G)$ is the space of all [*left-invariant vector fields*]{}, equipped with a bilinear operation called the Lie bracket.
A vector field on $G$ is the assignment of a tangent vector to each point of $G$. Left multiplication by group elements allows us to identify the tangent space at any point with the tangent space at the identity element $e \in G$. A vector field is called left-invariant if the assigned tangent vectors at all the points are the same (with the identification mentioned above). Thus one can identify the space of left-invariant vector fields with the tangent space at identity. A curve on $G$ is called an integral curve for a vector field if the tangent vector of the curve at every point agrees with the vector field. For a left-invariant vector field $M$, the curve ${\rm exp}(tM)$ is the unique integral curve for $M$ that passes through $e \in G$ at $t = 0$. In particular, $\frac{d}{dt} {\rm exp}(tM)|_{t=0} = M$ and this will be useful to us.
In the case of ${\operatorname{GL}}(V)$, let us reconcile the abstract definitions with the concrete ones in the above definition. Note that ${\operatorname{GL}}(V) = {\operatorname{GL}}_n$ is an open subset of ${\operatorname{Mat}}_{n}$. Hence, the tangent space at the identity can be identified with ${\operatorname{Mat}}_{n}$. Thus ${{\mathfrak{gl}}}(V)$, the Lie algebra of ${\operatorname{GL}}(V)$ can be identified with ${\operatorname{Mat}}_{n}$. The abstract exponential map coincides with the concrete description given in the example above.
For any Lie subgroup $H$ of any Lie group $G$, its Lie algebra is $${{\rm Lie}}(H) = \{M \in {{\rm Lie}}(G)\ |\ {\rm exp}(tM) \in H\ \forall t\}.$$
Many Lie groups occur as subgroups of ${\operatorname{GL}}(V)$, and these are often called matrix groups, and one can work extremely concretely in the setting of matrix groups. However, not every Lie group is a matrix group. For our purposes, the abstract point of view is elegant and helps us in the theoretical results, and the concrete description is more conducive for computations which is our main goal.
For any smooth action of an Lie group $G$ on a (finite-dimensional) vector space $W$ by linear transformations, we get a smooth morphism of Lie groups $\rho: G \rightarrow {\operatorname{GL}}(W)$. On differentiating, we get a morphism of Lie algebras $d\rho: {{\rm Lie}}(G) \rightarrow {{\mathfrak{gl}}}(W)$. In other words, we get an induced action of ${{\rm Lie}}(G)$ on $W$. Note that algebraic groups are Lie groups and algebraic actions of algebraic groups are smooth.
The exponential map commutes with this, i.e., for $M \in {{\rm Lie}}(G) \subseteq {{\mathfrak{gl}}}(V)$, we have ${\rm exp}(d\rho \cdot M) = \rho ({\rm exp} (M))$. We will simply write ${\rm exp}(M)$ for $\rho({\rm exp}(M))$ whenever there is no possibility of confusion.
The action of ${\operatorname{GL}}(V)$ on $V$ (by left multiplication) gives an action of ${\operatorname{GL}}(V)$ on ${{\mathbb C}}[V]$, ${{\mathbb C}}[V]_{\leq d}$ (polynomials of degree $\leq d$) and ${{\mathbb C}}[V]_a$ (homogeneous polynomials of degree $a$) by the formula $(g \cdot f) (v) = f(g^{-1} v)$. When we take $W = {{\mathbb C}}[V]_{\leq d}$ or ${{\mathbb C}}[V]_a$, the above discussion gives an action of the Lie algebra ${{\mathfrak{gl}}}(V)$. Thus, we have an action of ${{\mathfrak{gl}}}(V)$ on ${{\mathbb C}}[V]_a$ and hence on ${{\mathbb C}}[V] = \oplus_{a \in {{\mathbb N}}} {{\mathbb C}}[V]_a$. The Lie algebra ${{\mathfrak{gl}}}(V)$ acts on ${{\mathbb C}}[V]$ by derivations, and this we described explicitly in Section \[sec:explicit\]
### Computing the Lie algebra of symmetries
First a lemma. Suppose we have a Lie group $G$ acting on a vector space $W$ by linear transformations. Let $U$ be a linear subspace of $W$. Then let $\mathcal{G}_U := \{g \in G\ |\ g U = U\} = \{g \in G\ |\ gU \subseteq U\}$. The latter equality follows from Lemma \[repeat\]. We have:
The Lie algebra ${{\rm Lie}}(\mathcal{G}_U) = \{M \in {{\rm Lie}}(G)\ |\ M \cdot U \subseteq U\}$.
Suppose $M \in {{\rm Lie}}(\mathcal{G}_U)$. Let $u \in U$. Then ${\rm exp}(tM) \cdot u \in U$ for all $t$ because ${\rm exp}(tM) \in \mathcal{G}_U$. Thus ${\rm exp}(tM) \cdot u$ is a smooth curve in $U$. For the vector space $W$, the tangent space at any point is $W$. For any smooth curve completely contained in $U$, it is clear that the tangent vectors at any point of the curve is also in $U$. Thus, we have $M \cdot u = \frac{d}{dt} ({\rm exp}(tM) \cdot u)|_{t = 0} \in U$. So, we conclude that $M \cdot U \subseteq U$.
Conversely, suppose $M \cdot U \subseteq U$. Then for $u \in U$, we have ${\rm exp}(tM) \cdot u = \lim_{n \rightarrow \infty} (\sum_{i=0}^n \frac{M^i}{i!}) \cdot u$. Since each $(\sum_{i=0}^n \frac{M^i}{i!}) \cdot u \in U$, the limit is also in $U$. Thus ${\rm exp}(tM) \cdot U \subseteq U$, which means that ${\rm exp}(tM) \in \mathcal{G}_U$. Hence, $M \in {{\rm Lie}}(\mathcal{G}_U)$.
Applying the above lemma, we can formulate the Lie algebra versions of the results at the beginning of this appendix (i.e., Lemma \[L-ode\], Lemma \[asdfs\] and Lemma \[Lgos-graded\]).
\[P-Liealg-all2\] Let $S \subseteq V$ be an algebraic subset, and let $\mathcal{G}_S$ denote its group of symmetries, and ${\mathfrak{g}}_S$ its Lie algebra of symmetries. Let $I_S$ denote the ideal of polynomial functions in ${{\mathbb C}}[V]$ that vanish on $S$. Then we have $${\mathfrak{g}}_S = \{M \in {{\mathfrak{gl}}}(V)\ |\ M \cdot I_S \subseteq I_S\}.$$ Further, if $I_S$ is generated in degree $\leq d$, then we have $${\mathfrak{g}}_S = \{M \in {{\mathfrak{gl}}}(V)\ |\ M \cdot (I_S)_{\leq d} \subseteq (I_S)_{\leq d}\}.$$ Moreover, if $S$ is a cone, then $I_S$ is graded, and for any $a \in {{\mathbb N}}$, we have $${\mathfrak{g}}_S \subseteq \{M \in {{\mathfrak{gl}}}(V)\ |\ M \cdot (I_S)_a \subseteq (I_S)_a\}.$$ Finally, if the zero locus of $(I_S)_a$ is precisely the cone $S$, then we have equality.
The last part of the above proposition is precisely Proposition \[P-Liealg-all\].
Missing proofs for Section \[sec:van\] {#App.rep}
======================================
We will need the representation theory of ${\operatorname{GL}}_m \times {\operatorname{GL}}_n \times {\operatorname{GL}}_n$. In particular, an understanding of weights and highest weight vectors will be needed. We will recall the necessary background.
For this section, let $I \subseteq {{\mathbb C}}[{\operatorname{Mat}}_{n}]$ denote the ideal of polynomial functions that vanish on ${{\rm SING}}_{n,m}$. The first observation is that since ${{\rm SING}}_{n,m}$ is stable under the action of ${\operatorname{GL}}_m \times {\operatorname{GL}}_n \times {\operatorname{GL}}_n$, so is $I$. Let us now make this more precise.
We have an action of ${\operatorname{GL}}_m \times {\operatorname{GL}}_n \times {\operatorname{GL}}_n$ on ${\operatorname{Mat}}_{n}^m$ given by $$(A,B,C) \cdot (X_1,\dots,X_m) = (\sum_{j=1}^m a_{1j}BX_j C^{t}, \sum_j a_{2j} BX_j C^{t}, \dots, \sum_j a_{mj} BX_jC^t),$$ where $a_{ij}$ denotes the $(i,j)^{th}$ entry of $A$. While this is the most natural action, we will use a slight variant of this action which will make easier some later arguments. Consider the Cartan involution[^19] $\theta: {\operatorname{GL}}_k \rightarrow {\operatorname{GL}}_k$ given by $\theta(A) = (A^{-1})^t$. We will twist the above action with the Cartan involution of each of the ${\operatorname{GL}}$’s. In the below formula, we will write $A' = \theta(A)$, $B' = \theta(B)$ and $C' = \theta(C)$. Moreover, we will write $a'_{ij}$ to denote the $(i,j)^{th}$ entry of $A'$. The action of ${\operatorname{GL}}_m \times {\operatorname{GL}}_n \times {\operatorname{GL}}_n$ on ${\operatorname{Mat}}_{n}^m$ we will use is given by $$\label{dual-action}
(A,B,C) \cdot (X_1,\dots,X_m) = (\sum_{j=1}^m a'_{1j}B'X_j (C^{t})', \sum_j a'_{2j} B'X_j (C^{t})', \dots, \sum_j a'_{mj} B'X_j (C^t)'),$$
Let us now justify briefly why we use the second action instead of the first one. For $k \in {{\mathbb N}}$, there is a natural action of ${\operatorname{GL}}_k$ on ${{\mathbb C}}^k$ (viewed as column vectors) by left multiplication. This gives the contragredient action of ${\operatorname{GL}}_k$ on $({{\mathbb C}}^k)^*$ (for $g \in {\operatorname{GL}}_k$ and $\zeta \in ({{\mathbb C}}^k)^*$, the element $g \cdot \zeta \in ({{\mathbb C}}^k)^*$ is defined by $g \cdot \zeta (v) = \zeta(g^{-1} \cdot v)$ for $v \in {{\mathbb C}}^k$). This is the canonical action of ${\operatorname{GL}}_k$ on $({{\mathbb C}}^k)^*$. Thus, we have an action of ${\operatorname{GL}}_m \times {\operatorname{GL}}_n \times {\operatorname{GL}}_n$ on $({{\mathbb C}}^m)^* \otimes ({{\mathbb C}}^n)^* \otimes ({{\mathbb C}}^n)^*$ where each ${\operatorname{GL}}$ acts on the corresponding tensor factor.
Let $e_1,\dots,e_k$ denote the standard basis for ${{\mathbb C}}^k$ and let $e_1^*,\dots,e_k^*$ denote the corresponding dual basis of $({{\mathbb C}}^k)^*$. We identify ${\operatorname{Mat}}_{n}^m$ with $({{\mathbb C}}^m)^* \otimes ({{\mathbb C}}^n)^* \otimes ({{\mathbb C}}^n)^*$ as follows. $X = (X_1,\dots,X_m) \leftrightarrow \sum_{i,j,k} (X_i)_{j,k} e_i^* \otimes e_j^* \otimes e_k^*$. With this identification, the action of ${\operatorname{GL}}_m \times {\operatorname{GL}}_n \times {\operatorname{GL}}_n$ on $({{\mathbb C}}^m)^* \otimes ({{\mathbb C}}^n)^* \otimes ({{\mathbb C}}^n)^*$ agrees with the one in Equation \[dual-action\] above.
The advantage of this action is that the induced action on the coordinate ring ${{\mathbb C}}[{\operatorname{Mat}}_{n}^m]$ is a “polynomial” representation. This is notationally advantageous for many reasons – in particular that polynomial representations of ${\operatorname{GL}}_m \times {\operatorname{GL}}_n \times {\operatorname{GL}}_n$ are indexed by triples of partitions (more details later). The action on polynomial functions is as follows. For $g \in {\operatorname{GL}}_m \times {\operatorname{GL}}_n \times {\operatorname{GL}}_n$, and $f \in {{\mathbb C}}[{\operatorname{Mat}}_{n}^m]$, we have $g \cdot f$ is the polynomial defined by the formula $g \cdot f (X) = f(g^{-1} \cdot X)$ for $X \in {\operatorname{Mat}}_{n}^m$.
If $X = (X_1,\dots,X_m) \in {{\rm SING}}_{n,m}$, and $g \in {\operatorname{GL}}_m \times {\operatorname{GL}}_n \times {\operatorname{GL}}_n$, then $g \cdot X \in {{\rm SING}}_{n,m}$.
Let $g = (A,B,C)$ as above. We leave it to the reader to check that ${{\rm span}}(g \cdot X) = B' ({{\rm span}}(X)) (C^t)'$. Since $B$ and $C$ are invertible, ${{\rm span}}(X)$ contains a non-singular matrix if and only if ${{\rm span}}(g \cdot X)$ contains a non-singular matrix.
The ideal $I \subseteq {{\mathbb C}}[{\operatorname{Mat}}_{n}^m]$ is ${\operatorname{GL}}_m \times {\operatorname{GL}}_n \times {\operatorname{GL}}_n$ stable.
Suppose $f \in I$, and $g \in {\operatorname{GL}}_m \times {\operatorname{GL}}_n \times {\operatorname{GL}}_n$. Then for any $X \in {{\rm SING}}_{n,m}$, we have $(g \cdot f) (X) = f(g^{-1} \cdot X) = 0$ since $g^{-1} \cdot X \in {{\rm SING}}_{n,m}$ by the above lemma. Thus, $g \cdot f$ vanishes on ${{\rm SING}}_{n,m}$ and hence $g \cdot f \in I$ as required.
We want to prove Proposition \[P-ideal\]. The first part of the proposition follows since $S = {{\rm SING}}_{n,m}$ is a cone. For, the second part, it is simple to see that $I_d = 0$. Let $x^{(k)}_{i,j}$ denote the coordinate function corresponding to the $(i,j)^{th}$ entry of the $k^{th}$ matrix, so ${{\mathbb C}}[{\operatorname{Mat}}_{n}^m] = {{\mathbb C}}[x^{(k)}_{i,j}\ |\ 1 \leq i,j \leq n, 1 \leq k \leq m]$.
\[Proof of Proposition \[P-ideal\], part (2)\] Take $f \in {{\mathbb C}}[{\operatorname{Mat}}_{n}^m]_d$. Write $f$ as a sum of monomials. Suppose a monomial $m = \prod (x^{(k)}_{ij})^{a^{(k)}_{ij}}$ occurs in $f$ with non-zero coefficient (In particular, $\sum a^{(k)}_{ij} = d$). Consider the support of $m$, i.e., ${\rm Supp}(m) = \{(k,i,j)\ |\ a^{(k)}_{ij} >0\}$. The cardinality of ${\rm Supp}(m)$ is at most $d < n$. Let $X = (X_1,\dots,X_m)$ be such that $(X_k)_{ij} = 1$ if $(k,i,j) \in {\rm Supp}(m)$ and $0$ otherwise. Then the number of non-zero entries in any linear combination $\sum_i c_i X_i$ is at most $d$. Any matrix with at most $d$ non-zero entries is singular, so this means that $X \in {{\rm SING}}_{n,m}$[^20]. Moreover observe that by construction $f(X) = $ coefficient of $m$ in $f$ which is non-zero, so $f \notin I_d$.
The action of ${\operatorname{GL}}_m \times {\operatorname{GL}}_n \times {\operatorname{GL}}_n$ on polynomials preserve degree, so the homogeneous polynomials of degree $d$, i.e., ${{\mathbb C}}[{\operatorname{Mat}}_{n}^m]_d$ is a subrepresentation. From the above corollary, we get that $I_n \subseteq {{\mathbb C}}[{\operatorname{Mat}}_{n}^m]_n$ is a subrepresentation. The group ${\operatorname{GL}}_m \times {\operatorname{GL}}_n \times {\operatorname{GL}}_n$ is reductive, so its representations can be decomposed as a direct sum of irreducible representations. Thus to understand $I_n$, we will have to understand the irreducibles that make up $I_n$ and their multiplicities.We will need some representation theoretic results, and we will be very brief, picking up only those results that are necessary.
Irreducible polynomial representations of ${\operatorname{GL}}_m \times {\operatorname{GL}}_n \times {\operatorname{GL}}_n$ are indexed by triples of partitions $(\lambda,\mu,\nu)$ (where $\lambda$ has at most $m$ parts and $\mu,\nu$ have at most $n$ parts). A partition $\lambda$ is a (finite) decreasing sequence of positive numbers $(\lambda_1,\lambda_2,\dots,\lambda_k)$. We write $\lambda \vdash d$ if $\sum_i \lambda_i = d$. We will denote the irreducible representation corresponding to $(\lambda,\mu,\nu)$ by $S_{\lambda,\mu,\nu}$. An explicit description is given by Schur functors. For any partition $\pi$, denote by $S_{\pi}$ the Schur functor corresponding to $\pi$ as defined in [@Fulton]. $S_{\pi}$ is a functor from the category of vector spaces to itself. We refer to [@Fulton; @Weyman] for an extensive introduction. For us, it suffices to remark that $$S_{\lambda,\mu,\nu} = S_{\lambda}({{\mathbb C}}^m) \otimes S_{\mu}({{\mathbb C}}^n) \otimes S_{\nu}({{\mathbb C}}^n).$$
For a reductive group $G$, let $\{V_i:i \in I\}$ denote the irreducible representations. Then for any representation $V$, it can be decomposed as a direct sum of irreducibles. Such a decomposition is not always unique. Let $E_i$ denote the isotypic component w.r.t $V_i$, i.e., the sum of all subrepresentations of $V$ that are isomorphic to $V_i$. Then the isotypic decomposition $V = \oplus_{i \in I} E_i$ is unique. Further, each $E_i = V_i^{\oplus m_i}$, and $m_i$ is called the multiplicity of $V_i$ in $V$.
Consider the decomposition of ${{\mathbb C}}[{\operatorname{Mat}}_{n}^m]_d$ into isotypic components $${{\mathbb C}}[{\operatorname{Mat}}_{n}^m]_d = \bigoplus_{\lambda,\mu,\nu \vdash d} E_{\lambda,\mu,\nu}$$ where $E_{\lambda,\mu,\nu}$ is the isotypic component corresponding to $S_{\lambda,\mu,\nu}$. We have $E_{\lambda,\mu,\nu} = S_{\lambda,\mu,\nu}^{a_{\lambda,\mu,\nu}}$ where $a_{\lambda,\mu,\nu} \in {{\mathbb N}}$ are the celebrated Kronecker coefficients. This is actually one of many equivalent ways to define Kronecker coefficients. We now focus on degree $n$ polynomials.
\[L-kron1\] We have $$a_{\lambda, 1^n,1^n} = \begin{cases} 1 & \text{ if $\lambda = (n)$}; \\
0 & \text{ otherwise.}
\end{cases}$$
To see this, we have to understand Kronecker coefficients from the symmetric groups perspective. For a partition $\lambda \vdash n$, denote by $T_{\lambda}$ the corresponding representation of $S_n$. Then the decomposition of the tensor product $T_{\lambda} \otimes T_{\mu}$ into irreducibles is described by Kronecker coefficients, i.e.,
$$T_{\lambda} \otimes T_{\mu} = \bigoplus_{\nu} T_{\nu}^{a_{\lambda,\mu,\nu}}.$$
Note that for $\lambda = 1^n$, $T_{\lambda}$ corresponds to a $1$-dimensional representation which is called the sign representation. With this explicit description, one can deduce that $T_{\lambda} \otimes T_{1^n} = T_{\lambda^{\dag}}$ where $\lambda^{\dag}$ denotes the conjugate partition of $\lambda$. Thus, $a_{\lambda,1^n,1^n} = 0$ unless $\lambda = (1^n)^\dag = (n)$ and in the latter case, we have $a_{(n),1^n,1^n} = 1$.
For partitions $\lambda,\mu,\nu \vdash n$, the isotypic component $E_{\lambda,\mu,\nu} \cap I_n = \phi$ if $\mu \neq (1,1,\dots,1)$.
First, let $T_k \subseteq {\operatorname{GL}}_k$ denote the standard torus, i.e., the all invertible diagonal matrices. Then $T_k$ is a maximal torus for ${\operatorname{GL}}_k$. For our purposes, $T = T_m \times T_n \times T_n$ is a maximal torus of ${\operatorname{GL}}_m \times {\operatorname{GL}}_n \times {\operatorname{GL}}_n$.
A weight vector for the action of ${\operatorname{GL}}_m \times {\operatorname{GL}}_n \times {\operatorname{GL}}_n$ is simply a weight vector for $T$, which is a torus. We have already discussed weight vectors for actions of tori. Further, the characters of $T = T_m \times T_n \times T_n$ can be identified with ${{\mathbb Z}}^m \times {{\mathbb Z}}^n \times {{\mathbb Z}}^n$, and so a triple of partitions $(\lambda,\mu,\nu)$ (where $\lambda$ has at most $m$ parts and $\mu,\nu$ have at most $n$ parts) can be identified with a character.
If $E_{\lambda,\mu,\nu} \cap I_n \neq \phi$, then $I_n$ has a subrepresentation isomorphic to $S_{\lambda,\mu,\nu}$, then it has a highest weight vector of weight $(\lambda,\mu,\nu)$ (since $S_{\lambda,\mu,\nu}$ is generated by such a highest weight vector – a basic fact). Thus, to show that $E_{\lambda,\mu,\nu} \cap I_n = \phi$, it suffices to show that [*all*]{} weight vectors of weight $(\lambda,\mu,\nu)$ are not in $I_n$.
Let us consider all weight vectors of weight $(\lambda,\mu,\nu)$ in ${{\mathbb C}}[{\operatorname{Mat}}_{n}^m]_n$. Let $x^{(k)}_{ij}$ denote the coordinate function corresponding to the $(i,j)^{th}$ entry of the $k^{th}$ matrix $X_k$. Then its weight is $(\delta_k,\delta_i,\delta_j)$, where $\delta_a = (0,\dots,0,\underbrace{1}_a,0,\dots,0)$. Thus for a monomial $\prod (x^{(k)}_{ij})^{n^{(k)}_{ij}}$, its weight is $\sum n^{(k)}_{ij} (\delta_k,\delta_i,\delta_j)$.
The collection of all weight vectors of weight $(\lambda,\mu,\nu)$ in ${{\mathbb C}}[{\operatorname{Mat}}_{n}^m]$ is a linear subspace spanned by monomials of weight $(\lambda,\mu,\nu)$. Observe that since $\mu \vdash n$ and $\mu \neq (1,1,\dots,1)$, we have that $\mu_n = 0$. Thus, any monomial of degree $n$ whose weight is $(\lambda,\mu,\nu)$ does not depend on the last rows of the matrices.
Thus, any weight vector $f$ of weight $(\lambda,\mu,\nu)$ is a linear combination of monomials all of which do not involve the last rows of the matrices. Let $U \subseteq {\operatorname{Mat}}_{n}^m$ be the subspace of tuples of matrices whose last row is zero. Then the weight vector $f$ is a nonzero polynomial on $U$, and $U \subseteq {{\rm SING}}_{n,m}$ (in fact $U \subseteq {{\rm NSING}}_{n,m}$). Thus, the weight vectors of weight $(\lambda,\mu,\nu)$ are not in $I_n$.
The multiplicity of $S_{(n),1^n,1^n}$ in ${{\mathbb C}}[{\operatorname{Mat}}_{n}^m]_n$ is one, and this subrepresentation is equal to $I_n$.
First, we note that $I_n$ is a direct sum of irreducible subrepresentations of ${{\mathbb C}}[{\operatorname{Mat}}_{n}^m]$. Second, we note that all the isotypic components other than $E_{(n),1^n,1^n}$ do not intersect $I_n$. This is because, for any other choice of $(\lambda, \mu, \nu)$ for which $a_{\lambda,\mu,\nu} >0$, we have either $\mu \neq 1^n$ or $\nu \neq 1^n$ by Lemma \[L-kron1\]. If $\mu \neq 1^n$, then the above lemma tells us that the isotypic component does not intersect $I_n$. If $\nu \neq 1^n$, the argument is similar (In the proof of the above lemma, one would use the last column being zero rather than the last row). Thus $I_n \subseteq E_{(n),1^n,1^n}.$ Now, since $a_{(n),1^n,1^n} = 1$ by Lemma \[L-kron1\], we know that $E_{(n),1^n,1^n}$ is irreducible and has no proper subrepresentations. Clearly $I_n \neq \{0\}$ since $\det(X_1) \in I_n$, so $I_n = E_{(n),1^n,1^n}$, which comprises of the unique copy of $S_{(n),1^n,1^n}$ in ${{\mathbb C}}[{\operatorname{Mat}}_{n}^m]_n$.
\[Proof of Proposition \[P-ideal\], part (3)\] This can be seen in many ways, some very explicit. However, we will take a short route out by making the following observation. Consider the action of ${\operatorname{SL}}_n \times {\operatorname{SL}}_n \subseteq {\operatorname{GL}}_n \times {\operatorname{GL}}_n \subseteq {\operatorname{GL}}_m \times {\operatorname{GL}}_n \times {\operatorname{GL}}_n$ by left-right multiplication on ${\operatorname{Mat}}_{n}^m$. Then, the invariant polynomials of degree $n$ are precisely the isotypic component corresponding to $S_{(n),1^n,1^n}$ (see for example [@Visu Proposition 4.1]), which by the above corollary is precisely $I_n$.
There has been much work on the ring of invariants for the ${\operatorname{SL}}_n \times {\operatorname{SL}}_n$ action. It is a special case of a semi-invariant ring of quivers, in particular for the generalized Kronecker quiver. Such semi-invariants have explicit determinantal descriptions, an important and non-trivial result shown simultaneously and independently by three groups of researchers (see [@DW; @SVd; @DZ]). From this description, we get that the invariants of degree $n$ for the action of ${\operatorname{SL}}_n \times {\operatorname{SL}}_n$ are spanned by polynomials of the form $\det(\sum_i c_i X_i)$ (see [@DM; @IQS]). This completes the proof.
Ideal of polynomials vanishing on ${{\rm NSING}}_{n,m}$
-------------------------------------------------------
We note that all the arguments above for understanding the ideal of polynomials vanishing on ${{\rm SING}}_{n,m}$ work equally well for ${{\rm NSING}}_{n,m}$, and one obtains:
\[L-N-ideal\] Consider ${{\rm NSING}}_{n,m} \subseteq V = {\operatorname{Mat}}_{n}^m$, and let $J \subseteq {{\mathbb C}}[V]$ be the ideal of polynomial functions that vanish on ${{\rm NSING}}_{n,m}$. Then,
1. $J$ is graded;
2. $J_a$ is empty if $a < n$;
3. $J_n = {{\rm span}}(\det(\sum_i c_iX_i): c_i \in {{\mathbb C}}).$
Positive characteristic {#app.pos.char}
=======================
In this section, we will point out the parts of the paper that require characteristic zero, and how to make the requisite modifications for the statements to hold in. Let $K$ be an algebraically closed field of arbitrary characteristic. The first issue comes with the use of Lie algebras. Lie algebras are a little trickier to define because of the lack of derivatives when working over $K$. Nevertheless, one can define the Lie algebra ${{\rm Lie}}(G) = {\mathfrak{g}}$ of an algebraic group $G$ as the space of all derivations of $K[G]$. As a vector space one can identify this with the tangent space at the identity element. Next, one does not have an exponential map that allows us to pass back from the Lie algebra to the group. Now, let $V$ be a vector space over $K$ and let $S \subseteq V$ be a subvariety. Let $\mathcal{G}_S$ denote its group of symmetries, $\mathcal{G}_S^\circ$ its identity component and ${\mathfrak{g}}_S$ its Lie algebra. Let $I_S$ denote the ideal of polynomials in $K[V]$ vanishing on $S$. When we work over $K$ instead of ${{\mathbb C}}$, the proof of Lemma \[Lgos-graded\] clearly goes through and Proposition \[P-Liealg-all\] still remains true, for example by [@Milne Proposition 10.31].
The other main issue in implementing this strategy is to understand the degree $n$ component of the ideal of polynomials vanishing on ${{\rm SING}}_{n,m}$, as done in the previous appendix. Let $I = I_{{{\rm SING}}_{n,m}} \subseteq K[{\operatorname{Mat}}_{n}^m]$. On first glance, it looks like we used quite heavily the notion of complete reducibility for ${\operatorname{GL}}_m \times {\operatorname{GL}}_n \times {\operatorname{GL}}_n$ actions. But in fact, we can get away with far less. The first idea is to restrict our attention to ${\operatorname{SL}}_n \times {\operatorname{SL}}_n$. Clearly $I_n$ is an ${\operatorname{SL}}_n \times {\operatorname{SL}}_n$ subrepresentation. It need not break up as a direct sum of irreducibles, but will definitely have a composition series. Nevertheless, $I_n$ is a direct sum of weight spaces. We claim that the only highest weight vectors that can be in $I_n$ must have weight zero. Basically the argument we used in the previous section shows that any highest weight vector (for ${\operatorname{GL}}_n \times {\operatorname{GL}}_n$) in $I_n$ must have weight $((1,1,\dots,1), (1,1,\dots,1))$. Highest weight vectors for ${\operatorname{GL}}_n \times {\operatorname{GL}}_n$ are precisely the same as the highest weight vectors for ${\operatorname{SL}}_n \times {\operatorname{SL}}_n$, and the weight $((1,1,\dots,1), (1,1,\dots,1))$ for ${\operatorname{GL}}_n \times {\operatorname{GL}}_n$ corresponds to the zero weight for ${\operatorname{SL}}_n \times {\operatorname{SL}}_n$.
Now, let $\mathcal{X}$ denote the set of all weights for $I_n$ (w.r.t ${\operatorname{SL}}_n \times {\operatorname{SL}}_n)$ whose multiplicity is nonzero. It is well known (and easy to see) that the set of weights is stable under the action of the symmetric group $S_n$ (also known as the Weyl group). If $\mathcal{X}$ contains a non-zero weight, then it contains a non-zero dominant weight (using the action of $S_n$). Consider the collection of all non-zero dominant weights. Since this is a finite set, it has a maximal element w.r.t to the usual partial order ($\lambda \prec \mu$ if $\mu-\lambda$ is a sum of positive roots). Any weight vector for this maximal weight must be a highest weight vector! But this contradicts the discussion above, so $\mathcal{X}$ must be the singleton set $\{0\}$.
In other words, $I_n$ is an ${\operatorname{SL}}_n \times {\operatorname{SL}}_n$ stable subspace of the zero weight space (in the space of degree $n$ polynomials on ${\operatorname{Mat}}_n^m$). Irreducible representations of ${\operatorname{SL}}_n \times {\operatorname{SL}}_n$ are indexed by their highest weights (holds true in all characteristics), and so any composition series for $I_n$ must only contain trivial representations. However, trivial representations do not have any self-extensions, so $I_n$ must be a direct sum of trivial representations. In other words, $I_n$ must be a subspace of the ${\operatorname{SL}}_n \times {\operatorname{SL}}_n$ invariants. On the other hand, we know (by [@DW]) that ${\operatorname{SL}}_n \times {\operatorname{SL}}_n$ invariants are spanned by polynomials of the form $\det(\sum_i c_i X_i)$, which are all clearly in $I_n$. This shows that Proposition \[P-ideal\] continues to hold.
Now, armed with the above results, one sees that the computation of ${\mathfrak{g}}_S$ for $S = {{\rm SING}}_{n,m}$ is exactly the same, and we get ${\mathfrak{g}}_S = {{\rm Lie}}(G_{n,m})$. However, the subgroups-subalgebras correspondence is not necessarily true over fields of positive characteristic, so we cannot immediately conclude that $G_S^\circ = G_{n,m}$. However, it is definitely clear that $G_{n,m} \subseteq G_S^\circ$. Suppose $G_S^\circ$ were a strictly larger algebraic group, then $\dim G_S^\circ > \dim G_{n,m} = \dim ({{\rm Lie}}(G_{n,m})) = {\mathfrak{g}}_S$. Note that $\dim(G_{n,m}) = \dim({{\rm Lie}}(G_{n,m}))$ follows from the fact that $G_{n,m}$ is smooth (or one can simply compare the dimensions to those in characteristic zero). Thus, we have $\dim G_S^\circ > \dim {\mathfrak{g}}_S$, but this is a contradiction because $\dim {\mathfrak{g}}_S$ is the dimension of the tangent space at identity for $G_S^\circ$, which is always at least $\dim G_S^\circ$ (see [@Milne Proposition 1.37]). Hence, we have $G_S^\circ = G_{n,m}$. The computation of the entire group of symmetries follows verbatim. The same arguments also compute the group of symmetries for ${{\rm NSING}}_{n,m}$ just as in the characteristic zero case.
The rest of the arguments are effectively the same. The coordinate subspaces in ${{\rm NSING}}_{n,m}$ and ${{\rm SING}}_{n,m}$ have exactly the same descriptions in terms of permutation free supports, and so as long as ${{\rm NSING}}_{n,m} \subsetneq {{\rm SING}}_{n,m}$, the latter cannot be a null cone. Finally, to show that ${{\rm NSING}}_{n,m} \subsetneq {{\rm SING}}_{n,m}$ for $n,m \geq 3$, we relied on an explicit example of $3$-tuple of $3 \times 3$ matrices which was in ${{\rm SING}}_{n,m}$, but not in ${{\rm NSING}}_{n,m}$. This example continues to hold in positive characteristic as well, which can be explicitly checked (it can also be derived as a special case of [@DM-explicit Proposition 1.8] for $p = 1$).
[^1]: A technical term that includes all classical groups.
[^2]: where limits can be equivalently taken in the Euclidean or Zariski topology
[^3]: A canonical version of the Polynomial Identity Testing (PIT) problem.
[^4]: We do not require irreducibility in our definition of varieties.
[^5]: Reductivity is essential for this.
[^6]: For example, if we consider the parabola described as the zero locus of $y - x^2$ in ${{\mathbb C}}^2$, this is not a null cone (because null cones are stable under scalar multiplication, but the parabola isn’t. However, as a variety, this is just the affine line, which is definitely a null cone (for the action of ${{\mathbb C}}^*$ on ${{\mathbb C}}$ by multiplication).
[^7]: Any group $G$ acting on $V$ gives a map $\rho:G \rightarrow {\operatorname{GL}}(V)$. The null cone for $G$ is the same as the null cone for $\rho(G)$, so we can always restrict ourselves to subgroups of ${\operatorname{GL}}(V)$ when concerned about Problem \[prob.nullcone\]. Moreover, note that if $G$ is reductive, so is $\rho(G)$.
[^8]: The identity component is the connected component of $H$ that contains the identity element. It is always an algebraic subgroup.
[^9]: The free skew field is intuitively the natural non-commutative analog of ${{\mathbb C}}(t_1,\dots,t_m)$, namely may be viewed as the field of fractions completing non-commutative polynomials. However, we note that its very existence, let alone its construction is highly non-trivial, and was first established by Amitsur [@Am66] (see also [@Cohn]). For one illustration of the complexity of this field, it is easy to see that unlike in the commutative case, its elements cannot be represented as ratios of polynomials (or any finite number of inversions - an important result of [@Reutenauer]).
[^10]: It seems plausible that these polynomials generate the ideal of polynomials that vanish on ${{\rm SING}}_{n,m}$, but such questions can often be quite subtle to prove.
[^11]: An equivalent definition is that a linear algebraic group $G$ is an (affine) algebraic variety $G$ which is also a group such that the multiplication map $m: G \times G \rightarrow G$ and an inverse map $i:G \rightarrow G$ are morphisms of algebraic varieties. While this definition seems more general, it is a standard result that both definitions agree. For this reason, sometimes linear algebraic groups are also called affine algebraic groups.
[^12]: Note that morphism $\rho$ needs to be a regular morphism (and not a rational morphism) as it must be defined on all of $G$. In particular even though $\rho(X)$ is given by a matrix of rational functions, all these rational functions have to be defined on $G \subseteq {\operatorname{GL}}_n$, so their locus of indeterminacy must be away from $G$. A canonical example is the function $\frac{1}{\det}$ which is an honest ratio of polynomials that will be defined on $G$ (and indeed all of ${\operatorname{GL}}_n$). Also observe that $\frac{1}{\det}$ is a regular function on ${\operatorname{GL}}_n$ and hence on $G$ as well.
[^13]: The orbit closure can be taken in the Zariski topology or the analytic topology, since they are both the same.
[^14]: All maximal tori are conjugate. Moreover, the union of all maximal tori is dense in the identity component of $G$.
[^15]: This is a famous result due to Chevalley, and is often called the subgroups-subalgebras correspondence
[^16]: One way to compute the normalizer in this case is by understanding the automorphisms of the Dynkin diagram, see [@Guralnick]. However, we will give more concrete arguments as many of our readers may not possess an in depth knowledge of the theory of semisimple algebras.
[^17]: Component groups are always finite for linear algebraic groups.
[^18]: Clearly, for such groups the definition of a null cone must be take to be the analytic one.
[^19]: Differentiating this gives the Cartan involution on Lie algebras described in Section \[sec:explicit\].
[^20]: In fact $X \in {{\rm NSING}}_{n,m}$ – this is not hard (for example it follows from the shrunk subspace criterion, see [@GGOW16; @IQS]).
|
---
abstract: 'Despite its well-known shortcomings, $k$-means remains one of the most widely used approaches to data clustering. Current research continues to tackle its flaws while attempting to preserve its simplicity. Recently, the *power $k$-means* algorithm was proposed to avoid trapping in local minima by annealing through a family of smoother surfaces. However, the approach lacks theoretical justification and fails in high dimensions when many features are irrelevant. This paper addresses these issues by introducing *entropy regularization* to learn feature relevance while annealing. We prove consistency of the proposed approach and derive a scalable majorization-minimization algorithm that enjoys closed-form updates and convergence guarantees. In particular, our method retains the same computational complexity of $k$-means and power $k$-means, but yields significant improvements over both. Its merits are thoroughly assessed on a suite of real and synthetic data experiments.'
author:
- 'Saptarshi Chakraborty[^1]$\,\,^{1}$, Debolina Paul$^{\ast 1}$, Swagatam Das$^2$, and Jason Xu[^2]$\,\,^{3}$'
title: 'Entropy Regularized Power *k*-Means Clustering'
---
$^1$Indian Statistical Institute, Kolkata, India\
$^2$ Electronics and Communication Sciences Unit, Indian Statistical Institute, Kolkata, India\
$^3$ Department of Statistical Science, Duke University, Durham, NC, USA.
Introduction
============
Clustering is a fundamental task in unsupervised learning for partitioning data into groups based on some similarity measure. Perhaps the most popular approach is $k$-means clustering [@macqueen1967some]: given a dataset $\mathcal{X}=\{{\boldsymbol{x}}_1,\dots,{\boldsymbol{x}}_n\} \subset\mathbb{R}^p$, $\mathcal{X}$ is to be partitioned into $k$ mutually exclusive classes so that the variance within each cluster is minimized. The problem can be cast as minimization of the objective $$\label{obj1} P({\boldsymbol{\Theta}}) = \sum_{i=1}^n \min_{1 \le j \le k} \|{\boldsymbol{x}}_i-{\boldsymbol{\theta}}_j\|^2,$$ where ${\boldsymbol{\Theta}}=\{{{\boldsymbol{\theta}}}_1, {{\boldsymbol{\theta}}}_2, \dots , {{\boldsymbol{\theta}}}_k\}$ denotes the set of cluster centroids, and $\|{\boldsymbol{x}}_i-{\boldsymbol{\theta}}_j\|^2$ is the usual squared Euclidean distance metric. Lloyd’s algorithm [@lloyd1982least], which iterates between assigning points to their nearest centroid and updating each centroid by averaging over its assigned points, is the most frequently used heuristic to solve the preceding minimization problem. Such heuristics, however, suffer from several well-documented drawbacks. Because the task is NP-hard [@aloise2009np], Lloyd’s algorithm and its variants seek to approximately solve the problem and are prone to stopping at poor local minima, especially as the number of clusters $k$ and dimension $p$ grow. Many new variants have since contributed to a vast literature on the topic, including spectral clustering [@DBLP:conf/nips/NgJW01], Bayesian [@lock2013bayesian] and non-parametric methods [@DBLP:conf/icml/KulisJ12], subspace clustering [@DBLP:journals/spm/Vidal11], sparse clustering [@witten2010framework], and convex clustering [@chi2015splitting]; a more comprehensive overview can be found in [@jain2010data].
None of these methods have managed to supplant $k$-means clustering, which endures as the most widely used approach among practitioners due to its simplicity. Some work instead focuses on “drop-in" improvements of Lloyd’s algorithm. The most prevalent strategy is clever seeding: $k$-means++ [@arthur2007k; @ostrovsky2012effectiveness] is one such effective wrapper method in theory and practice, and proper initialization methods remain an active area of research [@celebi2013comparative; @bachem2016fast]. Geometric arguments have also been employed to overcome sensitivity to initialization. [@zhang1999k] proposed to replace the minimum function by the harmonic mean function to yield a smoother objective function landscape but retain a similar algorithm, though the strategy fails in all but very low dimensions. @xu2019power generalized this idea by using a sequence of successively smoother objectives via power means instead of the harmonic mean function to obtain better approximating functions in each iteration. The contribution of power $k$-means is algorithmic in nature—it effectively avoids local minima from an *optimization* perspective, and succeeds for large $p$ when the data points are well-separated. However, it does not address the *statistical* challenges in high-dimensional settings and performs as poorly as standard $k$-means in such settings. A meaningful similarity measure plays a key role in revealing clusters [@de2012minkowski; @chakraborty2017k], but pairwise Euclidean distances become decreasingly informative as the number of features grows due to the curse of dimensionality.
On the other hand, there is a rich literature on clustering in high dimensions, but standard approaches such as subspace clustering are not scalable due to the use of an affinity matrix pertaining to norm regularization [@ji2014efficient; @abcd]. For spectral clustering, even the creation of such a matrix quickly becomes intractable for modern, large-scale problems [@zhang2019neural]. Toward learning effective feature representations, [@huang2005automated] proposed weighted $k$-means clustering ($WK$-means), and sparse $k$-means [@witten2010framework] has become a benchmark feature selection algorithm, where selection is achieved by imposing $\ell_1$ and $\ell_2$ constraints on the feature weights. Further related developments can be found in the works of [@modha2003feature; @li2006novel; @huang2008weighting; @de2012minkowski; @jin2016influential]. These approaches typically lead to complex optimization problems in terms of transparency as well as computational efficiency—for instance, sparse $k$-means requires solving constrained sub-problems via bisection to find the necessary dual parameters $\lambda^\ast$ in evaluating the proximal map of the $\ell_1$ term. As they fail to retain the simplicity of Lloyd’s algorithm for $k$-means, they lose appeal to practitioners. Moreover, these works on feature weighing and selection do not benefit from recently algorithmic developments as mentioned above.
In this article, we propose a scalable clustering algorithm for high dimensional settings that leverages recent insights for avoiding poor local minima, performs adaptive feature weighing, and preserves the low complexity and transparency of $k$-means. Called Entropy Weighted Power $k$-means (EWP), we extend the merits of power $k$-means to the high-dimensional case by introducing feature weights together with entropy incentive terms. Entropy regularization is not only effective both theoretically and empirically, but leads to an elegant algorithm with closed form solution updates. The idea is to minimize along a continuum of smooth surrogate functions that gradually approach the $k$-means objective, while the feature space also gradually adapts so that clustering is driven by informative features. By transferring the task onto a sequence of better-behaved optimization landscapes, the algorithm fares better against the curse of dimensionality and against adverse initialization of the cluster centroids than existing methods.
[0.34]{} ![Peer methods fail to cluster in $100$ dimensions with $5$ effective features on illustrative example, while the proposed method achieves perfect separation. Solutions are visualized using `t-SNE`.[]{data-label="fig:eg1"}](images/d1.pdf "fig:"){height="\textwidth" width="\textwidth"}
[0.34]{} ![Peer methods fail to cluster in $100$ dimensions with $5$ effective features on illustrative example, while the proposed method achieves perfect separation. Solutions are visualized using `t-SNE`.[]{data-label="fig:eg1"}](images/d6.pdf "fig:"){height="\textwidth" width="\textwidth"}
[0.34]{} ![Peer methods fail to cluster in $100$ dimensions with $5$ effective features on illustrative example, while the proposed method achieves perfect separation. Solutions are visualized using `t-SNE`.[]{data-label="fig:eg1"}](images/d3.pdf "fig:"){height="\textwidth" width="\textwidth"}
[0.34]{} ![Peer methods fail to cluster in $100$ dimensions with $5$ effective features on illustrative example, while the proposed method achieves perfect separation. Solutions are visualized using `t-SNE`.[]{data-label="fig:eg1"}](images/d4.pdf "fig:"){height="\textwidth" width="\textwidth"}
[0.34]{} ![Peer methods fail to cluster in $100$ dimensions with $5$ effective features on illustrative example, while the proposed method achieves perfect separation. Solutions are visualized using `t-SNE`.[]{data-label="fig:eg1"}](images/d5.pdf "fig:"){height="\textwidth" width="\textwidth"}
The following summarizes our main contributions:
- We propose a clustering framework that automatically learns a weighted feature representation while simultaneously avoiding local minima through annealing.
- We develop a scalable Majorization-Minimization (MM) algorithm to minimize the proposed objective function.
- We establish descent and convergence properties of our method and prove the strong consistency of the global solution.
- Through an extensive empirical study on real and simulated data, we demonstrate the efficacy of our algorithm, finding that it outperforms comparable classical and state-of-the-art approaches.
The rest of the paper is organized as follows. After reviewing some necessary background, Section \[erpk\] formulates the Entropy Weighted Power $k$-means (EWP) objective and provides high-level intuition. Next, an MM algorithm to solve the resulting optimization problem is derived in Section \[optimization\]. Section \[theory\] establishes the theoretical properties of the EWP clustering. Detailed experiments on both real and simulated datasets are presented in Section \[experiment\], followed by a discussion of our contributions in Section \[discussion\].
#### Majorization-minimization
The principle of MM has become increasingly popular for large-scale optimization in statistical learning [@mairal2015incremental; @lange2016mm]. Rather than minimizing an objective of interest $f(\cdot)$ directly, an MM algorithm successively minimizes a sequence of simpler *surrogate functions* $g({\boldsymbol{\theta}}\mid {\boldsymbol{\theta}}_n)$ that *majorize* the original objective $f({\boldsymbol{\theta}})$ at the current estimate ${\boldsymbol{\theta}}_m$. Majorization requires two conditions: tangency $g({\boldsymbol{\theta}}_m \mid {\boldsymbol{\theta}}_m) = f({\boldsymbol{\theta}}_m)$ at the current iterate, and domination $g({\boldsymbol{\theta}}\mid {\boldsymbol{\theta}}_m) \geq f({\boldsymbol{\theta}})$ for all ${\boldsymbol{\theta}}$. The iterates of the MM algorithm are defined by the rule $$\label{eq:MMiter}
{\boldsymbol{\theta}}_{m+1} := \arg\min_{{\boldsymbol{\theta}}}\; g({\boldsymbol{\theta}}\mid {\boldsymbol{\theta}}_m)$$ which immediately implies the descent property $$\begin{aligned}
f({\boldsymbol{\theta}}_{m+1}) \, \leq \, g({\boldsymbol{\theta}}_{m+1} \mid {\boldsymbol{\theta}}_{m})
\, \le \, g({\boldsymbol{\theta}}_{m} \mid {\boldsymbol{\theta}}_{m})
\, = \, f({\boldsymbol{\theta}}_{m}). \label{eq:descent}\end{aligned}$$ That is, a decrease in $g$ results in a decrease in $f$. Note that $g({\boldsymbol{\theta}}_{m+1} \mid {\boldsymbol{\theta}}_{m} ) \le g({\boldsymbol{\theta}}_{m} \mid {\boldsymbol{\theta}}_{m})$ does not require ${\boldsymbol{\theta}}_{m+1}$ to minimize $g$ exactly, so that any descent step in $g$ suffices. The MM principle offers a general prescription for transferring a difficult optimization task onto a sequence of simpler problems [@LanHunYan2000], and includes the well-known EM algorithm for maximum likelihood estimation under missing data as a special case [@BecYanLan1997].
#### Power k-means
[@zhang1999k] attempt to reduce sensitivity to initialization in $k$-means by minimizing the criterion $$\begin{aligned}
\sum_{i=1}^n \Big(\frac{1}{k} \sum_{j=1}^k \|{\boldsymbol{x}}_i-{\boldsymbol{\theta}}_j\|^{-2} \Big)^{-1} := f_{-1}({\boldsymbol{\Theta}}). \label{KHM}\end{aligned}$$ Known as $k$-harmonic means, the method replaces the $\min$ appearing in by the harmonic average to yield a smoother optimization landscape, an effective approach in low dimensions. Recently, power $k$-means clustering extends this idea to work in higher dimensions where is no longer a good proxy for . Instead of considering only the closest centroid or the harmonic average, the *power mean* between each point and all $k$ centroids provides a family of successively smoother optimization landscapes. The power mean of a vector ${\boldsymbol{y}}$ is defined $M_s({\boldsymbol{y}}) = \left( \frac{1}{k}\sum_{i=1}^k y_i^s \right)^{1/s}. $ Within this class, $s>1$ corresponds to the usual $\ell_s$-norm of ${\boldsymbol{y}}$, $s=1$ to the arithmetic mean, and $s=-1$ to the harmonic mean.
Power means enjoy several nice properties that translate to algorithmic merits and are useful for establishing theoretical guarantees. They are monotonic, homogeneous, and differentiable with gradient $$\begin{aligned}
\frac{\partial}{\partial y_j} M_ s( {\boldsymbol{y}}) & =& \Big(\frac{1}{k}\sum_{i=1}^k y_i^s\Big)^{\frac{1}{s}-1} \frac{1}{k}y_j^{s-1} ,\label{power_mean_grad}\end{aligned}$$ and satisfy the limits
\[eq:limit\] $$\lim_{s \to -\infty}M_s({\boldsymbol{y}})=\min\{y_1,\ldots,y_k\}$$ $$\lim_{s \to \infty}M_s({\boldsymbol{y}})=\max\{y_1,\ldots,y_k\} .$$
Further, the well-known power mean inequality $M_s ({\boldsymbol{y}}) \le M_ t ({\boldsymbol{y}})$ for any $s \le t$ holds [@steele2004cauchy]. The power $k$-means objective function for a given power $s$ is given by the formula $$\label{eq:limit2}
f_s(\Theta )=\sum_{i=1}^n M_s(\|{\boldsymbol{x}}_i-{\boldsymbol{\theta}}_1\|^2,\ldots,\|{\boldsymbol{x}}_i-{\boldsymbol{\theta}}_k\|^2).$$ The algorithm then seeks to minimize $f_s$ iteratively while sending $s \rightarrow -\infty$. Doing so, the objective approaches $f_{-\infty}(\Theta)$ due to , coinciding with the original $k$-means objective and retaining its interpretation as minimizing within-cluster variance. The intermediate surfaces provide better optimization landscapes that exhibit fewer poor local optima than . Each minimization step is carried out via MM; see [@xu2019power] for details.
Entropy Weighted Power *k*-means
================================
#### A Motivating Example {#motivation}
We begin by considering a synthetic dataset with $k=20$ clusters, $n=1000$ points, and $p=100$. Of the $100$ features, only $5$ are relevant for distinguishing clusters, while the others are sampled from a standard normal distribution (further details are described later in Simulation 2 of Section \[set2\]). We compare standard $k$-means, $WK$-means, power $k$-means, and sparse $k$-means with our proposed method; sparse $k$-means is tuned using the gap statistic described in the original paper [@witten2010framework] as implemented in the `R` package, `sparcl`. Figure \[fig:eg1\] displays the solutions in a $t$-distributed Stochastic Neighbourhood Embedding (`t-SNE`) [@maaten2008visualizing] for easy visualization in two dimensions. It is evident that our EWP algorithm, formulated below, yields perfect recovery while the peer algorithms fail to do so. This transparent example serves to illustrate the need for an approach that simultaneously avoids poor local solutions while accommodating high dimensionality.
Problem Formulation {#erpk}
-------------------
Let ${\boldsymbol{x}}_1,\dots,{\boldsymbol{x}}_n \in \mathbb{R}^p$ denote the $n$ data points, and $\Theta_{k \times p}=[{\boldsymbol{\theta}}_1,\dots,{\boldsymbol{\theta}}_k]^\top$ denote the matrix whose rows contain the cluster centroids. We introduce a feature relevance vector ${\boldsymbol{w}}\in \mathbb{R}^p$ where $w_l$ contains the weight of the $l$-th feature, and require these weights to satisfy the constraints $$\tag{C}
\sum_{l=1}^p w_l=1; \qquad w_l \geq 0 \text{ for all } l=1,\dots,p .
\label{c1}$$ The EWP objective for a given $s$ is now given by $$\label{obj}
f_s(\Theta,{\boldsymbol{w}}) = \sum_{i=1}^n M_s(\|{\boldsymbol{x}}-{\boldsymbol{\theta}}_1\|_{{\boldsymbol{w}}}^2,\dots,\|{\boldsymbol{x}}-{\boldsymbol{\theta}}_k\|_{{\boldsymbol{w}}}^2) + \lambda \sum_{l=1}^p w_l \log w_l,$$ where the *weighted* norm $\|{\boldsymbol{y}}\|_{{\boldsymbol{w}}}^2=\sum_{l=1}^p w_l y_l^2$ now appears as arguments to the power mean $M_s$. The final term is the negative entropy of ${\boldsymbol{w}}$ [@jing2007entropy]. This entropy incentive is minimized when $w_l=1/p$ for all $l=1,\dots,p$; in this case, equation (\[obj\]) is equal to the power $k$-means objective, which in turn equals the $k$-means objective when $s\rightarrow -\infty$ (and coincides with KHM for $s=-1$). EWP thus generalizes these approaches, while newly allowing features to be adaptively weighed throughout the clustering algorithm. Moreover, we will see in Section \[optimization\] that entropy incentives are an ideal choice of regularizer in that they lead to closed form updates for ${\boldsymbol{w}}$ and ${\boldsymbol{\theta}}$ within an iterative algorithm.
#### Intuition and the curse of dimensionality {#intuition}
Power $k$-means combats the curse of dimensionality by providing smoothed *objective functions* that remain appropriate as dimension increases. Indeed, in practice the value of $s$ at convergence of power $k$-means becomes lower as the dimension increases, explaining its outperformance over $k$-harmonic means [@zhang1999k]— $f_{-1}$ deteriorates as a reasonable approximation of $f_{-\infty}$. However even if poor solutions are successfully avoided from the algorithmic perspective, the curse of dimensionality still affects the *arguments* to the objective. Minimizing within-cluster variance becomes less meaningful as pairwise Euclidean distances become uninformative in high dimensions [@aggarwal2001surprising]. It is therefore desirable to reduce the effective dimension in which distances are computed. While the entropy incentive term does not zero out variables, it weighs the dimensions according to how useful they are in driving clustering. When the data live in a high-dimensional space yet only a small number of features are relevant towards clustering, the optimal solution to our objective assigns non-negligible weights to only those few relevant features, while benefiting from annealing through the weighted power mean surfaces.
Optimization
------------
To optimize the EWP objective, we develop an MM algorithm [@lange2016mm] for sequentially minimizing (\[obj\]). As shown by [@xu2019power], $M_s({\boldsymbol{y}})$ is concave if $s<1$; in particular, it lies below its tangent plane. This observation provides the following inequality: denoting ${\boldsymbol{y}}_m$ the estimate of a variable ${\boldsymbol{y}}$ at iteration $m$, $$\label{mm1}
M_s({\boldsymbol{y}}) \leq M_s({\boldsymbol{y}}_m)+ \nabla_{{\boldsymbol{y}}} M_s({\boldsymbol{y}}_m)^\top ({\boldsymbol{y}}-{\boldsymbol{y}}_m)$$ Substituting $\|{\boldsymbol{x}}_i-{\boldsymbol{\theta}}_j\|_{{\boldsymbol{w}}}^2$ for $y_j$ and $\|{\boldsymbol{x}}_i-{\boldsymbol{\theta}}_{mj}\|_{{\boldsymbol{w}}_m}^2$ for $y_{mj}$ in equation (\[mm1\]) and summing over all $i$, we obtain [$$\begin{aligned}
&f_s({\boldsymbol{\Theta}},{\boldsymbol{w}}) \leq f_s({\boldsymbol{\Theta}}_m,{\boldsymbol{w}}_m)-\sum_{i=1}^n \sum_{j=1}^k \phi_{ij}^{(m)} \|{\boldsymbol{x}}_i-{\boldsymbol{\theta}}_{mj}\|_{{\boldsymbol{w}}_m}^2\\
&-\lambda \sum_{l=1}^p (w_{m,l} \log w_{m,l}-w_l \log w_l) + \sum_{i=1}^n \sum_{j=1}^k \phi_{ij}^{(m)}\|{\boldsymbol{x}}_i-{\boldsymbol{\theta}}_{j}\|_{{\boldsymbol{w}}}^2.\end{aligned}$$ ]{}Here the derivative expressions provide the values of the constants $$\phi_{ij}^{(m)}=\frac{\frac{1}{k}\|{\boldsymbol{x}}_i-{\boldsymbol{\theta}}_{m,j}\|_{{\boldsymbol{w}}_m}^{2(s-1)}}{\bigg(\frac{1}{k}\sum_{j=1}^k\|{\boldsymbol{x}}_i-{\boldsymbol{\theta}}_{m,j}\|_{{\boldsymbol{w}}_m}^{2s}\bigg)^{(1-\frac{1}{s})}}.$$ The right-hand side of the inequality above serves as a *surrogate function* majorizing $f_s({\boldsymbol{\Theta}}, {\boldsymbol{w}})$ at the current estimate ${\boldsymbol{\Theta}}_m$. Minimizing this surrogate amounts to minimizing the expression $$\label{eq7}
\sum_{i=1}^n \sum_{j=1}^k \phi_{ij}^{(m)} \|{\boldsymbol{x}}_i-{\boldsymbol{\theta}}_{j}\|_{{\boldsymbol{w}}}^2+\lambda\sum_{l=1}^p w_l \log w_l$$ subject to the constraints (\[c1\]). This problem admits closed form solutions: minimization over ${\boldsymbol{\Theta}}$ is straightforward, and the optimal solutions are given by $${\boldsymbol{\theta}}_{j}^* =\frac{\sum_{i=1}^n \phi_{ij} {\boldsymbol{x}}_i}{\sum_{i=1}^n \phi_{ij}}.$$ To minimize equation (\[eq7\]) in ${\boldsymbol{w}}$, we consider the Lagrangian $$\mathcal{L}= \sum_{i=1}^n \sum_{j=1}^k \phi_{ij} \|{\boldsymbol{x}}_i-{\boldsymbol{\theta}}_{j}\|_{{\boldsymbol{w}}}^2+\lambda\sum_{l=1}^p w_l \log w_l-\alpha (\sum_{l=1}^p w_l-1).$$ The optimality condition $\frac{\partial \mathcal{L}}{\partial w_l}=0$ implies $\sum_{i=1}^n \sum_{j=1}^k \phi_{ij} (x_{il}-\theta_{jl})^2+\lambda(1+ \log w_l)-\alpha=0. $ This further implies that
$$w_l^* \propto \exp{\bigg\{-\frac{\sum_{i=1}^n \sum_{j=1}^k \phi_{ij}(x_{il}-\theta_{jl})^2}{\lambda}\bigg\}}.$$ Now enforcing the constraint $\sum_{l=1}^p w_l =1$, we get $$w_l^* = \frac{\exp{\bigg\{-\frac{\sum_{i=1}^n \sum_{j=1}^k \phi_{ij}(x_{il}-\theta_{jl})^2}{\lambda}\bigg\}}}{\sum_{t=1}^p \exp{\bigg\{-\frac{\sum_{i=1}^n \sum_{j=1}^k \phi_{ij}(x_{it}-\theta_{jt})^2}{\lambda}\bigg\}}}.$$ Thus, the MM steps take a simple form and amount to two alternating updates: $$\begin{aligned}
{\boldsymbol{\theta}}_{m+1,j} & =\frac{\sum_{i=1}^n \phi_{ij}^{(m)} {\boldsymbol{x}}_i}{\sum_{i=1}^n \phi_{ij}^{(m)}}\\
w_{m+1,l} & = \frac{\exp{\bigg\{-\frac{\sum_{i=1}^n \sum_{j=1}^k \phi_{ij}^{(m)}(x_{il}-\theta_{jl})^2}{\lambda}\bigg\}}}{\sum_{t=1}^p \exp{\bigg\{-\frac{\sum_{i=1}^n \sum_{j=1}^k \phi_{ij}^{(m)}(x_{it}-\theta_{jt})^2}{\lambda}\bigg\}}}.\end{aligned}$$ The MM updates are similar to those in Lloyd’s algorithm [@lloyd1982least] in the sense that each step alternates between updating $\phi_{ij}$’s and updating ${\boldsymbol{\Theta}}$ and ${\boldsymbol{w}}$. These updates are summarised in Algorithm \[alg\]; though there are three steps rather than two, the overall per-iteration complexity of this algorithm is the same as that of $k$-means (and power $k$-means) at $\mathcal{O}(nkp)$ [@lloyd1982least]. We require the tuning parameter $\lambda>0$ to be specified, typically chosen via cross-validation detailed in Section \[simulation\]. It should be noted that the initial value $s_0$ and the constant $\eta$ do not require careful tuning: we fix them at $s_0=-1$ and $\eta=1.05$ across *all* real and simulated settings considered in this paper.
initialize $s_0<0$ and ${\boldsymbol{\Theta}}_0$\
**repeat**: [ $$\begin{aligned}
&\phi_{ij}^{(m)} \leftarrow\frac{1}{k}\|{\boldsymbol{x}}_i-{\boldsymbol{\theta}}_{m,j}\|_{{\boldsymbol{w}}_m}^{2(s_m-1)}
\bigg(\frac{1}{k}\sum_{j=1}^k\|{\boldsymbol{x}}_i-{\boldsymbol{\theta}}_{m,j}\|_{{\boldsymbol{w}}_m}^{2s_m}\bigg)^{(\frac{1}{s_m}-1)} \\
&{\boldsymbol{\theta}}_{m+1,j} \leftarrow \frac{\sum_{i=1}^n \phi_{ij}^{(m)} {\boldsymbol{x}}_i}{\sum_{i=1}^n \phi_{ij}^{(m)}}\\
&w_{m+1,l} \leftarrow \frac{\exp{\bigg\{-\frac{\sum_{i=1}^n \sum_{j=1}^k \phi_{ij}^{(m)}(x_{il}-\theta_{jl})^2}{\lambda}\bigg\}}}{\sum_{t=1}^p \exp{\bigg\{-\frac{\sum_{i=1}^n \sum_{j=1}^k \phi_{ij}^{(m)}(x_{it}-\theta_{jt})^2}{\lambda}\bigg\}}}\\
& s_{m+1} \leftarrow \eta s_m
\end{aligned}$$ ]{} **until** convergence
Theoretical Properties {#theory}
======================
We note that all iterates ${\boldsymbol{\theta}}_m$ in Algorithm 1 are defined within the convex hull of the data, all weight updates lie within $[0,1]$, and the procedure enjoys convergence guarantees as an MM algorithm [@lange2016mm]. Before we state and prove the main result of this section on strong consistency, we present results characterizing the sequence of minimizers. Theorems \[hull\] and \[uniform\] show that the minimizers of surfaces $f_s$ always lie in the convex hull of the data $C^k$, and converge uniformly to the minimizer of $f_{-\infty}$.
\[hull\] Let $s \leq 1$ also let $({\boldsymbol{\Theta}}_{n,s},{\boldsymbol{w}}_{n,s})$ be minimizer of $f_s({\boldsymbol{\Theta}},{\boldsymbol{w}})$. Then we have ${\boldsymbol{\Theta}}_{n,s} \in C^k$.
Let $P_C^{{\boldsymbol{w}}} ({\boldsymbol{\theta}})$ denote the projection of ${\boldsymbol{\theta}}$ onto $C$ w.r.t. the $\|\cdot\|_{{\boldsymbol{w}}}$ norm. Now for any ${\boldsymbol{v}}\in C$, using the obtise angle condition, we obtain, $\langle {\boldsymbol{\theta}}-P_C^{{\boldsymbol{w}}} ({\boldsymbol{\theta}}), {\boldsymbol{v}}-P_C^{{\boldsymbol{w}}} ({\boldsymbol{\theta}}) \rangle_{{\boldsymbol{w}}} \leq 0$. Since ${\boldsymbol{x}}_i \in C$, we obtain, $$\begin{aligned}
\|{\boldsymbol{x}}_i-{\boldsymbol{\theta}}_j\|^2_{{\boldsymbol{w}}} & = \|{\boldsymbol{x}}_i-P_C^{{\boldsymbol{w}}} ({\boldsymbol{\theta}}_j)\|^2_{{\boldsymbol{w}}} + \|P_C^{{\boldsymbol{w}}} ({\boldsymbol{\theta}}_j)-{\boldsymbol{\theta}}_j\|^2_{{\boldsymbol{w}}}\\
&-2 \langle {\boldsymbol{\theta}}-P_C^{{\boldsymbol{w}}} ({\boldsymbol{\theta}}_j), {\boldsymbol{x}}_i-P_C^{{\boldsymbol{w}}} ({\boldsymbol{\theta}}_j) \rangle_{{\boldsymbol{w}}}\\
& \geq \|{\boldsymbol{x}}_i-P_C^{{\boldsymbol{w}}} ({\boldsymbol{\theta}}_j)\|^2_{{\boldsymbol{w}}} + \|P_C^{{\boldsymbol{w}}} ({\boldsymbol{\theta}}_j)-{\boldsymbol{\theta}}_j\|^2_{{\boldsymbol{w}}}.\end{aligned}$$ Now since, $M_s(\cdot)$ is an increasing function in each of its argument, if we replace ${\boldsymbol{\theta}}_j$ by $P_C^{{\boldsymbol{w}}} ({\boldsymbol{\theta}}_j)$ in $M_s(\|{\boldsymbol{x}}_i-{\boldsymbol{\theta}}_1\|^2_{{\boldsymbol{w}}},\dots,\|{\boldsymbol{x}}_i-{\boldsymbol{\theta}}_k\|^2_{{\boldsymbol{w}}})$, the objective function value doesn’t go up. Thus we can effectively restrict our attention to $C^k$. Now since the function $f_s(\cdot,\cdot)$ is continuous on the compact set $C^k \times [0,1]^p$, it attains its minimum on $C^k \times [0,1]^p$. Thus, ${\boldsymbol{\Theta}}^* \in C^k$.
\[uniform\] For any decreasing sequence $\{s_m\}_{m=1}^\infty$ such that $s_1 \leq 1$ and $s_m \to -\infty$, $f_{s_m}({\boldsymbol{\Theta}},{\boldsymbol{w}})$ converges uniformly to $f_{-\infty}({\boldsymbol{\Theta}},{\boldsymbol{w}})$ on $C^k\times [0,1]^p$.
For any $({\boldsymbol{\Theta}},{\boldsymbol{w}}) \in C^k\times [0,1]^p$, $f_{s_m}({\boldsymbol{\Theta}},{\boldsymbol{w}})$ converges monotonically to $f_{-\infty}({\boldsymbol{\Theta}},{\boldsymbol{w}})$ (this is due to the power mean inequality). Since $C^k\times [0,1]^p$ is compact, the result follows immediately upon applying Dini’s theorem from real analysis.
Strong consistency is a fundamental requirement of any “good" estimator in the statistical sense: as the number of data points grows, one should be able to recover true parameters with arbitrary precision [@terada2014strong; @terada2015strong; @chakraborty2019strong]. The proof of our main result builds upon the core argument for $k$-means consistency by [@pollard1981strong], and extends the argument through novel arguments involving uniform convergence of the family of annealing functions.
Let ${\boldsymbol{x}}_1,\dots,{\boldsymbol{x}}_n \in \mathbb{R}^p$ be independently and identically distributed from distribution $P$ with support on a compact set $C \subset \mathbb{R}^p$. For notational convenience, we write $\mathcal{M}_s({\boldsymbol{x}},{\boldsymbol{\Theta}},{\boldsymbol{w}})$ for $M_s(\|{\boldsymbol{x}}-{\boldsymbol{\theta}}_1\|_{{\boldsymbol{w}}},\dots,\|{\boldsymbol{x}}-{\boldsymbol{\theta}}_1\|_{{\boldsymbol{w}}})$. We consider the following minimization problem $$\min_{{\boldsymbol{\Theta}},{\boldsymbol{w}}}\bigg\{\frac{1}{n} \sum_{i=1}^n \mathcal{M}_s({\boldsymbol{x}}_i,{\boldsymbol{\Theta}},{\boldsymbol{w}})+\lambda \sum_{l=1}^p w_l \log w_l\bigg\},$$ which is nothing but a scaled version of equation . Intuitively, as $n \to \infty$, $\frac{1}{n} \sum_{i=1}^n \mathcal{M}_s({\boldsymbol{x}}_i,{\boldsymbol{\Theta}},{\boldsymbol{w}})$ is very close to $\int \mathcal{M}_s({\boldsymbol{x}},{\boldsymbol{\Theta}},{\boldsymbol{w}})dP$ almost surely by appealing to the Strong Law of Large Numbers (SLLN). Together with , as $n \to \infty$ and $s \to -\infty$ we expect $$\label{former}\frac{1}{n} \sum_{i=1}^n \mathcal{M}_s({\boldsymbol{x}}_i,{\boldsymbol{\Theta}},{\boldsymbol{w}})+\lambda \sum_{l=1}^p w_l \log w_l$$ to be in close proximity of $$\label{latter}\int \min_{{\boldsymbol{\theta}}\in {\boldsymbol{\Theta}}} \|{\boldsymbol{x}}-{\boldsymbol{\theta}}\|_{{\boldsymbol{w}}}dP++\lambda \sum_{l=1}^p w_l \log w_l,$$ so that minimizers of the should be very close to the minimizers of under certain regularity conditions.
To formalize this intuition, let ${\boldsymbol{\Theta}}^*$, ${\boldsymbol{w}}^*$ be minimizers of $$\Phi({\boldsymbol{\Theta}},{\boldsymbol{w}})~=\int \min_{1\leq j \leq k} \|{\boldsymbol{x}}-{\boldsymbol{\theta}}_j\|^2_{{\boldsymbol{w}}}dP+\lambda \sum_{l=1}^p w_l\log w_,$$ and define ${\boldsymbol{\Theta}}_{n,s}$, ${\boldsymbol{w}}_{n,s}$ as the minimizers of $$\int \mathcal{M}_s({\boldsymbol{x}},{\boldsymbol{\Theta}},{\boldsymbol{w}})dP_n+\lambda \sum_{l=1}^p w_l\log w_l ,$$ where $P_n$ is the empirical measure. We will show that ${\boldsymbol{\Theta}}_{n,s} \xrightarrow{a.s.} {\boldsymbol{\Theta}}^* $ and ${\boldsymbol{w}}_{n,s} \xrightarrow{a.s.} {\boldsymbol{w}}^*$ as $n \to \infty$ and $s \to -\infty$ under the following identifiability assumption:
- For any neighbourhood $N$ of $({\boldsymbol{\Theta}}^*,{\boldsymbol{w}}^*)$, there exists $\eta>0$ such that if $({\boldsymbol{\Theta}}, {\boldsymbol{w}}) \not \in N$ implies that $\Phi({\boldsymbol{\Theta}}, {\boldsymbol{w}})> \Phi({\boldsymbol{\Theta}}^*, {\boldsymbol{w}}^*)+\eta$.
Theorem \[slln\] establishes a uniform SLLN, which plays a key role in the proof of the main result (Theorem \[main theorem\]).
\[slln\] (SLLN) Fix $s \leq 1$. Let $\mathcal{G}$ denote the family of functions $g_{{\boldsymbol{\Theta}},{\boldsymbol{w}}}({\boldsymbol{x}})=\mathcal{M}_s({\boldsymbol{x}},{\boldsymbol{\Theta}},{\boldsymbol{w}})$. Then $\sup_{g \in \mathcal{G}}|\int g dP_n -\int g dP|\to 0$ a.s. $[P]$.
Fix $\epsilon >0$. It is enough to find a finite family of functions $\mathcal{G}_\epsilon$ such that for all $g \in \mathcal{G}$, there exists $\Bar{g},\Dot{g} \in \mathcal{G}_\epsilon$ such that $\Dot{g} \leq g \leq \Bar{g}$ and $\int(\Bar{g}-\Dot{g})dP<\epsilon$.
Let us define $\phi(\cdot): \mathbb{R} \to \mathbb{R}$ such that $\phi(x)=\max\{0,x\}$. Since $C$ is compact, for every $\delta_1>0$, we can always construct a finite set $C_{\delta_1} \subset C$ such that if ${\boldsymbol{\theta}}\in C$, there exist ${\boldsymbol{\theta}}' \in C_{\delta_1}$ such that $\|{\boldsymbol{\theta}}-{\boldsymbol{\theta}}'\|<\delta_1$. Similarly, resorting to the compactness of $[0,1]^p$, for every $\delta_2>0$, we can always construct a finite set $W_{\delta_2} \subset [0,1]^p$ such that if ${\boldsymbol{w}}\in [0,1]^p$, there exist ${\boldsymbol{w}}' \in W_{\delta_2}$ such that $\|{\boldsymbol{w}}-{\boldsymbol{w}}'\|<\delta_2$. Consider the function $h({\boldsymbol{x}},{\boldsymbol{\Theta}},{\boldsymbol{w}})=M_s(\|{\boldsymbol{x}}-{\boldsymbol{\theta}}_1\|^2_{{\boldsymbol{w}}},\dots,\|{\boldsymbol{x}}-{\boldsymbol{\theta}}_k\|^2_{{\boldsymbol{w}}})$ on $C\times C^k \times [0,1]^p$. $h$, being continuous on the compact set $C\times C^k \times [0,1]^p$, is also uniformly continuous. Thus for all $\mathbf{x} \in C$, if $\|{\boldsymbol{w}}-{\boldsymbol{w}}'\|<\delta_2$ and $\|{\boldsymbol{\theta}}_j-{\boldsymbol{\theta}}'_j\|<\delta_1$ for all $j=1,\dots,k$ implies that $$\begin{aligned}
\label{eqq1}
\bigg|M_s(\|{\boldsymbol{x}}-{\boldsymbol{\theta}}_1\|^2_{{\boldsymbol{w}}},\dots,\|{\boldsymbol{x}}-{\boldsymbol{\theta}}_k\|^2_{{\boldsymbol{w}}}) -M_s(\|{\boldsymbol{x}}-{\boldsymbol{\theta}}_1'\|^2_{{\boldsymbol{w}}'},\dots,\|{\boldsymbol{x}}-{\boldsymbol{\theta}}'_k\|^2_{{\boldsymbol{w}}'})\bigg|< \epsilon/2\end{aligned}$$
We take $$\begin{aligned}
\mathcal{G}_\epsilon & =\{\phi(M_s(\|{\boldsymbol{x}}-{\boldsymbol{\theta}}_1'\|^2_{{\boldsymbol{w}}'},\dots,\|{\boldsymbol{x}}-{\boldsymbol{\theta}}'_k\|^2_{{\boldsymbol{w}}'})\pm \epsilon/2)\\
&: {\boldsymbol{\theta}}'_1,\dots,{\boldsymbol{\theta}}'_k \in C_{\delta_1} \text{ and } {\boldsymbol{w}}' \in W_{\delta_2}\}. \end{aligned}$$ Now if we take
$$\Bar{g}_{{\boldsymbol{\theta}},{\boldsymbol{w}}}=\phi(M_s(\|{\boldsymbol{x}}-{\boldsymbol{\theta}}_1'\|^2_{{\boldsymbol{w}}'},\dots,\|{\boldsymbol{x}}-{\boldsymbol{\theta}}'_k\|^2_{{\boldsymbol{w}}'})+ \epsilon/2)$$
and $$\Dot{g}_{{\boldsymbol{\theta}},{\boldsymbol{w}}}=\phi(M_s(\|{\boldsymbol{x}}-{\boldsymbol{\theta}}_1'\|^2_{{\boldsymbol{w}}'},\dots,\|{\boldsymbol{x}}-{\boldsymbol{\theta}}'_k\|^2_{{\boldsymbol{w}}'})- \epsilon/2),$$ where ${\boldsymbol{\theta}}'_j \in C_{\delta_1}$ and ${\boldsymbol{w}}\in W_{\delta_2}$ for $j=1,\dots,k$ such that $\|{\boldsymbol{\theta}}_j-{\boldsymbol{\theta}}'_j\|<\delta_1$ and $\|{\boldsymbol{w}}-{\boldsymbol{w}}'\|<\delta_2$. From equation (\[eqq1\]), we get, $\Dot{g} \leq g \leq \Bar{g}$. Now we need to show $\int(\Bar{g}-\Dot{g})dP<\epsilon$. This step is straight forward. $$\begin{aligned}
&\int(\Bar{g}-\Dot{g})dP\\
&= \bigg[ \phi(M_s(\|{\boldsymbol{x}}-{\boldsymbol{\theta}}_1'\|^2_{{\boldsymbol{w}}'},\dots,\|{\boldsymbol{x}}-{\boldsymbol{\theta}}'_k\|^2_{{\boldsymbol{w}}'})+ \epsilon/2)\\
& - \phi(M_s(\|{\boldsymbol{x}}-{\boldsymbol{\theta}}_1'\|^2_{{\boldsymbol{w}}'},\dots,\|{\boldsymbol{x}}-{\boldsymbol{\theta}}'_k\|^2_{{\boldsymbol{w}}'})- \epsilon/2)\bigg] dP\\
& \leq \epsilon \int dP = \epsilon.\end{aligned}$$ Hence the result.
We are now ready to establish the main consistency result, stated and proven below.
\[main theorem\] Under the condition A1, ${\boldsymbol{\Theta}}_{n,s} \xrightarrow{a.s.} {\boldsymbol{\Theta}}^* $ and ${\boldsymbol{w}}_{n,s} \xrightarrow{a.s.} {\boldsymbol{w}}^*$ as $n \to \infty$ and $s \to -\infty$.
It is enough to show that given any neighbourhood $N$ of $({\boldsymbol{\Theta}}^*,{\boldsymbol{w}}^*)$, there exists $M_1<0$ and $M_2>0$ such that if $s<M_1$ and $n>M_2$ such that $({\boldsymbol{\Theta}}, {\boldsymbol{w}}) \in N$ almost surely. By assumption A1, it is enough to show that for all $\eta>0$, there exists $M_1<0$ and $M_2>0$ such that if $s<M_1$ and $n>M_2$ such that $\Phi({\boldsymbol{\Theta}}, {\boldsymbol{w}})\leq \Phi({\boldsymbol{\Theta}}^*, {\boldsymbol{w}}^*)+\eta$ almost surely. For notational convenience, we write $\mathcal{M}_s({\boldsymbol{x}},{\boldsymbol{\Theta}},{\boldsymbol{w}}$) for $M_s(\|{\boldsymbol{x}}-{\boldsymbol{\theta}}_1\|^2_{{\boldsymbol{w}}},\dots,\|{\boldsymbol{x}}-{\boldsymbol{\theta}}_k\|^2_{{\boldsymbol{w}}})$ and $\alpha({\boldsymbol{w}})=\lambda \sum_{l=1}^pw_l \log w_l$. Now since $({\boldsymbol{\Theta}}_{n,s},{\boldsymbol{w}}_{n,s})$ is the minimizer for $\int \mathcal{M}_s({\boldsymbol{x}},{\boldsymbol{\Theta}},{\boldsymbol{w}})dP_n + \lambda \sum_{l=1}^pw_l \log w_l$, we get, $$\begin{aligned}
\label{eqq2}
&\int \mathcal{M}_s({\boldsymbol{x}},{\boldsymbol{\Theta}}_{n,s},{\boldsymbol{w}}_{n,s})dP_n +\lambda \alpha({\boldsymbol{w}}_{n,s}) \nonumber \\
& \leq \int \mathcal{M}_s({\boldsymbol{x}},{\boldsymbol{\Theta}}^*,{\boldsymbol{w}}^*)dP_n + \lambda \alpha({\boldsymbol{w}}^*).\end{aligned}$$ Now observe that $\Phi({\boldsymbol{\Theta}}_{n,s},{\boldsymbol{w}}_{n,s})-\Phi({\boldsymbol{\Theta}}^*,{\boldsymbol{w}}^*)=\xi_1+\xi_2+\xi_3$, where, $$\begin{aligned}
\xi_1 & =\Phi({\boldsymbol{\Theta}}_{n,s},{\boldsymbol{w}}_{n,s})-\int \mathcal{M}_s({\boldsymbol{x}},{\boldsymbol{\Theta}}_{n,s},{\boldsymbol{w}}_{n,s})dP-\lambda \alpha({\boldsymbol{w}}_{n,s}),\\
\xi_2 & = \int \mathcal{M}_s({\boldsymbol{x}},{\boldsymbol{\Theta}}_{n,s},{\boldsymbol{w}}_{n,s})dP-\int \mathcal{M}_s({\boldsymbol{x}},{\boldsymbol{\Theta}}_{n,s},{\boldsymbol{w}}_{n,s})dP_n,\\
\xi_3 & = \int \mathcal{M}_s({\boldsymbol{x}},{\boldsymbol{\Theta}}_{n,s},{\boldsymbol{w}}_{n,s})dP_n+\lambda \alpha({\boldsymbol{w}}_{n,s})- \Phi({\boldsymbol{\Theta}}^*,{\boldsymbol{w}}^*).\end{aligned}$$ We first choose $M_1<0$ such that if $s<M_1$ then $$\label{eqq3}
\bigg|\min_{1 \leq j \leq k} \|{\boldsymbol{x}}-{\boldsymbol{\theta}}_j\|_{{\boldsymbol{w}}}-\mathcal{M}_s({\boldsymbol{x}},{\boldsymbol{\Theta}},{\boldsymbol{w}})\bigg|<\eta/6$$ for all ${\boldsymbol{x}}\in C$, ${\boldsymbol{\Theta}}\in C^k$ and ${\boldsymbol{w}}\in [0,1]^p$. Thus for $s<M_1$, $\min_{1 \leq j \leq k} \|{\boldsymbol{x}}-{\boldsymbol{\theta}}_j\|_{{\boldsymbol{w}}}\leq \mathcal{M}_s({\boldsymbol{x}},{\boldsymbol{\Theta}},{\boldsymbol{w}})+\eta/6$ which in turn implies that $ \int\min_{1 \leq j \leq k} \|{\boldsymbol{x}}-{\boldsymbol{\theta}}_j\|_{{\boldsymbol{w}}}dP_n\leq \int\mathcal{M}_s({\boldsymbol{x}},{\boldsymbol{\Theta}},{\boldsymbol{w}})dP_n+\eta/3$. Substituting ${\boldsymbol{\Theta}}_{n,s}$ for $\Theta$ and ${\boldsymbol{w}}_{n,s}$ for ${\boldsymbol{w}}$ in the above expression and adding $\lambda \alpha({\boldsymbol{w}}_{n,s})$ to both sides, we get $\xi_1<\eta/6$. We also observe that the quantity $\xi_2$ can also be made smaller that $\eta/3$ by appealing to the uniform SLLN (Theorem 3). Now to bound $\xi_3$, we observe that $$\begin{aligned}
\xi_3 & \leq \int \mathcal{M}_s({\boldsymbol{x}},{\boldsymbol{\Theta}}^*,{\boldsymbol{w}}^*)dP_n+\lambda \alpha({\boldsymbol{w}}^*)- \Phi({\boldsymbol{\Theta}}^*,{\boldsymbol{w}}^*)\\
& = \int \mathcal{M}_s({\boldsymbol{x}},{\boldsymbol{\Theta}}^*,{\boldsymbol{w}}^*)dP_n - \int \min_{{\boldsymbol{\theta}}\in {\boldsymbol{\Theta}}^*} \|{\boldsymbol{x}}-{\boldsymbol{\theta}}\|_{{\boldsymbol{w}}^*}dP\end{aligned}$$ This inequality is obtained by appealing to equation (\[eqq2\]). Again appealing to the uniform SLLN, we get that for large enough $n$, $$\begin{aligned}
\xi_3 & \leq \int \mathcal{M}_s({\boldsymbol{x}},{\boldsymbol{\Theta}}^*,{\boldsymbol{w}}^*)dP - \int \min_{{\boldsymbol{\theta}}\in {\boldsymbol{\Theta}}^*}\|{\boldsymbol{x}}-{\boldsymbol{\theta}}\|_{{\boldsymbol{w}}^*}dP+\eta/6\\
& \leq \int [\min_{{\boldsymbol{\theta}}\in {\boldsymbol{\Theta}}^*}\|{\boldsymbol{x}}-{\boldsymbol{\theta}}\|_{{\boldsymbol{w}}^*}+\eta/6]dP - \int \min_{{\boldsymbol{\theta}}\in {\boldsymbol{\Theta}}^*}\|{\boldsymbol{x}}-{\boldsymbol{\theta}}\|_{{\boldsymbol{w}}^*}dP\\
& + \eta/6= \eta/3.\end{aligned}$$ The second inequality follows from equation (\[eqq3\]). Thus we get, $\Phi({\boldsymbol{\Theta}}_{n,s},{\boldsymbol{w}}_{n,s})-\Phi({\boldsymbol{\Theta}}^*,{\boldsymbol{w}}^*)=\xi_1+\xi_2+\xi_3 \leq \eta/6+\eta/3+\eta/3 < \eta$ almost surely. The result now follows.
Empirical Performance {#experiment}
=====================
We examine the performance of EWP on a variety of simulated and real datasets compared to classical and state-of-the-art peer algorithms. All the datasets and the codes pertaining to this paper are publicly available at <https://github.com/DebolinaPaul/EWP>. For evaluation purposes, we use the Normalized Mutual Information (NMI) [@vinh2010information] between the ground-truth partition and the partition obtained by each algorithm. A value of 1 indicates perfect clustering and a value of 0 indicates arbitrary labels. As our algorithm is meant to perform as a drop-in replacement to $k$-means, we focus comparisons to Lloyd’s classic algorithm [@lloyd1982least], $WK$-means [@huang2005automated], Power $k$-means [@xu2019power] and sparse $k$-means [@witten2010framework]. It should be noted that sparse $k$-means already entails higher computational complexity, and we do not exhaustively consider alternate methods which require orders of magnitude of higher complexity. In all cases, each algorithm is initiated with the same set of randomly chosen centroids.
Synthetic Experiments {#simulation}
---------------------
We now consider a suite of simulation studies to validate the proposed EWP algorithm.
#### Simulation 1 {#simulation 1}
[0.34]{} ![Solutions obtained by the peer algorithms for an example dataset with $k=100$ and $d=20$ in Simulation 1 (\[simulation 1\]). The obtained cluster centroids appear as black diamonds in the figure. []{data-label="fig: sim1"}](images/l2.pdf "fig:"){height="\textwidth" width="\textwidth"}
[0.34]{} ![Solutions obtained by the peer algorithms for an example dataset with $k=100$ and $d=20$ in Simulation 1 (\[simulation 1\]). The obtained cluster centroids appear as black diamonds in the figure. []{data-label="fig: sim1"}](images/l3.pdf "fig:"){height="\textwidth" width="\textwidth"}
[0.34]{} ![Solutions obtained by the peer algorithms for an example dataset with $k=100$ and $d=20$ in Simulation 1 (\[simulation 1\]). The obtained cluster centroids appear as black diamonds in the figure. []{data-label="fig: sim1"}](images/l4.pdf "fig:"){height="\textwidth" width="\textwidth"}
[0.34]{} ![Solutions obtained by the peer algorithms for an example dataset with $k=100$ and $d=20$ in Simulation 1 (\[simulation 1\]). The obtained cluster centroids appear as black diamonds in the figure. []{data-label="fig: sim1"}](images/l5.pdf "fig:"){height="\textwidth" width="\textwidth"}
[0.34]{} ![Solutions obtained by the peer algorithms for an example dataset with $k=100$ and $d=20$ in Simulation 1 (\[simulation 1\]). The obtained cluster centroids appear as black diamonds in the figure. []{data-label="fig: sim1"}](images/l1.pdf "fig:"){height="\textwidth" width="\textwidth"}
The first experiment assesses performance as the dimension and number of uninformative features grows. We generate $n=1000$ observations with $k=100$ clusters. Each observation has $p=d+2$ many features as $d$ varies between $5$ and $100$. The first two features reveal cluster structure, while the remaining $d$ variables are uninformative, generated independently from a $Unif(0,2)$ distribution. True centroids are spaced uniformly on a grid with $\theta_m=\frac{m-1}{10}$, and $x_{ij} \sim \frac{1}{10}\sum_{m=1}^{10} \mathcal{N}(\theta_m,0.15)$. Despite the simple data generating setup, clustering is difficult due to the low signal to noise ratio in this setting.
We report the average NMI values between the ground-truth partition and the partition obtained by each of the algorithms over $20$ trials in Table \[tab:s1\], with the standard deviations appearing in parentheses. The best performing algorithm in each column appears in bold, and the best solutions for $d=20$ are plotted in Figure \[fig: sim1\]. The benefits using EWP are visually stark, and Table \[tab:s1\] verifies in detail that EWP outperforms the classical $k$-means algorithm as well as the state-of-the-art sparse-$k$-means and the power $k$-means algorithms. The inability of $k$-means and Power $k$-means to properly learn the feature weights results in poor performance of these algorithms. On the other hand, although $WK$-means and sparse $k$-means can select features successfully, they fail from the optimization perspective when $k$ is large enough so that there are many local minima to trap the algorithm.
$d=5$ $d=10$ $d=20$ $d=50$ $d=100$
------------------ -------------------- -------------------- -------------------- -------------------- --------------------
$k$-means 0.3913 (0.002) 0.3701 (0.002) 0.3674 (0.003) 0.3629(0.002) 0.3517 (0.003)
$WK$-means 0.5144(0.002) 0.50446(0.003) 0.5050(0.003) 0.5026(0.005) 0.5029(0.003)
Power $k$-means 0.3924(0.001) 0.3873(0.002) 0.3722 (0.001) 0.3967 (0.003) 0.3871 (0.004)
Sparse $k$-means 0.3679 (0.002) 0.3677 (0.002) 0.3668 (0.001) 0.3675 (0.002) 0.3637 (0.002)
EWP-$k$-means **0.9641** (0.001) **0.9217** (0.001) **0.9139** (0.001) **0.9465** (0.001) **0.9082** (0.003)
: NMI values for Simulation 1, showing the effect of the number of unimportant features.[]{data-label="tab:s1"}
#### Simulation 2 {#set2}
We next examine the effect of $k$ on the performance, taking $n=100 \cdot k$ and $p=100$ while $k$ varies from $20$ to $500$. The matrix $\Theta_{k \times p}$, whose rows contain the cluster centroids, is generated as follows.
1. Select $5$ relevant features $l_1,\dots,l_5$ at random.
2. Simulate $\theta_{j,l_m} \sim Unif(0,1)$ for all $j=1,\dots,k$ and $m=1,\dots,5$.
3. Set $\theta_{j,l}=0$ for all $l \not\in \{l_1,\dots,l_5\}$ and all $j$.
After obtaining $\Theta$, $x_{il}$ is simulated as follows. $$\begin{aligned}
x_{il}\sim& \mathcal{N}(0,1) \text{ if } l \not\in \{l_1,\dots,l_5\}\\
x_{il}\sim& \frac{1}{k}\sum_{j=1}^k\mathcal{N}(\theta_{j,l},0.015) \text{ if } l \in \{l_1,\dots,l_5\}.\end{aligned}$$ We run each of the algorithms 20 times and report the average NMI values between the ground-truth partition an the partition obtained by each of the algorithms in Table \[tab:s2\]; with standard errors appearing in parentheses. Table \[tab:s2\] shows that $k$-means, $WK$-means, power $k$-means, and sparse $k$-means lead to almost the same result, while EWP outperforms all the peer algorithms for each $k$ as it narrows down the large number of features and avoids local minima from large $k$ simultaneously.
$k=20$ $k=100$ $k=200$ $k=500$
------------------ ------------------- -------------------- ------------------- --------------------
$k$-means 0.0674(0.001) 0.2502(0.021) 0.3399 (0.031) 0.3559 (0.014)
$WK$-means 0.0587(0.001) 0.2247(0.002) 0.3584(0.018) 0.3678(0.009)
Power $k$-means 0.0681(0.001) 0.2785(0.001) 0.3578 (0.002) 0.3867(0.001)
Sparse $k$-means 0.0679(0.001) 0.2490(0.058) 0.6705(0.007) 0.3537 (0.002)
EWP-$k$-means **0.9887**(0.001) **0.9844** (0.002) **0.9756**(0.001) **0.9908** (0.001)
: NMI values for Simulation 2, showing the effect of increasing number of clusters.[]{data-label="tab:s2"}
#### Feature Selection
We now examine the feature weighting properties of the EWP algorithm more closely. We take $n=1000$, $p=20$ and follow the same data generation procedure described in Simulation 2. For simplicity, in the first step of the simulation study, we select $l_i=i$ for $i=1,\dots,5$. We record the feature weights obtained by EWP, sparse $k$-means and $WK$-means over $100$ replicate datasets. The box-plot for these 100 optimal feature weights are shown in Figure \[boxplot\] for all the three algorithms. The proposed method successfully assigns almost all weight to relevant features 1 through 5, even though it does not make use of a sparsity-inducing penalty. Meanwhile, feature weights assigned by sparse $k$-means do not follow any clear pattern related to informative features, even though the ground truth is sparse in relevant features. The analogous plot for $WK$-means shows even worse performance than sparse $k$-means. The study clearly illustrates the necessity of feature weighing together with annealing for successful $k$-means clustering in high dimensional settings.
[0.34]{} ![Boxplots show that EWP consistently identifies true features while sparse $k$-means fails to do so. []{data-label="boxplot"}](images/box_e.pdf "fig:"){height="\textwidth" width="\textwidth"}
[0.34]{} ![Boxplots show that EWP consistently identifies true features while sparse $k$-means fails to do so. []{data-label="boxplot"}](images/box_sp.pdf "fig:"){height="\textwidth" width="\textwidth"}
[0.34]{} ![Boxplots show that EWP consistently identifies true features while sparse $k$-means fails to do so. []{data-label="boxplot"}](images/box_wk.pdf "fig:"){height="\textwidth" width="\textwidth"}
[0.34]{} ![`t-SNE` plots for the GLIOMA dataset, color-coded by the partitioning obtained at convergence by each peer algorithm.[]{data-label="tsne_glioma"}](images/gli_gt.pdf "fig:"){height="\textwidth" width="\textwidth"}
[0.34]{} ![`t-SNE` plots for the GLIOMA dataset, color-coded by the partitioning obtained at convergence by each peer algorithm.[]{data-label="tsne_glioma"}](images/gli_k.pdf "fig:"){height="\textwidth" width="\textwidth"}
[0.34]{} ![`t-SNE` plots for the GLIOMA dataset, color-coded by the partitioning obtained at convergence by each peer algorithm.[]{data-label="tsne_glioma"}](images/gli_wk.pdf "fig:"){height="\textwidth" width="\textwidth"}
[0.34]{} ![`t-SNE` plots for the GLIOMA dataset, color-coded by the partitioning obtained at convergence by each peer algorithm.[]{data-label="tsne_glioma"}](images/gli_pk.pdf "fig:"){height="\textwidth" width="\textwidth"}
[0.34]{} ![`t-SNE` plots for the GLIOMA dataset, color-coded by the partitioning obtained at convergence by each peer algorithm.[]{data-label="tsne_glioma"}](images/gli_sk.pdf "fig:"){height="\textwidth" width="\textwidth"}
[0.34]{} ![`t-SNE` plots for the GLIOMA dataset, color-coded by the partitioning obtained at convergence by each peer algorithm.[]{data-label="tsne_glioma"}](images/gli_ewp.pdf "fig:"){height="\textwidth" width="\textwidth"}
Case Study and Real Data {#glioma}
------------------------
We now assess performance on real data, beginning with a case study on Glioma. The GLIOMA dataset consists of 50 datapoints and is divided into 4 classes consisting of cancer glioblastomas (CG), noncancer glioblastomas (NG), cancer oligodendrogliomas (CO) and non-cancer oligodendrogliomas (NO). Each observation consists of 4434 features. The data were collected in the study by [@nutt2003gene], and are also available by [@li2018feature].In our experimental studies, we compare the EWP algorithm to the four peer algorithms considered in Section \[simulation\]. In order to visualize clustering solutions, we embed the data into the plane via `t-SNE` [@maaten2008visualizing]. The best partitioning obtained from each algorithm is shown in Figure \[tsne\_glioma\], which makes it visually clear that clustering under EWP more closely resembles the ground truth compared to competitors. This is detailed by average NMI values as well as standard deviations in parentheses listed in Table \[tab\_glioma\].
[|C[2.3cm]{}|C[2.3cm]{}|C[2.3cm]{}|C[2.3cm]{}|C[2.3cm]{}|]{} $k$-means & $WK$-means &Power & Sparse & EWP\
0.490 (0.040) & 0.427 (0.034) & 0.499 (0.020) & 0.108 (0.001) & **0.594** (0.001)\
#### Further real-data experiments
To further validate our method in various real data scenarios, we perform a series of experiments on 10 benchmark datasets collected from the UCI machine learning repository [@Dua:2019], Keel Repository [@alcala2011keel] and ASU repository [@li2018feature]. A brief description of these datasets can be found in Table \[data\_description\].
Average performances over $20$ independent trials are reported in Table \[tab\_real\]; the EWP algorithm outperforms by a large margin across all instances when compared to the other peer algorithms. To determine the statistical significance of the results, we employ Wilcoxon’s signed-rank test [@wasserman2006all] at the 5% level of significance. In Table \[tab\_real\], an entry marked with $+$ ($\simeq$) differs from the corresponding result of EWP with statistical significance. Finally, we emphasize that our results comprise a conservative comparison in that parameters $s_0=-1$ and $\eta=1.05$ are fixed across *all* settings. While this demonstrates that careful tuning of these parameters is not necessary for successful clustering, performance can be further improved by doing so [@xu2019power],
[|p[2.8cm]{}|C[2.8cm]{}|C[0.5cm]{}|C[0.7cm]{}|C[0.7cm]{}|]{} **Datasets** & **Source** & $\mathbf{k}$ & $\mathbf{n}$ & $\mathbf{p}$\
Iris & [Keel Repository](https://sci2s.ugr.es/keel/category.php?cat=clas) & 3 & 150 & 4\
Automobile & [Keel Repository](https://sci2s.ugr.es/keel/category.php?cat=clas) & 6 & 150 & 25\
Mammographic & [Keel Repository](https://sci2s.ugr.es/keel/category.php?cat=clas) & 2 & 830 & 5\
Newthyroid & [Keel Repository](https://sci2s.ugr.es/keel/category.php?cat=clas) & 3 & 215 & 5\
Wine & [Keel Repository](https://sci2s.ugr.es/keel/category.php?cat=clas) & 3 & 178 & 13\
WDBC & [Keel Repository](https://sci2s.ugr.es/keel/category.php?cat=clas) & 2 & 569 & 30\
Movement Libras & [Keel Repository](https://sci2s.ugr.es/keel/category.php?cat=clas) & 15 & 360 & 90\
Wall Robot 4 & [UCI Repository](https://archive.ics.uci.edu/ml/datasets.php) & 4 & 5456 & 4\
WarpAR10P & [ASU Repository](http://featureselection.asu.edu/datasets.php) & 10 & 130 & 2400\
WarpPIE10P & [ASU Repository](http://featureselection.asu.edu/datasets.php) & 10 & 210 & 2420\
[|C[3cm]{}|C[2.5cm]{}|C[2.5cm]{}|C[2.5cm]{}|C[2.5cm]{}|C[2.5cm]{}|]{} Datasets & $k$-means & Power $k$-means & $WK$-means & Sparse $k$-means & EWP-$k$-means\
Newthyroid & $0.403^+(0.002)$ & $0.262^+(0.002)$ & $0.407^+(0.004)$ & $0.102^+(0.002)$ & **0.5321**(0.003)\
Automobile & $0.165^+(0.009)$ & $0.203^+(0.010)$ & $0.168^+(0.005)$ & $0.168^+(0.007)$ & **0.311**(0.003)\
WarpAR10P & $0.170^+(0.042)$ & $0.233^+(0.031)$ & $0.201^+(0.019)$ & $0.185^+(0.008)$ & **0.350**(0.047)\
WarpPIE10P & $0.240^\simeq(0.031)$ & $0.240^\simeq(0.028)$ & $0.180^+(0.022)$ & $0.179^+(0.002)$ & **0.2761**(0.041)\
Iris & $0.758^+(0.003)$ & $0.788^+(0.005)$ & $0.741^+(0.005)$ & $0.813^\simeq(0.002)$ & **0.849**(0.005)\
Wine & $0.428^+(0.001)$ & $0.642^+(0.005)$ & $0.416^+(0.002)$ & $0.428^+(0.001)$ & **0.747**(0.003)\
Mammographic & $0.107^+(0.001)$ & $0.019^+(0.003)$ & $0.115^+(0.001)$ & $0.110^+(0.002)$ & **0.405**(0.002)\
WDBC & $0.463^+(0.002)$ & $0.005^+(0.005)$ & $0.464^+(0.002)$ & $0.467^+(0.003)$ & **0.656**(0.001)\
LIBRAS & $0.553^\simeq(0.017)$ & $0.339^+(0.020)$ & $0.461^+(0.021)$ & $0.254^+(0.014)$ & **0.575**(0.009)\
Wall Robot 4 & $0.167^+(0.027)$ & $0.183^+(0.013)$ & $0.171^+(0.030)$ & $0.186^+(0.012)$ & **0.234**(0.003)\
Discussion
==========
Despite decades of advancement on $k$-means clustering, Lloyd’s algorithm remains the most popular choice in spite of its well-known drawbacks. Extensions and variants that address these flaws fail to preserve its simplicity, scalability, and ease of use. Many of these methods still fall short at poor local optima or fail when data are high-dimensional with low signal-to-noise ratio, and few come with rigorous statistical guarantees such as consistency.
The contributions in this paper seek to fill this methodological gap, with a novel formulation that draws from good intuition in classic and recent developments. With emphasis on simplicity as a chief priority, we derive a method that can be seen as a drop-in replacement to Lloyd’s classic $k$-means algorithm, reaping large improvements in practice even when there are a large number of clusters or features in the data. By designing the algorithm from the perspective of MM, our method is robust as a descent algorithm and achieves an ideal $\mathcal{O}(nkp)$ complexity. In contrast to popular approaches such as sparse $k$-means and power $k$-means, the proposed approach is provably consistent. Extending the intuition to robust measures and other divergences in place of the Euclidean distance are warranted. Further, research toward finite-sample prediction error bounds or convergence rates relating to the annealing schedule will also be fruitful avenues for future work.
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[^1]: Joint first authors contributed equally to this work
[^2]: Corresponding author: jason.q.xu@duke.edu
|
---
abstract: 'In this paper we investigate the formation of Uranus and Neptune, according to the core-nucleated accretion model, considering formation locations ranging from 12 to 30 AU from the Sun, and with various disk solid-surface densities and core accretion rates. It is shown that in order to form Uranus-like and Neptune-like planets in terms of final mass [*and*]{} solid-to-gas ratio, very specific conditions are required. We also show that when recently proposed high solid accretion rates are assumed, along with solid surface densities about 10 times those in the minimum-mass solar nebula, the challenge in forming Uranus and Neptune at large radial distances is no longer the formation timescale, but is rather finding agreement with the final mass and composition of these planets. In fact, these conditions are more likely to lead to gas-giant planets. Scattering of planetesimals by the forming planetary core is found to be an important effect at the larger distances. Our study emphasizes how (even slightly) different conditions in the protoplanetary disk and the birth environment of the planetary embryos can lead to the formation of very different planets in terms of final masses and compositions (solid-to-gas ratios), which naturally explains the large diversity of intermediate-mass exoplanets.'
author:
- Ravit Helled$^1$ and Peter Bodenheimer$^2$
title: |
The Formation of Uranus & Neptune:\
Challenges and Implications For Intermediate-Mass Exoplanets
---
Introduction
============
The increasing number of detected exoplanets with masses similar to those of Uranus and Neptune emphasizes the need to better understand the formation process of the planetary class consisting of planets with rock-ice cores of up to $\approx$ 15 M$_\oplus$ and lower-mass hydrogen/helium envelopes, with a large range of solid-to-gas ratios. However, the formation mechanism for intermediate-mass planets is not well understood (e.g., Rogers et al. 2011), and even within our own solar system there are many open questions regarding the formation of Uranus and Neptune.
Uranus and Neptune have masses of about 14.5 and 17 M$_{\oplus}$, and are located at 19.2 and 30 AU, respectively. Their exact compositions are not known (e.g., Helled et al. 2011), but they are likely to consist mainly of rock and ices with smaller mass fractions of hydrogen-helium atmospheres (see Fortney & Nettelmann 2010; Nettelmann et al. 2013 and references therein). The estimated solid-to-gas mass ratios of Uranus and Neptune range between 2.3 and 19 (Guillot 2005; Helled et al. 2011). Hereafter when we refer to the solid-to-gas ratio it should be clear that the solid component consists of all elements heavier than hydrogen and helium, and that in fact, their physical state can differ from a solid state, while the gas corresponds solely to hydrogen and helium. It is commonly assumed that Uranus and Neptune have formed by the core accretion scenario, in which solid core formation accompanied by slow gas accretion is followed by more rapid gas accretion (e.g., Pollack et al. 1996). During the initial phases of planet formation, the slow accretion of gas is controlled by the growth rate of its core. As a result, the solid accretion rate essentially determines the formation timescale of an intermediate-mass planet (see D’Angelo et al. 2011 for a review). The (standard) solid accretion rate is given by (Safronov 1969): $$\dot M_\mathrm{core} = {\frac{dM_\mathrm{solid}}{dt}} =
\pi R_\mathrm{capt}^2
\sigma_s \Omega F_g,
\label{eq:accrete}$$ where $\pi R_\mathrm{capt}^2$ is the capture cross section for planetesimals, $\Omega$ is the orbital frequency, $\sigma_s$ is the solid surface density in the disk, and $F_g$ is the gravitational enhancement factor. As can be seen from Equation (1), the core accretion rate decreases with increasing radial distance, and as a result, the core formation timescale can be extremely long at radial distances larger than $\sim$ 10 AU. Using the model of the minimum mass solar nebula (MMSN, see Weidenschilling 1977 for details), the formation timescales for Uranus and Neptune exceed $10^9$ years (Safronov 1969). This estimate has introduced the [*formation timescale problem of Uranus and Neptune*]{}.
A detailed investigation of the formation of Uranus has been presented in Pollack et al. (1996). The authors have considered [*in situ*]{} formation and have shown that for $\sigma_s$ about twice that of the MMSN the approximate core mass and envelope mass of Uranus are reached in about 16 Myr. The timescale is considerably shorter than that of Safronov primarily because of the use of improved (and much higher) values for $F_g$. The core accretion rate, however, depends not only on $\sigma_s$ and $F_g$ but also on the sizes of the accreted planetesimals. Smaller planetesimals can be accreted more easily, and it was shown by Pollack et al. (1996) that when planetesimals are assumed to have sizes of 1 km (instead of the standard 100 km) the formation timescale of Uranus decreases to $\approx$ 2 Myr. This timescale is within the estimated range of lifetimes of gaseous protoplanetary disks but has been considered unrealistically short because of the use of a simplified core accretion rate in the model. Clearly, the formation timescale for Neptune under similar assumptions would be significantly longer. Due to the long accretion times at large radial distances, Uranus and Neptune are often referred to as “failed giant planets", because their formation process was too slow to reach runaway gas accretion before the disk gas had dissipated.
Goldreich et al. (2004) discuss in detail the problem of the accumulation of solid particles, particularly at large distances from the central star. Without considering the effects of the gas, they suggest that to form Uranus- or Neptune-like planets [*in situ*]{}, one requires, first, relatively small planetesimals, $< 1$ km in radius, and second, a value of $\sigma_s$ a few times that of the MMSN. In fact, they conclude that in order to form the planets within the lifetime of the gas disk, particles of only a few cm in size are needed. The small particles would be generated by collisions between the km-size objects. Numerous collisions among the small particles strongly damp the particle random velocities, resulting in a very cold disk. However, Levison & Morbidelli (2007), on the basis of N-body simulations, point out that this model is oversimplified and relies on numerous assumptions. They suggest that the main difficulty is the assumption that the surface density of the disk particles remains smooth and uniform. In fact the simulations show that the formation of rings and gaps actually dominates the dynamics.
The idea that the solar system was originally much more compact and that Uranus and Neptune were formed at smaller radial distances has been considered by a number of authors (e.g., Thommes et al. 1999; Tsiganis et al. 2005). The planets must arrive at their present locations post-formation, by gravitational scattering or by migration induced by a disk of planetesimals. The success of the “Nice Model” to explain many of the observed properties of the solar system led Dodson-Robinson & Bodenheimer (2010) to investigate the formation of Uranus and Neptune at radial distances of 12 and 15 AU, as suggested by that model. They adopted a disk model which accounts for disk evolution and disk chemistry (Dodson-Robinson et al. 2009), giving values of $\sigma_s$ at those distances an order of magnitude higher than in the MMSN. The planet-formation calculation was similar to that of Pollack et al. (1996). It was found that the formation timescales of both Uranus and Neptune fell in the range 4–6 Myr, and that in some cases the solid-to-gas ratio was similar to those in the present planets. In addition, the results are consistent with the observed carbon enhancement in the atmospheres of these planets. It was therefore concluded that, indeed, Uranus and Neptune could have formed at smaller radial distances as implied by the Nice model. There are, however, unsolved issues with this formation scenario— for example, there is no way to distinguish Uranus from Neptune, and more importantly, there must be a cutoff of both solid and gas accretion when the planets reach their current masses; otherwise the model predicts that they would continue to accrete to higher mass.
While the scenario in which Uranus and Neptune form at smaller radial distances is feasible and somewhat promising, there is no evidence for ruling out the possibility that these planets, and in particular extrasolar planets, could have formed at larger distances. Recent studies on the accretion rate of solids have provided new estimates for the rates in which solids are accreted to a planetary embryo in the core accretion paradigm (Rafikov 2011; Lambrechts & Johansen 2012). These studies suggest that the solid accretion rates can be significantly higher than previous estimates. With these high accretion rates the core formation timescale at large radial distances is significantly reduced. Although [*in situ*]{} formation of Neptune is considered unlikely (see below), [*in situ*]{} formation of Uranus as well as formation of extrasolar giant planets, or Neptune-sized planets, by core accretion at large radial distances might be feasible.
The aim of this paper is to re-investigate the formation of Uranus and Neptune (as representatives of intermediate-mass planets) and, in particular, to investigate the consequences of employing the high accretion rates mentioned in the previous paragraph. We account for various accretion rates, orbital locations, and disk properties. We employ a full core accretion–gas capture model. As we discuss below, formation of planets at 20 and 30 AU is, in principle, feasible, although their characteristics may not be those of Uranus and Neptune. We also suggest that a major challenge in forming Uranus and Neptune is to derive the correct final masses and the solid-to-gas ratios. However, the sensitivity of the properties of the forming planets to the assumed parameters provides a natural explanation for the diversity of extrasolar planets in the Uranus/Neptune mass regime.
The Formation Model and Its Parameters
======================================
To model the formation of the planets we use a standard core accretion model (e.g., Dodson-Robinson & Bodenheimer 2010; Lissauer et al. 2009), which combines a given core accretion rate with a detailed model of the structure of the gaseous envelope and for the interaction of incoming planetesimals with this envelope. The planetary formation calculation in core accretion theory usually begins with a small seed body surrounded by a swarm of planetesimals. Clearly, the formation of the initial core, even if small in mass, is not instant, and in some cases the formation time of the seed core can even be comparable to the planetary formation timescale. As a result, the formation timescales derived from most of our simulations, especially when considering large radial distances and/or low solid-surface densities, should be taken as lower bounds. In the calculations presented here, the initial core mass is about 1 M$_{\oplus}$, at which point a low-mass gaseous envelope has been accreted. These planetary embryos are assumed to form at given radial distances, and migration during the formation process is neglected. In some of the cases we do account for disk evolution, by gradually reducing the gas (and therefore also the small-dust) surface density with time. The planetesimals are assumed to be of a single size, which is a free parameter in the model. As discussed below, the planetesimal’s size has some impact on the core accretion rates, but other parameters are found to be more important.
The structure of the gaseous envelope is calculated according to the standard spherically symmetric equations of stellar structure. The radiative opacities for the dust in the envelope are assumed to be reduced by approximately a factor 50 relative to standard interstellar dust opacities (e.g. Pollack et al. 1994). This reduction roughly accounts for the settling and coagulation of dust grains in the envelope (Movshovitz et al. 2010). The energy source in the envelope is provided primarily by the accretion of planetesimals, although gravitational contraction is also included. The gas accretion rate into the envelope is determined by the requirement that the outer radius of the planet ($R_\mathrm{pl}$) match the modified accretion radius (Lissauer et al. 2009): $R_\mathrm{pl}^{-1} = R_B^{-1} + 4 R_H^{-1}$ where $R_H$ is the Hill radius (see below) and $R_B = GM_\mathrm{pl}/c_s^2$, the Bondi radius. Here $M_\mathrm{pl}$ is the total mass of the planet, and $c_s$ is the sound speed in the disk.
Core Accretion Rates
--------------------
The formation of the planet is essentially determined by the growth rate of the core. However, the growth rates for solid cores in the solar nebula are unknown, and must be inferred from models. Not surprisingly, the estimates for the core accretion rates have a large range, which can lead to rather different outcomes in terms of planetary formation. In order to investigate the sensitivity of the planetary formation to the assumed core accretion rate we consider three different accretion rates. The first corresponds to a dynamically cold planetesimal disk, providing high accretion rates, and is given by (Rafikov 2011): $$\begin{aligned}
\dot M_\mathrm{core}& \approx& 6.47 \Omega p^{1/2}\sigma_s R_H^2 \nonumber \\
&\approx& 8.5 \times 10^{-4} \sigma_s M_\mathrm{core}^{2/3}\end{aligned}$$ where $M_\mathrm{core}$ is the core mass, $\Omega$ is the orbital frequency, $p=R_\mathrm{core}/R_H$, and $R_H=a(M_\mathrm{pl}/(3 M_\odot))^{1/3}$, the Hill radius, where $a$ is the distance of the planet from the star. The second expression applies if $M_\mathrm{pl} \approx M_\mathrm{core}$ and under our standard assumption that the core is a sphere of constant density $\rho_\mathrm{core}=2$ g cm$^{-3}$. This accretion rate can be taken as the maximum accretion rate of solids according to Rafikov (2011). It should be noted that in this model the sizes of planetesimals are small, since large planetesimals are assumed to fragment into small pieces that are later affected by gas drag leading to the state of a dynamically cold planetesimal disk. Therefore, when this accretion rate is considered, the calculation is more consistent when the planetesimals are assumed to be small ($<<$ 100 km).
An alternative accretion rate was recently presented by Lambrechts & Johansen (2012) who find that accretion of pebbles (cm-sized particles) within the Hill radius is given by, $$\dot M_\mathrm{core} = 2 R_H \sigma_s v_H,$$ where $v_H$ is the relative velocity between the pebbles and the core, given by $v_H=\Omega R_H$. This expression applies when the core mass is higher than the transition mass (their Eq. 33) $$M_t \approx 3 \times 10^{-3} (\Delta/0.05)^3(a/5 AU)^{0.75} M_\oplus$$ where $\Delta= \Delta u_\phi/c_s$ represents the difference between the mean orbital gas flow and a pure Keplerian orbit ($c_s$ is the disk sound velocity). At 15 AU $M_t \approx 0.055$ M$_\oplus$; thus this accretion rate is valid for all computations reported here. These accretion rates are higher by about one order of magnitude than the maximum accretion rates derived from Equation (2). In our work, we will set the high accretion rate to the one given by Equation (2) unless differently stated, and refer to it as \[dM/dt\]$_{HIGH}$.
Finally, the lower-bound accretion rate used in our model is the one derived for the “transition case” (Rafikov 2011) in which $$\dot M_\mathrm{core} \sim (6-10)p R_H \sigma_s v_H,$$ where $p<<1$. This rate refers to a “warm” planetesimal disk, with planetesimal random velocity dispersion $s_p$ on the borderline between intermediate and high values ($s_p \approx \Omega R_H$). In the actual simulations we set the numerical coefficient equal to $2 \pi$. The rate is similar to the one used by Dodson-Robinson & Bodenheimer (2010) and is slightly lower than the accretion rates used by Pollack et al. (1996). Therefore, the rate derived from Equation (5) can be considered as the low (or standard) core accretion rate; we refer to it as \[dM/dt\]$_{LOW}$. Even lower rates are of course possible.
It should be noted, however, that core accretion rates are hard to estimate, and therefore the uncertainty in this quantity is rather large. As previously mentioned, detailed N-body simulations (Levison & Morbidelli 2007) suggest that the collisional damping scenario, needed to produce a planetesimal disk with low velocity dispersion, could be unreliable. Further N-body simulations (Levison et al. 2010) show that the gravitational interactions between the embryo and the planetesimals lead to the wholesale redistribution of material and to the opening of gaps near the embryos. If the region near the growing embryo is cleared of planetesimals before much growth can occur, the core accretion rate will decrease dramatically and will prevent the formation of a giant and/or intermediate mass planet. Nevertheless, the possibility of a high solid accretion rate cannot be excluded. Various authors suggest that high core accretion rates can be maintained also over long timescales due to fragmentation, even at large radial distance (Goldreich et al. 2004; Rafikov 2011 and references therein). In view of the uncertainties, we take the approach of selecting representative accretion rates, which cover a rather wide range. Thus we can quantify the effect of the assumed core accretion rate (as well as other model assumptions) on the planetary growth.
Formation Locations
-------------------
There is no simple way to constrain the original radial distances at which planets form. This is due to the fact that planets are likely to change their positions due to interactions with the disk (e.g., Kley & Nelson 2012) or with other planets (Thommes et al. 1999). Because of the location of the methane condensation front in the primitive solar nebula (Dodson-Robinson et al. 2009), Uranus and Neptune most probably formed beyond the orbit of Saturn, but whether they formed at the distances of $\sim$ 12-20 AU, as suggested by the Nice model, is yet to be determined. Formation of Neptune [*in situ*]{} at 30 AU is generally considered unlikely, and Malhotra (1995) and Hahn & Malhotra (1999) have shown that many properties of the Oort cloud and the Kuiper belt can be explained by the outward migration of Neptune, driven by a planetesimal disk. Nevertheless we present one test calculation involving formation at that distance. For our main results, we consider three formation locations of the planets: 12, 15 and 20 AU, radial distances that are predicted by and are consistent with the formation of Uranus and Neptune in the Nice Model. It should be noted, however, that within the framework of that model other orbital locations are possible as long as a separation of at least 2 AU between Uranus and Neptune is considered.
Solid Surface Densities
-----------------------
A critical unknown property in planet formation models is $\sigma_s$, the solid surface density in the disk. This parameter is important because it determines how much material is available for the formation of a core at the location where the planet is formed. In this work we consider various possibilities for the solid-surface density. In most of the cases we consider, the lower bound for $\sigma_s$ is similar to the one derived from the MMSM model (Weidenschilling 1977). The upper bound is typically about one order of magnitude higher than the MMSN model and is consistent with the solid surface densities derived by the disk model of Dodson-Robinson et al. (2009), in which condensation of various species of ices as well as disk evolution are included. Other solid-surface densities are considered as well. The $\sigma_s$ that are used at various radial distances are listed in Table 1.
Planetesimal Sizes and Enhanced Planetesimal Capture Cross Section
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As already discussed in Pollack et al. (1996), the planetary formation timescale decreases significantly when the accreted solids are small. Similarly to other disk properties, the sizes of planetesimals in the Solar Nebula are unknown, and in fact, they are likely to vary with radial distance and time. In addition, the planetesimals are expected to have a distribution of sizes. However, for simplicity, we consider single sizes for planetesimals. To investigate the effect of the planetesimal size on the planetary growth, we consider both small (1 km) and large (100 km) planetesimals. However the accretion rates given by Equations (2) and (3) imply that planetesimal sizes are smaller than 1 km. Therefore we have run one test with 50 meter planetesimals to investigate the effect of decreasing the size. Our experiments with a range of over three orders of magnitude in planetesimal size make it clear what the effect would be if even smaller sizes were taken, as discussed below.
The accretion rates defined by Equations (2) and (5) depend on the core radius $R_\mathrm{core}$ (note that Equation (3) does not depend on this quantity.) If the forming planet consists of a heavy-element core plus even a small amount of gas bound in an envelope, the capture rate of planetesimals is enhanced by the effects of gas drag and ablation as the accreting objects pass through the envelope. We account for the enhancement of the accretion rate by replacing $R_\mathrm{core}$ by $R_\mathrm{capt}$, the effective capture radius. This quantity is determined at every time step by integration of the orbits of planetesimals as they approach the planet at various impact parameters and pass through the envelope. The details of how the procedure works are described in Pollack et al. (1996) based on earlier work by Podolak et al. (1988). The capture radius increases noticeably as the assumed planetesimal size decreases, and even for 100 km planetesimals, the ratio $R_\mathrm{capt}/
R_\mathrm{core}$ can be a factor 10 if the envelope is sufficiently massive. This enhancement effect is included in most of the calculations reported here.
Planetesimal Ejection
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Planetesimals are not only accreted by the growing planet but can also be ejected from the planet’s feeding zone. The effect of planetesimal ejection is more profound at large radial distances and is taken into account in our model. Following Ida & Lin (2004) the planetesimal ejection rate is calculated according to $$f_\mathrm{cap}=\frac{\text{accretion rate}}{\text{ejection rate}}=
\bigg(\frac{2GM_{\odot}/a}{GM_\mathrm{core}/R_\mathrm{core}}\bigg)^2$$ where $M_{\odot}$ and $M_{\text{core}}$ are the masses of the Sun and planetary core, respectively (in this case, $R_\mathrm{core}$ is the actual core radius, not an enhanced capture radius). The probability that a planetesimal will be scattered, rather than accreted, is then $1/(f_\mathrm{cap} + 1)$. As Dodson-Robinson & Bodenheimer (2010) show, this probability approaches unity when $M_\mathrm{core} \approx
10$ M$_\oplus$ and $a > 15$ AU. Thus it is an important factor in limiting the final mass of the planet. In fact the use of Equation (6) gives a lower bound on the actual scattering rate, as it takes into account only very close encounters and scatterings directly to unbound states. Future work should take scattering into account in a more detailed way.
Available Formation Timescale
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The available time for forming a giant (or an [*icy*]{}) planet is typically taken to be the lifetime of the protoplanetary (gaseous) disk. At this point the gas disk dissipates and the planetary growth is terminated, at least, in terms of the gaseous component. The lifetimes of protoplanetary disks are derived from observations and are of the order of several million years. The median disk lifetime was found to be 3 Myr, a value we adopt in our model. However, some disks can survive on much longer timescales and can exist even up to 10 Myr (Hillenbrand 2008). In addition, the lifetime of gas in planetary disks can be longer than that of the dust, which can provide additional time for gas accretion. As a result, in some cases in which the planetary growth is limited by the disk’s lifetime, we allow the longer formation timescale and investigate the sensitivity of the formation model to the assumed available formation timescale. As we present below in several cases, the timing on which the gas disappears has a major impact on the final mass of the planets, and it can determine whether the planet will become a gas giant or not.
Solid-to-Gas Ratio
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In addition to the final mass of the planets, which should be the same as the masses of Uranus and Neptune, formation models should also be able to lead to the correct solid-to-gas ratios within the planets. Although there is a fairly large uncertainty in the compositions of Uranus and Neptune (e.g., Fortney & Nettelmann 2010), there are several constraints on the fraction of hydrogen and helium which they can contain. As discussed in Guillot (2005, and references therein), under the assumption that the envelopes of the planets consist of ices and rocks, upper limits for the hydrogen and helium masses in Uranus and Neptune can be derived, and those masses are found to be $\approx$ 4.2 M$_{\oplus}$ and $\approx$ 3.2 M$_{\oplus}$, respectively. A lower bound of $\approx$ 0.5 M$_{\oplus}$ for both Uranus and Neptune is inferred under the assumption that the outer envelope is pure hydrogen and helium.
Results
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The parameters assumed for the various runs are listed in Table 1. Selected results are shown in Table 2, and the planet evolutions for most cases are plotted in Figures 1–7. The starting times for the plots in all cases are arbitrarily set at $\approx 2 \times 10^5$ yr, corresponding to the time required to build the initial core. However, the actual starting times will depend on $a$ and $\sigma_s$. For example, Equation (8) of Rafikov (2011), which does not include the effects of scattering or enhancement of the capture radius by the presence of the envelope, would give starting times of $3 \times
10^4$ yr for Run 12UN2 (low $a$, high $\sigma_s$) to $4 \times 10^6$ yr for Run 30UN1 (high $a$, low $\sigma_s$). These time estimates are based on a solid accretion rate \[dM/dt\]$_{HIGH}$.
Formation at 20 AU
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We first model the formation of a planet at radial distance of 20 AU, which corresponds to [*in situ*]{} formation for Uranus or to Neptune’s formation within the framework of the Nice model (Run 20UN1). Based on the MMSN model, at 20 AU $\sigma_s \approx $ 0.35 g cm$^{-2}$. The planetary core accretion rate is taken to be the maximum accretion rate \[dM/dt\]$_{HIGH}$ (Eq. 2), the enhanced planetesimal capture radius is taken into account with 1 km planetesimals, and planetesimal scattering is included. The planetary growth under these assumptions is shown in Figure 1. The red, blue, and black curves represent the mass of the gaseous envelope (mostly hydrogen and helium), core (heavy elements), and total planetary mass, respectively.
The upper panel shows the formation up to about 3 Myr. Under these conditions, the planetary growth is slow and the final planetary mass is 5 M$_{\oplus}$, significantly smaller than the mass of Uranus. If the gas disk dissipates at 3 Myr, the forming planet will become a “mini-Neptune” with $M_\mathrm{core} = 3.6$ M$_\oplus$ and $M_\mathrm{env} = 1.4 $ M$_\oplus$ . However, if the disk lifetime is longer the planet can continue to grow. We then continue the formation to a longer timescale (see middle panel). As can be seen from the figure, at 8 Myr the planet can reach a mass exceeding 60 M$_{\oplus}$. Clearly, in this case the lifetime of the disk plays a crucial role in the formation process. The calculation illustrates the point that even with $M_\mathrm{core} < 4$ M$_\oplus$ the planet can reach crossover mass ($M_\mathrm{core} = M_\mathrm{env}$), given enough time. Crossover is actually obtained at 5.14 Myr at $M_\mathrm{core}=3.84 $ M$_\oplus$.
Finally, we search for the timescale for the formation of a planet with Uranus’s mass under these conditions. The result is shown in the bottom panel of the figure. To reach the mass of Uranus the gas disk has to dissipate at $\approx$ 7 Myr, and at $\approx$ 7.3 Myr in order to form a planet with Neptune’s mass. It should be noted, however, that the final composition of the planets, assuming gas dissipation at about 7 Myr, is significantly different from that of Uranus/Neptune. The heavy element mass is only 5 M$_{\oplus}$, while the rest is gas. Therefore, the forming planets will become something that is more like a “Mini-Saturn” rather than a Uranus/Neptune-like object.
As discussed in Dodson-Robinson & Bodenheimer (2010), the actual densities of the disk can be much higher than the ones derived from the MMSN model. In fact, the disk model of Dodson-Robinson et al. (2009) suggests that in the outer region of the disk, the solid surface densities are higher by a factor of 10. We therefore repeat the calculation at 20 AU (Run 20 UN2) but this time assume $\sigma_s=3.5$ g cm$^{-2}$ as derived by their disk model. Other parameters and assumptions are the same as in the run with $\sigma_s=0.35$ g cm$^{-2}$. The results are shown in the top panel of Figure 2. In that case the core accretion rate is higher which leads to a rapid growth. After only 0.55 Myr crossover mass is reached, at 16 M$_\oplus$, and the planet can grow rapidly in mass. The simulation is stopped when the planet has reached 38 M$_{\oplus}$ but gas accretion is expected to continue. We can conclude that the combination of high core accretion rates with relatively solid-rich disks leads to the formation of giant gaseous planets within relatively short timescales. Note, however, that $M_\mathrm{core}$ levels out at about 16 M$_\oplus$. The isolation mass, which gives approximately the amount of solid material in the feeding zone of the planet, is given by $M_\mathrm{iso} = 1.56 \times 10^{25}
(a^2 \sigma_s)^{3/2}$ g, where $a$ is given in AU and $\sigma_s$ in cgs units. In this case $M_\mathrm{iso} \approx 136$ M$_\oplus$, but scattering, particularly at high core mass, removes a large fraction of that material. In fact at the end of the simulation, practically all solid material available to the planet has either been accreted or scattered.
We next model (Run 20UN3) the formation at 20 AU with $\sigma_s=0.7$ g cm$^{-2}$, which is about twice the MMSN value. The core accretion rate is set to its maximum value, and both the effect of planetesimal scattering and cross section enhancement due to the planetary envelope are [*not*]{} included. The results are shown in the middle panel of Figure 2. In that case, a planet similar to Neptune can be formed within 1.3 Myr. At that point the core mass is $\sim$ 15 M$_{\oplus}$ with an envelope of 4 $M_{\oplus}$. If the simulation continues to longer times more gas is accreted and the forming planet will become a gas giant planet. Under these conditions, in order to form a Neptune-like planet the gas has to dissipate relatively early. Note, however, that the planetary growth is expected to be slower and the final heavy-element mass is expected to be smaller when planetesimal scattering is included. Finally, we model, in Run 20UN4, the planetary growth at 20 AU, $\sigma_s$ = 1.7 g cm$^{-2}$ but this time with the transitional core accretion rate $[dM/dt]_{LOW}$ (Eq. 5). The results are presented in the bottom panel of Figure 2. Even with the low accretion rate it is found that a planet of similar mass to that of Uranus can be formed within less than 2 Myr; however, the gaseous mass is significantly higher than that of Uranus.
Clearly, the properties of the planet can change significantly even due to changes in the solid-surface density alone. We can also conclude that the high accretion rates can now allow for [*in situ*]{} formation at 20 AU, even when the MMSN solid surface densities are used. However in that case, with very low surface density, the conclusion is marginal. First, as mentioned, the time required to reach Uranus mass is 7 Myr, which is beyond the lifetime of most gas disks. Second, to allow the use of the maximum accretion rate, the embryo has to grow big enough to excite the planetesimal velocities so that they collide, fragment, and produce the required small particles. Based on the work of Rafikov (2003, 2004), at 20 AU and in the MMSN, the required embryo mass is of order $10^{24}$ g, and the time to build that embryo is very roughly estimated at $10^6$ to $10^7$ yr. Therefore a detailed numerical simulation of the planetesimal dynamics, including fragmentation, is required to determine whether the use of the high accretion rate is justified in this particular case. At smaller distances from the Sun, or with higher values of $\sigma_s$, the corresponding time is considerably shorter.
Formation at 15 AU
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In this section we investigate the formation of a planet at 15 AU. First (Run 15UN1), we consider $\sigma_s$ = 0.55 g cm$^{-2}$, close to the value in the MMSN. The core accretion rate is set to \[dM/dt\]$_{HIGH}$, scattering of planetesimals is included, and envelope enhancement is included. The results are shown in the top panel of Figure 3. With this low surface density the planetary growth is slow and at $\sim$ 2.5 Myr the planet has a core mass of 3.6 M$_{\oplus}$ and an envelope mass of 0.8 M$_{\oplus}$. At 9.7 Myr crossover mass is still not reached, and the planet has a core mass of about 4 M$_{\oplus}$ and an envelope of 3.1 M$_{\oplus}$. Clearly, under these assumptions it is not possible to form a Neptune/Uranus mass planet. It should be noted, however, that at the point the simulation stops, the gas accretion rate is about 0.3 M$_{\oplus}$/Myr, therefore a slightly higher solid surface density and/or longer disk lifetime would probably lead to the formation of a gaseous planet.
We next model the planetary formation at 15 AU with $\sigma_s$ = 5.5 cm$^{-2}$ (Run 15UN2). Other assumptions are the same as in the previous case. It is found that the crossover mass is reached already after $\approx$ 6$\times$10$^5$ years when the core and envelope masses are 18.9 M$_{\oplus}$. The results for this case are shown in the bottom panel of Figure 3. Clearly, this combination of parameters is more preferable for giant planet formation than formation of intermediate-mass planets.
The use of \[dM/dt\]$_{HIGH}$ implies that small planetesimals, less than 100 meters in size, dominate the accretion (Rafikov 2004). Our calculations with 1 km planetesimals are not entirely consistent; therefore we run a calculation with planetesimal size 50 meters and otherwise with the parameters of Run 15UN1. In our calculation the only effect of decreasing the planetesimal size is to increase the enhancement of the cross section for planetesimal capture. Thus the the phase during which core accretion dominates is shortened; this phase is short in any case. Once the core mass approaches its limiting value, and envelope accretion is the dominant process, the effect of the smaller planetesimals is very small. In Run 15UN3, the limiting core mass is approached only 2.2 $\times 10^5$ years after the start, while in Run 15UN1 the corresponding time is 3.25 $\times 10^5$ years. In Run 15UN3 at 2.5 Myr, the core mass is 3.62 M$_\oplus$ and the envelope mass is 0.85 M$_\oplus$, practically the same as in Run 15UN1. The calculation in 15UN3 is run to 9.7 Myr, as in Run 15UN1, and at that time the core mass is 4.03 M$_\oplus$ and the envelope mass is 3.65 M$_\oplus$. These results give the same conclusion as in Run 15UN1: with these assumptions it is not possible to produce a Uranus or Neptune mass planet in a reasonable time.
Formation at 12 AU
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We next model the formation of a planet at 12 AU, the minimum radial distance for the formation of Uranus/Neptune as suggested by the Nice model. At 12 AU the solid surface density is relatively high, providing preferable conditions for planet formation.
First (Run 12UN1), we consider a solid surface density of $\sigma_s$ = 0.75 g cm$^{-2}$, which is approximately that of the MMSN at that radial distance, and assume the high solid accretion rate (Eq. 2). As in previous cases, the enhancement of the capture rate is included, with 1 km planetesimals. The results are shown in the top panel of Figure 4. The planetary growth is somewhat similar to that derived for the low-density case at 20 AU. The core mass reaches $\approx$ 3.5 M$_{\oplus}$ relatively fast and then levels off, while the envelope mass reaches 0.65 M$_{\oplus}$ at 2.5 Myr. Clearly, under these conditions it is not possible to form a planet which resembles Uranus/Neptune. In order to demonstrate whether it is possible to form a Uranus/Neptune-like planet we continue the simulation up to 10 Myr. At that time we obtain $M_\mathrm{core} = 3.8 $ M$_\oplus$ and $M_\mathrm{env} = 2.2 $ M$_\oplus$. We can therefore conclude that under such conditions it is not possible to form a planet of the order of 15 M$_{\oplus}$; instead a Super-Earth or Mini-Neptune planet is formed. Comparing Runs 20UN1 and 12UN1, both runs having low surface densities, the former case develops a somewhat higher core mass, because more solid material is available in the disk at the larger distance. The envelope accretion rate is strongly dependent on the core mass (Pollack et al. 1996), so 20UN1 is able to reach crossover in $< 10$ Myr, while 12UN1 is not.
We next consider planet formation with $\sigma_s$ = 7.5 g cm$^{-2}$ which corresponds to $\approx$ 10 times MMSN (Run 12UN2). Other assumptions are the same as in the previous case. The results are shown in the lower panel of Figure 4. Under these conditions the growth of the planet is rapid, and at 0.52 Myr it reaches the crossover mass of 21 M$_{\oplus}$ which leads to rapid gas accretion. The simulation was terminated at 0.56 Myr with a total mass of 48 M$_\oplus$ and an envelope mass of $\approx$ 26 M$_{\oplus}$. If the disk lifetime under these conditions is of the order of several Myr, a gaseous planet is expected to form. This simulation is similar to the one at 15 AU with the high solid-surface density. In both cases, a giant planet can be formed quickly. By the end of both runs most of the planetesimals in the feeding zone have either been accreted onto the core or scattered. However, closer to the Sun, a larger fraction of the planetesimals can be accreted; while at 12 AU $f_{cap}$ = 0.18 at the end of the simulation, at 15 AU, $f_{cap}$ = 0.13. Thus $M_\mathrm{core}$ at crossover is somewhat smaller at 15 AU (18.9 vs. 21 M$_\oplus$), and because of the lower surface density at 15 AU, the time to reach crossover is somewhat longer ($6.2 \times 10^5$ vs. 5.2 $\times 10^5$ yr.)
We next consider (Run 12UN3) the high solid surface density case (i.e., $\sigma_s$ = 7.5 g cm$^{-2}$) but this time using the “transition" accretion rate \[dM/dt\]$_{LOW}$ (Eq. 5). Other parameters are the same as in the previous case. The results are shown in the top panel of Figure 5. Even under these conditions the planetary growth is fairly rapid and after about 0.9 Myr the planet has reached a total mass of $\approx$ 45 M$_{\oplus}$ with an envelope mass of 24 M$_\oplus$. This case is very similar to the previous case using the maximum accretion rate, with the main difference being the timescale to reach the crossover mass. Nevertheless, due to the high solid surface density, a giant planet can be formed within the lifetime of most protoplanetary disks. In these cases, the challenge is to form planets with final masses similar to those of Uranus and Neptune. If Uranus/Neptune indeed formed at 12 and/or 15 AU, they can provide important constraints on $\sigma_s$ and the accretion rates at these locations, since according to our simulations, the growing planets can become gaseous planets fairly easily.
This case was recalculated with a planetesimal size of 100 km rather than 1 km, again using \[dM/dt\]$_{LOW}$ (not plotted or listed). The end result was practically the same, namely, the onset of rapid gas accretion, with a crossover mass of about 21 M$_\oplus$. The time to reach crossover, however was twice as long, 1.8 Myr instead of 0.9 Myr, as a result of the reduced enhancement of the planetesimal capture cross section with the larger planetesimals.
The case discussed in the previous paragraph is somewhat similar to a calculation done by Dodson-Robinson & Bodenheimer (2010), who considered planet formation at 12 AU with $\sigma_s=8.4$ g cm$^{-2}$. In their calculation, planetesimal scattering was included but the core accretion rate was a factor 2 lower than \[dM/dt\]$_{LOW}$. Their result shows that Uranus mass is reached at 4 Myr with $M_\mathrm{core} = 13.5$ M$_{\oplus}$ and $M_\mathrm{env} = 1.0$ M$_{\oplus}$. Neptune mass is reached at 4.2 Myr with $M_\mathrm{core} = 15$ M$_{\oplus}$ and $M_\mathrm{env} = 2$ M$_{\oplus}$. However, if the disk does not dissipate until later times, accretion can continue up to much higher masses. We have redone this simulation with the same parameters but with an improved version of the formation code. We take a planetesimal size of 100 km to be consistent with their assumptions. Our results show that Uranus mass is reached at 1.76 Myr with $M_\mathrm{core} = 13.9$ M$_{\oplus}$ and $M_\mathrm{env} = 0.6$ M$_{\oplus}$. Neptune mass is reached at 1.89 Myr with $M_\mathrm{core} = 16$ M$_{\oplus}$ and $M_\mathrm{env} = 1$ M$_{\oplus}$. Again, accretion continues to higher masses. Our times are about a factor 2 shorter than those of Dodson-Robinson & Bodenheimer (2010), mainly because the envelope enhancement of the planetesimal capture cross section is more accurately calculated in our case and turns out to be higher.
Finally (Run 12UN4), we model the formation of a planet at 12 AU, with the transitional accretion rate but with lower $\sigma_s$ than in Run 12UN3. The results are shown in the bottom panel of Figure 5. Although the formation timescale is longer compared to that in Run 12UN3, it is found that crossover mass can be reached within $\approx$ 2.5 Myr. The value of the crossover mass is, however, almost a factor 2 lower than in Run 12UN3. At this point the planetary mass is over 20 M$_{\oplus}$, and the solid-to-gas ratio is close to 1.
The Effect of Planetesimal Scattering - A Test At 30 AU
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The farthest formation location so far considered in this work is 20 AU, which is typically considered as an upper bound for Uranus and Neptune based on the Nice model. Formation at larger radial distances is however possible. Since the surface density decreases with radial distance the formation process is somewhat less efficient and the forming planets are expected to have smaller masses. Nevertheless, with high core accretion rates, and with sufficiently high $\sigma_s$ even at 30 AU planets could be formed, and even become gas giants. Our simulations, however, are limited to the case where $\sigma_s$ is close to the value for the MMSN.
It should be noted, however, that our simulations start with solid cores of $\approx$ 1.3 M$_{\oplus}$. At 30 AU, the timescale for forming such cores can be a few million years, which delays the formation process significantly. In addition, planetesimal scattering is more important for large radial distances. In this section we model planetary formation at 30 AU. In order to demonstrate the importance of planetesimal scattering and its crucial effect on the planetary growth, we compare two cases of planet formation at 30 AU (Runs 30UN1 and 30UN2), which differ solely by this assumption. The solid surface density is assumed to be low, 0.2 g cm$^{-2}$, and the high accretion rate \[dM/dt\]$_{HIGH}$ is used. The results are shown in Figure 6. The top panel presents the planetary growth when planetesimal scattering is included. It is found that after 3 Myr, a planet with a total mass of $\approx$ 5 M$_{\oplus}$ is formed. The gaseous mass by the end of the simulation is 1.6 M$_{\oplus}$. The planetary growth is somewhat similar to that in Run 20UN1, at 20 AU with the low $\sigma_s$ (Figure 1). If the planet is allowed to grow on a longer timescale, crossover mass is expected to be reached before 10 Myr, leading to the formation of a giant or intermediate mass planet within 10 Myr. The total mass of heavy elements is, however, small, and therefore, regardless of the final mass of the planet, it is expected to be gas-dominated. The results for the planetary growth when planetesimal scattering is [*not*]{} considered (Run 30UN2) are shown in the bottom panel. In this case the planet can grow faster, and crossover is reached after $\approx$ 2.5 Myr with a higher solid mass than in Run 30UN1. In that case Uranus/Neptune-mass planets can be formed even at 30 AU. For the case without planetesimal scattering, even if we add the $\approx$ 4.4 Myr that are required to build the seed core, intermediate-mass and gas giant planets can be formed at 30 AU when disk lifetimes are of the order of 10 Myr. The solid-to-gas ratio by the end of the simulation is close to one. Clearly, planetesimal scattering plays a major role in planet formation models, when relatively large radial distances are considered, and it has a crucial effect on both the formation timescale and the final composition of the planets.
However, at a distance of 12 AU the effect is much less important. A comparison to Run 12UN1 was run without scattering (not plotted or listed). The run without scattering also did not reach crossover; at 7 Myr it reached $M_\mathrm{core} = 5.05$ M$_\oplus$ and $M_\mathrm{env} =
2.84$ M$_\oplus$, in comparison with $M_\mathrm{core} = 3.7$ M$_\oplus$ and $M_\mathrm{env} = 1.62 $ M$_\oplus$ at the same time in Run 12UN1.
The Effect of the Planetesimal Size
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Clearly, since small planetesimals are more affected by the gaseous envelope, the planetary growth is more efficient when small planetesimals are assumed. We next present two cases (Runs 20UN5 and 12UN5) in which the planetesimals are assumed to be relatively large, i.e. 100 km (see also Section 3.2, Run 15UN3). Results for the two runs with 100 km-sized planetesimals are shown in Figure 7. The top panel (Run 20UN5) corresponds to formation at 20 AU with the maximum accretion rate and $\sigma_s=0.35$ g cm$^{-2}$, i.e., similar to the case presented in Figure 1 but assuming 100 km-sized planetesimals instead of 1 km. By comparing the two cases it can be concluded that although the growth is slightly slower in Run 20UN5, the overall growth of planet after 3 Myr is very similar. At 3 Myr the core mass is found to be 3.57 M$_{\oplus}$ and the envelope mass 1.6 M$_{\oplus}$, compared to $M_\mathrm{core}
= 3.55$ M$_\oplus$ and $M_\mathrm{env} = 1.43$ M$_\oplus$ in Run 20UN1 at the same time. The main difference between these two cases is that when larger planetesimals are used, it takes a somewhat longer time to build up the core during the earlier phases to the (scattering limited) mass of $\sim$ 3.5 M$_{\oplus}$.
The bottom panel of Figure 7 illustrates Run 12UN5, at 12 AU with $\sigma_s=3$ g cm$^{-2}$, \[dM/dt\]$_{LOW}$, and 100 km planetesimals. This case should be compared with Run 12UN4 (bottom panel of Figure 5), which has the same parameters except for a planetesimal size of 1 km; crossover mass of 11.6 M$_\oplus$ is reached after 2.3 Myr. With 100 km planetesimals, a Uranus-mass planet can be formed within about 3.3 Myr, with core mass 10.5 M$_\oplus$ and envelope mass 4.0 M$_\oplus$. After that point the gas accretion rate is significantly higher than the core accretion rate; as a result a planet with Neptune’s mass can be formed under these conditions after 3.6 Myr, with $M_\mathrm{core} = 10.9 $ M$_\oplus$ and $M_\mathrm{env} = 6.1 $ M$_\oplus$. The solid-to-gas ratio is too low compared with that of Neptune; however the case with the higher planetesimal size leads to a higher solid-to gas ratio at times around 3 Myr, since crossover mass is not reached by that time.
The Lambrechts–Johansen (LJ) Accretion Rate
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The LJ rate (Eq. 3) is higher than \[dM/dt\]$_{HIGH}$ by roughly one order of magnitude, depending on the stage of evolution. The cases with high $\sigma_s$, which include Runs 20UN2, 15UN2, 12UN2, and 12UN3, all reach crossover mass in relatively short times. If these runs were to be redone with the LJ rate, they would simply reach crossover mass faster, with similar core masses; the core masses are determined by the amount of solid material available and by the effects of scattering, not by the accretion rate. On the other hand, the low $\sigma_s$ cases, which include Runs 20UN1, 15UN1, 12UN1, 30UN1, and 20UN5, do not reach crossover in 3 Myr, although they could over longer times. The core masses, which fall in the range 3–4 M$_\oplus$ at 3 Myr, are limited by the isolation mass, which increases as $a^{3/4}$ in the MMSN, as reduced by the effects of scattering, which become more important at larger $a$. The gas accretion rates at these low core masses are slow, and are little affected by the core accretion rate, once the limiting core masses have been reached. Thus the use of a faster core accretion rate would simply speed up the time to approach the limiting core mass but would have little effect on the results at longer times.
A test calculation has been run at 20 AU and $\sigma_s = 0.35$ g cm$^{-2}$, with the LJ rate. A core mass of 3.2 M$_\oplus$ is reached after $5 \times 10^4$ years after the start, as compared with about $3 \times 10^5$ yr in the comparison run 20UN1 (top panel of Figure 1). The end result for the test case after 3 Myr is $M_\mathrm{core}= 3.65$ M$_\oplus$ and $M_\mathrm{env}= 2.13$ M$_\oplus$, while for Run 20UN1 the corresponding numbers are 3.59 and 1.72 M$_\oplus$, respectively. Thus the basic conclusions of this paper are only weakly affected by the choice of Equation (3) rather than Equation (2).
A problem may arise, however, in the use of the LJ theory. The time to reach the transition mass, Equation (4), can be quite long. In fact the LJ estimate of the time to reach this mass (their Eq. 42) exceeds 10$^8$ yr at 20 AU. Thus fast accretion by this mechanism requires, first, that a significant fraction of the solid material be in the form of “pebbles" (cm-size particles) close to the midplane, and second, that at least some planetesimals close to the transition mass ($\approx$ 1000 km in size) must have formed early by some independent process (Lambrechts & Johansen 2012).
Conclusions
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Understanding the formation of Uranus and Neptune is crucial for understanding the origin of our solar system, and in addition, the formation of intermediate-mass planets around other stars. Planets that are similar to Uranus and Neptune in terms of mass are likely to form by core accretion, i.e., from a growing solid core which accretes gas at a lower rate, although alternative mechanisms should not be excluded (Boss et al. 2002; Nayakshin 2011). If indeed formed by core accretion, such planets must form fast enough to ensure that gas is accreted onto the core, but at the same time slow enough, in order to remain small in mass and not become gas giant planets.
Our study shows that simulating the formation of Uranus and Neptune is not trivial, and that getting the correct masses and solid-to-gas ratio depends on the many (unknown) model parameters. Even small changes in the assumed parameters can lead to a very different planet. The core accretion rate and the disk’s properties such as solid-surface density and the planetesimals’ properties (sizes, dynamics, etc.) play a major role in the formation process, and even small changes in these parameters can influence the final masses and compositions of the planets considerably.
We have used high accretion rates for the solids and have investigated their impact on the planet formation process. With these high accretion rates and with values of $\sigma_s$ about 10 times those in the MMSN, the formation timescale problem for formation of Uranus or Neptune at 20 AU disappears. However, a new problem arises - the formation of the planets can be so efficient that instead of becoming failed giant planets, Uranus and Neptune would become giant planets, similar to Jupiter and Saturn. The situation becomes even worse with formation at smaller radial distance where the solid surface density is high. At radial distances such as 12 and 15 AU, the planets reach runaway gas accretion within a timescale which is shorter than the average lifetimes of protoplanetary disks, leading to the formation of gaseous giant planets.
However, if the values of $\sigma_s$ appropriate for the MMSN are used, a different problem arises. Even with high core accretion rates, the resulting solid masses are too low compared with those of Uranus/Neptune, [at all distances. Scattering of planetesimals is an important effect in this regard.]{} Our results suggest that intermediate values of $\sigma_s$, perhaps combined with relatively low core accretion rates (e.g. Run 12UN5), are needed to satisfy the joint constraints provided by disk lifetimes, total masses, and solid-to-gas ratios.
It should be noted that the “true" core accretion rate to form Uranus and Neptune is not known and could be different from the values we consider here. Our work shows that even with a relatively low accretion rate, combined with a relatively high solid surface density $\sigma_s$, it is possible to form a giant planet in a relatively short time (Run 12UN3). However, different, but still reasonable choices for these parameters can produce quite different results; for example Run 12UN1 produces a planet of less than half the mass of Uranus after 10 Myr. Finding the right set of parameters that will lead to the formation of planets with the final masses of Uranus and Neptune is challenging, and getting the correct gas masses is even harder. In some cases the gas accretion rate becomes high enough and an increase in mass to the Uranus/Neptune mass range occurs, but the solid-to-gas ratio in the models is inconsistent with that of Uranus and Neptune (see Table 2). The cases we present simply demonstrate the sensitivity of the planetary formation to the assumed parameters, and while they do provide possible scenarios for the formation of Uranus and Neptune, they are certainty not unique. Clearly, the core accretion rate is a major uncertainty in planet formation models, and a better determination of this property (and its time evolution) will have a significant impact on simulations of planetary growth of both terrestrial and giant planets.
Our work suggests that under the right conditions, [*in situ*]{} formation for Uranus at 20 AU, or formation of Neptune at about the same distance, is possible. At 30 AU, our run with $\sigma_s$ near the value for the MMSN and with planetesimal scattering, showed that a Neptune-mass planet was not formed. Although Rafikov (2011) shows that the use of the high accretion rate should result in evolution to rapid gas accretion out to 40–50 AU in a MMSN in 3 Myr, our result is different. The main reason is that planetesimal scattering limits the core mass to about 3.3 M$_\oplus$, leading to a slow accretion rate for the envelope. However that result could change if a higher value of $\sigma_s$ were taken.
[*In situ*]{} formation at relatively large radial distances seems to be possible for intermediate-mass planets in general. In addition, high solid surface density, which is expected in metal-rich environments, can lead to fast core formation, and therefore to the formation of giant planets instead of intermediate-mass planets. The latter provides a natural explanation to the correlation between stellar metallicity and the occurrence rate of gas giant planets. Nevertheless, we also find that giant planets can be formed in low-metallicity environments. As a result, giant planets around low-metallicity stars should not necessarily be associated with formation by gravitational instability. It is also concluded that the formation of Uranus/Neptune-mass planets is not always challenging in terms of the formation timescale but often, in terms of reducing the gas accretion in order to prevent the formation of gaseous planets.
Finally, while our work emphasizes once more the difficulty to simulate the formation of the solar-system planets very accurately, it provides a natural explanation for the diversity in planetary parameters in extrasolar planetary systems. Since planetary disks are expected to have different physical properties (e.g., surface densities, lifetimes) it is clear that the forming planets will have different growth histories, as well as different final masses and compositions.
Acknowledgments {#acknowledgments .unnumbered}
---------------
P. B. was supported in part by a grant from the NASA program “Origins of Solar Systems".
REFERENCES {#references .unnumbered}
==========
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[lc c c c c c c c|]{}\
RunName & Radial Distance (AU) & $\sigma_s$ (g cm$^{-2}$) & P-Size(km) & P-Scattering & $T_{\text {neb}} (K)$ & $\rho_{\text {neb}}$ (g cm$^{-3}$) & $\dot M_\mathrm{core}$\
20UN1 &20 & 0.35 & 1 & YES & 20 & $8 \times 10^{-13}$ & eq. (2)\
20UN2 & 20 & 3.5 & 1 & YES & 20 & $8 \times 10^{-12}$ & eq. (2)\
20UN3 & 20& 0.7 & 1 & NO & 20 & $1.6 \times 10^{-12}$ & eq. (2)\
20UN4 & 20 & 1.7 & 1 & YES & 20 & $4 \times 10^{-12}$ & eq. (5)\
15UN1 & 15 & 0.55 & 1& YES & 40 & $1.7 \times 10^{-12}$ & eq. (2)\
15UN2 & 15 & 5.5 & 1 & YES & 40 & $1.7 \times 10^{-11}$ & eq. (2)\
15UN3 & 15 & 0.55 & 0.05 & YES & 40 & $1.7 \times 10^{-11}$ & eq. (2)\
12UN1 & 12 & 0.75 & 1 & YES & 50 & $3 \times 10^{-12}$ & eq. (2)\
12UN2 & 12 & 7.5 & 1 & YES & 50 & $3 \times 10^{-11}$ & eq. (2)\
12UN3 & 12 & 7.5 & 1 & YES & 35 & $3 \times 10^{-11}$ & eq. (5)\
12UN4 & 12 & 3 & 1 & YES& 50 & $1.2 \times 10^{-11}$ & eq. (5)\
30UN1 & 30 & 0.2 & 1& YES & 20 & $3 \times 10^{-13}$ & eq. (2)\
30UN2 & 30 & 0.2 & 1 & NO & 20 & $3 \times 10^{-13}$ & eq. (2)\
20UN5 & 20 & 0.35 & 100 & YES & 20 & $8 \times 10^{-13}$ & eq. (2)\
12UN5 & 12 & 3 & 100 & YES & 50 & $1.2 \times 10^{-11}$ & eq. (5)\
[lc c c c c c c c c c|]{}\
RunName & $t_\mathrm{crossover}$ & $M_\mathrm{crossover} $ & $t_\mathrm{end}$ & $M_\mathrm{core,end}$ & $M_\mathrm{env,end}$ & $M_\mathrm{tot,end}$ & $f_{solid-to-gas}$ & $\dot M_\mathrm{core,end}$ & $\dot M_\mathrm{env,end}$\
& Myr& M$_\oplus$ & Myr & M$_\oplus$ & M$_\oplus$ & M$_\oplus$ & & M$_\oplus$ yr$^{-1}$ & M$_\oplus$ yr$^{-1}$\
\
20UN1 & 5.14 & 3.83 & 7.9 & 5.6 & 60 & 65.6 & 0.09 & $1.5 \times 10^{-5}$ & $1.9 \times 10^{-4}$\
20UN2 & 0.54 & 15.8 & 0.59 & 16.2 & 20.3 & 36.5 & 0.8 & $3.0 \times 10^{-5}$ & $6.0 \times 10^{-5}$\
20UN3 & $-$ & $-$ & 1.3& 14.3 & 3.9 & 18.2 & 3.67 & $3.3 \times 10^{-6}$& $8.8 \times 10^{-6}$\
20UN4 & $-$ & $-$& 1.67 & 9.8 & 4.8 & 14.6 & 2.04 &$2.0 \times 10^{-6}$ & $1.6 \times 10^{-5}$\
15UN1 & $-$ & $-$ & 9.7 & 4.0& 3.1 & 7.1& 1.29 & $3.2 \times 10^{-8}$ & $3.0 \times 10^{-7}$\
15UN2 & 0.62 & 18.9 &0.63 & 19.4 & 26.2 & 45.6 & 0.74 & $3.3 \times 10^{-5}$ & $1.3 \times 10^{-4}$\
15UN3 & - & - & 9.7 & 4.03 & 3.65 & 7.7 & 1.10 & $3.2 \times 10^{-8}$ & $4.0 \times 10^{-7}$\
12UN1 & $-$ & $-$ &10.0 & 3.8& 2.2 & 6.0 & 1.73 & $2.3 \times 10^{-8}$ & $1 \times 10^{-7}$\
12UN2 & 0.52 & 21.2 &0.56 &21.5 & 26.1 & 47.6 & 0.82 & $8.1 \times 10^{-5}$ & $4.2 \times 10^{-4}$\
12UN3 & 0.89 & 20.8 & 0.90& 21.0 & 23.9 & 44.9& 0.88 &$5.5 \times 10^{-5}$ & $4.1 \times 10^{-4}$\
12UN4 & 2.39 & 11.6 & 2.40 & 11.6& 11.7 & 23.3& 0.99 &$1.3 \times 10^{-6}$ & $5.4 \times 10^{-6}$\
30UN1 & $-$ & $-$ & 3.1 & 3.3 & 1.6 & 4.9& 2.06 &$1.9 \times 10^{-7}$ & $1.6 \times 10^{-5}$\
30UN2 & 2.6 & 9.45 & 2.6 & 9.4 & 10.1& 19.5 & 0.93 & $1.5 \times 10^{-7}$ & $4.7 \times 10^{-5}$\
20UN5 & $-$ & $-$ & 2.9 & 3.6 & 1.6 & 5.2 & 2.25 & $2.7 \times 10^{-7}$ & $1.5 \times 10^{-6}$\
12UN5 & $-$ & $-$& 3.7 & 11.0 & 6.8 & 17.8 & 1.62 &$1.3 \times 10^{-6}$ & $9.4 \times 10^{-6}$\
![Run 20UN1: planetary growth at 20 AU with solid surface density $\sigma_s$=0.35 g cm$^{-2}$ and high accretion rate - \[dM/dt\]$_{HIGH}$ (Eq. 2). The planetesimal size is 1 km. The red, blue, and black curves represent the mass of the gaseous envelope (hydrogen and helium), core (heavy elements), and total planetary mass, respectively. [**Top**]{}: planetary growth up to 3.7 Myr. [**Middle**]{}: planetary growth up to 8 Myr. [**Bottom:**]{} planetary growth up to $\approx$ 7.2 Myr, at which time the total mass is 17 M$_{\oplus}$. ](20AU_035_new1.pdf "fig:"){height="6.cm"} ![Run 20UN1: planetary growth at 20 AU with solid surface density $\sigma_s$=0.35 g cm$^{-2}$ and high accretion rate - \[dM/dt\]$_{HIGH}$ (Eq. 2). The planetesimal size is 1 km. The red, blue, and black curves represent the mass of the gaseous envelope (hydrogen and helium), core (heavy elements), and total planetary mass, respectively. [**Top**]{}: planetary growth up to 3.7 Myr. [**Middle**]{}: planetary growth up to 8 Myr. [**Bottom:**]{} planetary growth up to $\approx$ 7.2 Myr, at which time the total mass is 17 M$_{\oplus}$. ](20AU_035_long_new1.pdf "fig:"){height="6.cm"} ![Run 20UN1: planetary growth at 20 AU with solid surface density $\sigma_s$=0.35 g cm$^{-2}$ and high accretion rate - \[dM/dt\]$_{HIGH}$ (Eq. 2). The planetesimal size is 1 km. The red, blue, and black curves represent the mass of the gaseous envelope (hydrogen and helium), core (heavy elements), and total planetary mass, respectively. [**Top**]{}: planetary growth up to 3.7 Myr. [**Middle**]{}: planetary growth up to 8 Myr. [**Bottom:**]{} planetary growth up to $\approx$ 7.2 Myr, at which time the total mass is 17 M$_{\oplus}$. ](20AU_035_Neptune_new2.pdf "fig:"){height="6.cm"}
![Planetary growth at 20 AU, . [**Top:**]{} Same as Fig. 1 but with $\sigma_s$=3.5 g cm$^{-2}$ (Run 20UN2). [**Middle:**]{} Simulation without planetesimal scattering or envelope enhancement of the core accretion cross section, with $\sigma_s$=0.7 g cm$^{-2}$ and with the high core accretion rate \[dM/dt\]$_{HIGH}$, Eq. (2) (Run 20UN3). [**Bottom:**]{} Simulation with $\sigma_s$=1.7 g cm$^{-2}$ including planetesimal scattering and envelope enhancement of the cross section with 1 km-sized planetesimals. The core accretion rate is taken from Eq. (5), i.e., the transitional accretion rate (Run 20UN4).](20AU_35.pdf "fig:"){height="6.cm"} ![Planetary growth at 20 AU, . [**Top:**]{} Same as Fig. 1 but with $\sigma_s$=3.5 g cm$^{-2}$ (Run 20UN2). [**Middle:**]{} Simulation without planetesimal scattering or envelope enhancement of the core accretion cross section, with $\sigma_s$=0.7 g cm$^{-2}$ and with the high core accretion rate \[dM/dt\]$_{HIGH}$, Eq. (2) (Run 20UN3). [**Bottom:**]{} Simulation with $\sigma_s$=1.7 g cm$^{-2}$ including planetesimal scattering and envelope enhancement of the cross section with 1 km-sized planetesimals. The core accretion rate is taken from Eq. (5), i.e., the transitional accretion rate (Run 20UN4).](20AU_07.pdf "fig:"){height="6.cm"} ![Planetary growth at 20 AU, . [**Top:**]{} Same as Fig. 1 but with $\sigma_s$=3.5 g cm$^{-2}$ (Run 20UN2). [**Middle:**]{} Simulation without planetesimal scattering or envelope enhancement of the core accretion cross section, with $\sigma_s$=0.7 g cm$^{-2}$ and with the high core accretion rate \[dM/dt\]$_{HIGH}$, Eq. (2) (Run 20UN3). [**Bottom:**]{} Simulation with $\sigma_s$=1.7 g cm$^{-2}$ including planetesimal scattering and envelope enhancement of the cross section with 1 km-sized planetesimals. The core accretion rate is taken from Eq. (5), i.e., the transitional accretion rate (Run 20UN4).](20U14F.pdf "fig:"){height="6.cm"}
![[**Top:**]{} Run 15UN1– planetary growth at 15 AU with $\sigma_s$=0.55 g cm$^{-2}$ and accretion rate \[dM/dt\]$_{HIGH}$ (Eq. 2). [**Bottom:**]{} Run 15UN2– planetary growth at 15 AU with $\sigma_s$=5.5 g cm$^{-2}$ and accretion rate - \[dM/dt\]$_{HIGH}$ (Eq. 2).](15AU_055.pdf "fig:"){height="8cm"} ![[**Top:**]{} Run 15UN1– planetary growth at 15 AU with $\sigma_s$=0.55 g cm$^{-2}$ and accretion rate \[dM/dt\]$_{HIGH}$ (Eq. 2). [**Bottom:**]{} Run 15UN2– planetary growth at 15 AU with $\sigma_s$=5.5 g cm$^{-2}$ and accretion rate - \[dM/dt\]$_{HIGH}$ (Eq. 2).](15AU_55.pdf "fig:"){height="8cm"}
![[**Top:**]{} Run 12UN1– Planetary growth at 12 AU with $\sigma_s$=0.75 g cm$^{-2}$ and the high accretion rate \[dM/dt\]$_{HIGH}$ (Eq. 2). The planetesimal size is 1 km. The red, blue, and black curves represent the mass of the gaseous envelope (mostly hydrogen and helium), core (heavy elements), and total planetary mass, respectively. [**Bottom**]{}: Run 12UN2– planetary growth at 12 AU with $\sigma_s$=7.5 g cm$^{-2}$ and \[dM/dt\]$_{HIGH}$ (Eq. 2). ](12AU_075.pdf "fig:"){height="6.1cm"} ![[**Top:**]{} Run 12UN1– Planetary growth at 12 AU with $\sigma_s$=0.75 g cm$^{-2}$ and the high accretion rate \[dM/dt\]$_{HIGH}$ (Eq. 2). The planetesimal size is 1 km. The red, blue, and black curves represent the mass of the gaseous envelope (mostly hydrogen and helium), core (heavy elements), and total planetary mass, respectively. [**Bottom**]{}: Run 12UN2– planetary growth at 12 AU with $\sigma_s$=7.5 g cm$^{-2}$ and \[dM/dt\]$_{HIGH}$ (Eq. 2). ](12AU_75_U3.pdf "fig:"){height="6.1cm"}
![[**Top:**]{} Run 20UN3– planetary growth at 12 AU with $\sigma_s$=7.5 g cm$^{-2}$ and the transitional accretion rate \[dM/dt\]$_{LOW}$ (Eq. 5). [**Bottom:**]{} Run 20UN4–planetary growth at 12 AU with $\sigma_s$=3 g cm$^{-2}$ and the transitional accretion rate \[dM/dt\]$_{LOW}$ (Eq. 5).](12AU_75_new.pdf "fig:"){height="8cm"} ![[**Top:**]{} Run 20UN3– planetary growth at 12 AU with $\sigma_s$=7.5 g cm$^{-2}$ and the transitional accretion rate \[dM/dt\]$_{LOW}$ (Eq. 5). [**Bottom:**]{} Run 20UN4–planetary growth at 12 AU with $\sigma_s$=3 g cm$^{-2}$ and the transitional accretion rate \[dM/dt\]$_{LOW}$ (Eq. 5).](12AU_lowacc.pdf "fig:"){height="8cm"}
![[**Top:**]{} Run 30UN1– Planet formation at 30 AU assuming $\sigma_s$=0.2 g cm$^{-2}$ and \[dM/dt\]$_{HIGH}$ (Eq. 2). [**Bottom:**]{} Run 30UN2–Same as top panel but without planetesimal scattering. ](a30N2.pdf "fig:"){height="8cm"} ![[**Top:**]{} Run 30UN1– Planet formation at 30 AU assuming $\sigma_s$=0.2 g cm$^{-2}$ and \[dM/dt\]$_{HIGH}$ (Eq. 2). [**Bottom:**]{} Run 30UN2–Same as top panel but without planetesimal scattering. ](a30NF_new.pdf "fig:"){height="8cm"}
![ Planetary growth with 100 km-sized planetesimals. [**Top:**]{} Run 20UN5– planetary formation at 20 AU, $\sigma_s$=0.35 g cm$^{-2}$ with core accretion rate \[dM/dt\]$_{HIGH}$, i.e. Eq. (2). [**Bottom:**]{} Run 12UN5– planetary formation at 12 AU with $\sigma_s$=3 g cm$^{-2}$ and the low core accretion rate \[dM/dt\]$_{LOW}$ (Eq. 5).](20AU_35_100km.pdf "fig:"){height="8cm"} ![ Planetary growth with 100 km-sized planetesimals. [**Top:**]{} Run 20UN5– planetary formation at 20 AU, $\sigma_s$=0.35 g cm$^{-2}$ with core accretion rate \[dM/dt\]$_{HIGH}$, i.e. Eq. (2). [**Bottom:**]{} Run 12UN5– planetary formation at 12 AU with $\sigma_s$=3 g cm$^{-2}$ and the low core accretion rate \[dM/dt\]$_{LOW}$ (Eq. 5).](12_lowacc_100km.pdf "fig:"){height="8cm"}
|
---
abstract: 'We propose novel quantum antennas and metamaterials with strong *magnetic* response at optical frequencies. Our design is based on the arrangement of natural atoms with only *electric* dipole transition moments at distances smaller than a wavelength of light but much larger than their physical size. In particular, we show that an atomic dimer can serve as a magnetic antenna at its antisymmetric mode to enhance the decay rate of a magnetic transition in its vicinity by several orders of magnitude. Furthermore, we study metasurfaces composed of atomic bilayers with and without cavities and show that they can fully reflect the electric and magnetic fields of light, thus, forming nearly perfect electric/magnetic mirrors. The proposed quantum metamaterials can be fabricated with available state-of-the-art technologies and promise several applications both in classical optics and quantum engineering.'
address:
- '$^1$Max Planck Institute for the Science of Light, Erlangen 91058, Germany'
- '$^2$Department of Physics, University of Ottawa, Ottawa Q1N 6N5, Canada'
- '$^3$Department of Electrical Engineering and Computer Science, University of California, Irvine, CA 92617, USA'
- '$^{\ast}$Emails: rasoul.alaee@mpl.mpg.de and vahid.sandoghdar@mpl.mpg.de'
author:
- 'Rasoul Alaee,[$^{1,2,\ast}$]{} Burak Gurlek,[$^{1}$]{} Mohammad Albooyeh,[$^{3}$]{} Diego Martín-Cano,[$^{1}$]{} and Vahid Sandoghdar[$^{1,\ast}$]{}'
title: Quantum metamaterials with magnetic response at optical frequencies
---
Most natural materials interact weakly with the magnetic field of light at optical frequencies [@LandauLifshitz]. In fact, the magnetic interaction energy $\bm{-{\mu}\cdot\mathbf{B}}$ is typically about two orders of magnitude (i.e. order of fine-structure constant) smaller than its electric counterpart $\mathbf{-p\cdot\mathbf{E}}$, whereby $\mu\approx\mu_B$ and $p\approx ea_{0}$ represent the magnitude of the electric dipole moments, and $e$, $a_{0}$, $\mu_B$, denote the elementary charge, Bohr radius, and Bohr magneton, respectively [@Jackson1999]. However, two decades of progress in nano-optics has brought about novel electromagnetic properties that are not available in natural materials. In particular, “metamaterials” created through synthetic arrangement of subwavelength antennas [@engheta:2006; @Soukoulis:2011; @Yu2014; @Kuznetsov2016] can now generate magnetic functionalities at high frequencies [@Shalaev:Book2008]. Unfortunately, material absorption and limits in nanofabrication have hampered reaching a high performance in the optical regime.
Considering that natural atoms act as the smallest and most fundamental optical antennas , one can also envision the construction of “quantum” metamaterials by synthetically arranging natural atoms or molecules at distances smaller than an optical wavelength but much larger than the characteristic length of electronic orbitals. Indeed, a number of such proposals have emerged over the past few years [@meir2013Lambshift; @Bettles2016; @shahmoon2017; @Zhou2017; @mkhitaryan2018; @wild2018quantum; @Manzoni2018; @Genes2019; @plankensteiner2019enhanced; @grankin2018; @liberal2018; @Zoller2019; @bettles2019quantum; @Rui:2020], but these have only considered metamaterials with electric response. In this Letter, we show that a strong magnetic functionality can be obtained from conventional atoms at optical frequencies. In particular, we propose novel quantum antennas that can enhance the decay rate of a magnetic emitter (i.e. an emitter with magnetic dipole transitions) in their vicinity by several orders of magnitude. We demonstrate that a metasurface composed of the proposed antennas can act as nearly perfect electric and magnetic mirrors and can, moreover, strongly couple to a cavity mode independent of its position.
*Atomic dimer antenna.—*Let us first consider an atomic dimer consisting of two identical atoms with electric dipole transition moments placed at $\mathbf{r}_{u/d}=(0,0,\pm l/2)$ (see the inset in Fig. \[fig:AtomicDimer\] (a,b); u,d stand for up and down). We assume that the atomic response is isotropic and linear, i.e. we consider the weak-excitation limit. The electric polarizability of each atom amounts to $\alpha\left(\omega\right)=\frac{-\frac{\Gamma_{0}}{2}\alpha_{0}}{\delta+i\frac{\Gamma_{0}}{2}}$, where $\Gamma_{0}$ is the radiative linewidth of the atomic transition at frequency $\omega_{a}$ while $\delta=\omega-\omega_{a}\ll\omega_{a}$ represents the frequency detuning between the illumination and the atom, $\alpha_{0}=6\pi/k^{3}$ and $k$ is the wavenumber [@lambropoulos2007]. We note that our discussion can be readily generalized to any other quantum emitter such as molecules, color centers, quantum dots or ions with dipolar transitions.
The atomic dimer antenna is illuminated by an $x$-polarized plane wave $\mathbf{E}_{\mathrm{inc}}=E_{0}e^{ikz}\mathbf{e}_{x}$ propagating in the $z$ direction, where $\mathbf{e}_{x}$ is the unit vector in the $x$ direction, $E_{0}$ is the electric field amplitude, and $k$ is the wavenumber in free space. The induced volume current density for the dimer antenna can be written as (see the Supplementary Materials (SM) for a detailed derivation) $$\mathbf{J}\left(\mathbf{r},\omega\right) = -i\omega\epsilon_{0}E_{0}\left[\alpha_{d}\delta\left(\mathbf{r}-\mathbf{r}_{d}\right)+\alpha_{u}\delta\left(\mathbf{r}-\mathbf{r}_{u}\right)\right]\mathbf{e}_{x},\label{eq:Jind}$$ where $\delta\left(\mathbf{r}-\mathbf{r}_{u/d}\right)$ is the Dirac delta function and $\epsilon_{0}$ denotes the free-space permittivity. The quantities $\alpha_{u}$ and $\alpha_{d}$ are the effective electric polarizabilities of the upper and lower atoms according to $\alpha_{u/d}= \alpha\left[\frac{\mathrm{cos}\left(kl/2\right)}{D_{-}}\pm i\frac{\mathrm{sin}\left(kl/2\right)}{D_{+}}\right]$. Here, $D_{\pm}\equiv1\pm\epsilon_{0}\alpha G_{EE}^{xx}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)$ with $G_{EE}^{xx}\left(\mathbf{r}_{d},\mathbf{r}_{u}\right)$ signifying the scalar Greens function of the Helmholtz equation in free space. Using a multipole expansion of the induced current [@Alaee:2018] and Eq. (\[eq:Jind\]), we obtain the induced electric and magnetic dipole and quadrupole polarizabilities of the dimer antenna at $\mathbf{r}=0$ (see SM): $$\begin{aligned}
\alpha_{\rm ed} & = & \frac{2\alpha}{D_{-}}\left[j_{0}\left(kl/2\right)-\frac{j_{2}\left(kl/2\right)}{2}\right]\mathrm{cos}\left(kl/2\right),\nonumber \\
\alpha_{\rm md} & = &\frac{3\alpha}{D_{+}}j_{1}\left(kl/2\right)\mathrm{sin}\left(kl/2\right),\nonumber\end{aligned}$$ $$\begin{aligned}
\alpha_{\rm eq} & = &\frac{12}{k^{2}}\frac{\alpha}{D_{+}}\left[3j_{1}\left(kl/2\right)-2j_{3}\left(kl/2\right)\right]\mathrm{sin}\left(kl/2\right),\nonumber \\
\alpha_{\rm mq} & = &-\frac{60}{k^{2}}\frac{\alpha}{D_{-}}j_{2}\left(kl/2\right)\mathrm{cos}\left(kl/2\right),\label{eq:ME}\end{aligned}$$ where $j_{n}\left(r\right)$ is the spherical Bessel function and $\alpha_{\rm ed}$, $\alpha_{\rm md}$, $\alpha_{\rm eq}$, and $\alpha_{\rm mq}$ represent the electric dipole, magnetic dipole, electric quadrupole and magnetic quadrupole polarizabilities, respectively. For small separations ($l\ll \lambda$), the higher order spherical Bessel functions are negligible, i.e. $j_{2}\left(kl/2\right)\approx0$ and $j_{3}\left(kl/2\right)\approx0$. Thus, $\alpha_{\rm md}\approx k^2/12\alpha_{\rm eq}$ and the magnetic quadrupole polarizability can be neglected, i.e. $\alpha_{\rm mq}\approx0$. Once the induced dipole and quadrupole polarizabilities are obtained, the total scattering cross section ($C_{\mathrm{sca}}$) of the atomic dimer can also be calculated (see SM): $$\label{eq:C_sca}
C_{\mathrm{sca}}=\frac{k^{4}}{6\pi}\left(\left|\alpha_{{\rm ed}}\right|^{2}+\left|\alpha_{{\rm md}}\right|^{2}+\frac{3}{5}\left|\frac{k^{2}}{12}\alpha_{{\rm eq}}\right|^{2}\right).$$
Near-field coupling of the electric dipole transitions of two individual emitters has been explored in various systems [@DeVoe:1996; @Hettich:2002] and is known to lead to symmetric (superradiant) and antisymmetric (subradiant) states. The black curve in Fig. \[fig:AtomicDimer\](a) shows $C_{\mathrm{sca}}$ for the subradiant state as a function of the frequency detuning for two atoms separated by $l=\lambda_{a}/10$. In this case, the electric response of the dimer antenna becomes negligible, but it exhibits both magnetic dipolar (see right vertical axis) and electric quadrupolar responses with $\alpha_{\rm eq}\approx\frac{12}{k^{2}}\alpha_{\rm md}$ (see SM). The inset in Fig. \[fig:AtomicDimer\](a) illustrates the magnetic field distribution for this antisymmetric mode under plane wave illumination, where a strong magnetic field testifies to an optically induced magnetic response. The left vertical axis in Fig. \[fig:AtomicDimer\](b) shows $C_{\mathrm{sca}}$ as a function of the frequency detuning for the symmetric mode, where the two atoms oscillate in phase. The right vertical axis plots the electric dipolar response of the antenna structure, while the inset shows that the magnetic response is negligible in this scenario.
![*Atomic dimer antenna:* Optical response of two atoms placed at $z=\pm\frac{l}{2}$ (see insets) as a function of detuning at the antisymmetric (a) and symmetric (b) modes of the composite system, respectively. Left vertical axes (black): total scattering cross sections normalized to the free-space value $\frac{3}{2\pi}\lambda^{2}$ for a two-level atom on resonance. Right vertical axes (orange and blue): The real (solid curves) and imaginary (dashed curves) parts of the induced effective polarizabilities calculated using Eq. \[eq:ME\] and for $l=0.1\lambda_a$. Insets display normalized total magnetic field distribution for each case. The antisymmetric mode exhibits both magnetic and electric quadrupole response which lead to a larger total cross section than the symmetric mode [@Rahimzadegan:17].[]{data-label="fig:AtomicDimer"}](Figure1.pdf){width="50.00000%"}
*Enhancing magnetic transitions—* The strong magnetic field generated in the atomic dimer (see Fig. \[fig:AtomicDimer\](a)) prompts us to inquire whether it can act as a magnetic antenna to enhance the decay of a test magnetic dipole moment $\mu_{t}$ placed at the origin. Using the normalized local density of states (NLDOS) of the system (see SM), one can arrive at the antenna-modified decay rate $\Gamma_{\mathrm{\rm ant}}$ given by $$\begin{aligned}
\frac{\Gamma_{\mathrm{\rm ant}}}{\Gamma_{0}}=1-\epsilon_{0}^{2}\alpha_{0}\alpha\mathrm{Im}\left[\frac{g_{\rm EM}^{2}\left(\mathbf{r}_{0},\mathbf{r}_{u}\right)+g_{\rm EM}^{2}\left(\mathbf{r}_{0},\mathbf{r}_{d}\right)}{1+\epsilon_{0}\alpha G_{\rm EE}^{xx}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)}\right],
\label{eq:Decay}\end{aligned}$$ where $g_{\rm EM}\left(\mathbf{r},\mathbf{r}^{\prime}\right)=-\frac{3}{2\epsilon_{0}\alpha_{0}}e^{i\zeta}\left(\frac{1}{\zeta}+\frac{i}{\zeta^{2}}\right)$ is the scalar electro-magnetic Greens function in free space and $\zeta=k\left|\mathbf{r}-\mathbf{r}^{\prime}\right|$ (see SM). Figure \[fig:LDOS\](a) plots the calculated magnetic decay rate enhancement for both symmetric and antisymmetric modes. We find that the decay rate can be enhanced by *five* orders of magnitude at $l\approx\lambda_{a}/20$ for the antisymmetric mode. However, for the symmetric mode the decay rate is even slightly decreased below its unperturbed value (i.e., $\Gamma_{\mathrm{ant}}<\Gamma_{0}$) because of the weak magnetic response of this mode (see the inset in Fig. \[fig:AtomicDimer\](b)).
To achieve even larger enhancements, one can devise an atomic tetramer antenna, consisting of four identical atoms with electric polarizability $\alpha$ (see Fig. \[fig:LDOS\](c)). Figure \[fig:LDOS\](a) shows the enhancement of the magnetic transition rate for this case (see SM). We note that fabrication of quantum metamaterials in these configurations is readily within reach since the distances involved are well beyond atomic and molecular spacings in natural substances (e.g. $\lambda_{a}/20$ corresponds to several tens of nanometers). A particularly interesting class of materials for these applications are rare earth ions with weak magnetic dipole transition [@Karaveli:11; @Kasperczyk2015]. New efforts on the implantation of ions using ion traps or other bombardment strategies [@GrootBerning:2019; @Luhmann:2018] allow precise doping of various host materials.
![*Enhancing the decay rate of a magnetic transition*: (a) Enhanced decay rate of a magnetic dipole emitter placed in the middle of an atomic dimer antenna for the symmetric (blue), antisymmetric (red) modes and an atomic tetramer (purple) as a function of the antenna length $l$. The tetramer antenna is composed of four identical atoms placed at $\mathbf{r}_{1,2}=\mp l/2\,\mathbf{e}_{z}, \mathbf{r}_{3,4}=\pm l/2 \,\mathbf{e}_{x}$. (b,c) Schematics of an emitter with magnetic dipole moment $\mu_{t}$ placed in the middle of an atomic dimer (b) and tetramer (c). \[fig:LDOS\]](Figure2.pdf){width="48.00000%"}
![Atomic bilayer metasurface (ABM): (a) Schematics of an ABM. (b) The symmetric and antisymmetric modes corresponding to the electric and magnetic mirrors, respectively. (c) Intensity reflection coefficient $R$ as a function of layer separation $l$ and frequency detuning $\omega-\omega_a$ for an ABM with periodicity $\Lambda_{x}=\Lambda_{y}=\lambda_{a}/2$. (d) Intensity transmission and reflection coefficients corresponding to a cut through (c) at $l=0.1\lambda_a$. Solid curves: analytical results for an infinite array with plane wave illumination. Symbols: Numerical calculations for a finite array with $15\times15\times2$ atoms illuminated by a Gaussian beam. (e) The real and imaginary parts of the effective induced dipole moments of different layers. (f) The real and imaginary parts of the effective electric and magnetic polarizabilities. \[fig:Atomic\_Metasurface\]](Figure3.pdf){width="50.00000%"}
*Electric and magnetic mirrors based on atomic bilayer metasurfaces.—* It has been shown that optimal optical coupling to a two-level atom requires mode matching between the incident light and that of the field radiated by the atom [@Zumofen:2008]. It is, thus, found that a dipolar wave can be perfectly reflected by a single two-level atom with a dipolar transition. Similarly, it has been shown, both theoretically and experimentally, that a planar two-dimensional array of atoms acts as a nearly perfect electric mirror for a plane-wave illumination [@Bettles2016; @shahmoon2017; @Rui:2020]. Now, we show that a periodic planar arrangement of our dimer antennas, which we call atomic bilayer metasurfaces (ABM), can act as both electric and magnetic mirrors (see Fig. \[fig:Atomic\_Metasurface\](a,b)).
Let us illuminate an ABM by an $x$-polarized plane wave propagating in $z$-direction, $\mathbf{E}_{\mathrm{inc}}=E_{0}e^{ikz}\mathbf{e}_{x}$. The field reflection and transmission coefficients are given by (see SM) $$\begin{aligned}
r & = & \frac{ik}{2\Lambda^{2}\epsilon_{0}E_{0}}\left(p_{d}^{\mathrm{eff}}e^{-ikl/2}+p_{u}^{\mathrm{eff}}e^{ikl/2}\right),\nonumber \\
t & = & 1+\frac{ik}{2\Lambda^{2}\epsilon_{0}E_{0}}\left(p_{d}^{\mathrm{eff}}e^{ikl/2}+p_{u}^{\mathrm{eff}}e^{-ikl/2}\right),\label{eq:TR_2Layers}\end{aligned}$$ where $p_{u}^{\mathrm{eff}}$ and $p_{d}^{\mathrm{eff}}$ are the effective electric dipole moments of the upper and lower layers, respectively and can be calculated as $$\begin{aligned}
\left[\begin{array}{c}
p_{d}^{\mathrm{eff}}\\
p_{u}^{\mathrm{eff}}
\end{array}\right] & = & \left[\begin{array}{cc}
\frac{1}{\epsilon_0\alpha}-C_{dd} & -C_{du}\\
-C_{ud} & \frac{1}{\epsilon_0\alpha}-C_{uu}
\end{array}\right]^{-1}\left[\begin{array}{c}
E_{i}\left(\mathbf{r}_{d}\right)\\
E_{i}\left(\mathbf{r}_{u}\right)
\end{array}\right]\label{eq:Induced_ED_eff}\end{aligned}$$ with $$\begin{aligned}
C_{dd}=\sum_{n,\,n\neq0}G_{EE}^{xx}\left(\mathbf{r}_{d,0},\mathbf{r}_{d,n}\right) \\ C_{du}=\sum_{n}G_{EE}^{xx}\left(\mathbf{r}_{d,0},\mathbf{r}_{u,n}\right)\,,\label{eq:IC}\end{aligned}$$ denoting the interaction constants between atoms in the upper and lower layers, respectively. For identical atoms, the interaction constants are symmetric, i.e. $C_{ud}=C_{du}$ and $C_{uu}=C_{dd}$.
For lossless atoms (i.e. no nonradiative loss), the imaginary part of the interaction constants can be calculated exactly by using the conservation of energy principle \[see SM\], $$\begin{aligned}
\mathrm{Im}\left[C_{dd}\right] = \frac{k}{2\Lambda^{2}\epsilon_{0}}-\frac{1}{\epsilon_0\alpha_{0}}\\ \, \mathrm{Im}\left[C_{du}\right] = \frac{k}{2\Lambda^{2}\epsilon_{0}}\,\mathrm{cos}\left(kl\right),\end{aligned}$$ while their real parts are calculated numerically. Figure \[fig:Atomic\_Metasurface\](c) shows the reflectivity of an ABM as a function of the frequency detuning and the distance between the two layers. It can be seen that the array fully reflects the impinging light at both asymmetric and antisymmetric resonance frequencies. In Fig. \[fig:Atomic\_Metasurface\](d), we plot the reflection and transmission of a plane wave incident on an infinite array for $l=\lambda_{a}/10$ calculated using Eq. \[eq:TR\_2Layers\] (solid lines). The symbols in Fig. \[fig:Atomic\_Metasurface\](d) present the results obtained for a finite array of $15\times15\times2$ atoms and a Gaussian beam illumination (a possible experimental situation) by integrating the Poynting vector for the scattered and incident beam \[see SM\]. We find that the results for the two illuminations (i.e. Gaussian and plane wave) agree very well.
We now provide more insight into the working of the ABM in Fig. \[fig:Atomic\_Metasurface\](e,f). As displayed in Fig. \[fig:Atomic\_Metasurface\](e), the effective induced dipole moments of the upper and lower layers are out of phase at the antisymmetric mode, i.e. $p_{u}^{\mathrm{eff}}\approx-p_{d}^{\mathrm{eff}}\approx1.7 \epsilon_0\alpha_0E_0$. Figure \[fig:Atomic\_Metasurface\](f) displays the effective electric and magnetic polarizabilities of the atomic metasurface, which can be calculated by using $$\alpha_{\rm ed}^{\mathrm{eff}}=\frac{p_{d}^{\mathrm{eff}}+p_{u}^{\mathrm{eff}}}{\epsilon_0E_{0}}\mathrm{cos}\left(kl/2\right),
\alpha_{\rm md}^{\mathrm{eff}}=i\frac{p_{d}^{\mathrm{eff}}-p_{u}^{\mathrm{eff}}}{\epsilon_0E_{0}}\mathrm{sin}\left(kl/2\right).\nonumber\\$$ It follows that the the antisymmetric resonance of the ABM supports an effective magnetic response $\alpha_{\rm md}^{\mathrm{eff}}\approx i\alpha_0$, while $\alpha_{\rm ed}^{\mathrm{eff}}\approx0$. Together with a reflectivity $r\approx1$, this implies that the bilayer metasurface acts as a nearly perfect *atomic magnetic mirror* at its antisymmetric resonance. At the symmetric mode, however, the atoms in the upper and lower layers are in phase such that $p_{u}^{\mathrm{eff}}\approx p_{d}^{\mathrm{eff}}\approx0.5i\epsilon_0\alpha_0E_0$, leading to an effective electric response, i.e. $\alpha_{\rm ed}^{\mathrm{eff}}\approx i\alpha_0$, but $\alpha_{\rm md}^{\mathrm{eff}}\approx0$ (see Fig. \[fig:Atomic\_Metasurface\](f)). Therefore, the array acts as a nearly perfect *atomic electric mirror* with $r\approx-1$ (see SM for a Gaussian beam excitation).
![*Atomic metasurface inside a planar cavity*: (a) Schematics of a cavity and an atomic monlayer metasurface (AMM). (b) Transmission of the cavity loaded with the AMM as a function of its position inside the cavity ($D$) and detuning. (c,d) Same as (a,b) but for an atomic bilayer metasurface (ABM). The quality factor of the planar cavity is taken to be $Q\approx8.7\times10^{3}$, $\gamma_{c}\approx10^{3}\Gamma_{0}$ and $L_{c}=3\lambda_a$. \[fig:Cavity\] ](Figure4.pdf){width="48.00000%"}
*Atomic monolayer/bilayer metasurfaces in a cavity.—*Optical cavities are commonly used to enhance the interaction of light with matter. The maximum interaction occurs when an atom with an electric (magnetic) transition dipole moment is placed at the maximum electric (magnetic) field. In other words, the interaction strongly depends on the position of the atom inside the optical cavity. The proposed atomic bilayer metasurface can overcome this problem due to the combination of its strong electric and magnetic response to light.
Figure \[fig:Cavity\] depicts a planar cavity consisting of two mirrors separated by $L_{c}$, whereby the transmission of the bare cavity is assumed to reach unity at $\omega=\omega_{c}$, i.e., there are no absorption or scattering losses in the cavity. Now, we investigate the interaction of the cavity with an atomic metasurface placed at a distance $D$ from its center for i) an atomic monolayer metasurface (AMM), and ii) an ABM (see Fig. \[fig:Cavity\](a,c)). The cavity linewidth $\gamma_c$ is assumed to be much larger than that of the AMM resonance ($\gamma_{c}\approx10^{3}\Gamma_0$). The calculated transmission is plotted in Fig. \[fig:Cavity\](b) for the AMM as a function of the frequency detuning and metasurface position $D$. It is seen that one reaches strong coupling at $D=0$, but the resonance splitting decreases with increasing $D$. The transmission changes periodically by varying the position $D$ of the metasurface. At $D=\lambda_{c}/4$ there is no longer a splitting due to a vanishing electric field. At $\omega=\omega_c$, transmission is zero (see SM, section IV) because the first cavity mirror and the AMM (which is a nearly perfect electric mirror at the cavity frequency, i.e. $r_{\rm AMM}=-1$) form a new cavity.
For an ABM, however, strong coupling can be maintained at all positions inside the cavity (Fig. \[fig:Cavity\](d)). While at $D=0$, the cavity only interacts with the symmetric mode due to the maximum electric field inside cavity, at $D=\lambda_{c}/4$ the cavity only couples to the antisymmetric mode. At intermediate positions, where $0<D<\lambda_{c}/4$, the cavity couples to both symmetric and antisymmetric modes.
In conclusion, we have demonstrated that synthetic arrangements of natural atoms with only electric dipole transitions can support both electric and magnetic responses at optical frequencies. Since the interatomic distances required for our proposed designs are well above ten nanometers, our proposed quantum metasurfaces can be experimentally realized in both the gas and solid phases using a range of available methods in cold atom manipulation or implantation strategies. In particular, our proposal lends itself to applications based on natural species with weak magnetic dipole transitions, e.g., rare earth ions, (See Refs [@Dodson:2012; @van2018recent]). For instance, $\mathrm{Eu^{3+}}$ with a magnetic transition ($^{5}D_{0}\rightarrow^{7}F_{1}$) at 584nm and a linewidth of about 15Hz (see Table III Ref [@Dodson:2012]), is a suitable candidate for enhancement by an arrangement of different atoms such as Fe, Ar, Kr, N, Na with electric dipole transition at the same wavelength (see Ref [@van2018recent], J = 0 $\rightarrow$ J = 1 with an E1 transition). The predicted transition rate enhancements reaching $10^5$ would, thus, yield magnetic transitions with natural linewidths fully comparable to that of common electric dipole transitions. Such novel materials hold promise for the development of a range of technological applications and fundamental studies in quantum engineering and physics.
**Acknowledgments.—** This work was supported by the Max Planck Society. R.A. also acknowledges financial support provided by the Alexander von Humboldt Foundation. The authors warmly thank Claudiu Genes for helpful discussions.
Atomic dimer
============
Induced multipole moments
-------------------------
In this section, we derive induced multipole moments of an atomic dimer. Let us consider an atomic dimer consisting of two identical atoms placed at $\mathbf{r}_{u/d}=[0,0,\pm l/2]$ and illuminated by an $x$-polarized plane wave propagating in the $z$ direction, i.e. $\mathbf{E}_{\mathrm{inc}}=E_{0}e^{ikz}\mathbf{e}_{x}$ [\[]{}see Fig. \[fig:Geometry\_dimer\] (a)-(b)[\]]{}. We assume $e^{-i\omega t}$ time harmonic variation. $\mathbf{e}_{x}$ is the unit vector in the $x$ direction, and $E_{0}$ is the electric field amplitude, $k$ is the wave vector in free space. For a closed two level $J=0\rightarrow J=1$ atomic transition [@Bettles2016; @lambropoulos2007], the atomic response is isotropic and linear (far from saturation). The electric polarizability of each atom is defined as $\alpha=\frac{-\frac{\Gamma_{0}}{2}\alpha_{0}}{\delta+i\frac{\left(\Gamma_{0}+\Gamma_{nr}\right)}{2}}$ where $\delta=\omega-\omega_{a}\ll\omega_{a}$ is the detuning frequency between the light beam and the transition frequency of the atom and $\alpha_{0}=\frac{6\pi}{k^{3}}$. We assume elastic scattering events and therefore the non-radiative decay is zero, i.e. $\Gamma_{nr}=0$. The induced displacement volume current density for the atomic dimer can be written as $$\begin{aligned}
\mathbf{J}\left(\mathbf{r},\omega\right) & = & -i\omega\left[\hat{\mathbf{p}}_{d}\left(\mathbf{r}_{d}\right)\delta\left(\mathbf{r}-\mathbf{r}_{d}\right)+\hat{\mathbf{p}}_{u}\left(\mathbf{r}_{u}\right)\delta\left(\mathbf{r}-\mathbf{r}_{u}\right)\right],\label{eq:Current}\end{aligned}$$ where $\delta\left(\mathbf{r}-\mathbf{r}_{i}\right)$ is the Dirac delta function, $\epsilon_{0}$ is the permittivity of the free space. $\hat{\mathbf{p}}_{i}\left(\mathbf{r}_{i}\right)$ $\left(i=u\,\mathrm{and}\,d\right)$ is the induced electric dipole moment of the upper and lower dipole moments, respectively and for the two atoms under consideration reads [\[]{}Fig. \[fig:Geometry\_dimer\] (b)[\]]{} [@Abajo:2007] $$\begin{aligned}
\left[\begin{array}{c}
\hat{p}_{d}\\
\hat{p}_{u}
\end{array}\right] & = & \left[\begin{array}{cc}
\frac{1}{\epsilon_{0}\alpha} & -G_{EE}^{xx}\left(\mathbf{r}_{d},\mathbf{r}_{u}\right)\\
-G_{EE}^{xx}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right) & \frac{1}{\epsilon_{0}\alpha}
\end{array}\right]^{-1}\left[\begin{array}{c}
E_{0}e^{-ikl/2}\\
E_{0}e^{ikl/2}
\end{array}\right].\label{eq:ED_Matrix}\end{aligned}$$ Note that the induced moments have only an $x$-component ($\hat{\mathbf{p}}_{u/d}=\hat{p}_{u/d}\mathbf{e}_{x}$) due to an $x$-polarized plane wave illumination. $G_{EE}^{xx}\left(\mathbf{r}_{d},\mathbf{r}_{u}\right)=G_{EE}^{xx}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)$ is the Green function [\[]{}see the Green function section[\]]{} $$G_{EE}^{xx}\left(\mathbf{r}_{d},\mathbf{r}_{u}\right)=\frac{3}{2\alpha_{0}\epsilon_{0}}e^{i\zeta}\left(\frac{1}{\zeta}-\frac{1}{\zeta^{3}}+\frac{i}{\zeta^{2}}\right),\,\,\,\zeta=k\left|\mathbf{r}_{d}-\mathbf{r}_{u}\right|=kl.$$ Eq. \[eq:ED\_Matrix\] can be simplified as $$\begin{aligned}
\hat{p}_{d} & = & \epsilon_{0}\alpha_{d}E_{0}=\epsilon_{0}\alpha\frac{e^{-ikl/2}+\epsilon_{0}\alpha G_{EE}^{xx}e^{ikl/2}}{\left(1-\epsilon_{0}\alpha G_{EE}^{xx}\right)\left(1+\epsilon_{0}\alpha G_{EE}^{xx}\right)}E_{0},\nonumber \\
\hat{p}_{u} & = & \epsilon_{0}\alpha_{u}E_{0}=\epsilon_{0}\alpha\frac{e^{ikl/2}+\epsilon_{0}\alpha G_{EE}^{xx}e^{-ikl/2}}{\left(1-\epsilon_{0}\alpha G_{EE}^{xx}\right)\left(1+\epsilon_{0}\alpha G_{EE}^{xx}\right)}E_{0},\end{aligned}$$ where $\alpha_{d}$ and $\alpha_{u}$ defined as $$\begin{aligned}
\alpha_{d} & = & \alpha\left[\frac{\mathrm{cos}\left(kl/2\right)}{D_{-}}-i\frac{\mathrm{sin}\left(kl/2\right)}{D_{+}}\right],\nonumber \\
\alpha_{u} & = & \alpha\left[\frac{\mathrm{cos}\left(kl/2\right)}{D_{-}}+i\frac{\mathrm{sin}\left(kl/2\right)}{D_{+}}\right],\end{aligned}$$ where $D_{\pm}=1\pm\alpha G_{EE}^{xx}\left(\mathbf{r}_{d},\mathbf{r}_{u}\right)$ and $\alpha_{u/d}$ is related to effective polarizability of the dimer. Using above expressions, Eq. \[eq:Current\] can be written $$\begin{aligned}
\mathbf{J}\left(\mathbf{r}\right) & = & -i\omega\epsilon_{0}E_{0}\left[\alpha_{d}\delta\left(\mathbf{r}-\mathbf{r}_{d}\right)+\alpha_{u}\delta\left(\mathbf{r}-\mathbf{r}_{u}\right)\right]\mathbf{e}_{x},\label{eq:J_ind}\end{aligned}$$ which is the induced displacement volume current density for the atomic dimer when illuminated by a plane wave ( $\mathbf{E}_{\mathrm{inc}}=E_{0}e^{ikz}\mathbf{e}_{x}$). Now, we can use the multipole expansion [\[]{}see Ref. [@Alaee:2018], Table II[\]]{} and Eq. \[eq:J\_ind\] to obtain the induced multipole moments of the atomic dimer at the center $\mathbf{r}=0$.
![(a) An atomic dimer consisting of two identical atoms with electric polarizability $\alpha$ placed at $\mathbf{r}_{u/d}=[0,0,\pm l/2]$. (b) An atomic dimer when illuminated by an x-polarized plane wave propagating in the $z$ direction, i.e. $\mathbf{E}_{\mathrm{inc}}=E_{0}e^{ikz}\mathbf{e}_{x}$ and the induced upper and lower electric dipole moments (i.e. $\hat{p}_{u}$, $\hat{p}_{d}$), respectively. (c)-(d) Symmetric and antisymmetric modes and their equivalent multipole moments obtained by applying the multipole expansion at the center of the atomic dimer $\mathbf{r}=0$. Note that the magnetic quadrupole moment can be neglected for $\lambda\gg l$.\[fig:Geometry\_dimer\]](Figure1_SM.pdf){width="13cm"}
The induced *effective* electric dipole moment of the atomic dimer by applying the multipole expansion at the center of the dimer $\mathbf{r}=0$ [\[]{}Fig. \[fig:Geometry\_dimer\] (b)-(d)[\]]{} read as [@Alaee:2018] $$\begin{aligned}
p_{\beta} & = & -\frac{1}{i\omega}\left\{ \int dvJ_{\beta}j_{0}\left(kr\right)+\frac{k^{2}}{2}\int dv\left[3\left(\mathbf{r}\cdot\mathbf{J}\right)r_{\beta}-r^{2}J_{\beta}\right]\frac{j_{2}\left(kr\right)}{\left(kr\right)^{2}}\right\} ,\label{eq:p_ME}\end{aligned}$$ where $\beta=x,y,z$ and $j_{n}\left(kr\right)$ are spherical Bessel functions. Now by substituting Eq. \[eq:J\_ind\] into Eq. \[eq:p\_ME\], we have
$$\begin{aligned}
p_{x} & = & -\frac{1}{i\omega}\left\{ \int J_{x}j_{0}\left(kr\right)dv+\frac{k^{2}}{2}\int dv\left[3\left(\mathbf{r}\cdot\mathbf{J}\right)x-r^{2}J_{x}\right]\frac{j_{2}\left(kr\right)}{\left(kr\right)^{2}}\right\} ,\nonumber \\
& = & -\frac{1}{i\omega}\int J_{x}\left[j_{0}\left(kr\right)-\frac{1}{2}j_{2}\left(kr\right)\right]dv,\nonumber \\
& = & \epsilon_{0}E_{0}\left(\alpha_{d}+\alpha_{u}\right)\left[j_{0}\left(kl/2\right)-\frac{j_{2}\left(kl/2\right)}{2}\right].\nonumber \\
& = & \epsilon_{0}E_{0}\frac{2\alpha}{D_{-}}\left[j_{0}\left(kl/2\right)-\frac{j_{2}\left(kl/2\right)}{2}\right]\mathrm{cos}\left(kl/2\right).\label{eq:dipole_dimer}\end{aligned}$$
where $\mathbf{r}\cdot\mathbf{J}=0$, $\mathbf{J}\left(\mathbf{r}\right)=J_{x}(\mathbf{r})\mathbf{e}_{x}$ and $\mathbf{r}=\mathbf{r}_{u/d}=\pm l/2\mathbf{e}_{z}$. Note that $y$ and $z$ components of the electric dipole moments of the dimer are zero, i.e. $p_{y}=0,$ and $p_{z}=0$ [\[]{}see Eq. \[eq:J\_ind\] and the illumination direction in Fig. \[fig:Geometry\_dimer\] (b)[\]]{}. Next by using the definition of the electric dipole moment, i.e. $p_{x}=\epsilon_{0}\alpha_{\mathrm{ed}}E_{0}$, the *effective* electric polarizability of the atomic dimer can be defined
$$\boxed{\alpha_{\mathrm{ed}}=\frac{2\alpha}{D_{-}}\left[j_{0}\left(kl/2\right)-\frac{j_{2}\left(kl/2\right)}{2}\right]\mathrm{cos}\left(kl/2\right)}$$
The induced *effective* magnetic dipole moment at the center of the dimer $\mathbf{r}=0$ [\[]{}Fig. \[fig:Geometry\_dimer\] (b)-(d)[\]]{} read as [@Alaee:2018] $$\begin{aligned}
m_{\beta} & = & \frac{3}{2}\int dv\left(\mathbf{r}\times\mathbf{J}\right)_{\beta}\frac{j_{1}\left(kr\right)}{kr},\label{eq:m_ME}\end{aligned}$$ where $\beta=x,y,z$, by substituting Eq. \[eq:J\_ind\] into Eq. \[eq:m\_ME\], we obtain $$\begin{aligned}
m_{y} & = & \frac{3}{2}\int dv\left(\mathbf{r}\times\mathbf{J}\right)_{y}\frac{j_{1}\left(kr\right)}{kr}=\frac{3}{2}\int\left(zJ_{x}-xJ_{z}\right)\frac{j_{1}\left(kr\right)}{kr}dv,\nonumber \\
& = & \frac{3}{2k}\int zJ_{x}\frac{j_{1}\left(kr\right)}{r}dv,\nonumber \\
& = & \frac{3i}{2}\frac{E_{0}}{Z_{0}}\left(\alpha_{d}-\alpha_{u}\right)j_{1}\left(kl/2\right),\nonumber \\
& = & \left[\frac{3\alpha}{D_{+}}j_{1}\left(kl/2\right)\mathrm{sin}\left(kl/2\right)\right]\frac{E_{0}}{Z_{0}}.\label{eq:my_dimer}\end{aligned}$$ where $Z_{0}=\sqrt{\frac{\mu_{0}}{\epsilon_{0}}}$ is the intrinsic impedance of the free space. Note that we have only the *y*-component of the magnetic moment. Now, by using the definition of the magnetic dipole moment, i.e. $m_{y}=\alpha_{\mathrm{md}}H_{0}=\alpha_{\mathrm{md}}E_{0}/Z_{0}$, the *effective* magnetic polarizability can be defined $$\boxed{\alpha_{\mathrm{md}}=\frac{3\alpha}{D_{+}}j_{1}\left(kl/2\right)\mathrm{sin}\left(kl/2\right)}$$ The induced *effective* electric quadrupole moment at the center of the dimer $\mathbf{r}=0$ [\[]{}Fig. \[fig:Geometry\_dimer\] (b)-(d)[\]]{} read as [@Alaee:2018]
$$\begin{alignedat}{1}Q_{\mu\nu}^{\mathrm{e}} & =-\frac{3}{i\omega}\left\{ \int dv\left[3\left(r_{\nu}J_{\mu}+r_{\mu}J_{\nu}\right)-2\left(\mathbf{r}\cdot\mathbf{J}\right)\delta_{\mu\nu}\right]\frac{j_{1}\left(kr\right)}{kr}\right.\\
& \left.+2k^{2}\int dv\left[5r_{\mu}r_{\nu}\left(\mathbf{r}\cdot\mathbf{J}\right)-\left(r_{\mu}J_{\nu}+r_{\nu}J_{\mu}\right)r^{2}-r^{2}\left(\mathbf{r}\cdot\mathbf{J}\right)\delta_{\mu\nu}\right]\frac{j_{3}\left(kr\right)}{\left(kr\right)^{3}}\right\} ,
\end{alignedat}
\label{eq:Qe_ME}$$
where $\mu,\nu=x,y,z$, and $\delta_{\mu\nu}$ is the Kronecker delta. Next, by substituting Eq. \[eq:J\_ind\] into Eq. \[eq:Qe\_ME\], we have
$$\begin{alignedat}{1}Q_{xz}^{\mathrm{e}} & =-\frac{3}{i\omega}\left[\int dv3\left(zJ_{x}\right)\frac{j_{1}\left(kr\right)}{kr}-2\int dv\left(zJ_{x}\right)\frac{j_{3}\left(kr\right)}{kr}\right],\\
& =-\frac{3}{i\omega}\int dv\frac{zJ_{x}}{kr}\left[3j_{1}\left(kl/2\right)-2j_{3}\left(kl/2\right)\right],\\
& =\frac{3}{k}\epsilon_{0}E_{0}\left(\alpha_{u}-\alpha_{d}\right)\left[3j_{1}\left(kl/2\right)-2j_{3}\left(kl/2\right)\right],\\
& =\frac{6i}{k}\frac{\alpha}{D_{+}}\epsilon_{0}E_{0}\left[3j_{1}\left(kl/2\right)-2j_{3}\left(kl/2\right)\right]\mathrm{sin}\left(kl/2\right).
\end{alignedat}
\label{eq:Qe_dimer}$$
Note that other components of the tensor in Eq. \[eq:Qe\_ME\] are zero. Thus, by using the definition of the electric quadrupole moment [@alu2009], $Q_{xz}^{\mathrm{e}}=\frac{1}{2}\epsilon_{0}\alpha_{\mathrm{eq}}\left(\frac{\partial E_{x}}{\partial z}+\frac{\partial E_{z}}{\partial x}\right)=\frac{ik}{2}\epsilon_{0}E_{0}\alpha_{\mathrm{eq}}$, the *effective* electric quadrupole polarizability can be defined
$$\boxed{\alpha_{\mathrm{eq}}=\frac{12}{k^{2}}\frac{\alpha}{D_{+}}\left[3j_{1}\left(kl/2\right)-2j_{3}\left(kl/2\right)\right]\mathrm{sin}\left(kl/2\right)}$$
The induced *effective* magnetic quadrupole moment at the center of the dimer $\mathbf{r}=0$ [\[]{}Fig. \[fig:Geometry\_dimer\] (b)-(d)[\]]{} read as [@Alaee:2018]
$$\begin{aligned}
Q_{\mu\nu}^{m} & = & 15\int dv\left\{ r_{\mu}\left(\mathbf{r}\times\mathbf{J}\right)_{\nu}+r_{\nu}\left(\mathbf{r}\times\mathbf{J}\right)_{\mu}\right\} \frac{j_{2}\left(kr\right)}{\left(kr\right)^{2}},\label{eq:Qm_ME}\end{aligned}$$
where $\mu,\nu=x,y,z$, by substituting Eq. \[eq:J\_ind\] on Eq. \[eq:Qm\_ME\], we obtain $$\begin{aligned}
Q_{zy}^{m} & = & 15\int dv\left[z^{2}J_{x}\right]\frac{j_{2}\left(kr\right)}{\left(kr\right)^{2}},\nonumber \\
& = & \frac{15}{ik}\frac{E_{0}}{Z_{0}}\left(\alpha_{u}+\alpha_{d}\right)j_{2}\left(kl/2\right),\nonumber \\
& = & \frac{30}{ikD_{-}}\frac{E_{0}}{Z_{0}}\alpha j_{2}\left(kl/2\right)\mathrm{cos}\left(kl/2\right).\end{aligned}$$
Thus, by using the definition of the magnetic quadrupole moment [@alu2009], i.e. $Q_{zy}^{\mathrm{m}}=\frac{1}{2}\alpha_{\mathrm{mq}}\left(\frac{\partial H_{z}}{\partial y}+\frac{\partial H_{y}}{\partial z}\right)=\frac{ik}{2}\frac{E_{0}}{Z_{0}}\alpha_{\mathrm{mq}}$, the *effective* electric quadrupole polarizability read as
$$\boxed{\alpha_{\mathrm{mq}}=-\frac{60}{k^{2}}\frac{\alpha}{D_{-}}j_{2}\left(kl/2\right)\mathrm{cos}\left(kl/2\right)}$$
Scattering cross section
------------------------
In this section, we derive an expression to calculate the scattering cross section of an atomic dimer. Using the induced multipole moments, we can obtain the scattering cross section of the atomic dimer [@Alaee:2018] $$\begin{aligned}
C_{\mathrm{sca}} & \approx & \frac{k^{4}}{6\pi\epsilon_{0}^{2}\left|E_{0}\right|^{2}}\left(\left|p_{x}\right|^{2}+\left|\frac{m_{y}}{c}\right|^{2}+\frac{3}{5}\left|\frac{k}{6}Q_{xz}^{e}\right|^{2}+\frac{3}{5}\left|\frac{k}{6c}Q_{zy}^{m}\right|^{2}\right).\label{eq:CSca_ME1}\end{aligned}$$
Eq. \[eq:CSca\_ME1\] can be written as a function of polarizabilities
$$\begin{aligned}
C_{\mathrm{sca}} & = & \frac{k^{4}}{6\pi}\left[\left|\alpha_{\mathrm{ed}}\right|^{2}+\left|\alpha_{\mathrm{md}}\right|^{2}+\frac{3}{5}\left|\frac{k^{2}}{12}\alpha_{\mathrm{eq}}\right|^{2}+\frac{3}{5}\left|\frac{k^{2}}{12}\alpha_{\mathrm{mq}}\right|^{2}\right],\label{eq:Csca_ME}\end{aligned}$$
where we used the electric and magnetic multipole moments definitions $p_{x}=\epsilon_{0}\alpha_{\mathrm{ed}}E_{0}$, $m=\alpha_{\mathrm{md}}H_{0}$, $Q_{xz}^{e}=Q_{zx}^{e}=\frac{ik}{2}\alpha_{\mathrm{eq}}E_{0}$ and $Q_{zy}^{\mathrm{m}}=Q_{yz}^{\mathrm{m}}=\frac{ik}{2}\frac{E_{0}}{Z_{0}}\alpha_{\mathrm{mq}}$. For the atomic dimer with $\lambda\gg l$, magnetic quadrupole moment is negligible, i.e. $Q_{zy}^{m}\approx0$. Note that Eq. \[eq:Csca\_ME\] is not sufficient for $\lambda\ll l$ and one should consider higher order multipole moments. An alternative approach to obtain the scattering cross section of the atomic dimer is based on the coupled dipole theory \[see the coupled dipole theory section\]. We assume that the nonradiative losses is zero in the atomic dimer. Thus, according to the optical theorem, the extinction cross section is identical to the scattering cross section, i.e. $C_{\mathrm{sca}}=C_{\mathrm{ext}}$, and therefore, $$\begin{aligned}
C_{\mathrm{ext}} & = & \frac{k}{\epsilon_{0}\left|E_{0}\right|^{2}}\mathrm{Im}\left[p_{d}E_{\mathrm{inc}}^{*}\left(\mathbf{r}_{d}\right)+p_{u}E_{\mathrm{inc}}^{*}\left(\mathbf{r}_{u}\right)\right],\nonumber \\
& = & k\mathrm{Im}\left[\alpha_{d}e^{ikl/2}+\alpha_{u}e^{-ikl/2}\right]\\
& = & k\mathrm{Im}\left[2\alpha\frac{1+\epsilon_{0}\alpha\mathrm{G_{EE}^{xx}cos}\left(kl/2\right)}{D_{-}D_{+}}\right].\label{eq:Cext_exact}\end{aligned}$$ Eq. \[eq:Csca\_ME\] is identical to the Eq. $\ref{eq:Cext_exact}$ if an atomic dimer is without nonradiative losses and $\lambda\gg l$.
Radiation pattern
-----------------
Radiation pattern of an atomic dimer can be find by using the radiated far field [@Jackson1999; @campione2015; @Alaee_kerker:15] $$\begin{aligned}
\mathbf{E}_{ED} & = & \frac{k^{2}}{4\pi\epsilon_{0}}\frac{e^{ikr}}{r}p_{x}\left(-\mathrm{sin}\varphi\mathbf{e}_{\varphi}+\mathrm{cos}\theta\mathrm{cos}\varphi\mathbf{e}_{\theta}\right),\nonumber \\
\mathbf{E}_{MD} & = & \frac{k^{2}}{4\pi\epsilon_{0}}\frac{e^{ikr}}{r}\frac{m_{y}}{c}\left(-\mathrm{cos}\theta\mathrm{sin}\varphi\mathbf{e}_{\varphi}+\mathrm{cos}\varphi\mathbf{e}_{\theta}\right),\nonumber \\
\mathbf{E}_{EQ} & = & \frac{k^{2}}{4\pi\epsilon_{0}}\frac{e^{ikr}}{r}\frac{ik}{6}Q_{zx}^{e}\left[\mathrm{cos}\theta\mathrm{sin}\varphi\mathbf{e}_{\varphi}-\mathrm{\left(\mathrm{2cos^{2}}\theta-1\right)cos}\varphi\mathbf{e}_{\theta}\right],\nonumber \\
\mathbf{E}_{MQ} & = & \frac{k^{2}}{4\pi\epsilon_{0}}\frac{e^{ikr}}{r}\frac{ik}{6c}Q_{zy}^{m}\left[\left(\mathrm{2cos^{2}}\theta-1\right)\mathrm{sin}\varphi\mathbf{e}_{\varphi}-\mathrm{\mathrm{cos}\theta cos}\varphi\mathbf{e}_{\theta}\right],\end{aligned}$$
where $r,\theta,\varphi$ are the radial distance, polar angle, and azimuthal angle. In the xz-plane, i.e. $\varphi=0$, the radiation pattern considering the contribution from all multipole moments (up to magnetic quadrupole) can be written as
$$\begin{aligned}
\mathbf{E} & \approx & \frac{k^{2}}{4\pi\epsilon_{0}}\frac{e^{ikr}}{r}\left[p_{x}\mathrm{cos}\theta+\frac{m_{y}}{c}-\frac{ik}{6}Q_{xz}^{e}\left(\mathrm{2cos^{2}}\theta-1\right)-\frac{ik}{6c}Q_{zy}^{m}\mathrm{cos}\theta\right]\mathbf{e}_{\theta},\end{aligned}$$
Let us consider an atomic dimer with $\lambda\gg l$ that supports symmetric and antisymmetric modes. At the symmetric mode resonance frequency [\[]{}see Fig. \[fig:Geometry\_dimer\] (c)[\]]{}, $\left|p_{x}\right|\gg\left|\frac{m_{y}}{c}\right|$, and $\left|p_{x}\right|\gg\left|\frac{ik}{6}Q_{xz}^{e}\right|$ and the magnetic quadrupole moment is negligible, i.e. $Q_{zy}^{m}\approx0$. Thus, the radiation pattern is similar to a pure electric dipole moment, i.e. $$\begin{aligned}
\mathbf{E} & \approx & \frac{k^{2}}{4\pi\epsilon_{0}}\frac{e^{ikr}}{r}p_{x}\mathrm{cos}\theta\mathbf{e}_{\theta},\end{aligned}$$ where $p_{x}$ for the atomic dimer can be obtained using Eq. \[eq:dipole\_dimer\]. In the following section, we will show that an atomic bilayer composed of atomic dimer act as an atomic electric mirror at the symmetric mode resonance frequency. However, at the antisymmetric mode resonance frequency, $\left|p_{x}\right|\ll\left|\frac{m_{y}}{c}\right|$, and $\left|p_{x}\right|\ll\left|\frac{ik}{6}Q_{xz}^{e}\right|$ the radiation pattern is similar to a superposition of magnetic dipole and electric quadrupole moments, i.e. $$\begin{aligned}
\mathbf{E} & \approx & \frac{k^{2}}{4\pi\epsilon_{0}}\frac{e^{ikr}}{r}\left[\frac{m_{y}}{c}-\frac{ik}{6}Q_{xz}^{e}\left(\mathrm{2cos^{2}}\theta-1\right)\right]\mathbf{e}_{\theta},\nonumber \\
& \approx & \frac{k^{2}}{4\pi\epsilon_{0}}\frac{e^{ikr}}{r}\frac{m_{y}}{c}\left(\mathrm{2cos^{2}}\theta\right)\mathbf{e}_{\theta},\label{eq:E_AsyMode}\end{aligned}$$ where $m_{y}$ and $Q_{xz}^{e}$ for the atomic dimer can be obtained using Eq. \[eq:my\_dimer\] and Eq. \[eq:Qe\_dimer\], respectively [\[]{}see Fig. \[fig:Geometry\_dimer\] (d)[\]]{}. Note that for $\lambda\gg l$, we can show that $\frac{m_{y}}{c}\approx-\frac{ik}{6}Q_{xz}^{e}$ which is used in the derivation of Eq. \[eq:E\_AsyMode\]. The radiation patterns of both modes are plotted in Fig. \[fig:RadiationPattern\]. It can be seen that the symmetric mode resonance has an in phase radiation pattern in both forward and backward directions. Whereas for the anti-symmetric mode the radiated fields are out of phase in forward and backward directions. In the following section, we show that an atomic bilayer consist of atomic dimer acts as an electric or a magnetic mirror for symmetric or antisymmetric modes, respectively [\[]{}see Fig. \[fig:Geometry\_dimer\] and Fig. \[fig:GaussianResults\][\]]{}.
![An atomic dimer or a bilayer support both symmetric and antisymmetric modes. The symmetric mode acts as an electric mirror whereas the antisymmetric mode acts as a magnetic mirror.\[fig:RadiationPattern\]](Figure2_SM.pdf){width="13cm"}
Atomic monolayer metasurface (AMM): atomic electric mirrors
===========================================================
Reflection and transition coefficients
--------------------------------------
Let us consider an atomic monolayer composed of atoms with only electric dipole transition moments. The atoms are periodically arranged in $xy$-plane at $z=0$, $\mathbf{r}_{n}=\mathbf{r}_{n_{x},n_{y}}=\left(n_{x}\mathbf{e}_{x}+n_{y}\mathbf{e}_{y}\right)\Lambda$. $n_{x}$ and $n_{y}$ are integer numbers and $\Lambda$ is the periodicity in $x$ and $y$ directions. The reflected and transmitted electric fields ($\mathbf{E}_{r}$ and $\mathbf{E}_{t}$) at the interface when illuminated by a polarized plane wave propagating in the $z$ direction, i.e. $\mathbf{E}_{\mathrm{inc}}=E_{0}e^{ikz}\mathbf{e}_{x}$ defined as [@Tretyakov:03; @Abajo:2007; @shahmoon2017; @Alaee:2017Review] $$\begin{aligned}
\mathbf{E}_{r}=-\frac{1}{2}Z_{0}\mathbf{J}_{e},\,\,\,\,\,\,\,\,\,\mathbf{E}_{t} & = & \mathbf{E}_{\rm inc}-\frac{1}{2}Z_{0}\mathbf{J}_{e},\label{eq:Er_Et_ML}\end{aligned}$$ where $Z_{0}=\sqrt{\frac{\mu_{0}}{\epsilon_{0}}}$ is the impedance of the free space. $\mathbf{J}_{e}$ is the induced averaged surface electric current. $\mathbf{p}$ is the effective induced electric dipole moment of the atomic array and defined as $$\begin{aligned}
\mathbf{p}\left(\mathbf{r}_{0}\right) & = & \epsilon_{0}\alpha\left[\mathbf{E}_{\mathrm{inc}}+\underset{n,\,n\neq0}{\sum}\mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r}_{0},\mathbf{r}_{n}\right)\cdot\mathbf{p}\left(\mathbf{r}_{0}\right)\right],\end{aligned}$$ and using $\mathbf{E}_{\mathrm{inc}}=E_{0}e^{ikz}\mathbf{e}_{x}$, we get $$\begin{aligned}
\mathbf{p}\left(\mathbf{r}_{0}\right) & = & \epsilon_{0}E_{0}\alpha_{\mathrm{eff}}\mathbf{e}_{x},\,\,\,\,\alpha_{\mathrm{eff}}=\frac{\alpha}{1-\epsilon_{0}\alpha\underset{n,\,n\neq0}{\sum}G_{EE}^{xx}\left(\mathbf{r}_{0},\mathbf{r}_{n}\right)},\end{aligned}$$ where $\mathbf{r}_{0}=0$, $\underset{n\neq0}{\sum}G_{EE}^{xx}\left(\mathbf{r}_{0},\mathbf{r}_{n}\right)$ is the interaction constant (or the l*attice sum* of the dipolar interaction tensor). Now by substituting the induced averaged surface electric current $\mathbf{J}_{e}=-i\omega\frac{\mathbf{p}}{\Lambda^{2}}$ into Eq. \[eq:Er\_Et\_ML\], the reflection coefficient $r=\frac{E_{r}}{E_{0}}$ for an atomic monolayer metasurface can be obtained [@Tretyakov:03; @Abajo:2007; @Alaee:2017Review]
$$\begin{aligned}
r & = & \frac{ik}{2\Lambda^{2}}\left(\frac{1}{\frac{1}{\alpha}-\epsilon_{0}\underset{n,\,n\neq0}{\sum}G_{EE}^{xx}\left(0,\mathbf{r}_{n}\right)}\right)=\frac{ik}{2\Lambda^{2}}\alpha_{\mathrm{eff}},\nonumber \\
& = & \frac{ik}{2\Lambda^{2}}\left(\frac{-\frac{\Gamma_{0}}{2}\alpha_{0}}{\delta+i\frac{\Gamma_{0}}{2}+\frac{\Gamma_{0}}{2}\alpha_{0}\epsilon_{0}\underset{n,\,n\neq0}{\sum}G_{EE}^{xx}\left(0,\mathbf{r}_{n}\right)}\right)\nonumber \\
& = & \frac{ik}{2\Lambda^{2}}\left(\frac{-\alpha_{0}\frac{\Gamma_{0}}{2}}{\delta-\Delta+i\left(\frac{\Gamma_{0}}{2}+\frac{\Gamma}{2}\right)}\right),\label{eq:r_ML}\end{aligned}$$
where $\Delta$ and $\Gamma$ defined as $$\Delta=-\epsilon_{0}\alpha_{0}\frac{\Gamma_{0}}{2}\mathrm{Re}\left[\underset{n,\,n\neq0}{\sum}G_{EE}^{xx}\left(0,\mathbf{r}_{n}\right)\right],\,\,\,\,\,\,\,\,\,\frac{\Gamma}{2}=\epsilon_{0}\alpha_{0}\frac{\Gamma_{0}}{2}\mathrm{Im}\left[\underset{n,\,n\neq0}{\sum}G_{EE}^{xx}\left(0,\mathbf{r}_{n}\right)\right],$$ and transmission coefficient can be calculated by $t=1+r$. The reflection coefficient can be written as $$\begin{aligned}
r & = & \frac{-\frac{ik}{2\Lambda^{2}}\alpha_{0}\frac{\Gamma_{0}}{2}}{\delta-\Delta+i\frac{k}{2\Lambda^{2}}\alpha_{0}\frac{\Gamma_{0}}{2}},\label{eq:r_monolayer}\end{aligned}$$ where $\mathrm{Im}\left[\underset{n,\,n\neq0}{\sum}G_{EE}^{xx}\left(0,\mathbf{r}_{n}\right)\right]=-\frac{1}{\epsilon_{0}\alpha_{0}}+\frac{k}{2\Lambda^{2}\epsilon_{0}}$ which is obtained in the next section using the energy conservation. Note that the real part of the interaction constant (or lattice sum), i.e. $\mathrm{Re}\left[\underset{n,\,n\neq0}{\sum}G_{EE}^{xx}\left(0,\mathbf{r}_{n}\right)\right]$ can be calculated numerically [@Tretyakov:03; @Abajo:2007; @Alaee:2017Review].
Energy conservation and interaction constant
--------------------------------------------
The conservation of energy can be used to obtain an exact expression for the imaginary part of the the interaction constant (lattice sum), i.e. $\mathrm{Im}\left[C\right]=\mathrm{Im}\left[\underset{n\neq0}{\sum}G_{EE}^{xx}\left(\mathbf{r}=\mathbf{0},\mathbf{r}_{n}\right)\right]$, we assume the nonradiative losses is zero in the atomic monolayer and have $$\begin{aligned}
A=1-R-T & = & 1-\left|r\right|^{2}-\left|1+r\right|^{2}=0\label{eq:TR_EC}\end{aligned}$$ where $T=\left|1+r\right|^{2}$ , and $R=\left|r\right|^{2}$ are the transmission and reflection from the atomic monolayer. Using Eq. \[eq:TR\_EC\], it can be shown that $\frac{\mathrm{Re}\left(r\right)}{\left|r\right|^{2}}=\mathrm{Re}\left[\frac{1}{r}\right]=-1$ and we get $$\begin{aligned}
\mathrm{Re}\left[\frac{1}{r}\right] & = & -1,\nonumber \\
\mathrm{Re}\left(\frac{1}{\frac{ik}{2\Lambda^{2}}\alpha_{\mathrm{eff}}}\right) & = & -1,\nonumber \\
\mathrm{Im}\left(\frac{1}{\alpha_{\mathrm{eff}}}\right) & = & -\frac{k}{2\Lambda^{2}}.\label{eq:EC_E1}\end{aligned}$$ Now by using the definition of the effective polarizability $\alpha_{\mathrm{eff}}=\frac{\alpha}{1-\alpha\epsilon_{0}\underset{n,\,n\neq0}{\sum}G_{EE}^{xx}\left(0,\mathbf{r}_{n}\right)}$, we have $$\begin{aligned}
\mathrm{Im}\left[\frac{1}{\alpha_{\mathrm{eff}}}\right] & = & \mathrm{Im}\left(\frac{1}{\alpha}\right)-\mathrm{Im}\left[\epsilon_{0}\underset{n,\,n\neq0}{\sum}G_{EE}^{xx}\left(0,\mathbf{r}_{n}\right)\right].\label{eq:EC_E2}\end{aligned}$$ From the definition of polarizability $\alpha=\frac{-\frac{\Gamma_{0}}{2}\alpha_{0}}{\delta+i\frac{\Gamma_{0}}{2}}$, we obtain $\mathrm{Im}\left(\frac{1}{\alpha}\right)=-\frac{1}{\alpha_{0}}=\frac{k^{3}}{6\pi}$. Finally, by using Eqs. (\[eq:EC\_E1\]) and (\[eq:EC\_E2\]), the exact expression for the imaginary part of the interaction constant, i.e. $\mathrm{Im}\left[\epsilon_{0}\underset{n,\,n\neq0}{\sum}G_{EE}^{xx}\left(0,\mathbf{r}_{n}\right)\right]$ can be obtained as [@Tretyakov:03; @Abajo:2007; @Alaee:2017Review] $$\mathrm{Im}\left[\underset{n,\,n\neq0}{\sum}G_{EE}^{xx}\left(0,\mathbf{r}_{n}\right)\right]=-\frac{1}{\epsilon_{0}\alpha_{0}}+\frac{k}{2\Lambda^{2}\epsilon_{0}}.$$
Atomic bilayer metasurface (ABM): atomic electric and magnetic mirrors
======================================================================
Let us consider an atomic bilayer composed of atoms with *only* electric dipole transition moments (see Fig. \[fig:Geometry\_dimer\]) [@Tretyakov:03]. Atoms are periodically arranged in two layers in $xy$-planes, the position of the upper layer $\mathbf{r}_{u,n}=n_{x}\Lambda\mathbf{e}_{x}+n_{y}\Lambda\mathbf{e}_{y}+\frac{l}{2}\mathbf{e}_{z}$ and the position of the lower $\mathbf{r}_{d,n}=n_{x}\Lambda\mathbf{e}_{x}+n_{y}\Lambda\mathbf{e}_{y}-\frac{l}{2}\mathbf{e}_{z}$. $n_{x}$ and $n_{y}$ are integer numbers and $\Lambda$ is the periodicity in $x$ and $y$ directions. The atomic bilayer is illuminated by an $x$-polarized incident plane wave propagating in $z$-direction $\mathbf{E}_{\mathrm{inc}}=E_{0}e^{ikz}\mathbf{e}_{x}$ . Using $\mathbf{E}_{r}=-\frac{1}{2}Z_{0}\mathbf{J}=\frac{ik}{2\Lambda^{2}\epsilon_{0}}\mathbf{p}$, the incident and transmitted electric fields by the atomic bilayer metasurface at $z=0$ read as $$\begin{aligned}
E_{r} & = & \frac{ik}{2\Lambda^{2}\epsilon_{0}}\left[p_{d}^{\mathrm{eff}}e^{-ikl/2}+p_{u}^{\mathrm{eff}}e^{ikl/2}\right],\nonumber \\
E_{t} & = & E_{0}+\frac{ik}{2\Lambda^{2}\epsilon_{0}}\left[p_{d}^{\mathrm{eff}}e^{ikl/2}+p_{u}^{\mathrm{eff}}e^{-ikl/2}\right].\label{eq:Er_Et_BL}\end{aligned}$$ where $p_{u}^{\mathrm{eff}}$ and $p_{d}^{\mathrm{eff}}$ are the *effective* (collective) upper and lower electric dipole moments and $e^{\pm ikl/2}$ are the propagation terms. The reflection and transmission coefficients are given by $$\begin{aligned}
r & = & \frac{ik}{2\Lambda^{2}\epsilon_{0}E_{0}}\left(p_{d}^{\mathrm{eff}}e^{-ikl/2}+p_{u}^{\mathrm{eff}}e^{ikl/2}\right),\nonumber \\
t & = & 1+\frac{ik}{2\Lambda^{2}\epsilon_{0}E_{0}}\left(p_{d}^{\mathrm{eff}}e^{ikl/2}+p_{u}^{\mathrm{eff}}e^{-ikl/2}\right),\label{eq:TR_BL-SM}\end{aligned}$$ and defined as $$\begin{aligned}
p_{d}^{\mathrm{eff}} & = & \epsilon_{0}\alpha\left[E_{\mathrm{inc}}\left(\mathbf{r}_{d,0}\right)+\underset{n,\,n\neq0}{\sum}G_{EE}^{xx}\left(\mathbf{r}_{d,0},\mathbf{r}_{d,n}\right)p_{d}^{\mathrm{eff}}+\underset{n}{\sum}G_{EE}^{xx}\left(\mathbf{r}_{d,0},\mathbf{r}_{u,n}\right)p_{u}^{\mathrm{eff}}\right],\nonumber \\
p_{u}^{\mathrm{eff}} & = & \epsilon_{0}\alpha\left[E_{\mathrm{inc}}\left(\mathbf{r}_{u,0}\right)+\underset{n,\,n\neq0}{\sum}G_{EE}^{xx}\left(\mathbf{r}_{u,0},\mathbf{r}_{u,n}\right)p_{u}^{\mathrm{eff}}+\underset{n}{\sum}G_{EE}^{xx}\left(\mathbf{r}_{u,0},\mathbf{r}_{d,n}\right)p_{d}^{\mathrm{eff}}\right],\label{eq:TL_H1}\end{aligned}$$ and can be written as $$\begin{aligned}
\left[\begin{array}{c}
p_{d}^{\mathrm{eff}}\\
p_{u}^{\mathrm{eff}}
\end{array}\right] & = & \left[\begin{array}{cc}
\frac{1}{\epsilon_{0}\alpha}-C_{dd} & -C_{du}\\
-C_{ud} & \frac{1}{\epsilon_{0}\alpha}-C_{uu}
\end{array}\right]^{-1}\left[\begin{array}{c}
E_{\mathrm{inc}}\left(\mathbf{r}_{d,0}\right)\\
E_{\mathrm{inc}}\left(\mathbf{r}_{u,0}\right)
\end{array}\right].\label{eq:Induced_ED_eff-SM}\end{aligned}$$ For identical upper and lower atoms, i.e. $\alpha$, the interaction constants are symmetric, i.e. $C_{ud}=C_{du}$ and $C_{uu}=C_{dd}$ and defined as $$\begin{aligned}
C_{dd} & = & \underset{n,\,n\neq0}{\sum}G_{EE}^{xx}\left(\mathbf{r}_{d,0},\mathbf{r}_{d,n}\right),\nonumber \\
C_{du} & = & \underset{n}{\sum}G_{EE}^{xx}\left(\mathbf{r}_{d,0},\mathbf{r}_{u,n}\right).\end{aligned}$$ In general, the interaction constant can be numerically calculated. For a lossless system, the imaginary part of the interaction constant (lattice sum) can be exactly calculated by using the conservation of energy [@Tretyakov:03] $$\begin{aligned}
\mathrm{Im}\left[C_{dd}\right] & = & -\frac{1}{\epsilon_{0}\alpha_{0}}+\frac{k}{2\Lambda^{2}\epsilon_{0}},\nonumber \\
\mathrm{Im}\left[C_{du}\right] & = & \frac{k}{2\Lambda^{2}\epsilon_{0}}\mathrm{cos}\left(kl\right).\end{aligned}$$ The effective electric and magnetic polarizabilities of the atomic bilayer can be defined as $$\begin{aligned}
\alpha_{\mathrm{ed}}^{\mathrm{eff}} & = & \frac{p_{d}^{\mathrm{eff}}+p_{u}^{\mathrm{eff}}}{\epsilon_{0}E_{0}}\mathrm{cos}\left(kl/2\right),\nonumber \\
\alpha_{\mathrm{md}}^{\mathrm{eff}} & = & i\frac{p_{d}^{\mathrm{eff}}-p_{u}^{\mathrm{eff}}}{\epsilon_{0}E_{0}}\mathrm{sin}\left(kl/2\right).\label{eq:EM_polarizabilities-SM}\end{aligned}$$
At the symmetric mode resonance frequency the induced dipoles for both lower and upper layers are almost identical, i.e. $p_{d}^{\mathrm{eff}}\approx p_{u}^{\mathrm{eff}}$[\[]{}see Fig. 3 of the main manuscript[\]]{}, thus we get
$$\begin{aligned}
r & = & \frac{ik}{2\Lambda^{2}\epsilon_{0}E_{0}}p_{d}^{\mathrm{eff}}\left(e^{-ikl/2}+e^{ikl/2}\right),\nonumber \\
& = & \frac{ik}{\Lambda^{2}\epsilon_{0}E_{0}}p_{d}^{\mathrm{eff}}\mathrm{cos}\left(kl/2\right),\,\,\,\mathrm{cos}\left(kl/2\right)\approx1,\,\,\,\Lambda=\frac{\lambda}{2}=\frac{\pi}{k},\nonumber \\
& \approx & \left(\frac{p_{d}^{\mathrm{eff}}}{\alpha_{0}\epsilon_{0}E_{0}}\right)\frac{6i}{\pi},\end{aligned}$$
the total reflection, i.e. $r\approx-1$ occurs when $p_{d}^{\mathrm{eff}}\approx p_{u}^{\mathrm{eff}}=\left(\alpha_{0}\epsilon_{0}E_{0}\right)i\frac{\pi}{6}$ [\[]{}see Fig. 3 of the main manuscript[\]]{}. However, at the antisymmetric mode resonance frequency, i.e. $p_{d}^{\mathrm{eff}}\approx-p_{u}^{\mathrm{eff}}$ (see Fig. 3 of the main manuscript), we have $$\begin{aligned}
r & = & \frac{ik}{2\Lambda^{2}\epsilon_{0}E_{0}}p_{d}^{\mathrm{eff}}\left(e^{-ikl/2}-e^{ikl/2}\right),\nonumber \\
& = & \frac{k}{\Lambda^{2}\epsilon_{0}E_{0}}p_{d}^{\mathrm{eff}}\mathrm{sin}\left(kl/2\right),\,\,\,\mathrm{sin}\left(kl/2\right)\approx kl/2,\,\,\,\Lambda=\frac{\lambda}{2}=\frac{\pi}{k},\nonumber \\
& \approx & \left(\frac{p_{d}^{\mathrm{eff}}}{\alpha_{0}\epsilon_{0}E_{0}}\right)\frac{6l}{\lambda},\end{aligned}$$ the total reflection, i.e. $r\approx1$ occurs when $p_{d}^{\mathrm{eff}}\approx-p_{u}^{\mathrm{eff}}=\left(\alpha_{0}\epsilon_{0}E_{0}\right)\frac{\lambda}{6l}$.
Atomic monolayer/bilayer in a planar cavity
===========================================
Planar cavity
-------------
In this section, we consider a planar cavity consists of two identical distributed Bragg reflector (DBR) mirrors [\[]{}see Fig. 4 of the main manuscript[\]]{}. The mirrors are separated by a distance of $L_{c}$. The optical properties of the system is described by the transfer matrix product [@Saleh1991] $$\begin{aligned}
M_{\mathrm{Cavity}} & = & M_{\mathrm{DBR1}}M_{\mathrm{FS}}M_{\mathrm{DBR2}},\\
M_{\mathrm{FS}} & = & \left[\begin{array}{cc}
e^{i\varphi} & 0\\
0 & e^{-i\varphi}
\end{array}\right],\\
M_{\mathrm{DBR1}} & = & \frac{1}{t_{M}}\left[\begin{array}{cc}
t_{M}^{2}-r_{M,R}r_{M,L} & r_{M,R}\\
-r_{M,L} & 1
\end{array}\right],\\
M_{\mathrm{DBR2}} & = & \frac{1}{t_{M}}\left[\begin{array}{cc}
t_{M}^{2}-r_{M,R}r_{M,L} & r_{M,L}\\
-r_{M,R} & 1
\end{array}\right],\end{aligned}$$ where $\varphi=knd$, $n$ is the refractive index of the spacer and here is assumed to be the free space ($n=1$). $r_{M,L}$ and $r_{M,R}$ are reflection coefficients of the DBR mirrors when illuminated from left and right, respectively. For the reciprocal DBR mirrors, the transmission coefficients are identical, i.e. $t_{M}=t_{M,L}=t_{M,R}$. The transmission and reflection coefficients of the planar cavity are defined as $$\begin{aligned}
t_{c} & = & \frac{t_{M}^{2}e^{ikL_{\mathrm{c}}}}{1-e^{2ikL_{\mathrm{c}}}r_{M,R}^{2}},\nonumber \\
r_{c,L} & = & r_{c,R}=-\frac{r_{M,L}+e^{2ikL_{\mathrm{c}}}r_{M,R}\left(t_{M}^{2}-r_{M,R}r_{M,L}\right)}{1-e^{2ikL_{\mathrm{c}}}r_{M,R}^{2}},\end{aligned}$$ we assumed that there is no nonradiative losses, thus the cavity completely transmits the light at $\omega=\omega_{c}$. The finesse of the planer cavity can be defined as $$F=\frac{\pi\left|\sqrt{r_{M,R}^{2}}\right|}{1-\left|r_{M,R}^{2}\right|}.$$
Atomic monolayer/bilayer metasurface inside a planar cavity
-----------------------------------------------------------
In this section, we assume that the atomic monolayer is placed in a planar cavity with reflection and transmission coefficients $r_{c}$, $t_{c}$, respectively [\[]{}see Fig. 4 of the main manuscript[\]]{}. By using the transfer matrix approach, we can obtain the reflection and transmission coefficients of the atomic monolayer inside the planar cavity. The transfer matrix of the atomic monolayer/bilayer metasurface can be written as $$\begin{aligned}
M_{AL} & = & \frac{1}{t_{AL}}\left[\begin{array}{cc}
t_{AL}^{2}-r_{AL}^{2} & r_{AL}\\
-r_{AL} & 1
\end{array}\right],\end{aligned}$$ where $r_{AL}$ and $t_{AL}$ are the reflected and transmitted coefficients of the atomic metasurface, respectively [\[]{}see Eqs. \[eq:r\_ML\] and \[eq:TR\_BL-SM\][\]]{}. For the We assumed that the transmission and reflection coefficients are identical for forward and backward directions. The nonradiative losses is zero, therefore, the conservation of energy yields to following formulas $$\begin{aligned}
\left|r_{AL}\right|^{2}+\left|t_{AL}\right|^{2} & = & 1,\nonumber \\
\frac{t_{AL}}{t_{AL}^{*}} & = & -\frac{r_{AL}}{r_{AL}^{*}},\end{aligned}$$ thus the following relation holds for the transfer matrix, i.e. $\mathrm{det}\,\left(M\right)=1$, and $$\begin{aligned}
M_{AL} & = & \left[\begin{array}{cc}
\frac{1}{t_{AL}^{*}} & \frac{r_{AL}}{t_{AL}}\\
-\frac{r_{AL}^{*}}{t_{AL}^{*}} & \frac{1}{t_{AL}}
\end{array}\right].\end{aligned}$$ The transmission coefficient from an atomic monolayer/bilayer inside a cavity read as $$t=\frac{-t_{M}^{2}\left|t_{AL}\right|^{2}}{r_{M}^{2}t_{AL}e^{ikL_{\mathrm{c}}}+2r_{M}r_{AL}t_{AL}^{*}\mathrm{cos}\left(2kD\right)-t_{AL}^{*}e^{-ikL_{\mathrm{c}}}},\label{eq:t_Cavity_AL}$$ The reflection and transmission coefficients of an atomic monolayer metasurface (AMM) can be written as [\[]{}see Eq. \[eq:r\_monolayer\][\]]{} $$r_{AL}=\frac{-i\Gamma_{AL}/2}{\delta_{AL}+i\Gamma_{AL}/2},\,\,\,\,\frac{\Gamma_{AL}}{2}=\frac{k\alpha_{0}\Gamma_{0}}{4\Lambda^{2}},\,\,\,\,\delta_{AL}=\omega-\omega_{a}-\Delta,\label{eq:r_AL}$$ Now, by using $t_{AL}=1+r_{AL}=\frac{\delta_{AL}}{\delta_{AL}+i\Gamma_{AL}/2}$ and substituting Eq. \[eq:r\_AL\] into Eq. \[eq:t\_Cavity\_AL\], we obtain $$t_{\mathrm{AMM}}=\frac{t_{M}^{2}\delta_{AL}e^{ikL_{\mathrm{c}}}}{\delta_{AL}\left(1-e^{2ikL_{\mathrm{c}}}r_{M}^{2}\right)+i\frac{\Gamma_{a}}{2}\left[1+2r_{M}e^{ikL_{\mathrm{c}}}\mathrm{cos}\left(2kD\right)+r_{M}^{2}e^{2ikL_{\mathrm{c}}}\right]}.$$ Note that the atomic metasurface is placed at a distance $D$ (see Fig.4 of the main manuscript). The planar cavity consisting of two mirrors separated by $L_{c}.$ A similar expression can be obtained for an atomic bilayer metasurface (ABM) in a cavity $$t_{\mathrm{ABM}}=-\frac{r_{AL}t_{M}^{2}\left|r_{AL}+1\right|^{2}e^{ikL_{\mathrm{c}}}}{-\left|r_{AL}\right|^{2}-r_{AL}+r_{M}^{2}(r_{AL}^{2}+r_{AL})e^{2ikL_{\mathrm{c}}}-2ie^{ikL_{\mathrm{c}}}r_{21}r_{AL}\left[\left|r_{AL}\right|^{2}\mathrm{sin}\left(2kD\right)+\mathrm{Im}\left(r_{AL}^{*}e^{2ikD}\right)\right]}.$$ Note that $r_{AL}$ and $t_{AL}$ for the AMM and ABM can be obtained from Eq. \[eq:r\_monolayer\] and Eq. \[eq:TR\_BL-SM\], respectively.
Scattered fields from an atomic bilayer metasurface using a Gaussian beam excitation
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For an experimental realization limited number of atoms and a Gaussian beam excitation would be necessary. The Gaussian beam is defined as $$\begin{aligned}
\mathbf{E}_{\mathrm{inc}}\left(x,y,z\right) & = & \mathbf{e}_{x}E_{0}\frac{w_{0}}{w\left(z\right)}e^{ikz}e^{-i\varphi\left(z\right)}e^{-\frac{x^{2}+y^{2}}{w^{2}\left(z\right)}}e^{ik\frac{x^{2}+y^{2}}{2R\left(z\right)}},\label{eq:GaussianBeam}\end{aligned}$$ the beam parameters are defined as $$\begin{aligned}
w\left(z\right) & = & w_{0}\sqrt{1+\left(\frac{z}{z_{R}}\right)^{2}},\,\,\,\,z_{R}=\frac{\pi w_{0}^{2}}{\lambda},\nonumber \\
\varphi\left(z\right) & = & \mathrm{arctang}\left(\frac{z}{z_{R}}\right),\nonumber \\
R\left(z\right) & = & z\left[1+\left(\frac{z_{R}}{z}\right)^{2}\right],\end{aligned}$$ where $w_{0}$ is the beam waist at its focal point, $R\left(z\right)$ is the radius of curvature of the beam at $z$, and $\varphi\left(z\right)$ is the Gouy phase at $z$. The Gaussian beam is propagating along the $z$ direction. In our numerical calculations, we assumed the cross section of the Gaussian beam is smaller than the area of the finite atomic layer. The parameters of Gaussian beam is given in the caption of Fig. \[fig:GaussianResults\]. The incident and scattered electric fields for an electric and a magnetic mirror using such a Gaussian beam are demonstrated in Fig. \[fig:GaussianResults\]. For the symmetric mode ($z>0$), the scattered field is opposite in sign with respect to the incident (both real and imaginary). Thus, the transmitted field which is the sum of the incident and scattered electric fields is zero ($E_{t}=E_{\mathrm{inc}}+E_{\mathrm{sca}}=0$) [\[]{}Fig. \[fig:GaussianResults\], see the blue box[\]]{}. However, for ($z<0$), only the *real* part of the scattered field is opposite in sign with respect to the *real* part of the incident. Thus it acts as a electric mirror in the symmetric mode analogous to a perfect electric conductor. For the antisymmetric mode ($z>0$) similar to the symmetric mode ($z>0$), the scattered field is opposite in sign with respect to the incident (both real and imaginary). Thus, the transmitted field which is the sum of the incident and scattered electric fields is zero ($E_{t}=E_{\mathrm{inc}}+E_{\mathrm{sca}}=0$) [\[]{}Fig. \[fig:GaussianResults\], see the red box[\]]{}. However, for ($z<0$), the *imaginary* part of the scattered field is opposite in sign with respect to the *imaginary* part of the incident. Thus it acts as a magnetic mirror analogous to a perfect magnetic conductor. Using Gaussian beam, we also calculated transmission and reflection for limited number of atoms [\[]{}$15\times15\times2$ atoms[\]]{}. The results for the two illuminations (i.e. Gaussian and plane wave) are in a excellent agreement [\[]{}see the main manuscript Fig. 3[\]]{}.
![Atomic bilayer composed of $15\times15\times2$ atoms when illuminated by a Gaussian beam at the symmetric (electric mirror) and antisymmetric (magnetic mirror) modes. We considered a normal incidence where the cross section of the Gaussian beam is at the center of the atomic bilayer, z = 0 with $E_{0}=1$, and $W_{0}=\lambda_{a}$. $E_{\mathrm{inc}}$ and $E_{\mathrm{sca}}$ are the incident and scattered electric fields, respectively. It can be seen that the scattering and incident fields destructively interfere at the forward direction. \[fig:GaussianResults\]](Figure3_SM.pdf){width="14cm"}
Green functions and scattered fields
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In this section, we define the Green function which is used to calculation the interaction constant for the atomic dimer and atomic array. The electric and magnetic dyadic Green functions in free space, respectively, read [@tai1994dyadic] $$\bar{\bar{{\bf G}}}_{E}\left(\mathbf{r},\mathbf{r}^{\prime}\right)=\left({\bf \bar{\bar{I}}}+\frac{1}{k^{2}}\nabla\nabla\right)G_{0}\left(\mathbf{r},\mathbf{r}^{\prime}\right),$$ and $$\bar{\bar{{\bf G}}}_{M}\left(\mathbf{r},\mathbf{r}^{\prime}\right)=\nabla\times\left[{\bf \bar{\bar{I}}}G_{0}\left(\mathbf{r},\mathbf{r}^{\prime}\right)\right]=\nabla G_{0}\left(\mathbf{r},\mathbf{r}^{\prime}\right)\times{\bf \bar{\bar{I}}},\label{eq:magnetic dyadic}$$ where, $G_{0}\left(\mathbf{r},\mathbf{r}^{\prime}\right)={\rm e^{{\it ik\vert\mathbf{r}-\mathbf{r}^{\prime}\vert}}/\left(4\pi\vert\mathbf{r}-\mathbf{r}^{\prime}\vert\right)}$ is the scalar free space Green function and the identity $\nabla\times\left(G_{0}{\bf \bar{\bar{I}}}\right)=G_{0}\nabla\times{\bf \bar{\bar{I}}}+\nabla G_{0}\times{\bf \bar{\bar{I}}}$ (where ${\bf \bar{\bar{I}}}$ is the identity dyadic) is used in Eq. (\[eq:magnetic dyadic\]). The electric and magnetic fields in term of electric vector potential ${\bf A}$ read ${\bf E}=i\omega\left({\bf A}+\frac{1}{k^{2}}\nabla\nabla\cdot{\bf A}\right)$ and ${\bf H}=\frac{1}{\mu_{0}}\left(\nabla\times{\bf A}\right)$, respectively ($\nabla$ is taken over variable$\mathbf{r}$). For an electric dipole ${\bf p}$, the electric vector potential ${\bf A}$ reads [@Jackson1999] $${\bf A}\left(\mathbf{r}\right)=-i\omega\mu_{0}\frac{e^{{\it ik\vert\mathbf{r}-\mathbf{r}^{\prime}\vert}}}{4\pi\vert\mathbf{r}-\mathbf{r}^{\prime}\vert}{\bf p}\left(\mathbf{r}^{\prime}\right)=-i\omega\mu_{0}G_{0}\left(\mathbf{r},\mathbf{r}^{\prime}\right){\bf p}\left(\mathbf{r}^{\prime}\right).$$ Therefore, the electric and magnetic fields ${\bf E_{p}}$ and ${\bf H_{p}}$ of an electric dipole moment, respectively, read $${\bf E_{p}}\left(\mathbf{r}\right)=\omega^{2}\mu_{0}\bar{\bar{{\bf G}}}_{E}\left(\mathbf{r},\mathbf{r}^{\prime}\right)\cdot{\bf p}\left(\mathbf{r}^{\prime}\right),\label{e_filed_p}$$ $${\bf H_{p}\left(\mathbf{r}\right)}=-i\omega\bar{\bar{{\bf G}}}_{M}\left(\mathbf{r},\mathbf{r}^{\prime}\right)\cdot{\bf p}\left(\mathbf{r}^{\prime}\right).\label{H_filed_p}$$ For a magnetic dipole ${\bf m}$, the electric and magnetic fields ${\bf E_{m}}$ and ${\bf H_{m}}$, respectively, read $${\bf E_{m}}\left(\mathbf{r}\right)=i\omega Z_{0}\bar{\bar{{\bf G}}}_{M}\left(\mathbf{r},\mathbf{r}^{\prime}\right)\cdot\frac{{\bf m}\left(\mathbf{r}^{\prime}\right)}{c},\label{e_filed_m}$$ $${\bf H_{m}\left(\mathbf{r}\right)}=\omega^{2}\epsilon_{0}Z_{0}\bar{\bar{{\bf G}}}_{E}\left(\mathbf{r},\mathbf{r}^{\prime}\right)\cdot\frac{{\bf m}\left(\mathbf{r}^{\prime}\right)}{c},\label{h_filed_m}$$ since the electric vector potential for the dipole ${\bf m}$ reads [@Jackson1999] $${\bf A}\left(\mathbf{r}\right)=Z_{0}\nabla G_{0}\left(\mathbf{r},\mathbf{r}^{\prime}\right)\times\frac{{\bf m}\left(\mathbf{r}^{\prime}\right)}{c}.\label{eq:vector_pt_m}$$ Note that we have used [@Jackson1999] $${\bf A}\left(\mathbf{r}\right)=i\omega\mu_{0}\frac{e^{{\it ik\vert\mathbf{r}-\mathbf{r}^{\prime}\vert}}}{4\pi\vert\mathbf{r}-\mathbf{r}^{\prime}\vert}\left(1-\frac{1}{ik\vert\mathbf{r}-\mathbf{r}^{\prime}\vert}\right){\bf \left({\bf n}\times\frac{{\bf m}\left(\mathbf{r}^{\prime}\right)}{{\it c}}\right)},$$ and $$\nabla G_{0}\left(\mathbf{r},\mathbf{r}^{\prime}\right)=ikG_{0}\left(\mathbf{r},\mathbf{r}^{\prime}\right)\left(1-\frac{1}{ik\vert\mathbf{r}-\mathbf{r}^{\prime}\vert}\right){\bf \frac{\left(\mathbf{r}-\mathbf{r}^{\prime}\right)}{\vert\mathbf{r}-\mathbf{r}^{\prime}\vert}},\,\,\,\,{\bf n}=\frac{\left(\mathbf{r}-\mathbf{r}^{\prime}\right)}{\vert\mathbf{r}-\mathbf{r}^{\prime}\vert},$$ in Eq. (\[eq:vector\_pt\_m\]). Also note that we have used identity ${\bf a\times{\bf b}}={\bf a}\times\left(\bar{\bar{{\bf I}}}\cdot{\bf b}\right)=\left({\bf a}\times\bar{\bar{{\bf I}}}\right)\cdot{\bf b}$ in obtaining Eq.(\[e\_filed\_m\]). Finally, the fields created by an electric and a magnetic dipole from Eqs. (\[e\_filed\_p\])-(\[e\_filed\_m\]) read $$\begin{aligned}
\mathbf{E}\left(\mathbf{r}\right) & = & {\bf E_{p}}\left(\mathbf{r}\right)+{\bf E_{m}\left(\mathbf{r}\right)}=\mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r},\mathbf{r}^{\prime}\right)\cdot\mathbf{p}\left(\mathbf{r}^{\prime}\right)+{\bf \bar{\bar{G}}}_{EM}\left(\mathbf{r},\mathbf{r}^{\prime}\right)\cdot\frac{\mathbf{m}\left(\mathbf{r}^{\prime}\right)}{c},\nonumber \\
Z_{0}\mathbf{H}\left(\mathbf{r}\right) & = & Z_{0}\left[{\bf H_{p}}\left(\mathbf{r}\right)+{\bf H_{m}}\left(\mathbf{r}\right)\right]={\bf \bar{\bar{G}}}_{ME}\left(\mathbf{r},\mathbf{r}^{\prime}\right)\cdot\mathbf{p}\left(\mathbf{r}^{\prime}\right)+\mathbf{\bar{\bar{G}}}_{MM}\left(\mathbf{r},\mathbf{r}^{\prime}\right)\cdot\frac{\mathbf{m}\left(\mathbf{r}^{\prime}\right)}{c},\label{eq:EH_GreenFun}\end{aligned}$$ where,
$$\begin{aligned}
{\bf \bar{\bar{G}}}_{EE}\left(\mathbf{r},\mathbf{r}^{\prime}\right) & = & \omega^{2}\mu_{0}\bar{\bar{{\bf G}}}_{E}\left(\mathbf{r},\mathbf{r}^{\prime}\right),\,\,\,{\bf \bar{\bar{G}}}_{EM}\left(\mathbf{r},\mathbf{r}^{\prime}\right)=i\omega Z_{0}\bar{\bar{{\bf G}}}_{M}\left(\mathbf{r},\mathbf{r}^{\prime}\right),\nonumber \\
\mathbf{\bar{\bar{G}}}_{ME}\left(\mathbf{r},\mathbf{r}^{\prime}\right) & = & -{\bf \bar{\bar{G}}}_{EM}\left(\mathbf{r},\mathbf{r}^{\prime}\right),\,\,\,\mathbf{\bar{\bar{G}}}_{MM}\left(\mathbf{r},\mathbf{r}^{\prime}\right)={\bf \bar{\bar{G}}}_{EE}\left(\mathbf{r},\mathbf{r}^{\prime}\right),\end{aligned}$$
and can be written as
$${\bf \bar{\bar{G}}}_{EE}\left(\mathbf{r},\mathbf{r}^{\prime}\right)=\mathbf{\bar{\bar{G}}}_{MM}\left(\mathbf{r},\mathbf{r}^{\prime}\right)=\frac{3}{2\alpha_{0}\epsilon_{0}}e^{i\zeta}\left[\left(\frac{1}{\zeta}-\frac{1}{\zeta^{3}}+\frac{i}{\zeta^{2}}\right)\bar{\bar{{\bf I}}}+\left(-\frac{1}{\zeta}+\frac{3}{\zeta^{3}}-\frac{3i}{\zeta^{2}}\right)\mathbf{\mathbf{n}n}\right],$$
$${\bf \bar{\bar{G}}}_{EM}\left(\mathbf{r},\mathbf{r}^{\prime}\right)=-\mathbf{\bar{\bar{G}}}_{ME}\left(\mathbf{r},\mathbf{r}^{\prime}\right)=-\frac{3}{2\alpha_{0}\epsilon_{0}}e^{i\zeta}\left(\frac{1}{\zeta}-\frac{1}{i\zeta^{2}}\right)\mathbf{n}\times\bar{\bar{{\bf I}}},$$
where $\mathbf{n}=\frac{\mathbf{r}-\mathbf{r}^{\prime}}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}$, $\alpha_{0}=\frac{6\pi}{k^{3}}$ and $,\zeta=k\left(\mathbf{r}-\mathbf{r}^{\prime}\right)$ and can be also written as
$$\begin{aligned}
G_{EE}^{\alpha\beta}\left(\zeta=k\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right) & = & \frac{3}{2\alpha_{0}\epsilon_{0}}e^{i\zeta}\left[g_{1}\left(\zeta\right)\delta_{\alpha\beta}+g_{2}\left(\zeta\right)\frac{\zeta_{\alpha}\zeta_{\beta}}{\zeta^{2}}\right],\nonumber \\
g_{1}\left(\zeta\right) & = & \left(\frac{1}{\zeta}-\frac{1}{\zeta^{3}}+\frac{i}{\zeta^{2}}\right),\nonumber \\
g_{2}\left(\zeta\right) & = & \left(-\frac{1}{\zeta}+\frac{3}{\zeta^{3}}-\frac{3i}{\zeta^{2}}\right),\end{aligned}$$
Note that Eq. \[eq:EH\_GreenFun\] can be also written as
$$\begin{aligned}
\mathbf{E}\left(\mathbf{r}\right) & = & {\bf E_{p}}\left(\mathbf{r}\right)+{\bf E_{m}\left(\mathbf{r}\right)}=\mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r},\mathbf{r}^{\prime}\right)\cdot\mathbf{p}\left(\mathbf{r}^{\prime}\right)+g_{EM}\mathbf{n}\times\frac{\mathbf{m}\left(\mathbf{r}^{\prime}\right)}{c},\nonumber \\
Z_{0}\mathbf{H}\left(\mathbf{r}\right) & = & Z_{0}\left[{\bf H_{p}}\left(\mathbf{r}\right)+{\bf H_{m}}\left(\mathbf{r}\right)\right]=g_{ME}\mathbf{n}\times\mathbf{p}\left(\mathbf{r}^{\prime}\right)+\mathbf{\bar{\bar{G}}}_{MM}\left(\mathbf{r},\mathbf{r}^{\prime}\right)\cdot\frac{\mathbf{m}\left(\mathbf{r}^{\prime}\right)}{c},\label{eq:EH_GreenFunFinal}\end{aligned}$$
where $g_{ME}=-g_{EM}=\frac{3}{2\alpha_{0}\epsilon_{0}}e^{i\zeta}\left(\frac{1}{\zeta}-\frac{1}{i\zeta^{2}}\right)$.
Enhancing the decay rate of a magnetic emitter
==============================================
Coupled dipole theory
---------------------
Let us consider N atoms with electric dipole transition moments in free space. The self-consistent equation for the induced dipole moments of $i$th atom placed at $\mathbf{r}=\mathbf{r}_{i}$ read [@foldy1945; @Mulholland:94; @lagendijk1996; @Alaee:2017Review] $$\begin{aligned}
\mathbf{p}\left(\mathbf{r}_{i}\right) & = & \epsilon_{0}\alpha_{i}\left[\mathbf{E}_{\mathrm{inc}}\left(\mathbf{r}_{i}\right)+\underset{i\neq j}{\sum}\mathbf{E}_{\mathrm{sca}}\left(\left|\mathbf{r}_{j}-\mathbf{r}_{i}\right|\right)\right],\label{eq:CDT}\end{aligned}$$ where $\mathbf{E}_{\mathrm{inc}}\left(\mathbf{r}_{i}\right)$ is the incident field at the atom position, $\alpha_{i}$ is the atomic polarizability and $\underset{i\neq j}{\sum}\mathbf{E}_{\mathrm{sca}}\left(\left|\mathbf{r}_{j}-\mathbf{r}_{i}\right|\right)$ are the interaction fields created by the all atoms at $\mathbf{r}=\mathbf{r}_{i}$. Using Eq. \[eq:CDT\], we can obtain $$\begin{aligned}
\underset{i\neq j}{\sum}\mathbf{E}_{\mathrm{sca}}\left(\left|\mathbf{r}_{j}-\mathbf{r}_{i}\right|\right) & = & \mathbf{p}\left(\mathbf{r}_{i}\right)/\epsilon_{0}\alpha_{i}-\mathbf{E}_{\mathrm{inc}}\left(\mathbf{r}_{i}\right),\end{aligned}$$ Thus, the total field at $\mathbf{r}=\mathbf{r}_{i}$, can be calculated by using $$\begin{aligned}
\mathbf{E}_{\mathrm{tot}}\left(\mathbf{r}_{i}\right) & = & \mathbf{E}_{\mathrm{inc}}\left(\mathbf{r}_{i}\right)+\underset{i}{\sum}\mathbf{E}_{\mathrm{sca}}\left(\left|\mathbf{r}_{j}-\mathbf{r}_{i}\right|\right),\nonumber \\
& = & \mathbf{E}_{\mathrm{inc}}\left(\mathbf{r}_{i}\right)+\mathbf{E}_{\mathrm{sca}}\left(\left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|=0\right)+\underset{i\neq j}{\sum}\mathbf{E}_{\mathrm{sca}}\left(\left|\mathbf{r}_{j}-\mathbf{r}_{i}\right|\right),\nonumber \\
& = & \mathbf{E}_{\mathrm{inc}}\left(\mathbf{r}_{i}\right)+\mathbf{\bar{\bar{G}}}_{EE}\left(\left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|=0\right)\cdot\mathbf{p}\left(\mathbf{r}_{i}\right)+\frac{\mathbf{p}\left(\mathbf{r}_{i}\right)}{\epsilon_{0}\alpha_{i}}-\mathbf{E}_{\mathrm{inc}}\left(\mathbf{r}_{i}\right),\nonumber \\
& = & \left[\mathbf{\bar{\bar{G}}}_{EE}\left(\left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|=0\right)\cdot\mathbf{p}\left(\mathbf{r}_{i}\right)+\frac{1}{\epsilon_{0}\alpha_{i}}\mathbf{p}\left(\mathbf{r}_{i}\right)\right],\label{eq:Etot}\end{aligned}$$ where we used $\mathbf{E}_{\mathrm{sca}}\left(\left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|=0\right)=\mathbf{\bar{\bar{G}}}_{EE}\left(\left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|=0\right)\cdot\mathbf{p}\left(\mathbf{r}_{i}\right)$. Now by using the total field, we can compute the absorbed power $$\begin{aligned}
P_{\mathrm{abs}} & = & -\frac{\omega}{2}\mathrm{Im}\left[\underset{i}{\sum}\mathbf{p}^{*}\left(\mathbf{r}_{i}\right)\cdot\mathbf{E}_{\mathrm{tot}}\left(\mathbf{r}_{i}\right)\right],\nonumber \\
& = & -\frac{\omega}{2}\mathrm{Im}\left\{ \underset{i}{\sum}\mathbf{p}^{*}\left(\mathbf{r}_{i}\right)\cdot\left[\mathbf{\bar{\bar{G}}}_{EE}\left(\left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|=0\right)\cdot\mathbf{p}\left(\mathbf{r}_{i}\right)+\frac{1}{\epsilon_{0}\alpha_{i}}\mathbf{p}\left(\mathbf{r}_{i}\right)\right]\right\} ,\nonumber \\
& = & -\frac{\omega}{2\epsilon_{0}}\underset{i}{\sum}\left|\mathbf{p}\left(\mathbf{r}_{i}\right)\right|^{2}\left(\frac{1}{\alpha_{0}}+\mathrm{Im}\left[\frac{1}{\alpha_{i}}\right]\right),\label{eq:P_abs}\end{aligned}$$Note that $\mathrm{Im}\left[\mathbf{\bar{\bar{G}}}_{EE}\left(\left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|=0\right)\right]=\frac{1}{\epsilon_{0}\alpha_{0}}\bar{\bar{{\bf I}}}$ [@lagendijk1996; @Lagendijk1998; @Alaee:2017Review] and for lossless dipoles $\mathrm{Im}\left[\frac{1}{\alpha_{i}}\right]=-\frac{1}{\alpha_{0}}$, therefore $P_{\mathrm{abs}}=0$. The scattered power can be calculated as $$\begin{aligned}
P_{\mathrm{sca}} & = & \frac{\omega}{2}\mathrm{Im}\left[\underset{i}{\sum}\mathbf{p}^{*}\left(\mathbf{r}_{i}\right)\cdot\underset{j}{\sum}\mathbf{E}_{\mathrm{sca}}\left(\left|\mathbf{r}_{j}-\mathbf{r}_{i}\right|\right)\right],\nonumber \\
& = & \frac{\omega}{2}\mathrm{Im}\left\{ \underset{i}{\sum}\mathbf{p}^{*}\left(\mathbf{r}_{i}\right)\cdot\left[\mathbf{\bar{\bar{G}}}_{EE}\left(\left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|=0\right)\cdot\mathbf{p}\left(\mathbf{r}_{i}\right)+\mathbf{p}\left(\mathbf{r}_{i}\right)/\epsilon_{0}\alpha_{i}-\mathbf{E}_{\mathrm{inc}}\left(\mathbf{r}_{i}\right)\right]\right\} \nonumber \\
& = & \frac{\omega}{2}\mathrm{Im}\left\{ \underset{i}{\sum}\mathbf{p}_{i}^{*}\left(\mathbf{r}_{i}\right)\cdot\left[\frac{1}{\epsilon_{0}\alpha_{0}}\mathbf{p}_{i}\left(\mathbf{r}_{i}\right)+\frac{1}{\epsilon_{0}\alpha_{i}}\mathbf{p}_{i}\left(\mathbf{r}_{i}\right)-\mathbf{E}_{\mathrm{inc}}\left(\mathbf{r}_{i}\right)\right]\right\} \nonumber \\
& = & \frac{\omega}{2}\underset{i}{\sum}\left|\mathbf{p}\left(\mathbf{r}_{i}\right)\right|^{2}\left(\frac{1}{\epsilon_{0}\alpha_{0}}+\mathrm{Im}\left[\frac{1}{\epsilon_{0}\alpha_{i}}\right]\right)-\frac{\omega}{2}\underset{i}{\sum}\mathrm{Im}\left[\mathbf{p}^{*}\left(\mathbf{r}_{i}\right)\cdot\mathbf{E}_{\mathrm{inc}}\left(\mathbf{r}_{i}\right)\right],\nonumber \\
& = & -P_{\mathrm{abs}}-\frac{\omega}{2}\underset{i}{\sum}\mathrm{Im}\left[\mathbf{p}^{*}\left(\mathbf{r}_{i}\right)\cdot\mathbf{E}_{\mathrm{inc}}\left(\mathbf{r}_{i}\right)\right],\label{eq:Psca}\end{aligned}$$ now by using power conservation, the extracted power can be defined as
$$\begin{aligned}
P_{\mathrm{ext}} & = & -\frac{\omega}{2}\underset{i}{\sum}\mathrm{Im}\left[\mathbf{p}_{i}^{*}\left(\mathbf{r}_{i}\right)\cdot\mathbf{E}_{\mathrm{inc}}\left(\mathbf{r}_{i}\right)\right]=P_{\mathrm{sca}}+P_{\mathrm{abs}}.\label{eq:Pext}\end{aligned}$$
In conclusion, the absorbed, scattered, and extracted powers read as
$$P_{\mathrm{abs}}=-\frac{\omega}{2\epsilon_{0}}\underset{i}{\sum}\left|\mathbf{p}\left(\mathbf{r}_{i}\right)\right|^{2}\left(\frac{1}{\alpha_{0}}+\mathrm{Im}\left[\frac{1}{\alpha_{i}}\right]\right),$$
$$P_{\mathrm{sca}}=\frac{\omega}{2\epsilon_{0}}\underset{i}{\sum}\left|\mathbf{p}\left(\mathbf{r}_{i}\right)\right|^{2}\left(\frac{1}{\alpha_{0}}+\mathrm{Im}\left[\frac{1}{\alpha_{i}}\right]\right)-\frac{\omega}{2}\underset{i}{\sum}\mathrm{Im}\left[\mathbf{p}^{*}\left(\mathbf{r}_{i}\right)\cdot\mathbf{E}_{\mathrm{inc}}\left(\mathbf{r}_{i}\right)\right],$$
$$P_{\mathrm{ext}}=-\frac{\omega}{2}\underset{i}{\sum}\mathrm{Im}\left[\mathbf{p}^{*}\left(\mathbf{r}_{i}\right)\cdot\mathbf{E}_{\mathrm{inc}}\left(\mathbf{r}_{i}\right)\right]=\frac{\omega}{2}\underset{i}{\sum}\mathrm{Im}\left[\mathbf{p}\left(\mathbf{r}_{i}\right)\cdot\mathbf{E}^{*}_{\mathrm{inc}}\left(\mathbf{r}_{i}\right)\right].$$
Using duality ($\mathbf{p}\leftrightarrow\frac{\mathbf{m}}{c}$ and $\mathbf{E}\leftrightarrow\frac{\mathbf{H}}{Z_{0}}$ see Ref. [@Jackson1999]), similar results can be obtained for a magnetic dipole moment, i.e.
$$P_{\mathrm{abs}}=-\frac{\omega}{2\epsilon_{0}}\underset{i}{\sum}\left|\frac{\mathbf{m}\left(\mathbf{r}_{i}\right)}{c}\right|^{2}\left(\frac{1}{\alpha_{0}}+\mathrm{Im}\left[\frac{1}{\alpha_{i}}\right]\right),$$
$$P_{\mathrm{sca}}=\frac{\omega}{2\epsilon_{0}}\underset{i}{\sum}\left|\frac{\mathbf{m}\left(\mathbf{r}_{i}\right)}{c}\right|^{2}\left(\frac{1}{\alpha_{0}}+\mathrm{Im}\left[\frac{1}{\alpha_{i}}\right]\right)-\frac{\omega}{2}\underset{i}{\sum}\mathrm{Im}\left[\frac{\mathbf{m}^{*}\left(\mathbf{r}_{i}\right)}{c}\cdot\frac{\mathbf{H}_{\mathrm{inc}}\left(\mathbf{r}_{i}\right)}{Z_{0}}\right],$$
$$P_{\mathrm{ext}}=-\frac{\omega}{2}\underset{i}{\sum}\mathrm{Im}\left[\frac{\mathbf{m}^{*}\left(\mathbf{r}_{i}\right)}{c}\cdot\frac{\mathbf{H}_{\mathrm{inc}}\left(\mathbf{r}_{i}\right)}{Z_{0}}\right]=\frac{\omega}{2}\underset{i}{\sum}\mathrm{Im}\left[\frac{\mathbf{m}\left(\mathbf{r}_{i}\right)}{c}\cdot\frac{\mathbf{H}^{*}_{\mathrm{inc}}\left(\mathbf{r}_{i}\right)}{Z_{0}}\right].$$
In the next subsection, we used above expressions to obtain the radiated power of a test magnetic dipole emitter.
Emission rate enhancement of a magnetic emitter
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The power radiated by a test magnetic dipole emitter, i.e. $\bm{\mu}_{t}=\mu_{t}\mathbf{n}_{\mu}$ placed at $\mathbf{r}_{0}$ is given by [@Jackson1999] $$\begin{aligned}
P_{\mathrm{rad}}^{\mathrm{fs}} & = & \frac{ck^{4}}{12\pi\epsilon_{0}}\left|\frac{\bm{\mu}_{t}}{c}\right|^{2}=\frac{1}{2}\frac{\omega}{\alpha_{0}\epsilon_{0}}\left|\frac{\bm{\mu}_{t}}{c}\right|^{2},\label{eq:P_rad_MD_fs-1}\end{aligned}$$ where $\mathbf{n}_{\mu}$ is the unit vector in the direction of the dipole moment. The power radiated by a magnetic dipole emitter $\bm{\mu}_{t}=\bm{\mu}_{t}\left(\mathbf{r}_{0}\right)$ when placed close to a antenna consist of $N$ atoms at position $\mathbf{r}_{i}$ with *only* electric dipole moment $\mathbf{p}_{i}$, read as $$\begin{aligned}
P_{\mathrm{rad}}^{\mathrm{ant}} & = & \frac{\omega}{2}\mathrm{Im}\left[\frac{\bm{\mu}_{t}}{c}^{*}\cdot Z_{0}\mathbf{H}_{\mathrm{local}}\left(\mathbf{r}_{0}\right)\right],\label{eq:P_rad_MD_ant}\\
& = & \frac{\omega}{2}\left|\frac{\bm{\mu}_{t}}{c}\right|^{2}\mathrm{Im}\left[\mathbf{n}_{\mu}^{T}\mathbf{G}_{\mathrm{tot}}\left(\mathbf{r}_{0},\mathbf{r}_{0}\right)\mathbf{n}_{\mu}\right],\end{aligned}$$ where $\mathbf{G}_{\mathrm{tot}}\left(\mathbf{r}_{0},\mathbf{r}_{0}\right)$ is the total Green function at the position of the emitter, and the $\mathbf{H}\left(\mathbf{r}_{0}\right)$ is the electric field at the dipole position $\mathbf{r}_{0}$ can be obtained by $$\begin{aligned}
Z_{0}\mathbf{H}_{\mathrm{local}}\left(\mathbf{r}_{0}\right) & = & \mathbf{\bar{\bar{G}}}_{MM}\left(\mathbf{r}_{0},\mathbf{r}_{0}\right)\cdot\frac{\bm{\mu}_{t}}{c}+{\sum_{i=1}^N}g_{ME}\left(\mathbf{r}_{0},\mathbf{r}_{i}\right)\left[\mathbf{n}_{r_{0}r_{i}}\times\mathbf{p}\left(\mathbf{r}_{i}\right)\right],\label{eq:AA1}\end{aligned}$$ where $\mathbf{n}_{r_{0}r_{i}}=\frac{\mathbf{r}_{0}-\mathbf{r}_{i}}{\left|\mathbf{r}_{0}-\mathbf{r}_{i}\right|}$, $g_{ME}$ is defined in Eq. \[eq:EH\_GreenFunFinal\] and $\mathbf{p}\left(\mathbf{r}_{i}\right)$ can be calculated by using Eq. \[eq:EH\_GreenFunFinal\] $$\begin{aligned}
\mathbf{p}\left(\mathbf{r}_{i}\right) & = & \epsilon_{0}\alpha_{i}\mathbf{E}_{\mathrm{local}}\left(\mathbf{r}_{i}\right)\nonumber \\
& = & \epsilon_{0}\alpha_{i}\left[\mathbf{E}_{\mu_{t}}\left(\mathbf{r}_{i}\right)+{\sum_{j \neq i}^N}\left[\mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r}_{i},\mathbf{r}_{j}\right)\cdot\mathbf{p}\left(\mathbf{r}_{j}\right)\right]\right],\end{aligned}$$ where $\mathbf{E}_{\mu_{t}}\left(\mathbf{r}_{i}\right)\equiv g_{EM}\left(\mathbf{r}_{i},\mathbf{r}_{0}\right)\left(\mathbf{n}_{r_{i}r_{0}}\times\frac{\bm{\mu}_{t}}{c}\right)$. Thus it can be written in the following form $$\begin{aligned}
\mathbf{E}_{\mu_{t}}\left(\mathbf{r}_{i}\right) & = & \frac{1}{\epsilon_{0}\alpha_{i}}\mathbf{p}\left(\mathbf{r}_{i}\right)-{\sum_{j \neq i}^N}\left[\mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r}_{i},\mathbf{r}_{j}\right)\cdot\mathbf{p}\left(\mathbf{r}_{j}\right)\right],\end{aligned}$$ and we can write $$\begin{aligned}
\left[\overline{\mathbf{p}}\right]_{3N\times1} & = & \left[A\right]_{3N\times3N}\left[\overline{\mathbf{E}}_{m_{0}}\right]_{3N\times1},\label{eq:P_eff}\end{aligned}$$
where $\left(A^{-1}\right)_{ij}^{\mu\nu}=\frac{1}{\alpha\epsilon_{0}}\delta_{ij}\delta_{\mu\nu}-\left(1-\delta_{ij}\right)G_{EE}^{\mu\nu}\left(\mathbf{r}_{i},\mathbf{r}_{j}\right)$ and $G_{EE}^{\mu\nu}\left(\mathbf{r}_{i},\mathbf{r}_{j}\right)$ is the shorthand for the $(\mu,\nu)$th matrix element of $\mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r}_{i},\mathbf{r}_{j}\right)$ and $(\mu,\nu)\rightarrow\left(x,y,z\right)$. $\overline{\mathbf{E}}_{\mu_{t}}=\left[\begin{array}{cccccc}
g_{EM}\left(\mathbf{r}_{1},\mathbf{r}_{0}\right)\left(\mathbf{n}_{r_{1}r_{0}}\times\frac{\mathbf{m}_{0}}{c}\right), & g_{EM}\left(\mathbf{r}_{2},\mathbf{r}_{0}\right)\left(\mathbf{n}_{r_{2}r_{0}}\times\frac{\mathbf{m}_{0}}{c}\right), & & \cdots & , & g_{EM}\left(\mathbf{r}_{N},\mathbf{r}_{0}\right)\left(\mathbf{n}_{r_{N}r_{0}}\times\frac{\mathbf{m}_{0}}{c}\right)\end{array}\right]^{T}$ and $\overline{\mathbf{p}}=\left[\begin{array}{cccccc}
\mathbf{p}\left(\mathbf{r}_{1}\right), & \mathbf{p}\left(\mathbf{r}_{2}\right), & \mathbf{p}\left(\mathbf{r}_{1}\right), & \cdots & \mathbf{p}\left(\mathbf{r}_{N-1}\right), & \mathbf{p}\left(\mathbf{r}_{N}\right)\end{array}\right]^{T}$ are $3N\times1$ vectors. $\mathbf{p}\left(\mathbf{r}_{i}\right)=\left[\begin{array}{ccc}
p_{x}\left(\mathbf{r}_{i}\right), & p_{y}\left(\mathbf{r}_{i}\right), & p_{z}\left(\mathbf{r}_{i}\right)\end{array}\right]$ and $i=1,2,3,\ldots,N$. $A$ is the collective polarizability (a $3N\times3N$ matrix) and read as\
$$A^{-1} = \left[\begin{array}{ccccccc}
\frac{1}{\epsilon_{0}\alpha} & 0 & 0 & -G_{EE}^{xx}\left(\mathbf{r}_{1},\mathbf{r}_{2}\right) & -G_{EE}^{xy}\left(\mathbf{r}_{1},\mathbf{r}_{2}\right) & -G_{EE}^{xz}\left(\mathbf{r}_{1},\mathbf{r}_{2}\right)\\
0 & \frac{1}{\epsilon_{0}\alpha} & 0 & -G_{EE}^{yx}\left(\mathbf{r}_{1},\mathbf{r}_{2}\right) & -G_{EE}^{yy}\left(\mathbf{r}_{1},\mathbf{r}_{2}\right) & -G_{EE}^{yz}\left(\mathbf{r}_{1},\mathbf{r}_{2}\right) & \cdots\\
0 & 0 & \frac{1}{\epsilon_{0}\alpha} & -G_{EE}^{zx}\left(\mathbf{r}_{1},\mathbf{r}_{2}\right) & -G_{EE}^{zy}\left(\mathbf{r}_{1},\mathbf{r}_{2}\right) & -G_{EE}^{zz}\left(\mathbf{r}_{1},\mathbf{r}_{2}\right)\\
-G_{EE}^{xx}\left(\mathbf{r}_{2},\mathbf{r}_{1}\right) & -G_{EE}^{xy}\left(\mathbf{r}_{2},\mathbf{r}_{1}\right) & -G_{EE}^{xz}\left(\mathbf{r}_{2},\mathbf{r}_{1}\right) & \frac{1}{\epsilon_{0}\alpha} & 0 & 0 & 0\\
-G_{EE}^{yx}\left(\mathbf{r}_{2},\mathbf{r}_{1}\right) & -G_{EE}^{yy}\left(\mathbf{r}_{2},\mathbf{r}_{1}\right) & -G_{EE}^{yz}\left(\mathbf{r}_{2},\mathbf{r}_{1}\right) & 0 & \frac{1}{\epsilon_{0}\alpha} & 0 & \cdots\\
-G_{EE}^{zx}\left(\mathbf{r}_{2},\mathbf{r}_{1}\right) & -G_{EE}^{zy}\left(\mathbf{r}_{2},\mathbf{r}_{1}\right) & -G_{EE}^{zz}\left(\mathbf{r}_{2},\mathbf{r}_{1}\right) & 0 & 0 & \frac{1}{\epsilon_{0}\alpha}\\
& \vdots & & & \vdots
\end{array}\right]_{3N\times3N}$$ and can be written as $$\begin{aligned}
\left[\overline{\mathbf{p}}\right]_{i} & =\mathbf{p}\left(\mathbf{r}_{i}\right)= & \left[A\overline{\mathbf{E}}_{\mu_{t}}\right]_{i},\label{eq:P_eff-1}\end{aligned}$$ Now we can calculate $\mathbf{n}_{r_{0}r_{i}}\times\mathbf{p}\left(\mathbf{r}_{i}\right)$ $$\begin{aligned}
\mathbf{n}_{r_{0}r_{i}}\times\mathbf{p}\left(\mathbf{r}_{i}\right) & = & \mathbf{n}_{r_{0}r_{i}}\times\left[A\overline{\mathbf{E}}_{\mu_{t}}\right]_{i},\end{aligned}$$ and using Eq. \[eq:AA1\] and Eq. \[eq:P\_eff-1\] we get $$\begin{aligned}
Z_{0}\mathbf{H}_{\mathrm{local}}\left(\mathbf{r}_{0}\right) & = & \mathbf{\bar{\bar{G}}}_{MM}\left(\mathbf{r}_{0},\mathbf{r}_{0}\right)\cdot\frac{\bm{\mu}_{t}}{c}+{\sum_{i=1}^{N}}g_{ME}\left(\mathbf{r}_{0},\mathbf{r}_{i}\right)\left[\mathbf{n}_{r_{0}r_{i}}\times\mathbf{p}\left(\mathbf{r}_{i}\right)\right],\nonumber \\
& = & \mathbf{\bar{\bar{G}}}_{MM}\left(\mathbf{r}_{0},\mathbf{r}_{0}\right)\cdot\frac{\bm{\mu}_{t}}{c}+{\sum_{i=1}^{N}}g_{ME}\left(\mathbf{r}_{0},\mathbf{r}_{i}\right)\mathbf{n}_{r_{0}r_{i}}\times\left[A\overline{\mathbf{E}}_{\mu_{t}}\right]_{i},\end{aligned}$$ Now by using $g_{EM}\left(\mathbf{r}_{i},\mathbf{r}_{0}\right)=g_{EM}\left(\mathbf{r}_{0},\mathbf{r}_{i}\right)=-g_{ME}\left(\mathbf{r}_{0},\mathbf{r}_{i}\right)$, the radiated power read as $$\begin{aligned}
P_{\mathrm{rad}}^{\mathrm{ant}} & = & \frac{\omega}{2}\mathrm{Im}\left[\frac{\bm{\mu}_{t}}{c}^{*}\cdot Z_{0}\mathbf{H}_{\mathrm{local}}\left(\mathbf{r}_{0}\right)\right],\label{eq:P_rad_MD_ant-1}\\
& = & \frac{\omega}{2}\left|\frac{\bm{\mu}_{t}}{c}\right|^{2}\mathrm{Im}\left[\mathbf{n}_{\mu}^{T}\mathbf{\bar{\bar{G}}}_{MM}\left(\mathbf{r}_{0},\mathbf{r}_{0}\right)\mathbf{n}_{\mu}\right]+\nonumber \\
& & +\frac{\omega}{2}\mathrm{Im}{\sum_{i=1}^{N}}g_{ME}\left(\mathbf{r}_{0},\mathbf{r}_{i}\right)\frac{\bm{\mu}_{t}}{c}^{*}\cdot\left\{ \mathbf{n}_{r_{0}r_{i}}\times\left[A\overline{\mathbf{E}}_{\mu_{t}}\right]_{i}\right\} ,\end{aligned}$$ where $\mathrm{Im}\left[\mathbf{n}_{\mu}^{T}\mathbf{G}_{MM}\left(\mathbf{r}_{0},\mathbf{r}_{0}\right)\mathbf{n}_{\mu}\right]=\frac{1}{\epsilon_{0}\alpha_{0}}.$ Now, by using Eq. \[eq:P\_rad\_MD\_fs-1\] and Eq. \[eq:P\_rad\_MD\_ant-1\], the emission rate enhancement can be obtained as $$\begin{aligned}
\frac{P_{\mathrm{rad}}^{\mathrm{ant}}}{P_{\mathrm{rad}}^{\mathrm{fs}}} & = & 1+\frac{\epsilon_{0}\alpha_{0}}{\left|\frac{\bm{\mu}_{t}}{c}\right|}\mathrm{Im}{\sum_{i=1}^{N}}g_{ME}\left(\mathbf{r}_{0},\mathbf{r}_{i}\right)\mathbf{n}_{\mu}\cdot\left\{ \mathbf{n}_{r_{0}r_{i}}\times\left[A\overline{\mathbf{E}}_{\mu_{t}}\right]_{i}\right\} .\label{eq:PF_ED-1}\end{aligned}$$ In the following subsections, we consider a magnetic dipole emitter in the middle of i) an atomic dimer and ii) an atomic tetramer.
Atomic dimer
------------
Let us consider a test magnetic dipole emitter $\bm{\mu}_{t}=\mu_{t}\mathbf{e}_{y}$ placed at $\mathbf{r}_{0}=\mathbf{0}$ close to an antenna consist of two atoms with only electric dipole response (electric polarizability $\alpha$) at position$\mathbf{r}_{u/d}=\pm l/2\mathbf{e}_{z}$ [\[]{}see Fig. \[fig:AtomicDimerGeometry\] (a)[\]]{}. Now by using Eq. \[eq:PF\_ED-1\], the emission rate enhancement read as
$$\begin{aligned}
\frac{P_{\mathrm{rad}}^{\mathrm{ant}}}{P_{\mathrm{rad}}^{\mathrm{fs}}} & = & 1+\frac{\epsilon_{0}\alpha_{0}}{\left|\frac{\bm{\mu}_{t}}{c}\right|}\mathrm{Im}{\sum_{i=1}^{2}}\left\{ g_{ME}\left(\mathbf{r}_{0},\mathbf{r}_{i}\right)\mathbf{n}_{\mu}\cdot\left[\mathbf{n}_{r_{0}r_{i}}\times\mathbf{p}\left(\mathbf{r}_{i}\right)\right]\right\} ,\nonumber \\
& = & 1+\frac{\epsilon_{0}\alpha_{0}}{\left|\frac{\bm{\mu}_{t}}{c}\right|}\mathrm{Im}{\sum_{i=1}^{2}}\left\{ g_{ME}\left(\mathbf{r}_{0},\mathbf{r}_{u}\right)\mathbf{e}_{y}\cdot\left[\mathbf{n}_{r_{0}r_{u}}\times\mathbf{p}\left(\mathbf{r}_{u}\right)\right]+g_{ME}\left(\mathbf{r}_{0},\mathbf{r}_{d}\right)\mathbf{e}_{y}\cdot\left[\mathbf{n}_{r_{0}r_{d}}\times\mathbf{p}\left(\mathbf{r}_{d}\right)\right]\right\} \end{aligned}$$
where $\mathbf{n}_{r_{0}r_{u}}=-\mathbf{e}_{z}$, $\mathbf{n}_{r_{0}r_{d}}=\mathbf{e}_{z}$, $\mathbf{n}_{\mu}=\mathbf{e}_{y}$ and the collective polarizability read as
![A test magnetic dipole emitter placed in the middle of an atomic dimer (a) and tetramer (b) consisting of two/four identical atoms with *only* electric dipole moments, respectively. \[fig:AtomicDimerGeometry\]](Figure4_SM.pdf){width="10cm"}
$$A=\left[\begin{array}{cccccc}
\frac{\epsilon_{0}\alpha}{1-\epsilon_{0}^{2}\alpha^{2}G_{EE}^{xx}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)^{2}} & 0 & 0 & \frac{\alpha^{2}G_{EE}^{xx}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)}{1-\alpha^{2}G_{EE}^{xx}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)^{2}} & 0 & 0\\
0 & \frac{\epsilon_{0}\alpha}{1-\epsilon_{0}^{2}\alpha^{2}G_{EE}^{yy}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)^{2}} & 0 & 0 & \frac{\alpha^{2}G_{EE}^{yy}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)}{1-\alpha^{2}G_{EE}^{yy}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)^{2}} & 0\\
0 & 0 & \frac{\epsilon_{0}\alpha}{1-\epsilon_{0}^{2}\alpha^{2}G_{EE}^{zz}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)^{2}} & 0 & 0 & \frac{\alpha^{2}G_{EE}^{zz}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)}{1-\alpha^{2}G_{EE}^{zz}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)^{2}}\\
\frac{\alpha^{2}G_{EE}^{xx}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)}{1-\alpha^{2}G_{EE}^{xx}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)^{2}} & 0 & 0 & \frac{\epsilon_{0}\alpha}{1\epsilon_{0}^{2}\alpha^{2}G_{EE}^{xx}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)^{2}} & 0 & 0\\
0 & \frac{\alpha^{2}G_{EE}^{yy}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)}{1-\alpha^{2}G_{EE}^{yy}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)^{2}} & 0 & 0 & \frac{\epsilon_{0}\alpha}{1-\epsilon_{0}^{2}\alpha^{2}G_{EE}^{yy}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)^{2}} & 0\\
0 & 0 & \frac{\alpha^{2}G_{EE}^{zz}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)}{1-\alpha^{2}G_{EE}^{zz}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)^{2}} & 0 & 0 & \frac{\epsilon_{0}\alpha}{1-\epsilon_{0}^{2}\alpha^{2}G_{EE}^{zz}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)^{2}}
\end{array}\right].$$
Now by using $A$ and $\mathbf{E}_{\bm{\mu}_{t}}\left(\mathbf{r}_{i}\right)\equiv g_{EM}\left(\mathbf{r}_{i},\mathbf{r}_{0}\right)\left(\mathbf{n}_{r_{i}r_{0}}\times\frac{\bm{\mu}_{t}}{c}\right)$ we obtain the dipole moments
$$\begin{aligned}
\left[\begin{array}{c}
\mathbf{p}\left(\mathbf{r}_{u}\right)\\
\mathbf{p}\left(\mathbf{r}_{d}\right)
\end{array}\right] & = & A\left[\begin{array}{c}
g_{EM}\left(\mathbf{r}_{u},\mathbf{r}_{0}\right)\left(\mathbf{n}_{r_{u}r_{0}}\times\frac{\bm{\mu}_{t}}{c}\right)\\
g_{EM}\left(\mathbf{r}_{d},\mathbf{r}_{0}\right)\left(\mathbf{n}_{r_{d}r_{0}}\times\frac{\bm{\mu}_{t}}{c}\right)
\end{array}\right],\\
& = & \frac{\epsilon_{0}\alpha\left|\frac{\bm{\mu}_{t}}{c}\right|}{1+\epsilon_{0}\alpha G_{EE}^{xx}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)}\left[\begin{array}{c}
-g_{EM}\left(\mathbf{r}_{u},\mathbf{r}_{0}\right)\\
0\\
0\\
g_{EM}\left(\mathbf{r}_{d},\mathbf{r}_{0}\right)\\
0\\
0
\end{array}\right],\end{aligned}$$
we used $$\left[\begin{array}{c}
\mathbf{n}_{r_{u}r_{0}}\times\frac{\bm{\mu}_{t}}{c}\\
\mathbf{n}_{r_{d}r_{0}}\times\frac{\bm{\mu}_{t}}{c}
\end{array}\right]=\left[\begin{array}{cccccc}
-1 & 0 & 0 & 1 & 0 & 0\end{array}\right]^{T},$$ and we get $$\begin{aligned}
\left[\begin{array}{c}
\mathbf{n}_{r_{0}r_{u}}\times\mathbf{p}\left(\mathbf{r}_{u}\right)\\
\mathbf{n}_{r_{0}r_{d}}\times\mathbf{p}\left(\mathbf{r}_{d}\right)
\end{array}\right] & = & \frac{\epsilon_{0}\alpha\left|\frac{\bm{\mu}_{t}}{c}\right|}{1+\epsilon_{0}\alpha G_{EE}^{xx}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)}\left[\begin{array}{cccccc}
0 & g_{EM}\left(\mathbf{r}_{u},\mathbf{r}_{0}\right) & 0 & 0 & g_{EM}\left(\mathbf{r}_{d},\mathbf{r}_{0}\right) & 0\end{array}\right]^{T}.\end{aligned}$$ Finally, we obtain $$\begin{aligned}
\frac{P_{\mathrm{rad}}^{\mathrm{ant}}}{P_{\mathrm{rad}}^{\mathrm{fs}}} & = & 1+\frac{\epsilon_{0}\alpha_{0}}{\left|\frac{\bm{\mu}_{t}}{c}\right|}\mathrm{Im}{\sum_{i=1}^{2}}\left\{ g_{ME}\left(\mathbf{r}_{0},\mathbf{r}_{u}\right)\mathbf{e}_{y}\cdot\left[\mathbf{n}_{r_{0}r_{u}}\times\mathbf{p}\left(\mathbf{r}_{u}\right)\right]+g_{ME}\left(\mathbf{r}_{0},\mathbf{r}_{d}\right)\mathbf{e}_{y}\cdot\left[\mathbf{n}_{r_{0}r_{d}}\times\mathbf{p}\left(\mathbf{r}_{d}\right)\right]\right\} ,\nonumber \\
& = & 1+\epsilon_{0}\alpha_{0}\mathrm{Im}\left[\frac{g_{EM}\left(\mathbf{r}_{u},\mathbf{r}_{0}\right)g_{ME}\left(\mathbf{r}_{0},\mathbf{r}_{u}\right)+g_{EM}\left(\mathbf{r}_{d},\mathbf{r}_{0}\right)g_{ME}\left(\mathbf{r}_{0},\mathbf{r}_{d}\right)}{\frac{1}{\epsilon_{0}\alpha}+G_{EE}^{xx}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)}\right],\nonumber \\
& = & 1-2\epsilon_{0}^{2}\alpha_{0}\alpha\mathrm{Im}\left[\frac{g_{EM}^{2}\left(\mathbf{r}_{0},\mathbf{r}_{u}\right)}{1+\epsilon_{0}\alpha G_{EE}^{xx}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)}\right],\nonumber \\
& = & 1-2\epsilon_{0}^{2}\alpha_{0}\alpha\mathrm{Im}\left[\frac{g_{EM}^{2}\left(\mathbf{r}_{0},\mathbf{r}_{u}\right)}{D_{+}}\right],\label{Decay_dimer}\end{aligned}$$ where $D_{+}\equiv1+\epsilon_{0}\alpha G_{EE}^{xx}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right)$ and $g_{EM}\left(\mathbf{r}_{u},\mathbf{r}_{0}\right)=-g_{ME}\left(\mathbf{r}_{0},\mathbf{r}_{u}\right)$. We used Eq. \[Decay\_dimer\] to calculate the decay rate in Fig. 2 of the main manuscript. The Green functions for the atomic dimer (see Fig. \[fig:AtomicDimerGeometry\]) read as $$\begin{aligned}
G_{EE}^{xx}\left(\mathbf{r}_{u},\mathbf{r}_{d}\right) & = & \frac{3}{2\epsilon_{0}\alpha_{0}}e^{i\zeta}\left(\frac{1}{\zeta}-\frac{1}{\zeta^{3}}+\frac{i}{\zeta^{2}}\right),\,\,\,\zeta=k\left|\mathbf{r}_{u}-\mathbf{r}_{d}\right|=kl,\\
g_{ME}\left(\mathbf{r}_{0},\mathbf{r}_{u}\right) & = & g_{ME}\left(\mathbf{r}_{0},\mathbf{r}_{l}\right)=\frac{3}{2\epsilon_{0}\alpha_{0}}e^{i\zeta}\left(\frac{1}{\zeta}+\frac{i}{\zeta^{2}}\right)\,\,\,\,\zeta=k\left|\mathbf{r}_{0}-\mathbf{r}_{u}\right|=k\left|\mathbf{r}_{0}-\mathbf{r}_{d}\right|=kl/2.\nonumber \end{aligned}$$ Finally, we employ the link between the quantum (i.e. decay rate) and classical formalisms (i.e. radiated power) [@novotny2012], i.e. $$\boxed{\frac{\Gamma_{\mathrm{\rm ant}}}{\Gamma_{0}}=\frac{P_{\mathrm{rad}}^{\mathrm{ant}}}{P_{\mathrm{rad}}^{\mathrm{fs}}}= 1-2\epsilon_{0}^{2}\alpha_{0}\alpha\mathrm{Im}\left[\frac{g_{EM}^{2}\left(\mathbf{r}_{0},\mathbf{r}_{u}\right)}{D_{+}}\right]\label{eq:Decay_Rad_dimer}}$$
Atomic tetramer
---------------
Let us consider a magnetic dipole emitter $\bm{\mu}_{t}=\mu_{t}\mathbf{e}_{y}$ placed at $\mathbf{r}_{0}=\mathbf{0}$ close to four quantum antennas (i.e. electric dipole moments) with polarizability $\alpha$ at position $\mathbf{r}_{1,2}=\mp l/2\mathbf{e}_{z}$, $\mathbf{r}_{3,4}=\pm l/2\mathbf{e}_{x}$ [\[]{}see Fig. \[fig:Geometry\_dimer\] (b)[\]]{}. Now by using Eq. \[eq:PF\_ED-1\], the emission rate enhancement read as $$\begin{aligned}
\frac{P_{\mathrm{rad}}^{\mathrm{ant}}}{P_{\mathrm{rad}}^{\mathrm{fs}}} & = & 1+\frac{\epsilon_{0}\alpha_{0}}{\left|\frac{\bm{\mu}_{t}}{c}\right|}\mathrm{Im}\sum_{i=1}^{4}\left\{ g_{ME}\left(\mathbf{r}_{0},\mathbf{r}_{i}\right)\mathbf{e}_{y}\cdot\left[\mathbf{n}_{r_{0}r_{i}}\times\mathbf{p}\left(\mathbf{r}_{i}\right)\right]\right\} ,\\
& = & 1+\frac{\epsilon_{0}\alpha_{0}}{\left|\frac{\bm{\mu}_{t}}{c}\right|}\mathrm{Im}\sum_{i=1}^{4}g_{ME}\left(\mathbf{r}_{0},\mathbf{r}_{i}\right)\mathbf{n}_{\mu}\cdot\left\{ \mathbf{n}_{r_{0}r_{i}}\times\left[A\overline{\mathbf{E}}_{\mu_{t}}\right]_{i}\right\} \end{aligned}$$ where $\mathbf{n}_{\mu}=\mathbf{e}_{y}$ and the normal unit vectors read as $$\mathbf{n}_{r_{0}r_{1}}=\left[\begin{array}{c}
0\\
0\\
1
\end{array}\right],\,\,\,\,\mathbf{n}_{r_{0}r_{2}}=\left[\begin{array}{c}
0\\
0\\
-1
\end{array}\right],\,\,\,\,\mathbf{n}_{r_{0}r_{3}}=\left[\begin{array}{c}
-1\\
0\\
0
\end{array}\right],\,\,\,\,\mathbf{n}_{r_{0}r_{4}}=\left[\begin{array}{c}
1\\
0\\
0
\end{array}\right],$$ $A$ matrix read as $$A=\left[\begin{array}{cccc}
\mathbf{\bar{\bar{I}}}/\alpha & \mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r}_{1},\mathbf{r}_{2}\right) & \mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r}_{1},\mathbf{r}_{3}\right) & \mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r}_{1},\mathbf{r}_{4}\right)\\
\mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r}_{1},\mathbf{r}_{2}\right) & \mathbf{\bar{\bar{I}}}/\alpha & \mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r}_{2},\mathbf{r}_{3}\right) & \mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r}_{2},\mathbf{r}_{4}\right)\\
\mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r}_{1},\mathbf{r}_{3}\right) & \mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r}_{2},\mathbf{r}_{3}\right) & \mathbf{\bar{\bar{I}}}/\alpha & \mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r}_{3},\mathbf{r}_{4}\right)\\
\mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r}_{1},\mathbf{r}_{4}\right) & \mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r}_{2},\mathbf{r}_{4}\right) & \mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r}_{3},\mathbf{r}_{4}\right) & \mathbf{\bar{\bar{I}}}/\alpha
\end{array}\right],$$ where ${\bf \bar{\bar{I}}}$ is the identity dyadic
$$\mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r}_{1},\mathbf{r}_{2}\right)=\frac{3}{2\alpha_{0}\epsilon_{0}}e^{ikl}\left[\begin{array}{ccc}
g_{1}\left(kl\right) & 0 & 0\\
0 & g_{1}\left(kl\right) & 0\\
0 & 0 & g_{1}\left(kl\right)+g_{2}\left(kl\right)
\end{array}\right],$$
$$\mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r}_{3},\mathbf{r}_{4}\right)=\frac{3}{2\alpha_{0}\epsilon_{0}}e^{ikl}\left[\begin{array}{ccc}
g_{1}\left(kl\right)+g_{2}\left(kl\right) & 0 & 0\\
0 & g_{1}\left(kl\right) & 0\\
0 & 0 & g_{1}\left(kl\right)
\end{array}\right],$$
$$\mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r}_{1},\mathbf{r}_{3}\right)=\mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r}_{2},\mathbf{r}_{4}\right)=\frac{3}{2\alpha_{0}\epsilon_{0}}e^{i\frac{kl}{\sqrt{2}}}\left[\begin{array}{ccc}
g_{1}\left(\frac{kl}{\sqrt{2}}\right) & 0 & \frac{g_{2}\left(\frac{kl}{\sqrt{2}}\right)}{2}\\
0 & g_{1}\left(\frac{kl}{\sqrt{2}}\right) & 0\\
\frac{g_{2}\left(\frac{kl}{\sqrt{2}}\right)}{2} & 0 & g_{1}\left(\frac{kl}{\sqrt{2}}\right)
\end{array}\right],$$
$$\mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r}_{1},\mathbf{r}_{4}\right)=\mathbf{\bar{\bar{G}}}_{EE}\left(\mathbf{r}_{2},\mathbf{r}_{3}\right)=\frac{3}{2\alpha_{0}\epsilon_{0}}e^{i\frac{kl}{\sqrt{2}}}\left[\begin{array}{ccc}
g_{1}\left(\frac{kl}{\sqrt{2}}\right) & 0 & -\frac{g_{2}\left(\frac{kl}{\sqrt{2}}\right)}{2}\\
0 & g_{1}\left(\frac{kl}{\sqrt{2}}\right) & 0\\
-\frac{g_{2}\left(\frac{kl}{\sqrt{2}}\right)}{2} & 0 & g_{1}\left(\frac{kl}{\sqrt{2}}\right)
\end{array}\right],$$
we used the Green function definition, i.e.
$$\begin{aligned}
G_{EE}^{\alpha\beta}\left(\zeta=k\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right) & = & \frac{3}{2\alpha_{0}\epsilon_{0}}e^{iu}\left[g_{1}\left(\zeta\right)\delta_{\alpha\beta}+g_{2}\left(\zeta\right)\frac{\zeta_{\alpha}\zeta_{\beta}}{\zeta^{2}}\right],\nonumber \\
g_{1}\left(\zeta\right) & = & \left(\frac{1}{\zeta}-\frac{1}{\zeta^{3}}+\frac{i}{\zeta^{2}}\right),\nonumber \\
g_{2}\left(\zeta\right) & = & \left(-\frac{1}{\zeta}+\frac{3}{\zeta^{3}}-\frac{3i}{\zeta^{2}}\right).\end{aligned}$$
Thus the induced dipole moment read as $$\left[\begin{array}{c}
\mathbf{p}\left(\mathbf{r}_{1}\right)\\
\mathbf{p}\left(\mathbf{r}_{2}\right)\\
\mathbf{p}\left(\mathbf{r}_{3}\right)\\
\mathbf{p}\left(\mathbf{r}_{4}\right)
\end{array}\right]=A\left[\begin{array}{c}
g_{EM}\left(\mathbf{r}_{1},\mathbf{r}_{0}\right)\left(\mathbf{n}_{r_{1}r_{0}}\times\frac{\bm{\mu}_{t}}{c}\right)\\
g_{EM}\left(\mathbf{r}_{2},\mathbf{r}_{0}\right)\left(\mathbf{n}_{r_{2}r_{0}}\times\frac{\bm{\mu}_{t}}{c}\right)\\
g_{EM}\left(\mathbf{r}_{3},\mathbf{r}_{0}\right)\left(\mathbf{n}_{r_{3}r_{0}}\times\frac{\bm{\mu}_{t}}{c}\right)\\
g_{EM}\left(\mathbf{r}_{4},\mathbf{r}_{0}\right)\left(\mathbf{n}_{r_{4}r_{0}}\times\frac{\bm{\mu}_{t}}{c}\right)
\end{array}\right],$$ and we get $$\begin{aligned}
\mathbf{p}\left(\mathbf{r}_{1}\right) & = & \frac{\epsilon_{0}\alpha\left|\frac{\bm{\mu}_{t}}{c}\right|g_{EM}\left(\mathbf{r}_{1},\mathbf{r}_{0}\right)}{1+\frac{3\alpha}{2\alpha_{0}}g_{1}\left(kl\right)e^{ikl}-\frac{3\alpha}{2\alpha_{0}}g_{2}\left(\frac{kl}{\sqrt{2}}\right)e^{i\frac{kl}{\sqrt{2}}}}\mathbf{e}_{x},\nonumber \\
\mathbf{p}\left(\mathbf{r}_{2}\right) & = & -\frac{\epsilon_{0}\alpha\left|\frac{\bm{\mu}_{t}}{c}\right|g_{EM}\left(\mathbf{r}_{2},\mathbf{r}_{0}\right)}{1+\frac{3\alpha}{2\alpha_{0}}g_{1}\left(kl\right)e^{ikl}-\frac{3\alpha}{2\alpha_{0}}g_{2}\left(\frac{kl}{\sqrt{2}}\right)e^{i\frac{kl}{\sqrt{2}}}}\mathbf{e}_{x},\nonumber \\
\mathbf{p}\left(\mathbf{r}_{3}\right) & = & \frac{\epsilon_{0}\alpha\left|\frac{\bm{\mu}_{t}}{c}\right|g_{EM}\left(\mathbf{r}_{3},\mathbf{r}_{0}\right)}{1+\frac{3\alpha}{2\alpha_{0}}g_{1}\left(kl\right)e^{ikl}-\frac{3\alpha}{2\alpha_{0}}g_{2}\left(\frac{kl}{\sqrt{2}}\right)e^{i\frac{kl}{\sqrt{2}}}}\mathbf{e}_{z},\nonumber \\
\mathbf{p}\left(\mathbf{r}_{4}\right) & = & -\frac{\epsilon_{0}\alpha\left|\frac{\bm{\mu}_{t}}{c}\right|g_{EM}\left(\mathbf{r}_{4},\mathbf{r}_{0}\right)}{1+\frac{3\alpha}{2\alpha_{0}}g_{1}\left(kl\right)e^{ikl}-\frac{3\alpha}{2\alpha_{0}}g_{2}\left(\frac{kl}{\sqrt{2}}\right)e^{i\frac{kl}{\sqrt{2}}}}\mathbf{e}_{z},\end{aligned}$$ we get $$\begin{aligned}
\sum_{i=1}^{4}
\mathbf{e}_{y}\cdot\left[\mathbf{n}_{r_{0}r_{i}}\times\mathbf{p}\left(\mathbf{r}_{i}\right)\right] & = & \epsilon_{0}\alpha\left|\frac{\bm{\mu}_{t}}{c}\right|\frac{g_{EM}\left(\mathbf{r}_{1},\mathbf{r}_{0}\right)+g_{EM}\left(\mathbf{r}_{2},\mathbf{r}_{0}\right)+g_{EM}\left(\mathbf{r}_{3},\mathbf{r}_{0}\right)+g_{EM}\left(\mathbf{r}_{4},\mathbf{r}_{0}\right)}{1+\frac{3\alpha}{2\alpha_{0}}g_{1}\left(kl\right)e^{ikl}-\frac{3\alpha}{2\alpha_{0}}e^{i\frac{kl}{\sqrt{2}}}g_{2}\left(\frac{kl}{\sqrt{2}}\right)}.\end{aligned}$$ Using $g_{EM}\left(\mathbf{r}_{1},\mathbf{r}_{0}\right)=g_{EM}\left(\mathbf{r}_{2},\mathbf{r}_{0}\right)=g_{EM}\left(\mathbf{r}_{3},\mathbf{r}_{0}\right)=g_{EM}\left(\mathbf{r}_{4},\mathbf{r}_{0}\right)$, we obtain the emission rate enhancement $$\begin{aligned}
\frac{P_{\mathrm{rad}}^{\mathrm{ant}}}{P_{\mathrm{rad}}^{\mathrm{fs}}} & = & 1+\frac{\epsilon_{0}\alpha_{0}}{\left|\frac{\bm{\mu}_{t}}{c}\right|}\mathrm{Im}\sum_{i=1}^{4}\left\{ g_{ME}\left(\mathbf{r}_{0},\mathbf{r}_{i}\right)\mathbf{e}_{y}\cdot\left[\mathbf{n}_{r_{0}r_{i}}\times\mathbf{p}\left(\mathbf{r}_{i}\right)\right]\right\} ,\\
& = & 1-4\epsilon_{0}\alpha_{0}\mathrm{Im}\left[\frac{\epsilon_{0}\alpha g_{EM}^{2}\left(\mathbf{r}_{0},\mathbf{r}_{1}\right)}{1+\frac{3\alpha}{2\alpha_{0}}g_{1}\left(kl\right)e^{ikl}-\frac{3\alpha}{2\alpha_{0}}g_{2}\left(\frac{kl}{\sqrt{2}}\right)e^{i\frac{kl}{\sqrt{2}}}}\right].\end{aligned}$$ where $g_{ME}\left(\zeta=k\left|\mathbf{r}_{0}-\mathbf{r}_{1}\right|\right)=-g_{EM}\left(\zeta=k\left|\mathbf{r}_{0}-\mathbf{r}_{1}\right|\right)=\frac{3}{2\alpha_{0}\epsilon_{0}}e^{i\zeta}\left(\frac{1}{\zeta}-\frac{1}{i\zeta^{2}}\right)$, and $\zeta=k\left|\mathbf{r}_{0}-\mathbf{r}_{1}\right|=\frac{kl}{2}$. Finally, we employ the link between the quantum and classical formalisms [@novotny2012], i.e. $$\boxed{\frac{\Gamma_{\mathrm{\rm ant}}}{\Gamma_{0}}=1-4\epsilon_{0}\alpha_{0}\mathrm{Im}\left[\frac{\epsilon_{0}\alpha g_{EM}^{2}\left(\mathbf{r}_{0},\mathbf{r}_{1}\right)}{1+\frac{3\alpha}{2\alpha_{0}}g_{1}\left(kl\right)e^{ikl}-\frac{3\alpha}{2\alpha_{0}}g_{2}\left(\frac{kl}{\sqrt{2}}\right)e^{i\frac{kl}{\sqrt{2}}}}\right]\label{eq:Decay_Rad_tetramer}}$$\
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abstract: |
In this paper we present a Doob type maximal inequality for stochastic processes satisfying the conditional increment control condition. If we assume, in addition, that the margins of the process have uniform exponential tail decay, we prove that the supremum of the process decays exponentially in the same manner. Then we apply this result to the construction of the almost everywhere stochastic flow to stochastic differential equations with singular time dependent divergence-free drift.
*Key words*: Doob’s maximal inequality, Kolmogorov’s criteria, divergence-free, Aronson estimate
*MSC2010 subject classifications: 60E15, 60H10*
author:
- 'Xuan Liu[^1] $\ $and Guangyu Xi[^2]'
bibliography:
- 'MaximumInequality.bib'
title: On a maximal inequality and its application to SDEs with singular drift
---
Introduction
============
Let $\{\Omega,\{\mathcal{F}_{t}\}_{t\geq0},\mathcal{F},\mathbb{P}\}$ be a filtered probability space satisfying the usual conditions. Let $\{X_{t}\}_{t\in[0,T]}$ be an $\{\mathcal{F}_{t}\}$-adapted stochastic process. There has been abundant research on the distribution of the supremum $\sup_{t\in[0,T]}\vert X_{t}\vert$ since Doob’s martingale maximal inequality. See, for example, [@Bogachev1998; @McLeish1975; @Talagrand1996; @Talagrand2014]. Here we consider continuous processes $\{X_{t}\}_{t\in[0,T]}$ satisfying the conditional increment control condition as follows.
Let $\{X_{t}\}_{t\in[0,T]}$ be a continuous $\{\mathcal{F}_{t}\}$-adapted stochastic process, and let $p>1$, $0<h\leq1$, $ph>1$. $\{X_{t}\}_{t\in[0,T]}$ is said to satisfy the conditional increment control with parameter $(p,h)$ if $X_{t}\in L^{p}(\Omega,\rd)$ for all $t\in[0,T]$ and there exists a constant $A_{p,h}\geq0$ independent of $s$ and $t$ such that $$\mathbb{E}\left[\left|\mathbb{E}(X_{t}\vert\mathcal{F}_{s})-X_{s}\right|^{p}\right]\leq A_{p,h}\vert t-s\vert^{ph},\qquad\mbox{ for all }0\leq s<t\leq T.\label{eq: conditional increment control}$$
For processes satisfying condition (\[eq: conditional increment control\]), we prove a Doob type maximal inequality as follows.
\[Theorem: Doob Type Maximal Inequality\]Suppose $\{X_{t}\}_{t\in[0,T]}$ is a continuous $\{\mathcal{F}_{t}\}$-adapted process satisfying condition (\[eq: conditional increment control\]). Let $0\leq s_{0}<t_{0}\leq T$, and $X^{\ast}=\sup_{u\in[s_{0},t_{0}]}\vert X_{u}\vert$. Then for any $1<q\leq p$, $$\Vert X^{\ast}\Vert_{L^{q}}\leq\frac{q}{q-1}\left[C_{p,h}^{1/p}A_{p,h}^{1/p}\vert t_{0}-s_{0}\vert^{h}+\Vert X_{t_{0}}\Vert_{L^{q}}\right]$$ for some constant $C_{p,h}>0$.
Under condition (\[eq: conditional increment control\]), we further study the tail decay of $\sup_{t\in[0,T]}\vert X_{t}\vert$ when the margins of $\{X_{t}\}_{t\in[0,T]}$ have uniform $\alpha$-exponential decay for some $\alpha>0$, i.e. there exist $C_{1},C_{2}>0$ such that $$\mathbb{P}\left(\vert X_{t}\vert\geq\lambda\right)\leq C_{2}\exp\left(-C_{1}\lambda^{\alpha}\right),\qquad\mbox{ for all }\lambda>0\mbox{ and all }t\in[0,T].$$ In Theorem \[thm: Main theorem 1\], we prove that if a continuous process $\{X_{t}\}_{t\in[0,T]}$ has uniform $\alpha$-exponential marginal decay and satisfies the conditional increment control for $(p,h)$ with $p>1$, $0<h\leq1$ and $ph>1$, then its supremum $\sup_{t\in[0,T]}\vert X_{t}\vert$ decays in the same manner as its margins, i.e. $$\mathbb{P}\left(\sup_{t\in[0,T]}\vert X_{t}\vert\geq\lambda\right)\leq C\exp(-C\lambda^{\alpha}).$$
Our results here are closely related to the celebrated theorems of Kolmogorov and Doob. The conditional increment control condition can be easily deduced from Kolmogorov’s continuity and tightness criteria, which means our result can be applied to a large class of diffusion processes as well as Gaussian processes like fractional Brownian motion. Moreover, the conditional increment control condition is also satisfied by martingales, which makes Theorem \[Theorem: Doob Type Maximal Inequality\] a generalization of Doob’s maximal inequality. In addition to being mathematically interesting, our results also have practical significance since we have no structural assumption on the processes. We only assume the conditional increment control and exponential marginal decay, which can be directly verified using empirical data. In this article, we show an application of our results to the study of stochastic differential equations $$dX_{t}=b(t,X_{t})dt+dB_{t},\label{eq: SDE1}$$ where $b$ is a time dependent divergence-free vector field and $B_{t}$ is the standard Brownian motion on $\rd$. Our main result of this application is Theorem \[thm: SDE main theorem\], which states that there is unique almost everywhere stochastic flow $X(\omega,x):\Omega\times\rd\rightarrow C([0,T],\rd)$ to SDE (\[eq: SDE1\]) if $b\in L^{1}(0,T;W^{1,p}(\rd))\cap L^{l}(0,T;L^{q}(\rd))$ with $d\geq3$, $p\geq1$, $\frac{2}{l}+\frac{d}{q}\in[1,2)$. Here $X(\omega,x)$ is defined for almost every $(\omega,x)\in\Omega\times\rd$ under $\mathbb{P}\times m$, where $m$ is the Lebesgue measure on $\rd$.
For SDEs with singular drift, Aronson [@Aronson1968] proved that there is a unique weak solution to (\[eq: SDE1\]) when $b\in L^{l}(0,T;L^{q}(\rd))$ with $\frac{2}{l}+\frac{d}{q}<1$. Moreover, it was proved that the transition probability of $\{X_{t}\}_{t\in[0,T]}$ satisfies the Aronson estimate $$\frac{1}{C_{1}(t-\tau)^{d/2}}\exp\left(-C_{2}\left(\frac{\vert x-\xi\vert^{2}}{t-\tau}\right)\right)\leq\Gamma(t,x;\tau,\xi)\leq\frac{C}{(t-\tau)^{d/2}}\exp\left(-\frac{1}{C}\left(\frac{\vert x-\xi\vert^{2}}{t-\tau}\right)\right).$$ Actually the Aroson estimate is true in more general cases when the diffusion coefficient is uniformly elliptic and $b\in L^{l}(0,T;L^{q}(\rd))$ with $\frac{2}{l}+\frac{d}{q}\leq1$, $q>d$. With the same condition on $b$ as in Aronson [@Aronson1968], Krylov and Röckner [@KrylovRockner2005] proved that there exists a unique strong solution to (\[eq: SDE1\]). Regularity results about the strong solution are obtained in Fedrizzi and Flandoli [@FedrizziFlandoli2011; @FedrizziFlandoli2013]. If we assume boundedness of $\divg b$, since the introduction of the renormalized solutions by DiPerna and Lions [@DiPernaLions1989], there has been lots of work on ODEs $dX_{t}=b(t,X_{t})dt$. In particular, Crippa and De Lellis [@DelellisCrippa2008] developed new estimate on ODEs with Sobolev coefficient $b$ and gave a new approach to construct the Diperna-Lions flow. This idea is extended to solve SDEs in [@FangLuoThalmaier2010; @ZhangXicheng2009; @ZhangXicheng2010]. In Zhang [@ZhangXicheng2009], in addition to boundedness of $\divg b$, it assumes that $\nabla b\in L\log L(\rd)$ to control $X_{t}$ locally and that $\vert b\vert/(1+\vert x\vert)\in L^{\infty}(\rd)$ to control $X_{t}$ from explosion. Together with Sobolev condition on the diffusion coefficient, Zhang [@ZhangXicheng2009] proved existence of a unique almost everywhere stochastic flow to SDEs. Since it is harder to control the growth of solutions to SDEs than ODEs, the linear growth condition on $b$ is needed in Zhang [@ZhangXicheng2009], while it is not necessary in Crippa and De Lellis [@DelellisCrippa2008]. Fang, Luo and Thalmaier [@FangLuoThalmaier2010] extend it to SDEs in Gaussian space with Sobolev diffusion and drift coefficients. In Zhang [@ZhangXicheng2010], it relaxes the boundedness of $\divg b$ to only the negative part of $\divg b$, and proved large deviation principle for the corresponding SDEs.
In Section 3, we prove the existence of the almost everywhere stochastic flow to (\[eq: SDE1\]) through approximation. Take smooth approximation sequence $b_{n}\rightarrow b$, such that $\{b_{n}\}_{n\in\mathbb{N}}$ is uniformly bounded in $L^{1}(0,T;W^{1,p}(\rd))\cap L^{l}(0,T;L^{q}(\rd))$ and divergence-free. Using the Aronson type upper bound estimate of the transition probability proved in Qian and Xi [@qian2017parabolic Corollary 9], we can show that $\{X_{t}^{(n)}\}_{n\in\mathbb{N}}$ satisfies uniform conditional increment control and exponential marginal decay. Hence Theorem \[thm: Main theorem 1\] implies that $\{\sup_{t\in[0,T]}\vert X_{t}^{(n)}\vert\}_{n\in\mathbb{N}}$ can be controlled uniformly and allows us to remove the linear growth condition on $b$ used in Zhang [@ZhangXicheng2009]. Then following the idea of Zhang [@ZhangXicheng2009], in Theorem \[thm: SDE main theorem\] we prove that the sequence $\{X_{t}^{(n)}\}_{n\in\mathbb{N}}$ converges to a unique limit $X_{t}$, which is the unique almost everywhere stochastic flow to SDE (\[eq: SDE1\]). It worth noting that the proof of Theorem \[thm: SDE main theorem\] only uses the moment estimate of the supremum $\sup_{t\in[0,T]}\vert X_{t}\vert$ proved in Proposition \[prop: supremum estimate of diffusion\]. The moment estimate actually can be obtained from the Aronson estimate using Kolmogorov’s continuity theorem as well, while the exponential decay of the supremum proved in Proposition \[prop: supremum estimate of diffusion\] is new. In a special case, Theorem \[thm: SDE main theorem\] is true when divergence-free $b\in L^{2}(0,T;H^{1}(\rt))$, which is of particular interest since the Leray-Hopf weak solutions to the 3-dimensional Navier-Stokes equations are in this space. However, the existence of a unique almost everywhere stochastic flow does not imply the uniqueness of the weak solutions to the corresponding parabolic equations. Since the stochastic flow is defined for almost everywhere initial data $x\in\rd$, it actually disguises the “bad” points in the measure zero set. For more discussion on the non-uniqueness of parabolic equations, we refer to Modena and Székelyhidi [@StefanoSzekelyhidi2018]. For simplicity, in this article we only discuss the case when the drift is divergence-free and the diffusion is the standard Brownian motion. But actually, using the idea in [@qian2017parabolic Corollary 9], we can obtain the Aronson type estimate and hence extend the result in Theorem \[thm: SDE main theorem\] for $\divg b\in L^{l'}(0,T;L^{q'}(\rd))$ with $\frac{2}{l'}+\frac{d}{q'}<2$. Moreover, the diffusion coefficient also can be extended to be Sobolev as in Zhang [@ZhangXicheng2009].
Maximal Inequality
==================
In this section, we always assume that the process $\{X_{t}\}_{t\in[0,T]}$ satisfies the conditional increment control (\[eq: conditional increment control\]) with $p>1$, $0<h\leq1$ and $ph>1$. Under this condition, we prove the Doob type maximal inequality in Theorem \[Theorem: Doob Type Maximal Inequality\] and the exponential tail decay for the supremum $\sup_{t\in[0,T]}\vert X_{t}\vert$ in Theorem \[thm: Main theorem 1\]. Before starting the proof, we give below some examples of processes which satisfy the conditional increment control.
- Any continuous martingales. The conditional increment control is satisfied for any $(p,h)$ with $A_{p,h}=0$.
- Fractional Brownian motions with Hurst parameter $h\in(0,1)$.
- Let $\{X_{t}\}_{t\in[0,T]}$ be a continuous stochastic process satisfying $$\mathbb{E}\left(\left|X_{t}-X_{s}\right|^{p}\right)\leq A_{p,h}\vert t-s\vert^{ph},\qquad\mbox{ for all }0\leq s<t\leq T.\label{eq: conditional increment 2}$$ Using Jensen’s inequality, it is easy to see that the conditional increment control is satisfied with the same parameter $(p,h)$ and the same constant $A_{p,h}$. Processes satisfying (\[eq: conditional increment 2\]) are archetypal examples considered in the rough paths theory for which canonical constructions of associated geometric rough paths are available and well-studied (see [@MR2604669; @MR2036784]). This type of processes also arises as solutions to SDEs.
A Doob Type Maximal Inequality
------------------------------
Before proving the maximal inequality, we will need two lemmas. Firstly, we prove the following estimate for the supremum of the conditional increment.
\[lem: supremum p norm\]Suppose $\{X_{t}\}_{t\in[0,T]}$ is a continuous $\{\mathcal{F}_{t}\}$-adapted process satisfying conditional increment control with parameter $(p,h)$. For any $0\leq s_{0}<t_{0}\leq T$, there holds that $$\mathbb{E}\left(\sup_{s_{0}\leq s<t\leq t_{0}}\left|\mathbb{E}(X_{t}\vert\mathcal{F}_{s})-X_{s}\right|^{p}\right)\leq C_{p,h}A_{p,h}\vert t_{0}-s_{0}\vert^{ph},$$ where $$C_{p,h}=[2\zeta(\theta)]^{p-1}\left(\frac{p}{p-1}\right)^{p}\left(\frac{4}{ph-1}\right)^{\theta(p-1)+1}\Gamma[\theta(p-1)+1].$$ Here $\theta>1$ is an arbitrary constant, $\zeta(\theta):=\sum_{m=1}^{\infty}m^{-\theta}<\infty$ for all $\theta>1$ and $\Gamma(z)$ is the Gamma function.
Let $s,t\in[s_{0},t_{0}]$, $s<t$ be fixed temporarily. Denote the dyadic intervals $$I_{l}^{m}=[t_{l-1}^{m},t_{l}^{m}]=s_{0}+(t_{0}-s_{0})*\left[\frac{l-1}{2^{m}},\frac{l}{2^{m}}\right].$$ Then we will construct a sequence of intervals $\{J_{k}\}\subset\{I_{l}^{m}:1\leq l\leq2^{m},m\geq0\}$ which gives a partition to $[s,t]$ such that
[(i)]{}
: $J_{k}$, $k=1,2,\cdots$, have mutually disjoint interior;
[(ii)]{}
: For any $m\geq1$, there are at most two elements of $\{J_{k}\}$ with length $(t_{0}-s_{0})2^{-m}$;
[(iii)]{}
: $(s,t)\subset\cup_{k=1}^{\infty}J_{k}\subset[s,t]$.
Suppose $m_{0}=\min\{m\in\mathbb{N}:\exists1\leq l\leq2^{m}\text{\mbox{ such that }}I_{l}^{m}\subset[s,t]\}$. For $m_{0}$, either there is only one $1\leq l_{0}\leq2^{m_{0}}$ such that $I_{l_{0}}^{m_{0}}\subset[s,t]$ and $[s,t]=[s,t_{l_{0}-1}^{m_{0}}]\cup I_{l_{0}}^{m_{0}}\cup[t_{l_{0}}^{m_{0}},t]$, or there are two consecutive $1\leq l_{0}<l_{0}+1\leq2^{m_{0}}$ such that $\left(I_{l_{0}}^{m_{0}}\cup I_{l_{0}+1}^{m_{0}}\right)\subset[s,t]$ and $[s,t]=[s,t_{l_{0}-1}^{m_{0}}]\cup I_{l_{0}}^{m_{0}}\cup I_{l_{0}+1}^{m_{0}}\cup[t_{l_{0}+1}^{m_{0}},t]$. Here we only deal with the first case and the second case follows the same argument. Notice that $\vert s-t_{l_{0}-1}^{m_{0}}\vert$ and $\vert t_{l_{0}}^{m_{0}}-t\vert$ are smaller than $(t_{0}-s_{0})2^{-m_{0}}$. For $[t_{l_{0}}^{m_{0}},t]$, we set $m_{k+1}=\min\{m>m_{k}:\exists1\leq l\leq2^{m}\text{\mbox{ such that }}I_{l}^{m}\subset[t_{l_{k}}^{m_{k}},t]\}$. There is at most one $1\leq l_{k+1}\leq2^{m_{k+1}}$ such that $I_{l_{k+1}}^{m_{k+1}}\subset[t_{l_{k}}^{m_{k}},t]$ since $\vert t_{l_{k}}^{m_{k}}-t\vert<(t_{0}-s_{0})2^{-m_{k}}$. Then $\{I_{l_{k}}^{m_{k}}\}$ forms a dyadic partition to $[t_{l_{0}}^{m_{0}},t]$, together with $I_{l_{0}}^{m_{0}}$ and a dyadic partition to $[s,t_{l_{0}-1}^{m_{0}}]$ following similar argument, we obtained the collection of intervals $\{J_{k}\}$ satisfying (i)-(iii).
Suppose $J_{k}=[u_{k-1},u_{k}]$. Then $$\begin{aligned}
\left|\mathbb{E}(X_{t}\vert\mathcal{F}_{s})-X_{s}\right| & =\left|\sum_{k=1}^{\infty}\mathbb{E}(\Delta X_{J_{k}}\vert\mathcal{F}_{s})\right|\\
& \leq\sum_{k=1}^{\infty}\mathbb{E}\left[\left.\vert\mathbb{E}(\Delta X_{J_{k}}\vert\mathcal{F}_{u_{k-1}})\vert\right|\mathcal{F}_{s}\right]\\
& =\sum_{m=1}^{\infty}\sum_{\{J_{k}:\vert J_{k}\vert=(t_{0}-s_{0})2^{-m}\}}\mathbb{E}\left[\left.\vert\mathbb{E}(\Delta X_{J_{k}}\vert\mathcal{F}_{u_{k-1}})\vert\right|\mathcal{F}_{s}\right],\end{aligned}$$ where $\Delta X_{J_{k}}=X_{u_{k}}-X_{u_{k-1}}$. Let $\xi_{l}^{m}=\mathbb{E}(\Delta X_{I_{l}^{m}}\vert\mathcal{F}_{t_{l-1}^{m}})$ for $1\leq l\leq2^{m}$, $m=1,2,\cdots$. For any $\theta>1$, take $\zeta(\theta)=\sum_{m=1}^{\infty}m^{-\theta}$ to be the Riemann zeta function. By Jensen’s inequality, we have $$\begin{aligned}
\left|\mathbb{E}(X_{t}\vert\mathcal{F}_{s})-X_{s}\right|^{p} & \leq\left(\sum_{m=1}^{\infty}\frac{1}{\zeta(\theta)m^{\theta}}\zeta(\theta)m^{\theta}\sum_{\{J_{k}:\vert J_{k}\vert=(t_{0}-s_{0})2^{-m}\}}\mathbb{E}\left[\left.\vert\mathbb{E}(\Delta X_{J_{k}}\vert\mathcal{F}_{u_{k-1}})\vert\right|\mathcal{F}_{s}\right]\right)^{p}\\
& \leq\sum_{m=1}^{\infty}\frac{1}{\zeta(\theta)m^{\theta}}\left(\zeta(\theta)m^{\theta}\sum_{\{J_{k}:\vert J_{k}\vert=(t_{0}-s_{0})2^{-m}\}}\mathbb{E}\left[\left.\vert\mathbb{E}(\Delta X_{J_{k}}\vert\mathcal{F}_{u_{k-1}})\vert\right|\mathcal{F}_{s}\right]\right)^{p}\\
& =\zeta(\theta)^{p-1}\sum_{m=1}^{\infty}m^{\theta(p-1)}\left(\sum_{\{J_{k}:\vert J_{k}\vert=(t_{0}-s_{0})2^{-m}\}}\mathbb{E}\left[\left.\vert\mathbb{E}(\Delta X_{J_{k}}\vert\mathcal{F}_{u_{k-1}})\vert\right|\mathcal{F}_{s}\right]\right)^{p}\\
& \leq[2\zeta(\theta)]^{p-1}\sum_{m=1}^{\infty}m^{\theta(p-1)}\sum_{\{J_{k}:\vert J_{k}\vert=(t_{0}-s_{0})2^{-m}\}}\left(\mathbb{E}\left[\left.\vert\mathbb{E}(\Delta X_{J_{k}}\vert\mathcal{F}_{u_{k-1}})\vert\right|\mathcal{F}_{s}\right]\right)^{p}\\
& \le[2\zeta(\theta)]^{p-1}\sum_{m=1}^{\infty}m^{\theta(p-1)}\sum_{l=1}^{2^{m}}\sup_{r\in[s_{0},t_{0}]}\left[\mathbb{E}\left(\left.\vert\xi_{l}^{m}\vert\right|\mathcal{F}_{r}\right)\right]^{p},\end{aligned}$$ where the inequality in the fourth line is due to property (ii) of $\{J_{k}\}$. Notice that for $s\leq t_{2}\leq t_{1}$, by Jensen’s inequality we have $$\begin{aligned}
\mathbb{E}\left|\mathbb{E}(X_{t_{1}}\vert\mathcal{F}_{s})-\mathbb{E}(X_{t_{2}}\vert\mathcal{F}_{s})\right|^{p} & =\mathbb{E}\left|\mathbb{E}\left(\left.[\mathbb{E}(X_{t_{1}}\vert\mathcal{F}_{t_{2}})-X_{t_{2}}]\right|\mathcal{F}_{s}\right)\right|^{p}\\
& \leq\mathbb{E}\left(\mathbb{E}\left(\left.\vert\mathbb{E}(X_{t_{1}}\vert\mathcal{F}_{t_{2}})-X_{t_{2}}\vert^{p}\right|\mathcal{F}_{s}\right)\right)\\
& =\mathbb{E}\left[\vert\mathbb{E}(X_{t_{1}}\vert\mathcal{F}_{t_{2}})-X_{t_{2}}\vert^{p}\right]\\
& \leq A_{p,h}\vert t_{1}-t_{2}\vert^{ph}.\end{aligned}$$ By Kolmogorov’s continuity theorem, for any fixed $s\in[s_{0},t_{0}]$, $\mathbb{E}(X_{t}\vert\mathcal{F}_{s})$ is a continuous process with respect to $t$. Recall that the filtration satisfies the usual conditions. $\mathbb{E}(X_{t}\vert\mathcal{F}_{s})$ can be regarded as a process of $s$ for fixed $t$ and it has a càdlàg modification by Doob’s regularization theorem. Hence, $\sup_{s_{0}\leq s<t\leq t_{0}}\left|\mathbb{E}(X_{t}\vert\mathcal{F}_{s})-X_{s}\right|^{p}$ is measurable with respect to $\mathcal{F}$ and we have $$\sup_{s_{0}\leq s<t\leq t_{0}}\left|\mathbb{E}(X_{t}\vert\mathcal{F}_{s})-X_{s}\right|^{p}\leq[2\zeta(\theta)]^{p-1}\sum_{m=1}^{\infty}m^{\theta(p-1)}\sum_{l=1}^{2^{m}}\sup_{r\in[s_{0},t_{0}]}\left[\mathbb{E}\left(\left.\vert\xi_{l}^{m}\vert\right|\mathcal{F}_{r}\right)\right]^{p}.$$ By Doob’s maximal inequality for martingales, $$\begin{aligned}
\mathbb{E}\left(\sup_{s_{0}\leq s<t\leq t_{0}}\left|\mathbb{E}(X_{t}\vert\mathcal{F}_{s})-X_{s}\right|^{p}\right) & \leq[2\zeta(\theta)]^{p-1}\sum_{m=1}^{\infty}m^{\theta(p-1)}\sum_{l=1}^{2^{m}}\mathbb{E}\left(\sup_{r\in[s_{0},t_{0}]}\left[\mathbb{E}\left(\left.\vert\xi_{l}^{m}\vert\right|\mathcal{F}_{r}\right)\right]^{p}\right)\\
& \leq[2\zeta(\theta)]^{p-1}\sum_{m=1}^{\infty}m^{\theta(p-1)}\sum_{l=1}^{2^{m}}\left(\frac{p}{p-1}\right)^{p}\mathbb{E}\left(\left[\mathbb{E}\left(\left.\vert\xi_{l}^{m}\vert\right|\mathcal{F}_{t_{0}}\right)\right]^{p}\right)\\
& \leq[2\zeta(\theta)]^{p-1}\left(\frac{p}{p-1}\right)^{p}\sum_{m=1}^{\infty}m^{\theta(p-1)}\sum_{l=1}^{2^{m}}\mathbb{E}\left(\vert\xi_{l}^{m}\vert^{p}\right)\\
& \leq A_{p,h}[2\zeta(\theta)]^{p-1}\left(\frac{p}{p-1}\right)^{p}\sum_{m=1}^{\infty}m^{\theta(p-1)}2^{m}\left(\frac{\vert t_{0}-s_{0}\vert}{2^{m}}\right)^{ph}\\
& =C_{p,h}A_{p,h}\vert t_{0}-s_{0}\vert^{ph},\end{aligned}$$ where $$C_{p,h}=[2\zeta(\theta)]^{p-1}\left(\frac{p}{p-1}\right)^{p}\sum_{m=1}^{\infty}m^{\theta(p-1)}2^{-m(ph-1)}.$$ Notice that $$\begin{aligned}
\sum_{m=1}^{\infty}m^{\theta(p-1)}2^{-m(ph-1)} & \leq\sum_{m=1}^{\infty}\left(e^{\theta(p-1)}\int_{m-1}^{m}r^{\theta(p-1)}dr\right)2^{-m(ph-1)}\\
& \leq\sum_{m=1}^{\infty}e^{\theta(p-1)}\int_{m-1}^{m}r^{\theta(p-1)}e^{-r(ph-1)\ln2}dr\\
& \leq\left(\frac{e}{(ph-1)\ln2}\right)^{\theta(p-1)+1}\int_{0}^{\infty}r^{\theta(p-1)}e^{-r}dr\\
& \leq\left(\frac{4}{ph-1}\right)^{\theta(p-1)+1}\Gamma[\theta(p-1)+1].\end{aligned}$$ Now the proof is complete.
Using Lemma \[lem: supremum p norm\], we show a Doob type inequality for processes satisfying condition (\[eq: conditional increment control\]). To this end, we shall need the following elementary result.
\[lem: lemma inequalities\]Let $\{Y_{t}\}_{t\in[0,T]}$ be a continuous stochastic process such that $\mathbb{E}(\vert Y_{t}\vert)<\infty$ for all $t\in[0,T]$. Let $0\leq s_{0}<t_{0}\leq T$. Then
[(1)]{}
: For any stopping time $\tau$ with $s_{0}\leq\tau\leq t_{0}$, we have $$\left|\mathbb{E}(Y_{t_{0}}\vert\mathcal{F}_{\tau})-Y_{\tau}\right|\leq\mathbb{E}\left[\left.\sup_{u\in[s_{0},t_{0}]}\vert\mathbb{E}(Y_{t_{0}}\vert\mathcal{F}_{u})-Y_{u}\vert\right|\mathcal{F_{\tau}}\right].\label{eq: lemma inequality 1}$$
[(2)]{}
: For any $\lambda>0$, we have $$\mathbb{P}\left(\sup_{u\in[s_{0},t_{0}]}\vert Y_{u}\vert\geq\lambda\right)\leq\frac{1}{\lambda}\int_{\{\sup_{u\in[s_{0},t_{0}]}\vert Y_{u}\vert\geq\lambda\}}\left[\sup_{u\in[s_{0},t_{0}]}\vert\mathbb{E}(Y_{t_{0}}\vert\mathcal{F}_{u})-Y_{u}\vert+\vert Y_{t_{0}}\vert\right]d\mathbb{P}.\label{eq: Maximal inequality loca inequality}$$
\(1) By the right continuity of $Y_{t}$ and $\mathbb{E}(Y_{t_{0}}\vert\mathcal{F}_{t})$, we may assume that $\tau$ takes only countably many values $\{u_{k}:k=1,2,\cdots\}\subset[s_{0},t_{0}].$ Then $$\begin{aligned}
\left|\mathbb{E}(Y_{t_{0}}\vert\mathcal{F}_{\tau})-Y_{\tau}\right| & =\sum_{k=1}^{\infty}\left|\mathbb{E}(Y_{t_{0}}\vert\mathcal{F}_{\tau})-Y_{\tau}\right|1_{\{\tau=u_{k}\}}\\
& =\sum_{k=1}^{\infty}\left|\mathbb{E}\left[\left.(Y_{t_{0}}-Y_{\tau})1_{\{\tau=u_{k}\}}\right|\sigma(\mathcal{F}_{\tau}\cap\{\tau=u_{k}\})\right]\right|\\
& =\sum_{k=1}^{\infty}\left|\mathbb{E}\left[\left.\mathbb{E}\left((Y_{t_{0}}-Y_{u_{k}})\vert\mathcal{F}_{u_{k}}\right)1_{\{\tau=u_{k}\}}\right|\sigma(\mathcal{F}_{\tau}\cap\{\tau=u_{k}\})\right]\right|\\
& \leq\sum_{k=1}^{\infty}\left|\mathbb{E}\left[\left.\left(\sup_{u\in[s_{0},t_{0}]}\vert\mathbb{E}(Y_{t_{0}}\vert\mathcal{F}_{u})-Y_{u})\vert\right)1_{\{\tau=u_{k}\}}\right|\sigma(\mathcal{F}_{\tau}\cap\{\tau=u_{k}\})\right]\right|\\
& =\sum_{k=1}^{\infty}\mathbb{E}\left[\left.\sup_{u\in[s_{0},t_{0}]}\vert\mathbb{E}(Y_{t_{0}}\vert\mathcal{F}_{u})-Y_{u})\vert\right|\mathcal{F}_{\tau}\right]1_{\{\tau=u_{k}\}}\\
& =\mathbb{E}\left[\left.\sup_{u\in[s_{0},t_{0}]}\vert\mathbb{E}(Y_{t_{0}}\vert\mathcal{F}_{u})-Y_{u})\vert\right|\mathcal{F}_{\tau}\right].\end{aligned}$$
\(2) Let $\tau=\inf\{u\in[s_{0},t_{0}]:\vert Y_{u}\vert\geq\lambda\}\wedge T$. Then $\{\sup_{u\in[s_{0},t_{0}]}\vert Y_{u}\vert\geq\lambda\}=\{\tau<T\}\cup\{\tau=t_{0},\vert Y_{t_{0}}\vert\geq\lambda\}\in\mathcal{F}_{\tau}$. Therefore, by (\[eq: lemma inequality 1\]) we have $$\begin{aligned}
\int_{\{\sup_{u\in[s_{0},t_{0}]}\vert Y_{u}\vert\geq\lambda\}}\vert Y_{\tau}\vert d\mathbb{P} & \leq\int_{\{\sup_{u\in[s_{0},t_{0}]}\vert Y_{u}\vert\geq\lambda\}}\vert\mathbb{E}(Y_{t_{0}}\vert\mathcal{F}_{\tau})-Y_{\tau}\vert d\mathbb{P}+\int_{\{\sup_{u\in[s_{0},t_{0}]}\vert Y_{u}\vert\geq\lambda\}}\vert\mathbb{E}(Y_{t_{0}}\vert\mathcal{F}_{\tau})\vert d\mathbb{P}\\
& \leq\int_{\{\sup_{u\in[s_{0},t_{0}]}\vert Y_{u}\vert\geq\lambda\}}\mathbb{E}\left[\left.\sup_{u\in[s_{0},t_{0}]}\vert\mathbb{E}(Y_{t_{0}}\vert\mathcal{F}_{u})-Y_{u})\vert\right|\mathcal{F}_{\tau}\right]+\mathbb{E}\left(\left.\vert Y_{t_{0}}\vert\right|\mathcal{F}_{\tau}\right)d\mathbb{P}\\
& \leq\int_{\{\sup_{u\in[s_{0},t_{0}]}\vert Y_{u}\vert\geq\lambda\}}\left[\sup_{u\in[s_{0},t_{0}]}\vert\mathbb{E}(Y_{t_{0}}\vert\mathcal{F}_{u})-Y_{u}\vert+\vert Y_{t_{0}}\vert\right]d\mathbb{P}.\end{aligned}$$ Finally, using $$\mathbb{P}\left(\sup_{u\in[s_{0},t_{0}]}\vert Y_{u}\vert\geq\lambda\right)\leq\frac{1}{\lambda}\int_{\{\sup_{u\in[s_{0},t_{0}]}\vert Y_{u}\vert\geq\lambda\}}\vert Y_{\tau}\vert d\mathbb{P},$$ and we complete the proof
Now we are ready to prove Theorem \[Theorem: Doob Type Maximal Inequality\].
\[Proof of Theorem \[Theorem: Doob Type Maximal Inequality\]\]Denote $Z=\sup_{u\in[s_{0},t_{0}]}\vert\mathbb{E}(X_{t_{0}}\vert\mathcal{F}_{u})-X_{u}\vert+\vert X_{t_{0}}\vert$ and fix a $q\in(1,p]$. By (\[eq: Maximal inequality loca inequality\]) and Lemma \[lem: supremum p norm\]
$$\begin{aligned}
\Vert X^{\ast}\Vert_{L^{q}}^{q} & =q\int_{0}^{\infty}\lambda^{q-1}\mathbb{P}(X^{\ast}\geq\lambda)d\lambda\\
& \leq q\int_{0}^{\infty}\lambda^{q-2}\int_{\{X^{\ast}\geq\lambda\}}Zd\mathbb{P}d\lambda\\
& =q\int_{\Omega}\left(\int_{0}^{X^{\ast}}\lambda^{q-2}d\lambda\right)Zd\mathbb{P}\\
& =\frac{q}{q-1}\int_{\Omega}\vert X^{\ast}\vert^{q-1}Zd\mathbb{P}\\
& \leq\frac{q}{q-1}\Vert X^{\ast}\Vert_{L^{q}}^{q-1}\Vert Z\Vert_{L^{q}}.\end{aligned}$$
Therefore, $$\begin{aligned}
\Vert X^{\ast}\Vert_{L^{q}} & \leq\frac{q}{q-1}\Vert Z\Vert_{L^{q}}\\
& \leq\frac{q}{q-1}\left[\left\Vert \sup_{u\in[s_{0},t_{0}]}\vert\mathbb{E}(X_{t_{0}}\vert\mathcal{F}_{u})-X_{u}\vert\right\Vert _{L^{q}}+\left\Vert X_{t_{0}}\right\Vert _{L^{q}}\right]\\
& \leq\frac{q}{q-1}\left[C_{p,h}^{1/P}A_{p,h}^{1/p}\vert t_{0}-s_{0}\vert^{h}+\left\Vert X_{t_{0}}\right\Vert _{L^{q}}\right].\end{aligned}$$
Tail decay for the supremum
---------------------------
Here, we shall show that the distribution of the supremum $\sup_{t\in[0,T]}\vert X_{t}\vert$ has $\alpha$-exponential decay if the margins of $\{X_{t}\}_{t\in[0,T]}$ have uniform $\alpha$-exponential decay as in the following definition.
Given $\alpha>0$, a continuous stochastic process $\{X_{t}\}_{t\in[0,T]}$ is said to have uniform $\alpha$-exponential marginal decay if there exists constants $C_{1},C_{2}>0$ such that $$\mathbb{P}\left(\vert X_{t}\vert\geq\lambda\right)\leq C_{2}\exp\left(-C_{1}\lambda^{\alpha}\right),\qquad\mbox{ for all }\lambda>0\mbox{ and all }t\in[0,T].\label{eq: alpha exponential decay}$$
Suppose $$\mathbb{P}\left(\vert X_{t}\vert\geq\lambda\right)\leq C_{2}\exp\left(-C_{1}\lambda^{\alpha}\right),\qquad\mbox{ for all }\lambda>M\label{eq:alpha exponential decay 2}$$ with $M$ being a large enough constant. Since we always have $\mathbb{P}\left(\vert X_{t}\vert\geq\lambda\right)\leq1$, actually (\[eq:alpha exponential decay 2\]) implies (\[eq: alpha exponential decay\]) for another pair of constants $(C_{1},C_{2})$. In the following, we will always use (\[eq: alpha exponential decay\]).
Notice that if $\{X_{t}\}_{t\in[0,T]}$ satisfies (\[eq: alpha exponential decay\]), one has $$\mathbb{E}(\vert X_{t}\vert^{q})\leq C_{2}C_{1}^{-q/\alpha}\Gamma(\frac{q}{\alpha}+1)\label{eq: Lq norm control}$$ for any $q>0$. Now we state our theorem.
\[thm: Main theorem 1\]Suppose stochastic process $\{X_{t}\}_{t\in[0,T]}$ satisfies condition (\[eq: conditional increment control\]) with parameter $(p,h)$ and has uniform $\alpha$-exponential marginal decay. Then $$\mathbb{P}\left(\sup_{t\in[0,T]}\vert X_{t}\vert\geq\lambda\right)\leq C\lambda^{-1/h}\exp\left[-\frac{C_{1}}{2^{\alpha+2}}\left(1-\frac{1}{ph}\right)\lambda^{\alpha}\right]$$ for large enough $\lambda$, where $C$ depends on $(C_{1},C_{2},\alpha,p,h,A_{p,h}).$
For $N\in\mathbb{N}_{+}$, let $I_{n}=[t_{n-1},t_{n}]=[(n-1)T/N,nT/N]$, $1\leq n\leq N$. Then $$\left\{ \sup_{t\in[0,T]}\vert X_{t}\vert\geq2\lambda\right\} \subset\bigcup_{n=1}^{N}\left(\left\{ \sup_{t\in I_{n}}\vert\mathbb{E}(X_{t_{n}}\vert\mathcal{F}_{t})-X_{t}\vert\geq\lambda\right\} \bigcup\left\{ \sup_{t\in I_{n}}\vert\mathbb{E}(X_{t_{n}}\vert\mathcal{F}_{t})\vert\geq\lambda\right\} \right).$$ Therefore $$\begin{aligned}
\mathbb{P}\left(\sup_{t\in[0,T]}\vert X_{t}\vert\geq2\lambda\right) & \leq\sum_{n=1}^{N}\mathbb{P}\left(\sup_{t\in I_{n}}\vert\mathbb{E}(X_{t_{n}}\vert\mathcal{F}_{t})-X_{t}\vert\geq\lambda\right)+\sum_{n=1}^{N}\mathbb{P}\left(\sup_{t\in I_{n}}\vert\mathbb{E}(X_{t_{n}}\vert\mathcal{F}_{t})\vert\geq\lambda\right).\label{eq: main theorem local 1}\end{aligned}$$ By Lemma \[lem: supremum p norm\], $$\mathbb{P}\left(\sup_{t\in I_{n}}\vert\mathbb{E}(X_{t_{n}}\vert\mathcal{F}_{t})-X_{t}\vert\geq\lambda\right)\leq C_{p,h}A_{p,h}\lambda^{-p}\left(\frac{T}{N}\right)^{ph}.\label{eq: main theorem local 2}$$ Next we need to estimate $\mathbb{P}\left(\sup_{t\in I_{n}}\vert\mathbb{E}(X_{t_{n}}\vert\mathcal{F}_{t})\vert\geq\lambda\right)$. Notice that $$\mathbb{E}\left[\exp\left(\frac{C_{1}}{4}\sup_{t\in I_{n}}\vert\mathbb{E}(X_{t_{n}}\vert\mathcal{F}_{t})\vert^{\alpha}\right)\right]=\sum_{q=0}^{\infty}\frac{(C_{1}/4)^{q}}{q!}\mathbb{E}\left(\sup_{t\in I_{n}}\vert\mathbb{E}(X_{t_{n}}\vert\mathcal{F}_{t})\vert^{\alpha q}\right).$$ Here we fix an arbitrary constant $\beta>1$. When $\alpha q\leq\beta$, by Doob’s maximal inequality and (\[eq: Lq norm control\]), we have that $$\begin{aligned}
\mathbb{E}\left(\sup_{t\in I_{n}}\vert\mathbb{E}(X_{t_{n}}\vert\mathcal{F}_{t})\vert^{\alpha q}\right) & \leq\mathbb{E}\left(\sup_{t\in I_{n}}\vert\mathbb{E}(X_{t_{n}}\vert\mathcal{F}_{t})\vert^{\beta}\right)^{\frac{\alpha q}{\beta}}\\
& \leq\left(\frac{\beta}{\beta-1}\right)^{\beta\cdot\frac{\alpha q}{\beta}}\mathbb{E}\left(\vert X_{t_{n}}\vert^{\beta}\right)^{\frac{\alpha q}{\beta}}\\
& \leq\left(\frac{\beta}{\beta-1}\right)^{\alpha q}\left(C_{2}C_{1}^{-\beta/\alpha}\Gamma(\frac{\beta}{\alpha}+1)\right)^{\frac{\alpha q}{\beta}}.\end{aligned}$$ When $\alpha q>\beta$, again by Doob’s maximal inequality and (\[eq: Lq norm control\]), we have that $$\begin{aligned}
\mathbb{E}\left(\sup_{t\in I_{n}}\vert\mathbb{E}(X_{t_{n}}\vert\mathcal{F}_{t})\vert^{\alpha q}\right) & \leq\mathbb{E}\left(\sup_{t\in I_{n}}\vert\mathbb{E}(\vert X_{t_{n}}\vert^{\frac{\alpha q}{\beta}}\vert\mathcal{F}_{t})\vert^{\beta}\right)\\
& \leq\left(\frac{\beta}{\beta-1}\right)^{\beta}\mathbb{E}\left(\vert X_{t_{n}}\vert^{\alpha q}\right)\\
& \leq\left(\frac{\beta}{\beta-1}\right)^{\beta}C_{2}C_{1}^{-q}\Gamma(q+1).\end{aligned}$$ Therefore $$\begin{aligned}
\mathbb{E}\left[\exp\left(\frac{C_{1}}{4}\sup_{t\in I_{n}}\vert\mathbb{E}(X_{t_{n}}\vert\mathcal{F}_{t})\vert^{\alpha}\right)\right] & \leq\sum_{q=0}^{\lfloor\frac{\beta}{\alpha}\rfloor}\frac{(C_{1}/4)^{q}}{q!}\left(\frac{\beta}{\beta-1}\right)^{\alpha q}\left(C_{2}C_{1}^{-\beta/\alpha}\Gamma(\frac{\beta}{\alpha}+1)\right)^{\frac{\alpha q}{\beta}}\\
& \quad+\sum_{q=\lfloor\frac{\beta}{\alpha}\rfloor+1}^{\infty}\frac{(C_{1}/4)^{q}}{q!}\left(\frac{\beta}{\beta-1}\right)^{\beta}C_{2}C_{1}^{-q}\Gamma(q+1)\\
& \leq C+\sum_{q=\lfloor\frac{\beta}{\alpha}\rfloor+1}^{\infty}4^{-q}\left(\frac{\beta}{\beta-1}\right)^{\beta}C_{2}\\
& \leq C,\end{aligned}$$ where the constant C depends on $(\alpha,\beta,C_{1},C_{2})$. By Chebyshev’s inequality, $$\mathbb{P}\left(\sup_{t\in I_{n}}\vert\mathbb{E}(X_{t_{n}}\vert\mathcal{F}_{t})\vert\geq\lambda\right)\leq C\exp\left(-\frac{C_{1}}{4}\lambda^{\alpha}\right).\label{eq: main theorem local 3}$$ Hence, for any $N\in\mathbb{N}_{+}$, by (\[eq: main theorem local 2\]) and (\[eq: main theorem local 3\]), we have that $$\begin{aligned}
\mathbb{P}\left(\sup_{t\in[0,T]}\vert X_{t}\vert\geq2\lambda\right) & \leq NC_{p,h}A_{p,h}\lambda^{-p}\left(\frac{T}{N}\right)^{ph}+NC\exp\left(-\frac{C_{1}}{4}\lambda^{\alpha}\right)\\
& =:EN^{1-ph}+FN,\end{aligned}$$ where $E=C_{p,h}A_{p,h}T^{ph}\lambda^{-p}$ and $F=C\exp\left(-\frac{C_{1}}{4}\lambda^{\alpha}\right)$. When $\lambda$ is large enough, notice that $E\gg F$ and $E/F$ is large enough. Then we can set $N$ to be the greatest integer less than or equal to $\left(\frac{E}{F}\right)^{\frac{1}{ph}}$ to obtain that $$\begin{aligned}
\mathbb{P}\left(\sup_{t\in[0,T]}\vert X_{t}\vert\geq2\lambda\right) & \leq CE^{\frac{1}{ph}}F^{1-\frac{1}{ph}}\\
& =CC_{p,h}^{\frac{1}{ph}}A_{p,h}^{\frac{1}{ph}}T\lambda^{-\frac{1}{h}}\exp\left(-\frac{C_{1}}{4}\left(1-\frac{1}{ph}\right)\lambda^{\alpha}\right),\end{aligned}$$ where $C$ depends on $C_{1}$, $C_{2}$, $p$, $h$ and $\alpha$. Finally, replace $2\lambda$ by $\lambda$ and the proof is complete.
SDEs with singular drift\[sec:SDEs\]
====================================
In this section, we apply the results in the previous section to solve SDEs $$dX_{t}=b(t,X_{t})dt+dB_{t},\label{eq: SDE}$$ where divergence-free vector field $b\in L^{l}(0,T;L^{q}(\rd))\cap L^{1}(0,T;W^{1,p}(\rd))$ with $\frac{2}{l}+\frac{d}{q}=\gamma\in[1,2)$, $d\geq3$ and $p\geq1$. We use the idea of approximation and *a priori* estimates to show that the approximation sequence converges in probability. Together with the $L^{k}(\Omega\times B_{r};C([0,T]))$ bound of the approximation sequence, we have that the sequence converges in $L^{k}(\Omega\times B_{r};C([0,T]))$ for any $k\geq1$. Here $B_{r}$ is the ball in $\rd$ of radius $r$ and center at the origin.
Supremum of solutions to SDEs
-----------------------------
To control the supremum $\sup_{t\in[0,T]}\vert X_{t}\vert$, we will need the following Aronson type estimate of the transition probability of $X_{t}$ proved in [@qian2017parabolic Corollary 9].
Suppose $b$ is divergence-free and $b\in L^{l}(0,T;L^{q}(\rd))$ for some $d\geq3$, $l>1$ and $q>\frac{d}{2}$ such that $\frac{2}{l}+\frac{d}{q}=\gamma\in[1,2)$. In addition, we assume that $b$ is smooth with bounded derivatives. If $\mu:=\frac{2}{2-\gamma+\frac{2}{l}}>1$, the transition probability has upper bound $$\Gamma(t,x;\tau,\xi)\leq\begin{cases}
\frac{C_{1}}{(t-\tau)^{d/2}}\exp\left(-\frac{1}{C_{2}}\left(\frac{\vert x-\xi\vert^{2}}{t-\tau}\right)\right) & \frac{\vert x-\xi\vert^{\mu-2}}{(t-\tau)^{\mu-\nu-1}}<1\\
\frac{C_{1}}{(t-\tau)^{d/2}}\exp\left(-\frac{1}{C_{2}}\left(\frac{\vert x-\xi\vert^{\mu}}{(t-\tau)^{\nu}}\right)^{\frac{1}{\mu-1}}\right) & \frac{\vert x-\xi\vert^{\mu-2}}{(t-\tau)^{\mu-\nu-1}}\geq1,
\end{cases}$$ where $\nu=\frac{2-\gamma}{2-\gamma-\frac{2}{l}}$, $\Lambda=\Vert b\Vert_{L^{l}(0,T;L^{q}(\rd))}$, $C_{1}=C_{1}(l,q,d)$, $C_{2}=C_{2}(l,q,d,\Lambda)$. If $\mu=1$, which implies $q=\infty$, we have $$\Gamma(t,x;\tau,\xi)\leq\frac{C_{1}}{(t-\tau)^{d/2}}\exp\left(-\frac{(C_{1}\Lambda(t-\tau)^{\nu}-\vert x-\xi\vert)^{2}}{4C_{1}(t-\tau)}\right).$$
This theorem is actually true for diffusion processes corresponding to parabolic equations $$\partial_{t}u(t,x)-\sum_{i,j=1}^{d}\partial_{j}(a_{ij}(t,x)\partial_{i}u(t,x))+\sum_{i=1}^{d}b_{i}(t,x)\partial_{i}u(t,x)=0$$ for uniformly elliptic $\{a_{ij}\}$, $b\in L^{l}(0,T;L^{q}(\rd))$ with $\frac{2}{l}+\frac{d}{q}=\gamma\in[1,2)$ and $\divg b\in L^{l'}(0,T;L^{q'}(\rd))$ with $\frac{2}{l'}+\frac{d}{q'}=\gamma'\in[1,2)$. The proof follows the idea in [@qian2017parabolic] with small modification.
This upper bound estimate of the transition probability implies the following.
\[prop: lp norm and exponential decay\]Suppose $b$ is a smooth divergence-free vector field with bounded derivatives and $b\in L^{l}(0,T;L^{q}(\rd))$ for some $d\geq3$, $l>1$ and $q>\frac{d}{2}$ such that $1\leq\gamma<2$. Then the solution $\{X_{t}\}_{t\in[0,T]}$ to (\[eq: SDE\]) satisfies that $$\mathbb{E}\vert X_{t}-X_{s}\vert^{p}\leq C\vert t-s\vert^{\frac{(2-\gamma)(p+d)-d}{2}}\label{eq: moment estimate}$$ for $0\leq s<t\leq T$. Moreover, there exists a constant $\alpha>1$ depending on $(l,q,d)$ such that $$\mathbb{P}(\vert X_{t}-X_{s}\vert>\lambda)\leq C\exp(-C\lambda^{\alpha})\label{eq: exponential decay}$$ for large enough $\lambda$ and $0\leq s<t\leq T$. The constant $C$ depends on $(l,q,d,\Lambda)$.
Without loss of generality, we may take $s=0$ and $X_{0}=0$. When $\mu>1$, we have $$\begin{aligned}
\mathbb{E}\vert X_{t}-X_{0}\vert^{p} & =\int_{\rd}\vert x\vert^{p}\Gamma(t,x;0,0)dx\\
& \leq\int_{\rd}\vert x\vert^{p}\frac{C_{1}}{t^{d/2}}\exp\left(-\frac{1}{C_{2}}\left(\frac{\vert x\vert^{2}}{t}\right)\right)dx+\int_{\rd}\vert x\vert^{p}\frac{C_{1}}{t^{d/2}}\exp\left(-\frac{1}{C_{2}}\left(\frac{\vert x\vert^{\mu}}{t^{\nu}}\right)^{\frac{1}{\mu-1}}\right)dx\\
& \leq C\left(t^{\frac{p}{2}}+t^{\frac{(2-\gamma)(p+d)-d}{2}}\right)\\
& \leq Ct^{\frac{(2-\gamma)(p+d)-d}{2}}.\end{aligned}$$ When $\mu=1$, following similar argument, we have that $$\begin{aligned}
\mathbb{E}\vert X_{t}-X_{0}\vert^{p} & =\int_{\rd}\vert x\vert^{p}\Gamma(t,x;0,0)dx\\
& \leq\int_{\vert x\vert\geq Ct^{\nu}}\vert x\vert^{p}\Gamma(t,x;0,0)dx+\int_{\vert x\vert<Ct^{\nu}}\vert x\vert^{p}\Gamma(t,x;0,0)dx\\
& \leq\int_{\rd}\vert x\vert^{p}\frac{C_{1}}{t^{d/2}}\exp\left(-\frac{C\vert x\vert^{2}}{4C_{1}t}\right)dx+\int_{\vert x\vert<Ct^{\nu}}\vert x\vert^{p}\frac{C_{1}}{t^{d/2}}dx\\
& \leq C\left(t^{\frac{p}{2}}+t^{\frac{(2-\gamma)(p+d)-d}{2}}\right)\\
& \leq Ct^{\frac{(2-\gamma)(p+d)-d}{2}}.\end{aligned}$$ Now we prove the uniform $\alpha$-exponential marginal decay for $X_{t}$. When $\mu>1$, we have that $$\begin{aligned}
\mathbb{P}(\vert X_{t}\vert>\lambda) & \leq\int_{\vert x\vert>\lambda}\frac{C_{1}}{t^{d/2}}\exp\left(-\frac{1}{C_{2}}\left(\frac{\vert x\vert^{2}}{t}\right)\right)dx+\int_{\vert x\vert>\lambda}\frac{C_{1}}{t^{d/2}}\exp\left(-\frac{1}{C_{2}}\left(\frac{\vert x\vert^{\mu}}{t^{\nu}}\right)^{\frac{1}{\mu-1}}\right)dx\\
& =C_{1}\int_{\vert x\vert>\lambda t^{-\frac{1}{2}}}\exp\left(-\frac{\vert x\vert^{2}}{C_{2}}\right)dx+\frac{C_{1}}{t^{d(\gamma-1)/2}}\int_{\vert x\vert>\lambda t^{-\frac{2-\gamma}{2}}}\exp\left(-\frac{\vert x\vert^{\frac{\mu}{\mu-1}}}{C_{2}}\right)dx\\
& \leq C\exp(-C(\lambda t^{-\frac{1}{2}})^{2})+\frac{C}{t^{d(\gamma-1)/2}}\exp\left(-C(\lambda t^{-\frac{2-\gamma}{2}})^{\frac{\mu}{\mu-1}}\right)\\
& \leq C\exp(-C\lambda^{2})+C\exp(-C\lambda^{\frac{\mu}{\mu-1}})\end{aligned}$$ for $0<t\leq T$ and large enough $\lambda$. Similarly, when $\mu=1$, we have $$\begin{aligned}
\mathbb{P}(\vert X_{t}\vert>\lambda) & \leq\int_{\vert x\vert>\lambda}\frac{C_{1}}{t^{d/2}}\exp\left(-\frac{(C_{1}\Lambda t^{\nu}-\vert x\vert)^{2}}{4C_{1}t}\right)dx\\
& \leq C_{1}\int_{\vert x\vert>\lambda t^{-\frac{1}{2}}}\exp\left(-C(Ct^{\nu-\frac{1}{2}}-\vert x\vert)^{2}\right)dx.\end{aligned}$$ Recall that $\mu=1$ implies $q=\infty$ and hence $\nu=\frac{2-\gamma}{2}\in(0,\frac{1}{2}]$. For large enough $\lambda$, we have $\vert x\vert>\lambda t^{-\frac{1}{2}}\gg Ct^{\nu-\frac{1}{2}}$ and $$\begin{aligned}
\mathbb{P}(\vert X_{t}\vert>\lambda) & \leq C_{1}\int_{\vert x\vert>\lambda t^{-\frac{1}{2}}}\exp\left(-C\vert x\vert^{2}\right)\\
& \leq C\exp(-C\lambda^{2}).\end{aligned}$$
Now we can apply Theorem \[thm: Main theorem 1\] to obtain the following result.
\[prop: supremum estimate of diffusion\]Suppose $b$ is a smooth divergence-free vector field with bounded derivatives and $b\in L^{l}(0,T;L^{q}(\rd))$ for some $d\geq3$, $l>1$ and $q>\frac{d}{2}$ such that $1\leq\gamma<2$. Then the solution $\{X_{t}\}_{t\in[0,T]}$ to (\[eq: SDE\]) satisfies that $$\mathbb{P}\left(\sup_{t\in[0,T]}\vert X_{t}-X_{0}\vert>\lambda\right)\leq C\exp(-C\lambda^{\alpha})\label{eq: exponential decay supremum}$$ with the same $\alpha$ as in Proposition \[prop: lp norm and exponential decay\]. Moreover, for any $p\geq1$ we have that $$\mathbb{E}\left[\sup_{t\in[0,T]}\vert X_{t}-X_{0}\vert^{p}\right]<C,\label{eq: moment estimate supremum}$$ where $C$ depends on $l$, $q$, $p$, $d$ and $\Vert b\Vert_{L^{l}(0,T;L^{q}(\rd))}$.
Using Kolmogorov’s continuity theorem, for any large enough $p$ such that $\frac{(2-\gamma)(p+d)-d}{2}>1$, inequality (\[eq: moment estimate\]) implies that there exists $\alpha>0$ such that $$\vert X_{t}-X_{s}\vert\leq K\vert t-s\vert^{\alpha}\qquad\mbox{for all }0\leq s<t\leq T.$$ Here $K$ is a random variable satisfying $\mathbb{E}[K^{p}]<C$. Since this is true for any large enough $p$, in this way we can also prove the moment estimate (\[eq: moment estimate supremum\]) for any $p\geq1$, although it can not prove the exponential decay (\[eq: exponential decay supremum\]). But actually Theorem \[thm: SDE main theorem\] below uses only the moment estimate (\[eq: moment estimate supremum\]). This provides an alternative proof to Theorem \[thm: SDE main theorem\].
Existence and uniqueness of strong solutions
--------------------------------------------
Now we can construct a unique almost everywhere stochastic flow to (\[eq: SDE\]) using approximation. The argument essentially follows Zhang [@ZhangXicheng2009]. Firstly, we define almost everywhere stochastic flow as follows.
\[def: almost everywhere stochastic flow\]Suppose $\{X_{t}\}_{t\in[0,T]}$ is a $\rd$-valued stochastic process defined on $\Omega\times\rd\times[0,T]$. We say that $\{X_{t}\}_{t\in[0,T]}$ is an almost everywhere stochastic flow to (\[eq: SDE\]) if
[(1)]{}
: for $\mathbb{P}\times m$-almost all $(\omega,x)\in\Omega\times\rd$, $t\rightarrow X_{t}(\omega,x)$ is a $\rd$-continuous function on $[0,T]$;
[(2)]{}
: for $\mathbb{P}$-almost all $\omega\in\Omega$, under mapping $x\rightarrow X_{t}(\omega,x)$, the push-forward of the Lebesgue measure $m$ restricted to any Borel set $A\subset\rd$ has density, i.e. $(m1_{A})\circ X_{t}^{-1}(\omega)=\rho_{t}(\omega,A,x)dx$, where the density satisfies that $\rho_{t}(\omega,A,\cdot)\leq1$ for all $x\in\rd$ and $\int_{\rd}\rho_{t}(\omega,A,x)dx=m(A)$;
[(3)]{}
: for any $t\in[0,T]$, we have $$X_{t}(x)=x+\int_{0}^{t}b(s,X_{s}(x))ds+\int_{0}^{t}dB_{s}$$ for $\mathbb{P}\times m$-almost all $(\omega,x)\in\Omega\times\rd$.
When the vector field $b$ is smooth and divergence-free, the strong solution $X_{t}$ to (\[eq: SDE\]) preserves the Lebesgue measure in the sense that $$\mathbb{P}\left[\omega\in\Omega:m(X_{t}(\omega,A))=m(A)\right]=1,\label{eq: preserve lebesgue measure}$$ where $X_{t}(\omega,A)$ is the image of any Borel set $A\in\rd$ under mapping $x\rightarrow X_{t}(\omega,x)$. Clearly $X_{t}$ satisfies Definition \[def: almost everywhere stochastic flow\].
We first recall the following lemma in Crippa and De Lellis [@DelellisCrippa2008 Lemma A.3].
\[lem: local maximal estimate\]Let $M_{R}f$ be the local maximal function of locally integrable function $f$ defined as $$M_{R}f(x)=\sup_{0<r<R}\frac{1}{\vert B_{r}\vert}\int_{B_{r}(x)}f(y)dy.$$ Suppose $f\in BV_{loc}(\rd)$, then $$\vert f(x)-f(y)\vert\leq C\vert x-y\vert\left[M_{R}\vert\nabla f\vert(x)-M_{R}\vert\nabla f\vert(y)\right]\label{eq: local maximal inequality 1}$$ for $x,y\in\rd\backslash N$, where $N$ is a negligible set in $\rd$, $R=\vert x-y\vert$ is the distance between $x$ and $y$, and constant $C$ depends only on the dimension $d$.
We denote by $Mf$ the maximal function $$Mf(x)=\sup_{0<r<\infty}\frac{1}{\vert B_{r}\vert}\int_{B_{r}(x)}f(y)dy$$ and clearly inequality (\[eq: local maximal inequality 1\]) is also true if we replace $M_{R}\vert\nabla f\vert$ with $M\vert\nabla f\vert$.
\[lem: the log estiamte\]Suppose $X_{t}(x)$ and $\tilde{X}_{t}(x)$ are almost everywhere stochastic flows to SDE (\[eq: SDE\]) driven by the same Brownian motion, with initial data $x$ and drifts $b$ and $\tilde{b}$ in $L^{1}(0,T;W^{1,p}(\rd))$, $p\geq1$ respectively. Then for any $r>0$ and $\theta>0$, $$\mathbb{E}\left[\int_{B_{r}}\log\left(\frac{\sup_{0\leq t\leq T}\vert X_{t}(x)-\tilde{X}_{t}(x)\vert^{2}}{\theta^{2}}+1\right)dx\right]\leq C\left(\Vert\nabla b\Vert_{L^{1}(0,T;L^{p}(\rd))}+\frac{1}{\theta}\Vert b-\tilde{b}\Vert_{L^{1}(0,T;L^{p}(\rd))}\right),$$ where the constant $C$ depends on $(r,p,d)$.
Consider $$\begin{aligned}
\frac{d}{dt}\log\left(\frac{\vert X_{t}(x)-\tilde{X}_{t}(x)\vert^{2}}{\theta^{2}}+1\right) & \leq\frac{\vert X_{t}(x)-\tilde{X}_{t}(x)\vert\vert b(t,X_{t}(x))-\tilde{b}(t,\tilde{X}_{t}(x))\vert}{\vert X_{t}(x)-\tilde{X}_{t}(x)\vert^{2}+\theta^{2}}\\
& \leq\frac{\vert b(t,X_{t}(x))-b(t,\tilde{X}_{t}(x))\vert}{\sqrt{\vert X_{t}(x)-\tilde{X}_{t}(x)\vert^{2}+\theta^{2}}}+\frac{\vert b(t,\tilde{X}_{t}(x))-\tilde{b}(t,\tilde{X}_{t}(x))\vert}{\sqrt{\vert X_{t}(x)-\tilde{X}_{t}(x)\vert^{2}+\theta^{2}}}\\
& =g_{1}(x)+g_{2}(x).\end{aligned}$$ Integrate both sides on $B_{r}$ and take expectation, then by Lemma \[lem: local maximal estimate\] we have that $$\begin{aligned}
\mathbb{E}\left[\int_{B_{r}}g_{1}(x)dx\right] & \leq\mathbb{E}\left[\int_{B_{r}}\frac{C\vert X_{t}(x)-\tilde{X}_{t}(x)\vert(M\vert\nabla b\vert(t,X_{t}(x))+M\vert\nabla b\vert(t,\tilde{X}_{t}(x))}{\sqrt{\vert X_{t}(x)-\tilde{X}_{t}(x)\vert^{2}+\theta^{2}}}dx\right]\\
& \leq C\int_{\Omega}\left(\int_{B_{r}}M\vert\nabla b\vert(t,X_{t}(\omega,x))dx+\int_{B_{r}}M\vert\nabla b\vert(t,\tilde{X}_{t}(\omega,x))dx\right)d\mathbb{P}(\omega).\\
& =C\int_{\Omega}\left(\int_{\rd}M\vert\nabla b\vert(t,x)\rho_{t}(\omega,B_{r},x)dx+\int_{\rd}M\vert\nabla b\vert(t,x)\tilde{\rho}_{t}(\omega,B_{r},x)dx\right)d\mathbb{P}(\omega).\end{aligned}$$ Here $\Vert M\vert\nabla b\vert\Vert_{L^{p}(\rd)}\leq C\Vert\nabla b\Vert_{L^{p}(\rd)}$ and $L^{p}(\rd)\subset L^{1}(\rd)+L^{\infty}(\rd)$. For any $f\in L^{p}(\rd)$, we have $f=f_{1}+f_{2}$, where $f_{1}=f1_{\{f<\Vert f\Vert_{L^{p}}\}}$ and $f_{2}=f1_{\{f\geq\Vert f\Vert_{L^{p}}\}}$. It is easy to verify that $\Vert f_{1}\Vert_{L^{\infty}(\rd)}\leq\Vert f\Vert_{L^{p}(\rd)}$ and $\Vert f_{2}\Vert_{L^{1}(\rd)}\leq\Vert f\Vert_{L^{p}(\rd)}$. By (2) in Definition \[def: almost everywhere stochastic flow\], we have that $\Vert\rho_{t}(\omega,B_{r},\cdot)\Vert_{L^{\infty}(\rd)}\leq1$, $\Vert\rho_{t}(\omega,B_{r},\cdot)\Vert_{L^{1}(\rd)}\leq\vert B_{r}\vert$ and the same is true for $\tilde{\rho}_{t}(\omega,B_{r}\cdot)$. Hence $$\begin{aligned}
\mathbb{E}\left[\int_{B_{r}}g_{1}(x)dx\right] & \leq2C\int_{\Omega}(1+\vert B_{r}\vert)\Vert\nabla b\Vert_{L^{p}(\rd)}d\mathbb{P}(\omega)\\
& =2C(1+\vert B_{r}\vert)\Vert\nabla b\Vert_{L^{p}(\rd)}.\end{aligned}$$ Similarly, we have $$\begin{aligned}
\mathbb{E}\left[\int_{B_{r}}g_{2}(x)dx\right] & \leq\frac{1}{\theta}\mathbb{E}\left[\int_{B_{r}}\vert b(t,\tilde{X}_{t}(x))-\tilde{b}(t,\tilde{X}_{t}(x))\vert dx\right]\\
& \leq\frac{1}{\theta}\int_{\Omega}\int_{\rd}\vert b-\tilde{b}\vert(t,x)\tilde{\rho}_{t}(\omega,B_{r},x)dxd\mathbb{P}(\omega)\\
& \leq\frac{1}{\theta}(1+\vert B_{r}\vert)\Vert b-\tilde{b}\Vert_{L^{p}(\rd)}.\end{aligned}$$ Finally, we integrate in $t$, and take supremum over time $t$ for $\log\left(\frac{\vert X_{t}(x)-\tilde{X}_{t}(x)\vert^{2}}{\theta^{2}}+1\right)$ and the proof is complete.
Now we are ready to prove the main result in this section.
\[thm: SDE main theorem\]Given a divergence-free vector field $b\in L^{1}(0,T;W^{1,p}(\rd))\cap L^{l}(0,T;L^{q}(\rd))$ with $d\geq3$, $p\geq1$, $\frac{2}{l}+\frac{d}{q}\in[1,2)$, there is a unique almost everywhere stochastic flow $X(\omega,x):\Omega\times\rd\rightarrow C([0,T],\rd)$ to $$dX_{t}(\omega,x)=b(t,X_{t}(\omega,x))dt+dB_{t}(\omega),\qquad X_{0}(\omega,x)=x$$ in space $L^{k}(\Omega\times B_{r};C([0,T],\rd))$ for any $k\geq1$ and $r>0$.
Step 1: We prove the existence of solution $X_{t}$ using approximation. By cut-off and mollification, we can find a sequence of divergence-free $b^{(n)}\in C([0,T],C_{0}^{\infty}(\rd))$ such that $b^{(n)}\rightarrow b$ in $L^{1}(0,T;W^{1,p}(\rd))\cap L^{l}(0,T;L^{q}(\rd))$ and denote by $X_{t}^{(n)}$ the corresponding solution. We first prove that $X_{t}^{(n)}$ is a Cauchy sequence in space $L^{k}(\Omega\times B_{r};C([0,T]))$ for any $k\geq1$ and $r>0$. Denote $$O_{n,m}^{R}(\omega)=\left\{ x\in\rd:\sup_{0\leq t\leq T}\vert X_{t}^{(n)}(\omega,x)\vert<R,\sup_{0\leq t\leq T}\vert X_{t}^{(m)}(\omega,x)\vert<R\right\}$$ and by Proposition \[prop: supremum estimate of diffusion\] we have that for any fixed $r>0$, $$\lim_{R\rightarrow\infty}\sup_{n,m}\sup_{x\in B_{r}}\mathbb{P}(\omega:x\notin O_{n,m}^{R}(\omega))=0.\label{eq: theorem local inequality 1}$$ Set $S_{T}^{(n,m)}(\omega,x)=\sup_{0\leq t\leq T}\vert X_{t}^{(n)}(\omega,x)-X_{t}^{(m)}(\omega,x)\vert^{2}$, then for any fixed $\delta>0$ we have $$\begin{aligned}
\mathbb{P}\left(\omega:\int_{B_{r}}S_{T}^{(n,m)}(\omega,x)dx\geq2\delta\right) & \leq\mathbb{P}\left(\omega:\int_{B_{r}\cap O_{n,m}^{R}(\omega)}S_{T}^{(n,m)}(\omega,x)dx\geq\delta\right)\\
& \quad+\mathbb{P}\left(\omega:\int_{B_{r}\backslash O_{n,m}^{R}(\omega)}S_{T}^{(n,m)}(\omega,x)dx\geq\delta\right)\\
& =I_{1}^{(n,m)}+I_{2}^{(n,m)}.\end{aligned}$$ To show convergence in probability of $X_{t}^{(n)}$, for any $\epsilon>0$, we find $R$ and large enough $n,m$ such that $I_{i}^{n,m}\leq\epsilon$, $i=1,2$. We first estimate the second term $I_{2}^{(n,m)}$ $$\begin{aligned}
I_{2}^{(n,m)} & \leq\frac{1}{\delta}\mathbb{E}\left[\int_{B_{r}\backslash O_{n,m}^{R}(\omega)}\sup_{0\leq t\leq T}\vert X_{t}^{(n)}(\omega,x)-X_{t}^{(m)}(\omega,x)\vert^{2}dx\right]\\
& \leq\frac{1}{\delta}\int_{B_{r}}\int_{\Omega}\sup_{0\leq t\leq T}\vert X_{t}^{(n)}(\omega,x)-X_{t}^{(m)}(\omega,x)\vert^{2}1_{\{\omega:x\notin O_{n,m}^{R}(\omega)\}}d\mathbb{P}(\omega)dx\\
& \leq\frac{1}{\delta}\int_{B_{r}}4\mathbb{E}\left[\sup_{0\leq t\leq T}\vert X_{t}^{(n)}(\omega,x)-x\vert^{4}+\sup_{0\leq s\leq t}\vert X_{t}^{(m)}(\omega,x)-x\vert^{4}\right]^{\frac{1}{2}}\mathbb{P}(\omega:x\notin O_{n,m}^{R}(\omega))^{\frac{1}{2}}dx\\
& \leq\epsilon\end{aligned}$$ for large enough $R>M$ by (\[eq: theorem local inequality 1\]) and Proposition \[prop: supremum estimate of diffusion\]. Here the choice of $M$ is independent of $(n,m)$. To obtain the estimate that $I_{2}^{(n,m)}\leq\epsilon$, we fixed an $R$. With the same $R$, next we estimate $I_{1}^{(n,m)}$. For any $\omega\in\Omega$, if $$\int_{B_{r}}\log\left(\frac{S_{T}^{(n,m)}(\omega,x)}{\theta^{2}}+1\right)dx\leq L,$$ we have that $\vert\{S_{T}^{(n,m)}(\omega,x)\geq\theta^{2}(e^{L^{2}}-1)\}\vert\leq\frac{1}{L}$, which implies $$\begin{aligned}
\int_{B_{r}\cap O_{n,m}^{R}(\omega)}S_{T}^{n,m}(\omega,x)dx & =\int_{B_{r}\cap O_{n,m}^{R}(\omega)}S_{T}^{(n,m)}(\omega,x)1_{\{S_{T}^{(n,m)}(x)\geq\theta^{2}(e^{L^{2}}-1)\}}dx\\
& \quad+\int_{B_{r}\cap O_{n,m}^{R}(\omega)}S_{T}^{(n,m)}(\omega,x)1_{\{S_{T}^{(n,m)}(x)<\theta^{2}(e^{L^{2}}-1)\}}dx\\
& \leq\theta^{2}(e^{L^{2}}-1)\vert B_{r}\vert+4R^{2}\frac{1}{L}.\end{aligned}$$ Now we set $\theta^{(n,m)}=\Vert b^{(n)}-b^{(m)}\Vert_{L_{t}^{1}L_{x}^{p}}$ to obtain $$\sup_{n,m}\mathbb{E}\left[\int_{B_{r}}\log\left(\frac{S_{T}^{(n,m)}}{(\theta^{(n,m)})^{2}}+1\right)dx\right]\leq C$$ by Lemma \[lem: the log estiamte\], which implies that $$\sup_{n,m}\mathbb{P}\left(\int_{B_{r}}\log\left(\frac{S_{T}^{(n,m)}}{(\theta^{(n,m)})^{2}}+1\right)dx\geq L\right)\leq\frac{C}{L}.$$ For fixed $\delta>0$ and the fixed $R$ obtained from the estimate of $I_{2}^{(n,m)}$, we can first choose $L$ large enough and them choose $(n,m)$ large enough, which means $\theta^{(n,m)}$ is small enough, such that $$(\theta^{(n,m)})^{2}(e^{L^{2}}-1)\vert B_{r}\vert+4R^{2}\frac{1}{L}<\delta\quad\mbox{and}\quad\frac{C}{L}\leq\epsilon.$$ Hence $$\mathbb{P}\left(\omega:\int_{B_{r}\cap O_{n,m}^{R}(\omega)}S_{T}^{n,m}(\omega,x)dx\geq\delta,\int_{B_{r}}\log\left(\frac{S_{T}^{(n,m)}(\omega,x)}{(\theta^{(n,m)})^{2}}+1\right)dx\leq L\right)=0,$$ which implies that $$\begin{aligned}
I_{1}^{(n,m)} & =\mathbb{P}\left(\omega:\int_{B_{r}\cap O_{n,m}^{R}(\omega)}S_{T}^{n,m}(\omega,x)dx\geq\delta,\int_{B_{r}}\log\left(\frac{S_{T}^{(n,m)}(\omega,x)}{(\theta^{(n,m)})^{2}}+1\right)dx>L\right)\\
& \leq\mathbb{P}\left(\omega:\int_{B_{r}}\log\left(\frac{S_{T}^{(n,m)}(\omega,x)}{(\theta^{(n,m)})^{2}}+1\right)dx>L\right)\\
& \leq\epsilon.\end{aligned}$$ Now we have that for any $\epsilon>0$, there is $(n,m)$ large enough such that $$\mathbb{P}\left(\omega:\int_{B_{r}}S_{T}^{(n,m)}(\omega,x)dx\geq2\delta\right)\leq2\epsilon.$$ Hence $$\lim_{n,m\rightarrow\infty}\mathbb{P}\left(\omega:\int_{B_{r}}S_{T}^{(n,m)}(\omega,x)dx\geq2\delta\right)=0$$ for any fixed $\delta$. This implies that for any fixed $r>0$, $\{X_{t}^{(n)}\}$ converges in probability under the finite measure $\mathbb{P}\times m1_{B_{r}}$ as functions $X^{(n)}:\Omega\times B_{r}\rightarrow C([0,T],\rd)$. Recall that for any $k\geq1$ we have $$\sup_{n,x\in B_{r}}\mathbb{E}\left[\sup_{0\leq s\leq t}\vert X_{s}^{(n)}(x)\vert^{k}\right]<\infty$$ by Proposition \[prop: supremum estimate of diffusion\], which means that for any fixed $k\geq1$, $\sup_{0\leq t\leq T}\vert X_{t}^{(n)}(x)\vert^{k}$ are uniformly integrable. This implies that for any fixed $r>0$, $\{X_{t}^{(n)}\}_{n}$ is a Cauchy sequence in $L^{k}(\Omega\times B_{r};C([0,T]))$ and we denote the limit as $X_{t}$. Since each $X_{t}^{(n)}$ satisfies Definition \[def: almost everywhere stochastic flow\], it is easy to verify that their limit $X_{t}$ also satisfies (1) and (2) of Definition \[def: almost everywhere stochastic flow\].
Step 2: Now we verify that the limit $X_{t}$ satisfies (3) of Definition \[def: almost everywhere stochastic flow\], i.e. if we define $$Y_{t}(x)=x+\int_{0}^{t}b(s,X_{s}(x))ds+\int_{0}^{t}dB_{s},$$ then $X_{t}=Y_{t}$ in space $L^{1}(\Omega\times B_{r};C([0,T]))$. Consider $$\sup_{0\leq t\leq T}\vert X_{t}^{n}(x)-Y_{t}(x)\vert\leq\int_{0}^{T}\vert b^{n}(t,X_{t}^{(n)}(x))-b(t,X_{t}(x))\vert dt.$$ Then integrate both sides on $B_{r}$ and take expectation, we have that $$\begin{aligned}
\mathbb{E}\left[\int_{B_{r}}\sup_{0\leq t\leq T}\vert X_{t}^{n}(x)-Y_{t}(x)\vert dx\right] & \leq\mathbb{E}\left[\int_{B_{r}}\int_{0}^{T}\vert b^{n}(t,X_{t}^{(n)}(x))-b(t,X_{t}^{(n)}(x))\vert dtdx\right]\\
& \quad+\mathbb{E}\left[\int_{B_{r}}\int_{0}^{T}\vert b(t,X_{t}^{(n)}(x))-b(t,X_{t}(x))\vert dtdx\right]\\
& =I_{1}+I_{2}.\end{aligned}$$ Again by (2) in Definition \[def: almost everywhere stochastic flow\], we have that $I_{1}\leq(1+\vert B_{r}\vert)\Vert b-b^{(n)}\Vert_{L_{t}^{1}L_{x}^{p}}$. For the second term, we will find another smooth $b_{\epsilon}$ such that $\Vert b_{\epsilon}-b\Vert_{L_{t,x}^{p}}\leq\epsilon$ and then separate $I_{2}$ into three parts $$\begin{aligned}
I_{2} & \leq\mathbb{E}\left[\int_{B_{r}}\int_{0}^{T}\vert b_{\epsilon}(t,X_{t}^{(n)}(x))-b_{\epsilon}(t,X_{t}(x))\vert dtdx\right]\\
& \quad+\mathbb{E}\left[\int_{B_{r}}\int_{0}^{T}\vert b_{\epsilon}(t,X_{t}^{(n)}(x))-b(t,X_{t}^{(n)}(x))\vert dtdx\right]\\
& \quad+\mathbb{E}\left[\int_{B_{r}}\int_{0}^{T}\vert b_{\epsilon}(t,X_{t}(x))-b(t,X_{t}(x))\vert dtdx\right].\end{aligned}$$ For the second and the third terms, we control them just as $I_{1}$. The first term converges to $0$ as $n\rightarrow0$ since $X_{t}^{(n)}\rightarrow X_{t}$ in $L^{k}(\Omega\times B_{r};C([0,T]))$ and now we can conclude that $X_{t}=Y_{t}$ $\mathbb{P}\times m$-almost everywhere.
Step 3: Finally we prove that the solution is unique. Suppose that we have two almost everywhere stochastic flows $X_{t}$ and $\tilde{X}_{t}$ corresponding to the same $b\in L^{1}(0,T;W^{1,p}(\rd))\cap L^{l}(0,T;L^{q}(\rd))$ and we apply Lemma \[lem: the log estiamte\] to them to deduce that $$\mathbb{E}\left[\int_{B_{r}}\log\left(\frac{\sup_{0\leq t\leq T}\vert X_{t}(x)-\tilde{X}_{t}(x)\vert^{2}}{\theta^{2}}+1\right)dx\right]\leq C\Vert\nabla b\Vert_{L_{t}^{1}L_{x}^{p}},$$ which is uniform for all $\theta>0$. Hence we can take $\theta\rightarrow0$ and now the proof is complete.
**Acknowledgment:** The Authors would like to thank our supervisor Professor Zhongmin Qian for bringing these questions to our attention and discussing constantly with us. We also want to thank the reviewers for pointing out an alternative proof of Theorem \[thm: SDE main theorem\] using Kolmogorov’s continuity theorem.
[^1]: Normura International, 30/FL Two International Finance Centre, Hong Kong. Email: chamonixliu@163.com
[^2]: Department of Mathematics, University of Maryland, College Park, MD 20742, USA. Email: gxi@umd.edu
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abstract: 'We introduce and study Maker/Breaker-type positional games on random graphs. Our main concern is to determine the threshold probability $p_{\cf}$ for the existence of Maker’s strategy to claim a member of $\cf$ in the unbiased game played on the edges of random graph $G(n,p)$, for various target families $\cf$ of winning sets. More generally, for each probability above this threshold we study the smallest bias $b$ such that Maker wins the $(1\:b)$ biased game. We investigate these functions for a number of basic games, like the connectivity game, the perfect matching game, the clique game and the Hamiltonian cycle game.'
author:
- 'Miloš Stojaković [^1] [^2]'
- Tibor Szabó
title: Positional games on random graphs
---
Introduction {#s:intro}
============
#### (Un)biased positional games.
Let $X$ be a finite nonempty set and $\cf\subseteq 2^X$. The pair $(X, \cf)$ is a [*positional game*]{} on $X$. The game is played by two players Maker and Breaker, where in each move Maker claims one previously unclaimed element of $X$ and then Breaker claims one previously unclaimed element of $X$. Maker wins if he claims all the elements of some set in $\cf$, otherwise Breaker wins. The set $X$ will be referred to as the board, and the set $\cf$ as the set of winning sets. Whenever there is no confusion about what the board is, we may refer to the game $(X,\cf)$ as just $\cf$.
Unless otherwise stated, we assume that Maker starts the game. We note, however, that the asymptotic statements discussed in the paper are not influenced by which player makes the first move. For technical reasons we still have to talk about games in which Breaker starts. So in order to avoid confusion, the positional game with board $X$ and set of winning sets $\cf$ in which Breaker makes the first move is denoted by $(\widehat X, \cf)$.
The set of all positional games could be partitioned into two classes. The game $(X,\cf)$ is called a [*Maker’s win*]{} if Maker has a winning strategy, that is, playing against an arbitrary strategy Maker can occupy a member of $\cf$. Clearly, if $(X,\cf )$ is [*not*]{} a Maker’s win, then Breaker is able to prevent any opponent from occupying a winning set. Such a positional game is called a [*Breaker’s win*]{}.
Typical, well-studied examples of such positional games are played on the edges of a complete graph, i.e. $X=E(K_n)$. Maker’s goal usually is to build a graph theoretic structure – like a spanning tree, a perfect matching, a Hamiltonian cycle, or a clique of fixed size. It turns out that all these games are won easily by Maker if $n$ is sufficiently big, so in order to make things more fair (if such thing exists; actually no game of perfect information is [*fair*]{} as the winner—in theory—is known in the beginning of the game) one could give Breaker extra power by allowing him to claim more than $1$ edge in each move.
If $X$ is a finite nonempty set, $\cf\subseteq 2^X$ and $a,b$ are positive integers (possibly functions of the board size), then the 4-tuple $(X, \cf, a, b)$ is a [*biased $(a\:b)$ game*]{}. In a biased $(a\:b)$ game, Maker claims $a$ elements (instead of 1) and Breaker claims $b$ elements (instead of 1) in each move. Recall, that unless otherwise stated Maker starts the game. The biased game in which Breaker starts is denoted by $(\widehat X, \cf, a, b)$. Note that $a$ is always the bias of Maker, independently from who is the first player to move.
For a family $\cf$ the smallest integer $b_{\cf}$ is sought (and sometimes found; see [@bec; @be2; @be3; @bl1; @bl2; @ce]) for which Breaker wins the $(1:b_{\cf})$ game.
In the [*connectivity game*]{} Maker’s goal is to build a connected spanning subgraph; i.e. in this game the family of winning sets is the family $\ct=\ct_n$ of all spanning trees on $n$ vertices. Chv' atal and Erdős proved [@ce] that $b_\ct=\Theta (\frac{n}{\log n} )$.
Beck [@bec] established $b_\ch=\Theta(\frac{n}{\log n})$, where $\ch=\ch_n$ is the family of all Hamiltonian cycles on $n$ vertices.
For the family $\ckk_k=\ckk_{k,n}$ of all $k$-cliques on $n$ vertices, Bednarska and Łuczak [@bl1] showed that $b_{\ckk_k}=\Theta (n^{\frac{2}{k+1}})$. More generally, they proved that in the game in which Maker’s goal is to claim an arbitrary fixed graph $G$, the threshold bias is $\Theta (n^{1/m'(G)})$. (Here $m'(G)$ is the maximum of $\frac{e(H)-1}{v(H)-2}$ over all subgraphs $H$ of $G$ with at least $3$ vertices.)
#### Playing on a random board.
In the present paper we introduce another approach to even out the advantage Maker has in a $(1\: 1)$ game, by randomly reducing the board size and keeping only those winning sets which survive this thinning intact.
\[df:1\] Let $(X, \cf, a, b)$ be a biased game. [*Random game*]{} $(X_p, \cf_p,a,b)$ is a probability space of games where each $x\in X$ is independently included in $X_p$ with probability $p$, and $\cf_p=\{W\in \cf {:\,}W\subseteq X_p\}$.
Apart from the trivial case $\emptyset \in \cf$, Breaker surely wins when $p=0$. On the other hand, the unbiased version of all the graph games that we consider are (easy) Maker’s wins, when $p=1$ and the board is sufficiently large. For any other probability $p$, $0<p<1$, we cannot be sure who (Maker or Breaker) wins the random game $\cf_p$. The best we can conclude is that Maker (or Breaker) wins a.s. (almost surely), i.e. the probability that Maker (Breaker) wins tends to 1 if the board size tends to infinity. (So we actually talk about an infinite family of probability spaces of games …)
Let $(X,\cf)$ be a particular sequence of games, where $\emptyset\notin \cf$, the board size tends to infinity, and $(X,\cf,1,1)$ is won by Maker provided $|X|$ is big enough. The first natural question to ask is: What is the threshold probability $p_{\cf}$ at which an almost sure Breaker’s win turns into an almost sure Maker’s win. More precisely we would like to determine $p_\cf$ for which
- ${{\rm Pr}\hspace{-0.2ex}\left[ (X_p,\cf_p,1,1) \mbox{ is a Breaker's win}\right]}\rightarrow 1$ for $p=o(p_{\cf})$, and\
- ${{\rm Pr}\hspace{-0.2ex}\left[ (X_p,\cf_p,1,1) \mbox{ is a Maker's win}\right]}\rightarrow 1$ for $p=\omega(p_{\cf})$.
Such a threshold $p_\cf$ exists [@bt], since being a Maker’s win is an [*increasing property*]{}.
The main goal of this paper is to establish a connection between the natural threshold values, $b_\cf$ and $p_\cf$, corresponding to the two different weakenings of Maker’s power: bias and random thinning, respectively. We find that there is an intriguing reciprocal connection between these two thresholds in a number of well-studied games on graphs.
Recall the notations $\ct$, $\ch$, and $\ckk_k$, and let us denote by $\cm$ the set of all perfect matchings on the graph $K_n$.
\[main1\] For positional games, played on $E(K_n)$, we have
- $p_\ct = \frac{\log n}{n} $,\
- $p_\cm = \frac{\log n}{n}$,\
- $\frac{\log n}{n} \leq p_\ch \leq
\frac{\log n}{\sqrt{n}}$,\
- $n^{-\frac{2}{k+1}-\varepsilon} \leq
p_{\ckk_k} \leq n^{-\frac{2}{k+1}}$, for every integer $k\geq 4$ and every constant $\varepsilon>0$.\
- $p_{\ckk_3} = n^{-\frac{5}{9}}$.
For the connectivity game $\ct$ an even more precise statement is true. In Corollary \[connectivity-precise\] we observe that Maker starts to win a.s. at the very moment when the last vertex of a random graph process picks up its second incident edge.
More generally, for every $p$ we would like to find the smallest bias $b^p_{\cf}$ such that Breaker wins the random game $(X_p, \cf_p, 1, b^p_\cf)$ a.s. Note that by definition $b_\cf=b^1_\cf$. Another trivial observation is that $b^p_\cf=0$ provided $p$ is less than the threshold for the appearance of the first element of $\cf$ in the random graph.
We obtain the following.
\[main2\] There exist constants $C_1, C_2, C_3$, such that
- $b_\ct^p = \Theta \left(pb_\ct\right) =
\Theta \left( p\frac{n}{\log n}\right)$, provided $p\geq
C_1\frac{1}{b_{\ct}}$,\
- $b_\cm^p = \Theta \left(p b_\cm\right) =
\Theta \left(p\frac{n}{\log n}\right)$, provided $p\geq
C_2\frac{1}{b_{\cm}}$,\
- $\Omega\left(p\frac{\sqrt{n}}{\log n}\right)
\leq b_\ch^p \leq O \left(p\frac{n}{\log n}\right) $, provided $p\geq
C_3\frac{\log n}{\sqrt{n}}$,\
- There exists $c_k>0$, such that $b_{\ckk_k}^p = \Theta\left( pb_{\ckk_k}\right)
= \Theta \left( p n^{\frac{2}{k+1}} \right)$, provided $p= \Omega\left(\frac{\log^{c_k} n}{b_{\ckk_k}}\right)$.
One can see that $b_\cf^p$ is of order $p/p_\cf=pb_{\cf}$ for the connectivity game and the perfect matching game, provided $p\geq Cp_\cf$ for some constant $C$. In particular for these games $p_\cf=\Theta(1/b_\cf )$. In part $(iv)$ of Theorem \[main2\], generalizing the arguments of Bednarska and Łuczak [@bl1] we show that one can estimate $b_{\ckk_k}^p$ up to a constant factor, for all probabilities down to a polylogarithmic factor away from the critical probability $1/b_{\ckk_k}=n^{-\frac{2}{k+1}}$. On the other hand Theorem \[main1\] part $(v)$ shows that in the case $k=3$ we cannot get arbitrarily close to probability $1/b_{\ckk_k}$, since Maker [*can win*]{} even for probabilities below $1/b_{\ckk_3}=n^{-1/2}$.
Nevertheless we think the Hamiltonian cycle game behaves “nicely”, i.e. the same way as the connectivity game and the perfect matching game.
Let $\ch$ be the set of Hamiltonian cycles in $K_n$. There exists a constant $C$ such that $$b_{\ch}^p = \Theta \left( p\frac{n}{\log n}\right), \mbox{ provided
$p\geq C\frac{\log n}{n}$}.$$ In particular, $$p_{\ch} = \frac{\log n}{n}.$$
Observe that the validity of the conjecture would mean that in a random graph with edge probability $p\geq C\frac{\log n}{n}$ Maker could build a Hamiltonian cycle. So Pósa’s Theorem (which only proves the existence of a Hamiltonian cycle) would be true constructively even if an adversary is playing against us.
The paper is organized as follows. In Section \[s:criterion\] we prove a general criterion for Breaker’s win in a different, auxiliary random game. In Section \[s:games\], the analysis of four biased random games is presented. In particular, in Subsections \[ss:conn\], \[ss:ham\], \[ss:pm\] and \[ss:clique\] we look at the connectivity game, the Hamiltonian cycle game, the perfect matching game and the clique game, respectively. In Section \[s:unbiased\] we analyze more precisely a couple of $(1\:1)$ games – the connectivity game (Subsection \[ss:con1on1\]) and the clique game (Subsection \[ss:clique1on1\]). Finally, in Section \[s:open\] we give a collection of open questions and conjectures.
[**Notation.**]{}
For a graph $G$, $e(G)$ and $v(G)$ denote the number of edges and vertices (respectively) of $G$, $\delta (G)$ denotes the minimum degree of $G$, and $E(G)$ and $V(G)$ denote the sets of edges and vertices (respectively). If $C\subseteq V(G)$ and $v\in V(G)$, then $N_C(v)$ denotes the set of neighbors of $v$ in $C$. The logarithm $\log n$ in this paper is always of natural base. For functions $f(n), g(n)\geq 0$, we say that $f=O(g)$ if there are constants $C$ and $K$, such that $f(n) \leq Cg(n)$ for $n\geq K$; $f=\Omega( g)$ if $g=O(f)$; $f=\Theta (g)$ if $f=O(g)$ and $f=\Omega (g)$; $f=o(g)$ if $f(n)/g(n)\rightarrow 0$ when $n\rightarrow \infty$; $f=\omega (g)$ if $g=o(f)$.
A criterion {#s:criterion}
===========
One of few general, but still very applicable results to decide the winner of biased positional games is the biased version of the Erdős–Selfridge Theorem [@es; @be2]. It provides a criterion for Breaker to win, applicable on any game.
(Beck, [@be2]) \[t:es\] If $$\sum_{A\in \cf} (1+b)^{-|A|/a} < 1,$$ then Breaker has a winning strategy in the $(\widehat{X},\cf,a,b)$ game.
If Maker plays the first move then the $1$ on the right hand side of the criterion is to be replaced by the fraction $\frac{1}{1+b}$.
We will also need the following extension.
([@be2; @bl1]) \[t:gen-es\] If for a positive integer $c$ we have $$\sum_{A\in \cf} (1+b)^{-|A|/a} < c\frac{1}{1+b},$$ then Breaker has a winning strategy in the $(X,\{\cup_{B\in F} B {:\,}F\in {\cf \choose c} \},a,b)$ game.
In this section we give an adaptation of the first criterion which proves to be very useful in dealing with positional games on a random board. We need the following technical definition.
Let $(X, \cf, a, b)$ be a biased game. [*Random game $(X_p, \cf_p^\cap,a,b)$ with induced set of winning sets*]{} is a probability space of games, where $X_p$ is defined as in Definition \[df:1\] and $\cf_p^\cap=\{W{:\,}\exists F\in \cf,\, W=F\cap X_p\}$.
The following statement is the randomized version of Theorem \[t:es\]. It is stated for the biased $(b\:1)$ game in which Breaker is the first player, because this is the version we will need in our applications.
\[t:es\_cap\] Let $\cf$ be a set of winning sets on $X$ with $$\begin{aligned}
\sum_{A\in\cf} 2^{-\frac{|A|}{b}} <1 \label{c:es}\end{aligned}$$ (i.e. the condition of the Erdős–Selfridge Theorem holds for the $(\widehat X, \cf, b, 1)$ game), and $$\begin{aligned}
\lim_{n\rightarrow\infty} \min_{A\in\cf} \frac{|A|}{b} =
\infty. \label{c:ws_large}\end{aligned}$$
If $p$ and $\delta >0$ are chosen so that $p>\frac{4\log 2}{\delta^2 b}$ holds, then the game $(\widehat{X}_p, \cf_{p}^\cap,
(1-\delta)pb, 1)$ is a Breaker’s win a.s.
For each $A\in\cf$ and its corresponding set ${A'} \in\cf_{p}^\cap$ we have ${{\hbox{\bf E}}\hspace{-0.3ex}\left[|{A'}|\right]} = p|A|$. If all winning sets $A'\in\cf_{p}^\cap$ have size at least $(1-\delta) p|A|$, then $$\sum_{{A'}\in \cf_{p}^\cap} 2^{-\frac{|A'|}{(1-\delta)pb}}
\leq
\sum_{{A}\in \cf} 2^{-\frac{(1-\delta)p|A|}{(1-
\delta)pb}} =
\sum_{{A}\in \cf} 2^{-\frac{|A|}{b}}
<1.$$ Using the Erdős–Selfridge theorem we obtain that Breaker wins the $(\widehat{X}_p, \cf_{p}^\cap,(1-\delta)pb, 1)$ game, provided $|A'|\geq (1-\delta) p|A|$ for all $A'\in \cf_{p}^\cap$.
Next we check that this condition holds almost surely. Using a Chernoff bound, we obtain that $${{\rm Pr}\hspace{-0.2ex}\left[ \exists A\in\cf {:\,}|A'|\leq (1-\delta) p|A|\right]} \leq
\sum_{A\in\cf}
e^{-\frac{\delta^2p |A|}{2}}.$$
If we denote $\min_{A\in\cf} \frac{|A|}{b} $ by $m_n$, then we have $$\begin{aligned}
\sum_{A\in\cf}
e^{-\frac{\delta^2p |A|}{2}} \leq
\sum_{A\in\cf}
2^{-2\frac{|A|}{b}} \leq \sum_{A\in\cf}
2^{-m_n} 2^{-\frac{|A|}{b}} <
2^{-m_n} \rightarrow 0,\end{aligned}$$ and therefore all winning sets $A'\in\cf_{p}^\cap$ have size at least $(1-\delta) p|A|$ a.s.
Games {#s:games}
=====
Connectivity game {#ss:conn}
-----------------
The first game we study is a random version of the biased connectivity game $(E(K_n), \ct, 1,b)$ on a complete graph on $n$ vertices $K_n$. Maker’s goal is to build a spanning, connected subgraph, i.e. $\ct$ is the set of all spanning trees on $n$ vertices.
It is obvious that $p_{\ct}=\Omega (\frac{\log n}{n})$, since for lower probabilities the random graph is a.s. not connected, and Breaker wins even if he does not claim any edges.
First we generalize this for arbitrary probability $p$ by providing Breaker with a strategy to isolate a vertex. One of our main tools is the following winning criterion of Chv' atal and Erdős on games with disjoint winning sets.
\[t:C-E\] [@ce] In a biased $(b\:1)$ game with $k$ disjoint winning sets of size $s$ Maker wins if $$\label{e:C-E}
s\leq (b-1)\sum_{i=1}^{k-1} \frac{1}{i}.$$
\[c:C-E\] In a biased $(b\:2)$ game with $k+1$ disjoint winning sets of size at most $s$ Maker wins if $$s\leq \left(\left\lfloor\frac{b}{2}\right\rfloor-1\right)\sum_{i=1}^{k-1}
\frac{1}{i}.$$
[**Proof of Corollary.**]{} Recall that as a default Maker starts the game in the Theorem and the Corollary as well. Now Theorem \[t:C-E\] obviously remains true (i.e. Maker wins) even if Breaker starts, provided there are $k+1$ disjoint winning sets instead of $k$. This implies that when Breaker starts, the bias is $(2b:2)$, there are $k+1$ winning sets and (\[e:C-E\]) holds, then Maker still wins. Indeed, since the winning sets are disjoint, after Breaker’s move Maker can just pretend to play a $(b:1)$ game and answer with his first $b$ moves to one of the two selections of Breaker, and answer with his second $b$ moves to the other move of Breaker, both according to the $(b:1)$ strategy. Now the Corollary follows, since starting instead of being second player cannot hurt Maker.
\[t:cg-b\] There exists $K_0>0$ so that for arbitrary $p\in [0,1]$ and $b\geq K_0 p \frac{n}{\log n}$ Breaker, playing the $(1:b)$ game on the edges of random graph $G(n,p)$, can achieve that Maker’s graph has an isolated vertex a.s.
Let us fix $b=\lfloor K_0pn/\log n \rfloor $, where $K_0$ is a constant to be determined later. Note that we can assume $p>\log n/2n$, since otherwise the random graph does have an isolated vertex a.s., thus Breaker achieves his goal without having to play any moves.
We present a strategy for Breaker to claim all the edges incident to some vertex of $G(n,p)$. If successful, this strategy prevents Maker from building a connected subgraph. Similar strategy was introduced by Chv' atal and Erdős [@ce] for solving the problem on the complete graph.
Let $C$ be an arbitrary subset of the vertex set of cardinality $\lfloor
n/\log n \rfloor$. Breaker will claim all the edges incident to some vertex $v\in C$ (thus preventing Maker from claiming any edge incident to $v$). We would like to use the game from Corollary \[c:C-E\], with the winning sets being the $\lfloor n/\log n\rfloor$ stars of size at most $n-1$ whose center is in $C$. Since these stars are not necessarily disjoint, formally we will talk about ordered pairs of vertices: the winning sets are denoted by $W_v=\{ (v, u): u\in V\}$, $v\in C$. We call this game [*Box*]{}. To avoid confusion with Maker and Breaker of the game from Theorem \[t:cg-b\], the players from Corollary \[c:C-E\] will be called BoxMaker and BoxBreaker. Recall that in Box the bias is $(b:2)$.
Breaker will utilize the strategy of BoxMaker from Corollary \[c:C-E\] to achieve his goal. How? He will play a game of Box in such a way that a win for BoxMaker automatically implies a win for Breaker. When Maker selects an edge $uv$, Breaker interprets it as BoxBreaker claimed the elements $(u,v)$ and $(v,u)$ in Box. Whenever Breaker would like to make a move, he looks at the current move of BoxMaker in Box, and takes those edges which correspond to the $b$ ordered pairs BoxMaker selected. If he is supposed to select an edge which has already been selected by him, he selects an arbitrary unoccupied edge. Note that the above strategy never calls for Breaker to select an edge which has already been selected by Maker.
It is also obvious, that if BoxMaker wins Box, then Breaker occupied all incident edges of a vertex from $C$.
In order to apply Corollary \[c:C-E\] it is enough then to show that the size $d(v)$ of each winning set is appropriately bounded from above, i.e. for each $v\in C$ we have $d(v) \leq \frac{K_0}{8} pn\leq
\left(\left\lfloor\frac{b}{2}\right\rfloor-1\right) \sum_{i=1}^{k-
1} \frac{1}{i}$ a.s.
Indeed, using a Chernoff bound and a large enough $K_0$, we obtain that for every $v\in C$ $${{\rm Pr}\hspace{-0.2ex}\left[ d (v)>\frac{K_0}{8} pn\right]} \leq
e^{-\frac{K_0pn}{8}}\leq n^{-\frac{K_0}{16}}.$$ Therefore we have $$\begin{aligned}
{{\rm Pr}\hspace{-0.2ex}\left[ \exists v\in C {:\,}d(v) >
\frac{K_0}{8} pn\right]} \leq
n\cdot n^{-\frac{K_0}{16}}\rightarrow 0,
\end{aligned}$$ provided $K_0$ is large enough. Then Corollary \[c:C-E\] guarantees BoxMaker’s win, thus Breaker’s win a.s., and the proof of Theorem \[t:cg-b\] is complete.
Next we give a winning strategy for Maker in the connectivity game, thus determining the threshold bias $b_{\cal T}^p$ up to a constant factor.
Obviously, Breaker wins if and only if he claims all the edges of a cut, i.e. all the edges connecting some set of vertices with its complement. In order to win Maker has to claim one edge in each of the cuts. This observation enables us to formulate the connectivity game in a different way, where winning sets are cuts and roles of players are exchanged – Breaker wants to occupy a cut and Maker wants to prevent Breaker from doing so. To avoid confusion we refer to the players of this “cut-game” by CutMaker and CutBreaker.
This new point of view enables us to give Maker a winning strategy using Theorem \[t:es\_cap\], which is a criterion for CutBreaker’s win. Observe, that in this “cut-game” CutBreaker (alias Maker) only cares about occupying the existing edges of a cut, that’s why we are going to look at the family $\cf_p^{\cap}$ instead of $\cf_p$.
\[t:cg-m\] There exists $k_0>0$, so that for $p>\frac{32\log n}{n}$ and $b\leq k_0 p\frac{n}{\log n}$ Maker wins the random connectivity game $(E(K_n)_p, \ct_p, 1, b)$ a.s.
For $b_0=\frac{\log 2}{2}\cdot\frac{n}{\log n}$ we are going to prove that the conditions of Theorem \[t:es\_cap\] are satisfied if $\cf$ is the set of all cuts in a complete graph with $n$ vertices.
On one hand, Beck [@be2] showed $\sum_{k=1}^{n/2} {n \choose k} 2^{-\frac{k(n-k)}{b_0}}\rightarrow 0$, which means that condition (\[c:es\]) holds in this setting.
On the other hand, for a cut $A\in\cf$ we have $|A|\geq n-1$ which implies condition (\[c:ws\_large\]). If we set $\delta=1/2$ we can apply Theorem \[t:es\_cap\] which gives that $(\widehat{E(K_n)}_p, \cf_p^\cap, \frac{\log 2}{4}p\frac{n}{\log
n}, 1)$ is a CutBreaker’s win a.s. The statement of the theorem immediately follows.
Theorem \[t:cg-b\] and Theorem \[t:cg-m\] together imply part $(i)$ of both Theorem \[main1\] and \[main2\].
Hamiltonian cycle game {#ss:ham}
----------------------
Here we investigate the random version of the $(1\:b)$ biased game $(E(K_n), \ch, 1,b)$ on the complete graph $K_n$, where $\ch$ is the set of all Hamiltonian cycles. Maker’s goal is to occupy all edges of a Hamiltonian cycle, while Breaker wants to prevent that. Breaker can obviously win when Maker is not able to claim a connected graph and thus from Theorem \[t:cg-b\] we obtain the following corollary.
\[t:hg-b\] There exists $H_0>0$ so that for every $p\in [0,1]$ and $b\geq H_0 p\frac{n}{\log n}$ Breaker wins the random Hamiltonian cycle game $(E(K_n)_p, \ch_p, 1, b)$ a.s.
The next Theorem describes Maker’s strategy.
\[t:hg-m\] There exists $h_0>0$, so that for $p>\frac{32\log n}{\sqrt{n}}$ and $b\leq h_0 p\frac{\sqrt{n}}{\log n}$ Maker wins the random Hamiltonian cycle game $(E(K_n)_p, \ch_p, 1, b)$ a.s.
Maker wins, if at the end of the game the subgraph $G_M$ (containing the edges claimed by Maker) has connectivity $\kappa(G_M)$ greater or equal than independence number $\alpha(G_M)$. Indeed, from the criterion of Chv' atal and Erdős for Hamiltonicity [@ce2], we obtain that $G_M$ then contains a Hamiltonian cycle.
We show that Maker, using only his odd moves, can ensure that the connectivity of his graph at the end of the game is greater then $k=\sqrt{n}/2$ and, using his even moves, can make the independence number at the end of the game smaller then $k=\sqrt{n}/2$. In other words we will look at two separate games where in each of them Maker plays one move against Breaker’s $2b$ moves. This is a correct strategy, because moves of Maker made in one of these games cannot hurt him in the other.
We first look at the odd Maker’s moves. To ensure that $\kappa(G_M)\geq k$, Maker has to claim one edge in every cut of a graph obtained from the initial graph by removing some $k$ vertices. More precisely, we are going to prove the conditions of Theorem \[t:es\_cap\] for the biased $(b'\:1)$ game, where $b'=\frac{\log 2}{2}\cdot\frac{\sqrt{n}}{\log n}$ and $$\cf = \bigg\{ \{v_1 v_2 {:\,}v_1\in V_1,\, v_2\in V_2\} {:\,}V(K_n)= V_0 \cupdis V_1 \cupdis V_2,\, |V_0|=k,\, V_1,V_2\not=
\emptyset \bigg\}.$$ That is, Maker plays the role of “CutBreaker” by trying to break all the cuts in $\cf$.
Since the size of each of the sets in $\cf$ is at least $n-k-1$ we have $$\lim_{n\rightarrow\infty} \min_{A\in\cf} \frac{|A|}{b'} =
\lim_{n\rightarrow\infty}\frac{2\log n (n-\sqrt{n}/2-1)}{\log 2\, \sqrt{n}} =
\infty,$$ and the condition (\[c:ws\_large\]) holds. Next, we have $$\begin{aligned}
\sum_{A\in\cf} 2^{-\frac{|A|}{b'}}
&=& \sum_{i=1}^\frac{n-k}{2} {n \choose i} {n-i \choose k}
2^{-\frac{i(n-i-k)}{b'}} \\
&<& \sum_{i=1}^{k} n^{2k} 2^{-\frac{n-k-1}{b'}}
+ \sum_{i=k+1}^\frac{n-k}{2}
2^{2n-\frac{k(n-2k)}{b'}}\\
&<& k\cdot n^{-\sqrt{n}} +
n\cdot n^{-n}\rightarrow 0,\end{aligned}$$ which gives the condition (\[c:es\]). Therefore, CutBreaker (alias Maker) wins the game $(\widehat{E(K_n)}_p, \cf^\cap_p,
\frac{\log 2}{4}p\frac{\sqrt{n}}{\log n}, 1)$ a.s., provided $p\geq \frac{32 \log n}{\sqrt{n}}$.
In the other part of the game using even moves Maker has to ensure that $\alpha(G_M)\leq k=\sqrt{n}/2$. That is going to be true if Maker manages to claim at least one edge in every clique of $k$ elements. To prove that it is possible we again use Theorem \[t:es\_cap\] for a biased $(b'\:1)$ game with the same value of $b'= \frac{\log 2}{2}\cdot
\frac{\sqrt{n}}{\log n}$. But now $\cf$ is the family of the edgesets of all cliques of size $k$ and Maker will play the role of “CliqueBreaker” in this game.
We have $$\lim_{n\rightarrow\infty} \min_{A\in\cf} \frac{|A|}{b'} =
\lim_{n\rightarrow\infty}\frac{2\log n {\frac{\sqrt{n}}{2} \choose
2}}{\log 2\, \sqrt{n}} = \infty,$$ and the condition (\[c:ws\_large\]) is satisfied. It remains to prove that the condition (\[c:es\]) holds. $$\begin{aligned}
\sum_{A\in\cf} 2^{-\frac{|A|}{b'}} &=&{n \choose k}
2^{-\frac{{k\choose 2}}{b'}}<
\left( \frac{ne}{k}2^{- \frac{k-1}{2b'}}\right)^k \\
&<& 2^{-\sqrt{n}} \rightarrow 0. \end{aligned}$$ Therefore, CliqueBreaker wins the game $(\widehat{E(K_n)}_p, \cf^\cap_p,
\frac{\log 2}{4}p\frac{\sqrt{n}}{\log n}, 1)$ a.s., provided $p\geq \frac{32\log n}{\sqrt{n}}$.
Putting the two parts of the game together we have that Maker wins $(E(K_n)_p, \ch_p, 1, \frac{1}{16} p\frac{\sqrt{n}}{\log
n})$ a.s.
Combining the statements of Corollary \[t:hg-b\] and Theorem \[t:hg-m\] we obtain part $(iii)$ of both Theorems \[main1\] and \[main2\].
Perfect matching game {#ss:pm}
---------------------
The upper and lower bounds obtained in the previous subsection for the threshold bias of the random Hamiltonian cycle game are not tight. We firmly believe that our strategy for Maker in that game is not optimal. The game we consider next is simpler for Maker, and for that we are able to obtain bounds optimal up to a constant factor.
Recall that $\cm$ is the set of all perfect matchings on $K_n$. We will assume that $n$ is even. In the game $(E(K_n), \cm, 1,b)$ Maker’s goal is to occupy all edges of a perfect matching, while Breaker wants to prevent that.
The following theorem provides the winning strategy in the random perfect matching game for Maker.
\[t:mg-m\] There exists $m_0>0$, so that for $p>64 \frac{\log n }{n}$ and $b\leq m_0 p\frac{n}{\log n}$ Maker wins the random perfect matching game $(E(K_n)_p, \cm_p, 1, b)$ a.s.
We can show that Maker can win in a slightly harder game. More precisely, if the set of vertices of $K_n$ is partitioned into two sets $A$ and $B$ of equal size before the game starts, we are going to show that Maker can claim a perfect matching with edges going only between $A$ and $B$.
For disjoint sets $X,Y\subset V(K_n)$, we define $E(X,Y)$ to be the set of edges between $X$ and $Y$. Let $\cf$ be a family of sets of edges, $$\cf = \{ E(X,Y) {:\,}\emptyset \not= X \subset A,\,
\emptyset \not= Y \subset B,\, |X|+|Y|=\frac{n}{2} +1 \}.$$
Suppose that at the end of the game Maker has not claimed all edges of any perfect matching between $A$ and $B$. Hall’s necessary and sufficient condition for existence of a perfect matching implies that there exist sets $X_0 \subset A$ and $Y_0 \subset B$ such that $|X_0|>|Y_0|$ and all edges in $E(K_n)_p \cap
E(X_0, B\setminus Y_0)$ were claimed by Breaker.
Therefore, in order to win, Maker has to claim at least one edge in each of the sets from $\cf$, i.e. the game $(\widehat{E(K_n)}_p,
\cf^\cap_p, b, 1)$, which we call [*Hall*]{}, should be a HallBreaker’s win.
To prove that HallBreaker wins we are going to use Theorem \[t:es\_cap\]. We set $\delta = 1/2$ and $b_0=\frac{\log 2}{4}\cdot \frac{n}{\log n} $ .
First we show that condition (\[c:es\]) holds. We have $$\begin{aligned}
\sum_{k=1}^{n/2} {n/2 \choose k} {n/2 \choose n/2 -k+1}
2^{-\frac{k(n/2-k+1)}{b_0}} &<& 2\sum_{k=1}^{\lfloor n/4 \rfloor}
{n/2 \choose k}^2 2^{-\frac{k(n/2-k+1)}{b_0}} \\
&<& 2\sum_{k=1}^{\lfloor n/4 \rfloor}
\left( e^{2\log (n/2) - 2\log n }\right)^k \\
&=& 2\sum_{k=1}^{\lfloor n/4 \rfloor} \left( \frac{1}{4} \right)^k< 1.\end{aligned}$$
Since $$\lim_{n\rightarrow\infty} \min_{A\in\cf} \frac{|A|}{b_0}>
\lim_{n\rightarrow\infty} \log n = \infty,$$ the condition (\[c:ws\_large\]) is also satisfied and we can apply Theorem \[t:es\_cap\] proving that HallBreaker wins the random game $(\widehat{E(K_n)}_p, \cf^\cap_p, \frac{\log 2}{8}
p\frac{n}{\log
n}, 1)$ a.s., provided $p>64\log n/n$.
This immediately implies that Maker wins $(E(K_n)_p, \cm_p, 1, b)$ a.s.
Theorem \[t:cg-b\] ensures a win for Breaker in the perfect matching game, if $b> K_0pn/\log n$. This, together with the above Theorem \[t:mg-m\] proves part $(ii)$ of Theorems \[main1\] and \[main2\].
Clique game {#ss:clique}
-----------
Here we look at the random version of the $(1\:b)$ biased clique game $(E(K_n), \ckk_k, 1,b)$ on a complete graph $K_n$, where $\ckk_k$ is the set of all cliques of constant size $k$. Maker’s goal is to occupy all edges of a clique of size $k$ while Breaker wants to prevent that.
The deterministic clique game was extensively studied by Bednarska and Łuczak in [@bl1]. They proved a more general result by determining the order of the threshold bias for the whole family of games in which Maker’s goal is to claim an arbitrary fixed graph $H$. In this section, we will largely rely on the constructions and ideas from their paper.
If $\{F_1,\dots ,F_t\}$ is a family of $k$-cliques having two common vertices, and $e_i\in E(F_i)$, $i=1,\dots ,t$ are distinct edges, then we call the graph $\cup_{i=1}^t F_i$ a [*$t$-2-cluster*]{} and the graph $\cup_{i=1}^t (F_i - e_i)$ a [*$t$-fan*]{}. If furthermore the $k$-cliques have three vertices in common, then a $t$-2-cluster is called a [*$t$-3-cluster*]{} and a $t$-fan is called a [*$t$-flower*]{}. A $t$-fan or a $t$-2-cluster is said to be [*simple*]{}, if the pairwise intersections (of any two $k$-cliques) have size exactly $2$.
In order to prevent Maker to occupy a clique $K_k$, Breaker will play two auxiliary games. In the first one he prevents Maker from occupying a 3-cluster of constant size.
\[l:3-clusters\] There exists $t=t(k)$, so that for $\eps=\frac{1}{2(k+2)}$, $p=\omega (n^{-\frac{2}{k+1}})$ and $b>pn^\frac{2(1-\eps)}{k+1}$ Breaker wins the game $(E(K_n)_p,
\mbox{$t$-3-clusters}, 1, b)$ a.s.
To apply Theorem \[t:es\], it is enough to check that there exists $t$ such that for the random variable $$Y:=\sum_{\mbox{\scriptsize $t$-3-cluster $C$ in $G(n,p)$}} (1+b)^{-
e(C)},$$ $Y<\frac{1}{b+1}$ holds a.s.
We have $${{\hbox{\bf E}}\hspace{-0.3ex}\left[Y\right]}
= \sum_{\mbox{\scriptsize $t$-3-cluster $C$ in $K_n$}}
\left(\frac{p}{1+b}\right)^{e(C)}.$$ Let $b_1=\frac{b+1}{p}-1$. In [@bl1], it is shown that there exists $t$ for which $$\sum_{\mbox{\scriptsize $t$-3-cluster $C$ in $K_n$}}
\left(\frac{p}{1+b}\right)^{e(C)} \leq K_0\frac{1}{b^{1+k_0}_1},$$ where $k_0,K_0>0$ are constants depending on $k$. This implies ${{\hbox{\bf E}}\hspace{-0.3ex}\left[Y\right]}=o\left(
\frac{1}{b+1}\right)$, and by Markov inequality we get that $Y<\frac{1}{b+1}$ a.s.
During a game, a $t$-fan (or $t$-flower) is said to be [*dangerous*]{} if all the $t$ edges missing from the cliques that make up the $t$-fan are present in the graph on which the game is played, but not yet claimed by any of the players. Note that if at any moment of the game $(E(K_n)_p,
\mbox{$t$-3-clusters}, 1, b)$ Maker claimed a dangerous $(b+1)t$-flower, then he could win since he could claim a $t$-3-cluster in his next $t$ moves by simply claiming missing edges, one by one. Hence, Lemma \[l:3-clusters\] implies the following.
\[c:flowers\] There exists $t=t(k)$ so that for $\eps=\frac{1}{2(k+2)}$ and $p=\omega(n^{-\frac{2}{k+1}})$, Breaker playing a $(1\: pn^\frac{2(1-\eps)}{k+1})$ game on edges of random graph $E(K_n)_p$ can make sure that Maker does not claim a dangerous $\left( pn^\frac{2(1-\eps)}{k+1}t\right)$-flower at any moment of the game.
Next we deal with the second auxiliary game of Breaker; in this game he prevents the appearance of too many simple $b^\varepsilon$-fans.
\[l:simple\_fans\] There exists $C_0>0$, such that for $\eps_1=\frac{1}{6(k+2)}$, $p\geq n^{-\frac{2}{k+1}}\log^{1/\eps_1} n$, $b>C_0 p
n^\frac{2}{k+1}$ and $s=b^{\eps_1}$ Breaker wins the game $(E(K_n)_p, \mbox{unions of }
\frac{1}{2}{b \choose s}$ $\mbox{simple
$s$-fans},$ $1, b/2)$ a.s.
Let $c_s(n)$ be the number of simple $s$-2-clusters contained in $K_n$, and let $X_s$ be the random variable counting the number of simple $s$-2-clusters contained in $G(n,p)$. Using the first moment method we get $${{\rm Pr}\hspace{-0.2ex}\left[ X_s\geq {{\hbox{\bf E}}\hspace{-0.3ex}\left[X_s\right]}\log n\right]} \leq \frac{1}{\log n} \longrightarrow 0,$$ and using this, a.s. we have that $$\begin{aligned}
& &\sum_{\mbox{\scriptsize dangerous simple} \atop
\mbox{\scriptsize $s$-fan $C$ in $G(n,p)$}}
(1+b/2)^{-e(C)} \\
&\leq &
\sum_{\mbox{\scriptsize simple $s$-2-cluster $K$}\atop
\mbox{\scriptsize in $G(n,p)$}} {k\choose 2}^s
(1+b/2)^{-s({k \choose 2}-2)-1} \\
&\leq & {k \choose 2}^s \log n \cdot c_s(n) p^{s({k \choose 2}-1)+1}
2^{sk^2} b^{-s({k \choose 2}-2)-1} \\
&\leq& \log n\cdot C_1^s {n \choose 2}\frac{{n \choose k-2}^s}{s!}
\left(\frac{p}{b}\right)^{s({k \choose 2}-1)+1} b^s\\
&\leq& n^3 \cdot C_1^s n^{(k-2)s}
\left(\frac{1}{C_0n^{\frac{2}{k+1}}}\right)^{s(k+1)(k-2)/2+1}
\frac{b^s}{s!}\\
&\leq& n^3 \cdot \left(\frac{C_1}{C_0^{{k\choose 2}-1}}\right)^s
\left(\frac{1}{C_0n^{\frac{2}{k+1}}}\right)
\frac{b^s}{s!} < \frac{1}{2}{b\choose s}\frac{1}{b+1},\end{aligned}$$ where $C_1=C_1(k)$ is a constant. The last inequality is valid since $p\geq n^{-\frac{2}{k+1}}\log^{1/\eps_1} n$, and for $C_0$ large enough $\left( C_1/C_0^{{k\choose 2}-1} \right)^s\leq n^{-5}$. This enables us to apply Theorem \[t:gen-es\], and the statement of the lemma is proved.
Now we are ready to state and prove the theorem ensuring Breaker’s win in the clique game on the random graph. In the proof, we are going to use this result of Bednarska and Łuczak.
[@bl1] \[l:ind\_sets\] For every $0<\eps <1$ there exists $b_0$ so that every graph with $b>b_0$ vertices and at most $b^{2-\eps}$ edges has at least $\frac{1}{2} {b \choose b^{\eps /3}}$ independent sets of size $b^{\eps /3}$.
\[t:B\_clique\] There exists $C_0>0$ so that for $p\geq
n^{-\frac{2}{k+1}} \log^{6k+12} n$ and $b\geq C_0 p n^\frac{2}{k+1}$ Breaker wins the random clique game $(E(K_n)_p, (\ckk_k)_p, 1,b)$ a.s.
Breaker will use $b/2$ of his moves to defend “immediate threats”, i.e. to claim the remaining edge in all $k$-cliques in which Maker occupied all but one edge. In order to be able to do this Breaker must ensure that he never has to block more than $b/2$ immediate threats, that is, there is no dangerous $b/2$-fan.
He will use his other $b/2$ moves to prevent Maker from creating a dangerous $(b/2)$-fan.
From Corollary \[c:flowers\] we get that Breaker can prevent Maker from claiming a dangerous $f$-flower (where $f=tpn^\frac{2(1-\eps)}{k+1}$, $\eps=\frac{1}{2(k+2)}$ and $t$ is a positive constant) using less than $b/4$ edges per move. On the other hand, from Lemma \[l:simple\_fans\] we have that if $C_0$ is large enough Breaker can prevent Maker from claiming $\frac{1}{2}{b/2 \choose s}$ simple $s$-fans using $b/4$ edges per move, where $s=(b/2)^{\eps/3}$.
Suppose that Maker managed to claim a dangerous $(b/2)$-fan. We define an auxiliary graph $G'$ with the vertex set being the set of all $b/2$ $k$-cliques of this dangerous fan, and two $k$-cliques being connected with an edge if they have at least $3$ vertices in common. Since there is no dangerous $f$-flower in Maker’s graph, the degree of each of the vertices of the graph $G'$ is at most $fk$ and therefore $e(G')<\frac{bfk}{2}\leq
\left(\frac{b}{2}\right)^{2-\eps}$. On the other hand, the number of independent sets in $G'$ of size $s$ cannot be more than $\frac{1}{2}{b/2 \choose
s}$, since each of the independent sets in $G'$ corresponds to a simple $s$-fan in Maker’s graph.
Since the last two facts are obviously in contradiction with Lemma \[l:ind\_sets\], Maker cannot claim a dangerous $b/2$-fan and the statement of the theorem is proved.
To prove the theorem for Maker’s win, we need the following lemma which is a slight modification of a result from [@bl1]. Let $G(n,M)$ denote the graph obtained by choosing a graph on $n$ vertices with $M$ edges uniformly at random.
\[l:bl1\] There exists $0<\delta_k<1$, such that for $M=2
\lfloor n^{2-2/(k+1)} \rfloor$ a.s. each subgraph of $G(n,M)$ with $\lfloor (1-\delta_k) M\rfloor$ edges contains a copy of $K_k$.
For $0<\delta_k<1$, we call a subgraph $F$ of $K_n$ bad, if $F$ has $M$ edges and it contains a subgraph $F'$ with $\lfloor (1-\delta_k) M\rfloor$ edges that does not contain a copy of $K_k$. In [@bl1], it is proved that there exist constants $0<\delta_k<1$ and $c'_1>0$ such that the number of bad subgraphs of $K_n$ is bounded from above by $$e^{-c'_1 M/6} {{n \choose 2}\choose M } = o(1) {{n \choose
2}\choose M }.$$
Using the last lemma we can prove a theorem for Maker’s win in the random clique game.
\[t:M\_clique\] There exists $c_0>0$ so that for $p>\frac{1}{c_0}
n^{-\frac{2}{k+1}}$ and $b\leq c_0 p n^\frac{2}{k+1}$ Maker wins the random clique game $(E(K_n)_p, (\ckk_k)_p, 1,b)$ a.s.
We will follow the analysis of the random Maker’s strategy proposed in [@bl1], looking at $G(n,M')$, where $M'=p{n\choose 2}$. We will prove that the $k$-clique game on $G(n,M')$ is a Maker’s win a.s., which implies that the same is true on $G(n,p)$, as being a Maker’s win is a monotone property [@bol Chapter 2].
In each of his moves Maker chooses one of the edges of $G(n,M')$ that was not previously claimed by him, uniformly at random. If the edge is free he claims it and we call that a successful Maker’s move. If the edge was already claimed by Breaker, then Maker skips his move (e.g. claims an arbitrary free edge, and that edge we will not encounter for the future analysis).
Let $0<\delta_k<1$ be chosen so that the conditions of Lemma \[l:bl1\] are satisfied. We look at the course of game after $M=2\lfloor n^{2-2/(k+1)} \rfloor$ moves.
By choosing $c_0\leq \delta_k/12$, we have $$\begin{aligned}
M &\leq& \frac{\delta_k}{6 c_0} \lfloor n^{2-2/(k+1)} \rfloor \\
&\leq& \frac{\delta_k}{2} \frac{1}{b+1} p{n \choose 2}.\end{aligned}$$ That means that only at most $\delta_k /2$ fraction of the total number of elements of the board $E(G(n,M'))$ is claimed (by both players) after move $M$. Therefore, the probability that the edge randomly chosen in Maker’s $m$th move, $m\leq M$, is already claimed by Breaker is bounded from above by $\delta_k/2$. That means that Maker has at least $(1-\delta_k)M$ successful moves a.s.
Since in each of his moves Maker has chosen edges uniformly at random (without repetition) from $E(G(n,M'))$, the graph containing edges chosen by Maker in his first $M$ moves (both successful and unsuccessful) actually is a random graph $G(n,M)$. Applying Lemma \[l:bl1\], we get that the graph containing edges claimed by Maker in his successful moves contains a clique of size $k$ a.s., which means that a.s. there exists a non-randomized winning strategy for Maker.
Combining the statements of Theorem \[t:B\_clique\] and Theorem \[t:M\_clique\] we obtain part $(iv)$ of Theorem \[main2\].
Unbiased games {#s:unbiased}
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Connectivity one-on-one {#ss:con1on1}
-----------------------
A theorem of Lehman enables us to determine the threshold probability $p_{\ct}$ with extraordinary precision. Namely, Lehman [@L] proved that the unbiased connectivity game is won by Maker (now as a second player!) if and only if the underlying graph contains two edge-disjoint spanning trees. The threshold for the appearance of two edge-disjoint spanning trees was determined exactly by Palmer and Spencer [@PS].
To formulate the consequence of these two results we need the concept of [*graph process*]{}. Let $e_1,\ldots e_m$ be the edges of $K_n$, where $m={n\choose 2}$. Choose a permutation $\pi\in S_m$ uniformly at random and define an increasing sequence of subgraphs $(G_i)$ where $V(G_i)=V(K_n)$ and $E(G_i)=\{e_{\pi(1)},\ldots , e_{\pi (i)}\}$. It is clear that $G_i$ is an $n$-vertex graph with $i$ edges, selected uniformly at random from all $n$-vertex graphs with $i$ edges.
Given a particular graph process $(G_i)$ and a graph property $\cp$ possessed by $K_n$, the [*hitting time*]{} $\tau (\cp)=\tau (\cp, (G_i))$ is the minimal $i$ for which $G_i$ has property $\cp$.
The consequence of the theorems of Lehman, and Palmer and Spencer is that the very moment the last vertex receives its second adjacent edge, the unbiased connectivity game is won by Maker a.s. More precisely, the following is true.
\[connectivity-precise\] For the unbiased connectivity game we have that a.s. $$\tau(\mbox{Maker wins $\ct$})=\tau(\mbox{$\exists$ two edge-disjoint spanning trees})=
\tau (\delta(G)\geq 2).$$
In particular, for edge-probability $p=(\log n + \log\log n +g(n))/n$, where $g(n)$ tends to infinity arbitrarily slowly, Maker wins the unbiased connectivity game a.s., while if $g (n)\rightarrow -\infty$, then Breaker wins a.s.
[**Remark.**]{}
The assumption that Maker is the second player is just technical, for the sake of smooth applicability of Lehman’s Theorem. If Maker is the first player, then from the proof of Lehman’s Theorem one can infer that Maker wins if and only if the base graph contains a spanning tree and a spanning forest of two components, which are edge-disjoint. This property has the same sharp threshold as the presence of two edge-disjoint spanning trees, and the hitting time should be the same when the next to last vertex receives its second incident edge.
$k$-cliques one-on-one {#ss:clique1on1}
----------------------
Let us fix $k$ and let $(F_1,\ldots , F_s)$ be a sequence of $k$-cliques. Then $F=\cup_{i=1}^sF_i$ is called an [*$s$-bunch*]{} if $V(F_i)\setminus (\cup_{j=1}^{i-1} V(F_j))\not= \emptyset$ and $|V(F_i)\cap (\cup_{j<i}V(F_j))|\geq 2$, for each $i=2,\ldots , s$. Recall that an $s$-bunch in which the pairwise intersection of any two cliques is the same two vertices, was called a [*simple $s$-$2$-cluster*]{}. Let us denote the simple $s$-$2$-cluster by $C_s$.
For a graph $G$, the [*density*]{} of $G$ is defined as $d(G)=\frac{e(G)}{v(G)}$, and the [*maximum density*]{} of $G$ is defined as $m(G)=\max_{H\subseteq G} d(H)$. A graph $G$ with $m(G)=d(G)$ is called [*balanced*]{}. The maximum density of a graph $G$ determines the threshold probability for the appearance of $G$ in the random graph. More precisely, $(i)$ if $p=o(n^{-1/m(G)})$, then $G(n,p)$ does not contain $G$ a.s., and $(ii)$ if $p=\omega(n^{-1/m(G)})$, then $G(n,p)$ does contain $G$ a.s.
We need two properties of simple $s$-$2$-clusters and $s$-bunches.
\[l:mcs\] For every positive integer $s$, $C_s$ is balanced and has maximum density $m(C_s)=d(C_s)=\frac{k+1}{2}-\frac{k}{sk-2s+2}$.
It is easy to check that $v(C_s)=s(k-2)+2$, $e(C_s)=s{k\choose 2} -s+1$, and thus $d(C_s)=\frac{e(C_s)}{v(C_s)}=\frac{k+1}{2}-\frac{k}{sk-2s+2}$.
Let $T$ be a subgraph of $C_s$. We want to prove $d(T)\leq d(C_s)$. Since $C_s$ is the union of $k$-cliques, $C_s=\cup_{i=1}^s F_i$, if we set $E_i=F_i\cap T$ we have that $T=\cup_{i=1}^s E_i$, and we can assume that each $E_i$ is a clique of order $k_i\leq k$.
We can also assume that the two vertices in $\cap_{i=1}^s V(F_i)$ are in $T$, since otherwise their inclusion would increase the density. This implies $k_i\geq 2$ for $i=1,\ldots ,s$.
Let us relabel the cliques in such a way that $E_i\neq F_i$ if and only if $i=1,\ldots , s_1$. Then $$\frac{e(C_s)}{v(C_s)} \geq \frac{e(T)}{v(T)}=
\frac{e(C_s)-\sum_{i=1}^{s_1} \left( {k\choose 2}
-{k_i\choose 2}\right)}{v(C_s)-\sum_{i=1}^{s_1} (k-k_i)},$$ since $$\frac{e(C_s)}{v(C_s)}<\frac{k+1}{2}\leq
\frac{\sum_{i=1}^{s_1} \left( k-k_i\right)
\frac{k+k_i-1}{2}}{\sum_{i=1}^{s_1} (k-k_i)}.$$ The last inequality is true since the last fraction is the weighted average of the numbers $(k+k_i-1)/2$, each of them being at least $(k+1)/2$.
\[megegy\] Let $s\geq 3$ be a positive integer. No $s$-bunch has smaller maximum density than the simple $s$-$2$-cluster.
When $k=3$, the $s$ bunch is a union of triangles. Then any $s$-bunch has the same number of vertices as the simple $s$-$2$-cluster, while the number of edges, and thus the density is minimized for the simple $s$-$2$-cluster.
From now on let us assume that $k\geq 4$. Let $s\geq 3$, and let $(F_1, F_2,\dots , F_s)$ be the sequence of $k$-cliques of an arbitrary $s$-bunch $B_s=\cup_{i=1}^s F_i$. For every $i\in\{2,3,\dots ,s\}$, let $F'_i= \left( \cup_{j=1}^{i-1}
F_j \right) \cap F_i$. Then, we have $$\begin{aligned}
d(B_{s}) &=& \frac{s{k \choose 2} - \sum_{i=2}^s e(F'_i)}{
sk-\sum_{i=2}^s v(F'_i)} \\
&=& \frac{e(C_s) - \sum_{i=2}^s (e(F'_i) -1)}{
v(C_s)-\sum_{i=2}^s (v(F'_i) -2)} \\
&\geq & \frac{e(C_s) - \sum_{i=2}^s ({v(F'_i) \choose 2} -1)}{
v(C_s)-\sum_{i=2}^s (v(F'_i) -2)} \\
&\geq & \frac{e(C_s)}{v(C_s)}.\end{aligned}$$
In the last inequality the terms with $v(F_i')=2$ disappear, and otherwise we use that $v(F'_i)\leq k-1$ for every $i$, so $\frac{{v(F'_i) \choose 2} -1}{v(F'_i) -2} \leq \frac{k}{2} \leq
\frac{e(C_s)}{v(C_s)}$.
Hence, simple $s$-$2$-clusters have the smallest density among all $s$-bunches. For any $s$-bunch $B_s$ and the simple $s$-$2$-cluster $C_s$ we immediately obtain $$m(B_s)\geq d(B_s)\geq d(C_s) = m(C_s),$$ and the lemma is proved.
[**Remark.**]{}
The previous lemma is of course true for $s=1$, but not for $s=2$.
As a consequence of the last two lemmas we get a strategy for Breaker in the $(1:1)$ clique game.
Let $H$ be a graph and consider the auxiliary graph $G_H$ with vertices corresponding to the $k$-cliques of $H$, two vertices being adjacent if the corresponding cliques have at least two vertices in common. Let $F_1,\ldots, F_s$ be the cliques corresponding to a connected component of $G_H$. Then the graph $\cup_{i=1}^sF_i$ is called an [*$s$-collection*]{} or just a [*collection*]{} of $H$. Note that the edgeset of any $H$ is uniquely partitioned into sets $N$ and $E(A_i)$, where $N$ contains the edges which do not participate in a $k$-clique, while the $A_i$ are the collections of $H$.
\[p:kk\_1on1\] For every $k\geq 4$ and $\varepsilon >0$, $p_{\ckk_k} \geq n^{-\frac{2}{k+1}-\varepsilon}$. For $k=3$, we have that $p_{\ckk_3} \geq n^{-\frac{5}{9}}$.
First we give a strategy for Breaker to win $\ckk_k$ if the game is played on the edgeset of a $(2k-4)$-degenerate graph $L$. Consider the ordering $v_1,\ldots , v_{v(L)}$ of $V(L)$, such that $|N_{V_j}(v_{j+1})|\leq 2k-4$ for $j=1,\ldots , v(L)-1$, where $V_j=\{ v_1,\ldots , v_j\}$. Then Breaker’s strategy is the following: if Maker takes an edge connecting $v_{j+1}$ to $V_j$, then Breaker takes another one also connecting $v_{j+1}$ to $V_j$. If there is no such edge available, then Breaker takes an arbitrary edge. Suppose for a contradiction that Maker managed to occupy a $k$-clique $v_{i_1}, \ldots, v_{i_k}$ against this strategy, where $i_1 < \cdots < i_k$. This is impossible, since Maker could have never claimed $k-1$ of the edges $v_{j}v_{i_k}$, $j<i_k$.
Let $E(K_n)_p=N\cupdis E(A_1)\cupdis\dots \cupdis E(A_{h})$ be the partition of the edges, such that $N$ contains all edges that do not participate in any $k$-clique, and each $A_i$ is a collection of $k$-cliques. (Corresponding to the connected components of the auxiliary graph $G_{G(n,p)}$ defined on the set of $k$-cliques of $G(n,p)$.)
Breaker can play the game $(E(K_n)_p, (\ckk_k)_p,1,1)$ by playing separately on each of the sets $E(A_i)$. More precisely, whenever Maker claims an edge which is in some $E(A_i)$, Breaker can play according to a strategy restricted just to $E(A_i)$. Since, crucially, the edgeset of each $k$-clique is completely contained in exactly one of the $E(A_i)$, Maker can only win the game on $E(K_n)_p$ if he wins on one of the $E(A_i)$.
Now we are going to show that every collection $A$ on $v(A)=v$ vertices contains a $\lceil \frac{v-2}{k-2} \rceil$-bunch. We take an arbitrary $k$-clique $F_1$ from $A$, and build a bunch recursively as follows. If we picked $k$-cliques $F_1, \dots, F_i$ then we choose $F_{i+1}$ such that $|V(F_{i+1})\cap (\cup_{j=1}^i V(F_j) )| \geq 2$ and $V(F_{i+1})\setminus (\cup_{j=1}^i V(F_j) ) \not= \emptyset$. Note that this means that $\cup_{j=1}^{i+1} F_j$ is an $(i+1)$-bunch. Since the auxiliary graph $G_{A}$ of the collection is connected we can keep doing this until $V(A)=\cup_{j=1}^{i_0}
V(F_j)$ for some $i_0$. Knowing that $v(F_i)=k$ for all $i\leq i_0$, we have $i_0\geq 1+ \frac{v-k}{k-2} = \frac{v-2}{k-2}$. So there exists an $\lceil\frac{v-2}{k-2}\rceil$-bunch which is a subgraph of $A$.
We first look at the case $k\geq 4$. Let $\varepsilon>0$ be a constant. From Lemma \[l:mcs\] it follows that there exists an integer $v$ such that for $s_0=\lceil\frac{v-2}{k-2}\rceil$ we have $m(C_{s_0})\geq \frac{k+1}{2}-\frac{k}{v}
> \left( \frac{2}{k+1} + \varepsilon \right)^{-1}$. Then for $p = O( n^{-\frac{2}{k+1}-\varepsilon})$ it follows that there is no $s_0$-bunch in $G(n,p)$ a.s., since we have have that the first $s_0$-bunch that appears in the random graph is the one of the minimum maximum density, which, by Lemma \[megegy\], is the simple $s_0$-$2$-cluster. Note here that there is a constant (depending on $k$ and $\varepsilon$) number of nonisomorphic $s_0$-bunches.
Since in $G(n,p)$ there are no $s_0$-bunches a.s., there are also no collections on $v$ vertices a.s.
Finally, all the collections $A_i$ are $(2k-4)$-degenerate a.s., since graphs which are not $(2k-4)$-degenerate have maximum density at least $\frac{2k-3}{2}\geq \frac{k+1}{2}$, provided $k\geq 4$. Note that we know already that a.s. all collections have order at most $v$ and thus there are at most a constant (depending on $k$ and $\epsilon$) number of nonisomorphic non-$(2k-4)$-degenerate graphs.
This proves that Breaker has a winning strategy a.s., if $k\geq 4$ and $p = O( n^{-\frac{2}{k+1}-\varepsilon})$.
Next, we look at the case $k=3$. As we saw, any collection of triangles on $v$ vertices contains a $(v-2)$-bunch. Thus for $p=o(n^{-5/9})$, no $v$-collection with $v\geq 15$ will appear in $G(n,p)$ a.s., since it would contain a $13$-bunch, whose maximum density is at least $m(C_{13})= 2 -\frac{3}{15}$. This observation makes the problem finite: one has to check who wins on collections up to $14$ vertices.
Suppose that Maker can win the triangle game on some collection of triangles on $v\leq 14$ vertices and with maximum density less then $9/5$. Let $A$ be a minimal such collection (Maker cannot win on any proper subcollection of $A$).
If there was a vertex $w\in V(A)$ with $d_A(w)\leq 2$, the minimality of $A$ would imply that Breaker has a winning strategy on $A$. Indeed, Breaker plays according to his strategy on $A-w$, and as soon as Maker claims one edge adjacent to $w$ Breaker claims the other edge adjacent to $w$ (if that edge exists otherwise he does not move). This would mean that Breaker can win on $A$, a contradiction. Thus, $\delta_A \geq 3$.
Let $B$ be a $(v-2)$-bunch contained in $A$, with $V(A)=V(B)$. Since $\delta_B = 2$, we have $e(A)\not=
e(B)$. Then $$2-\frac{3}{v}=m(C_{v-2})\leq \frac{e(B)}{v} < \frac{e(A)}{v}
<\frac{9}{5},$$ and $$2v-3 = e(C_{v-2}) \leq e(B) < e(A) < \frac{9v}{5}.$$ It is easy to check that Maker cannot win the game on a graph with less then $5$ vertices, thus $v>4$, so $e(B)=e(C_{v-2})$ and $e(A)-e(B) = 1$.
Let $\{e\} = E(A)\setminus E(B)$, and let $T_1,\dots, T_{v-2}$ be the sequence of triangles whose union is the $(v-2)$-bunch $B$. Since $e(B)=e(C_{v-2})$, for every $i=2, \dots, v-2$ we have that $T_i$ has a common edge with $\cup_{j=1}^{i-1} T_j$. Then $B$ must have at least $2$ vertices of degree $2$. From $\delta_{B\cup\{e\}} = \delta_{A} =3$ we obtain that $B$ has exactly two vertices $b_1,b_2$ with $d_B(b_1)=d_B(b_2)=2$, and moreover $e=\{b_1, b_2\}$. Since $e$ has to participate in at least one triangle of the collection $A$, $b_1$ and $b_2$ have to be connected with a $2$-path in $B$, which is possible only if all $T_1,\dots,T_{v-2}$ share a vertex. That means that $A$ is a $(v-1)$-wheel and it is easy to see that Breaker can win the triangle game on a wheel of arbitrary size by a simple pairing strategy.
This contradiction proves that for $p=o(n^{-5/9})$, a.s. there is no triangle collection in $G(n, p)$ on which Maker can win, which means that Breaker a.s. wins the game on the whole graph.
From Theorem \[t:M\_clique\] we get that Maker can win the game $(E(K_n)_p, (\ckk_k)_p,1,1)$ for $p=\Theta
(n^{-\frac{2}{k+1}})$ and thus we immediately obtain $p_{\ckk_k}=O
(n^{-\frac{2}{k+1}})$. For the triangle game $\ckk_3$ a stronger upper bound can be found.
\[p:triangle\] The game $(E(K_n)_p, (\ckk_3)_p,1,1)$ is a Maker’s win a.s., provided $p=\omega(n^{-\frac{5}{9}}$).
It is easy to check that Maker can claim a triangle in the $(1\: 1)$ game if the board on which the game is played is $K_5$ minus an edge. Therefore, as soon as the graph $G(n,p)$ contains $K_5-e$ a.s., the initial game can be won by Maker a.s.
Theorem \[t:M\_clique\], Theorem \[p:kk\_1on1\] and Proposition \[p:triangle\] imply parts $(iv)$ and $(v)$ of Theorem \[main1\].
Open questions {#s:open}
==============
[**More sharp thresholds?**]{}
We saw in the previous section that the connectivity game has a sharp threshold, and even more. We think that both the perfect matching game and the Hamiltonian cycle game have the same sharp threshold $\frac{\log n}{n}$, and maybe even more… It would be very interesting to decide whether the following conjectures are true.
$$\begin{aligned}
& (i) & \tau(\mbox{Maker wins $\cm$}) = \tau(\delta(G)\geq 2),\\
& (ii) & \tau(\mbox{Maker wins $\ch$}) = \tau(\delta(G)\geq 4).\end{aligned}$$
[**Clique game/$H$-game.**]{}
The exact determination of the threshold $p_{\ckk_k}$ for the $k$-clique game remains outstanding.
Decide whether $p_{\ckk_k}=n^{-\frac{2}{k+1}}$ for $k\geq 4$.
The arguments of Bednarska and Łuczak [@bl1] could be extended to full generality to positional games on random graphs along the lines of Section \[ss:clique\]. More precisely, the following is true. Let $\ckk_H$ be the family of subgraphs of $K_n$, isomorphic to $H$. Then for any fixed graph $H$ there is a constant $c(H)$, such that $$b_{\ckk_H}^p=\Theta\left (pb_{\ckk_H}\right) = \Theta \left(p n^{-1/m'(H)}
\right),$$ provided $p\geq \Omega\left( \frac{\log ^{c(H)} n}{n^{1/m'(H)}}
\right).$
Concerning the one-on-one game, it would be desirable to determine those graphs for which an extension of the low-density Maker’s win, à la Proposition \[p:triangle\], exists.
Characterize those graphs $H$ for which there exists a constant $\epsilon(H) >0$, such that the unbiased game $\ckk_H$ is a.s. a Maker’s win if $p=n^{-1/m'(H)-\epsilon (H)}$.
For such graphs the determination of the threshold $p_{\ckk_H}$ is a finite problem, in a way similar to the case $H=K_3$.
[**Relationships between thresholds.**]{}
It is an intriguing task to understand under what circumstances the following is true.
\[p:gen1\] Characterize those games $(X, \cf)$ for which $$p_\cf=\frac{1}{b_\cf}.$$ More generally, characterize the games for which $$b_\cf^p = \Theta \left(pb_\cf\right),$$ for every $p = \omega \left( \frac{1}{b_\cf}\right)$.
This is not true in general as the triangle game shows. What is the reason it is true for the connectivity game and the perfect matching game? Is it because the appearance of these properties has a sharp threshold in $G(n,p)$? Or because the winning sets are not of constant size?
\[p:gen2\] Suppose $p_\cf=1/b_\cf$. Is it true that for every $p\geq p_\cf$, $b_{\cf}^p=\Theta \left( pb_\cf\right)$?
It would be very interesting to relate the thresholds $b_\cf$ and $p_\cf$ to some thresholds of the family $\cf$ in the random graph $G(n,p)$ (or, more generally, in the random set $X_p$). It seems to us that if the family $\cf_p$ is quite dense and well-distributed in $X$, then Maker still wins the $(1\:1)$ game.
Characterize those games $(X,\cf )$ for which there exists a constant $K$, such that for any probability $p$ with ${{\rm Pr}\hspace{-0.2ex}\left[ \min_{x\in X_p}|\{ F\in \cf_p : x\in
F\}|>K\right]}\longrightarrow 1$, we have $p_\cf =O(p)$ and/or $b_\cf =\Omega (1/p)$.
#### Acknowledgments.
We would like to thank the anonymous referees for their thorough work. Their numerous suggestions and simplifications greatly improved the presentation of the paper. We are also indebted to Małgorzata Bednarska and Oleg Pikhurko whose remarks about Theorem \[p:kk\_1on1\] not only shortened the proof, but also improved the statement.
[99]{}
J. Beck: Random graphs and positional games on the complete graph, [*Ann. Discrete Math.*]{} 28(1985), 7–13.
J. Beck: Remarks on positional games, [*Acta Math. Acad. Sci. Hungar.*]{} 40(1982), 65–71.
J. Beck: Positional games and the second moment method, [*Combinatorica*]{} 22(2002), 169–216.
M. Bednarska, T. Łuczak: Biased positional games for which random strategies are nearly optimal, [*Combinatorica*]{} 20(2000), 477–488.
M. Bednarska, T. Łuczak: Biased positional games and the phase transition, [*Random Structures & Algorithms*]{} 18(2001), 141–152.
B. Bollob' as, A. Thomason: Threshold functions, [*Combinatorica*]{} 7(1987), 35–38.
B. Bollob' as: [*Random graphs*]{}, Cambridge Unversity Press, 1985.
V. Chv' atal, P. Erdős: Biased positional games, [*Annals of Discrete Math.*]{} 2(1978), 221–228.
V. Chv' atal, P. Erdős: A note on Hamiltonian circuits, [*Discrete Math.*]{} 2(1972), 111–113.
P. Erdős, J. Selfridge: On a combinatorial game, [*J. Combinatorial Theory*]{} 14(1973), 298–301.
A. Lehman, A solution of the Shannon switching game, [*J. Soc. Indust. Appl. Math.*]{} 12(1964) 687–725.
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[^1]: Institute of Theoretical Computer Science, ETH Zurich, CH-8092 Switzerland. Email addresses: $\{$smilos, szabo$\}$@inf.ethz.ch
[^2]: Supported by the joint Berlin/Zurich graduate program Combinatorics, Geometry and Computation, financed by ETH Zurich and the German Science Foundation (DFG).
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[Yoonbai Kim$^{\ast}$]{}\
\[2mm\] [*Department of Physics, Nagoya University, Nagoya 464-01, Japan*]{}
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\#1[preprint[\#1]{}]{} ‘=12
[**Abstract**]{}\
The first-order phase transition of $O(3)$ symmetric models is considered in the limit of high temperature. It is shown that this model supports a new bubble solution where the global monopole is formed at the center of the bubble in addition to the ordinary $O(3)$ bubble. Though the free energy of it is larger than that of normal bubble, the production rate can considerably be large at high temperatures.
Since a general theory of the decay of the metastable phase was developed [@Lan; @Col1], the study of first-order phase transitions have attracted the attention due to their possible relevance to the physics of the early universe. The semiclassical expression of the tunneling rate from false vacuum to true vacuum is given by the bubble solution of lowest (Euclidean) action both for the first-order phase transitions at zero temperature field theories [@Col1] and those at finite temperature [@Lin; @KT]. However, the above case does not include the possibility that there exist various decay modes between two given classical vacua. In this note, we explore such problem by examining a scalar model of internal $O(3)$ symmetry at finite temperature and examine a new bubble solution with a global monopole.
Suppose that the model of our interest contains a series of stationary points which support bubble solutions, a decay probability per unit time per unit volume is given by \[decay\] /V=\_[n]{}A\_[n]{}e\^[-B\_[n]{}/]{}, where $n$ represents the $n$-th local minimum. For each $n$, $B_{n}$ is given by a value of Euclidean action for $n$-th bubble solution and $A_{n}$ is estimated by integrating out the fluctuations around a given $n$-th configuration.
We choose the $O(3)$-symmetric scalar models described in terms of an isovector field $\phi^{a}$ $(a=1,2,3)$ as the first possibility for sample calculations. At finite temperature the Euclidean action is \[action\] S\_[E]{}=\^\_[0]{} dt\_[E]{} { ()\^[2]{} +(\^[a]{})\^[2]{}+V(\^[a]{}) }, where $\beta=\hbar /k_{B}T$. Although our argument followed is general and does scarcely depend on the detailed form of scalar potential if it has the vacuum structure with both a true and a false vacuum, we will consider a specific model, [*i.e.*]{} a $\phi^{6}$-potential V()=(\^[2]{}+v\^[2]{})(\^[2]{}-v\^[2]{})\^[2]{} +V\_[0]{} where $\phi$ denotes the amplitude of scalar fields $\phi^{a}$ defined by $\phi=\sqrt{\phi^{a}\phi^{a}}$. Here we consider the transition from the symmetric vacuum to the broken vacuum, [*i.e.*]{} $0<\alpha<1/2$, and choose $V_{0}$ as $-\lambda\alpha v^{4}$ in order to make the value of $S_{E}$ coinside with $B_{n}$ in Eq.(\[decay\]). This does not lose generality since the case of the transition from the broken vacuum to the symmetric one ($-1<\alpha<0$) also possesses the same type of bubble solutions if we replace $\phi$ to $v-\phi$.
It has been known that the rate of vacuum transitions at finite temperature has relevant amount of contributions from $O(3)$-symmetric sphaleron-type bubbles [@Lin; @KT]. In high temperature limit this contribution dominates and then we can neglect the dependence along $t_{E}$-axis. Here we are interested in the regime where thermal fluctuations are much larger than quantum fluctuations and address the problem for different kind of finite temperature bubbles which connect global $O(3)$ internal symmetry to that of spatial rotation. We now ask for a solution of the field equations that is time-independent and spherically symmetric, apart from the angle dependence due to the mapping between the $(\theta,\varphi)$ angles in space and those of isovector space $\hat{\phi}^{a}_{n}
(\equiv \phi^{a}/\phi)$ such as \[angle\] \^[a]{}\_[n]{}=(nn,nn, n), where the allowed $n$ is 0 or 1 in order to render the scalar amplitude $\phi$ a function of $r$ only. Under this assumption, the Euler-Lagrange equation becomes \[equation\] + -\_[n1]{}=, where $\delta_{n1}$ in the third term of Eq.(\[equation\]) denotes Kronecker delta for $n=0$ or 1.
The condition that the theory should be in false vacuum at spatial infinity fixes the boundary value of field, [*i.e.*]{} $\lim_{r\rightarrow\infty}\phi\rightarrow 0$. To be nonsingular solution at the origin of coordinates, the boundary condition is {
[ll]{} .|\_[r=0]{}=0 &\
(0)=0 &
. When $n=0$, it is well-known bubble solution at finite temperature [@Lin]. We will analyze $n=1$ case and show that there always be $n=1$ solution if the equation contains $n=0$ solution. A brief argument of the existence of $n=1$ solution is as follows.
If we regard the radius $r$ as time and the scalar amplitude $\phi (r)$ as the coordinate of a particle, Eq.(\[equation\]) describes a one-dimensional motion of a unit-mass hypothetical particle under the conserved force due to the potential $-V(\phi)$ and two nonconservative forces, [*i.e.*]{} one is the friction of time-dependent coefficient $-\frac{2}{r}\frac{d\phi}{dr}$ and the other is time-dependent repulsion $\frac{2}{r^{2}}\phi$. Hence, in the terminology of Newton equation, $n=1$ solution in Fig. 1 is interpreted as the motion of a particle that starts at time zero at the origin $(\phi(0)=0)$, turns at an appropriate nonzero position at a certain time $(\phi(t_{turn})=\phi_{turn})$, and stops at the origin at infinite time $(\phi(\infty)=0)$.
At first, if one considers a hypothetical particle at the origin at time zero, it is accelerated by the time-dependent repulsion of which coefficient is divergently large for small $r$. Since one initial condition is fixed by the starting point $(\phi(0)=0)$, such motions near the origin are characterized by another initial condition $C$ \[phi0\] (r)C{ r+[O]{}(r\^[3]{})+}, where $C$ is the initial velocity of a particle which should be tuned by the proper boundary condition at infinite time $(\phi(\infty)=0)$. From now on let us consider a set of solutions specified by a real parameter $C$ and show that there always exists the unique motion of $\phi(\infty)=0$ for an appropriate $C$. When $C$ is sufficiently large, the acceleration near the origin due to time-dependent repulsion and conservative force is too strong and then the particle overshoots the top of potential $-V(\phi=v)$ and goes to infinity $(\phi(\infty)=\infty)$, despite the deceleration due to the friction. Since the solution of our interest is the motion which includes a return, we will look at the motions of which $C$ is smaller than a critical value $C_{top}$. Here $C_{top}$ gives the motion that the particle stops at the hilltop of the potential $-V$ at infinite time $\phi(\infty)=v$. Suppose that $C$ be too small, [*i.e.*]{} $C$ is smaller than another critical value $C_{0}\; (C_{0} <C_{top})$, particle turns at a point smaller than $\phi_{0}$ ($0<\phi_{0}<v$) where $-V(\phi=\phi_{0})=0$ and thereby it can not return to the origin. Moreover, the particle which turns at a point too close to $v$ arrives at the origin at a finite time, since the time-dependent friction and repulsion during returning do not play a role due to the vanishing of their coefficients $1/r$ and $1/r^{2}$ while the particle stays near $v$ for sufficiently long time. By continuity for an appropriate $C\;(C_{0}<C<C_{top})$ there is a solution which describes the motion that starts at time zero at the origin turns at a position $\phi_{turn}$ between $\phi_{0}$ and $v$, and then stops at the origin at infinite time. Here for the existence proof of $n=1$ solutions, we have used the properties of scalar potential $V(\phi)$ that it has a local minimum at $\phi=0$, a local maximum at $\phi_{bottom}$ ($0<\phi_{bottom}<\phi_{0}$) and an absolute minimum at $v$. It is exactly the same condition as that for the existence of $n=0$ solution. Hence we completes our argument that the equation always contains $n=1$ solutions when it has $n=0$ solution. A rigorous proof for the existence of $n=1$ solution is demonstrated in ref.[@KKK]. We have no analytic form of solutions, so numerical solutions for $n=0$ and $n=1$ bubbles are given in Fig. 1.
Suppose that each classical solution describes a critical bubble corresponding to a different decay channel, from the profile of energy density $T^{0}_{\;0}$ (see Fig. 2) we can read the following characteristics of $n=1$ bubbles. First, the boundary value of scalar field at the origin always has that of false vacuum, [*i.e.*]{} $\phi(0)=0$, so this implies that there remains a false-vacuum core inside the true-vacuum region of the bubble due to the winding between $O(3)$ internal symmetry and spatial rotation. Second, the value of energy density has a local minimum at the origin, [*i.e.*]{} $3C^{2}/2+V(0)$ in which $C$ is a constant appeared in Eq.(\[phi0\]), increases to the maximum at $R_{m}$ in Fig. 2 and decays below $V(0)$. This implies that a matter droplet is created inside the bubble and is surrounded by inner bubble wall with size of order $R_{m}\sim 1/m_{Higgs}
=\sqrt{4(3+2\alpha)\lambda}v$. Third, if we read the expression of energy density when the scalar amplitude has maximum value $\phi=\phi_{turn}$, its derivative term vanishes and then it becomes \[turn\] T\^[0]{}\_[0]{}=+V(\_[turn]{}). The potential term $V(\phi_{turn})$ can be neglected in thin-wall limit, so the object inside the bubble has a long-range hair which penetrates the inner bubble wall. Since the central region of this matter is in false symmetric vacuum due to the hedgehog ansatz of scalar field in Eq.(\[angle\]) and for large $r$ the scalar field goes to the true broken vacuum but the long-range tail of energy proportional to $r$ has a cutoff at the outer bubble wall at $R_{n=1}$ in Fig. 2, we can interpret the matter aggregate inside the $n=1$ bubble as a global monopole of size $R_{m}$ [@Vil]. Even though there is a no-go theorem for static scalar objects in spacetime dimensions more than two [@Der], the global monopole of this case can be supported inside the Euclidean bubble configuration as a smooth finite-energy configuration due to a natural cutoff introduced by the outer bubble wall. Fig. 2 shows that the radius of $n=1$ bubble is larger than that of $n=0$ bubble. It can be easily understood by the conservation of energy, [*i.e.*]{} the additional energy used to make a matter aggregate is equal to the loss of energy due to the increase of the radius of bubble.
Though we do not have any analytic solution, we can estimate $B_{n}$ in Eq.(\[decay\]) by use of the obtained bubble configurations through numerical analysis for given parameters of theory. It has already been proved that $n=0$ solution describes the nontrivial solution of lowest action [@CGM] and from Fig. 3 we read that the value of action for $n=1$ solution, $vB^{'}_{1}/T$, is larger than that of $n=0$, $vB^{'}_{0}/T$, irrespective of the shape of scalar potential. Fig. 3 also shows that the ratio $B^{'}_{1}/B^{'}_{0}$ becomes small while the difference $B^{'}_{1}-
B^{'}_{0}$ increases as the size of bubble becomes large in comparison with the mass scale of theory, [*i.e.*]{} thin-wall limit. This can be understood as follows; the amount of energy consumed to support a global monopole at the center of bubble is proportional to the radius of bubble due to its long-range tail (see Eq.(\[turn\])), however the free energy to support the bubble itself, $vB^{'}_{0}$ or $vB^{'}_{1}$, is proportional to the cubic of bubble radius in thin-wall limit.
The next task is to estimate the pre-exponential factor $A_{n}$ in Eq.(\[decay\]). The scheme of computing $A_{0}$ for $n=0$ bubble solution which is the lowest action solution with one negative mode was given in the second paper of Ref.[@Col1]. Here, let us attempt to calculate the pre-exponential factor $A_{1}$ for $n=1$ bubble. Considering the small fluctuation around $n=1$ bubble $\delta\phi^{a}_{n=1}=\sum_{k}c^{a}_{k}
\psi^{a}_{k}$, we obtain a Schrödinger-type equation for three particles in three dimensions \[sch\] (-\^[2]{}\^[ab]{}+\^[a]{}\^[b]{} |\_[\^[a]{}\_[n=1]{}]{}+(\^[ab]{}-\^[a]{}\^[b]{}) |\_[\^[a]{}\_[n=1]{}]{})\^[b]{}\_[k]{}=\_[k]{} \^[a]{}\_[k]{}. It looks too difficult to solve this equation since it includes $\theta$ and $\varphi$ dependent off-diagonal terms in its Hamiltonian and the potential form is only determined numerically, however the eigenfunctions of six zero modes due to three translations and three rotations are explicitly given in a form \[zero\] \^[a]{}\_[k, i]{}&=&N\_[t]{}\_[i]{}\^[a]{}\_[n=1]{}\
&=&N\_[t]{}{\_[i]{}\^[a]{} +(\_[i]{}\^[a]{}-\_[i]{}\^[a]{}) }, and \[azero\] \^[a]{}\_[k, i]{}=N\_[r]{}\_[ijk]{}x\^[j]{}\^[k]{}\^[a]{}\_[n=1]{} =N\_[r]{}\_[ija]{}\^[j]{}\_[n=1]{}, where $N_{t}$ and $N_{r}$ are normalization constants. Here we give few comments on the number of negative modes. First, every component of translational zero-mode eigenfunction in Eq.(\[zero\]) has $(\theta,\varphi)$-dependence where its radial part, $\frac{d\phi_{n=1}}{dr}-\frac{\phi_{n=1}}{r}$, has single node at the origin, and each $a=i$ component includes an additional $(\theta,\varphi)$-independent part, $\frac{\phi_{n=1}}{r}$, which has no node. Every $a\neq i$ component of rotational ones, $\phi_{n=1}$, has one node at the origin. For the sake of simplicity, let us examine the problem for the perturbation with specific direction; $(i)$ one for amplitude ($\delta\phi^{a}_{n=1}=
\hat{r}^{a}\delta\phi_{n=1})$ and $(ii)$ two for transverse ($\delta\phi^{a}_{n=1}=(\delta^{ab}-\hat{r}^{a}\hat{r}^{b})
\delta\phi^{b}_{n=1}$). Under these fluctuations Eq.(\[sch\]) reduces to \[nonz\] (-\^[2]{}+U(r))\^[a]{}\_[k]{}=\_[k]{}\^[a]{}\_[k]{}, where $U(r)=\left.\frac{d^{2}V}{d\phi^{2}}\right|_{\phi_{n=1}}$ for the first case $(i)$ and $U(r)=\left.\frac{1}{\phi}\frac{dV}{d\phi}
\right|_{\phi_{n=1}}$ for the second case $(ii)$. When $
U(r)=\left.\frac{d^{2}V}{d\phi^{2}}\right|_{\phi_{n=0}}$, each single ‘$a$’ component of Eq.(\[nonz\]) is nothing but the fluctuation equation for $n=0$ bubble which contains a nodless $s$-wave mode as the unique negative mode [@Col1; @SCol]. For the fluctuation of scalar amplitude for $n=1$ bubble, the lowest mode can not be nodless $s$-wave mode but $l=1$ mode with single node at $r=0$ since $\psi^{a}_{k}$ should have $\theta$ and $\varphi$ dependence proportional to $\hat{r}^{a}$ even though the operator in Eq.(\[nonz\]) is a scalar operator. It is analogous for the perturbation to the transversal directions. It implies that the eigenfunction of negative mode may take a somewhat complicated form which depends on angels. Second, the $n=1$ solution which is a parity-odd bounce solution, $\phi(r=0)=\phi(r=\infty)$, is supported by assuming the winding between three spatial rotations and those in internal space. It seems that the argument for the system of quantum mechanics with one time variable in Ref.[@SCol] does not forbid directly the existence of $n=1$ solution as a bubble configuration. We know very little on the counting of negative modes, and then there should be further work on this issue. Since the operator in Eq.(\[sch\]) is even under parity transformation and is covariant under the rotation, [*i.e.*]{} $x^{i}\rightarrow O^{ij}x^{j}$ and $\psi^{a}_{k}(x^{'})=O^{ab}\psi^{b}_{k}(x)$ where both $O^{ij}$ and $O^{ab}$ are the elements of $O(3)$ group, vector spherical harmonics is a method to investigate the modes of which the eigenfunctions can be chosen to be either even or odd in parity [@Tam].
In order to compute the decay rate accurately, we should calculate the nonzero modes $\lambda_{k}$, which is extremely difficult even for $n=0$ bubbles. However, the dimensional estimate may reach a rough result such as [@Lin] \~( )\^. At high temperature limit, $T>>v$, both decay rates for $n=0$ and 1 bubbles, of course, increase and, moreover, the relative decay rate of $n=1$ bubble to that of $n=0$ is also enhanced exponentially. When $v/T\sim 10^{-1}$, the order of $\Gamma^{(1)}/\Gamma^{(0)}$ is around 0.17 in a thin-wall case ($\lambda=1$, $\alpha=0.12$) and it is around 28 in a thick-wall case ($\lambda=1$, $\alpha=0.47$). Therefore, at high temperatures and in thick-wall case, this $n=1$ bubbles can be preferred to those of $n=0$, and obviously the existence of another decay channel can enhance considerably the total nucleation rate of bubbles, $\Gamma=\Gamma^{(0)}+\Gamma^{(1)}$, except for the bubbles with extremely thin wall.
The main consideration of the paper was to gain an understanding as to how bubbles with global monopoles nucleated in first-order phase transitions at high temperatures and the next question will be how the high-temperature bubbles grow [@Ste], particularly at the site of global monopole. Once we assume these bubbles in early universe where the gravity effect should be included, the bubbles with solitons may result in interesting phenomena [@KMS] in relation with inflationary models, [*i.e.*]{} the inflation in the cores of topological defects [@Vil2].
The author would like to express deep gratitude to Jooyoo Hong, K. Ishikawa, Chanju Kim, Kimyeong Lee, K. Maeda, V. P. Nair, Q. Park, N. Sakai, A. I. Sanda, S. Tanimura, E. J. Weinberg, Y. S. Wu and K. Yamawaki for valuable discussions and also thanks Kyung Hee University and Hanyang University for their hospitality. This work is supported by JSPS under $\#$93033.
\#1
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**Figure Captions** {#figure-captions .unnumbered}
===================
FIG. 1. Plot of bubble solutions. $n=0$ and $n=1$ configurations are shown as dotted and solid lines, respectively. The parameters chosen in the figures are: $\lambda=1$, $\alpha=0.12$, and $V_{0}=-0.12v^{4}$.
FIG. 2. Plot of energy density $T^{0}_{0}$. The parameters chosen in the figures are: $\lambda=1$, $\alpha=0.12$, and $V_{0}=-0.12v^{4}$ which is the minimum of energy density.
FIG. 3. Plot of action $S_{E}$ (or equivalently $B_{n}$) as a function of $\alpha$. Another parameters are chosen as $\lambda=1$ and $V_{0}=-\lambda\alpha v^{4}$.
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(1321,23)[(0,0)[10]{}]{} (1321.0,857.0)
------------------------------------------------------------------------
(176.0,68.0)
------------------------------------------------------------------------
(1436.0,68.0)
------------------------------------------------------------------------
(176.0,877.0)
------------------------------------------------------------------------
(176.0,68.0)
------------------------------------------------------------------------
(1306,812)[(0,0)\[r\][$n=1$]{}]{} (1328.0,812.0)
------------------------------------------------------------------------
(176,68) (176.58,68.00)(0.492,1.581)[19]{}
------------------------------------------------------------------------
(175.17,68.00)(11.000,31.226)[2]{}
------------------------------------------------------------------------
(187.58,102.00)(0.492,1.487)[21]{}
------------------------------------------------------------------------
(186.17,102.00)(12.000,32.371)[2]{}
------------------------------------------------------------------------
(199.58,137.00)(0.492,1.628)[19]{}
------------------------------------------------------------------------
(198.17,137.00)(11.000,32.151)[2]{}
------------------------------------------------------------------------
(210.58,172.00)(0.492,1.487)[21]{}
------------------------------------------------------------------------
(209.17,172.00)(12.000,32.371)[2]{}
------------------------------------------------------------------------
(222.58,207.00)(0.492,1.722)[19]{}
------------------------------------------------------------------------
(221.17,207.00)(11.000,34.000)[2]{}
------------------------------------------------------------------------
(233.58,244.00)(0.492,1.616)[21]{}
------------------------------------------------------------------------
(232.17,244.00)(12.000,35.163)[2]{}
------------------------------------------------------------------------
(245.58,282.00)(0.492,1.817)[19]{}
------------------------------------------------------------------------
(244.17,282.00)(11.000,35.849)[2]{}
------------------------------------------------------------------------
(256.58,321.00)(0.492,1.703)[21]{}
------------------------------------------------------------------------
(255.17,321.00)(12.000,37.025)[2]{}
------------------------------------------------------------------------
(268.58,361.00)(0.492,1.864)[19]{}
------------------------------------------------------------------------
(267.17,361.00)(11.000,36.773)[2]{}
------------------------------------------------------------------------
(279.58,401.00)(0.492,1.746)[21]{}
------------------------------------------------------------------------
(278.17,401.00)(12.000,37.956)[2]{}
------------------------------------------------------------------------
(291.58,442.00)(0.492,1.817)[19]{}
------------------------------------------------------------------------
(290.17,442.00)(11.000,35.849)[2]{}
------------------------------------------------------------------------
(302.58,481.00)(0.492,1.817)[19]{}
------------------------------------------------------------------------
(301.17,481.00)(11.000,35.849)[2]{}
------------------------------------------------------------------------
(313.58,520.00)(0.492,1.530)[21]{}
------------------------------------------------------------------------
(312.17,520.00)(12.000,33.302)[2]{}
------------------------------------------------------------------------
(325.58,556.00)(0.492,1.581)[19]{}
------------------------------------------------------------------------
(324.17,556.00)(11.000,31.226)[2]{}
------------------------------------------------------------------------
(336.58,590.00)(0.492,1.272)[21]{}
------------------------------------------------------------------------
(335.17,590.00)(12.000,27.717)[2]{}
------------------------------------------------------------------------
(348.58,620.00)(0.492,1.251)[19]{}
------------------------------------------------------------------------
(347.17,620.00)(11.000,24.755)[2]{}
------------------------------------------------------------------------
(359.58,647.00)(0.492,1.013)[21]{}
------------------------------------------------------------------------
(358.17,647.00)(12.000,22.132)[2]{}
------------------------------------------------------------------------
(371.58,671.00)(0.492,0.920)[19]{}
------------------------------------------------------------------------
(370.17,671.00)(11.000,18.283)[2]{}
------------------------------------------------------------------------
(382.58,691.00)(0.492,0.712)[21]{}
------------------------------------------------------------------------
(381.17,691.00)(12.000,15.616)[2]{}
------------------------------------------------------------------------
(394.58,708.00)(0.492,0.637)[19]{}
------------------------------------------------------------------------
(393.17,708.00)(11.000,12.736)[2]{}
------------------------------------------------------------------------
(405.00,722.58)(0.496,0.492)[21]{}
------------------------------------------------------------------------
(405.00,721.17)(10.962,12.000)[2]{}
------------------------------------------------------------------------
(417.00,734.58)(0.547,0.491)[17]{}
------------------------------------------------------------------------
(417.00,733.17)(9.879,10.000)[2]{}
------------------------------------------------------------------------
(428.00,744.59)(0.692,0.488)[13]{}
------------------------------------------------------------------------
(428.00,743.17)(9.651,8.000)[2]{}
------------------------------------------------------------------------
(439.00,752.59)(0.874,0.485)[11]{}
------------------------------------------------------------------------
(439.00,751.17)(10.369,7.000)[2]{}
------------------------------------------------------------------------
(451.00,759.59)(1.155,0.477)[7]{}
------------------------------------------------------------------------
(451.00,758.17)(8.966,5.000)[2]{}
------------------------------------------------------------------------
(462.00,764.59)(1.267,0.477)[7]{}
------------------------------------------------------------------------
(462.00,763.17)(9.800,5.000)[2]{}
------------------------------------------------------------------------
(474.00,769.60)(1.505,0.468)[5]{}
------------------------------------------------------------------------
(474.00,768.17)(8.509,4.000)[2]{}
------------------------------------------------------------------------
(485.00,773.61)(2.472,0.447)[3]{}
------------------------------------------------------------------------
(485.00,772.17)(8.472,3.000)[2]{}
------------------------------------------------------------------------
(497.00,776.61)(2.248,0.447)[3]{}
------------------------------------------------------------------------
(497.00,775.17)(7.748,3.000)[2]{}
------------------------------------------------------------------------
(508,779.17)
------------------------------------------------------------------------
(508.00,778.17)(6.811,2.000)[2]{}
------------------------------------------------------------------------
(520,781.17)
------------------------------------------------------------------------
(520.00,780.17)(6.226,2.000)[2]{}
------------------------------------------------------------------------
(531,782.67)
------------------------------------------------------------------------
(531.00,782.17)(6.000,1.000)[2]{}
------------------------------------------------------------------------
(543,784.17)
------------------------------------------------------------------------
(543.00,783.17)(6.226,2.000)[2]{}
------------------------------------------------------------------------
(554,785.67)
------------------------------------------------------------------------
(554.00,785.17)(5.500,1.000)[2]{}
------------------------------------------------------------------------
(565,786.67)
------------------------------------------------------------------------
(565.00,786.17)(6.000,1.000)[2]{}
------------------------------------------------------------------------
(577,787.67)
------------------------------------------------------------------------
(577.00,787.17)(5.500,1.000)[2]{}
------------------------------------------------------------------------
(588,788.67)
------------------------------------------------------------------------
(588.00,788.17)(6.000,1.000)[2]{}
------------------------------------------------------------------------
(600,789.67)
------------------------------------------------------------------------
(600.00,789.17)(5.500,1.000)[2]{}
------------------------------------------------------------------------
(611,790.67)
------------------------------------------------------------------------
(611.00,790.17)(6.000,1.000)[2]{}
------------------------------------------------------------------------
(634,791.67)
------------------------------------------------------------------------
(634.00,791.17)(6.000,1.000)[2]{}
------------------------------------------------------------------------
(623.0,792.0)
------------------------------------------------------------------------
(669,792.67)
------------------------------------------------------------------------
(669.00,792.17)(5.500,1.000)[2]{}
------------------------------------------------------------------------
(646.0,793.0)
------------------------------------------------------------------------
(726,792.67)
------------------------------------------------------------------------
(726.00,793.17)(5.500,-1.000)[2]{}
------------------------------------------------------------------------
(680.0,794.0)
------------------------------------------------------------------------
(749,791.67)
------------------------------------------------------------------------
(749.00,792.17)(5.500,-1.000)[2]{}
------------------------------------------------------------------------
(760,790.67)
------------------------------------------------------------------------
(760.00,791.17)(6.000,-1.000)[2]{}
------------------------------------------------------------------------
(772,789.17)
------------------------------------------------------------------------
(772.00,790.17)(6.226,-2.000)[2]{}
------------------------------------------------------------------------
(783,787.17)
------------------------------------------------------------------------
(783.00,788.17)(6.811,-2.000)[2]{}
------------------------------------------------------------------------
(795.00,785.95)(2.248,-0.447)[3]{}
------------------------------------------------------------------------
(795.00,786.17)(7.748,-3.000)[2]{}
------------------------------------------------------------------------
(806.00,782.94)(1.505,-0.468)[5]{}
------------------------------------------------------------------------
(806.00,783.17)(8.509,-4.000)[2]{}
------------------------------------------------------------------------
(817.00,778.93)(1.033,-0.482)[9]{}
------------------------------------------------------------------------
(817.00,779.17)(10.132,-6.000)[2]{}
------------------------------------------------------------------------
(829.00,772.93)(0.943,-0.482)[9]{}
------------------------------------------------------------------------
(829.00,773.17)(9.270,-6.000)[2]{}
------------------------------------------------------------------------
(840.00,766.93)(0.669,-0.489)[15]{}
------------------------------------------------------------------------
(840.00,767.17)(10.685,-9.000)[2]{}
------------------------------------------------------------------------
(852.00,757.92)(0.496,-0.492)[19]{}
------------------------------------------------------------------------
(852.00,758.17)(9.962,-11.000)[2]{}
------------------------------------------------------------------------
(863.58,745.65)(0.492,-0.582)[21]{}
------------------------------------------------------------------------
(862.17,746.82)(12.000,-12.824)[2]{}
------------------------------------------------------------------------
(875.58,731.02)(0.492,-0.779)[19]{}
------------------------------------------------------------------------
(874.17,732.51)(11.000,-15.509)[2]{}
------------------------------------------------------------------------
(886.58,713.68)(0.492,-0.884)[21]{}
------------------------------------------------------------------------
(885.17,715.34)(12.000,-19.340)[2]{}
------------------------------------------------------------------------
(898.58,691.96)(0.492,-1.109)[19]{}
------------------------------------------------------------------------
(897.17,693.98)(11.000,-21.981)[2]{}
------------------------------------------------------------------------
(909.58,667.57)(0.492,-1.229)[21]{}
------------------------------------------------------------------------
(908.17,669.79)(12.000,-26.786)[2]{}
------------------------------------------------------------------------
(921.58,637.75)(0.492,-1.486)[19]{}
------------------------------------------------------------------------
(920.17,640.38)(11.000,-29.377)[2]{}
------------------------------------------------------------------------
(932.58,605.30)(0.492,-1.628)[19]{}
------------------------------------------------------------------------
(931.17,608.15)(11.000,-32.151)[2]{}
------------------------------------------------------------------------
(943.58,570.47)(0.492,-1.573)[21]{}
------------------------------------------------------------------------
(942.17,573.23)(12.000,-34.233)[2]{}
------------------------------------------------------------------------
(955.58,532.85)(0.492,-1.769)[19]{}
------------------------------------------------------------------------
(954.17,535.92)(11.000,-34.924)[2]{}
------------------------------------------------------------------------
(966.58,495.19)(0.492,-1.659)[21]{}
------------------------------------------------------------------------
(965.17,498.09)(12.000,-36.094)[2]{}
------------------------------------------------------------------------
(978.58,455.85)(0.492,-1.769)[19]{}
------------------------------------------------------------------------
(977.17,458.92)(11.000,-34.924)[2]{}
------------------------------------------------------------------------
(989.58,418.47)(0.492,-1.573)[21]{}
------------------------------------------------------------------------
(988.17,421.23)(12.000,-34.233)[2]{}
------------------------------------------------------------------------
(1001.58,381.45)(0.492,-1.581)[19]{}
------------------------------------------------------------------------
(1000.17,384.23)(11.000,-31.226)[2]{}
------------------------------------------------------------------------
(1012.58,348.16)(0.492,-1.358)[21]{}
------------------------------------------------------------------------
(1011.17,350.58)(12.000,-29.579)[2]{}
------------------------------------------------------------------------
(1024.58,316.21)(0.492,-1.345)[19]{}
------------------------------------------------------------------------
(1023.17,318.60)(11.000,-26.604)[2]{}
------------------------------------------------------------------------
(1035.58,287.85)(0.492,-1.142)[21]{}
------------------------------------------------------------------------
(1034.17,289.92)(12.000,-24.924)[2]{}
------------------------------------------------------------------------
(1047.58,260.96)(0.492,-1.109)[19]{}
------------------------------------------------------------------------
(1046.17,262.98)(11.000,-21.981)[2]{}
------------------------------------------------------------------------
(1058.58,237.41)(0.492,-0.967)[19]{}
------------------------------------------------------------------------
(1057.17,239.21)(11.000,-19.207)[2]{}
------------------------------------------------------------------------
(1069.58,216.96)(0.492,-0.798)[21]{}
------------------------------------------------------------------------
(1068.17,218.48)(12.000,-17.478)[2]{}
------------------------------------------------------------------------
(1081.58,198.17)(0.492,-0.732)[19]{}
------------------------------------------------------------------------
(1080.17,199.58)(11.000,-14.585)[2]{}
------------------------------------------------------------------------
(1092.58,182.51)(0.492,-0.625)[21]{}
------------------------------------------------------------------------
(1091.17,183.75)(12.000,-13.755)[2]{}
------------------------------------------------------------------------
(1104.58,167.62)(0.492,-0.590)[19]{}
------------------------------------------------------------------------
(1103.17,168.81)(11.000,-11.811)[2]{}
------------------------------------------------------------------------
(1115.00,155.92)(0.543,-0.492)[19]{}
------------------------------------------------------------------------
(1115.00,156.17)(10.887,-11.000)[2]{}
------------------------------------------------------------------------
(1127.00,144.92)(0.547,-0.491)[17]{}
------------------------------------------------------------------------
(1127.00,145.17)(9.879,-10.000)[2]{}
------------------------------------------------------------------------
(1138.00,134.93)(0.758,-0.488)[13]{}
------------------------------------------------------------------------
(1138.00,135.17)(10.547,-8.000)[2]{}
------------------------------------------------------------------------
(1150.00,126.93)(0.692,-0.488)[13]{}
------------------------------------------------------------------------
(1150.00,127.17)(9.651,-8.000)[2]{}
------------------------------------------------------------------------
(1161.00,118.93)(0.874,-0.485)[11]{}
------------------------------------------------------------------------
(1161.00,119.17)(10.369,-7.000)[2]{}
------------------------------------------------------------------------
(1173.00,111.93)(1.155,-0.477)[7]{}
------------------------------------------------------------------------
(1173.00,112.17)(8.966,-5.000)[2]{}
------------------------------------------------------------------------
(1184.00,106.93)(1.155,-0.477)[7]{}
------------------------------------------------------------------------
(1184.00,107.17)(8.966,-5.000)[2]{}
------------------------------------------------------------------------
(1195.00,101.93)(1.267,-0.477)[7]{}
------------------------------------------------------------------------
(1195.00,102.17)(9.800,-5.000)[2]{}
------------------------------------------------------------------------
(1207.00,96.94)(1.505,-0.468)[5]{}
------------------------------------------------------------------------
(1207.00,97.17)(8.509,-4.000)[2]{}
------------------------------------------------------------------------
(1218.00,92.95)(2.472,-0.447)[3]{}
------------------------------------------------------------------------
(1218.00,93.17)(8.472,-3.000)[2]{}
------------------------------------------------------------------------
(1230.00,89.95)(2.248,-0.447)[3]{}
------------------------------------------------------------------------
(1230.00,90.17)(7.748,-3.000)[2]{}
------------------------------------------------------------------------
(1241,86.17)
------------------------------------------------------------------------
(1241.00,87.17)(6.811,-2.000)[2]{}
------------------------------------------------------------------------
(1253.00,84.95)(2.248,-0.447)[3]{}
------------------------------------------------------------------------
(1253.00,85.17)(7.748,-3.000)[2]{}
------------------------------------------------------------------------
(1264,81.67)
------------------------------------------------------------------------
(1264.00,82.17)(6.000,-1.000)[2]{}
------------------------------------------------------------------------
(1276,80.17)
------------------------------------------------------------------------
(1276.00,81.17)(6.226,-2.000)[2]{}
------------------------------------------------------------------------
(1287,78.17)
------------------------------------------------------------------------
(1287.00,79.17)(6.811,-2.000)[2]{}
------------------------------------------------------------------------
(1299,76.67)
------------------------------------------------------------------------
(1299.00,77.17)(5.500,-1.000)[2]{}
------------------------------------------------------------------------
(1310,75.67)
------------------------------------------------------------------------
(1310.00,76.17)(5.500,-1.000)[2]{}
------------------------------------------------------------------------
(1321,74.67)
------------------------------------------------------------------------
(1321.00,75.17)(6.000,-1.000)[2]{}
------------------------------------------------------------------------
(1333,73.67)
------------------------------------------------------------------------
(1333.00,74.17)(5.500,-1.000)[2]{}
------------------------------------------------------------------------
(1344,72.67)
------------------------------------------------------------------------
(1344.00,73.17)(6.000,-1.000)[2]{}
------------------------------------------------------------------------
(737.0,793.0)
------------------------------------------------------------------------
(1367,71.67)
------------------------------------------------------------------------
(1367.00,72.17)(6.000,-1.000)[2]{}
------------------------------------------------------------------------
(1356.0,73.0)
------------------------------------------------------------------------
(1390,70.67)
------------------------------------------------------------------------
(1390.00,71.17)(6.000,-1.000)[2]{}
------------------------------------------------------------------------
(1379.0,72.0)
------------------------------------------------------------------------
(1413,69.67)
------------------------------------------------------------------------
(1413.00,70.17)(6.000,-1.000)[2]{}
------------------------------------------------------------------------
(1402.0,71.0)
------------------------------------------------------------------------
(1425.0,70.0)
------------------------------------------------------------------------
(1306,767)[(0,0)\[r\][$n=0$]{}]{} (1328,767)(20.756,0.000)[4]{} (1394,767) (176,803) (176.00,803.00) (196.76,803.00) (199,803)(20.756,0.000)[0]{} (217.51,803.00) (222,803)(20.756,0.000)[0]{} (238.27,803.00) (245,803)(20.756,0.000)[0]{} (259.02,803.00) (268,803)(20.756,0.000)[0]{} (279.78,803.00) (300.53,803.00) (302,803)(20.756,0.000)[0]{} (321.29,803.00) (325,803)(20.756,0.000)[0]{} (342.04,803.00) (348,803)(20.756,0.000)[0]{} (362.80,803.00) (371,803)(20.756,0.000)[0]{} (383.55,803.00) (404.31,803.00) (405,803)(20.756,0.000)[0]{} (425.07,803.00) (428,803)(20.756,0.000)[0]{} (445.82,803.00) (451,803)(20.756,0.000)[0]{} (466.58,803.00) (474,803)(20.756,0.000)[0]{} (487.33,803.00) (497,803)(20.756,0.000)[0]{} (508.09,803.00) (528.84,803.00) (531,803)(20.756,0.000)[0]{} (549.60,803.00) (554,803)(20.670,-1.879)[0]{} (570.31,802.00) (577,802)(20.670,-1.879)[0]{} (591.02,801.00) (600,801)(20.670,-1.879)[0]{} (611.73,799.94) (632.40,798.15) (634,798)(20.473,-3.412)[0]{} (652.74,794.16) (657,793)(20.136,-5.034)[0]{} (672.73,788.64) (680,786)(18.895,-8.589)[0]{} (691.81,780.53) (709.04,769.06) (724.58,755.31) (726,754)(12.823,-16.320)[0]{} (737.59,739.17) (749,723)(9.631,-18.386)[2]{} (768.39,685.21) (776.50,666.13) (783,649)(7.288,-19.434)[2]{} (795,617)(6.223,-19.801)[2]{} (806,582)(5.771,-19.937)[2]{} (817,544)(6.104,-19.838)[2]{} (829,505)(5.634,-19.976)[2]{} (840,466)(6.104,-19.838)[2]{} (857.40,408.84) (863,390)(6.732,-19.634)[2]{} (875,355)(6.563,-19.690)[2]{} (890.72,310.60) (898,293)(7.831,-19.222)[2]{} (915.29,253.42) (924.57,234.86) (934.08,216.41) (944.77,198.64) (957.19,182.02) (969.97,165.70) (984.31,150.69) (999.84,136.96) (1001,136)(16.064,-13.143)[0]{} (1016.35,124.46) (1034.04,113.61) (1035,113)(18.564,-9.282)[0]{} (1052.65,104.43) (1058,102)(19.506,-7.093)[0]{} (1072.01,97.00) (1091.88,91.03) (1092,91)(20.136,-5.034)[0]{} (1111.97,85.83) (1115,85)(20.473,-3.412)[0]{} (1132.36,82.03) (1138,81)(20.684,-1.724)[0]{} (1152.94,79.47) (1161,78)(20.684,-1.724)[0]{} (1173.51,76.95) (1194.19,75.07) (1195,75)(20.684,-1.724)[0]{} (1214.86,73.29) (1218,73)(20.756,0.000)[0]{} (1235.58,72.49) (1241,72)(20.684,-1.724)[0]{} (1256.27,71.00) (1264,71)(20.756,0.000)[0]{} (1277.03,70.91) (1297.74,70.00) (1299,70)(20.756,0.000)[0]{} (1318.50,70.00) (1321,70)(20.684,-1.724)[0]{} (1339.21,69.00) (1344,69)(20.756,0.000)[0]{} (1359.96,69.00) (1367,69)(20.756,0.000)[0]{} (1380.72,69.00) (1401.48,69.00) (1402,69)(20.756,0.000)[0]{} (1422.20,68.23) (1425,68)(20.756,0.000)[0]{} (1436,68)
(1500,900)(0,0) =cmr10 at 10pt
(-40,480)[(0,0)\[1\]]{} (800,-100)[(0,0)[$rv$]{}]{} (290,80)[(0,0)[$\cdot$]{}]{} (290,100)[(0,0)[$\cdot$]{}]{} (290,120)[(0,0)[$\cdot$]{}]{} (290,140)[(0,0)[$\cdot$]{}]{} (290,160)[(0,0)[$\cdot$]{}]{} (290,180)[(0,0)[$\cdot$]{}]{} (290,200)[(0,0)[$\cdot$]{}]{} (290,220)[(0,0)[$\cdot$]{}]{} (290,240)[(0,0)[$\cdot$]{}]{} (290,260)[(0,0)[$\cdot$]{}]{} (290,280)[(0,0)[$\cdot$]{}]{} (290,300)[(0,0)[$\cdot$]{}]{} (290,320)[(0,0)[$\cdot$]{}]{} (290,340)[(0,0)[$\cdot$]{}]{} (290,360)[(0,0)[$\cdot$]{}]{} (290,380)[(0,0)[$\cdot$]{}]{} (290,400)[(0,0)[$\cdot$]{}]{} (290,420)[(0,0)[$\cdot$]{}]{} (290,440)[(0,0)[$\cdot$]{}]{} (290,460)[(0,0)[$\cdot$]{}]{} (290,480)[(0,0)[$\cdot$]{}]{} (290,500)[(0,0)[$\cdot$]{}]{} (290,520)[(0,0)[$\cdot$]{}]{} (290,540)[(0,0)[$\cdot$]{}]{} (290,560)[(0,0)[$\cdot$]{}]{} (290,580)[(0,0)[$\cdot$]{}]{} (290,600)[(0,0)[$\cdot$]{}]{} (290,620)[(0,0)[$\cdot$]{}]{} (290,640)[(0,0)[$\cdot$]{}]{} (290,660)[(0,0)[$\cdot$]{}]{} (290,680)[(0,0)[$\cdot$]{}]{} (290,700)[(0,0)[$\cdot$]{}]{} (290,720)[(0,0)[$\cdot$]{}]{} (290,740)[(0,0)[$\cdot$]{}]{} (290,760)[(0,0)[$\cdot$]{}]{} (290,780)[(0,0)[$\cdot$]{}]{} (290,800)[(0,0)[$\cdot$]{}]{} (290,820)[(0,0)[$\cdot$]{}]{} (290,-30)[(0,0)[$R_{m}$]{}]{} (845,80)[(0,0)[$\cdot$]{}]{} (845,100)[(0,0)[$\cdot$]{}]{} (845,120)[(0,0)[$\cdot$]{}]{} (845,140)[(0,0)[$\cdot$]{}]{} (845,160)[(0,0)[$\cdot$]{}]{} (845,180)[(0,0)[$\cdot$]{}]{} (845,200)[(0,0)[$\cdot$]{}]{} (845,220)[(0,0)[$\cdot$]{}]{} (845,240)[(0,0)[$\cdot$]{}]{} (845,260)[(0,0)[$\cdot$]{}]{} (845,280)[(0,0)[$\cdot$]{}]{} (845,300)[(0,0)[$\cdot$]{}]{} (845,320)[(0,0)[$\cdot$]{}]{} (845,340)[(0,0)[$\cdot$]{}]{} (845,360)[(0,0)[$\cdot$]{}]{} (845,380)[(0,0)[$\cdot$]{}]{} (845,400)[(0,0)[$\cdot$]{}]{} (845,420)[(0,0)[$\cdot$]{}]{} (845,440)[(0,0)[$\cdot$]{}]{} (845,460)[(0,0)[$\cdot$]{}]{} (845,480)[(0,0)[$\cdot$]{}]{} (845,500)[(0,0)[$\cdot$]{}]{} (845,520)[(0,0)[$\cdot$]{}]{} (845,540)[(0,0)[$\cdot$]{}]{} (845,560)[(0,0)[$\cdot$]{}]{} (845,580)[(0,0)[$\cdot$]{}]{} (845,600)[(0,0)[$\cdot$]{}]{} (830,-30)[(0,0)[$R_{n=0}$]{}]{} (980,80)[(0,0)[$\cdot$]{}]{} (980,100)[(0,0)[$\cdot$]{}]{} (980,120)[(0,0)[$\cdot$]{}]{} (980,140)[(0,0)[$\cdot$]{}]{} (980,160)[(0,0)[$\cdot$]{}]{} (980,180)[(0,0)[$\cdot$]{}]{} (980,200)[(0,0)[$\cdot$]{}]{} (980,220)[(0,0)[$\cdot$]{}]{} (980,240)[(0,0)[$\cdot$]{}]{} (980,260)[(0,0)[$\cdot$]{}]{} (980,280)[(0,0)[$\cdot$]{}]{} (980,300)[(0,0)[$\cdot$]{}]{} (980,320)[(0,0)[$\cdot$]{}]{} (980,340)[(0,0)[$\cdot$]{}]{} (980,360)[(0,0)[$\cdot$]{}]{} (980,380)[(0,0)[$\cdot$]{}]{} (980,400)[(0,0)[$\cdot$]{}]{} (980,420)[(0,0)[$\cdot$]{}]{} (980,440)[(0,0)[$\cdot$]{}]{} (980,460)[(0,0)[$\cdot$]{}]{} (980,480)[(0,0)[$\cdot$]{}]{} (980,500)[(0,0)[$\cdot$]{}]{} (980,520)[(0,0)[$\cdot$]{}]{} (980,540)[(0,0)[$\cdot$]{}]{} (980,560)[(0,0)[$\cdot$]{}]{} (980,580)[(0,0)[$\cdot$]{}]{} (980,600)[(0,0)[$\cdot$]{}]{} (995,-30)[(0,0)[$R_{n=1}$]{}]{} (800,-400)[(0,0)[Figure 2]{}]{}
(176.0,289.0)
------------------------------------------------------------------------
(176.0,68.0)
------------------------------------------------------------------------
(176.0,68.0)
------------------------------------------------------------------------
(154,68)[(0,0)\[r\][-0.15]{}]{} (1416.0,68.0)
------------------------------------------------------------------------
(176.0,142.0)
------------------------------------------------------------------------
(154,142)[(0,0)\[r\][-0.1]{}]{} (1416.0,142.0)
------------------------------------------------------------------------
(176.0,215.0)
------------------------------------------------------------------------
(154,215)[(0,0)\[r\][-0.05]{}]{} (1416.0,215.0)
------------------------------------------------------------------------
(176.0,289.0)
------------------------------------------------------------------------
(154,289)[(0,0)\[r\][0]{}]{} (1416.0,289.0)
------------------------------------------------------------------------
(176.0,362.0)
------------------------------------------------------------------------
(154,362)[(0,0)\[r\][0.05]{}]{} (1416.0,362.0)
------------------------------------------------------------------------
(176.0,436.0)
------------------------------------------------------------------------
(154,436)[(0,0)\[r\][0.1]{}]{} (1416.0,436.0)
------------------------------------------------------------------------
(176.0,509.0)
------------------------------------------------------------------------
(154,509)[(0,0)\[r\][0.15]{}]{} (1416.0,509.0)
------------------------------------------------------------------------
(176.0,583.0)
------------------------------------------------------------------------
(154,583)[(0,0)\[r\][0.2]{}]{} (1416.0,583.0)
------------------------------------------------------------------------
(176.0,656.0)
------------------------------------------------------------------------
(154,656)[(0,0)\[r\][0.25]{}]{} (1416.0,656.0)
------------------------------------------------------------------------
(176.0,730.0)
------------------------------------------------------------------------
(154,730)[(0,0)\[r\][0.3]{}]{} (1416.0,730.0)
------------------------------------------------------------------------
(176.0,803.0)
------------------------------------------------------------------------
(154,803)[(0,0)\[r\][0.35]{}]{} (1416.0,803.0)
------------------------------------------------------------------------
(176.0,877.0)
------------------------------------------------------------------------
(154,877)[(0,0)\[r\][0.4]{}]{} (1416.0,877.0)
------------------------------------------------------------------------
(176.0,68.0)
------------------------------------------------------------------------
(176,23)[(0,0)[0]{}]{} (176.0,857.0)
------------------------------------------------------------------------
(405.0,68.0)
------------------------------------------------------------------------
(405,23)[(0,0)[2]{}]{} (405.0,857.0)
------------------------------------------------------------------------
(634.0,68.0)
------------------------------------------------------------------------
(634,23)[(0,0)[4]{}]{} (634.0,857.0)
------------------------------------------------------------------------
(863.0,68.0)
------------------------------------------------------------------------
(863,23)[(0,0)[6]{}]{} (863.0,857.0)
------------------------------------------------------------------------
(1092.0,68.0)
------------------------------------------------------------------------
(1092,23)[(0,0)[8]{}]{} (1092.0,857.0)
------------------------------------------------------------------------
(1321.0,68.0)
------------------------------------------------------------------------
(1321,23)[(0,0)[10]{}]{} (1321.0,857.0)
------------------------------------------------------------------------
(176.0,68.0)
------------------------------------------------------------------------
(1436.0,68.0)
------------------------------------------------------------------------
(176.0,877.0)
------------------------------------------------------------------------
(176.0,68.0)
------------------------------------------------------------------------
(1306,812)[(0,0)\[r\][$n=1$]{}]{} (1328.0,812.0)
------------------------------------------------------------------------
(176,604) (176.00,604.60)(1.505,0.468)[5]{}
------------------------------------------------------------------------
(176.00,603.17)(8.509,4.000)[2]{}
------------------------------------------------------------------------
(187.58,608.00)(0.492,0.539)[21]{}
------------------------------------------------------------------------
(186.17,608.00)(12.000,11.893)[2]{}
------------------------------------------------------------------------
(199.58,621.00)(0.492,0.967)[19]{}
------------------------------------------------------------------------
(198.17,621.00)(11.000,19.207)[2]{}
------------------------------------------------------------------------
(210.58,642.00)(0.492,1.186)[21]{}
------------------------------------------------------------------------
(209.17,642.00)(12.000,25.855)[2]{}
------------------------------------------------------------------------
(222.58,670.00)(0.492,1.534)[19]{}
------------------------------------------------------------------------
(221.17,670.00)(11.000,30.302)[2]{}
------------------------------------------------------------------------
(233.58,703.00)(0.492,1.530)[21]{}
------------------------------------------------------------------------
(232.17,703.00)(12.000,33.302)[2]{}
------------------------------------------------------------------------
(245.58,739.00)(0.492,1.628)[19]{}
------------------------------------------------------------------------
(244.17,739.00)(11.000,32.151)[2]{}
------------------------------------------------------------------------
(256.58,774.00)(0.492,1.272)[21]{}
------------------------------------------------------------------------
(255.17,774.00)(12.000,27.717)[2]{}
------------------------------------------------------------------------
(268.58,804.00)(0.492,0.873)[19]{}
------------------------------------------------------------------------
(267.17,804.00)(11.000,17.358)[2]{}
------------------------------------------------------------------------
(279.00,823.60)(1.651,0.468)[5]{}
------------------------------------------------------------------------
(279.00,822.17)(9.302,4.000)[2]{}
------------------------------------------------------------------------
(291.58,824.32)(0.492,-0.684)[19]{}
------------------------------------------------------------------------
(290.17,825.66)(11.000,-13.660)[2]{}
------------------------------------------------------------------------
(302.58,806.60)(0.492,-1.534)[19]{}
------------------------------------------------------------------------
(301.17,809.30)(11.000,-30.302)[2]{}
------------------------------------------------------------------------
(313.58,771.67)(0.492,-2.133)[21]{}
------------------------------------------------------------------------
(312.17,775.33)(12.000,-46.333)[2]{}
------------------------------------------------------------------------
(325.58,719.38)(0.492,-2.854)[19]{}
------------------------------------------------------------------------
(324.17,724.19)(11.000,-56.188)[2]{}
------------------------------------------------------------------------
(336.58,658.31)(0.492,-2.866)[21]{}
------------------------------------------------------------------------
(335.17,663.16)(12.000,-62.157)[2]{}
------------------------------------------------------------------------
(348.58,590.62)(0.492,-3.090)[19]{}
------------------------------------------------------------------------
(347.17,595.81)(11.000,-60.811)[2]{}
------------------------------------------------------------------------
(359.58,526.14)(0.492,-2.607)[21]{}
------------------------------------------------------------------------
(358.17,530.57)(12.000,-56.572)[2]{}
------------------------------------------------------------------------
(371.58,465.58)(0.492,-2.477)[19]{}
------------------------------------------------------------------------
(370.17,469.79)(11.000,-48.792)[2]{}
------------------------------------------------------------------------
(382.58,414.36)(0.492,-1.918)[21]{}
------------------------------------------------------------------------
(381.17,417.68)(12.000,-41.679)[2]{}
------------------------------------------------------------------------
(394.58,370.00)(0.492,-1.722)[19]{}
------------------------------------------------------------------------
(393.17,373.00)(11.000,-34.000)[2]{}
------------------------------------------------------------------------
(405.58,334.43)(0.492,-1.272)[21]{}
------------------------------------------------------------------------
(404.17,336.72)(12.000,-27.717)[2]{}
------------------------------------------------------------------------
(417.58,304.96)(0.492,-1.109)[19]{}
------------------------------------------------------------------------
(416.17,306.98)(11.000,-21.981)[2]{}
------------------------------------------------------------------------
(428.58,281.72)(0.492,-0.873)[19]{}
------------------------------------------------------------------------
(427.17,283.36)(11.000,-17.358)[2]{}
------------------------------------------------------------------------
(439.58,263.37)(0.492,-0.669)[21]{}
------------------------------------------------------------------------
(438.17,264.69)(12.000,-14.685)[2]{}
------------------------------------------------------------------------
(451.58,247.62)(0.492,-0.590)[19]{}
------------------------------------------------------------------------
(450.17,248.81)(11.000,-11.811)[2]{}
------------------------------------------------------------------------
(462.00,235.92)(0.543,-0.492)[19]{}
------------------------------------------------------------------------
(462.00,236.17)(10.887,-11.000)[2]{}
------------------------------------------------------------------------
(474.00,224.93)(0.611,-0.489)[15]{}
------------------------------------------------------------------------
(474.00,225.17)(9.778,-9.000)[2]{}
------------------------------------------------------------------------
(485.00,215.93)(0.669,-0.489)[15]{}
------------------------------------------------------------------------
(485.00,216.17)(10.685,-9.000)[2]{}
------------------------------------------------------------------------
(497.00,206.93)(0.798,-0.485)[11]{}
------------------------------------------------------------------------
(497.00,207.17)(9.488,-7.000)[2]{}
------------------------------------------------------------------------
(508.00,199.93)(1.033,-0.482)[9]{}
------------------------------------------------------------------------
(508.00,200.17)(10.132,-6.000)[2]{}
------------------------------------------------------------------------
(520.00,193.93)(1.155,-0.477)[7]{}
------------------------------------------------------------------------
(520.00,194.17)(8.966,-5.000)[2]{}
------------------------------------------------------------------------
(531.00,188.93)(1.267,-0.477)[7]{}
------------------------------------------------------------------------
(531.00,189.17)(9.800,-5.000)[2]{}
------------------------------------------------------------------------
(543.00,183.93)(1.155,-0.477)[7]{}
------------------------------------------------------------------------
(543.00,184.17)(8.966,-5.000)[2]{}
------------------------------------------------------------------------
(554.00,178.94)(1.505,-0.468)[5]{}
------------------------------------------------------------------------
(554.00,179.17)(8.509,-4.000)[2]{}
------------------------------------------------------------------------
(565.00,174.95)(2.472,-0.447)[3]{}
------------------------------------------------------------------------
(565.00,175.17)(8.472,-3.000)[2]{}
------------------------------------------------------------------------
(577.00,171.94)(1.505,-0.468)[5]{}
------------------------------------------------------------------------
(577.00,172.17)(8.509,-4.000)[2]{}
------------------------------------------------------------------------
(588.00,167.95)(2.472,-0.447)[3]{}
------------------------------------------------------------------------
(588.00,168.17)(8.472,-3.000)[2]{}
------------------------------------------------------------------------
(600.00,164.95)(2.248,-0.447)[3]{}
------------------------------------------------------------------------
(600.00,165.17)(7.748,-3.000)[2]{}
------------------------------------------------------------------------
(611,161.17)
------------------------------------------------------------------------
(611.00,162.17)(6.811,-2.000)[2]{}
------------------------------------------------------------------------
(623.00,159.95)(2.248,-0.447)[3]{}
------------------------------------------------------------------------
(623.00,160.17)(7.748,-3.000)[2]{}
------------------------------------------------------------------------
(634,156.17)
------------------------------------------------------------------------
(634.00,157.17)(6.811,-2.000)[2]{}
------------------------------------------------------------------------
(646,154.17)
------------------------------------------------------------------------
(646.00,155.17)(6.226,-2.000)[2]{}
------------------------------------------------------------------------
(657,152.17)
------------------------------------------------------------------------
(657.00,153.17)(6.811,-2.000)[2]{}
------------------------------------------------------------------------
(669,150.17)
------------------------------------------------------------------------
(669.00,151.17)(6.226,-2.000)[2]{}
------------------------------------------------------------------------
(680,148.67)
------------------------------------------------------------------------
(680.00,149.17)(5.500,-1.000)[2]{}
------------------------------------------------------------------------
(691,147.17)
------------------------------------------------------------------------
(691.00,148.17)(6.811,-2.000)[2]{}
------------------------------------------------------------------------
(703,145.67)
------------------------------------------------------------------------
(703.00,146.17)(5.500,-1.000)[2]{}
------------------------------------------------------------------------
(714,144.17)
------------------------------------------------------------------------
(714.00,145.17)(6.811,-2.000)[2]{}
------------------------------------------------------------------------
(726,142.67)
------------------------------------------------------------------------
(726.00,143.17)(5.500,-1.000)[2]{}
------------------------------------------------------------------------
(737,141.67)
------------------------------------------------------------------------
(737.00,142.17)(6.000,-1.000)[2]{}
------------------------------------------------------------------------
(749,140.67)
------------------------------------------------------------------------
(749.00,141.17)(5.500,-1.000)[2]{}
------------------------------------------------------------------------
(772,139.67)
------------------------------------------------------------------------
(772.00,140.17)(5.500,-1.000)[2]{}
------------------------------------------------------------------------
(760.0,141.0)
------------------------------------------------------------------------
(795,139.67)
------------------------------------------------------------------------
(795.00,139.17)(5.500,1.000)[2]{}
------------------------------------------------------------------------
(806,141.17)
------------------------------------------------------------------------
(806.00,140.17)(6.226,2.000)[2]{}
------------------------------------------------------------------------
(817.00,143.60)(1.651,0.468)[5]{}
------------------------------------------------------------------------
(817.00,142.17)(9.302,4.000)[2]{}
------------------------------------------------------------------------
(829.00,147.59)(0.798,0.485)[11]{}
------------------------------------------------------------------------
(829.00,146.17)(9.488,7.000)[2]{}
------------------------------------------------------------------------
(840.00,154.58)(0.600,0.491)[17]{}
------------------------------------------------------------------------
(840.00,153.17)(10.796,10.000)[2]{}
------------------------------------------------------------------------
(852.58,164.00)(0.492,0.779)[19]{}
------------------------------------------------------------------------
(851.17,164.00)(11.000,15.509)[2]{}
------------------------------------------------------------------------
(863.58,181.00)(0.492,1.056)[21]{}
------------------------------------------------------------------------
(862.17,181.00)(12.000,23.063)[2]{}
------------------------------------------------------------------------
(875.58,206.00)(0.492,1.581)[19]{}
------------------------------------------------------------------------
(874.17,206.00)(11.000,31.226)[2]{}
------------------------------------------------------------------------
(886.58,240.00)(0.492,1.961)[21]{}
------------------------------------------------------------------------
(885.17,240.00)(12.000,42.610)[2]{}
------------------------------------------------------------------------
(898.58,286.00)(0.492,2.713)[19]{}
------------------------------------------------------------------------
(897.17,286.00)(11.000,53.415)[2]{}
------------------------------------------------------------------------
(909.58,344.00)(0.492,2.780)[21]{}
------------------------------------------------------------------------
(908.17,344.00)(12.000,60.295)[2]{}
------------------------------------------------------------------------
(921.58,409.00)(0.492,3.137)[19]{}
------------------------------------------------------------------------
(920.17,409.00)(11.000,61.736)[2]{}
------------------------------------------------------------------------
(932.58,476.00)(0.492,2.901)[19]{}
------------------------------------------------------------------------
(931.17,476.00)(11.000,57.113)[2]{}
------------------------------------------------------------------------
(943.58,538.00)(0.492,2.047)[21]{}
------------------------------------------------------------------------
(942.17,538.00)(12.000,44.472)[2]{}
------------------------------------------------------------------------
(955.58,586.00)(0.492,1.345)[19]{}
------------------------------------------------------------------------
(954.17,586.00)(11.000,26.604)[2]{}
------------------------------------------------------------------------
(966.00,615.59)(0.758,0.488)[13]{}
------------------------------------------------------------------------
(966.00,614.17)(10.547,8.000)[2]{}
------------------------------------------------------------------------
(978.00,621.92)(0.496,-0.492)[19]{}
------------------------------------------------------------------------
(978.00,622.17)(9.962,-11.000)[2]{}
------------------------------------------------------------------------
(989.58,607.99)(0.492,-1.099)[21]{}
------------------------------------------------------------------------
(988.17,609.99)(12.000,-23.994)[2]{}
------------------------------------------------------------------------
(1001.58,580.45)(0.492,-1.581)[19]{}
------------------------------------------------------------------------
(1000.17,583.23)(11.000,-31.226)[2]{}
------------------------------------------------------------------------
(1012.58,546.33)(0.492,-1.616)[21]{}
------------------------------------------------------------------------
(1011.17,549.16)(12.000,-35.163)[2]{}
------------------------------------------------------------------------
(1024.58,507.85)(0.492,-1.769)[19]{}
------------------------------------------------------------------------
(1023.17,510.92)(11.000,-34.924)[2]{}
------------------------------------------------------------------------
(1035.58,470.88)(0.492,-1.444)[21]{}
------------------------------------------------------------------------
(1034.17,473.44)(12.000,-31.440)[2]{}
------------------------------------------------------------------------
(1047.58,437.06)(0.492,-1.392)[19]{}
------------------------------------------------------------------------
(1046.17,439.53)(11.000,-27.528)[2]{}
------------------------------------------------------------------------
(1058.58,407.66)(0.492,-1.203)[19]{}
------------------------------------------------------------------------
(1057.17,409.83)(11.000,-23.830)[2]{}
------------------------------------------------------------------------
(1069.58,382.68)(0.492,-0.884)[21]{}
------------------------------------------------------------------------
(1068.17,384.34)(12.000,-19.340)[2]{}
------------------------------------------------------------------------
(1081.58,362.02)(0.492,-0.779)[19]{}
------------------------------------------------------------------------
(1080.17,363.51)(11.000,-15.509)[2]{}
------------------------------------------------------------------------
(1092.58,345.79)(0.492,-0.539)[21]{}
------------------------------------------------------------------------
(1091.17,346.89)(12.000,-11.893)[2]{}
------------------------------------------------------------------------
(1104.00,333.92)(0.496,-0.492)[19]{}
------------------------------------------------------------------------
(1104.00,334.17)(9.962,-11.000)[2]{}
------------------------------------------------------------------------
(1115.00,322.93)(0.758,-0.488)[13]{}
------------------------------------------------------------------------
(1115.00,323.17)(10.547,-8.000)[2]{}
------------------------------------------------------------------------
(1127.00,314.93)(0.943,-0.482)[9]{}
------------------------------------------------------------------------
(1127.00,315.17)(9.270,-6.000)[2]{}
------------------------------------------------------------------------
(1138.00,308.93)(1.267,-0.477)[7]{}
------------------------------------------------------------------------
(1138.00,309.17)(9.800,-5.000)[2]{}
------------------------------------------------------------------------
(1150.00,303.94)(1.505,-0.468)[5]{}
------------------------------------------------------------------------
(1150.00,304.17)(8.509,-4.000)[2]{}
------------------------------------------------------------------------
(1161.00,299.95)(2.472,-0.447)[3]{}
------------------------------------------------------------------------
(1161.00,300.17)(8.472,-3.000)[2]{}
------------------------------------------------------------------------
(1173,296.17)
------------------------------------------------------------------------
(1173.00,297.17)(6.226,-2.000)[2]{}
------------------------------------------------------------------------
(1184,294.17)
------------------------------------------------------------------------
(1184.00,295.17)(6.226,-2.000)[2]{}
------------------------------------------------------------------------
(1195,292.67)
------------------------------------------------------------------------
(1195.00,293.17)(6.000,-1.000)[2]{}
------------------------------------------------------------------------
(1207,291.67)
------------------------------------------------------------------------
(1207.00,292.17)(5.500,-1.000)[2]{}
------------------------------------------------------------------------
(1218,290.67)
------------------------------------------------------------------------
(1218.00,291.17)(6.000,-1.000)[2]{}
------------------------------------------------------------------------
(1230,289.67)
------------------------------------------------------------------------
(1230.00,290.17)(5.500,-1.000)[2]{}
------------------------------------------------------------------------
(783.0,140.0)
------------------------------------------------------------------------
(1264,288.67)
------------------------------------------------------------------------
(1264.00,289.17)(6.000,-1.000)[2]{}
------------------------------------------------------------------------
(1241.0,290.0)
------------------------------------------------------------------------
(1276.0,289.0)
------------------------------------------------------------------------
(1306,767)[(0,0)\[r\][$n=0$]{}]{} (1328,767)(20.756,0.000)[4]{} (1394,767) (176,112) (176.00,112.00) (196.76,112.00) (199,112)(20.756,0.000)[0]{} (217.51,112.00) (222,112)(20.756,0.000)[0]{} (238.27,112.00) (245,112)(20.756,0.000)[0]{} (259.02,112.00) (268,112)(20.756,0.000)[0]{} (279.78,112.00) (300.53,112.00) (302,112)(20.756,0.000)[0]{} (321.29,112.00) (325,112)(20.756,0.000)[0]{} (342.04,112.00) (348,112)(20.756,0.000)[0]{} (362.80,112.00) (371,112)(20.756,0.000)[0]{} (383.55,112.00) (404.31,112.00) (405,112)(20.756,0.000)[0]{} (425.07,112.00) (428,112)(20.756,0.000)[0]{} (445.82,112.00) (451,112)(20.756,0.000)[0]{} (466.58,112.00) (474,112)(20.756,0.000)[0]{} (487.33,112.00) (497,112)(20.756,0.000)[0]{} (508.09,112.00) (528.84,112.00) (531,112)(20.756,0.000)[0]{} (549.60,112.00) (554,112)(20.756,0.000)[0]{} (570.35,112.00) (577,112)(20.756,0.000)[0]{} (591.11,112.00) (600,112)(20.756,0.000)[0]{} (611.86,112.07) (632.58,113.00) (634,113)(20.756,0.000)[0]{} (653.30,113.66) (657,114)(20.473,3.412)[0]{} (673.71,117.28) (680,119)(19.506,7.093)[0]{} (693.18,124.27) (709.64,136.64) (722.41,152.92) (732.06,171.22) (737,182)(6.563,19.690)[2]{} (749,218)(4.730,20.209)[2]{} (760,265)(4.137,20.339)[3]{} (772,324)(3.314,20.489)[3]{} (783,392)(3.459,20.465)[4]{} (795,463)(3.412,20.473)[3]{} (806,529)(4.218,20.322)[3]{} (822.49,596.65) (830.57,615.57) (845.70,620.25) (857.06,603.50) (863,590)(6.732,-19.634)[2]{} (875,555)(5.771,-19.937)[2]{} (886,517)(6.250,-19.792)[2]{} (898,479)(6.065,-19.850)[2]{} (915.68,426.29) (921,413)(8.087,-19.115)[2]{} (941.15,369.53) (952.30,352.05) (965.26,335.87) (966,335)(15.300,-14.025)[0]{} (980.70,322.04) (998.38,311.31) (1001,310)(18.895,-8.589)[0]{} (1017.45,303.18) (1024,301)(20.024,-5.461)[0]{} (1037.42,297.60) (1057.86,294.02) (1058,294)(20.670,-1.879)[0]{} (1078.54,292.21) (1081,292)(20.670,-1.879)[0]{} (1099.21,290.40) (1104,290)(20.756,0.000)[0]{} (1119.95,290.00) (1127,290)(20.670,-1.879)[0]{} (1140.66,289.00) (1150,289)(20.756,0.000)[0]{} (1161.42,289.00) (1182.17,289.00) (1184,289)(20.756,0.000)[0]{} (1202.93,289.00) (1207,289)(20.756,0.000)[0]{} (1223.68,289.00) (1230,289)(20.756,0.000)[0]{} (1244.44,289.00) (1253,289)(20.756,0.000)[0]{} (1265.20,289.00) (1285.95,289.00) (1287,289)(20.756,0.000)[0]{} (1306.71,289.00) (1310,289)(20.756,0.000)[0]{} (1327.46,289.00) (1333,289)(20.756,0.000)[0]{} (1348.22,289.00) (1356,289)(20.756,0.000)[0]{} (1368.97,289.00) (1389.73,289.00) (1390,289)(20.756,0.000)[0]{} (1410.48,289.00) (1413,289)(20.756,0.000)[0]{} (1431.24,289.00) (1436,289)
(1500,900)(0,0) =cmr10 at 10pt
(0,500)[(0,0)\[1\]]{} (0,420)[(0,0)\[1\]]{} (800,-40)[(0,0)[$\alpha$]{}]{} (800,-400)[(0,0)[Figure 3]{}]{}
(176.0,68.0)
------------------------------------------------------------------------
(176.0,68.0)
------------------------------------------------------------------------
(154,68)[(0,0)\[r\][0]{}]{} (1416.0,68.0)
------------------------------------------------------------------------
(176.0,158.0)
------------------------------------------------------------------------
(154,158)[(0,0)\[r\][10]{}]{} (1416.0,158.0)
------------------------------------------------------------------------
(176.0,248.0)
------------------------------------------------------------------------
(154,248)[(0,0)\[r\][20]{}]{} (1416.0,248.0)
------------------------------------------------------------------------
(176.0,338.0)
------------------------------------------------------------------------
(154,338)[(0,0)\[r\][30]{}]{} (1416.0,338.0)
------------------------------------------------------------------------
(176.0,428.0)
------------------------------------------------------------------------
(154,428)[(0,0)\[r\][40]{}]{} (1416.0,428.0)
------------------------------------------------------------------------
(176.0,517.0)
------------------------------------------------------------------------
(154,517)[(0,0)\[r\][50]{}]{} (1416.0,517.0)
------------------------------------------------------------------------
(176.0,607.0)
------------------------------------------------------------------------
(154,607)[(0,0)\[r\][60]{}]{} (1416.0,607.0)
------------------------------------------------------------------------
(176.0,697.0)
------------------------------------------------------------------------
(154,697)[(0,0)\[r\][70]{}]{} (1416.0,697.0)
------------------------------------------------------------------------
(176.0,787.0)
------------------------------------------------------------------------
(154,787)[(0,0)\[r\][80]{}]{} (1416.0,787.0)
------------------------------------------------------------------------
(176.0,877.0)
------------------------------------------------------------------------
(154,877)[(0,0)\[r\][90]{}]{} (1416.0,877.0)
------------------------------------------------------------------------
(176.0,68.0)
------------------------------------------------------------------------
(176,23)[(0,0)[0.1]{}]{} (176.0,857.0)
------------------------------------------------------------------------
(334.0,68.0)
------------------------------------------------------------------------
(334,23)[(0,0)[0.15]{}]{} (334.0,857.0)
------------------------------------------------------------------------
(491.0,68.0)
------------------------------------------------------------------------
(491,23)[(0,0)[0.2]{}]{} (491.0,857.0)
------------------------------------------------------------------------
(648.0,68.0)
------------------------------------------------------------------------
(648,23)[(0,0)[0.25]{}]{} (648.0,857.0)
------------------------------------------------------------------------
(806.0,68.0)
------------------------------------------------------------------------
(806,23)[(0,0)[0.3]{}]{} (806.0,857.0)
------------------------------------------------------------------------
(963.0,68.0)
------------------------------------------------------------------------
(963,23)[(0,0)[0.35]{}]{} (963.0,857.0)
------------------------------------------------------------------------
(1121.0,68.0)
------------------------------------------------------------------------
(1121,23)[(0,0)[0.4]{}]{} (1121.0,857.0)
------------------------------------------------------------------------
(1278.0,68.0)
------------------------------------------------------------------------
(1278,23)[(0,0)[0.45]{}]{} (1278.0,857.0)
------------------------------------------------------------------------
(1436.0,68.0)
------------------------------------------------------------------------
(1436,23)[(0,0)[0.5]{}]{} (1436.0,857.0)
------------------------------------------------------------------------
(176.0,68.0)
------------------------------------------------------------------------
(1436.0,68.0)
------------------------------------------------------------------------
(176.0,877.0)
------------------------------------------------------------------------
(176.0,68.0)
------------------------------------------------------------------------
(1306,812)[(0,0)\[r\][$n=0$]{}]{} (1328,812)(20.756,0.000)[4]{} (1394,812) (239,518) (239,518)(8.827,-18.785)[5]{} (278,435)(11.125,-17.522)[4]{} (318,372)(12.925,-16.240)[3]{} (357,323)(15.237,-14.094)[2]{} (397,286)(16.156,-13.029)[2]{} (428,261)(17.184,-11.641)[2]{} (459,240)(18.090,-10.176)[2]{} (506.71,213.89) (522,206)(19.229,-7.812)[2]{} (554,193)(19.434,-7.288)[2]{} (602.99,175.52) (617,171)(19.932,-5.787)[2]{} (662.70,157.87) (680,153)(20.276,-4.435)[2]{} (723.33,143.81) (743,140)(20.377,-3.944)[2]{} (774,134)(20.400,-3.825)[2]{} (825.46,125.57) (838,124)(20.491,-3.305)[2]{} (887.08,116.74) (901,115)(20.585,-2.656)[2]{} (948.91,109.36) (963,108)(20.665,-1.937)[2]{} (1010.90,103.51) (1027,102)(20.659,-1.999)[2]{} (1072.88,97.56) (1089,96)(20.715,-1.295)[2]{} (1134.98,93.10) (1152,92)(20.692,-1.617)[3]{} (1216,87)(20.712,-1.336)[2]{} (1259.19,84.21) (1278,83)(20.715,-1.295)[2]{} (1321.33,80.27) (1341,79)(20.715,-1.295)[2]{} (1373,77) (1306,767)[(0,0)\[r\][$n=1$]{}]{} (1328.0,767.0)
------------------------------------------------------------------------
(239,818) (239.58,812.55)(0.497,-1.527)[59]{}
------------------------------------------------------------------------
(238.17,815.28)(31.000,-91.275)[2]{}
------------------------------------------------------------------------
(270.58,719.59)(0.497,-1.210)[61]{}
------------------------------------------------------------------------
(269.17,721.79)(32.000,-74.795)[2]{}
------------------------------------------------------------------------
(302.58,643.32)(0.497,-0.989)[61]{}
------------------------------------------------------------------------
(301.17,645.16)(32.000,-61.158)[2]{}
------------------------------------------------------------------------
(334.58,580.80)(0.497,-0.841)[59]{}
------------------------------------------------------------------------
(333.17,582.40)(31.000,-50.400)[2]{}
------------------------------------------------------------------------
(365.58,529.25)(0.497,-0.704)[61]{}
------------------------------------------------------------------------
(364.17,530.62)(32.000,-43.625)[2]{}
------------------------------------------------------------------------
(397.58,484.55)(0.497,-0.613)[59]{}
------------------------------------------------------------------------
(396.17,485.77)(31.000,-36.775)[2]{}
------------------------------------------------------------------------
(428.58,446.82)(0.497,-0.531)[59]{}
------------------------------------------------------------------------
(427.17,447.91)(31.000,-31.909)[2]{}
------------------------------------------------------------------------
(459.00,414.92)(0.551,-0.497)[55]{}
------------------------------------------------------------------------
(459.00,415.17)(30.876,-29.000)[2]{}
------------------------------------------------------------------------
(491.00,385.92)(0.620,-0.497)[47]{}
------------------------------------------------------------------------
(491.00,386.17)(29.763,-25.000)[2]{}
------------------------------------------------------------------------
(522.00,360.92)(0.697,-0.496)[43]{}
------------------------------------------------------------------------
(522.00,361.17)(30.637,-23.000)[2]{}
------------------------------------------------------------------------
(554.00,337.92)(0.804,-0.496)[37]{}
------------------------------------------------------------------------
(554.00,338.17)(30.464,-20.000)[2]{}
------------------------------------------------------------------------
(586.00,317.92)(0.866,-0.495)[33]{}
------------------------------------------------------------------------
(586.00,318.17)(29.363,-18.000)[2]{}
------------------------------------------------------------------------
(617.00,299.92)(0.977,-0.494)[29]{}
------------------------------------------------------------------------
(617.00,300.17)(29.184,-16.000)[2]{}
------------------------------------------------------------------------
(648.00,283.92)(1.079,-0.494)[27]{}
------------------------------------------------------------------------
(648.00,284.17)(30.021,-15.000)[2]{}
------------------------------------------------------------------------
(680.00,268.92)(1.250,-0.493)[23]{}
------------------------------------------------------------------------
(680.00,269.17)(29.749,-13.000)[2]{}
------------------------------------------------------------------------
(712.00,255.92)(1.315,-0.492)[21]{}
------------------------------------------------------------------------
(712.00,256.17)(28.648,-12.000)[2]{}
------------------------------------------------------------------------
(743.00,243.92)(1.315,-0.492)[21]{}
------------------------------------------------------------------------
(743.00,244.17)(28.648,-12.000)[2]{}
------------------------------------------------------------------------
(774.00,231.92)(1.642,-0.491)[17]{}
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(774.00,232.17)(29.136,-10.000)[2]{}
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(806.00,221.92)(1.642,-0.491)[17]{}
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(806.00,222.17)(29.136,-10.000)[2]{}
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(838.00,211.93)(1.776,-0.489)[15]{}
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(838.00,212.17)(27.933,-9.000)[2]{}
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(869.00,202.93)(2.079,-0.488)[13]{}
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(869.00,203.17)(28.472,-8.000)[2]{}
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(901.00,194.93)(2.013,-0.488)[13]{}
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(901.00,195.17)(27.575,-8.000)[2]{}
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(932.00,186.93)(2.013,-0.488)[13]{}
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(932.00,187.17)(27.575,-8.000)[2]{}
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(963.00,178.93)(2.399,-0.485)[11]{}
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(963.00,179.17)(27.997,-7.000)[2]{}
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(995.00,171.93)(2.399,-0.485)[11]{}
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(995.00,172.17)(27.997,-7.000)[2]{}
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(1027.00,164.93)(2.323,-0.485)[11]{}
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(1027.00,165.17)(27.116,-7.000)[2]{}
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(1058.00,157.93)(2.751,-0.482)[9]{}
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(1058.00,158.17)(26.503,-6.000)[2]{}
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(1089.00,151.93)(2.841,-0.482)[9]{}
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(1089.00,152.17)(27.365,-6.000)[2]{}
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(1121.00,145.93)(2.751,-0.482)[9]{}
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(1121.00,146.17)(26.503,-6.000)[2]{}
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(1152.00,139.93)(2.841,-0.482)[9]{}
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(1184.00,133.93)(2.841,-0.482)[9]{}
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(1184.00,134.17)(27.365,-6.000)[2]{}
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(1216.00,127.93)(2.751,-0.482)[9]{}
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(1247.00,121.93)(2.751,-0.482)[9]{}
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(1247.00,122.17)(26.503,-6.000)[2]{}
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(1278.00,115.93)(2.841,-0.482)[9]{}
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|
---
author:
- 'Suhail G. Masda, Mashhoor A. Al-Wardat, Ralph Neuhäuser, Hamid M. Al-Naimiy'
title: 'Physical and Geometrical Parameters of CVBS X: The Spectroscopic Binary Gliese 762.1'
---
INTRODUCTION
=============
The importance of the study of binary stars arises from the fact that more than 50% among nearby solar-type main-sequence stars are binary or multiple stellar systems (the fraction is 42% among nearby M stars ) and several astronomical phenomena occur only in binary stars. They provide a source of direct measurements of stellar parameters or galactic quantities. Stellar physics needs masses, luminosities and radii obtained through the studies of binary stars. Galactic physics benefits also from these studies, e.g. the galactic potential can be tested using wide binaries, and the chemical evolution depends on binaries through the supernovae Ia process [@2002EAS.....2..155A]. The mass-luminosity relation of low-mass main-sequence stars in the solar neighborhood are known with much lower accuracy than those of massive early-type stars , this requires precise determination of their masses and luminosities.
The close visual binary stars (CVBS) are visually close enough to be resolved except by special techniques like speckle interferometry or by deducing their duplicity using high resolution spectroscopy. So, the case is a bit complicated and needs indirect methods to estimate their physical and geometrical parameters.
Combining observational measurements with stellar theoretical models is the most powerful indirect method to analyze such binary and multiple systems. This was implemented in Al-Wardat’s complex method [@2007AN....328...63A], which combines magnitude difference measurements of speckle interferometry, entire spectral energy distribution (SED) of spectrophotometry and radial velocity measurements, all along with atmospheres modeling to estimate the individual physical parameters. In coordination with these physical parameters, the geometrical parameters and their errors are calculated using a modern version of Tokovinin’s ORBITX program, which depends on the standard least-squares method [@1992ASPC...32..573T].
The method was firstly introduced by [@2002BSAO...53...51A; @2007AN....328...63A](henceforth paper I and paper II in this series respectively), where it was applied to the analysis of the quadruple hierarchical system ADS11061 and the two binary systems COU1289 and COU1291 . Later on, the method was successfully applied to several solar-type and subgiant binary systems: Hip11352, Hip11253 and Hip689 [@2009AN....330..385A; @2009AstBu..64..365A; @2012PASA...29..523A] (henceforth papers III, IV and V in this series respectively).
The method was then developed to a complex one by combining the physical solution with the geometrical one represented by the orbital solution of the system and applied to the systems: HD25811, HD375, Gliese 150.5 and HD6009 [@2014AstBu..69...58A; @2014AstBu..69..198A; @2014PASA...31....5A; @2014AstBu..69..454A] (henceforth papers VI, VII, VIII and IX in this series respectively).
In order to be analyzed using Al-Wardat’s method, the binary system should have a magnitude difference measurement, an observational entire SED covering the optical range, and a precise entire optical (UBV) photometrical magnitude measurement. The procedure starts with calculating the individual flux of each component using the magnitude difference with the entire photometrical magnitude, then estimating their preliminary effective temperatures and gravity accelerations in order to build their SED using grids of [@1994KurCD..19.....K] blanketed models (ATLAS9). These two models in their turn are used to build a synthetic entire SED for the system, which compares with the observational one in an iterative way until the best fit is achieved. Of course there should be a coincidence between the masses calculated using the physical solution and those calculated using the orbital one, otherwise a new set of parameters would be tested.
This is the tenth paper in this series, which gives a complete analysis of the CVBS Gliese 762.1 using Al-Wardat’s complex method.
Gliese 762.1 (MCA 56 AB = WDS J19311+5835 = GJ 762.1 = HD 184467 = HIP 95995) was first visually resolved by [@1983ApJS...51..309M]. It is a well known double-lined spectroscopic binary (SB2) , with an orbital period of 1.35 year [@1989PDAO...17....1B]. It shines at an apparent visual magnitude of $ m_v=6\fm60$ and the spectral types of both components are catalogued as K2V and K4V for the primary and secondary components respectively [@2010AJ....139.2308F]. Hipparcos trigonometric parallax measurement of the system as $\pi=58.96\pm0.65$ mas places this system at a distance of 16.96 pc, being one of the nearby K-type stars.
Table \[tlab2\] contains basic data of the system Gliese 762.1 from SIMBAD, NASA/IPAC, Hipparcos and Tycho Catalogues [@1997yCat.1239....0E].
GJ 762.1 source of data
---------------------------- ----------------------- ----------------
$\alpha_{2000}$ $\dagger$ $19^h 31^m 07\fs974$ SIMBAD
$\delta_{2000}$ $\ddagger$ $+58\degr35' 09.''64$ -
HIP 95995 -
Sp. Typ. K1V -
$E(B-V)^{*}$ $0.07\pm 0.002$ NASA/IPAC
$A_v^{*}$ $0\fm21$ NASA/IPAC
$B_J(Hip)$ $7\fm46$ Hipparcos
$V_J(Hip)$ $6\fm60$ -
$R_J(Hip)$ $6\fm10$ -
$(B-V)_J(Hip)$ $0\fm86\pm0.001$ -
$(U-B)_{J}$ $0\fm52\pm0.001$ -
$B_T$ $7\fm71\pm0.006$ Tycho
$V_T$ $6\fm71\pm0.005$ -
$(B-V)_J(Tyc)$ $0\fm87\pm0.006$ -
$\pi_{Hip}$ (mas) $59.84\pm0.64$ Hipparcos
$\pi_{Tyc}$ (mas) $58.00\pm2.90$ Tycho
$\pi^{**}_{Hip}$ (mas) $58.96\pm0.65$ New Hipparcos
: Basic data of the system Gliese 762.1 from SIMBAD, Hipparcos and Tycho Catalogues[]{data-label="tlab2"}
\
$\dagger$ Right Ascention, $\ddagger$ Declination\
$^{*}$ http://irsa.ipac.caltech.edu, $^{**}$
ANALYSIS OF THE SYSTEM {#orbital_elements}
======================
Atmospheres modelling and the estimation of the physical parameters {#m1}
-------------------------------------------------------------------
In spite of the fact that the duplicity of the system was detected using high resolution spectroscopy and speckle interferometry, the system is seen as a single star even with the aid of the biggest telescopes. So, in order to estimate the physical parameters of the individual components of the system, we followed Al-Wardat’s complex method for analyzing CVBS. The synthetic spectral energy distributions (SEDs) of the individual components of the system are computed using ATLAS9 line blanketed model atmospheres of [@1994KurCD..19.....K] using a special subroutine. In order to build the model atmospheres of each components, we need preliminary input parameters ($T_{eff}$ and $\log g$), these are calculated as follows:
Using the apparent visual magnitude of the system $ m_v=6\fm60$ from the previous data (Table \[tlab2\]), and the visual magnitude difference $\triangle m=0\fm 33\pm0.06$ between the two components as the average of fifteen $\triangle m$ measurements of the filters $ \lambda$ $503$ -$ 850 $ (Table \[deltam1\]), we calculated a preliminary individual $m_v$ for each component using the following equations: $$\begin{aligned}
\label{Ma}
m_A=m_v+2.5\log(1+10^{-0.4\Delta m}),\\
m_B=m_A+\Delta m,\end{aligned}$$ which give:
$m_v^A=7\fm20\pm0.03 , m_v^B=7\fm53\pm0.07$.
Combining these magnitudes with the Hipparcos trigonometric parallax $(\pi_{Hip})$ from , we can derive the preliminary absolute magnitudes for the components using the following relation: $$\begin{aligned}
\label{eq3}
\ M_V=m_v+5-5\log(d)-A_v\end{aligned}$$ as follows: $M_V^A=5\fm85\pm0.04$ and $M_V^B=6\fm18\pm0.07$, where $\ A_v$ is the interstellar reddening which was taken from NASA/IPAC (See Table \[tlab2\]).
$\triangle m $ [$\sigma_{\Delta m}$]{} filter ($\lambda/\Delta\lambda$) references
---------------- ------------------------- ---------------------------------- ------------
$0\fm26$ 0.05 $545nm/30 $ 1
$0\fm33$ 0.07 $545nm/30 $ 2
$0\fm27$ 0.04 $610nm/20 $ 3
$0\fm27$ 0.15 $648nm/41$ 4
$0\fm32$ 0.15 $503nm/40 $ 4
$0\fm24$ 0.02 $545nm/30 $ 5
$0\fm25$ 0.19 $600nm/30 $ 5
$0\fm24$ 0.31 $850nm/75 $ 5
$0\fm41 $ - $698nm/39 $ 6
$0\fm29$ 0.03 $600nm/30 $ 7
$0\fm73$ - $698nm/39 $ 6
$0\fm53$ - $550nm/40 $ 6
$0\fm27$ - $754nm/44 $ 6
$0\fm19$ - $562nm/40 $ 8
$0\fm28$ - $692nm/40 $ 8
: Magnitude difference between the components of the system Gliese 762.1, along with filters used to obtain the observations. []{data-label="deltam1"}
\
$^1$, $^2$, $^3$, $^4$[[@2004AJ....127.1727H]]{}, $^5$[[@2006BSAO...59...20B]]{}, $^6$[[@2008AJ....136..312H]]{}, $^7$[[@2007AstBu..62..339B]]{}, $^8$[[@2011AJ....141...45H]]{}.
\[po\]
(deg) (arcsec) References
----------- ------- ---------- --------------
1980.4797 254.2 0.117 McA1983 (1)
1980.7228 226.7 0.106 McA1983 (1)
1981.4736 310.4 0.081 McA1984a (2)
1983.4175 225.8 0.115 McA1987b (3)
1984.7039 235.7 0.112 McA1987b (3)
1985.4900 322.4 0.066 McA1987b (3)
1985.7390 273.1 0.104 Tok1988 (5)
1986.8883 308.5 0.065 McA1989(6)
1987.7618 172.7 0.067 McA1989(6)
1989.7059 286.0 0.093 Hrt1992b(7)
1989.8041 272.2 0.100 Bag1994(8)
1989.8096 268.2 0.102 Bag1994(8)
1990.4322 171.6 0.062 Ism1992(9)
1991.25 284.0 0.100 HIP1997a(10)
1993.8438 271.3 0.102 Bag1994(8)
1994.7080 292.1 0.062 Hrt2000a(11)
1995.4397 244.8 0.113 Hrt1997(12)
1995.7621 201.9 0.086 Hrt1997(12)
1996.6903 255.0 0.111 Hrt1997(12)
1999.8179 204.1 0.086 Bag2002(13)
2000.6166 271.9 0.099 Bag2004(14)
2000.7640 253.8 0.111 Hor2002a(15)
2001.7550 134.6 0.065 Bag2006b(16)
2003.6339 235.6 0.114 Hor2008(17)
2004.8150 75.0 0.110 Bag2007b(18)
2005.7662 153.5 0.054 CIA2010(19)
2006.4381 44.9 0.104 Bag2013(20)
2006.5198 211.3 0.098 Hor2008(17)
2006.5227 213.8 0.095 Hor2008(17)
2007.8011 46.0 0.112 Hrt2009(21)
2010.4816 227.6 0.106 Hor2011(22)
: Relative position measurements obtained using different methods, which are used to build the orbit of the system. These points are taken from the Fourth Catalog of Interferometric Measurements of Binary Stars.
\
$^1$[[@1983ApJS...51..309M]]{}, $^2$[[@1984ApJS...54..251M]]{}, $^3$[[@1987AJ.....93..688M]]{}, $^4$, $^5$, $^6$[[@1989AJ.....97..510M]]{}, $^7$[[@1992AJ....104..810H]]{}, $^8$, $^9$, $^{10}$[[@1997yCat.1239....0E]]{}, $^{11}$[[@2000AJ....119.3084H]]{}, $^{12}$[[@1997AJ....114.1639H]]{}, $^{13}$, $^{14}$, $^{15}$[[@2002AJ....123.3442H]]{}, $^{16}$[[@2006BSAO...59...20B]]{}, $^{17}$[[@2008AJ....136..312H]]{}, $^{18}$[[@2007AstBu..62..339B]]{}, $^{19}$[[@2010AJ....139.2308F]]{}, $^{20}$[[@2013AstBu..68...53B]]{}, $^{21}$[[@2009AJ....138..813H]]{}, $^{22}$[[@2011AJ....141...45H]]{}.
The bolometric corrections$(B.C.)$, bolometric magnitudes and the stellar luminosities of the system were taken from [@1992adps.book.....L] and [@2005oasp.book.....G]. These values, along with the following two equations: $$\begin{aligned}
\label{eq8}
\log(R/R_\odot)= 0.5 \log(L/L_\odot)-2\log(T_{eff}/T_\odot),\\
\label{eq5}
\log g = \log(M/M_\odot)- 2\log(R/R_\odot) + 4.43\end{aligned}$$
were used to calculate the preliminary input parameters as: $T_{eff}^A=5300K, T_{eff}^B=5050K$, $\log g_{A}=4.56, \log g_{B}=4.54$ and $R_{A}=0.815R_\odot$, $R_{B}=0.806R_\odot$. $T_\odot$ were taken as $5777\rm{K}$.
The entire synthetic SED as if it is received from the system and measured above the earth’s atmosphere is calculated using the following equations: $$\begin{aligned}
\label{eq6}
F_\lambda \cdot d^2 = H_\lambda ^A \cdot R_{A} ^2 + H_\lambda ^B
\cdot R_{B} ^2,\end{aligned}$$ from which $$\begin{aligned}
\label{eq7}
F_\lambda = (R_{A} /d)^2(H_\lambda ^A + H_\lambda ^B \cdot(R_{B}/R_{A})^2) ,\end{aligned}$$ where $ R_{A}$ and $ R_{B}$ are the radii of the primary and secondary components of the system in solar units, $H_\lambda ^A $ and $H_\lambda ^B$ are the fluxes at the surface of the star and $F_\lambda$ is the flux for the entire SED of the system above the Earth’s atmosphere which is located at a distance d (pc) from the system.
The exact physical parameters of the components of the system are those which lead to the best fit between the entire synthetic SED and the observational one, which was taken form [@2002BSAO...54...29A]. The observational spectrum (Fig. \[fig1\]) was obtained using a low resolution grating ($325/4^{\circ}$ grooves/mm, [Å]{}/px reciprocal dispersion) within the UAGS spectrograph at the 1m (Zeiss-1000) SAO-Russian telescope.
Beside the visual best fit between the two spectra, the synthetic magnitudes, color indices and line profiles especially those of Hydrogen $\ H_{\beta}$(4861.33Å), $\ H_{\gamma}$(4340.5Å) and $\ H_{\delta}$(4101Å) should fit the observational ones. Otherwise, a new set of parameters would be tested in iterated way until the best fit is reached.
The best fit (Fig. \[fig1\]) was achieved using the parameters shown in Table \[tablef1\]. The luminosities and masses of the components were calculated using Equs. \[eq8\] and \[eq5\], and the spectral types of the components of the system were derived from [@1992adps.book.....L] empirical $Sp-M_V$ relation for main sequence stars.
![Best fit between the observed entire spectrum (dotted line) which was taken from [@2002BSAO...54...29A] and the synthetic entire SED (solid line) for the system Gliese 762.1. Individual synthetic SEDs were computed using $T_{\rm eff}^A =5300\pm50$K, log $g_A=4.56\pm0.10, R_A=0.815\pm0.09R_\odot$, $T_{\rm eff}^B =5150\pm50$K, log $g_B=4.54\pm0.10$ and $R_B=0.806\pm0.10 R_\odot $, with $d=16.96$ pc ($\pi=58.96\pm0.65 $ mas).[]{data-label="fig1"}](ms2015-0025fig1.eps){width="14cm"}
[lcc]{} Parameters & Comp. A & Comp. B\
$T_{\rm eff}$(K) & $5300\pm50$ & $5150\pm50$\
Radius (R$_{\odot}$) & $0.845\pm0.09$ & $0.795\pm0.10$\
$\log g$ & $4.52\pm0.10$ & $4.54\pm0.15$\
$L (L_\odot)$ & $0.51\pm0.08 $ & $0.40\pm0.07$\
$M_{bol}$ & $5\fm48\pm0.90$ & $5\fm74\pm1.02$\
$M_{V}$ & $5\fm85\pm0.80$ & $6\fm18\pm0.85$\
Mass ($M_{\odot})^{*}$& $0.89 \pm0.08$ & $0.83 \pm0.07$\
Sp. Type$^{**}$ & K0 & K1.5\
&\
&\
&\
\
$^{*}$[depending on the equation \[eq5\]]{},\
$^{**}$[depending on the tables of [@1992adps.book.....L]]{},\
$^{***}$[depending on the orbital solution]{}.
Orbital solution and Masses {#2.2}
---------------------------
Once available, the orbital elements of a binary system would enhance and help in examining the physical parameters of its individual components. The mass sum of the two components given by equations \[eq31\] and \[eq32\] should coincide with that estimated from their positions on the evolutionary tracks and that calculated using the empirical equations and standard tables.
We followed Tokovinin’s method [@1992ASPC...32..573T] to calculate the orbital elements. The method performs a least-squares adjustment to all available radial velocity and relative position observations, with weights inversely proportional to the square of their standard errors. The orbital solution involves: the orbital period, P; the semi-amplitudes of the primary and secondary velocities, K1 and K2; the eccentricity, e; the semi-major axis, a; the center of mass velocity, $\gamma$; and the time of primary minimum, $\ T_0$. The radial velocities for the system were taken from [@1983PASP...95..201M].
Table \[orbit1\] lists the results of the radial-velocity solution (Fig. \[fig2\]). The best orbit passes through the relative position measurements is shown in Fig. \[fig3\] and the resulting orbital elements are compared with earlier studies in Table \[orbit\].
Parameters [@1983PASP...95..201M] This paper
------------------------- ------------------------ ---------------------
$P$, yr $1.3477 \pm 0.0038$ $1.3534 \pm 0.0010$
$T_0,$ MJD 44194.3 $\pm 3.2$ $47670.20 \pm 2.01$
$e$ $0.416 \pm 0.016$ $0.40 \pm 0.013$
$\omega$, deg $177.1 \pm 2.70$ $179.0 \pm 2.24$
$ K1$, km$s^{-1}$ $ 9.52\pm0.16$ $ 9.40\pm0.12$
$ K2$, km$s^{-1}$ $ 10.46\pm0.19$ $ 10.35\pm0.19$
$ V_\gamma$, km$s^{-1}$ $ 11.41\pm0.014$ $ 11.31\pm0.12$
: Orbital solution of the system using the velocity curves Fig. \[fig2\].[]{data-label="orbit1"}
\
Parameters [@2000IAUS..200P.135A] [@2000AAS..145..215P] [@2010AJ....139.2308F] This work
--------------- ------------------------ ------------------------ ------------------------ ----------------------
$P$, yr $1.35458 \pm 0.00131$ $1.352776 \pm 0.00071$ $1.35297 \pm 0.00159$ $1.3534 \pm 0.00075$
$T_0,$ MJD $48641.21 \pm 3.10$ $46164.9 \pm 1.66$ $46671.4 \pm 8.5$ $47670.20 \pm 2.37$
$e$ $0.340 \pm 0.013$ $0.3600 \pm 0.0078$ $0.371 \pm 0.006$ $0.36 \pm 0.020$
$a, $ arcsec $0.084\pm 0.003 $ $0.0860\pm 0.0014 $ $0.0842 \pm 0.3 $ $0.0865 \pm 0.010 $
$i $, deg $ 144.6 \pm 1.7$ $ 144 \pm 2.4$ $ 144.0 \pm 1.29$ $ 140.0 \pm 2.00$
$\omega$, deg $177.8 \pm 2.1$ $356 \pm 2.1$ $16.57 \pm 4.1$ $198.0 \pm 4.4$
$\Omega$, deg $74.6 \pm 6.8$ $243\pm 1.5$ $256.9 \pm 2.666$ $253 \pm 6.55$
$\pi$, mas $ 57.99\pm0.57$ $ 57.3\pm0.3$ $ 59.2\pm2.04$ $ 58.96\pm0.65^{a}$
$ M, M_\odot$ $1.62\pm0.18$ $1.67\pm0.83$ $1.59\pm0.18$ $1.72\pm0.60$
\
${^a}$ New Hipparcos (See Table \[tlab2\]).
![ Spectroscopic orbital solution for Gliese 762.1 in Table \[orbit1\] and radial velocities. Triangles represent radial velocities of the primary and squares represent radial velocities of the secondary component. The dotted line in the figure represents the center of mass velocity ( $ V_\gamma$=$ 11.31\pm0.12$ km$s^{-1}$). []{data-label="fig2"}](ms2015-0025fig2.eps){width="14cm"}
![The best visual orbit of the system with the relative position measurements from the Fourth Catalog of Interferometric Measurements of Binary Stars. The squares represent the position of the primary component.[]{data-label="fig3"}](ms2015-0025fig3.eps){width="14cm"}
The estimated orbital elements, semi-major axis, orbital period (see Table \[orbit\]), Hipparcos parallax of as $\pi=58.96\pm0.65$ mas, along with Kepler’s third law: $$\begin{aligned}
\label{eq31}
\ M_A +M_B=(\frac{a^3}{\pi^3P^2})\ M_\odot\end{aligned}$$ $$\begin{aligned}
\label{eq32}
\frac{\sigma_M }{M} =\sqrt{(3\frac{\sigma_\pi}{\pi})^2+(3\frac{\sigma_a}{a})^2+(2\frac{\sigma_p}{p})^2}\end{aligned}$$ yield a mass sum with its corresponding error for the system as $\ M_A +M_B$=$1.72\pm0.60M_\odot$. Using atmospheric modeling equation \[eq5\], the total mass of the system is $1.69\pm0.22M_\odot$.
SYNTHETIC PHOTOMETRY
====================
The entire and individual synthetic magnitudes are calculated by integrating the model fluxes over each bandpass of the system calibrated to the reference star (Vega) using the following equation [@2007ASPC..364..227M; @2012PASA...29..523A]: $$m_p[F_{\lambda,s}(\lambda)] = -2.5 \log \frac{\int P_{p}(\lambda)F_{\lambda,s}(\lambda)\lambda{\rm d}\lambda}{\int P_{p}(\lambda)F_{\lambda,r}(\lambda)\lambda{\rm d}\lambda}+ {\rm ZP}_p\,,$$ where $m_p$ is the synthetic magnitude of the passband $p$, $P_p(\lambda)$ is the dimensionless sensitivity function of the passband $p$, $F_{\lambda,s}(\lambda)$ is the synthetic SED of the object and $F_{\lambda,r}(\lambda)$ is the SED of Vega. Zero points (ZP$_p$) from [@2007ASPC..364..227M] (and references there in) were adopted.
The results of the calculated magnitudes and color indices (Johnson: $U$, $B$, $ V$, $R$, $U-B$, $B-V$, $V-R$; Strömgren: $u$, $v$, $b$, $y$, $u-v$, $v-b$, $b-y$ and Tycho: $B_{T}$, $ V_{T}$, $B_{T}-V_{T}$) of the entire system and individual components, in different photometrical systems, are shown in Table \[synth1\].
[lcccc]{} Sys. & Filter & entire & comp.& comp.\
& & $\sigma=\pm0.02$& A & B\
Joh- & $U$ & 7.99 & 8.53 & 9.01\
Cou. & $B$ & 7.47 & 8.05 & 8.43\
& $V$ & 6.60 & 7.20 & 7.53\
& $R$ & 6.12 & 6.74 & 7.03\
&$U-B$ & 0.52 & 0.48 & 0.58\
&$B-V$ & 0.87 & 0.85 & 0.90\
&$V-R$ & 0.47 & 0.46 & 0.50\
Ström. & $u$ & 9.15 & 9.69 & 10.18\
& $v$ & 7.96 & 8.52 & 8.94\
& $b$ & 7.06 & 7.65 & 8.00\
& $y$ & 6.56 & 7.16 & 7.48\
&$u-v$ & 1.19 & 1.16 & 1.24\
&$v-b$ & 0.90 & 0.87 & 0.94\
&$b-y$ & 0.50 & 0.49 & 0.52\
Tycho &$B_T$ & 7.71 & 8.28 & 8.68\
&$V_T$ & 6.70 & 7.29 & 7.63\
&$B_T-V_T$ & 1.02 & 0.99 & 1.05\
[lcc]{} & Observed $^\dag$ & Synthetic (This work)\
$V_{J}$ & $6\fm60$ & $6\fm60\pm0.02$\
$B_{J}$ & $7\fm46$ & $7\fm47\pm0.02$\
$R_{J}$ & $6\fm10$ & $6\fm12\pm0.02$\
$B_T$ & $7\fm71\pm0.01$ &$7\fm71\pm0.02$\
$V_T$ & $6\fm71\pm0.01$ &$6\fm70\pm0.02$\
$(B-V)_{J}$&$ 0\fm86\pm0.01$ &$ 0\fm87\pm0.02$\
$(U-B)_{J}$&$ 0\fm52\pm0.02$ &$ 0\fm52\pm0.02$\
$\triangle m$ &$ 0\fm33^{\ddag}\pm0.06$ &$ 0\fm33\pm0.04$\
\
$\dag$ See Table \[tlab2\]\
$\ddag$ Average value for fifteen $\triangle m$ measurements (See Table \[deltam1\]).
![The systems’ components on the evolutionary tracks of masses ( 0.5, 0.6, 0.7,...., 1.1 $M_\odot$) of [@2000yCat..41410371G]. []{data-label="fig4"}](ms2015-0025fig4.eps){width="14cm"}
RESULTS AND DISCUSSION
======================
Table \[synth3565\] shows a high consistency between the synthetic magnitudes and colors and the observational ones. This gives a good indication about the reliability of the estimated parameters listed in Table \[tablef1\]. Also, the resulted magnitude difference, individual magnitudes and absolute magnitudes (Tables \[tablef1\]& \[synth1\]) are consistent with the calculated ones as preliminary input parameters.
The positions of the system’s components on the evolutionary tracks of (Fig. \[fig4\]) show that both components, of mass between $0.8$ and $0.9 M_\odot$ for each of them, belong to the main-sequence stars, but both show a slight displacement from the zero-age main-sequence upwards. And their positions on isochrones for low- and intermediate-mass stars of different metallicities and that of the solar composition \[$Z=0.019, Y=0.273$\] are shown in Figs \[fig5\] & \[fig6\], which give an age of the system around $9\pm 1$ Gy.
The spectral types and luminosity classes of both components are assigned as K0V and K1.5V for the primary and secondary components respectively, and their positions on the evolutionary tracks are showed in Fig. \[fig4\], which are brighter than those given by [@2010AJ....139.2308F] as K2V and K4V. The estimated orbital elements of the system (Tables \[orbit1\] & \[orbit\]) are consistent with previous works. The orbit of the system was solved using a combination of the relative position measurements and the radial velocity curves, which gives more reliable and accurate results.
The mass sum of the system components and the individual masses were calculated and estimated in three different ways; using the physical parameters and standard relations as $M_a=0.89\pm0.08M_\odot, M_b=0.83\pm0.07M_\odot$, using the orbital elements with Hipparcos parallax as $M_a+M_b=1.72\pm0.60M_\odot$ and depending on the positions of the system’s components on the evolutionary tracks of (Fig. \[fig4\]) which coincide with the calculated ones.
Depending on the estimated parameters of the system’s components and their positions on the evolutionary tracks, fragmentation is a possible process for the formation of the system. Where [@1994MNRAS.269..837B] concludes that fragmentation of a rotating disk around an incipient central protostar is possible, as long as there is continuing infall. [@2001IAUS..200.....Z] pointed out that hierarchical fragmentation during rotational collapse has been invoked to produce binaries and multiple systems.
It is worthwhile to mention here that the system Gliese 762.1 is a detached one with an orbital period of $1.3534\pm0.0010$yr. So, this system is a visually close but not a contact binary, and if we compare it with other extremely K-type close binary systems like BI Vulpeculae [@2013ApJS..209...13Q], AD Cancri [@2007ApJ...671..811Q] and PY Virginis [@2013AJ....145...39Z], which have shorter orbital periods ( days) and lower angular momentum among K-type binary stars, we find that the orbital evolution of such systems are affected by the existence of a third component by removing angular momentum from the central binary system during the early stellar formation process or/and later dynamical interactions. However, as for Gliese 762.1, such dynamical interactions may not exist. It may form directly from stellar formation process because the orbital separation between the two components is much larger.
CONCLUSIONS
===========
We present the results of the complex analysis of the double-lined spectroscopic binary system Gliese 762.1. We were able to achieve the best fit between the entire synthetic SED’s and the observational one (Fig. \[fig1\]) by producing and calibrating synthetic SED of the individual components in an iterated method. The orbit of the system and its radial velocities were also solved to estimate reliable orbital elements consistent with the physical ones of atmospheres modeling.
We relayed on Hipparcos parallax ($58.96 \pm 0.65 $ mas, $d=16.96$pc ) for the calculations of the entire SED and the masses of the system. The Hipparcos parallax was not that far from the dynamical parallaxes introduced in previous orbital solutions (see Table \[orbit\]).
This research has made use of SAO/NASA, SIMBAD database, Fourth Catalog of Interferometric Measurements of Binary Stars, IPAC data systems and CHORIZOS code of photometric and spectrophotometric data analysis. We would like to thank Jürgen Weiprecht from Astrophysikalisches Institut und Universitäts-Sternwarte, FSU Jena for his help. S. Masda would like to thank ministry of higher education and scientific research in Yemen for the scholarship as well as S. Masda would like to thank Hadhramout University (Faculty of Al-Mahra Education) in Yemen for facilitate some things.
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---
abstract: 'We discuss the thermoelectrical properties of nanowires hosting Majorana edge states. For a Majorana nanowire directly coupled to two normal reservoirs the thermopower always vanishes regardeless of the value of the Majorana hybridization. This situation changes drastically if we insert a quantum dot. Then, the dot Majorana side coupled system exhibits a different behavior for the thermopower depending on the Majorana hybridization parameter $\varepsilon_M$. Thermopower reverses its sign when the half fermionic state is fully developed, i.e., when $\varepsilon_M=0$. As long as $\varepsilon_M$ becomes finite the Seebeck coefficient behaves similarly to a resonant level system. The sign change of the thermopower when Majorana physics takes place and the fact that both, the electrical and thermal conductances reach, their half fermionic value could serve as a proof of the existence of Majorana edge states in nanowires. Finally, we perform some predictions about the gate dependence of the Seebeck coefficient when Kondo correlations are present in the dot.'
author:
- Rosa López
- Minchul Lee
- Llorens Serra
- Jong Soo Lim
title: Thermoelectrical detection of Majorana states
---
Introduction
============
Nowadays there is a lot of interest in the interplay between heat and charge flows in nanostructures. [@Dhar08; @Dubi11] Thermovoltages generated in response to a temperature gradient have been shown to be much bigger at the nanoscale due to the peculiar properties of quantum systems. [@Butcher90; @Mabesoone92; @Dzurak97; @Godjin99; @Matthews12] For example, delta like density of states occurring in confined nanostructures like quantum wells, [@Molenkamp92] alter dramatically their thermolectrical properties. The main utility of thermoelectrical devices is the heat-to- electricity conversion processes. However, from a more fundamental point of view, both thermal and electrical transport reveal information on the intrinsic nature of a quantum system. An instance is the departure of the Wiedemann Franz law attributed to the non Fermi liquid behavior. [@Coleman05] In addition, thermoelectric transport measurements are able to distinguish between distinct types of carriers, like electrons and holes in Andreev systems [@Jacquod10; @Balachandran12] and molecular junctions. [@Reddy07]
![(a) Majorana nanowire tunnel coupled to two normal contacts by tunneling barriers of probability $\Gamma$. Here, $\eta_1$, and $\eta_2$ denote the two Majorana ends states at the semiconductor nanowire. Left(right) metallic contact is electrical and thermal biased with $V_{L(R)}$, and $\theta_{L(R)}$. (b) A quantum dot is inserted and symmetrically coupled to the metallic reservoirs with tunneling rate $\gamma$. The dot is side coupled to the Majorana nanowire, such coupling is characterized by the parameter $\zeta$.[]{data-label="figure1"}](scheme.eps){width="40.00000%"}
Our motivation is to address to what extent Majorana physics can be reflected in the thermoelectrical transport properties of a system. The unambiguous detection of Majorana fermions in solid state devices is still a discussional issue. Majorana physics, in the low energy domain, was predicted to occur as quasiparticle excitations. [@Wilczek09] The first proposals suggested their observation in quantum Hall states, the Moore Read state at filling factor $\nu=5/2$. [@Read91] Then, other suggestions considered some exotic superconductors like Sr$_2$RuO$_4$ or $p$-wave superconductors. [@Kitaev01; @Ivanov01; @Sarma06; @Linder10a] Later on, the pioneering work by Fu and Kane [@Ku08] demonstrated that such quasiparticles could be created in a topological insulator brought in close proximity with a superconductivity source. However, the Majorana search has been very prolific in the realm of quasi one dimensional semiconductor nanowires [@Yuval10; @Alicea10; @Lutchyn10; @Linder10b; @Potter11], and in particularly in large $g$ factor materials like InAs and InSb. Most of the experiments designed to detect these elusive quasipartices have been performed via electrical transport measurements [@Mourik12; @Deng12; @Heiblum12; @Churchill12; @Finck13] by tunnel spectroscopy. A voltage shift, $\delta V$, is applied to the nanowire edges that generates an electrical current $I$. The Majorana signature appears as a zero bias anomaly in the nonlinear conductance $dI/dV$.[@Liu12; @Pientka12; @Elsa12a; @Lim12] In semiconductor nanowires, Majorana quasiparticles arise when superconductivity (source of electrons and holes), strong spin orbit interaction, and magnetic field work together. Then, under certain conditions the nanowire enters in the named topological phase and shows up spinless, chargeless zero energy states, very elusive quasiparticle excitations. We refer to this as Majorana nanowire. However, the presence of a zero bias anomaly in the nonlinear conductance does not warrant the presence of Majorana quasiparticles. Kondo physics can be observed in normal superconductor nanowires as well. [@Chang13; @Eduardo12] Furthermore, nearly zero energy Andreev states [@Kells12; @Eduardo13] or weak antilocalization [@Pikulin12a] effects are possible sources of zero bias anomaly in normal superconductor nanowires. There are other suggestions to detect Majorana zero energy states in Josephson junctions and rings. [@Kwon03; @Fu09; @Tanaka09; @Ioselevich11; @Jiang11; @Pikulin12b; @Elsa12b; @Fernando12] The Josephson current displays an anomalous periodicity of $4\pi$ if Majorana physics takes place. However, so far the experimental verification is not yet definitive. [@Rokhinson12]
Our goal consists in utilizing the thermoelectrical properties as a tool to detect the presence of Majorana edge states formed in normal superconductor nanowires. The only attempt to study similar issues has done in $p$-wave superconductors. [@Refael13] Here, we propose a way of detecting Majorana edge states in semiconductor nanowires when a temperature gradient ($\delta\theta=\theta_L-\theta_R$) is applied and an induced electrical shift ($\delta V=V_L-V_R$) is generated. We analyze a two terminal device as depicted in Fig. \[figure1\](a) and determine both the electrical and energy currents. Here, the Majorana nanowire is contacted to two normal reservoirs. In general, the linear response electric $I$ and energy $J$ currents can be expressed as $$\label{eq_matrix}
\begin{pmatrix}
I\\ J
\end{pmatrix}
=
\begin{pmatrix}
G& L\\ M& K
\end{pmatrix}
\begin{pmatrix}
\delta V\\ \delta \theta
\end{pmatrix} \,.$$ The $2\times 2$ matrix is the Onsager matrix that includes diagonal elements,the electric $G$ and thermal $K$ conductances, and non diagonal coefficients, the thermoelectric $L$ and electrothermal $M$ conductances. The two latter are related due to microreversibility condition. [@Onsager31; @Casimir45] More specifically, we are interested in the determination of the Seebeck coefficient or thermopower that measures how efficient is the conversion of heat into electricity in a thermoelectrical machine. The larger the Seebeck coefficient, the more efficient this conversion is. Seebeck coefficient is easily determined from the relation: $S=-\delta V/\delta\theta=L/G$.
Our results for a two terminal Majorana nanowire \[see Fig. \[figure1\](a)\] show that both, the electrical and heat conductances reach their maximum value *only* when Majorana edge states do not overlap. On the contrary, the thermoelectrical(electrothermal) response always vanishes irrespectively of the Majorana hybridization. As a result, the Seebeck coefficient vanishes owing to the intrinsic particle hole symmetry of the system under consideration. However, this physical scenario can be dramatically altered by inserting a quantum dot in between the two normal contacts and side coupled to the Majorana nanowire. [@Flensberg11; @Dong11; @Minchul12] Figure. \[figure1\](b) illustrates the sample configuration. In this arrangement, the Seebeck coefficient can be tuned by gating the dot i.e., $S=S(\varepsilon_d)$ with $\varepsilon_d$ the dot level position.
General formalism
=================
We present our theory for the thermoelectrical transport by employing the nonequilibrium Keldysh Green function framework. We consider a semiconductor nanowire with strong Rashba spin orbit interaction with proximity induced $s$-wave superconductivity, and a applied magnetic field $B$. We assume a sufficiently long wire to neglect charging effects. The magnetic field is such that the wire is in the topological phase, $\Delta_Z>\sqrt{\Delta^2+\mu^2}$, with $\Delta_Z=g\mu_B B/ 2$, and $\mu$ the wire chemical potential. Then, isolated Majorana zero energy states $\eta_1=f+f\dagger$, and $\eta_2=i(f^\dagger-f)$ (in terms of $f$ Dirac fermions) are formed at the nanowire ends points. We consider that two normal contacts are tunnel coupled to the wire ends as shown in Fig. \[figure1\](a). The Hamiltonian describing this system is given by these three contributions: $\mathcal{H} = \mathcal{H}_C + \mathcal{H}_M + \mathcal{H}_T$, where $$\begin{aligned}
\label{hamiltonian1}
\mathcal{H}_C &= &\sum_{\alpha,k} \varepsilon_{\alpha k}
c_{\alpha k}^{\dagger} c_{\alpha k},\\ \nonumber
\mathcal{H}_M &=& \frac{i}{2} \varepsilon_M \eta_1 \eta_2 \,,\\ \nonumber
\mathcal{H}_T &=&\mathcal{H}_{TL}+\mathcal{H}_{TR}=\sum_{\alpha,k;\beta} \left[V_{\alpha k,\beta}
^{\ast} c_{\alpha k}^{\dagger} \eta_{\beta} +
V_{\alpha k,\beta} \eta_{\beta} c_{\alpha k}\right] \nonumber\,.\end{aligned}$$ Here, $\mathcal{H}_C$ describes the two normal leads, with $c^\dagger_{\alpha k}(c_{\alpha k})$ being the creation (annihilation) operator for an electron with wavevector $k$ in the lead $\alpha$. Note that the spin degree of freedom is omitted. This can be understood considering that we need to apply a large magnetic field to observe the edge Majoranas, so that only one kind of spin is effectively involved. $\mathcal{H}_{M}$ characterizes the coupling between the two end Majorana states where $\varepsilon_M \sim f(B,\Delta)e^{-L/\xi_0}$ with $L$ the length of the wire and $\xi_0$ the superconducting coherence length. $f(B,\Delta)$ is a complicated function of $B$ and $\Delta$ that determines $\varepsilon_M$. For our purpose we assume that $\varepsilon_M$ is a parameter. The last contribution, $\mathcal{H}_T$ corresponds to the tunnel Hamiltonian between normal leads and the Majorana end states. Below, the tunnel amplitude $V_{\alpha k,\beta}$ is taken as $V_0$ for $\alpha=\beta$ and zero for $\alpha\neq \beta$. This defines $\Gamma=\pi V_0^2 \rho_0$, with $\rho_0$ the contact density of states.
The charge and energy currents have the Landauer and Büttiker form $$I=\frac{e}{h}\int d\omega \mathcal{T}(\omega)[f_{L}(\omega)-f_R(\omega)]\,,$$ and $$J=\frac{1}{h}\int d\omega \omega\mathcal{T}(\omega)[f_{L}(\omega)-f_R(\omega)]\,,$$ with a transmission coefficient given by $$\mathcal{T}(\omega)=\frac{4\Gamma^2\left(\omega^2
+4\Gamma^2+\varepsilon_M^2\right)}{\left(\omega^2
+4\Gamma^2\right)^2+\varepsilon_M^2\left[\varepsilon_M^2
-2\left(\omega^2-4\Gamma^2\right)\right]}\,.$$
Here $f_L=1/[1+\exp{(\omega-(\mu+V_L))/k_B\theta_L}+1]$ ($k_B$ Boltzamnn constant) and $f_R=1/[1+\exp{(\omega-(\mu+V_R)/k_B\theta_R}+1]$ are the Fermi Dirac distribution function for the left and right contacts respectively with $V_{L,R}=\pm\delta V/2$, and $\theta_{L,R}=T_b\pm\delta \theta/2$.
The linear conductances are (we take $\mu=0$) $$\begin{aligned}
G&=&\frac{e^2}{h}\int d\omega\mathcal{T}(\omega) \left[-\frac{\partial f_{eq}}{\partial \omega}\right], \\
L&=&\frac{e}{hT_b}\int d\omega \omega \mathcal{T}(\omega) \left[-\frac{\partial f_{eq}}{\partial \varepsilon}\right], \\
M&=&\frac{e}{h}\int d\omega \omega\mathcal{T}(\omega) \left[-\frac{\partial f_{eq}}{\partial \omega}\right], \\
K&=&\frac{1}{hT_b}\int d\omega \omega^2 \mathcal{T}(\omega) \left[-\frac{\partial f_{eq}}{\partial \omega}\right] ,\end{aligned}$$ where $f_{eq}$ is the equilibrium Fermi Dirac distribution function when $\delta T=0$ and $\delta V=0$. In a Sommerfeld expansion, at sufficiently low temperatures, the linear response conductances $G$, and $K$ have the same behavior with the transmission coefficient up to a proportionality factor: $G_0$, and $K_0$. Thus, $$G(K)\! =\!\lim_{\delta V\rightarrow 0} \frac{dI}{dV} \left(\lim_{\delta\theta\rightarrow 0}\frac{dJ}{d\theta}\right)\!\!=\!\!
G_0(K_0)\frac{4\Gamma^2}{\varepsilon_M^2+4\Gamma^2}.
\label{eq:LinearG0}$$ with $G_0=e^2/h$ (quantum electrical conductance), and $K_0= \pi^2 k_B^2 T_b/3h$ (quantum thermal conductance). They take their maximum value $G_0$, and $K_0$, respectively when $\varepsilon_M=0$, otherwise, they vanish as $\varepsilon_M$ grows. Importantly, the off diagonal conductances are always zero, $L=L_0\partial \mathcal{T}(\omega)/\partial\omega|_{\omega=0}$ with $L_0=e\pi^2 k_B^2 T_b/3h$ (and $M=L /T_b$). The vanishing value of the $L(M)$ has profound consequences in the thermopower or Seebeck coefficient (we recall that $S=L/G$). The Seebeck coefficient vanishes regardless of the value of $\varepsilon_M$. The reason for this result lies in the inherent particle hole symmetry of our system, there is no electrical response to a thermal gradient.
Asymmetry in the particle and hole subspaces can happen if we insert a quantum dot between the two normal contacts. Here the dot is side coupled to the Majorana as illustrated in Fig. \[figure1\](b). The thermoelectrical transport through the dot Majorana system shows a non zero value for the off diagonal Onsager conductances when the dot is off resonance, i.e., a nonzero Seebeck coefficient. Importantly, we can tune the Seebeck coefficient from zero when the dot is on resonance to large values when is off resonance. Besides, the behavior of the Seebeck coefficient with the dot level is quite different depending on the value of the Majorana hybridization parameter, $\varepsilon_M$. Thus, Seebeck coefficient might allow us to detect truly zero energy Majorana states for which $\varepsilon_M$ is negligible .
Side tunel coupled dot Majorana system
======================================
In order to include the quantum dot we need to reformulate the Hamiltonian as follows. First, we consider the dot Hamiltonian $$\mathcal{H}_d=\sum\varepsilon_d d^\dagger d\,,$$ where $d(d^\dagger)$ operator annihilates(creates) an electron on the dot site. We consider a single dot level with energy $\varepsilon_d$. The dot is connected to the left and right normal contacts by tunnel barriers $$\mathcal{H}_{Td}=\sum_{\alpha k} (W_{\alpha } c^\dagger_{\alpha k} d + h.c)\,.$$ We consider symmetrically dot coupling to the normal contacts with a common tunneling rate: $\gamma=\pi W^2\rho_0$, with $W=W_{L}=W_{R}$. The dot is side coupled to the Majorana nanowire as $$\mathcal{H}_{TM}=\sum_{\beta} \zeta (d^\dagger \eta_\beta+\eta_\beta d)\,,$$ with $\beta=1,2$. Here, we assume that only the closest Majorana state to the dot is coupled, say $\eta_1$. The total Hamiltonian is the sum of all these contributions, and the contact and Majorana Hamiltonians \[$\mathcal{H}_C$, and $\mathcal{H}_M$, see Eq. (\[hamiltonian1\])\]: $\mathcal{H}=\mathcal{H}_C+\mathcal{H}_d+\mathcal{H}_{M}+\mathcal{H}_{Td}+ \mathcal{H}_{TM}$. Now, the charge and energy flows can be expressed in terms of the dot transmission (see Ref. \[\] for details) $$\label{transmission}
\mathcal{T}_d(\omega)=-\frac{1}{2}\frac{\gamma}{\pi} \rm{Im} \mathcal{G}^r_{d}(\omega) \,,$$ where $\mathcal{G}^r_{d}$ is the retarded dot Green function $$\mathcal{G}^r_{d}(\omega)=\frac{1}{\omega-\varepsilon_{d}+i\frac{\gamma}{2} - B(\omega)\left[1+\tilde{B}(\omega)\right]}\,,$$ with $$\begin{aligned}
\tilde{B}(\omega)=\frac{B(\omega)}{\omega+\varepsilon_{d}+i\frac{\gamma}{2}- B(\omega)}.\end{aligned}$$ The parameter $\zeta$ in Eq. (\[transmission\]) characterizes the dot Majorana coupling where $B(\omega)=|\zeta|^2/(\omega-\varepsilon_M^2/\omega)$ being the dot Majorana selfenergy coupling.
Discussion
==========
Before starting the discussion of the thermoelectrical properties in the dot Majorana system it is worth to revisit the behavior of the dot transmission with the system parameters, $\varepsilon_M$, $\varepsilon_d$, $\zeta$ and $\gamma$. [@Dong11] Hereafter, we employ $D=50$ for the contact bandwidth that determines our energy unit. The dependence of $\mathcal{T}_d(\omega)$ with $\zeta$, and $\gamma$ is illustrated in Fig. \[figure2\] when the dot is on resonance and no Majorana overlap occurs ($\varepsilon_d=0$, and $\varepsilon_M=0$). For the uncoupled Majorana situation the transmission corresponds to the resonant level model with unitary transmission. As $\zeta$ is turn on two peaks at $\omega=\pm\zeta$ appear due to the dot Majorana finite coupling. Now, keeping fixed $\zeta$ and tuning $\gamma$ the dot transmission shows a three peak structure when $\gamma\approx \zeta$ in which the zero energy peak is the signature of the presence of Majorana edge states \[see Fig. \[figure2\](b)\] . In all cases, when $\zeta\neq 0$, the dot transmission is always half fermionic. [@Dong11; @Minchul12]
![Dot transmission $\mathcal{T}_d(\omega)$ for (a) various $\zeta$ values as indicated and $\gamma=0.25$; (b) for different $\gamma$ values and $\zeta=0.05$. Parameters: $\varepsilon_d=0.0$, $\varepsilon_M=0$.[]{data-label="figure2"}](fig1.eps){width="45.00000%"}
When $\varepsilon_M$ acquires a finite value, $\mathcal{T}_d$ becomes unitary, as shown in Fig. \[figure3\](a). For large $\varepsilon_M$, $\mathcal{T}_d$ corresponds to the one for a resonant level mode, with resonances at $\omega\pm\epsilon_M$ due to the coupling of the dot state with the $f$ Dirac fermions in the wire (resulting from the large Majorana hybridization).
Thermoelectrical effects appears when the transmission becomes asymmetric. In order to observe such asymmetric transmission for $\omega<0$, and $\omega>0$ the dot level must be positioned off resonance, i.e., $\varepsilon_d\neq 0$. This situation is presented in Fig. \[figure3\](b) for several values of the Majorana hybridization parameter when $\varepsilon_d=0.12$. Note that, the transmission is asymmetric even for $\varepsilon_M=0$ although is still half fermionic. For a nonzero Majorana overlap, the transmission depends strongly on the dot gate value leading to a non unitary electrical(thermal) conductance.
The dot gate dependence of $\mathcal{T}_d(\omega)$ for an ideal Majorana nanowire ($\varepsilon_M=0$) is depicted in Fig. \[figure4\](a) and its energy derivative in Fig. \[figure4\](b). These curves shown that the transmission at zero energy is always half fermionic as should be for $\varepsilon_M=0$, regardless of the dot gate value. However, it is interesting to observe that the energy derivative of the transmission at zero energy acquires some dot gate dependence reflecting the asymmetry between the particle and hole sectors. This result is important for the thermoelectrical conductance $L$,we recall that $L=L_0\partial T_b(\omega)/\partial\omega|_{\omega=0}$ implying that $L$ becomes gate dependent. Whereas the diagonal conductances are not sensitive to the particle hole asymmetry introduced by $\varepsilon_d\neq 0$, the off diagonal conductances show a dot gate dependence with important consequences in the thermoelectrical transport.
![Dot transmission, $\mathcal{T}_d(\omega)$ for different values of the Majorana overlap $\varepsilon_M$ (a) for $\varepsilon_d=0$, and (b) for $\varepsilon_d=0.12$. Parameters: $\gamma=0.25$, $\zeta=0.15$.[]{data-label="figure3"}](fig2.eps){width="45.00000%"}
Our previous analysis for the dot transmission explains the curves for the conductances illustrated in Fig. \[figure5\]. Both, the electrical and thermal conductances, $G$, and $K$ depend strongly on $\varepsilon_d$ whenever the two end Majorana states overlap. Otherwise, in the ideal situation where $\varepsilon_M=0$, $G$, and $K$ take its maximum value and they becomes half fermionic. [@Beenakker11; @Dong11; @Minchul12] This important result it serves to us to detect the presence of Majorana edge states in side coupled dot nanowires systems. However, the previous results are applicable only for purely electrical or thermal transport measurements. Here, we are interested more in the thermolectrical signatures of the Majorana edge states. For that purpose, we analyze how the off diagonal conductances behave with the dot gate values. We find, that when Majorana edge states have negligible overlap ( i.e., $\varepsilon_M=0$) the off diagonal conductance $L(M)$ reverses it sign in comparison with a situation with finite overlap, i.e., $\varepsilon_M\neq 0$. Our results show that for zero Majorana overlap $\varepsilon_M=0$, the thermoelectrical conductance $L$ depends linearly with $\varepsilon_d $ with a negative slope $-1/2\zeta^2$ that depends inversely on the dot Majorana strength. However, for a finite Majorana overlap, when $\varepsilon_M\neq 0$ the thermoelectrical conductance $L/L_0=[\varepsilon_d/ (4\varepsilon_d^2+\gamma^2)^2][8\gamma^2(\varepsilon_M^2+\zeta^2)/\varepsilon_M^2] $, displays two extrema at $\varepsilon_d=\pm\gamma/2$. In this case, $L$ behaves similarly to the resonant level model. Importantly, the different behavior found for the gate dependence of the thermoelectrical conductance $L$ could be utilized as an *smooking gun* for the Majorana detection in thermoelectrical transport measurements.
![ (a) Dot transmission $\mathcal{A}_d(\omega)$ and (b) its derivative $\partial \mathcal{T}_d(\omega)/\partial \omega$ for the indicated $\varepsilon_d$ values and $\varepsilon_M=0$. Parameters: $\gamma=0.25$, $\zeta=0.15$. []{data-label="figure4"}](fig3.eps){width="45.00000%"}
Using the previous results, we discuss the gate dependence of the thermopower $S=L/G=-\delta V/\delta \theta$, where $S=(\pi^2 k_B^2T_b/3e) d\ln\mathcal{T}(\omega)/d\omega|_{\omega=0}$ is the Mott formula. We define $S_0=\pi^2 k_B^2T_b/3e$. For the dot Majorana uncoupled case, $\zeta=0$, the thermopower $S/S_0=8\varepsilon_d/(4\varepsilon_d^2+\gamma^2)$ vanishes when $\varepsilon_d=0$ and follows the resonant level model as expected. For the coupled system, when $\zeta\neq 0$, the thermopower $S$ versus the dot gate position is plotted in Fig. \[figure6\]. Remarkably, the thermopower is linear with $\varepsilon_d$ for zero Majorana overlap: $S/S_0=-\varepsilon_d/\zeta^2$ when $\varepsilon_M=0$ and $\zeta\neq 0$. The dot gate dependence of $S$ is due to the particle hole asymmetry introduced when $\varepsilon_d$ is tuned from the *on* to the *off* resonance situation. The way to understand this result is by the addition of two effects. First, the Majorana state contributes to the thermopower in a rigid way with a constant term $-1/\zeta^2$. Second, the particle hole asymmetry grows as $\varepsilon_d$ does and this explains why the thermopower grows with $\varepsilon_d$. Then, both features add up and produce a linear dependence of the Seebeck coefficient with the dot gate with a negative slope that depends on the inverse of the dot Majorana coupling $\zeta$.
Figure \[figure6\] displays our results for the thermopower for various values of $\varepsilon_M$. For $\varepsilon_M=0$, Fig. \[figure6\] shows that the thermopower is positive(negative) for negative(positive) $\varepsilon_d$ having $\delta V<0$ by heating up(cooling down) the left contact. The thermopower sign dependence with $\varepsilon_d$ is inverted when the Majorana overlap is finite. Here, for $\varepsilon_M$ finite the thermopower is: $S/S_0= [\varepsilon_d/(4\gamma^2+\varepsilon_d^2)][8(\varepsilon_M^2+\zeta^2)/\varepsilon_M^2)]$. This means that when $\varepsilon_d<0(\varepsilon_d>0)$ the heating(cooling) of the left contact induced a positive(negative) voltage difference. Here, the Seebeck coefficient follows the behavior for a resonant model with two extrema at $\varepsilon_d=\pm \gamma/2$. All these differences for $S(\varepsilon_d)$ depending on $\varepsilon_M$ it allows us to distinguish situations where nanowires can host truly Majorana edge states or not.
![ (a) Dot gate dependence of the linear electrical(thermal) conductance $G(K)$ (with $G_0=e^2/h$, $K_0=\pi^2k_B^2 T_b/3h$) for zero $\varepsilon_M=0$ and finite Majorana overlap $\varepsilon_M\neq 0$. (b) Thermoelectrical conductance $L$ versus $\varepsilon_d$ at different Majorana overlaps $\varepsilon_M$. The case $\varepsilon_M=0$ has been multiplied by a factor $20$ for comparison purposes. Parameters: $\gamma=0.25$, $\zeta=0.15$, and $T_b=0.025$.[]{data-label="figure5"}](fig4.eps){width="45.00000%"}
Some of the previous results allow us to predict the dot gate dependence of the Seebeck coefficient, when Coulomb interactions take place. A quantum dot with a free local moment is able to form a Kondo singlet with the delocalized electrons in the normal reservoirs when is strongly tunnel coupled to them. Then, at temperatures much lower than the Kondo scale $T_K$ the dot physics can be explained within the Fermi Liquid theory. [@sbmft] In this scenario, both the dot gate position $\tilde{\varepsilon}_d\rightarrow \varepsilon_d+\lambda$, and the lead dot tunneling rate $\Gamma\rightarrow \tilde{\Gamma}$ are renormalized by Kondo correlations as $\lambda=-\varepsilon_d$, and $\tilde{\Gamma}=T_K$. Under this situation, the Seebeck coefficient, in the Kondo regime is zero (with $T_K$ larger that the dot Majorana coupling selfenergy [@Minchul12], i.e., in the Kondo dominant regime). In the pure Kondo regime spin fluctuations carry the charge and energy transport in a particle and hole symmetric situation, then, it quite reasonable to expect a vanishing Seebeck coefficient no matter the Majorana overlap is. For more exotic Kondo effects in which particle hole symmetry breaks down, like in the SU(4) Kondo effect (recently observed in carbon nanotube quantum dots [@jarillo05; @rosa05]) a nonvanishing Seebeck effect is expected. Here, within the Fermi Liquid description we have $\tilde{\varepsilon}_d\approx T_K^{SU(4)}$, and $\tilde{\Gamma}=T_K^{SU(4)}$ \[with $T_K^{SU(4)}$ as the Kondo scale for the SU(4) case\]. These two renormalized parameters produce a nonzero, but constant Seebeck coefficients: $S(\varepsilon_d)\approx -T_K^{SU(4)}/\zeta^2$ when $\varepsilon_M=0$ and $S(\varepsilon_d)= c/T_K^{SU(4)}$ ($c>0$) when $\varepsilon_M$ is finite. The richness of the Kondo behavior when Majorana physics occurs has been detailed discussed in Ref. \[\] by some of the authors but only for the electrical transport. The understanding of the thermoelectrical properties for the different range of parameters, i.e., in the Kondo and Majorana dominant regimes, requires further analysis with more powerful theoretical techniques [@progress]).
Conclusion
==========
We have investigated the linear response conductances to a thermal and electrical voltage shift in a two terminal geometry with normal superconductor nanowires showing Majorana physics. Firstly, we have considered a nanowire directly coupled to two normal reservoirs. Due to the intrinsic particle hole symmetry this system exhibits a null thermopower, no voltage is generated in response to a thermal gradient. Then, we insert a quantum dot between the two normal contacts which is side coupled to the Majorana nanowire. With this arrangement the detection of the Majorana edge states can be performed by looking at the sign of the thermoelectrical conductance or the thermopower $S$. Besides, we show that both, the electrical and thermal conductances take their half fermionic values whenever a true Majorana fermion state is formed, when $\varepsilon_M=0$. Finally, we make some predictions for the gate dependence of the Seebeck coefficient for interacting dots in the Kondo regime. We believe that our results could serve as an unambiguous tool for the detection of Majorana edge states in semiconductor nanowires.
![Thermopower $S$ versus $\varepsilon_d$ for various $\varepsilon_M$ values. The curve corresponding to $\varepsilon_M=0$ has been enlarged by a factor $20$ for comparison purposes. Parameters: $\gamma=0.25$, $\zeta=0.15$, and $T_b=0.025$.[]{data-label="figure6"}](fig5.eps){width="45.00000%"}
*Note added—*During the completion of this paper we become aware of a related work dealing with thermolectric transport in normal-dot-Majorana nanowires systems. The difference is that we consider thermal and electrical bias applied to the normal contacts, in Ref. \[\] the thermoelectrical forces are applied to the normal and Majorana parts.
Acknowledgement
===============
We thank David Sánchez for useful discussions. Work supported by MINECO Grant No. FIS2011-23526. This research was supported in part by the Kavli Institute for Theoretical Physics through NSF grant PHY11-25915.
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